LMFDB - Embedded newform 4050.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(1,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4050 = 2 \cdot 3^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4050.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:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 6$ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 450)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	4050.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.449490 q^{7} -1.00000 q^{8} -4.89898 q^{11} -0.449490 q^{13} +0.449490 q^{14} +1.00000 q^{16} +4.89898 q^{17} +7.44949 q^{19} +4.89898 q^{22} -2.44949 q^{23} +0.449490 q^{26} -0.449490 q^{28} -2.44949 q^{29} +4.44949 q^{31} -1.00000 q^{32} -4.89898 q^{34} -11.3485 q^{37} -7.44949 q^{38} +9.00000 q^{41} +2.55051 q^{43} -4.89898 q^{44} +2.44949 q^{46} -10.8990 q^{47} -6.79796 q^{49} -0.449490 q^{52} -3.55051 q^{53} +0.449490 q^{56} +2.44949 q^{58} -5.44949 q^{59} +8.00000 q^{61} -4.44949 q^{62} +1.00000 q^{64} +0.348469 q^{67} +4.89898 q^{68} +13.3485 q^{71} -1.00000 q^{73} +11.3485 q^{74} +7.44949 q^{76} +2.20204 q^{77} +16.6969 q^{79} -9.00000 q^{82} -5.44949 q^{83} -2.55051 q^{86} +4.89898 q^{88} -9.00000 q^{89} +0.202041 q^{91} -2.44949 q^{92} +10.8990 q^{94} +8.79796 q^{97} +6.79796 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 10 q^{19} - 4 q^{26} + 4 q^{28} + 4 q^{31} - 2 q^{32} - 8 q^{37} - 10 q^{38} + 18 q^{41} + 10 q^{43} - 12 q^{47} + 6 q^{49}+ \cdots - 6 q^{98}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 + 4 * q^13 - 4 * q^14 + 2 * q^16 + 10 * q^19 - 4 * q^26 + 4 * q^28 + 4 * q^31 - 2 * q^32 - 8 * q^37 - 10 * q^38 + 18 * q^41 + 10 * q^43 - 12 * q^47 + 6 * q^49 + 4 * q^52 - 12 * q^53 - 4 * q^56 - 6 * q^59 + 16 * q^61 - 4 * q^62 + 2 * q^64 - 14 * q^67 + 12 * q^71 - 2 * q^73 + 8 * q^74 + 10 * q^76 + 24 * q^77 + 4 * q^79 - 18 * q^82 - 6 * q^83 - 10 * q^86 - 18 * q^89 + 20 * q^91 + 12 * q^94 - 2 * q^97 - 6 * q^98

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

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	−0.449490	−0.124666	−0.0623330	−	0.998055i	$-0.519854\pi$
$13$	−0.449490	−0.124666	−0.0623330	+	0.998055i	$0.519854\pi$
$14$	0.449490	0.120131
$15$	0	0
$16$	1.00000	0.250000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	7.44949	1.70903	0.854515	−	0.519427i	$-0.173854\pi$
$19$	7.44949	1.70903	0.854515	+	0.519427i	$0.173854\pi$
$20$	0	0
$21$	0	0
$22$	4.89898	1.04447
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	0	0
$26$	0.449490	0.0881522
$27$	0	0
$28$	−0.449490	−0.0849456
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	4.44949	0.799152	0.399576	−	0.916700i	$-0.369157\pi$
$31$	4.44949	0.799152	0.399576	+	0.916700i	$0.369157\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.89898	−0.840168
$35$	0	0
$36$	0	0
$37$	−11.3485	−1.86568	−0.932838	−	0.360295i	$-0.882676\pi$
$37$	−11.3485	−1.86568	−0.932838	+	0.360295i	$0.882676\pi$
$38$	−7.44949	−1.20847
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	2.55051	0.388949	0.194475	−	0.980908i	$-0.437700\pi$
$43$	2.55051	0.388949	0.194475	+	0.980908i	$0.437700\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	2.44949	0.361158
$47$	−10.8990	−1.58978	−0.794890	−	0.606754i	$-0.792471\pi$
$47$	−10.8990	−1.58978	−0.794890	+	0.606754i	$0.792471\pi$
$48$	0	0
$49$	−6.79796	−0.971137
$50$	0	0
$51$	0	0
$52$	−0.449490	−0.0623330
$53$	−3.55051	−0.487700	−0.243850	−	0.969813i	$-0.578410\pi$
$53$	−3.55051	−0.487700	−0.243850	+	0.969813i	$0.578410\pi$
$54$	0	0
$55$	0	0
$56$	0.449490	0.0600656
$57$	0	0
$58$	2.44949	0.321634
$59$	−5.44949	−0.709463	−0.354732	−	0.934968i	$-0.615428\pi$
$59$	−5.44949	−0.709463	−0.354732	+	0.934968i	$0.615428\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−4.44949	−0.565086
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0.348469	0.0425723	0.0212861	−	0.999773i	$-0.493224\pi$
$67$	0.348469	0.0425723	0.0212861	+	0.999773i	$0.493224\pi$
$68$	4.89898	0.594089
$69$	0	0
$70$	0	0
$71$	13.3485	1.58417	0.792086	−	0.610410i	$-0.208995\pi$
$71$	13.3485	1.58417	0.792086	+	0.610410i	$0.208995\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	11.3485	1.31923
$75$	0	0
$76$	7.44949	0.854515
$77$	2.20204	0.250946
$78$	0	0
$79$	16.6969	1.87855	0.939276	−	0.343162i	$-0.111498\pi$
