LMFDB - Embedded newform 4080.2.m.n.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(2449,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4080.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	4080.2449
Dual form			4080.2.m.n.2449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{13} +2.23607i q^{15} +1.00000i q^{17} -4.23607 q^{19} -1.00000 q^{21} -0.472136i q^{23} +5.00000 q^{25} +1.00000i q^{27} +6.23607 q^{29} -2.47214 q^{31} -3.00000i q^{33} +2.23607i q^{35} -4.70820i q^{37} +2.00000 q^{39} -3.76393 q^{41} -8.47214i q^{43} +2.23607 q^{45} -12.7082i q^{47} +6.00000 q^{49} +1.00000 q^{51} -3.00000i q^{53} -6.70820 q^{55} +4.23607i q^{57} -8.94427 q^{59} -10.0000 q^{61} +1.00000i q^{63} -4.47214i q^{65} +5.52786i q^{67} -0.472136 q^{69} -12.9443 q^{71} +1.76393i q^{73} -5.00000i q^{75} -3.00000i q^{77} +2.00000 q^{79} +1.00000 q^{81} +4.94427i q^{83} -2.23607i q^{85} -6.23607i q^{87} +12.0000 q^{89} +2.00000 q^{91} +2.47214i q^{93} +9.47214 q^{95} +13.4164i q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 12 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 8 q^{31} + 8 q^{39} - 24 q^{41} + 24 q^{49} + 4 q^{51} - 40 q^{61} + 16 q^{69} - 16 q^{71} + 8 q^{79} + 4 q^{81} + 48 q^{89} + 8 q^{91} + 20 q^{95} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 12 * q^11 - 8 * q^19 - 4 * q^21 + 20 * q^25 + 16 * q^29 + 8 * q^31 + 8 * q^39 - 24 * q^41 + 24 * q^49 + 4 * q^51 - 40 * q^61 + 16 * q^69 - 16 * q^71 + 8 * q^79 + 4 * q^81 + 48 * q^89 + 8 * q^91 + 20 * q^95 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	2.23607i	0.577350i
$16$	0	0
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	0	0
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	− 0.472136i	− 0.0984472i	−0.998788	−	0.0492236i	$-0.984325\pi$
$23$	− 0.472136i	− 0.0984472i	0.998788	−	0.0492236i	$-0.0156747\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	2.23607i	0.377964i
$36$	0	0
$37$	− 4.70820i	− 0.774024i	−0.922075	−	0.387012i	$-0.873507\pi$
$37$	− 4.70820i	− 0.774024i	0.922075	−	0.387012i	$-0.126493\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−3.76393	−0.587827	−0.293914	−	0.955832i	$-0.594958\pi$
$41$	−3.76393	−0.587827	−0.293914	+	0.955832i	$0.594958\pi$
$42$	0	0
$43$	− 8.47214i	− 1.29199i	−0.763342	−	0.645994i	$-0.776443\pi$
$43$	− 8.47214i	− 1.29199i	0.763342	−	0.645994i	$-0.223557\pi$
$44$	0	0
$45$	2.23607	0.333333
$46$	0	0
$47$	− 12.7082i	− 1.85368i	−0.375454	−	0.926841i	$-0.622513\pi$
$47$	− 12.7082i	− 1.85368i	0.375454	−	0.926841i	$-0.377487\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	0	0
$55$	−6.70820	−0.904534
$56$	0	0
$57$	4.23607i	0.561081i
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	− 4.47214i	− 0.554700i
$66$	0	0
$67$	5.52786i	0.675336i	0.941265	+	0.337668i	$0.109638\pi$
$67$	5.52786i	0.675336i	−0.941265	+	0.337668i	$0.890362\pi$
$68$	0	0
$69$	−0.472136	−0.0568385
$70$	0	0
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	0	0
$73$	1.76393i	0.206453i	0.994658	+	0.103226i	$0.0329166\pi$
$73$	1.76393i	0.206453i	−0.994658	+	0.103226i	$0.967083\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.94427i	0.542704i	0.962480	+	0.271352i	$0.0874708\pi$
$83$	4.94427i	0.542704i	−0.962480	+	0.271352i	$0.912529\pi$
$84$	0	0
$85$	− 2.23607i	− 0.242536i
$86$	0	0
$87$	− 6.23607i	− 0.668577i
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	2.47214i	0.256349i
$94$	0	0
$95$	9.47214	0.971821
$96$	0	0
$97$	13.4164i	1.36223i	0.732177	+	0.681115i	$0.238505\pi$
$97$	13.4164i	1.36223i	−0.732177	+	0.681115i	$0.761495\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−14.4721	−1.44003	−0.720016	−	0.693958i	$-0.755865\pi$
$101$	−14.4721	−1.44003	−0.720016	+	0.693958i	$0.755865\pi$
$102$	0	0
$103$	10.9443i	1.07837i	0.842187	+	0.539186i	$0.181268\pi$
$103$	10.9443i	1.07837i	−0.842187	+	0.539186i	$0.818732\pi$
$104$	0	0
$105$	2.23607	0.218218
$106$	0	0
$107$	6.47214i	0.625685i	0.949805	+	0.312842i	$0.101281\pi$
$107$	6.47214i	0.625685i	−0.949805	+	0.312842i	$0.898719\pi$
$108$	0	0
$109$	−2.47214	−0.236788	−0.118394	−	0.992967i	$-0.537775\pi$
$109$	−2.47214	−0.236788	−0.118394	+	0.992967i	$0.537775\pi$
$110$	0	0
$111$	−4.70820	−0.446883
$112$	0	0
$113$	− 18.9443i	− 1.78213i	−0.453878	−	0.891064i	$-0.649960\pi$
$113$	− 18.9443i	− 1.78213i	0.453878	−	0.891064i	$-0.350040\pi$
$114$	0	0
