LMFDB - Embedded newform 4560.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} -2.60555 q^{11} -2.60555 q^{13} -1.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -4.60555 q^{21} +2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.60555 q^{29} -4.00000 q^{31} -2.60555 q^{33} +4.60555 q^{35} +10.6056 q^{37} -2.60555 q^{39} -0.605551 q^{41} -3.39445 q^{43} -1.00000 q^{45} -6.00000 q^{47} +14.2111 q^{49} +2.00000 q^{51} +2.60555 q^{55} +1.00000 q^{57} +9.21110 q^{59} +7.21110 q^{61} -4.60555 q^{63} +2.60555 q^{65} -4.00000 q^{67} +2.00000 q^{69} -5.21110 q^{71} +6.00000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -3.21110 q^{83} -2.00000 q^{85} -4.60555 q^{87} -0.605551 q^{89} +12.0000 q^{91} -4.00000 q^{93} -1.00000 q^{95} +9.39445 q^{97} -2.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 2 q^{45} - 12 q^{47} + 14 q^{49} + 4 q^{51} - 2 q^{55} + 2 q^{57} + 4 q^{59} - 2 q^{63} - 2 q^{65} - 8 q^{67} + 4 q^{69} + 4 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 2 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93} - 2 q^{95} + 26 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 8 * q^31 + 2 * q^33 + 2 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 2 * q^45 - 12 * q^47 + 14 * q^49 + 4 * q^51 - 2 * q^55 + 2 * q^57 + 4 * q^59 - 2 * q^63 - 2 * q^65 - 8 * q^67 + 4 * q^69 + 4 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 - 2 * q^87 + 6 * q^89 + 24 * q^91 - 8 * q^93 - 2 * q^95 + 26 * q^97 + 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	0	0
$13$	−2.60555	−0.722650	−0.361325	−	0.932440i	$-0.617675\pi$
$13$	−2.60555	−0.722650	−0.361325	+	0.932440i	$0.617675\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.60555	−1.00501
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.60555	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$29$	−4.60555	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−2.60555	−0.453568
$34$	0	0
$35$	4.60555	0.778480
$36$	0	0
$37$	10.6056	1.74354	0.871771	−	0.489914i	$-0.162972\pi$
$37$	10.6056	1.74354	0.871771	+	0.489914i	$0.162972\pi$
$38$	0	0
$39$	−2.60555	−0.417222
$40$	0	0
$41$	−0.605551	−0.0945712	−0.0472856	−	0.998881i	$-0.515057\pi$
$41$	−0.605551	−0.0945712	−0.0472856	+	0.998881i	$0.515057\pi$
$42$	0	0
$43$	−3.39445	−0.517649	−0.258824	−	0.965924i	$-0.583335\pi$
$43$	−3.39445	−0.517649	−0.258824	+	0.965924i	$0.583335\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	2.60555	0.351332
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	9.21110	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$59$	9.21110	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	−4.60555	−0.580245
$64$	0	0
$65$	2.60555	0.323179
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−5.21110	−0.618444	−0.309222	−	0.950990i	$-0.600069\pi$
$71$	−5.21110	−0.618444	−0.309222	+	0.950990i	$0.600069\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.21110	−0.352464	−0.176232	−	0.984349i	$-0.556391\pi$
$83$	−3.21110	−0.352464	−0.176232	+	0.984349i	$0.556391\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−4.60555	−0.493767
$88$	0	0
$89$	−0.605551	−0.0641883	−0.0320942	−	0.999485i	$-0.510218\pi$
$89$	−0.605551	−0.0641883	−0.0320942	+	0.999485i	$0.510218\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	9.39445	0.953862	0.476931	−	0.878941i	$-0.341749\pi$
$97$	9.39445	0.953862	0.476931	+	0.878941i	$0.341749\pi$
$98$	0	0
$99$	−2.60555	−0.261868
$100$	0	0
$101$	7.21110	0.717532	0.358766	−	0.933428i	$-0.383198\pi$
$101$	7.21110	0.717532	0.358766	+	0.933428i	$0.383198\pi$
$102$	0	0
$103$	10.4222	1.02693	0.513465	−	0.858110i	$-0.328362\pi$
$103$	10.4222	1.02693	0.513465	+	0.858110i	$0.328362\pi$
$104$	0	0
$105$	4.60555	0.449456
$106$	0	0
$107$	10.4222	1.00755	0.503776	−	0.863834i	$-0.331944\pi$
$107$	10.4222	1.00755	0.503776	+	0.863834i	$0.331944\pi$
$108$	0	0
$109$	3.21110	0.307568	0.153784	−	0.988105i	$-0.450854\pi$
$109$	3.21110	0.307568	0.153784	+	0.988105i	$0.450854\pi$
$110$	0	0
$111$	10.6056	1.00663
$112$	0	0
$113$	13.2111	1.24280	0.621398	−	0.783495i	$-0.286565\pi$