$79$	16.6969	1.87855	0.939276	+	0.343162i	$0.111498\pi$
$80$	0	0
$81$	0	0
$82$	−9.00000	−0.993884
$83$	−5.44949	−0.598159	−0.299080	−	0.954228i	$-0.596680\pi$
$83$	−5.44949	−0.598159	−0.299080	+	0.954228i	$0.596680\pi$
$84$	0	0
$85$	0	0
$86$	−2.55051	−0.275029
$87$	0	0
$88$	4.89898	0.522233
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	0.202041	0.0211797
$92$	−2.44949	−0.255377
$93$	0	0
$94$	10.8990	1.12414
$95$	0	0
$96$	0	0
$97$	8.79796	0.893297	0.446649	−	0.894709i	$-0.352617\pi$
$97$	8.79796	0.893297	0.446649	+	0.894709i	$0.352617\pi$
$98$	6.79796	0.686698
$99$	0	0
$100$	0	0
$101$	8.44949	0.840756	0.420378	−	0.907349i	$-0.361898\pi$
$101$	8.44949	0.840756	0.420378	+	0.907349i	$0.361898\pi$
$102$	0	0
$103$	16.6969	1.64520	0.822599	−	0.568622i	$-0.192523\pi$
$103$	16.6969	1.64520	0.822599	+	0.568622i	$0.192523\pi$
$104$	0.449490	0.0440761
$105$	0	0
$106$	3.55051	0.344856
$107$	9.24745	0.893985	0.446992	−	0.894538i	$-0.352495\pi$
$107$	9.24745	0.893985	0.446992	+	0.894538i	$0.352495\pi$
$108$	0	0
$109$	5.55051	0.531642	0.265821	−	0.964022i	$-0.414357\pi$
$109$	5.55051	0.531642	0.265821	+	0.964022i	$0.414357\pi$
$110$	0	0
$111$	0	0
$112$	−0.449490	−0.0424728
$113$	−4.10102	−0.385792	−0.192896	−	0.981219i	$-0.561788\pi$
$113$	−4.10102	−0.385792	−0.192896	+	0.981219i	$0.561788\pi$
$114$	0	0
$115$	0	0
$116$	−2.44949	−0.227429
$117$	0	0
$118$	5.44949	0.501666
$119$	−2.20204	−0.201861
$120$	0	0
$121$	13.0000	1.18182
$122$	−8.00000	−0.724286
$123$	0	0
$124$	4.44949	0.399576
$125$	0	0
$126$	0	0
$127$	3.34847	0.297129	0.148564	−	0.988903i	$-0.452535\pi$
$127$	3.34847	0.297129	0.148564	+	0.988903i	$0.452535\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−3.79796	−0.331829	−0.165915	−	0.986140i	$-0.553058\pi$
$131$	−3.79796	−0.331829	−0.165915	+	0.986140i	$0.553058\pi$
$132$	0	0
$133$	−3.34847	−0.290349
$134$	−0.348469	−0.0301032
$135$	0	0
$136$	−4.89898	−0.420084
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	−13.3485	−1.12018
$143$	2.20204	0.184144
$144$	0	0
$145$	0	0
$146$	1.00000	0.0827606
$147$	0	0
$148$	−11.3485	−0.932838
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−7.44949	−0.604233
$153$	0	0
$154$	−2.20204	−0.177446
$155$	0	0
$156$	0	0
$157$	−19.7980	−1.58005	−0.790025	−	0.613075i	$-0.789932\pi$
$157$	−19.7980	−1.58005	−0.790025	+	0.613075i	$0.789932\pi$
$158$	−16.6969	−1.32834
$159$	0	0
$160$	0	0
$161$	1.10102	0.0867726
$162$	0	0
$163$	7.44949	0.583489	0.291745	−	0.956496i	$-0.405764\pi$
$163$	7.44949	0.583489	0.291745	+	0.956496i	$0.405764\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	5.44949	0.422962
$167$	19.5959	1.51638	0.758189	−	0.652035i	$-0.226085\pi$
$167$	19.5959	1.51638	0.758189	+	0.652035i	$0.226085\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	2.55051	0.194475
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	0	0
$175$	0	0
$176$	−4.89898	−0.369274
$177$	0	0
$178$	9.00000	0.674579
$179$	9.24745	0.691187	0.345593	−	0.938384i	$-0.387678\pi$
$179$	9.24745	0.691187	0.345593	+	0.938384i	$0.387678\pi$
$180$	0	0
$181$	17.7980	1.32291	0.661456	−	0.749984i	$-0.269939\pi$
$181$	17.7980	1.32291	0.661456	+	0.749984i	$0.269939\pi$
$182$	−0.202041	−0.0149763
$183$	0	0
$184$	2.44949	0.180579
$185$	0	0
$186$	0	0
$187$	−24.0000	−1.75505
$188$	−10.8990	−0.794890
$189$	0	0
$190$	0	0
$191$	−1.10102	−0.0796670	−0.0398335	−	0.999206i	$-0.512683\pi$
$191$	−1.10102	−0.0796670	−0.0398335	+	0.999206i	$0.512683\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	−8.79796	−0.631657
$195$	0	0
$196$	−6.79796	−0.485568
$197$	−0.247449	−0.0176300	−0.00881500	−	0.999961i	$-0.502806\pi$
$197$	−0.247449	−0.0176300	−0.00881500	+	0.999961i	$0.502806\pi$
$198$	0	0
$199$	−13.7980	−0.978111	−0.489056	−	0.872253i	$-0.662659\pi$
$199$	−13.7980	−0.978111	−0.489056	+	0.872253i	$0.662659\pi$
$200$	0	0
$201$	0	0
$202$	−8.44949	−0.594504
$203$	1.10102	0.0772765
$204$	0	0
$205$	0	0
$206$	−16.6969	−1.16333
$207$	0	0
$208$	−0.449490	−0.0311665
$209$	−36.4949	−2.52440
$210$	0	0
$211$	−3.44949	−0.237473	−0.118736	−	0.992926i	$-0.537884\pi$
$211$	−3.44949	−0.237473	−0.118736	+	0.992926i	$0.537884\pi$
$212$	−3.55051	−0.243850
$213$	0	0
$214$	−9.24745	−0.632143
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	−5.55051	−0.375928
$219$	0	0
$220$	0	0
$221$	−2.20204	−0.148125
$222$	0	0
$223$	17.7980	1.19184	0.595920	−	0.803044i	$-0.296788\pi$