$115$	1.05573i	0.0984472i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	3.76393i	0.339382i
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	0	0
$129$	−8.47214	−0.745930
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	4.23607i	0.367314i
$134$	0	0
$135$	− 2.23607i	− 0.192450i
$136$	0	0
$137$	− 10.4164i	− 0.889934i	−0.895547	−	0.444967i	$-0.853215\pi$
$137$	− 10.4164i	− 0.889934i	0.895547	−	0.444967i	$-0.146785\pi$
$138$	0	0
$139$	4.94427	0.419368	0.209684	−	0.977769i	$-0.432757\pi$
$139$	4.94427	0.419368	0.209684	+	0.977769i	$0.432757\pi$
$140$	0	0
$141$	−12.7082	−1.07022
$142$	0	0
$143$	6.00000i	0.501745i
$144$	0	0
$145$	−13.9443	−1.15801
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	−10.4721	−0.857911	−0.428955	−	0.903326i	$-0.641118\pi$
$149$	−10.4721	−0.857911	−0.428955	+	0.903326i	$0.641118\pi$
$150$	0	0
$151$	−18.2361	−1.48403	−0.742015	−	0.670383i	$-0.766130\pi$
$151$	−18.2361	−1.48403	−0.742015	+	0.670383i	$0.766130\pi$
$152$	0	0
$153$	− 1.00000i	− 0.0808452i
$154$	0	0
$155$	5.52786	0.444009
$156$	0	0
$157$	7.41641i	0.591894i	0.955204	+	0.295947i	$0.0956351\pi$
$157$	7.41641i	0.591894i	−0.955204	+	0.295947i	$0.904365\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−0.472136	−0.0372095
$162$	0	0
$163$	10.4164i	0.815876i	0.913010	+	0.407938i	$0.133752\pi$
$163$	10.4164i	0.815876i	−0.913010	+	0.407938i	$0.866248\pi$
$164$	0	0
$165$	6.70820i	0.522233i
$166$	0	0
$167$	− 9.52786i	− 0.737288i	−0.929571	−	0.368644i	$-0.879822\pi$
$167$	− 9.52786i	− 0.737288i	0.929571	−	0.368644i	$-0.120178\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.23607	0.323940
$172$	0	0
$173$	− 24.3607i	− 1.85211i	−0.377391	−	0.926054i	$-0.623179\pi$
$173$	− 24.3607i	− 1.85211i	0.377391	−	0.926054i	$-0.376821\pi$
$174$	0	0
$175$	− 5.00000i	− 0.377964i
$176$	0	0
$177$	8.94427i	0.672293i
$178$	0	0
$179$	−10.4721	−0.782724	−0.391362	−	0.920237i	$-0.627996\pi$
$179$	−10.4721	−0.782724	−0.391362	+	0.920237i	$0.627996\pi$
$180$	0	0
$181$	−21.8885	−1.62696	−0.813481	−	0.581591i	$-0.802430\pi$
$181$	−21.8885	−1.62696	−0.813481	+	0.581591i	$0.802430\pi$
$182$	0	0
$183$	10.0000i	0.739221i
$184$	0	0
$185$	10.5279i	0.774024i
$186$	0	0
$187$	3.00000i	0.219382i
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	4.47214	0.323592	0.161796	−	0.986824i	$-0.448271\pi$
$191$	4.47214	0.323592	0.161796	+	0.986824i	$0.448271\pi$
$192$	0	0
$193$	17.4164i	1.25366i	0.779156	+	0.626830i	$0.215648\pi$
$193$	17.4164i	1.25366i	−0.779156	+	0.626830i	$0.784352\pi$
$194$	0	0
$195$	−4.47214	−0.320256
$196$	0	0
$197$	13.5279i	0.963820i	0.876221	+	0.481910i	$0.160057\pi$
$197$	13.5279i	0.963820i	−0.876221	+	0.481910i	$0.839943\pi$
$198$	0	0
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	5.52786	0.389905
$202$	0	0
$203$	− 6.23607i	− 0.437686i
$204$	0	0
$205$	8.41641	0.587827
$206$	0	0
$207$	0.472136i	0.0328157i
$208$	0	0
$209$	−12.7082	−0.879045
$210$	0	0
$211$	−26.3607	−1.81474	−0.907372	−	0.420328i	$-0.861915\pi$
$211$	−26.3607	−1.81474	−0.907372	+	0.420328i	$0.861915\pi$
$212$	0	0
$213$	12.9443i	0.886927i
$214$	0	0
$215$	18.9443i	1.29199i
$216$	0	0
$217$	2.47214i	0.167820i
$218$	0	0
$219$	1.76393	0.119195
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	− 25.4164i	− 1.70201i	−0.525159	−	0.851004i	$-0.675994\pi$
$223$	− 25.4164i	− 1.70201i	0.525159	−	0.851004i	$-0.324006\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	− 20.9443i	− 1.39012i	−0.718952	−	0.695060i	$-0.755378\pi$
$227$	− 20.9443i	− 1.39012i	0.718952	−	0.695060i	$-0.244622\pi$
$228$	0	0
$229$	−10.4164	−0.688336	−0.344168	−	0.938908i	$-0.611839\pi$
$229$	−10.4164	−0.688336	−0.344168	+	0.938908i	$0.611839\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	28.4164i	1.85368i
$236$	0	0
$237$	− 2.00000i	− 0.129914i
$238$	0	0
$239$	−8.47214	−0.548017	−0.274008	−	0.961727i	$-0.588350\pi$
$239$	−8.47214	−0.548017	−0.274008	+	0.961727i	$0.588350\pi$
$240$	0	0
$241$	−12.9443	−0.833814	−0.416907	−	0.908949i	$-0.636886\pi$
$241$	−12.9443	−0.833814	−0.416907	+	0.908949i	$0.636886\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	−13.4164	−0.857143
$246$	0	0
$247$	− 8.47214i	− 0.539069i
$248$	0	0
$249$	4.94427	0.313331
$250$	0	0
$251$	−23.8885	−1.50783	−0.753916	−	0.656971i	$-0.771837\pi$
$251$	−23.8885	−1.50783	−0.753916	+	0.656971i	$0.771837\pi$
$252$	0	0
$253$	− 1.41641i	− 0.0890488i