$113$	13.2111	1.24280	0.621398	+	0.783495i	$0.286565\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−2.60555	−0.240883
$118$	0	0
$119$	−9.21110	−0.844380
$120$	0	0
$121$	−4.21110	−0.382828
$122$	0	0
$123$	−0.605551	−0.0546007
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.4222	0.924821	0.462411	−	0.886666i	$-0.346985\pi$
$127$	10.4222	0.924821	0.462411	+	0.886666i	$0.346985\pi$
$128$	0	0
$129$	−3.39445	−0.298865
$130$	0	0
$131$	22.6056	1.97506	0.987528	−	0.157443i	$-0.0503250\pi$
$131$	22.6056	1.97506	0.987528	+	0.157443i	$0.0503250\pi$
$132$	0	0
$133$	−4.60555	−0.399352
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	16.4222	1.40304	0.701522	−	0.712648i	$-0.252504\pi$
$137$	16.4222	1.40304	0.701522	+	0.712648i	$0.252504\pi$
$138$	0	0
$139$	5.21110	0.442000	0.221000	−	0.975274i	$-0.429068\pi$
$139$	5.21110	0.442000	0.221000	+	0.975274i	$0.429068\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	6.78890	0.567716
$144$	0	0
$145$	4.60555	0.382470
$146$	0	0
$147$	14.2111	1.17211
$148$	0	0
$149$	−12.4222	−1.01767	−0.508833	−	0.860865i	$-0.669923\pi$
$149$	−12.4222	−1.01767	−0.508833	+	0.860865i	$0.669923\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	0.788897	0.0629609	0.0314804	−	0.999504i	$-0.489978\pi$
$157$	0.788897	0.0629609	0.0314804	+	0.999504i	$0.489978\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.21110	−0.725937
$162$	0	0
$163$	−19.0278	−1.49037	−0.745184	−	0.666858i	$-0.767639\pi$
$163$	−19.0278	−1.49037	−0.745184	+	0.666858i	$0.767639\pi$
$164$	0	0
$165$	2.60555	0.202842
$166$	0	0
$167$	21.2111	1.64136	0.820682	−	0.571385i	$-0.193594\pi$
$167$	21.2111	1.64136	0.820682	+	0.571385i	$0.193594\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	5.21110	0.396193	0.198096	−	0.980183i	$-0.436524\pi$
$173$	5.21110	0.396193	0.198096	+	0.980183i	$0.436524\pi$
$174$	0	0
$175$	−4.60555	−0.348147
$176$	0	0
$177$	9.21110	0.692349
$178$	0	0
$179$	6.78890	0.507426	0.253713	−	0.967280i	$-0.418348\pi$
$179$	6.78890	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$180$	0	0
$181$	−15.2111	−1.13063	−0.565316	−	0.824874i	$-0.691246\pi$
$181$	−15.2111	−1.13063	−0.565316	+	0.824874i	$0.691246\pi$
$182$	0	0
$183$	7.21110	0.533060
$184$	0	0
$185$	−10.6056	−0.779736
$186$	0	0
$187$	−5.21110	−0.381074
$188$	0	0
$189$	−4.60555	−0.335005
$190$	0	0
$191$	−15.8167	−1.14445	−0.572226	−	0.820096i	$-0.693920\pi$
$191$	−15.8167	−1.14445	−0.572226	+	0.820096i	$0.693920\pi$
$192$	0	0
$193$	6.60555	0.475478	0.237739	−	0.971329i	$-0.423594\pi$
$193$	6.60555	0.475478	0.237739	+	0.971329i	$0.423594\pi$
$194$	0	0
$195$	2.60555	0.186587
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	−2.78890	−0.197700	−0.0988498	−	0.995102i	$-0.531516\pi$
$199$	−2.78890	−0.197700	−0.0988498	+	0.995102i	$0.531516\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	21.2111	1.48873
$204$	0	0
$205$	0.605551	0.0422935
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−2.60555	−0.180230
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−5.21110	−0.357059
$214$	0	0
$215$	3.39445	0.231499
$216$	0	0
$217$	18.4222	1.25058
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−5.21110	−0.350537
$222$	0	0
$223$	−18.4222	−1.23364	−0.616821	−	0.787103i	$-0.711580\pi$
$223$	−18.4222	−1.23364	−0.616821	+	0.787103i	$0.711580\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.21110	−0.0803837	−0.0401918	−	0.999192i	$-0.512797\pi$
$227$	−1.21110	−0.0803837	−0.0401918	+	0.999192i	$0.512797\pi$
$228$	0	0
$229$	11.2111	0.740851	0.370425	−	0.928862i	$-0.379212\pi$
$229$	11.2111	0.740851	0.370425	+	0.928862i	$0.379212\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	12.4222	0.813806	0.406903	−	0.913471i	$-0.366609\pi$
$233$	12.4222	0.813806	0.406903	+	0.913471i	$0.366609\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−4.18335	−0.270598	−0.135299	−	0.990805i	$-0.543200\pi$
$239$	−4.18335	−0.270598	−0.135299	+	0.990805i	$0.543200\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.2111	−0.907914
$246$	0	0
$247$	−2.60555	−0.165787
$248$	0	0
