LMFDB - Embedded newform 2280.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.52444 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -2.52444 q^{7} +1.00000 q^{9} -3.10278 q^{11} +6.72999 q^{13} -1.00000 q^{15} -2.57834 q^{17} -1.00000 q^{19} +2.52444 q^{21} -4.57834 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.10278 q^{29} +7.83276 q^{31} +3.10278 q^{33} -2.52444 q^{35} -4.52444 q^{37} -6.72999 q^{39} +6.15165 q^{41} -7.68111 q^{43} +1.00000 q^{45} -3.42166 q^{47} -0.627213 q^{49} +2.57834 q^{51} +2.57834 q^{53} -3.10278 q^{55} +1.00000 q^{57} -10.2056 q^{59} -14.8816 q^{61} -2.52444 q^{63} +6.72999 q^{65} +8.41110 q^{67} +4.57834 q^{69} +9.45998 q^{71} -9.25443 q^{73} -1.00000 q^{75} +7.83276 q^{77} +1.00000 q^{81} -4.47054 q^{83} -2.57834 q^{85} +1.10278 q^{87} -14.5628 q^{89} -16.9894 q^{91} -7.83276 q^{93} -1.00000 q^{95} -12.9355 q^{97} -3.10278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{15} - 6 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 8 q^{37} - 14 q^{43} + 3 q^{45} - 12 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} - 2 q^{55} + 3 q^{57} - 16 q^{59} - 6 q^{61} - 2 q^{63} - 4 q^{67} + 12 q^{69} - 12 q^{71} - 2 q^{73} - 3 q^{75} - 4 q^{77} + 3 q^{81} - 4 q^{83} - 6 q^{85} - 4 q^{87} + 4 q^{89} - 20 q^{91} + 4 q^{93} - 3 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^15 - 6 * q^17 - 3 * q^19 + 2 * q^21 - 12 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 - 8 * q^37 - 14 * q^43 + 3 * q^45 - 12 * q^47 + 11 * q^49 + 6 * q^51 + 6 * q^53 - 2 * q^55 + 3 * q^57 - 16 * q^59 - 6 * q^61 - 2 * q^63 - 4 * q^67 + 12 * q^69 - 12 * q^71 - 2 * q^73 - 3 * q^75 - 4 * q^77 + 3 * q^81 - 4 * q^83 - 6 * q^85 - 4 * q^87 + 4 * q^89 - 20 * q^91 + 4 * q^93 - 3 * q^95 - 4 * 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
$7$	−2.52444	−0.954148	−0.477074	−	0.878863i	$-0.658303\pi$
$7$	−2.52444	−0.954148	−0.477074	+	0.878863i	$0.658303\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.10278	−0.935522	−0.467761	−	0.883855i	$-0.654939\pi$
$11$	−3.10278	−0.935522	−0.467761	+	0.883855i	$0.654939\pi$
$12$	0	0
$13$	6.72999	1.86656	0.933281	−	0.359146i	$-0.116932\pi$
$13$	6.72999	1.86656	0.933281	+	0.359146i	$0.116932\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.57834	−0.625339	−0.312669	−	0.949862i	$-0.601223\pi$
$17$	−2.57834	−0.625339	−0.312669	+	0.949862i	$0.601223\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.52444	0.550878
$22$	0	0
$23$	−4.57834	−0.954649	−0.477325	−	0.878727i	$-0.658393\pi$
$23$	−4.57834	−0.954649	−0.477325	+	0.878727i	$0.658393\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.10278	−0.204780	−0.102390	−	0.994744i	$-0.532649\pi$
$29$	−1.10278	−0.204780	−0.102390	+	0.994744i	$0.532649\pi$
$30$	0	0
$31$	7.83276	1.40681	0.703403	−	0.710791i	$-0.251663\pi$
$31$	7.83276	1.40681	0.703403	+	0.710791i	$0.251663\pi$
$32$	0	0
$33$	3.10278	0.540124
$34$	0	0
$35$	−2.52444	−0.426708
$36$	0	0
$37$	−4.52444	−0.743813	−0.371907	−	0.928270i	$-0.621296\pi$
$37$	−4.52444	−0.743813	−0.371907	+	0.928270i	$0.621296\pi$
$38$	0	0
$39$	−6.72999	−1.07766
$40$	0	0
$41$	6.15165	0.960726	0.480363	−	0.877070i	$-0.340505\pi$
$41$	6.15165	0.960726	0.480363	+	0.877070i	$0.340505\pi$
$42$	0	0
$43$	−7.68111	−1.17136	−0.585679	−	0.810543i	$-0.699172\pi$
$43$	−7.68111	−1.17136	−0.585679	+	0.810543i	$0.699172\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−3.42166	−0.499101	−0.249550	−	0.968362i	$-0.580283\pi$
$47$	−3.42166	−0.499101	−0.249550	+	0.968362i	$0.580283\pi$
$48$	0	0
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	2.57834	0.361039
$52$	0	0
$53$	2.57834	0.354162	0.177081	−	0.984196i	$-0.443335\pi$
$53$	2.57834	0.354162	0.177081	+	0.984196i	$0.443335\pi$
$54$	0	0
$55$	−3.10278	−0.418378
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−10.2056	−1.32865	−0.664325	−	0.747444i	$-0.731281\pi$
$59$	−10.2056	−1.32865	−0.664325	+	0.747444i	$0.731281\pi$
$60$	0	0
$61$	−14.8816	−1.90540	−0.952699	−	0.303914i	$-0.901706\pi$
$61$	−14.8816	−1.90540	−0.952699	+	0.303914i	$0.901706\pi$
$62$	0	0
$63$	−2.52444	−0.318049
$64$	0	0
$65$	6.72999	0.834752
$66$	0	0
$67$	8.41110	1.02758	0.513790	−	0.857916i	$-0.328241\pi$
$67$	8.41110	1.02758	0.513790	+	0.857916i	$0.328241\pi$
$68$	0	0
$69$	4.57834	0.551167
$70$	0	0
$71$	9.45998	1.12269	0.561346	−	0.827581i	$-0.310284\pi$
$71$	9.45998	1.12269	0.561346	+	0.827581i	$0.310284\pi$
$72$	0	0
$73$	−9.25443	−1.08315	−0.541574	−	0.840653i	$-0.682172\pi$
$73$	−9.25443	−1.08315	−0.541574	+	0.840653i	$0.682172\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	7.83276	0.892626