$254$	0	0
$255$	−2.23607	−0.140028
$256$	0	0
$257$	− 30.9443i	− 1.93025i	−0.261788	−	0.965125i	$-0.584312\pi$
$257$	− 30.9443i	− 1.93025i	0.261788	−	0.965125i	$-0.415688\pi$
$258$	0	0
$259$	−4.70820	−0.292554
$260$	0	0
$261$	−6.23607	−0.386003
$262$	0	0
$263$	− 9.65248i	− 0.595197i	−0.954691	−	0.297599i	$-0.903814\pi$
$263$	− 9.65248i	− 0.595197i	0.954691	−	0.297599i	$-0.0961857\pi$
$264$	0	0
$265$	6.70820i	0.412082i
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	−15.1803	−0.925562	−0.462781	−	0.886473i	$-0.653148\pi$
$269$	−15.1803	−0.925562	−0.462781	+	0.886473i	$0.653148\pi$
$270$	0	0
$271$	13.8885	0.843669	0.421834	−	0.906673i	$-0.361386\pi$
$271$	13.8885	0.843669	0.421834	+	0.906673i	$0.361386\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	0	0
$275$	15.0000	0.904534
$276$	0	0
$277$	− 2.58359i	− 0.155233i	−0.996983	−	0.0776165i	$-0.975269\pi$
$277$	− 2.58359i	− 0.155233i	0.996983	−	0.0776165i	$-0.0247310\pi$
$278$	0	0
$279$	2.47214	0.148003
$280$	0	0
$281$	30.3607	1.81117	0.905583	−	0.424169i	$-0.139434\pi$
$281$	30.3607	1.81117	0.905583	+	0.424169i	$0.139434\pi$
$282$	0	0
$283$	− 8.52786i	− 0.506929i	−0.967345	−	0.253464i	$-0.918430\pi$
$283$	− 8.52786i	− 0.506929i	0.967345	−	0.253464i	$-0.0815701\pi$
$284$	0	0
$285$	− 9.47214i	− 0.561081i
$286$	0	0
$287$	3.76393i	0.222178i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	13.4164	0.786484
$292$	0	0
$293$	26.8885i	1.57085i	0.618960	+	0.785423i	$0.287554\pi$
$293$	26.8885i	1.57085i	−0.618960	+	0.785423i	$0.712446\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	0	0
$297$	3.00000i	0.174078i
$298$	0	0
$299$	0.944272	0.0546087
$300$	0	0
$301$	−8.47214	−0.488326
$302$	0	0
$303$	14.4721i	0.831402i
$304$	0	0
$305$	22.3607	1.28037
$306$	0	0
$307$	18.8328i	1.07485i	0.843313	+	0.537423i	$0.180602\pi$
$307$	18.8328i	1.07485i	−0.843313	+	0.537423i	$0.819398\pi$
$308$	0	0
$309$	10.9443	0.622598
$310$	0	0
$311$	−19.4721	−1.10416	−0.552082	−	0.833790i	$-0.686166\pi$
$311$	−19.4721	−1.10416	−0.552082	+	0.833790i	$0.686166\pi$
$312$	0	0
$313$	6.70820i	0.379170i	0.981864	+	0.189585i	$0.0607143\pi$
$313$	6.70820i	0.379170i	−0.981864	+	0.189585i	$0.939286\pi$
$314$	0	0
$315$	− 2.23607i	− 0.125988i
$316$	0	0
$317$	− 20.4721i	− 1.14983i	−0.818213	−	0.574915i	$-0.805035\pi$
$317$	− 20.4721i	− 1.14983i	0.818213	−	0.574915i	$-0.194965\pi$
$318$	0	0
$319$	18.7082	1.04746
$320$	0	0
$321$	6.47214	0.361239
$322$	0	0
$323$	− 4.23607i	− 0.235701i
$324$	0	0
$325$	10.0000i	0.554700i
$326$	0	0
$327$	2.47214i	0.136709i
$328$	0	0
$329$	−12.7082	−0.700626
$330$	0	0
$331$	30.1246	1.65580	0.827899	−	0.560877i	$-0.189536\pi$
$331$	30.1246	1.65580	0.827899	+	0.560877i	$0.189536\pi$
$332$	0	0
$333$	4.70820i	0.258008i
$334$	0	0
$335$	− 12.3607i	− 0.675336i
$336$	0	0
$337$	0.347524i	0.0189308i	0.999955	+	0.00946542i	$0.00301298\pi$
$337$	0.347524i	0.0189308i	−0.999955	+	0.00946542i	$0.996987\pi$
$338$	0	0
$339$	−18.9443	−1.02891
$340$	0	0
$341$	−7.41641	−0.401621
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	1.05573	0.0568385
$346$	0	0
$347$	1.52786i	0.0820200i	0.999159	+	0.0410100i	$0.0130576\pi$
$347$	1.52786i	0.0820200i	−0.999159	+	0.0410100i	$0.986942\pi$
$348$	0	0
$349$	5.58359	0.298883	0.149441	−	0.988771i	$-0.452252\pi$
$349$	5.58359	0.298883	0.149441	+	0.988771i	$0.452252\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	25.4721i	1.35574i	0.735179	+	0.677872i	$0.237098\pi$
$353$	25.4721i	1.35574i	−0.735179	+	0.677872i	$0.762902\pi$
$354$	0	0
$355$	28.9443	1.53620
$356$	0	0
$357$	− 1.00000i	− 0.0529256i
$358$	0	0
$359$	23.8885	1.26079	0.630395	−	0.776275i	$-0.282893\pi$
$359$	23.8885	1.26079	0.630395	+	0.776275i	$0.282893\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	0	0
$363$	2.00000i	0.104973i
$364$	0	0
$365$	− 3.94427i	− 0.206453i
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	3.76393	0.195942
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	8.47214i	0.438671i	0.975650	+	0.219335i	$0.0703889\pi$
$373$	8.47214i	0.438671i	−0.975650	+	0.219335i	$0.929611\pi$
$374$	0	0
$375$	11.1803i	0.577350i
$376$	0	0
$377$	12.4721i	0.642348i
$378$	0	0
$379$	−8.36068	−0.429459	−0.214730	−	0.976674i	$-0.568887\pi$
$379$	−8.36068	−0.429459	−0.214730	+	0.976674i	$0.568887\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	− 25.6525i	− 1.31078i	−0.755291	−	0.655390i	$-0.772504\pi$