$249$	−3.21110	−0.203495
$250$	0	0
$251$	27.8167	1.75577	0.877886	−	0.478870i	$-0.158953\pi$
$251$	27.8167	1.75577	0.877886	+	0.478870i	$0.158953\pi$
$252$	0	0
$253$	−5.21110	−0.327619
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	17.2111	1.07360	0.536800	−	0.843710i	$-0.319633\pi$
$257$	17.2111	1.07360	0.536800	+	0.843710i	$0.319633\pi$
$258$	0	0
$259$	−48.8444	−3.03504
$260$	0	0
$261$	−4.60555	−0.285076
$262$	0	0
$263$	7.21110	0.444656	0.222328	−	0.974972i	$-0.428634\pi$
$263$	7.21110	0.444656	0.222328	+	0.974972i	$0.428634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.605551	−0.0370591
$268$	0	0
$269$	−24.6056	−1.50023	−0.750113	−	0.661309i	$-0.770001\pi$
$269$	−24.6056	−1.50023	−0.750113	+	0.661309i	$0.770001\pi$
$270$	0	0
$271$	31.6333	1.92159	0.960793	−	0.277266i	$-0.0894282\pi$
$271$	31.6333	1.92159	0.960793	+	0.277266i	$0.0894282\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	−2.60555	−0.157121
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−32.2389	−1.92321	−0.961605	−	0.274439i	$-0.911508\pi$
$281$	−32.2389	−1.92321	−0.961605	+	0.274439i	$0.911508\pi$
$282$	0	0
$283$	11.3944	0.677330	0.338665	−	0.940907i	$-0.390025\pi$
$283$	11.3944	0.677330	0.338665	+	0.940907i	$0.390025\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	2.78890	0.164623
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	9.39445	0.550712
$292$	0	0
$293$	−23.6333	−1.38067	−0.690336	−	0.723489i	$-0.742537\pi$
$293$	−23.6333	−1.38067	−0.690336	+	0.723489i	$0.742537\pi$
$294$	0	0
$295$	−9.21110	−0.536291
$296$	0	0
$297$	−2.60555	−0.151189
$298$	0	0
$299$	−5.21110	−0.301366
$300$	0	0
$301$	15.6333	0.901089
$302$	0	0
$303$	7.21110	0.414267
$304$	0	0
$305$	−7.21110	−0.412907
$306$	0	0
$307$	−31.6333	−1.80541	−0.902704	−	0.430262i	$-0.858421\pi$
$307$	−31.6333	−1.80541	−0.902704	+	0.430262i	$0.858421\pi$
$308$	0	0
$309$	10.4222	0.592899
$310$	0	0
$311$	−9.39445	−0.532710	−0.266355	−	0.963875i	$-0.585819\pi$
$311$	−9.39445	−0.532710	−0.266355	+	0.963875i	$0.585819\pi$
$312$	0	0
$313$	15.2111	0.859782	0.429891	−	0.902881i	$-0.358552\pi$
$313$	15.2111	0.859782	0.429891	+	0.902881i	$0.358552\pi$
$314$	0	0
$315$	4.60555	0.259493
$316$	0	0
$317$	22.4222	1.25936	0.629678	−	0.776856i	$-0.283187\pi$
$317$	22.4222	1.25936	0.629678	+	0.776856i	$0.283187\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	10.4222	0.581711
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−2.60555	−0.144530
$326$	0	0
$327$	3.21110	0.177574
$328$	0	0
$329$	27.6333	1.52347
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	10.6056	0.581181
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−33.0278	−1.79914	−0.899568	−	0.436780i	$-0.856119\pi$
$337$	−33.0278	−1.79914	−0.899568	+	0.436780i	$0.856119\pi$
$338$	0	0
$339$	13.2111	0.717529
$340$	0	0
$341$	10.4222	0.564394
$342$	0	0
$343$	−33.2111	−1.79323
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	21.6333	1.16134	0.580668	−	0.814140i	$-0.302791\pi$
$347$	21.6333	1.16134	0.580668	+	0.814140i	$0.302791\pi$
$348$	0	0
$349$	−22.8444	−1.22283	−0.611417	−	0.791309i	$-0.709400\pi$
$349$	−22.8444	−1.22283	−0.611417	+	0.791309i	$0.709400\pi$
$350$	0	0
$351$	−2.60555	−0.139074
$352$	0	0
$353$	−28.4222	−1.51276	−0.756381	−	0.654132i	$-0.773034\pi$
$353$	−28.4222	−1.51276	−0.756381	+	0.654132i	$0.773034\pi$
$354$	0	0
$355$	5.21110	0.276577
$356$	0	0
$357$	−9.21110	−0.487503
$358$	0	0
$359$	−7.81665	−0.412547	−0.206274	−	0.978494i	$-0.566134\pi$
$359$	−7.81665	−0.412547	−0.206274	+	0.978494i	$0.566134\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.21110	−0.221026
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	11.0278	0.575644	0.287822	−	0.957684i	$-0.407069\pi$
$367$	11.0278	0.575644	0.287822	+	0.957684i	$0.407069\pi$
$368$	0	0
$369$	−0.605551	−0.0315237
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.39445	0.486426	0.243213	−	0.969973i	$-0.421799\pi$
$373$	9.39445	0.486426	0.243213	+	0.969973i	$0.421799\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	2.42221	0.124420	0.0622102	−	0.998063i	$-0.480185\pi$