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.47054	−0.490705	−0.245353	−	0.969434i	$-0.578904\pi$
$83$	−4.47054	−0.490705	−0.245353	+	0.969434i	$0.578904\pi$
$84$	0	0
$85$	−2.57834	−0.279660
$86$	0	0
$87$	1.10278	0.118230
$88$	0	0
$89$	−14.5628	−1.54365	−0.771824	−	0.635836i	$-0.780656\pi$
$89$	−14.5628	−1.54365	−0.771824	+	0.635836i	$0.780656\pi$
$90$	0	0
$91$	−16.9894	−1.78098
$92$	0	0
$93$	−7.83276	−0.812220
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−12.9355	−1.31340	−0.656702	−	0.754150i	$-0.728049\pi$
$97$	−12.9355	−1.31340	−0.656702	+	0.754150i	$0.728049\pi$
$98$	0	0
$99$	−3.10278	−0.311841
$100$	0	0
$101$	−11.4600	−1.14031	−0.570155	−	0.821537i	$-0.693117\pi$
$101$	−11.4600	−1.14031	−0.570155	+	0.821537i	$0.693117\pi$
$102$	0	0
$103$	14.5089	1.42960	0.714800	−	0.699329i	$-0.246518\pi$
$103$	14.5089	1.42960	0.714800	+	0.699329i	$0.246518\pi$
$104$	0	0
$105$	2.52444	0.246360
$106$	0	0
$107$	−15.2544	−1.47470	−0.737351	−	0.675510i	$-0.763923\pi$
$107$	−15.2544	−1.47470	−0.737351	+	0.675510i	$0.763923\pi$
$108$	0	0
$109$	16.6167	1.59159	0.795793	−	0.605568i	$-0.207054\pi$
$109$	16.6167	1.59159	0.795793	+	0.605568i	$0.207054\pi$
$110$	0	0
$111$	4.52444	0.429441
$112$	0	0
$113$	−4.37279	−0.411357	−0.205679	−	0.978620i	$-0.565940\pi$
$113$	−4.37279	−0.411357	−0.205679	+	0.978620i	$0.565940\pi$
$114$	0	0
$115$	−4.57834	−0.426932
$116$	0	0
$117$	6.72999	0.622188
$118$	0	0
$119$	6.50885	0.596665
$120$	0	0
$121$	−1.37279	−0.124799
$122$	0	0
$123$	−6.15165	−0.554676
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.2544	−1.70855	−0.854277	−	0.519818i	$-0.826000\pi$
$127$	−19.2544	−1.70855	−0.854277	+	0.519818i	$0.826000\pi$
$128$	0	0
$129$	7.68111	0.676284
$130$	0	0
$131$	11.4061	0.996554	0.498277	−	0.867018i	$-0.333966\pi$
$131$	11.4061	0.996554	0.498277	+	0.867018i	$0.333966\pi$
$132$	0	0
$133$	2.52444	0.218897
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.74557	−0.234570	−0.117285	−	0.993098i	$-0.537419\pi$
$137$	−2.74557	−0.234570	−0.117285	+	0.993098i	$0.537419\pi$
$138$	0	0
$139$	5.79445	0.491479	0.245739	−	0.969336i	$-0.420969\pi$
$139$	5.79445	0.491479	0.245739	+	0.969336i	$0.420969\pi$
$140$	0	0
$141$	3.42166	0.288156
$142$	0	0
$143$	−20.8816	−1.74621
$144$	0	0
$145$	−1.10278	−0.0915805
$146$	0	0
$147$	0.627213	0.0517317
$148$	0	0
$149$	−4.09775	−0.335701	−0.167850	−	0.985812i	$-0.553683\pi$
$149$	−4.09775	−0.335701	−0.167850	+	0.985812i	$0.553683\pi$
$150$	0	0
$151$	−6.67609	−0.543292	−0.271646	−	0.962397i	$-0.587568\pi$
$151$	−6.67609	−0.543292	−0.271646	+	0.962397i	$0.587568\pi$
$152$	0	0
$153$	−2.57834	−0.208446
$154$	0	0
$155$	7.83276	0.629143
$156$	0	0
$157$	−19.4600	−1.55308	−0.776538	−	0.630071i	$-0.783026\pi$
$157$	−19.4600	−1.55308	−0.776538	+	0.630071i	$0.783026\pi$
$158$	0	0
$159$	−2.57834	−0.204475
$160$	0	0
$161$	11.5577	0.910877
$162$	0	0
$163$	12.7300	0.997090	0.498545	−	0.866864i	$-0.333868\pi$
$163$	12.7300	0.997090	0.498545	+	0.866864i	$0.333868\pi$
$164$	0	0
$165$	3.10278	0.241551
$166$	0	0
$167$	−20.8816	−1.61587	−0.807935	−	0.589272i	$-0.799415\pi$
$167$	−20.8816	−1.61587	−0.807935	+	0.589272i	$0.799415\pi$
$168$	0	0
$169$	32.2927	2.48406
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−0.372787	−0.0283425	−0.0141712	−	0.999900i	$-0.504511\pi$
$173$	−0.372787	−0.0283425	−0.0141712	+	0.999900i	$0.504511\pi$
$174$	0	0
$175$	−2.52444	−0.190830
$176$	0	0
$177$	10.2056	0.767096
$178$	0	0
$179$	1.04888	0.0783967	0.0391983	−	0.999231i	$-0.487520\pi$
$179$	1.04888	0.0783967	0.0391983	+	0.999231i	$0.487520\pi$
$180$	0	0
$181$	5.36222	0.398571	0.199285	−	0.979941i	$-0.436138\pi$
$181$	5.36222	0.398571	0.199285	+	0.979941i	$0.436138\pi$
$182$	0	0
$183$	14.8816	1.10008
$184$	0	0
$185$	−4.52444	−0.332643
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	2.52444	0.183626
$190$	0	0
$191$	−25.7194	−1.86099	−0.930496	−	0.366302i	$-0.880624\pi$
$191$	−25.7194	−1.86099	−0.930496	+	0.366302i	$0.880624\pi$
$192$	0	0
$193$	7.77886	0.559935	0.279967	−	0.960009i	$-0.409676\pi$
$193$	7.77886	0.559935	0.279967	+	0.960009i	$0.409676\pi$
$194$	0	0
$195$	−6.72999	−0.481944
$196$	0	0
$197$	−5.42166	−0.386277	−0.193139	−	0.981171i	$-0.561867\pi$
$197$	−5.42166	−0.386277	−0.193139	+	0.981171i	$0.561867\pi$
$198$	0	0
$199$	14.2056	1.00700	0.503502	−	0.863994i	$-0.332045\pi$
$199$	14.2056	1.00700	0.503502	+	0.863994i	$0.332045\pi$
$200$	0	0
$201$	−8.41110	−0.593273
$202$	0	0
$203$	2.78389	0.195391
$204$	0	0
$205$	6.15165	0.429650
$206$	0	0
$207$	−4.57834	−0.318216