$383$	− 25.6525i	− 1.31078i	0.755291	−	0.655390i	$-0.227496\pi$
$384$	0	0
$385$	6.70820i	0.341882i
$386$	0	0
$387$	8.47214i	0.430663i
$388$	0	0
$389$	−16.3607	−0.829519	−0.414760	−	0.909931i	$-0.636134\pi$
$389$	−16.3607	−0.829519	−0.414760	+	0.909931i	$0.636134\pi$
$390$	0	0
$391$	0.472136	0.0238769
$392$	0	0
$393$	− 0.944272i	− 0.0476322i
$394$	0	0
$395$	−4.47214	−0.225018
$396$	0	0
$397$	− 11.7639i	− 0.590415i	−0.955433	−	0.295207i	$-0.904611\pi$
$397$	− 11.7639i	− 0.590415i	0.955433	−	0.295207i	$-0.0953888\pi$
$398$	0	0
$399$	4.23607	0.212069
$400$	0	0
$401$	24.7082	1.23387	0.616934	−	0.787015i	$-0.288374\pi$
$401$	24.7082	1.23387	0.616934	+	0.787015i	$0.288374\pi$
$402$	0	0
$403$	− 4.94427i	− 0.246292i
$404$	0	0
$405$	−2.23607	−0.111111
$406$	0	0
$407$	− 14.1246i	− 0.700131i
$408$	0	0
$409$	17.8328	0.881776	0.440888	−	0.897562i	$-0.354664\pi$
$409$	17.8328	0.881776	0.440888	+	0.897562i	$0.354664\pi$
$410$	0	0
$411$	−10.4164	−0.513804
$412$	0	0
$413$	8.94427i	0.440119i
$414$	0	0
$415$	− 11.0557i	− 0.542704i
$416$	0	0
$417$	− 4.94427i	− 0.242122i
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−22.4164	−1.09251	−0.546254	−	0.837619i	$-0.683947\pi$
$421$	−22.4164	−1.09251	−0.546254	+	0.837619i	$0.683947\pi$
$422$	0	0
$423$	12.7082i	0.617894i
$424$	0	0
$425$	5.00000i	0.242536i
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	18.5279	0.892456	0.446228	−	0.894919i	$-0.352767\pi$
$431$	18.5279	0.892456	0.446228	+	0.894919i	$0.352767\pi$
$432$	0	0
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	0	0
$435$	13.9443i	0.668577i
$436$	0	0
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	−25.4164	−1.21306	−0.606529	−	0.795061i	$-0.707439\pi$
$439$	−25.4164	−1.21306	−0.606529	+	0.795061i	$0.707439\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	12.9443i	0.615001i	0.951548	+	0.307500i	$0.0994926\pi$
$443$	12.9443i	0.615001i	−0.951548	+	0.307500i	$0.900507\pi$
$444$	0	0
$445$	−26.8328	−1.27200
$446$	0	0
$447$	10.4721i	0.495315i
$448$	0	0
$449$	25.4164	1.19947	0.599737	−	0.800197i	$-0.295272\pi$
$449$	25.4164	1.19947	0.599737	+	0.800197i	$0.295272\pi$
$450$	0	0
$451$	−11.2918	−0.531710
$452$	0	0
$453$	18.2361i	0.856805i
$454$	0	0
$455$	−4.47214	−0.209657
$456$	0	0
$457$	− 34.0000i	− 1.59045i	−0.606313	−	0.795226i	$-0.707352\pi$
$457$	− 34.0000i	− 1.59045i	0.606313	−	0.795226i	$-0.292648\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−14.9443	−0.696024	−0.348012	−	0.937490i	$-0.613143\pi$
$461$	−14.9443	−0.696024	−0.348012	+	0.937490i	$0.613143\pi$
$462$	0	0
$463$	23.4164i	1.08825i	0.839003	+	0.544126i	$0.183139\pi$
$463$	23.4164i	1.08825i	−0.839003	+	0.544126i	$0.816861\pi$
$464$	0	0
$465$	− 5.52786i	− 0.256349i
$466$	0	0
$467$	10.2361i	0.473669i	0.971550	+	0.236834i	$0.0761099\pi$
$467$	10.2361i	0.473669i	−0.971550	+	0.236834i	$0.923890\pi$
$468$	0	0
$469$	5.52786	0.255253
$470$	0	0
$471$	7.41641	0.341730
$472$	0	0
$473$	− 25.4164i	− 1.16865i
$474$	0	0
$475$	−21.1803	−0.971821
$476$	0	0
$477$	3.00000i	0.137361i
$478$	0	0
$479$	6.11146	0.279240	0.139620	−	0.990205i	$-0.455412\pi$
$479$	6.11146	0.279240	0.139620	+	0.990205i	$0.455412\pi$
$480$	0	0
$481$	9.41641	0.429351
$482$	0	0
$483$	0.472136i	0.0214829i
$484$	0	0
$485$	− 30.0000i	− 1.36223i
$486$	0	0
$487$	14.8328i	0.672139i	0.941837	+	0.336070i	$0.109098\pi$
$487$	14.8328i	0.672139i	−0.941837	+	0.336070i	$0.890902\pi$
$488$	0	0
$489$	10.4164	0.471046
$490$	0	0
$491$	−20.8328	−0.940172	−0.470086	−	0.882621i	$-0.655777\pi$
$491$	−20.8328	−0.940172	−0.470086	+	0.882621i	$0.655777\pi$
$492$	0	0
$493$	6.23607i	0.280858i
$494$	0	0
$495$	6.70820	0.301511
$496$	0	0
$497$	12.9443i	0.580630i
$498$	0	0
$499$	6.36068	0.284743	0.142372	−	0.989813i	$-0.454527\pi$
$499$	6.36068	0.284743	0.142372	+	0.989813i	$0.454527\pi$
$500$	0	0
$501$	−9.52786	−0.425674
$502$	0	0
$503$	− 1.88854i	− 0.0842060i	−0.999113	−	0.0421030i	$-0.986594\pi$
$503$	− 1.88854i	− 0.0842060i	0.999113	−	0.0421030i	$-0.0134058\pi$
$504$	0	0
$505$	32.3607	1.44003
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	36.9443	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$509$	36.9443	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$510$	0	0
$511$	1.76393	0.0780318
$512$	0	0
$513$	− 4.23607i	− 0.187027i
$514$	0	0
$515$	− 24.4721i	− 1.07837i
$516$	0	0