$379$	2.42221	0.124420	0.0622102	+	0.998063i	$0.480185\pi$
$380$	0	0
$381$	10.4222	0.533946
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−3.39445	−0.172550
$388$	0	0
$389$	7.57779	0.384209	0.192105	−	0.981374i	$-0.438469\pi$
$389$	7.57779	0.384209	0.192105	+	0.981374i	$0.438469\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	22.6056	1.14030
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−36.4222	−1.82798	−0.913989	−	0.405739i	$-0.867014\pi$
$397$	−36.4222	−1.82798	−0.913989	+	0.405739i	$0.867014\pi$
$398$	0	0
$399$	−4.60555	−0.230566
$400$	0	0
$401$	32.6056	1.62824	0.814122	−	0.580694i	$-0.197219\pi$
$401$	32.6056	1.62824	0.814122	+	0.580694i	$0.197219\pi$
$402$	0	0
$403$	10.4222	0.519167
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−27.6333	−1.36973
$408$	0	0
$409$	−27.2111	−1.34550	−0.672751	−	0.739869i	$-0.734888\pi$
$409$	−27.2111	−1.34550	−0.672751	+	0.739869i	$0.734888\pi$
$410$	0	0
$411$	16.4222	0.810048
$412$	0	0
$413$	−42.4222	−2.08746
$414$	0	0
$415$	3.21110	0.157627
$416$	0	0
$417$	5.21110	0.255189
$418$	0	0
$419$	−29.0278	−1.41810	−0.709049	−	0.705159i	$-0.750876\pi$
$419$	−29.0278	−1.41810	−0.709049	+	0.705159i	$0.750876\pi$
$420$	0	0
$421$	20.4222	0.995317	0.497659	−	0.867373i	$-0.334193\pi$
$421$	20.4222	0.995317	0.497659	+	0.867373i	$0.334193\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−33.2111	−1.60720
$428$	0	0
$429$	6.78890	0.327771
$430$	0	0
$431$	−5.21110	−0.251010	−0.125505	−	0.992093i	$-0.540055\pi$
$431$	−5.21110	−0.251010	−0.125505	+	0.992093i	$0.540055\pi$
$432$	0	0
$433$	−9.39445	−0.451468	−0.225734	−	0.974189i	$-0.572478\pi$
$433$	−9.39445	−0.451468	−0.225734	+	0.974189i	$0.572478\pi$
$434$	0	0
$435$	4.60555	0.220819
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−5.57779	−0.266214	−0.133107	−	0.991102i	$-0.542495\pi$
$439$	−5.57779	−0.266214	−0.133107	+	0.991102i	$0.542495\pi$
$440$	0	0
$441$	14.2111	0.676719
$442$	0	0
$443$	38.8444	1.84555	0.922777	−	0.385335i	$-0.125914\pi$
$443$	38.8444	1.84555	0.922777	+	0.385335i	$0.125914\pi$
$444$	0	0
$445$	0.605551	0.0287059
$446$	0	0
$447$	−12.4222	−0.587550
$448$	0	0
$449$	32.2389	1.52145	0.760723	−	0.649077i	$-0.224845\pi$
$449$	32.2389	1.52145	0.760723	+	0.649077i	$0.224845\pi$
$450$	0	0
$451$	1.57779	0.0742955
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−40.0555	−1.87372	−0.936859	−	0.349708i	$-0.886281\pi$
$457$	−40.0555	−1.87372	−0.936859	+	0.349708i	$0.886281\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	34.8444	1.62287	0.811433	−	0.584445i	$-0.198688\pi$
$461$	34.8444	1.62287	0.811433	+	0.584445i	$0.198688\pi$
$462$	0	0
$463$	3.02776	0.140712	0.0703559	−	0.997522i	$-0.477587\pi$
$463$	3.02776	0.140712	0.0703559	+	0.997522i	$0.477587\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	34.8444	1.61241	0.806204	−	0.591638i	$-0.201519\pi$
$467$	34.8444	1.61241	0.806204	+	0.591638i	$0.201519\pi$
$468$	0	0
$469$	18.4222	0.850658
$470$	0	0
$471$	0.788897	0.0363505
$472$	0	0
$473$	8.84441	0.406666
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.3944	−0.977537	−0.488769	−	0.872413i	$-0.662554\pi$
$479$	−21.3944	−0.977537	−0.488769	+	0.872413i	$0.662554\pi$
$480$	0	0
$481$	−27.6333	−1.25997
$482$	0	0
$483$	−9.21110	−0.419120
$484$	0	0
$485$	−9.39445	−0.426580
$486$	0	0
$487$	−43.6333	−1.97721	−0.988607	−	0.150520i	$-0.951905\pi$
$487$	−43.6333	−1.97721	−0.988607	+	0.150520i	$0.951905\pi$
$488$	0	0
$489$	−19.0278	−0.860465
$490$	0	0
$491$	−38.2389	−1.72570	−0.862848	−	0.505464i	$-0.831321\pi$
$491$	−38.2389	−1.72570	−0.862848	+	0.505464i	$0.831321\pi$
$492$	0	0
$493$	−9.21110	−0.414847
$494$	0	0
$495$	2.60555	0.117111
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	−15.6333	−0.699843	−0.349921	−	0.936779i	$-0.613792\pi$
$499$	−15.6333	−0.699843	−0.349921	+	0.936779i	$0.613792\pi$
$500$	0	0
$501$	21.2111	0.947642
$502$	0	0
$503$	−27.2111	−1.21328	−0.606642	−	0.794975i	$-0.707484\pi$
$503$	−27.2111	−1.21328	−0.606642	+	0.794975i	$0.707484\pi$
$504$	0	0
$505$	−7.21110	−0.320890
$506$	0	0