$208$	0	0
$209$	3.10278	0.214623
$210$	0	0
$211$	6.09775	0.419787	0.209893	−	0.977724i	$-0.432688\pi$
$211$	6.09775	0.419787	0.209893	+	0.977724i	$0.432688\pi$
$212$	0	0
$213$	−9.45998	−0.648187
$214$	0	0
$215$	−7.68111	−0.523848
$216$	0	0
$217$	−19.7733	−1.34230
$218$	0	0
$219$	9.25443	0.625356
$220$	0	0
$221$	−17.3522	−1.16723
$222$	0	0
$223$	−24.8222	−1.66222	−0.831109	−	0.556110i	$-0.812293\pi$
$223$	−24.8222	−1.66222	−0.831109	+	0.556110i	$0.812293\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−22.7839	−1.51222	−0.756110	−	0.654445i	$-0.772902\pi$
$227$	−22.7839	−1.51222	−0.756110	+	0.654445i	$0.772902\pi$
$228$	0	0
$229$	6.47054	0.427585	0.213793	−	0.976879i	$-0.431418\pi$
$229$	6.47054	0.427585	0.213793	+	0.976879i	$0.431418\pi$
$230$	0	0
$231$	−7.83276	−0.515358
$232$	0	0
$233$	−19.3522	−1.26780	−0.633902	−	0.773414i	$-0.718548\pi$
$233$	−19.3522	−1.26780	−0.633902	+	0.773414i	$0.718548\pi$
$234$	0	0
$235$	−3.42166	−0.223205
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.9250	1.54758	0.773789	−	0.633443i	$-0.218359\pi$
$239$	23.9250	1.54758	0.773789	+	0.633443i	$0.218359\pi$
$240$	0	0
$241$	−8.09775	−0.521622	−0.260811	−	0.965390i	$-0.583990\pi$
$241$	−8.09775	−0.521622	−0.260811	+	0.965390i	$0.583990\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.627213	−0.0400712
$246$	0	0
$247$	−6.72999	−0.428219
$248$	0	0
$249$	4.47054	0.283309
$250$	0	0
$251$	1.00502	0.0634365	0.0317183	−	0.999497i	$-0.489902\pi$
$251$	1.00502	0.0634365	0.0317183	+	0.999497i	$0.489902\pi$
$252$	0	0
$253$	14.2056	0.893095
$254$	0	0
$255$	2.57834	0.161462
$256$	0	0
$257$	−16.7839	−1.04695	−0.523475	−	0.852041i	$-0.675365\pi$
$257$	−16.7839	−1.04695	−0.523475	+	0.852041i	$0.675365\pi$
$258$	0	0
$259$	11.4217	0.709708
$260$	0	0
$261$	−1.10278	−0.0682601
$262$	0	0
$263$	−22.0383	−1.35894	−0.679470	−	0.733703i	$-0.737790\pi$
$263$	−22.0383	−1.35894	−0.679470	+	0.733703i	$0.737790\pi$
$264$	0	0
$265$	2.57834	0.158386
$266$	0	0
$267$	14.5628	0.891226
$268$	0	0
$269$	32.3572	1.97285	0.986427	−	0.164202i	$-0.0525050\pi$
$269$	32.3572	1.97285	0.986427	+	0.164202i	$0.0525050\pi$
$270$	0	0
$271$	−23.1466	−1.40606	−0.703029	−	0.711161i	$-0.748169\pi$
$271$	−23.1466	−1.40606	−0.703029	+	0.711161i	$0.748169\pi$
$272$	0	0
$273$	16.9894	1.02825
$274$	0	0
$275$	−3.10278	−0.187104
$276$	0	0
$277$	30.8222	1.85193	0.925963	−	0.377614i	$-0.123255\pi$
$277$	30.8222	1.85193	0.925963	+	0.377614i	$0.123255\pi$
$278$	0	0
$279$	7.83276	0.468935
$280$	0	0
$281$	−3.94610	−0.235405	−0.117702	−	0.993049i	$-0.537553\pi$
$281$	−3.94610	−0.235405	−0.117702	+	0.993049i	$0.537553\pi$
$282$	0	0
$283$	17.4756	1.03881	0.519407	−	0.854527i	$-0.326153\pi$
$283$	17.4756	1.03881	0.519407	+	0.854527i	$0.326153\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−15.5295	−0.916675
$288$	0	0
$289$	−10.3522	−0.608952
$290$	0	0
$291$	12.9355	0.758295
$292$	0	0
$293$	2.05944	0.120314	0.0601568	−	0.998189i	$-0.480840\pi$
$293$	2.05944	0.120314	0.0601568	+	0.998189i	$0.480840\pi$
$294$	0	0
$295$	−10.2056	−0.594190
$296$	0	0
$297$	3.10278	0.180041
$298$	0	0
$299$	−30.8122	−1.78191
$300$	0	0
$301$	19.3905	1.11765
$302$	0	0
$303$	11.4600	0.658358
$304$	0	0
$305$	−14.8816	−0.852120
$306$	0	0
$307$	10.2056	0.582462	0.291231	−	0.956653i	$-0.405935\pi$
$307$	10.2056	0.582462	0.291231	+	0.956653i	$0.405935\pi$
$308$	0	0
$309$	−14.5089	−0.825380
$310$	0	0
$311$	−1.20053	−0.0680756	−0.0340378	−	0.999421i	$-0.510837\pi$
$311$	−1.20053	−0.0680756	−0.0340378	+	0.999421i	$0.510837\pi$
$312$	0	0
$313$	−7.45998	−0.421663	−0.210831	−	0.977522i	$-0.567617\pi$
$313$	−7.45998	−0.421663	−0.210831	+	0.977522i	$0.567617\pi$
$314$	0	0
$315$	−2.52444	−0.142236
$316$	0	0
$317$	−4.48059	−0.251655	−0.125827	−	0.992052i	$-0.540159\pi$
$317$	−4.48059	−0.251655	−0.125827	+	0.992052i	$0.540159\pi$
$318$	0	0
$319$	3.42166	0.191576
$320$	0	0
$321$	15.2544	0.851419
$322$	0	0
$323$	2.57834	0.143463
$324$	0	0
$325$	6.72999	0.373313
$326$	0	0
$327$	−16.6167	−0.918903
$328$	0	0
$329$	8.63778	0.476216
$330$	0	0
$331$	24.2439	1.33256	0.666282	−	0.745700i	$-0.267885\pi$
$331$	24.2439	1.33256	0.666282	+	0.745700i	$0.267885\pi$
$332$	0	0
$333$	−4.52444	−0.247938
$334$	0	0
$335$	8.41110	0.459547
$336$	0	0
$337$	23.4444	1.27710	0.638549	−	0.769581i	$-0.279535\pi$
$337$	23.4444	1.27710	0.638549	+	0.769581i	$0.279535\pi$
$338$	0	0
$339$	4.37279	0.237497
$340$	0	0
$341$	−24.3033	−1.31610
$342$	0	0
$343$	19.2544	1.03964
$344$	0	0
$345$	4.57834	0.246489
$346$	0	0
$347$	0.881639	0.0473289	0.0236644	−	0.999720i	$-0.492467\pi$