$517$	− 38.1246i	− 1.67672i
$518$	0	0
$519$	−24.3607	−1.06932
$520$	0	0
$521$	−20.2361	−0.886558	−0.443279	−	0.896384i	$-0.646185\pi$
$521$	−20.2361	−0.886558	−0.443279	+	0.896384i	$0.646185\pi$
$522$	0	0
$523$	40.8328i	1.78549i	0.450558	+	0.892747i	$0.351225\pi$
$523$	40.8328i	1.78549i	−0.450558	+	0.892747i	$0.648775\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	− 2.47214i	− 0.107688i
$528$	0	0
$529$	22.7771	0.990308
$530$	0	0
$531$	8.94427	0.388148
$532$	0	0
$533$	− 7.52786i	− 0.326068i
$534$	0	0
$535$	− 14.4721i	− 0.625685i
$536$	0	0
$537$	10.4721i	0.451906i
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	23.4164	1.00675	0.503375	−	0.864068i	$-0.332091\pi$
$541$	23.4164	1.00675	0.503375	+	0.864068i	$0.332091\pi$
$542$	0	0
$543$	21.8885i	0.939327i
$544$	0	0
$545$	5.52786	0.236788
$546$	0	0
$547$	− 0.416408i	− 0.0178043i	−0.999960	−	0.00890216i	$-0.997166\pi$
$547$	− 0.416408i	− 0.0178043i	0.999960	−	0.00890216i	$-0.00283368\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−26.4164	−1.12538
$552$	0	0
$553$	− 2.00000i	− 0.0850487i
$554$	0	0
$555$	10.5279	0.446883
$556$	0	0
$557$	8.11146i	0.343693i	0.985124	+	0.171847i	$0.0549734\pi$
$557$	8.11146i	0.343693i	−0.985124	+	0.171847i	$0.945027\pi$
$558$	0	0
$559$	16.9443	0.716666
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	20.1246i	0.848151i	0.905627	+	0.424076i	$0.139401\pi$
$563$	20.1246i	0.848151i	−0.905627	+	0.424076i	$0.860599\pi$
$564$	0	0
$565$	42.3607i	1.78213i
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	−8.11146	−0.340050	−0.170025	−	0.985440i	$-0.554385\pi$
$569$	−8.11146	−0.340050	−0.170025	+	0.985440i	$0.554385\pi$
$570$	0	0
$571$	−40.8328	−1.70880	−0.854400	−	0.519616i	$-0.826075\pi$
$571$	−40.8328	−1.70880	−0.854400	+	0.519616i	$0.826075\pi$
$572$	0	0
$573$	− 4.47214i	− 0.186826i
$574$	0	0
$575$	− 2.36068i	− 0.0984472i
$576$	0	0
$577$	− 33.3050i	− 1.38650i	−0.720696	−	0.693252i	$-0.756177\pi$
$577$	− 33.3050i	− 1.38650i	0.720696	−	0.693252i	$-0.243823\pi$
$578$	0	0
$579$	17.4164	0.723801
$580$	0	0
$581$	4.94427	0.205123
$582$	0	0
$583$	− 9.00000i	− 0.372742i
$584$	0	0
$585$	4.47214i	0.184900i
$586$	0	0
$587$	2.23607i	0.0922924i	0.998935	+	0.0461462i	$0.0146940\pi$
$587$	2.23607i	0.0922924i	−0.998935	+	0.0461462i	$0.985306\pi$
$588$	0	0
$589$	10.4721	0.431497
$590$	0	0
$591$	13.5279	0.556462
$592$	0	0
$593$	17.4721i	0.717495i	0.933435	+	0.358747i	$0.116796\pi$
$593$	17.4721i	0.717495i	−0.933435	+	0.358747i	$0.883204\pi$
$594$	0	0
$595$	−2.23607	−0.0916698
$596$	0	0
$597$	13.4164i	0.549097i
$598$	0	0
$599$	4.47214	0.182727	0.0913633	−	0.995818i	$-0.470878\pi$
$599$	4.47214	0.182727	0.0913633	+	0.995818i	$0.470878\pi$
$600$	0	0
$601$	−0.583592	−0.0238052	−0.0119026	−	0.999929i	$-0.503789\pi$
$601$	−0.583592	−0.0238052	−0.0119026	+	0.999929i	$0.503789\pi$
$602$	0	0
$603$	− 5.52786i	− 0.225112i
$604$	0	0
$605$	4.47214	0.181818
$606$	0	0
$607$	− 46.8885i	− 1.90315i	−0.307421	−	0.951574i	$-0.599466\pi$
$607$	− 46.8885i	− 1.90315i	0.307421	−	0.951574i	$-0.400534\pi$
$608$	0	0
$609$	−6.23607	−0.252698
$610$	0	0
$611$	25.4164	1.02824
$612$	0	0
$613$	− 42.9443i	− 1.73450i	−0.497870	−	0.867251i	$-0.665885\pi$
$613$	− 42.9443i	− 1.73450i	0.497870	−	0.867251i	$-0.334115\pi$
$614$	0	0
$615$	− 8.41641i	− 0.339382i
$616$	0	0
$617$	11.8885i	0.478615i	0.970944	+	0.239307i	$0.0769204\pi$
$617$	11.8885i	0.478615i	−0.970944	+	0.239307i	$0.923080\pi$
$618$	0	0
$619$	−7.88854	−0.317067	−0.158534	−	0.987354i	$-0.550677\pi$
$619$	−7.88854	−0.317067	−0.158534	+	0.987354i	$0.550677\pi$
$620$	0	0
$621$	0.472136	0.0189462
$622$	0	0
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	12.7082i	0.507517i
$628$	0	0
$629$	4.70820	0.187728
$630$	0	0
$631$	−8.12461	−0.323436	−0.161718	−	0.986837i	$-0.551703\pi$
$631$	−8.12461	−0.323436	−0.161718	+	0.986837i	$0.551703\pi$
$632$	0	0
$633$	26.3607i	1.04774i
$634$	0	0
$635$	− 13.4164i	− 0.532414i
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	12.9443	0.512067
$640$	0	0
$641$	−23.2918	−0.919971	−0.459985	−	0.887927i	$-0.652145\pi$
$641$	−23.2918	−0.919971	−0.459985	+	0.887927i	$0.652145\pi$
$642$	0	0
$643$	− 49.2492i	− 1.94220i	−0.238673	−	0.971100i	$-0.576713\pi$
$643$	− 49.2492i	− 1.94220i	0.238673	−	0.971100i	$-0.423287\pi$
$644$	0	0
$645$	18.9443	0.745930
$646$	0	0