$223$	17.7980	1.19184	0.595920	+	0.803044i	$0.296788\pi$
$224$	0.449490	0.0300328
$225$	0	0
$226$	4.10102	0.272796
$227$	13.6515	0.906084	0.453042	−	0.891489i	$-0.350339\pi$
$227$	13.6515	0.906084	0.453042	+	0.891489i	$0.350339\pi$
$228$	0	0
$229$	13.1464	0.868740	0.434370	−	0.900734i	$-0.356971\pi$
$229$	13.1464	0.868740	0.434370	+	0.900734i	$0.356971\pi$
$230$	0	0
$231$	0	0
$232$	2.44949	0.160817
$233$	−23.6969	−1.55244	−0.776219	−	0.630463i	$-0.782865\pi$
$233$	−23.6969	−1.55244	−0.776219	+	0.630463i	$0.782865\pi$
$234$	0	0
$235$	0	0
$236$	−5.44949	−0.354732
$237$	0	0
$238$	2.20204	0.142737
$239$	14.4495	0.934660	0.467330	−	0.884083i	$-0.345216\pi$
$239$	14.4495	0.934660	0.467330	+	0.884083i	$0.345216\pi$
$240$	0	0
$241$	−3.20204	−0.206262	−0.103131	−	0.994668i	$-0.532886\pi$
$241$	−3.20204	−0.206262	−0.103131	+	0.994668i	$0.532886\pi$
$242$	−13.0000	−0.835672
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	−3.34847	−0.213058
$248$	−4.44949	−0.282543
$249$	0	0
$250$	0	0
$251$	−0.550510	−0.0347479	−0.0173739	−	0.999849i	$-0.505531\pi$
$251$	−0.550510	−0.0347479	−0.0173739	+	0.999849i	$0.505531\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−3.34847	−0.210102
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.7980	−0.798315	−0.399157	−	0.916882i	$-0.630697\pi$
$257$	−12.7980	−0.798315	−0.399157	+	0.916882i	$0.630697\pi$
$258$	0	0
$259$	5.10102	0.316962
$260$	0	0
$261$	0	0
$262$	3.79796	0.234639
$263$	20.4495	1.26097	0.630485	−	0.776202i	$-0.282856\pi$
$263$	20.4495	1.26097	0.630485	+	0.776202i	$0.282856\pi$
$264$	0	0
$265$	0	0
$266$	3.34847	0.205308
$267$	0	0
$268$	0.348469	0.0212861
$269$	14.4495	0.881001	0.440500	−	0.897752i	$-0.354801\pi$
$269$	14.4495	0.881001	0.440500	+	0.897752i	$0.354801\pi$
$270$	0	0
$271$	15.3485	0.932353	0.466177	−	0.884692i	$-0.345631\pi$
$271$	15.3485	0.932353	0.466177	+	0.884692i	$0.345631\pi$
$272$	4.89898	0.297044
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	−1.55051	−0.0931611	−0.0465806	−	0.998915i	$-0.514832\pi$
$277$	−1.55051	−0.0931611	−0.0465806	+	0.998915i	$0.514832\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	19.1010	1.13947	0.569736	−	0.821828i	$-0.307046\pi$
$281$	19.1010	1.13947	0.569736	+	0.821828i	$0.307046\pi$
$282$	0	0
$283$	−13.2474	−0.787479	−0.393740	−	0.919222i	$-0.628819\pi$
$283$	−13.2474	−0.787479	−0.393740	+	0.919222i	$0.628819\pi$
$284$	13.3485	0.792086
$285$	0	0
$286$	−2.20204	−0.130209
$287$	−4.04541	−0.238793
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	16.0454	0.937383	0.468691	−	0.883362i	$-0.344726\pi$
$293$	16.0454	0.937383	0.468691	+	0.883362i	$0.344726\pi$
$294$	0	0
$295$	0	0
$296$	11.3485	0.659616
$297$	0	0
$298$	6.00000	0.347571
$299$	1.10102	0.0636737
$300$	0	0
$301$	−1.14643	−0.0660790
$302$	−20.0000	−1.15087
$303$	0	0
$304$	7.44949	0.427258
$305$	0	0
$306$	0	0
$307$	22.6969	1.29538	0.647691	−	0.761903i	$-0.275735\pi$
$307$	22.6969	1.29538	0.647691	+	0.761903i	$0.275735\pi$
$308$	2.20204	0.125473
$309$	0	0
$310$	0	0
$311$	−1.10102	−0.0624331	−0.0312166	−	0.999513i	$-0.509938\pi$
$311$	−1.10102	−0.0624331	−0.0312166	+	0.999513i	$0.509938\pi$
$312$	0	0
$313$	−5.89898	−0.333430	−0.166715	−	0.986005i	$-0.553316\pi$
$313$	−5.89898	−0.333430	−0.166715	+	0.986005i	$0.553316\pi$
$314$	19.7980	1.11726
$315$	0	0
$316$	16.6969	0.939276
$317$	17.1464	0.963039	0.481520	−	0.876435i	$-0.340085\pi$
$317$	17.1464	0.963039	0.481520	+	0.876435i	$0.340085\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	0	0
$322$	−1.10102	−0.0613575
$323$	36.4949	2.03063
$324$	0	0
$325$	0	0
$326$	−7.44949	−0.412589
$327$	0	0
$328$	−9.00000	−0.496942
$329$	4.89898	0.270089
$330$	0	0
$331$	6.34847	0.348943	0.174472	−	0.984662i	$-0.444178\pi$
$331$	6.34847	0.348943	0.174472	+	0.984662i	$0.444178\pi$
$332$	−5.44949	−0.299080
$333$	0	0
$334$	−19.5959	−1.07224
$335$	0	0
$336$	0	0
$337$	−20.8990	−1.13844	−0.569220	−	0.822185i	$-0.692755\pi$
$337$	−20.8990	−1.13844	−0.569220	+	0.822185i	$0.692755\pi$
$338$	12.7980	0.696117
$339$	0	0
$340$	0	0
$341$	−21.7980	−1.18043
$342$	0	0
$343$	6.20204	0.334879
$344$	−2.55051	−0.137514
$345$	0	0
$346$	−9.79796	−0.526742
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	4.89898	0.261116
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	0	0
$356$	−9.00000	−0.476999
$357$	0	0