$347$	0.881639	0.0473289	0.0236644	+	0.999720i	$0.492467\pi$
$348$	0	0
$349$	−1.66553	−0.0891536	−0.0445768	−	0.999006i	$-0.514194\pi$
$349$	−1.66553	−0.0891536	−0.0445768	+	0.999006i	$0.514194\pi$
$350$	0	0
$351$	−6.72999	−0.359220
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	9.45998	0.502083
$356$	0	0
$357$	−6.50885	−0.344485
$358$	0	0
$359$	−17.7194	−0.935196	−0.467598	−	0.883941i	$-0.654880\pi$
$359$	−17.7194	−0.935196	−0.467598	+	0.883941i	$0.654880\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.37279	0.0720526
$364$	0	0
$365$	−9.25443	−0.484399
$366$	0	0
$367$	−6.93554	−0.362032	−0.181016	−	0.983480i	$-0.557939\pi$
$367$	−6.93554	−0.362032	−0.181016	+	0.983480i	$0.557939\pi$
$368$	0	0
$369$	6.15165	0.320242
$370$	0	0
$371$	−6.50885	−0.337923
$372$	0	0
$373$	3.25997	0.168795	0.0843973	−	0.996432i	$-0.473104\pi$
$373$	3.25997	0.168795	0.0843973	+	0.996432i	$0.473104\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−7.42166	−0.382235
$378$	0	0
$379$	14.1461	0.726637	0.363318	−	0.931665i	$-0.381644\pi$
$379$	14.1461	0.726637	0.363318	+	0.931665i	$0.381644\pi$
$380$	0	0
$381$	19.2544	0.986434
$382$	0	0
$383$	28.0766	1.43465	0.717324	−	0.696739i	$-0.245367\pi$
$383$	28.0766	1.43465	0.717324	+	0.696739i	$0.245367\pi$
$384$	0	0
$385$	7.83276	0.399195
$386$	0	0
$387$	−7.68111	−0.390453
$388$	0	0
$389$	35.3522	1.79243	0.896213	−	0.443623i	$-0.146307\pi$
$389$	35.3522	1.79243	0.896213	+	0.443623i	$0.146307\pi$
$390$	0	0
$391$	11.8045	0.596979
$392$	0	0
$393$	−11.4061	−0.575360
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.2544	−1.06673	−0.533365	−	0.845885i	$-0.679073\pi$
$397$	−21.2544	−1.06673	−0.533365	+	0.845885i	$0.679073\pi$
$398$	0	0
$399$	−2.52444	−0.126380
$400$	0	0
$401$	23.8272	1.18987	0.594937	−	0.803772i	$-0.297177\pi$
$401$	23.8272	1.18987	0.594937	+	0.803772i	$0.297177\pi$
$402$	0	0
$403$	52.7144	2.62589
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	14.0383	0.695853
$408$	0	0
$409$	15.9900	0.790652	0.395326	−	0.918541i	$-0.370632\pi$
$409$	15.9900	0.790652	0.395326	+	0.918541i	$0.370632\pi$
$410$	0	0
$411$	2.74557	0.135429
$412$	0	0
$413$	25.7633	1.26773
$414$	0	0
$415$	−4.47054	−0.219450
$416$	0	0
$417$	−5.79445	−0.283755
$418$	0	0
$419$	1.30833	0.0639159	0.0319579	−	0.999489i	$-0.489826\pi$
$419$	1.30833	0.0639159	0.0319579	+	0.999489i	$0.489826\pi$
$420$	0	0
$421$	−39.2333	−1.91211	−0.956057	−	0.293181i	$-0.905286\pi$
$421$	−39.2333	−1.91211	−0.956057	+	0.293181i	$0.905286\pi$
$422$	0	0
$423$	−3.42166	−0.166367
$424$	0	0
$425$	−2.57834	−0.125068
$426$	0	0
$427$	37.5678	1.81803
$428$	0	0
$429$	20.8816	1.00818
$430$	0	0
$431$	27.5577	1.32741	0.663705	−	0.747995i	$-0.268983\pi$
$431$	27.5577	1.32741	0.663705	+	0.747995i	$0.268983\pi$
$432$	0	0
$433$	3.58336	0.172205	0.0861027	−	0.996286i	$-0.472559\pi$
$433$	3.58336	0.172205	0.0861027	+	0.996286i	$0.472559\pi$
$434$	0	0
$435$	1.10278	0.0528740
$436$	0	0
$437$	4.57834	0.219012
$438$	0	0
$439$	5.68665	0.271409	0.135705	−	0.990749i	$-0.456670\pi$
$439$	5.68665	0.271409	0.135705	+	0.990749i	$0.456670\pi$
$440$	0	0
$441$	−0.627213	−0.0298673
$442$	0	0
$443$	0.362741	0.0172343	0.00861716	−	0.999963i	$-0.497257\pi$
$443$	0.362741	0.0172343	0.00861716	+	0.999963i	$0.497257\pi$
$444$	0	0
$445$	−14.5628	−0.690341
$446$	0	0
$447$	4.09775	0.193817
$448$	0	0
$449$	0.799473	0.0377295	0.0188647	−	0.999822i	$-0.493995\pi$
$449$	0.799473	0.0377295	0.0188647	+	0.999822i	$0.493995\pi$
$450$	0	0
$451$	−19.0872	−0.898781
$452$	0	0
$453$	6.67609	0.313670
$454$	0	0
$455$	−16.9894	−0.796477
$456$	0	0
$457$	24.2056	1.13229	0.566144	−	0.824306i	$-0.308435\pi$
$457$	24.2056	1.13229	0.566144	+	0.824306i	$0.308435\pi$
$458$	0	0
$459$	2.57834	0.120346
$460$	0	0
$461$	23.7633	1.10677	0.553383	−	0.832927i	$-0.313337\pi$
$461$	23.7633	1.10677	0.553383	+	0.832927i	$0.313337\pi$
$462$	0	0
$463$	−4.62219	−0.214811	−0.107406	−	0.994215i	$-0.534254\pi$
$463$	−4.62219	−0.214811	−0.107406	+	0.994215i	$0.534254\pi$
$464$	0	0
$465$	−7.83276	−0.363236
$466$	0	0
$467$	−4.98944	−0.230884	−0.115442	−	0.993314i	$-0.536828\pi$
$467$	−4.98944	−0.230884	−0.115442	+	0.993314i	$0.536828\pi$
$468$	0	0
$469$	−21.2333	−0.980463
$470$	0	0
$471$	19.4600	0.896668
$472$	0	0
$473$	23.8328	1.09583
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	2.57834	0.118054
$478$	0	0
$479$	12.9739	0.592790	0.296395	−	0.955065i	$-0.404215\pi$
$479$	12.9739	0.592790	0.296395	+	0.955065i	$0.404215\pi$
$480$	0	0
$481$	−30.4494	−1.38837
$482$	0	0
$483$	−11.5577	−0.525895
$484$	0	0
$485$	−12.9355	−0.587373
$486$	0	0