$647$	− 36.9443i	− 1.45243i	−0.687468	−	0.726215i	$-0.741278\pi$
$647$	− 36.9443i	− 1.45243i	0.687468	−	0.726215i	$-0.258722\pi$
$648$	0	0
$649$	−26.8328	−1.05328
$650$	0	0
$651$	2.47214	0.0968906
$652$	0	0
$653$	− 33.3050i	− 1.30332i	−0.758510	−	0.651662i	$-0.774072\pi$
$653$	− 33.3050i	− 1.30332i	0.758510	−	0.651662i	$-0.225928\pi$
$654$	0	0
$655$	−2.11146	−0.0825014
$656$	0	0
$657$	− 1.76393i	− 0.0688175i
$658$	0	0
$659$	26.8328	1.04526	0.522629	−	0.852560i	$-0.324951\pi$
$659$	26.8328	1.04526	0.522629	+	0.852560i	$0.324951\pi$
$660$	0	0
$661$	−1.58359	−0.0615946	−0.0307973	−	0.999526i	$-0.509805\pi$
$661$	−1.58359	−0.0615946	−0.0307973	+	0.999526i	$0.509805\pi$
$662$	0	0
$663$	2.00000i	0.0776736i
$664$	0	0
$665$	− 9.47214i	− 0.367314i
$666$	0	0
$667$	− 2.94427i	− 0.114003i
$668$	0	0
$669$	−25.4164	−0.982655
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	− 3.52786i	− 0.135989i	−0.997686	−	0.0679946i	$-0.978340\pi$
$673$	− 3.52786i	− 0.135989i	0.997686	−	0.0679946i	$-0.0216601\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	13.0557i	0.501772i	0.968017	+	0.250886i	$0.0807220\pi$
$677$	13.0557i	0.501772i	−0.968017	+	0.250886i	$0.919278\pi$
$678$	0	0
$679$	13.4164	0.514874
$680$	0	0
$681$	−20.9443	−0.802586
$682$	0	0
$683$	− 29.7771i	− 1.13939i	−0.821857	−	0.569694i	$-0.807062\pi$
$683$	− 29.7771i	− 1.13939i	0.821857	−	0.569694i	$-0.192938\pi$
$684$	0	0
$685$	23.2918i	0.889934i
$686$	0	0
$687$	10.4164i	0.397411i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	46.2492	1.75940	0.879702	−	0.475526i	$-0.157742\pi$
$691$	46.2492	1.75940	0.879702	+	0.475526i	$0.157742\pi$
$692$	0	0
$693$	3.00000i	0.113961i
$694$	0	0
$695$	−11.0557	−0.419368
$696$	0	0
$697$	− 3.76393i	− 0.142569i
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	19.9443i	0.752212i
$704$	0	0
$705$	28.4164	1.07022
$706$	0	0
$707$	14.4721i	0.544281i
$708$	0	0
$709$	37.4164	1.40520	0.702601	−	0.711584i	$-0.252022\pi$
$709$	37.4164	1.40520	0.702601	+	0.711584i	$0.252022\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	1.16718i	0.0437114i
$714$	0	0
$715$	− 13.4164i	− 0.501745i
$716$	0	0
$717$	8.47214i	0.316398i
$718$	0	0
$719$	28.3050	1.05560	0.527798	−	0.849370i	$-0.323018\pi$
$719$	28.3050	1.05560	0.527798	+	0.849370i	$0.323018\pi$
$720$	0	0
$721$	10.9443	0.407586
$722$	0	0
$723$	12.9443i	0.481403i
$724$	0	0
$725$	31.1803	1.15801
$726$	0	0
$727$	27.3050i	1.01268i	0.862333	+	0.506342i	$0.169003\pi$
$727$	27.3050i	1.01268i	−0.862333	+	0.506342i	$0.830997\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.47214	0.313353
$732$	0	0
$733$	− 14.4721i	− 0.534541i	−0.963622	−	0.267270i	$-0.913878\pi$
$733$	− 14.4721i	− 0.534541i	0.963622	−	0.267270i	$-0.0861216\pi$
$734$	0	0
$735$	13.4164i	0.494872i
$736$	0	0
$737$	16.5836i	0.610864i
$738$	0	0
$739$	14.8328	0.545634	0.272817	−	0.962066i	$-0.412045\pi$
$739$	14.8328	0.545634	0.272817	+	0.962066i	$0.412045\pi$
$740$	0	0
$741$	−8.47214	−0.311232
$742$	0	0
$743$	− 19.8885i	− 0.729640i	−0.931078	−	0.364820i	$-0.881131\pi$
$743$	− 19.8885i	− 0.729640i	0.931078	−	0.364820i	$-0.118869\pi$
$744$	0	0
$745$	23.4164	0.857911
$746$	0	0
$747$	− 4.94427i	− 0.180901i
$748$	0	0
$749$	6.47214	0.236487
$750$	0	0
$751$	−4.11146	−0.150029	−0.0750146	−	0.997182i	$-0.523900\pi$
$751$	−4.11146	−0.150029	−0.0750146	+	0.997182i	$0.523900\pi$
$752$	0	0
$753$	23.8885i	0.870547i
$754$	0	0
$755$	40.7771	1.48403
$756$	0	0
$757$	− 4.58359i	− 0.166593i	−0.996525	−	0.0832967i	$-0.973455\pi$
$757$	− 4.58359i	− 0.166593i	0.996525	−	0.0832967i	$-0.0265449\pi$
$758$	0	0
$759$	−1.41641	−0.0514123
$760$	0	0
$761$	−5.05573	−0.183270	−0.0916350	−	0.995793i	$-0.529209\pi$
$761$	−5.05573	−0.183270	−0.0916350	+	0.995793i	$0.529209\pi$
$762$	0	0
$763$	2.47214i	0.0894973i
$764$	0	0
$765$	2.23607i	0.0808452i
$766$	0	0
$767$	− 17.8885i	− 0.645918i
$768$	0	0
$769$	−14.0557	−0.506863	−0.253431	−	0.967353i	$-0.581559\pi$
$769$	−14.0557	−0.506863	−0.253431	+	0.967353i	$0.581559\pi$
$770$	0	0
$771$	−30.9443	−1.11443
$772$	0	0
$773$	− 20.7771i	− 0.747300i	−0.927570	−	0.373650i	$-0.878106\pi$
$773$	− 20.7771i	− 0.747300i	0.927570	−	0.373650i	$-0.121894\pi$
$774$	0	0
$775$	−12.3607	−0.444009
$776$	0	0
$777$	4.70820i	0.168906i
$778$	0	0
$779$	15.9443	0.571263
$780$	0	0
$781$	−38.8328	−1.38955
$782$	0	0
$783$	6.23607i	0.222859i
$784$	0	0
$785$	− 16.5836i	− 0.591894i
$786$	0	0