$507$	−6.21110	−0.275845
$508$	0	0
$509$	29.4500	1.30535	0.652673	−	0.757639i	$-0.273647\pi$
$509$	29.4500	1.30535	0.652673	+	0.757639i	$0.273647\pi$
$510$	0	0
$511$	−27.6333	−1.22243
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−10.4222	−0.459257
$516$	0	0
$517$	15.6333	0.687552
$518$	0	0
$519$	5.21110	0.228742
$520$	0	0
$521$	−25.8167	−1.13105	−0.565524	−	0.824732i	$-0.691326\pi$
$521$	−25.8167	−1.13105	−0.565524	+	0.824732i	$0.691326\pi$
$522$	0	0
$523$	10.7889	0.471766	0.235883	−	0.971782i	$-0.424202\pi$
$523$	10.7889	0.471766	0.235883	+	0.971782i	$0.424202\pi$
$524$	0	0
$525$	−4.60555	−0.201003
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	9.21110	0.399728
$532$	0	0
$533$	1.57779	0.0683419
$534$	0	0
$535$	−10.4222	−0.450591
$536$	0	0
$537$	6.78890	0.292963
$538$	0	0
$539$	−37.0278	−1.59490
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−15.2111	−0.652771
$544$	0	0
$545$	−3.21110	−0.137549
$546$	0	0
$547$	37.2111	1.59103	0.795516	−	0.605933i	$-0.207200\pi$
$547$	37.2111	1.59103	0.795516	+	0.605933i	$0.207200\pi$
$548$	0	0
$549$	7.21110	0.307762
$550$	0	0
$551$	−4.60555	−0.196203
$552$	0	0
$553$	36.8444	1.56678
$554$	0	0
$555$	−10.6056	−0.450181
$556$	0	0
$557$	24.4222	1.03480	0.517401	−	0.855743i	$-0.326900\pi$
$557$	24.4222	1.03480	0.517401	+	0.855743i	$0.326900\pi$
$558$	0	0
$559$	8.84441	0.374079
$560$	0	0
$561$	−5.21110	−0.220013
$562$	0	0
$563$	11.6333	0.490285	0.245143	−	0.969487i	$-0.421165\pi$
$563$	11.6333	0.490285	0.245143	+	0.969487i	$0.421165\pi$
$564$	0	0
$565$	−13.2111	−0.555795
$566$	0	0
$567$	−4.60555	−0.193415
$568$	0	0
$569$	−25.4500	−1.06692	−0.533459	−	0.845826i	$-0.679108\pi$
$569$	−25.4500	−1.06692	−0.533459	+	0.845826i	$0.679108\pi$
$570$	0	0
$571$	35.6333	1.49121	0.745604	−	0.666390i	$-0.232161\pi$
$571$	35.6333	1.49121	0.745604	+	0.666390i	$0.232161\pi$
$572$	0	0
$573$	−15.8167	−0.660750
$574$	0	0
$575$	2.00000	0.0834058
$576$	0	0
$577$	−8.78890	−0.365887	−0.182943	−	0.983123i	$-0.558562\pi$
$577$	−8.78890	−0.365887	−0.182943	+	0.983123i	$0.558562\pi$
$578$	0	0
$579$	6.60555	0.274517
$580$	0	0
$581$	14.7889	0.613547
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.60555	0.107726
$586$	0	0
$587$	16.4222	0.677817	0.338908	−	0.940819i	$-0.389942\pi$
$587$	16.4222	0.677817	0.338908	+	0.940819i	$0.389942\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	−32.4222	−1.33142	−0.665710	−	0.746210i	$-0.731871\pi$
$593$	−32.4222	−1.33142	−0.665710	+	0.746210i	$0.731871\pi$
$594$	0	0
$595$	9.21110	0.377618
$596$	0	0
$597$	−2.78890	−0.114142
$598$	0	0
$599$	20.8444	0.851680	0.425840	−	0.904799i	$-0.359979\pi$
$599$	20.8444	0.851680	0.425840	+	0.904799i	$0.359979\pi$
$600$	0	0
$601$	19.2111	0.783637	0.391819	−	0.920042i	$-0.371846\pi$
$601$	19.2111	0.783637	0.391819	+	0.920042i	$0.371846\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	4.21110	0.171206
$606$	0	0
$607$	−19.6333	−0.796891	−0.398446	−	0.917192i	$-0.630450\pi$
$607$	−19.6333	−0.796891	−0.398446	+	0.917192i	$0.630450\pi$
$608$	0	0
$609$	21.2111	0.859517
$610$	0	0
$611$	15.6333	0.632456
$612$	0	0
$613$	19.2111	0.775929	0.387965	−	0.921674i	$-0.373178\pi$
$613$	19.2111	0.775929	0.387965	+	0.921674i	$0.373178\pi$
$614$	0	0
$615$	0.605551	0.0244182
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	0	0
$619$	47.6333	1.91454	0.957272	−	0.289189i	$-0.0933855\pi$
$619$	47.6333	1.91454	0.957272	+	0.289189i	$0.0933855\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	2.78890	0.111735
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.60555	−0.104056
$628$	0	0
$629$	21.2111	0.845742
$630$	0	0
$631$	20.8444	0.829803	0.414901	−	0.909866i	$-0.363816\pi$
$631$	20.8444	0.829803	0.414901	+	0.909866i	$0.363816\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−10.4222	−0.413593
$636$	0	0
$637$	−37.0278	−1.46709
$638$	0	0
$639$	−5.21110	−0.206148
$640$	0	0
$641$	−29.4500	−1.16320	−0.581602	−	0.813474i	$-0.697574\pi$
$641$	−29.4500	−1.16320	−0.581602	+	0.813474i	$0.697574\pi$
$642$	0	0
$643$	−12.6056	−0.497114	−0.248557	−	0.968617i	$-0.579956\pi$