$358$	−9.24745	−0.488743
$359$	−14.2020	−0.749555	−0.374778	−	0.927115i	$-0.622281\pi$
$359$	−14.2020	−0.749555	−0.374778	+	0.927115i	$0.622281\pi$
$360$	0	0
$361$	36.4949	1.92078
$362$	−17.7980	−0.935440
$363$	0	0
$364$	0.202041	0.0105898
$365$	0	0
$366$	0	0
$367$	−12.6969	−0.662775	−0.331387	−	0.943495i	$-0.607517\pi$
$367$	−12.6969	−0.662775	−0.331387	+	0.943495i	$0.607517\pi$
$368$	−2.44949	−0.127688
$369$	0	0
$370$	0	0
$371$	1.59592	0.0828559
$372$	0	0
$373$	−23.5959	−1.22175	−0.610875	−	0.791727i	$-0.709182\pi$
$373$	−23.5959	−1.22175	−0.610875	+	0.791727i	$0.709182\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	10.8990	0.562072
$377$	1.10102	0.0567054
$378$	0	0
$379$	−8.89898	−0.457110	−0.228555	−	0.973531i	$-0.573400\pi$
$379$	−8.89898	−0.457110	−0.228555	+	0.973531i	$0.573400\pi$
$380$	0	0
$381$	0	0
$382$	1.10102	0.0563331
$383$	21.5505	1.10118	0.550590	−	0.834776i	$-0.314403\pi$
$383$	21.5505	1.10118	0.550590	+	0.834776i	$0.314403\pi$
$384$	0	0
$385$	0	0
$386$	−20.0000	−1.01797
$387$	0	0
$388$	8.79796	0.446649
$389$	25.5959	1.29776	0.648882	−	0.760889i	$-0.275237\pi$
$389$	25.5959	1.29776	0.648882	+	0.760889i	$0.275237\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	6.79796	0.343349
$393$	0	0
$394$	0.247449	0.0124663
$395$	0	0
$396$	0	0
$397$	−17.5959	−0.883114	−0.441557	−	0.897233i	$-0.645574\pi$
$397$	−17.5959	−0.883114	−0.441557	+	0.897233i	$0.645574\pi$
$398$	13.7980	0.691629
$399$	0	0
$400$	0	0
$401$	38.6969	1.93243	0.966216	−	0.257732i	$-0.0829752\pi$
$401$	38.6969	1.93243	0.966216	+	0.257732i	$0.0829752\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	8.44949	0.420378
$405$	0	0
$406$	−1.10102	−0.0546427
$407$	55.5959	2.75579
$408$	0	0
$409$	0.101021	0.00499514	0.00249757	−	0.999997i	$-0.499205\pi$
$409$	0.101021	0.00499514	0.00249757	+	0.999997i	$0.499205\pi$
$410$	0	0
$411$	0	0
$412$	16.6969	0.822599
$413$	2.44949	0.120532
$414$	0	0
$415$	0	0
$416$	0.449490	0.0220380
$417$	0	0
$418$	36.4949	1.78502
$419$	−26.1464	−1.27734	−0.638668	−	0.769482i	$-0.720514\pi$
$419$	−26.1464	−1.27734	−0.638668	+	0.769482i	$0.720514\pi$
$420$	0	0
$421$	−26.0454	−1.26938	−0.634688	−	0.772769i	$-0.718871\pi$
$421$	−26.0454	−1.26938	−0.634688	+	0.772769i	$0.718871\pi$
$422$	3.44949	0.167919
$423$	0	0
$424$	3.55051	0.172428
$425$	0	0
$426$	0	0
$427$	−3.59592	−0.174019
$428$	9.24745	0.446992
$429$	0	0
$430$	0	0
$431$	25.3485	1.22099	0.610496	−	0.792019i	$-0.290970\pi$
$431$	25.3485	1.22099	0.610496	+	0.792019i	$0.290970\pi$
$432$	0	0
$433$	9.59592	0.461150	0.230575	−	0.973055i	$-0.425939\pi$
$433$	9.59592	0.461150	0.230575	+	0.973055i	$0.425939\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	5.55051	0.265821
$437$	−18.2474	−0.872894
$438$	0	0
$439$	3.34847	0.159814	0.0799069	−	0.996802i	$-0.474538\pi$
$439$	3.34847	0.159814	0.0799069	+	0.996802i	$0.474538\pi$
$440$	0	0
$441$	0	0
$442$	2.20204	0.104740
$443$	8.20204	0.389691	0.194845	−	0.980834i	$-0.437579\pi$
$443$	8.20204	0.389691	0.194845	+	0.980834i	$0.437579\pi$
$444$	0	0
$445$	0	0
$446$	−17.7980	−0.842758
$447$	0	0
$448$	−0.449490	−0.0212364
$449$	−28.5959	−1.34952	−0.674762	−	0.738035i	$-0.735754\pi$
$449$	−28.5959	−1.34952	−0.674762	+	0.738035i	$0.735754\pi$
$450$	0	0
$451$	−44.0908	−2.07616
$452$	−4.10102	−0.192896
$453$	0	0
$454$	−13.6515	−0.640698
$455$	0	0
$456$	0	0
$457$	13.6969	0.640716	0.320358	−	0.947297i	$-0.396197\pi$
$457$	13.6969	0.640716	0.320358	+	0.947297i	$0.396197\pi$
$458$	−13.1464	−0.614292
$459$	0	0
$460$	0	0
$461$	4.65153	0.216643	0.108322	−	0.994116i	$-0.465452\pi$
$461$	4.65153	0.216643	0.108322	+	0.994116i	$0.465452\pi$
$462$	0	0
$463$	−5.34847	−0.248564	−0.124282	−	0.992247i	$-0.539663\pi$
$463$	−5.34847	−0.248564	−0.124282	+	0.992247i	$0.539663\pi$
$464$	−2.44949	−0.113715
$465$	0	0
$466$	23.6969	1.09774
$467$	−10.3485	−0.478870	−0.239435	−	0.970912i	$-0.576962\pi$
$467$	−10.3485	−0.478870	−0.239435	+	0.970912i	$0.576962\pi$
$468$	0	0
$469$	−0.156633	−0.00723266
$470$	0	0
$471$	0	0
$472$	5.44949	0.250833
$473$	−12.4949	−0.574516
$474$	0	0
$475$	0	0
$476$	−2.20204	−0.100930
$477$	0	0
$478$	−14.4495	−0.660904
$479$	0.247449	0.0113062	0.00565311	−	0.999984i	$-0.498201\pi$
$479$	0.247449	0.0113062	0.00565311	+	0.999984i	$0.498201\pi$
$480$	0	0
$481$	5.10102	0.232587
$482$	3.20204	0.145849
$483$	0	0