$787$	19.5836i	0.698080i	0.937108	+	0.349040i	$0.113492\pi$
$787$	19.5836i	0.698080i	−0.937108	+	0.349040i	$0.886508\pi$
$788$	0	0
$789$	−9.65248	−0.343637
$790$	0	0
$791$	−18.9443	−0.673581
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	0	0
$795$	6.70820	0.237915
$796$	0	0
$797$	− 39.0000i	− 1.38145i	−0.723117	−	0.690725i	$-0.757291\pi$
$797$	− 39.0000i	− 1.38145i	0.723117	−	0.690725i	$-0.242709\pi$
$798$	0	0
$799$	12.7082	0.449584
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	5.29180i	0.186743i
$804$	0	0
$805$	1.05573	0.0372095
$806$	0	0
$807$	15.1803i	0.534373i
$808$	0	0
$809$	−7.52786	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$809$	−7.52786	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	− 13.8885i	− 0.487092i
$814$	0	0
$815$	− 23.2918i	− 0.815876i
$816$	0	0
$817$	35.8885i	1.25558i
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	7.52786	0.262724	0.131362	−	0.991334i	$-0.458065\pi$
$821$	7.52786	0.262724	0.131362	+	0.991334i	$0.458065\pi$
$822$	0	0
$823$	7.00000i	0.244005i	0.992530	+	0.122002i	$0.0389315\pi$
$823$	7.00000i	0.244005i	−0.992530	+	0.122002i	$0.961068\pi$
$824$	0	0
$825$	− 15.0000i	− 0.522233i
$826$	0	0
$827$	− 0.111456i	− 0.00387571i	−0.999998	−	0.00193786i	$-0.999383\pi$
$827$	− 0.111456i	− 0.00387571i	0.999998	−	0.00193786i	$-0.000616839\pi$
$828$	0	0
$829$	−21.3607	−0.741887	−0.370944	−	0.928655i	$-0.620966\pi$
$829$	−21.3607	−0.741887	−0.370944	+	0.928655i	$0.620966\pi$
$830$	0	0
$831$	−2.58359	−0.0896238
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	21.3050i	0.737288i
$836$	0	0
$837$	− 2.47214i	− 0.0854495i
$838$	0	0
$839$	−16.4164	−0.566757	−0.283379	−	0.959008i	$-0.591455\pi$
$839$	−16.4164	−0.566757	−0.283379	+	0.959008i	$0.591455\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	− 30.3607i	− 1.04568i
$844$	0	0
$845$	−20.1246	−0.692308
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	−8.52786	−0.292676
$850$	0	0
$851$	−2.22291	−0.0762005
$852$	0	0
$853$	24.2492i	0.830278i	0.909758	+	0.415139i	$0.136267\pi$
$853$	24.2492i	0.830278i	−0.909758	+	0.415139i	$0.863733\pi$
$854$	0	0
$855$	−9.47214	−0.323940
$856$	0	0
$857$	− 54.3607i	− 1.85693i	−0.371426	−	0.928463i	$-0.621131\pi$
$857$	− 54.3607i	− 1.85693i	0.371426	−	0.928463i	$-0.378869\pi$
$858$	0	0
$859$	37.1803	1.26858	0.634288	−	0.773097i	$-0.281293\pi$
$859$	37.1803	1.26858	0.634288	+	0.773097i	$0.281293\pi$
$860$	0	0
$861$	3.76393	0.128274
$862$	0	0
$863$	32.2361i	1.09733i	0.836043	+	0.548664i	$0.184863\pi$
$863$	32.2361i	1.09733i	−0.836043	+	0.548664i	$0.815137\pi$
$864$	0	0
$865$	54.4721i	1.85211i
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	−11.0557	−0.374609
$872$	0	0
$873$	− 13.4164i	− 0.454077i
$874$	0	0
$875$	11.1803i	0.377964i
$876$	0	0
$877$	54.1246i	1.82766i	0.406099	+	0.913829i	$0.366889\pi$
$877$	54.1246i	1.82766i	−0.406099	+	0.913829i	$0.633111\pi$
$878$	0	0
$879$	26.8885	0.906928
$880$	0	0
$881$	7.06888	0.238157	0.119078	−	0.992885i	$-0.462006\pi$
$881$	7.06888	0.238157	0.119078	+	0.992885i	$0.462006\pi$
$882$	0	0
$883$	− 56.8328i	− 1.91258i	−0.292426	−	0.956288i	$-0.594462\pi$
$883$	− 56.8328i	− 1.91258i	0.292426	−	0.956288i	$-0.405538\pi$
$884$	0	0
$885$	− 20.0000i	− 0.672293i
$886$	0	0
$887$	− 31.3050i	− 1.05112i	−0.850757	−	0.525559i	$-0.823856\pi$
$887$	− 31.3050i	− 1.05112i	0.850757	−	0.525559i	$-0.176144\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	53.8328i	1.80145i
$894$	0	0
$895$	23.4164	0.782724
$896$	0	0
$897$	− 0.944272i	− 0.0315283i
$898$	0	0
$899$	−15.4164	−0.514166
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	8.47214i	0.281935i
$904$	0	0
$905$	48.9443	1.62696
$906$	0	0
$907$	− 13.5836i	− 0.451036i	−0.974239	−	0.225518i	$-0.927593\pi$
$907$	− 13.5836i	− 0.451036i	0.974239	−	0.225518i	$-0.0724074\pi$
$908$	0	0
$909$	14.4721	0.480010
$910$	0	0
$911$	49.2492	1.63170	0.815850	−	0.578264i	$-0.196270\pi$
$911$	49.2492	1.63170	0.815850	+	0.578264i	$0.196270\pi$
$912$	0	0
$913$	14.8328i	0.490895i
$914$	0	0
$915$	− 22.3607i	− 0.739221i
$916$	0	0
$917$	− 0.944272i	− 0.0311826i
$918$	0	0
$919$	−18.7082	−0.617127	−0.308563	−	0.951204i	$-0.599848\pi$
$919$	−18.7082	−0.617127	−0.308563	+	0.951204i	$0.599848\pi$
$920$	0	0
$921$	18.8328	0.620562
$922$	0	0
$923$	− 25.8885i	− 0.852132i
$924$	0	0
$925$	− 23.5410i	− 0.774024i
$926$	0	0
$927$	− 10.9443i	− 0.359457i