$643$	−12.6056	−0.497114	−0.248557	+	0.968617i	$0.579956\pi$
$644$	0	0
$645$	3.39445	0.133656
$646$	0	0
$647$	45.6333	1.79403	0.897015	−	0.442000i	$-0.145731\pi$
$647$	45.6333	1.79403	0.897015	+	0.442000i	$0.145731\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	18.4222	0.722023
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	−22.6056	−0.883272
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−16.7889	−0.653012	−0.326506	−	0.945195i	$-0.605871\pi$
$661$	−16.7889	−0.653012	−0.326506	+	0.945195i	$0.605871\pi$
$662$	0	0
$663$	−5.21110	−0.202382
$664$	0	0
$665$	4.60555	0.178596
$666$	0	0
$667$	−9.21110	−0.356655
$668$	0	0
$669$	−18.4222	−0.712244
$670$	0	0
$671$	−18.7889	−0.725337
$672$	0	0
$673$	29.3944	1.13307	0.566536	−	0.824037i	$-0.308283\pi$
$673$	29.3944	1.13307	0.566536	+	0.824037i	$0.308283\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−18.4222	−0.708023	−0.354011	−	0.935241i	$-0.615183\pi$
$677$	−18.4222	−0.708023	−0.354011	+	0.935241i	$0.615183\pi$
$678$	0	0
$679$	−43.2666	−1.66042
$680$	0	0
$681$	−1.21110	−0.0464096
$682$	0	0
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	−16.4222	−0.627460
$686$	0	0
$687$	11.2111	0.427730
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−30.0555	−1.14337	−0.571683	−	0.820475i	$-0.693709\pi$
$691$	−30.0555	−1.14337	−0.571683	+	0.820475i	$0.693709\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	−5.21110	−0.197668
$696$	0	0
$697$	−1.21110	−0.0458738
$698$	0	0
$699$	12.4222	0.469851
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	10.6056	0.399996
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	−33.2111	−1.24903
$708$	0	0
$709$	17.6333	0.662233	0.331116	−	0.943590i	$-0.392575\pi$
$709$	17.6333	0.662233	0.331116	+	0.943590i	$0.392575\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−6.78890	−0.253890
$716$	0	0
$717$	−4.18335	−0.156230
$718$	0	0
$719$	34.2389	1.27689	0.638447	−	0.769666i	$-0.279577\pi$
$719$	34.2389	1.27689	0.638447	+	0.769666i	$0.279577\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	−4.60555	−0.171046
$726$	0	0
$727$	24.6056	0.912569	0.456285	−	0.889834i	$-0.349180\pi$
$727$	24.6056	0.912569	0.456285	+	0.889834i	$0.349180\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.78890	−0.251096
$732$	0	0
$733$	4.42221	0.163338	0.0816689	−	0.996660i	$-0.473975\pi$
$733$	4.42221	0.163338	0.0816689	+	0.996660i	$0.473975\pi$
$734$	0	0
$735$	−14.2111	−0.524184
$736$	0	0
$737$	10.4222	0.383907
$738$	0	0
$739$	−9.57779	−0.352325	−0.176162	−	0.984361i	$-0.556368\pi$
$739$	−9.57779	−0.352325	−0.176162	+	0.984361i	$0.556368\pi$
$740$	0	0
$741$	−2.60555	−0.0957173
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	12.4222	0.455114
$746$	0	0
$747$	−3.21110	−0.117488
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	16.8444	0.614661	0.307331	−	0.951603i	$-0.400564\pi$
$751$	16.8444	0.614661	0.307331	+	0.951603i	$0.400564\pi$
$752$	0	0
$753$	27.8167	1.01370
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	36.0555	1.31046	0.655230	−	0.755429i	$-0.272572\pi$
$757$	36.0555	1.31046	0.655230	+	0.755429i	$0.272572\pi$
$758$	0	0
$759$	−5.21110	−0.189151
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	−14.7889	−0.535394
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	8.42221	0.303712	0.151856	−	0.988403i	$-0.451475\pi$
$769$	8.42221	0.303712	0.151856	+	0.988403i	$0.451475\pi$
$770$	0	0
$771$	17.2111	0.619843
$772$	0	0
$773$	43.2666	1.55619	0.778096	−	0.628145i	$-0.216186\pi$
$773$	43.2666	1.55619	0.778096	+	0.628145i	$0.216186\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−48.8444	−1.75228
$778$	0	0
$779$	−0.605551	−0.0216961
$780$	0	0
$781$	13.5778	0.485852
$782$	0	0
$783$	−4.60555	−0.164589
$784$	0	0
$785$	−0.788897	−0.0281570
$786$	0	0
$787$	−23.6333	−0.842436	−0.421218	−	0.906959i	$-0.638397\pi$
$787$	−23.6333	−0.842436	−0.421218	+	0.906959i	$0.638397\pi$
$788$	0	0
$789$	7.21110	0.256722
$790$	0	0
$791$	−60.8444	−2.16338
$792$	0	0
$793$	−18.7889	−0.667213
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.78890	0.0987878	0.0493939	−	0.998779i	$-0.484271\pi$