$484$	13.0000	0.590909
$485$	0	0
$486$	0	0
$487$	−24.4495	−1.10791	−0.553956	−	0.832546i	$-0.686882\pi$
$487$	−24.4495	−1.10791	−0.553956	+	0.832546i	$0.686882\pi$
$488$	−8.00000	−0.362143
$489$	0	0
$490$	0	0
$491$	−27.2474	−1.22966	−0.614830	−	0.788660i	$-0.710775\pi$
$491$	−27.2474	−1.22966	−0.614830	+	0.788660i	$0.710775\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	3.34847	0.150655
$495$	0	0
$496$	4.44949	0.199788
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−8.34847	−0.373729	−0.186864	−	0.982386i	$-0.559833\pi$
$499$	−8.34847	−0.373729	−0.186864	+	0.982386i	$0.559833\pi$
$500$	0	0
$501$	0	0
$502$	0.550510	0.0245705
$503$	21.5505	0.960890	0.480445	−	0.877025i	$-0.340475\pi$
$503$	21.5505	0.960890	0.480445	+	0.877025i	$0.340475\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	3.34847	0.148564
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0.449490	0.0198843
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.7980	0.564494
$515$	0	0
$516$	0	0
$517$	53.3939	2.34826
$518$	−5.10102	−0.224126
$519$	0	0
$520$	0	0
$521$	−29.3939	−1.28777	−0.643885	−	0.765123i	$-0.722678\pi$
$521$	−29.3939	−1.28777	−0.643885	+	0.765123i	$0.722678\pi$
$522$	0	0
$523$	−20.3485	−0.889776	−0.444888	−	0.895586i	$-0.646757\pi$
$523$	−20.3485	−0.889776	−0.444888	+	0.895586i	$0.646757\pi$
$524$	−3.79796	−0.165915
$525$	0	0
$526$	−20.4495	−0.891640
$527$	21.7980	0.949534
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	−3.34847	−0.145175
$533$	−4.04541	−0.175226
$534$	0	0
$535$	0	0
$536$	−0.348469	−0.0150516
$537$	0	0
$538$	−14.4495	−0.622962
$539$	33.3031	1.43446
$540$	0	0
$541$	−37.7980	−1.62506	−0.812531	−	0.582919i	$-0.801911\pi$
$541$	−37.7980	−1.62506	−0.812531	+	0.582919i	$0.801911\pi$
$542$	−15.3485	−0.659273
$543$	0	0
$544$	−4.89898	−0.210042
$545$	0	0
$546$	0	0
$547$	15.6515	0.669211	0.334606	−	0.942358i	$-0.391397\pi$
$547$	15.6515	0.669211	0.334606	+	0.942358i	$0.391397\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	0	0
$551$	−18.2474	−0.777367
$552$	0	0
$553$	−7.50510	−0.319149
$554$	1.55051	0.0658749
$555$	0	0
$556$	−4.00000	−0.169638
$557$	41.3939	1.75391	0.876957	−	0.480568i	$-0.159570\pi$
$557$	41.3939	1.75391	0.876957	+	0.480568i	$0.159570\pi$
$558$	0	0
$559$	−1.14643	−0.0484887
$560$	0	0
$561$	0	0
$562$	−19.1010	−0.805728
$563$	−23.9444	−1.00914	−0.504568	−	0.863372i	$-0.668348\pi$
$563$	−23.9444	−1.00914	−0.504568	+	0.863372i	$0.668348\pi$
$564$	0	0
$565$	0	0
$566$	13.2474	0.556832
$567$	0	0
$568$	−13.3485	−0.560089
$569$	33.7980	1.41688	0.708442	−	0.705769i	$-0.249398\pi$
$569$	33.7980	1.41688	0.708442	+	0.705769i	$0.249398\pi$
$570$	0	0
$571$	25.9444	1.08574	0.542869	−	0.839817i	$-0.317338\pi$
$571$	25.9444	1.08574	0.542869	+	0.839817i	$0.317338\pi$
$572$	2.20204	0.0920720
$573$	0	0
$574$	4.04541	0.168852
$575$	0	0
$576$	0	0
$577$	40.3939	1.68162	0.840810	−	0.541331i	$-0.182079\pi$
$577$	40.3939	1.68162	0.840810	+	0.541331i	$0.182079\pi$
$578$	−7.00000	−0.291162
$579$	0	0
$580$	0	0
$581$	2.44949	0.101622
$582$	0	0
$583$	17.3939	0.720381
$584$	1.00000	0.0413803
$585$	0	0
$586$	−16.0454	−0.662830
$587$	−26.6969	−1.10190	−0.550950	−	0.834538i	$-0.685735\pi$
$587$	−26.6969	−1.10190	−0.550950	+	0.834538i	$0.685735\pi$
$588$	0	0
$589$	33.1464	1.36577
$590$	0	0
$591$	0	0
$592$	−11.3485	−0.466419
$593$	−7.89898	−0.324372	−0.162186	−	0.986760i	$-0.551854\pi$
$593$	−7.89898	−0.324372	−0.162186	+	0.986760i	$0.551854\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−1.10102	−0.0450241
$599$	−22.6515	−0.925516	−0.462758	−	0.886485i	$-0.653140\pi$
$599$	−22.6515	−0.925516	−0.462758	+	0.886485i	$0.653140\pi$
$600$	0	0
$601$	−16.4949	−0.672841	−0.336420	−	0.941712i	$-0.609216\pi$
$601$	−16.4949	−0.672841	−0.336420	+	0.941712i	$0.609216\pi$
$602$	1.14643	0.0467249
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	3.34847	0.135910	0.0679551	−	0.997688i	$-0.478353\pi$
$607$	3.34847	0.135910	0.0679551	+	0.997688i	$0.478353\pi$
$608$	−7.44949	−0.302117
$609$	0	0
$610$	0	0
$611$	4.89898	0.198191
$612$	0	0
$613$	12.0454	0.486509	0.243255	−	0.969962i	$-0.421785\pi$
$613$	12.0454	0.486509	0.243255	+	0.969962i	$0.421785\pi$
$614$	−22.6969	−0.915974
$615$	0	0
$616$	−2.20204	−0.0887228
$617$	44.3939	1.78723	0.893615	−	0.448834i	$-0.148161\pi$
$617$	44.3939	1.78723	0.893615	+	0.448834i	$0.148161\pi$
$618$	0	0