$487$	7.14663	0.323845	0.161922	−	0.986804i	$-0.448231\pi$
$487$	7.14663	0.323845	0.161922	+	0.986804i	$0.448231\pi$
$488$	0	0
$489$	−12.7300	−0.575670
$490$	0	0
$491$	22.1416	0.999237	0.499618	−	0.866246i	$-0.333474\pi$
$491$	22.1416	0.999237	0.499618	+	0.866246i	$0.333474\pi$
$492$	0	0
$493$	2.84333	0.128057
$494$	0	0
$495$	−3.10278	−0.139459
$496$	0	0
$497$	−23.8811	−1.07121
$498$	0	0
$499$	−4.51890	−0.202294	−0.101147	−	0.994872i	$-0.532251\pi$
$499$	−4.51890	−0.202294	−0.101147	+	0.994872i	$0.532251\pi$
$500$	0	0
$501$	20.8816	0.932923
$502$	0	0
$503$	−8.35166	−0.372382	−0.186191	−	0.982514i	$-0.559614\pi$
$503$	−8.35166	−0.372382	−0.186191	+	0.982514i	$0.559614\pi$
$504$	0	0
$505$	−11.4600	−0.509962
$506$	0	0
$507$	−32.2927	−1.43417
$508$	0	0
$509$	4.28057	0.189733	0.0948666	−	0.995490i	$-0.469758\pi$
$509$	4.28057	0.189733	0.0948666	+	0.995490i	$0.469758\pi$
$510$	0	0
$511$	23.3622	1.03348
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	14.5089	0.639336
$516$	0	0
$517$	10.6167	0.466920
$518$	0	0
$519$	0.372787	0.0163635
$520$	0	0
$521$	40.1305	1.75815	0.879075	−	0.476683i	$-0.158161\pi$
$521$	40.1305	1.75815	0.879075	+	0.476683i	$0.158161\pi$
$522$	0	0
$523$	20.5189	0.897229	0.448614	−	0.893725i	$-0.351918\pi$
$523$	20.5189	0.897229	0.448614	+	0.893725i	$0.351918\pi$
$524$	0	0
$525$	2.52444	0.110176
$526$	0	0
$527$	−20.1955	−0.879730
$528$	0	0
$529$	−2.03883	−0.0886448
$530$	0	0
$531$	−10.2056	−0.442883
$532$	0	0
$533$	41.4005	1.79326
$534$	0	0
$535$	−15.2544	−0.659506
$536$	0	0
$537$	−1.04888	−0.0452623
$538$	0	0
$539$	1.94610	0.0838245
$540$	0	0
$541$	−15.6867	−0.674422	−0.337211	−	0.941429i	$-0.609484\pi$
$541$	−15.6867	−0.674422	−0.337211	+	0.941429i	$0.609484\pi$
$542$	0	0
$543$	−5.36222	−0.230115
$544$	0	0
$545$	16.6167	0.711779
$546$	0	0
$547$	23.8922	1.02156	0.510778	−	0.859712i	$-0.329357\pi$
$547$	23.8922	1.02156	0.510778	+	0.859712i	$0.329357\pi$
$548$	0	0
$549$	−14.8816	−0.635133
$550$	0	0
$551$	1.10278	0.0469798
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.52444	0.192052
$556$	0	0
$557$	22.8222	0.967008	0.483504	−	0.875342i	$-0.339364\pi$
$557$	22.8222	0.967008	0.483504	+	0.875342i	$0.339364\pi$
$558$	0	0
$559$	−51.6938	−2.18641
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	−15.6061	−0.657718	−0.328859	−	0.944379i	$-0.606664\pi$
$563$	−15.6061	−0.657718	−0.328859	+	0.944379i	$0.606664\pi$
$564$	0	0
$565$	−4.37279	−0.183965
$566$	0	0
$567$	−2.52444	−0.106016
$568$	0	0
$569$	−0.691675	−0.0289965	−0.0144983	−	0.999895i	$-0.504615\pi$
$569$	−0.691675	−0.0289965	−0.0144983	+	0.999895i	$0.504615\pi$
$570$	0	0
$571$	−44.7910	−1.87445	−0.937223	−	0.348730i	$-0.886613\pi$
$571$	−44.7910	−1.87445	−0.937223	+	0.348730i	$0.886613\pi$
$572$	0	0
$573$	25.7194	1.07444
$574$	0	0
$575$	−4.57834	−0.190930
$576$	0	0
$577$	3.45998	0.144041	0.0720203	−	0.997403i	$-0.477055\pi$
$577$	3.45998	0.144041	0.0720203	+	0.997403i	$0.477055\pi$
$578$	0	0
$579$	−7.77886	−0.323279
$580$	0	0
$581$	11.2856	0.468205
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	6.72999	0.278251
$586$	0	0
$587$	27.8328	1.14878	0.574391	−	0.818581i	$-0.305239\pi$
$587$	27.8328	1.14878	0.574391	+	0.818581i	$0.305239\pi$
$588$	0	0
$589$	−7.83276	−0.322743
$590$	0	0
$591$	5.42166	0.223017
$592$	0	0
$593$	−17.5889	−0.722290	−0.361145	−	0.932510i	$-0.617614\pi$
$593$	−17.5889	−0.722290	−0.361145	+	0.932510i	$0.617614\pi$
$594$	0	0
$595$	6.50885	0.266837
$596$	0	0
$597$	−14.2056	−0.581394
$598$	0	0
$599$	−19.4700	−0.795524	−0.397762	−	0.917489i	$-0.630213\pi$
$599$	−19.4700	−0.795524	−0.397762	+	0.917489i	$0.630213\pi$
$600$	0	0
$601$	40.4777	1.65112	0.825560	−	0.564315i	$-0.190860\pi$
$601$	40.4777	1.65112	0.825560	+	0.564315i	$0.190860\pi$
$602$	0	0
$603$	8.41110	0.342526
$604$	0	0
$605$	−1.37279	−0.0558117
$606$	0	0
$607$	4.71440	0.191352	0.0956758	−	0.995413i	$-0.469499\pi$
$607$	4.71440	0.191352	0.0956758	+	0.995413i	$0.469499\pi$
$608$	0	0
$609$	−2.78389	−0.112809
$610$	0	0
$611$	−23.0278	−0.931603
$612$	0	0
$613$	−9.48110	−0.382938	−0.191469	−	0.981499i	$-0.561325\pi$
$613$	−9.48110	−0.382938	−0.191469	+	0.981499i	$0.561325\pi$
$614$	0	0
$615$	−6.15165	−0.248059
$616$	0	0
$617$	7.73501	0.311400	0.155700	−	0.987804i	$-0.450237\pi$
$617$	7.73501	0.311400	0.155700	+	0.987804i	$0.450237\pi$
$618$	0	0
$619$	6.12892	0.246342	0.123171	−	0.992385i	$-0.460694\pi$
$619$	6.12892	0.246342	0.123171	+	0.992385i	$0.460694\pi$
$620$	0	0
$621$	4.57834	0.183722
$622$	0	0
$623$	36.7628	1.47287
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.10278	−0.123913