$797$	2.78890	0.0987878	0.0493939	+	0.998779i	$0.484271\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−0.605551	−0.0213961
$802$	0	0
$803$	−15.6333	−0.551687
$804$	0	0
$805$	9.21110	0.324649
$806$	0	0
$807$	−24.6056	−0.866156
$808$	0	0
$809$	−19.5778	−0.688319	−0.344159	−	0.938911i	$-0.611836\pi$
$809$	−19.5778	−0.688319	−0.344159	+	0.938911i	$0.611836\pi$
$810$	0	0
$811$	−16.8444	−0.591487	−0.295744	−	0.955267i	$-0.595567\pi$
$811$	−16.8444	−0.591487	−0.295744	+	0.955267i	$0.595567\pi$
$812$	0	0
$813$	31.6333	1.10943
$814$	0	0
$815$	19.0278	0.666513
$816$	0	0
$817$	−3.39445	−0.118757
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	28.7889	1.00474	0.502370	−	0.864653i	$-0.332462\pi$
$821$	28.7889	1.00474	0.502370	+	0.864653i	$0.332462\pi$
$822$	0	0
$823$	26.1833	0.912694	0.456347	−	0.889802i	$-0.349158\pi$
$823$	26.1833	0.912694	0.456347	+	0.889802i	$0.349158\pi$
$824$	0	0
$825$	−2.60555	−0.0907137
$826$	0	0
$827$	−15.6333	−0.543623	−0.271812	−	0.962350i	$-0.587623\pi$
$827$	−15.6333	−0.543623	−0.271812	+	0.962350i	$0.587623\pi$
$828$	0	0
$829$	0.788897	0.0273995	0.0136998	−	0.999906i	$-0.495639\pi$
$829$	0.788897	0.0273995	0.0136998	+	0.999906i	$0.495639\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	28.4222	0.984771
$834$	0	0
$835$	−21.2111	−0.734040
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	15.6333	0.539722	0.269861	−	0.962899i	$-0.413022\pi$
$839$	15.6333	0.539722	0.269861	+	0.962899i	$0.413022\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	0	0
$843$	−32.2389	−1.11037
$844$	0	0
$845$	6.21110	0.213668
$846$	0	0
$847$	19.3944	0.666401
$848$	0	0
$849$	11.3944	0.391056
$850$	0	0
$851$	21.2111	0.727107
$852$	0	0
$853$	−13.6333	−0.466796	−0.233398	−	0.972381i	$-0.574984\pi$
$853$	−13.6333	−0.466796	−0.233398	+	0.972381i	$0.574984\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−22.4222	−0.765928	−0.382964	−	0.923763i	$-0.625097\pi$
$857$	−22.4222	−0.765928	−0.382964	+	0.923763i	$0.625097\pi$
$858$	0	0
$859$	48.8444	1.66655	0.833275	−	0.552859i	$-0.186463\pi$
$859$	48.8444	1.66655	0.833275	+	0.552859i	$0.186463\pi$
$860$	0	0
$861$	2.78890	0.0950454
$862$	0	0
$863$	22.4222	0.763261	0.381630	−	0.924315i	$-0.375363\pi$
$863$	22.4222	0.763261	0.381630	+	0.924315i	$0.375363\pi$
$864$	0	0
$865$	−5.21110	−0.177183
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	20.8444	0.707098
$870$	0	0
$871$	10.4222	0.353143
$872$	0	0
$873$	9.39445	0.317954
$874$	0	0
$875$	4.60555	0.155696
$876$	0	0
$877$	−9.02776	−0.304846	−0.152423	−	0.988315i	$-0.548708\pi$
$877$	−9.02776	−0.304846	−0.152423	+	0.988315i	$0.548708\pi$
$878$	0	0
$879$	−23.6333	−0.797132
$880$	0	0
$881$	44.7889	1.50898	0.754488	−	0.656314i	$-0.227885\pi$
$881$	44.7889	1.50898	0.754488	+	0.656314i	$0.227885\pi$
$882$	0	0
$883$	−6.18335	−0.208086	−0.104043	−	0.994573i	$-0.533178\pi$
$883$	−6.18335	−0.208086	−0.104043	+	0.994573i	$0.533178\pi$
$884$	0	0
$885$	−9.21110	−0.309628
$886$	0	0
$887$	−36.8444	−1.23711	−0.618557	−	0.785740i	$-0.712282\pi$
$887$	−36.8444	−1.23711	−0.618557	+	0.785740i	$0.712282\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−2.60555	−0.0872893
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−6.78890	−0.226928
$896$	0	0
$897$	−5.21110	−0.173994
$898$	0	0
$899$	18.4222	0.614415
$900$	0	0
$901$	0	0
$902$	0	0
$903$	15.6333	0.520244
$904$	0	0
$905$	15.2111	0.505634
$906$	0	0
$907$	42.0555	1.39643	0.698215	−	0.715888i	$-0.253978\pi$
$907$	42.0555	1.39643	0.698215	+	0.715888i	$0.253978\pi$
$908$	0	0
$909$	7.21110	0.239177
$910$	0	0
$911$	18.4222	0.610355	0.305177	−	0.952296i	$-0.401284\pi$
$911$	18.4222	0.610355	0.305177	+	0.952296i	$0.401284\pi$
$912$	0	0
$913$	8.36669	0.276897
$914$	0	0
$915$	−7.21110	−0.238392
$916$	0	0
$917$	−104.111	−3.43805
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−31.6333	−1.04235
$922$	0	0
$923$	13.5778	0.446919
$924$	0	0
$925$	10.6056	0.348708
$926$	0	0
$927$	10.4222	0.342310
$928$	0	0
$929$	−2.36669	−0.0776487	−0.0388243	−	0.999246i	$-0.512361\pi$