$619$	−31.7423	−1.27583	−0.637916	−	0.770106i	$-0.720203\pi$
$619$	−31.7423	−1.27583	−0.637916	+	0.770106i	$0.720203\pi$
$620$	0	0
$621$	0	0
$622$	1.10102	0.0441469
$623$	4.04541	0.162076
$624$	0	0
$625$	0	0
$626$	5.89898	0.235771
$627$	0	0
$628$	−19.7980	−0.790025
$629$	−55.5959	−2.21675
$630$	0	0
$631$	−6.20204	−0.246899	−0.123450	−	0.992351i	$-0.539396\pi$
$631$	−6.20204	−0.246899	−0.123450	+	0.992351i	$0.539396\pi$
$632$	−16.6969	−0.664169
$633$	0	0
$634$	−17.1464	−0.680972
$635$	0	0
$636$	0	0
$637$	3.05561	0.121068
$638$	−12.0000	−0.475085
$639$	0	0
$640$	0	0
$641$	−14.3939	−0.568524	−0.284262	−	0.958747i	$-0.591749\pi$
$641$	−14.3939	−0.568524	−0.284262	+	0.958747i	$0.591749\pi$
$642$	0	0
$643$	−17.6515	−0.696108	−0.348054	−	0.937474i	$-0.613157\pi$
$643$	−17.6515	−0.696108	−0.348054	+	0.937474i	$0.613157\pi$
$644$	1.10102	0.0433863
$645$	0	0
$646$	−36.4949	−1.43587
$647$	24.2474	0.953266	0.476633	−	0.879102i	$-0.341857\pi$
$647$	24.2474	0.953266	0.476633	+	0.879102i	$0.341857\pi$
$648$	0	0
$649$	26.6969	1.04795
$650$	0	0
$651$	0	0
$652$	7.44949	0.291745
$653$	18.2474	0.714078	0.357039	−	0.934090i	$-0.383786\pi$
$653$	18.2474	0.714078	0.357039	+	0.934090i	$0.383786\pi$
$654$	0	0
$655$	0	0
$656$	9.00000	0.351391
$657$	0	0
$658$	−4.89898	−0.190982
$659$	9.85357	0.383841	0.191920	−	0.981411i	$-0.438528\pi$
$659$	9.85357	0.383841	0.191920	+	0.981411i	$0.438528\pi$
$660$	0	0
$661$	7.39388	0.287588	0.143794	−	0.989608i	$-0.454070\pi$
$661$	7.39388	0.287588	0.143794	+	0.989608i	$0.454070\pi$
$662$	−6.34847	−0.246740
$663$	0	0
$664$	5.44949	0.211481
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	19.5959	0.758189
$669$	0	0
$670$	0	0
$671$	−39.1918	−1.51298
$672$	0	0
$673$	22.6969	0.874903	0.437451	−	0.899242i	$-0.355881\pi$
$673$	22.6969	0.874903	0.437451	+	0.899242i	$0.355881\pi$
$674$	20.8990	0.804999
$675$	0	0
$676$	−12.7980	−0.492229
$677$	−25.8434	−0.993241	−0.496621	−	0.867968i	$-0.665426\pi$
$677$	−25.8434	−0.993241	−0.496621	+	0.867968i	$0.665426\pi$
$678$	0	0
$679$	−3.95459	−0.151763
$680$	0	0
$681$	0	0
$682$	21.7980	0.834687
$683$	41.9444	1.60496	0.802479	−	0.596681i	$-0.203514\pi$
$683$	41.9444	1.60496	0.802479	+	0.596681i	$0.203514\pi$
$684$	0	0
$685$	0	0
$686$	−6.20204	−0.236795
$687$	0	0
$688$	2.55051	0.0972373
$689$	1.59592	0.0607996
$690$	0	0
$691$	39.0454	1.48536	0.742679	−	0.669648i	$-0.233555\pi$
$691$	39.0454	1.48536	0.742679	+	0.669648i	$0.233555\pi$
$692$	9.79796	0.372463
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	44.0908	1.67006
$698$	−14.0000	−0.529908
$699$	0	0
$700$	0	0
$701$	33.7980	1.27653	0.638266	−	0.769816i	$-0.279652\pi$
$701$	33.7980	1.27653	0.638266	+	0.769816i	$0.279652\pi$
$702$	0	0
$703$	−84.5403	−3.18850
$704$	−4.89898	−0.184637
$705$	0	0
$706$	9.00000	0.338719
$707$	−3.79796	−0.142837
$708$	0	0
$709$	4.44949	0.167104	0.0835520	−	0.996503i	$-0.473374\pi$
$709$	4.44949	0.167104	0.0835520	+	0.996503i	$0.473374\pi$
$710$	0	0
$711$	0	0
$712$	9.00000	0.337289
$713$	−10.8990	−0.408170
$714$	0	0
$715$	0	0
$716$	9.24745	0.345593
$717$	0	0
$718$	14.2020	0.530015
$719$	7.95459	0.296656	0.148328	−	0.988938i	$-0.452611\pi$
$719$	7.95459	0.296656	0.148328	+	0.988938i	$0.452611\pi$
$720$	0	0
$721$	−7.50510	−0.279505
$722$	−36.4949	−1.35820
$723$	0	0
$724$	17.7980	0.661456
$725$	0	0
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−0.202041	−0.00748814
$729$	0	0
$730$	0	0
$731$	12.4949	0.462140
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	12.6969	0.468653
$735$	0	0
$736$	2.44949	0.0902894
$737$	−1.70714	−0.0628834
$738$	0	0
$739$	3.04541	0.112027	0.0560136	−	0.998430i	$-0.482161\pi$
$739$	3.04541	0.112027	0.0560136	+	0.998430i	$0.482161\pi$
$740$	0	0
$741$	0	0
$742$	−1.59592	−0.0585880
$743$	49.3485	1.81042	0.905210	−	0.424965i	$-0.139714\pi$
$743$	49.3485	1.81042	0.905210	+	0.424965i	$0.139714\pi$
$744$	0	0
$745$	0	0
$746$	23.5959	0.863908
$747$	0	0
$748$	−24.0000	−0.877527
$749$	−4.15663	−0.151880
$750$	0	0
$751$	−5.95459	−0.217286	−0.108643	−	0.994081i	$-0.534651\pi$
$751$	−5.95459	−0.217286	−0.108643	+	0.994081i	$0.534651\pi$
$752$	−10.8990	−0.397445
$753$	0	0
$754$	−1.10102	−0.0400968
$755$	0	0
$756$	0	0
$757$	−38.0454	−1.38278	−0.691392	−	0.722480i	$-0.743002\pi$
$757$	−38.0454	−1.38278	−0.691392	+	0.722480i	$0.743002\pi$
$758$	8.89898	0.323225