$628$	0	0
$629$	11.6655	0.465135
$630$	0	0
$631$	4.19550	0.167020	0.0835102	−	0.996507i	$-0.473387\pi$
$631$	4.19550	0.167020	0.0835102	+	0.996507i	$0.473387\pi$
$632$	0	0
$633$	−6.09775	−0.242364
$634$	0	0
$635$	−19.2544	−0.764089
$636$	0	0
$637$	−4.22114	−0.167247
$638$	0	0
$639$	9.45998	0.374231
$640$	0	0
$641$	−47.3850	−1.87159	−0.935797	−	0.352541i	$-0.885318\pi$
$641$	−47.3850	−1.87159	−0.935797	+	0.352541i	$0.885318\pi$
$642$	0	0
$643$	21.3678	0.842662	0.421331	−	0.906907i	$-0.361563\pi$
$643$	21.3678	0.842662	0.421331	+	0.906907i	$0.361563\pi$
$644$	0	0
$645$	7.68111	0.302443
$646$	0	0
$647$	9.29274	0.365335	0.182668	−	0.983175i	$-0.441527\pi$
$647$	9.29274	0.365335	0.182668	+	0.983175i	$0.441527\pi$
$648$	0	0
$649$	31.6655	1.24298
$650$	0	0
$651$	19.7733	0.774978
$652$	0	0
$653$	−21.9094	−0.857381	−0.428690	−	0.903451i	$-0.641025\pi$
$653$	−21.9094	−0.857381	−0.428690	+	0.903451i	$0.641025\pi$
$654$	0	0
$655$	11.4061	0.445672
$656$	0	0
$657$	−9.25443	−0.361050
$658$	0	0
$659$	−8.52998	−0.332281	−0.166140	−	0.986102i	$-0.553130\pi$
$659$	−8.52998	−0.332281	−0.166140	+	0.986102i	$0.553130\pi$
$660$	0	0
$661$	5.36222	0.208566	0.104283	−	0.994548i	$-0.466745\pi$
$661$	5.36222	0.208566	0.104283	+	0.994548i	$0.466745\pi$
$662$	0	0
$663$	17.3522	0.673903
$664$	0	0
$665$	2.52444	0.0978935
$666$	0	0
$667$	5.04888	0.195493
$668$	0	0
$669$	24.8222	0.959682
$670$	0	0
$671$	46.1744	1.78254
$672$	0	0
$673$	−2.31889	−0.0893866	−0.0446933	−	0.999001i	$-0.514231\pi$
$673$	−2.31889	−0.0893866	−0.0446933	+	0.999001i	$0.514231\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−35.9305	−1.38092	−0.690461	−	0.723370i	$-0.742592\pi$
$677$	−35.9305	−1.38092	−0.690461	+	0.723370i	$0.742592\pi$
$678$	0	0
$679$	32.6550	1.25318
$680$	0	0
$681$	22.7839	0.873080
$682$	0	0
$683$	−5.15667	−0.197315	−0.0986573	−	0.995121i	$-0.531455\pi$
$683$	−5.15667	−0.197315	−0.0986573	+	0.995121i	$0.531455\pi$
$684$	0	0
$685$	−2.74557	−0.104903
$686$	0	0
$687$	−6.47054	−0.246866
$688$	0	0
$689$	17.3522	0.661065
$690$	0	0
$691$	19.9688	0.759650	0.379825	−	0.925058i	$-0.375984\pi$
$691$	19.9688	0.759650	0.379825	+	0.925058i	$0.375984\pi$
$692$	0	0
$693$	7.83276	0.297542
$694$	0	0
$695$	5.79445	0.219796
$696$	0	0
$697$	−15.8610	−0.600779
$698$	0	0
$699$	19.3522	0.731967
$700$	0	0
$701$	−39.2333	−1.48182	−0.740911	−	0.671604i	$-0.765606\pi$
$701$	−39.2333	−1.48182	−0.740911	+	0.671604i	$0.765606\pi$
$702$	0	0
$703$	4.52444	0.170642
$704$	0	0
$705$	3.42166	0.128867
$706$	0	0
$707$	28.9300	1.08802
$708$	0	0
$709$	36.8605	1.38433	0.692163	−	0.721741i	$-0.256658\pi$
$709$	36.8605	1.38433	0.692163	+	0.721741i	$0.256658\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−35.8610	−1.34301
$714$	0	0
$715$	−20.8816	−0.780929
$716$	0	0
$717$	−23.9250	−0.893495
$718$	0	0
$719$	−17.8383	−0.665256	−0.332628	−	0.943058i	$-0.607935\pi$
$719$	−17.8383	−0.665256	−0.332628	+	0.943058i	$0.607935\pi$
$720$	0	0
$721$	−36.6267	−1.36405
$722$	0	0
$723$	8.09775	0.301159
$724$	0	0
$725$	−1.10278	−0.0409560
$726$	0	0
$727$	−2.00554	−0.0743813	−0.0371907	−	0.999308i	$-0.511841\pi$
$727$	−2.00554	−0.0743813	−0.0371907	+	0.999308i	$0.511841\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.8045	0.732496
$732$	0	0
$733$	−43.4288	−1.60408	−0.802040	−	0.597271i	$-0.796252\pi$
$733$	−43.4288	−1.60408	−0.802040	+	0.597271i	$0.796252\pi$
$734$	0	0
$735$	0.627213	0.0231351
$736$	0	0
$737$	−26.0978	−0.961323
$738$	0	0
$739$	−20.6066	−0.758026	−0.379013	−	0.925391i	$-0.623736\pi$
$739$	−20.6066	−0.758026	−0.379013	+	0.925391i	$0.623736\pi$
$740$	0	0
$741$	6.72999	0.247232
$742$	0	0
$743$	−47.4499	−1.74077	−0.870385	−	0.492373i	$-0.836130\pi$
$743$	−47.4499	−1.74077	−0.870385	+	0.492373i	$0.836130\pi$
$744$	0	0
$745$	−4.09775	−0.150130
$746$	0	0
$747$	−4.47054	−0.163568
$748$	0	0
$749$	38.5089	1.40708
$750$	0	0
$751$	−7.83276	−0.285822	−0.142911	−	0.989736i	$-0.545646\pi$
$751$	−7.83276	−0.285822	−0.142911	+	0.989736i	$0.545646\pi$
$752$	0	0
$753$	−1.00502	−0.0366251
$754$	0	0
$755$	−6.67609	−0.242968
$756$	0	0
$757$	−43.4600	−1.57958	−0.789790	−	0.613378i	$-0.789810\pi$
$757$	−43.4600	−1.57958	−0.789790	+	0.613378i	$0.789810\pi$
$758$	0	0
$759$	−14.2056	−0.515629
$760$	0	0
$761$	−2.41110	−0.0874023	−0.0437012	−	0.999045i	$-0.513915\pi$
$761$	−2.41110	−0.0874023	−0.0437012	+	0.999045i	$0.513915\pi$
$762$	0	0
$763$	−41.9477	−1.51861
$764$	0	0
$765$	−2.57834	−0.0932200
$766$	0	0
$767$	−68.6832	−2.48001
$768$	0	0
$769$	−8.09775	−0.292012	−0.146006	−	0.989284i	$-0.546642\pi$