$929$	−2.36669	−0.0776487	−0.0388243	+	0.999246i	$0.512361\pi$
$930$	0	0
$931$	14.2111	0.465750
$932$	0	0
$933$	−9.39445	−0.307560
$934$	0	0
$935$	5.21110	0.170421
$936$	0	0
$937$	−4.78890	−0.156446	−0.0782232	−	0.996936i	$-0.524925\pi$
$937$	−4.78890	−0.156446	−0.0782232	+	0.996936i	$0.524925\pi$
$938$	0	0
$939$	15.2111	0.496396
$940$	0	0
$941$	20.2389	0.659768	0.329884	−	0.944021i	$-0.392990\pi$
$941$	20.2389	0.659768	0.329884	+	0.944021i	$0.392990\pi$
$942$	0	0
$943$	−1.21110	−0.0394389
$944$	0	0
$945$	4.60555	0.149819
$946$	0	0
$947$	−7.21110	−0.234329	−0.117165	−	0.993113i	$-0.537381\pi$
$947$	−7.21110	−0.234329	−0.117165	+	0.993113i	$0.537381\pi$
$948$	0	0
$949$	−15.6333	−0.507479
$950$	0	0
$951$	22.4222	0.727090
$952$	0	0
$953$	−9.21110	−0.298377	−0.149188	−	0.988809i	$-0.547666\pi$
$953$	−9.21110	−0.298377	−0.149188	+	0.988809i	$0.547666\pi$
$954$	0	0
$955$	15.8167	0.511815
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	−75.6333	−2.44233
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	10.4222	0.335851
$964$	0	0
$965$	−6.60555	−0.212640
$966$	0	0
$967$	−24.2389	−0.779469	−0.389735	−	0.920927i	$-0.627433\pi$
$967$	−24.2389	−0.779469	−0.389735	+	0.920927i	$0.627433\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	−2.60555	−0.0834444
$976$	0	0
$977$	−22.4222	−0.717350	−0.358675	−	0.933463i	$-0.616771\pi$
$977$	−22.4222	−0.717350	−0.358675	+	0.933463i	$0.616771\pi$
$978$	0	0
$979$	1.57779	0.0504265
$980$	0	0
$981$	3.21110	0.102523
$982$	0	0
$983$	−5.21110	−0.166208	−0.0831042	−	0.996541i	$-0.526483\pi$
$983$	−5.21110	−0.166208	−0.0831042	+	0.996541i	$0.526483\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	27.6333	0.879578
$988$	0	0
$989$	−6.78890	−0.215874
$990$	0	0
$991$	−21.5778	−0.685441	−0.342721	−	0.939437i	$-0.611348\pi$
$991$	−21.5778	−0.685441	−0.342721	+	0.939437i	$0.611348\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	2.78890	0.0884140
$996$	0	0
$997$	−18.3667	−0.581679	−0.290839	−	0.956772i	$-0.593935\pi$
$997$	−18.3667	−0.581679	−0.290839	+	0.956772i	$0.593935\pi$
$998$	0	0
$999$	10.6056	0.335545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bl.1.1		2
4.3	odd	2		1140.2.a.e.1.2	✓	2
12.11	even	2		3420.2.a.i.1.2		2
20.3	even	4		5700.2.f.n.3649.1		4
20.7	even	4		5700.2.f.n.3649.4		4
20.19	odd	2		5700.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.e.1.2	✓	2	4.3	odd	2
3420.2.a.i.1.2		2	12.11	even	2
4560.2.a.bl.1.1		2	1.1	even	1	trivial
5700.2.a.u.1.1		2	20.19	odd	2
5700.2.f.n.3649.1		4	20.3	even	4
5700.2.f.n.3649.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} -2.60555 q^{11} -2.60555 q^{13} -1.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -4.60555 q^{21} +2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.60555 q^{29} -4.00000 q^{31} -2.60555 q^{33} +4.60555 q^{35} +10.6056 q^{37} -2.60555 q^{39} -0.605551 q^{41} -3.39445 q^{43} -1.00000 q^{45} -6.00000 q^{47} +14.2111 q^{49} +2.00000 q^{51} +2.60555 q^{55} +1.00000 q^{57} +9.21110 q^{59} +7.21110 q^{61} -4.60555 q^{63} +2.60555 q^{65} -4.00000 q^{67} +2.00000 q^{69} -5.21110 q^{71} +6.00000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -3.21110 q^{83} -2.00000 q^{85} -4.60555 q^{87} -0.605551 q^{89} +12.0000 q^{91} -4.00000 q^{93} -1.00000 q^{95} +9.39445 q^{97} -2.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 2 q^{45} - 12 q^{47} + 14 q^{49} + 4 q^{51} - 2 q^{55} + 2 q^{57} + 4 q^{59} - 2 q^{63} - 2 q^{65} - 8 q^{67} + 4 q^{69} + 4 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 2 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93} - 2 q^{95} + 26 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 8 * q^31 + 2 * q^33 + 2 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 2 * q^45 - 12 * q^47 + 14 * q^49 + 4 * q^51 - 2 * q^55 + 2 * q^57 + 4 * q^59 - 2 * q^63 - 2 * q^65 - 8 * q^67 + 4 * q^69 + 4 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 - 2 * q^87 + 6 * q^89 + 24 * q^91 - 8 * q^93 - 2 * q^95 + 26 * q^97 + 2 * q^99

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