$759$	0	0
$760$	0	0
$761$	13.8990	0.503838	0.251919	−	0.967748i	$-0.418938\pi$
$761$	13.8990	0.503838	0.251919	+	0.967748i	$0.418938\pi$
$762$	0	0
$763$	−2.49490	−0.0903214
$764$	−1.10102	−0.0398335
$765$	0	0
$766$	−21.5505	−0.778652
$767$	2.44949	0.0884459
$768$	0	0
$769$	−28.1918	−1.01662	−0.508312	−	0.861173i	$-0.669730\pi$
$769$	−28.1918	−1.01662	−0.508312	+	0.861173i	$0.669730\pi$
$770$	0	0
$771$	0	0
$772$	20.0000	0.719816
$773$	−10.4041	−0.374209	−0.187104	−	0.982340i	$-0.559910\pi$
$773$	−10.4041	−0.374209	−0.187104	+	0.982340i	$0.559910\pi$
$774$	0	0
$775$	0	0
$776$	−8.79796	−0.315828
$777$	0	0
$778$	−25.5959	−0.917658
$779$	67.0454	2.40215
$780$	0	0
$781$	−65.3939	−2.33998
$782$	12.0000	0.429119
$783$	0	0
$784$	−6.79796	−0.242784
$785$	0	0
$786$	0	0
$787$	34.6969	1.23681	0.618406	−	0.785859i	$-0.287779\pi$
$787$	34.6969	1.23681	0.618406	+	0.785859i	$0.287779\pi$
$788$	−0.247449	−0.00881500
$789$	0	0
$790$	0	0
$791$	1.84337	0.0655426
$792$	0	0
$793$	−3.59592	−0.127695
$794$	17.5959	0.624456
$795$	0	0
$796$	−13.7980	−0.489056
$797$	−30.2474	−1.07142	−0.535710	−	0.844402i	$-0.679956\pi$
$797$	−30.2474	−1.07142	−0.535710	+	0.844402i	$0.679956\pi$
$798$	0	0
$799$	−53.3939	−1.88894
$800$	0	0
$801$	0	0
$802$	−38.6969	−1.36644
$803$	4.89898	0.172881
$804$	0	0
$805$	0	0
$806$	2.00000	0.0704470
$807$	0	0
$808$	−8.44949	−0.297252
$809$	−6.30306	−0.221604	−0.110802	−	0.993843i	$-0.535342\pi$
$809$	−6.30306	−0.221604	−0.110802	+	0.993843i	$0.535342\pi$
$810$	0	0
$811$	−28.5505	−1.00254	−0.501272	−	0.865290i	$-0.667134\pi$
$811$	−28.5505	−1.00254	−0.501272	+	0.865290i	$0.667134\pi$
$812$	1.10102	0.0386382
$813$	0	0
$814$	−55.5959	−1.94864
$815$	0	0
$816$	0	0
$817$	19.0000	0.664726
$818$	−0.101021	−0.00353210
$819$	0	0
$820$	0	0
$821$	27.1918	0.949002	0.474501	−	0.880255i	$-0.342629\pi$
$821$	27.1918	0.949002	0.474501	+	0.880255i	$0.342629\pi$
$822$	0	0
$823$	−17.1010	−0.596104	−0.298052	−	0.954550i	$-0.596337\pi$
$823$	−17.1010	−0.596104	−0.298052	+	0.954550i	$0.596337\pi$
$824$	−16.6969	−0.581665
$825$	0	0
$826$	−2.44949	−0.0852286
$827$	−35.9444	−1.24991	−0.624954	−	0.780661i	$-0.714882\pi$
$827$	−35.9444	−1.24991	−0.624954	+	0.780661i	$0.714882\pi$
$828$	0	0
$829$	−46.7423	−1.62343	−0.811714	−	0.584055i	$-0.801465\pi$
$829$	−46.7423	−1.62343	−0.811714	+	0.584055i	$0.801465\pi$
$830$	0	0
$831$	0	0
$832$	−0.449490	−0.0155833
$833$	−33.3031	−1.15388
$834$	0	0
$835$	0	0
$836$	−36.4949	−1.26220
$837$	0	0
$838$	26.1464	0.903213
$839$	37.3485	1.28941	0.644706	−	0.764430i	$-0.276980\pi$
$839$	37.3485	1.28941	0.644706	+	0.764430i	$0.276980\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	26.0454	0.897584
$843$	0	0
$844$	−3.44949	−0.118736
$845$	0	0
$846$	0	0
$847$	−5.84337	−0.200780
$848$	−3.55051	−0.121925
$849$	0	0
$850$	0	0
$851$	27.7980	0.952902
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	3.59592	0.123050
$855$	0	0
$856$	−9.24745	−0.316071
$857$	−49.8990	−1.70452	−0.852258	−	0.523121i	$-0.824768\pi$
$857$	−49.8990	−1.70452	−0.852258	+	0.523121i	$0.824768\pi$
$858$	0	0
$859$	−20.3485	−0.694281	−0.347140	−	0.937813i	$-0.612847\pi$
$859$	−20.3485	−0.694281	−0.347140	+	0.937813i	$0.612847\pi$
$860$	0	0
$861$	0	0
$862$	−25.3485	−0.863372
$863$	−43.8434	−1.49245	−0.746223	−	0.665696i	$-0.768135\pi$
$863$	−43.8434	−1.49245	−0.746223	+	0.665696i	$0.768135\pi$
$864$	0	0
$865$	0	0
$866$	−9.59592	−0.326083
$867$	0	0
$868$	−2.00000	−0.0678844
$869$	−81.7980	−2.77481
$870$	0	0
$871$	−0.156633	−0.00530732
$872$	−5.55051	−0.187964
$873$	0	0
$874$	18.2474	0.617229
$875$	0	0
$876$	0	0
$877$	−31.7980	−1.07374	−0.536870	−	0.843665i	$-0.680394\pi$
$877$	−31.7980	−1.07374	−0.536870	+	0.843665i	$0.680394\pi$
$878$	−3.34847	−0.113005
$879$	0	0
$880$	0	0
$881$	38.6969	1.30373	0.651866	−	0.758334i	$-0.273986\pi$
$881$	38.6969	1.30373	0.651866	+	0.758334i	$0.273986\pi$
$882$	0	0
$883$	47.7980	1.60853	0.804265	−	0.594271i	$-0.202559\pi$
$883$	47.7980	1.60853	0.804265	+	0.594271i	$0.202559\pi$
$884$	−2.20204	−0.0740627
$885$	0	0
$886$	−8.20204	−0.275553
$887$	5.50510	0.184843	0.0924216	−	0.995720i	$-0.470539\pi$
$887$	5.50510	0.184843	0.0924216	+	0.995720i	$0.470539\pi$
$888$	0	0
$889$	−1.50510	−0.0504795
$890$	0	0
$891$	0	0
$892$	17.7980	0.595920
$893$	−81.1918	−2.71698
$894$	0	0
$895$	0	0
$896$	0.449490	0.0150164
$897$	0	0