$769$	−8.09775	−0.292012	−0.146006	+	0.989284i	$0.546642\pi$
$770$	0	0
$771$	16.7839	0.604457
$772$	0	0
$773$	13.6172	0.489775	0.244888	−	0.969551i	$-0.421249\pi$
$773$	13.6172	0.489775	0.244888	+	0.969551i	$0.421249\pi$
$774$	0	0
$775$	7.83276	0.281361
$776$	0	0
$777$	−11.4217	−0.409750
$778$	0	0
$779$	−6.15165	−0.220406
$780$	0	0
$781$	−29.3522	−1.05030
$782$	0	0
$783$	1.10278	0.0394100
$784$	0	0
$785$	−19.4600	−0.694556
$786$	0	0
$787$	36.9099	1.31570	0.657848	−	0.753151i	$-0.271467\pi$
$787$	36.9099	1.31570	0.657848	+	0.753151i	$0.271467\pi$
$788$	0	0
$789$	22.0383	0.784585
$790$	0	0
$791$	11.0388	0.392496
$792$	0	0
$793$	−100.153	−3.55655
$794$	0	0
$795$	−2.57834	−0.0914442
$796$	0	0
$797$	−26.4705	−0.937635	−0.468817	−	0.883295i	$-0.655320\pi$
$797$	−26.4705	−0.937635	−0.468817	+	0.883295i	$0.655320\pi$
$798$	0	0
$799$	8.82220	0.312107
$800$	0	0
$801$	−14.5628	−0.514550
$802$	0	0
$803$	28.7144	1.01331
$804$	0	0
$805$	11.5577	0.407356
$806$	0	0
$807$	−32.3572	−1.13903
$808$	0	0
$809$	29.2544	1.02853	0.514265	−	0.857631i	$-0.328065\pi$
$809$	29.2544	1.02853	0.514265	+	0.857631i	$0.328065\pi$
$810$	0	0
$811$	38.9200	1.36666	0.683332	−	0.730108i	$-0.260530\pi$
$811$	38.9200	1.36666	0.683332	+	0.730108i	$0.260530\pi$
$812$	0	0
$813$	23.1466	0.811788
$814$	0	0
$815$	12.7300	0.445912
$816$	0	0
$817$	7.68111	0.268728
$818$	0	0
$819$	−16.9894	−0.593659
$820$	0	0
$821$	−36.6167	−1.27793	−0.638965	−	0.769236i	$-0.720637\pi$
$821$	−36.6167	−1.27793	−0.638965	+	0.769236i	$0.720637\pi$
$822$	0	0
$823$	41.5522	1.44842	0.724209	−	0.689580i	$-0.242205\pi$
$823$	41.5522	1.44842	0.724209	+	0.689580i	$0.242205\pi$
$824$	0	0
$825$	3.10278	0.108025
$826$	0	0
$827$	25.0972	0.872716	0.436358	−	0.899773i	$-0.356268\pi$
$827$	25.0972	0.872716	0.436358	+	0.899773i	$0.356268\pi$
$828$	0	0
$829$	16.9511	0.588737	0.294368	−	0.955692i	$-0.404891\pi$
$829$	16.9511	0.588737	0.294368	+	0.955692i	$0.404891\pi$
$830$	0	0
$831$	−30.8222	−1.06921
$832$	0	0
$833$	1.61717	0.0560315
$834$	0	0
$835$	−20.8816	−0.722639
$836$	0	0
$837$	−7.83276	−0.270740
$838$	0	0
$839$	−29.6555	−1.02382	−0.511910	−	0.859039i	$-0.671062\pi$
$839$	−29.6555	−1.02382	−0.511910	+	0.859039i	$0.671062\pi$
$840$	0	0
$841$	−27.7839	−0.958065
$842$	0	0
$843$	3.94610	0.135911
$844$	0	0
$845$	32.2927	1.11090
$846$	0	0
$847$	3.46552	0.119077
$848$	0	0
$849$	−17.4756	−0.599760
$850$	0	0
$851$	20.7144	0.710081
$852$	0	0
$853$	36.4777	1.24897	0.624486	−	0.781036i	$-0.285308\pi$
$853$	36.4777	1.24897	0.624486	+	0.781036i	$0.285308\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−24.8716	−0.849597	−0.424799	−	0.905288i	$-0.639655\pi$
$857$	−24.8716	−0.849597	−0.424799	+	0.905288i	$0.639655\pi$
$858$	0	0
$859$	15.0388	0.513118	0.256559	−	0.966529i	$-0.417411\pi$
$859$	15.0388	0.513118	0.256559	+	0.966529i	$0.417411\pi$
$860$	0	0
$861$	15.5295	0.529243
$862$	0	0
$863$	−40.4877	−1.37822	−0.689109	−	0.724658i	$-0.741998\pi$
$863$	−40.4877	−1.37822	−0.689109	+	0.724658i	$0.741998\pi$
$864$	0	0
$865$	−0.372787	−0.0126751
$866$	0	0
$867$	10.3522	0.351578
$868$	0	0
$869$	0	0
$870$	0	0
$871$	56.6066	1.91804
$872$	0	0
$873$	−12.9355	−0.437802
$874$	0	0
$875$	−2.52444	−0.0853416
$876$	0	0
$877$	−40.0822	−1.35348	−0.676739	−	0.736223i	$-0.736608\pi$
$877$	−40.0822	−1.35348	−0.676739	+	0.736223i	$0.736608\pi$
$878$	0	0
$879$	−2.05944	−0.0694631
$880$	0	0
$881$	−21.0278	−0.708443	−0.354221	−	0.935162i	$-0.615254\pi$
$881$	−21.0278	−0.708443	−0.354221	+	0.935162i	$0.615254\pi$
$882$	0	0
$883$	−0.287716	−0.00968241	−0.00484121	−	0.999988i	$-0.501541\pi$
$883$	−0.287716	−0.00968241	−0.00484121	+	0.999988i	$0.501541\pi$
$884$	0	0
$885$	10.2056	0.343056
$886$	0	0
$887$	−20.4111	−0.685338	−0.342669	−	0.939456i	$-0.611331\pi$
$887$	−20.4111	−0.685338	−0.342669	+	0.939456i	$0.611331\pi$
$888$	0	0
$889$	48.6066	1.63021
$890$	0	0
$891$	−3.10278	−0.103947
$892$	0	0
$893$	3.42166	0.114502
$894$	0	0
$895$	1.04888	0.0350601
$896$	0	0
$897$	30.8122	1.02879
$898$	0	0
$899$	−8.63778	−0.288086
$900$	0	0
$901$	−6.64782	−0.221471
$902$	0	0
$903$	−19.3905	−0.645275
$904$	0	0
$905$	5.36222	0.178246
$906$	0	0
$907$	−16.3799	−0.543887	−0.271943	−	0.962313i	$-0.587666\pi$
$907$	−16.3799	−0.543887	−0.271943	+	0.962313i	$0.587666\pi$
$908$	0	0
$909$	−11.4600	−0.380103
$910$	0	0
$911$	−50.5855	−1.67597	−0.837986	−	0.545692i	$-0.816267\pi$
$911$	−50.5855	−1.67597	−0.837986	+	0.545692i	$0.816267\pi$
$912$	0	0
$913$	13.8711	0.459066
$914$	0	0
$915$	14.8816	0.491972
$916$	0	0
$917$	−28.7939	−0.950859
$918$	0	0
$919$	−28.7456	−0.948229	−0.474114	−	0.880463i	$-0.657232\pi$