$928$	0	0
$929$	32.4853	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$929$	32.4853	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$930$	0	0
$931$	−25.4164	−0.832989
$932$	0	0
$933$	19.4721i	0.637489i
$934$	0	0
$935$	− 6.70820i	− 0.219382i
$936$	0	0
$937$	− 25.8885i	− 0.845742i	−0.906190	−	0.422871i	$-0.861022\pi$
$937$	− 25.8885i	− 0.845742i	0.906190	−	0.422871i	$-0.138978\pi$
$938$	0	0
$939$	6.70820	0.218914
$940$	0	0
$941$	−12.4721	−0.406580	−0.203290	−	0.979119i	$-0.565163\pi$
$941$	−12.4721	−0.406580	−0.203290	+	0.979119i	$0.565163\pi$
$942$	0	0
$943$	1.77709i	0.0578699i
$944$	0	0
$945$	−2.23607	−0.0727393
$946$	0	0
$947$	− 28.4721i	− 0.925220i	−0.886562	−	0.462610i	$-0.846913\pi$
$947$	− 28.4721i	− 0.925220i	0.886562	−	0.462610i	$-0.153087\pi$
$948$	0	0
$949$	−3.52786	−0.114519
$950$	0	0
$951$	−20.4721	−0.663854
$952$	0	0
$953$	39.3607i	1.27502i	0.770443	+	0.637509i	$0.220035\pi$
$953$	39.3607i	1.27502i	−0.770443	+	0.637509i	$0.779965\pi$
$954$	0	0
$955$	−10.0000	−0.323592
$956$	0	0
$957$	− 18.7082i	− 0.604750i
$958$	0	0
$959$	−10.4164	−0.336363
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	− 6.47214i	− 0.208562i
$964$	0	0
$965$	− 38.9443i	− 1.25366i
$966$	0	0
$967$	46.0000i	1.47926i	0.673014	+	0.739630i	$0.265000\pi$
$967$	46.0000i	1.47926i	−0.673014	+	0.739630i	$0.735000\pi$
$968$	0	0
$969$	−4.23607	−0.136082
$970$	0	0
$971$	21.3050	0.683708	0.341854	−	0.939753i	$-0.388945\pi$
$971$	21.3050	0.683708	0.341854	+	0.939753i	$0.388945\pi$
$972$	0	0
$973$	− 4.94427i	− 0.158506i
$974$	0	0
$975$	10.0000	0.320256
$976$	0	0
$977$	− 13.0557i	− 0.417690i	−0.977949	−	0.208845i	$-0.933030\pi$
$977$	− 13.0557i	− 0.417690i	0.977949	−	0.208845i	$-0.0669704\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	2.47214	0.0789292
$982$	0	0
$983$	33.5279i	1.06937i	0.845051	+	0.534686i	$0.179570\pi$
$983$	33.5279i	1.06937i	−0.845051	+	0.534686i	$0.820430\pi$
$984$	0	0
$985$	− 30.2492i	− 0.963820i
$986$	0	0
$987$	12.7082i	0.404507i
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	− 30.1246i	− 0.955976i
$994$	0	0
$995$	30.0000	0.951064
$996$	0	0
$997$	34.5967i	1.09569i	0.836580	+	0.547845i	$0.184552\pi$
$997$	34.5967i	1.09569i	−0.836580	+	0.547845i	$0.815448\pi$
$998$	0	0
$999$	4.70820	0.148961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.n.2449.1		4
4.3	odd	2		255.2.b.b.154.4	yes	4
5.4	even	2	inner	4080.2.m.n.2449.3		4
12.11	even	2		765.2.b.b.154.1		4
20.3	even	4		1275.2.a.m.1.2		2
20.7	even	4		1275.2.a.i.1.1		2
20.19	odd	2		255.2.b.b.154.1	✓	4
60.23	odd	4		3825.2.a.q.1.1		2
60.47	odd	4		3825.2.a.w.1.2		2
60.59	even	2		765.2.b.b.154.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.b.b.154.1	✓	4	20.19	odd	2
255.2.b.b.154.4	yes	4	4.3	odd	2
765.2.b.b.154.1		4	12.11	even	2
765.2.b.b.154.4		4	60.59	even	2
1275.2.a.i.1.1		2	20.7	even	4
1275.2.a.m.1.2		2	20.3	even	4
3825.2.a.q.1.1		2	60.23	odd	4
3825.2.a.w.1.2		2	60.47	odd	4
4080.2.m.n.2449.1		4	1.1	even	1	trivial
4080.2.m.n.2449.3		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{13} +2.23607i q^{15} +1.00000i q^{17} -4.23607 q^{19} -1.00000 q^{21} -0.472136i q^{23} +5.00000 q^{25} +1.00000i q^{27} +6.23607 q^{29} -2.47214 q^{31} -3.00000i q^{33} +2.23607i q^{35} -4.70820i q^{37} +2.00000 q^{39} -3.76393 q^{41} -8.47214i q^{43} +2.23607 q^{45} -12.7082i q^{47} +6.00000 q^{49} +1.00000 q^{51} -3.00000i q^{53} -6.70820 q^{55} +4.23607i q^{57} -8.94427 q^{59} -10.0000 q^{61} +1.00000i q^{63} -4.47214i q^{65} +5.52786i q^{67} -0.472136 q^{69} -12.9443 q^{71} +1.76393i q^{73} -5.00000i q^{75} -3.00000i q^{77} +2.00000 q^{79} +1.00000 q^{81} +4.94427i q^{83} -2.23607i q^{85} -6.23607i q^{87} +12.0000 q^{89} +2.00000 q^{91} +2.47214i q^{93} +9.47214 q^{95} +13.4164i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 12 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 8 q^{31} + 8 q^{39} - 24 q^{41} + 24 q^{49} + 4 q^{51} - 40 q^{61} + 16 q^{69} - 16 q^{71} + 8 q^{79} + 4 q^{81} + 48 q^{89} + 8 q^{91} + 20 q^{95} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 12 * q^11 - 8 * q^19 - 4 * q^21 + 20 * q^25 + 16 * q^29 + 8 * q^31 + 8 * q^39 - 24 * q^41 + 24 * q^49 + 4 * q^51 - 40 * q^61 + 16 * q^69 - 16 * q^71 + 8 * q^79 + 4 * q^81 + 48 * q^89 + 8 * q^91 + 20 * q^95 - 12 * q^99

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