$919$	−28.7456	−0.948229	−0.474114	+	0.880463i	$0.657232\pi$
$920$	0	0
$921$	−10.2056	−0.336284
$922$	0	0
$923$	63.6655	2.09558
$924$	0	0
$925$	−4.52444	−0.148763
$926$	0	0
$927$	14.5089	0.476533
$928$	0	0
$929$	−49.7733	−1.63301	−0.816505	−	0.577339i	$-0.804091\pi$
$929$	−49.7733	−1.63301	−0.816505	+	0.577339i	$0.804091\pi$
$930$	0	0
$931$	0.627213	0.0205561
$932$	0	0
$933$	1.20053	0.0393035
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	34.3033	1.12064	0.560320	−	0.828276i	$-0.310678\pi$
$937$	34.3033	1.12064	0.560320	+	0.828276i	$0.310678\pi$
$938$	0	0
$939$	7.45998	0.243447
$940$	0	0
$941$	34.6705	1.13023	0.565114	−	0.825013i	$-0.308832\pi$
$941$	34.6705	1.13023	0.565114	+	0.825013i	$0.308832\pi$
$942$	0	0
$943$	−28.1643	−0.917157
$944$	0	0
$945$	2.52444	0.0821200
$946$	0	0
$947$	5.96169	0.193729	0.0968644	−	0.995298i	$-0.469119\pi$
$947$	5.96169	0.193729	0.0968644	+	0.995298i	$0.469119\pi$
$948$	0	0
$949$	−62.2822	−2.02177
$950$	0	0
$951$	4.48059	0.145293
$952$	0	0
$953$	9.52946	0.308690	0.154345	−	0.988017i	$-0.450673\pi$
$953$	9.52946	0.308690	0.154345	+	0.988017i	$0.450673\pi$
$954$	0	0
$955$	−25.7194	−0.832261
$956$	0	0
$957$	−3.42166	−0.110607
$958$	0	0
$959$	6.93103	0.223815
$960$	0	0
$961$	30.3522	0.979103
$962$	0	0
$963$	−15.2544	−0.491567
$964$	0	0
$965$	7.77886	0.250410
$966$	0	0
$967$	−19.9844	−0.642655	−0.321328	−	0.946968i	$-0.604129\pi$
$967$	−19.9844	−0.642655	−0.321328	+	0.946968i	$0.604129\pi$
$968$	0	0
$969$	−2.57834	−0.0828281
$970$	0	0
$971$	18.3900	0.590162	0.295081	−	0.955472i	$-0.404653\pi$
$971$	18.3900	0.590162	0.295081	+	0.955472i	$0.404653\pi$
$972$	0	0
$973$	−14.6277	−0.468943
$974$	0	0
$975$	−6.72999	−0.215532
$976$	0	0
$977$	−38.4394	−1.22978	−0.614892	−	0.788611i	$-0.710800\pi$
$977$	−38.4394	−1.22978	−0.614892	+	0.788611i	$0.710800\pi$
$978$	0	0
$979$	45.1849	1.44412
$980$	0	0
$981$	16.6167	0.530529
$982$	0	0
$983$	54.5260	1.73911	0.869555	−	0.493836i	$-0.164406\pi$
$983$	54.5260	1.73911	0.869555	+	0.493836i	$0.164406\pi$
$984$	0	0
$985$	−5.42166	−0.172749
$986$	0	0
$987$	−8.63778	−0.274943
$988$	0	0
$989$	35.1667	1.11824
$990$	0	0
$991$	2.91995	0.0927553	0.0463777	−	0.998924i	$-0.485232\pi$
$991$	2.91995	0.0927553	0.0463777	+	0.998924i	$0.485232\pi$
$992$	0	0
$993$	−24.2439	−0.769356
$994$	0	0
$995$	14.2056	0.450346
$996$	0	0
$997$	−36.2822	−1.14907	−0.574534	−	0.818481i	$-0.694817\pi$
$997$	−36.2822	−1.14907	−0.574534	+	0.818481i	$0.694817\pi$
$998$	0	0
$999$	4.52444	0.143147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.s.1.2	✓	3
3.2	odd	2		6840.2.a.bf.1.2		3
4.3	odd	2		4560.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.s.1.2	✓	3	1.1	even	1	trivial
4560.2.a.bv.1.2		3	4.3	odd	2
6840.2.a.bf.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.52444 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.52444 q^{7} +1.00000 q^{9} -3.10278 q^{11} +6.72999 q^{13} -1.00000 q^{15} -2.57834 q^{17} -1.00000 q^{19} +2.52444 q^{21} -4.57834 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.10278 q^{29} +7.83276 q^{31} +3.10278 q^{33} -2.52444 q^{35} -4.52444 q^{37} -6.72999 q^{39} +6.15165 q^{41} -7.68111 q^{43} +1.00000 q^{45} -3.42166 q^{47} -0.627213 q^{49} +2.57834 q^{51} +2.57834 q^{53} -3.10278 q^{55} +1.00000 q^{57} -10.2056 q^{59} -14.8816 q^{61} -2.52444 q^{63} +6.72999 q^{65} +8.41110 q^{67} +4.57834 q^{69} +9.45998 q^{71} -9.25443 q^{73} -1.00000 q^{75} +7.83276 q^{77} +1.00000 q^{81} -4.47054 q^{83} -2.57834 q^{85} +1.10278 q^{87} -14.5628 q^{89} -16.9894 q^{91} -7.83276 q^{93} -1.00000 q^{95} -12.9355 q^{97} -3.10278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{15} - 6 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 8 q^{37} - 14 q^{43} + 3 q^{45} - 12 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} - 2 q^{55} + 3 q^{57} - 16 q^{59} - 6 q^{61} - 2 q^{63} - 4 q^{67} + 12 q^{69} - 12 q^{71} - 2 q^{73} - 3 q^{75} - 4 q^{77} + 3 q^{81} - 4 q^{83} - 6 q^{85} - 4 q^{87} + 4 q^{89} - 20 q^{91} + 4 q^{93} - 3 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^15 - 6 * q^17 - 3 * q^19 + 2 * q^21 - 12 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 - 8 * q^37 - 14 * q^43 + 3 * q^45 - 12 * q^47 + 11 * q^49 + 6 * q^51 + 6 * q^53 - 2 * q^55 + 3 * q^57 - 16 * q^59 - 6 * q^61 - 2 * q^63 - 4 * q^67 + 12 * q^69 - 12 * q^71 - 2 * q^73 - 3 * q^75 - 4 * q^77 + 3 * q^81 - 4 * q^83 - 6 * q^85 - 4 * q^87 + 4 * q^89 - 20 * q^91 + 4 * q^93 - 3 * q^95 - 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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