$898$	28.5959	0.954258
$899$	−10.8990	−0.363501
$900$	0	0
$901$	−17.3939	−0.579474
$902$	44.0908	1.46806
$903$	0	0
$904$	4.10102	0.136398
$905$	0	0
$906$	0	0
$907$	36.3485	1.20693	0.603466	−	0.797389i	$-0.293786\pi$
$907$	36.3485	1.20693	0.603466	+	0.797389i	$0.293786\pi$
$908$	13.6515	0.453042
$909$	0	0
$910$	0	0
$911$	−7.34847	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$911$	−7.34847	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$912$	0	0
$913$	26.6969	0.883540
$914$	−13.6969	−0.453054
$915$	0	0
$916$	13.1464	0.434370
$917$	1.70714	0.0563748
$918$	0	0
$919$	3.34847	0.110456	0.0552279	−	0.998474i	$-0.482411\pi$
$919$	3.34847	0.110456	0.0552279	+	0.998474i	$0.482411\pi$
$920$	0	0
$921$	0	0
$922$	−4.65153	−0.153190
$923$	−6.00000	−0.197492
$924$	0	0
$925$	0	0
$926$	5.34847	0.175762
$927$	0	0
$928$	2.44949	0.0804084
$929$	−51.1918	−1.67955	−0.839775	−	0.542935i	$-0.817313\pi$
$929$	−51.1918	−1.67955	−0.839775	+	0.542935i	$0.817313\pi$
$930$	0	0
$931$	−50.6413	−1.65970
$932$	−23.6969	−0.776219
$933$	0	0
$934$	10.3485	0.338612
$935$	0	0
$936$	0	0
$937$	7.20204	0.235280	0.117640	−	0.993056i	$-0.462467\pi$
$937$	7.20204	0.235280	0.117640	+	0.993056i	$0.462467\pi$
$938$	0.156633	0.00511426
$939$	0	0
$940$	0	0
$941$	53.6413	1.74866	0.874329	−	0.485334i	$-0.161302\pi$
$941$	53.6413	1.74866	0.874329	+	0.485334i	$0.161302\pi$
$942$	0	0
$943$	−22.0454	−0.717897
$944$	−5.44949	−0.177366
$945$	0	0
$946$	12.4949	0.406244
$947$	26.1464	0.849645	0.424822	−	0.905277i	$-0.360337\pi$
$947$	26.1464	0.849645	0.424822	+	0.905277i	$0.360337\pi$
$948$	0	0
$949$	0.449490	0.0145911
$950$	0	0
$951$	0	0
$952$	2.20204	0.0713686
$953$	2.20204	0.0713311	0.0356656	−	0.999364i	$-0.488645\pi$
$953$	2.20204	0.0713311	0.0356656	+	0.999364i	$0.488645\pi$
$954$	0	0
$955$	0	0
$956$	14.4495	0.467330
$957$	0	0
$958$	−0.247449	−0.00799471
$959$	−1.34847	−0.0435443
$960$	0	0
$961$	−11.2020	−0.361356
$962$	−5.10102	−0.164464
$963$	0	0
$964$	−3.20204	−0.103131
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−13.0000	−0.417836
$969$	0	0
$970$	0	0
$971$	−40.8434	−1.31073	−0.655363	−	0.755314i	$-0.727484\pi$
$971$	−40.8434	−1.31073	−0.655363	+	0.755314i	$0.727484\pi$
$972$	0	0
$973$	1.79796	0.0576399
$974$	24.4495	0.783412
$975$	0	0
$976$	8.00000	0.256074
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	0	0
$979$	44.0908	1.40915
$980$	0	0
$981$	0	0
$982$	27.2474	0.869501
$983$	−13.1010	−0.417858	−0.208929	−	0.977931i	$-0.566998\pi$
$983$	−13.1010	−0.417858	−0.208929	+	0.977931i	$0.566998\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	−3.34847	−0.106529
$989$	−6.24745	−0.198657
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−4.44949	−0.141271
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	−8.04541	−0.254801	−0.127400	−	0.991851i	$-0.540663\pi$
$997$	−8.04541	−0.254801	−0.127400	+	0.991851i	$0.540663\pi$
$998$	8.34847	0.264266
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.br.1.1		2
3.2	odd	2		4050.2.a.by.1.1		2
5.2	odd	4		4050.2.c.y.649.1		4
5.3	odd	4		4050.2.c.y.649.4		4
5.4	even	2		4050.2.a.bu.1.2		2
9.2	odd	6		1350.2.e.k.901.2		4
9.4	even	3		450.2.e.m.151.2	yes	4
9.5	odd	6		1350.2.e.k.451.2		4
9.7	even	3		450.2.e.m.301.1	yes	4
15.2	even	4		4050.2.c.w.649.3		4
15.8	even	4		4050.2.c.w.649.2		4
15.14	odd	2		4050.2.a.bl.1.2		2
45.2	even	12		1350.2.j.g.199.3		8
45.4	even	6		450.2.e.l.151.1	✓	4
45.7	odd	12		450.2.j.f.49.1		8
45.13	odd	12		450.2.j.f.349.1		8
45.14	odd	6		1350.2.e.n.451.1		4
45.22	odd	12		450.2.j.f.349.4		8
45.23	even	12		1350.2.j.g.1099.3		8
45.29	odd	6		1350.2.e.n.901.1		4
45.32	even	12		1350.2.j.g.1099.2		8
45.34	even	6		450.2.e.l.301.2	yes	4
45.38	even	12		1350.2.j.g.199.2		8
45.43	odd	12		450.2.j.f.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.e.l.151.1	✓	4	45.4	even	6
450.2.e.l.301.2	yes	4	45.34	even	6
450.2.e.m.151.2	yes	4	9.4	even	3
450.2.e.m.301.1	yes	4	9.7	even	3
450.2.j.f.49.1		8	45.7	odd	12
450.2.j.f.49.4		8	45.43	odd	12
450.2.j.f.349.1		8	45.13	odd	12
450.2.j.f.349.4		8	45.22	odd	12
1350.2.e.k.451.2		4	9.5	odd	6
1350.2.e.k.901.2		4	9.2	odd	6
1350.2.e.n.451.1		4	45.14	odd	6
1350.2.e.n.901.1		4	45.29	odd	6
1350.2.j.g.199.2		8	45.38	even	12
1350.2.j.g.199.3		8	45.2	even	12
1350.2.j.g.1099.2		8	45.32	even	12
1350.2.j.g.1099.3		8	45.23	even	12
4050.2.a.bl.1.2		2	15.14	odd	2
4050.2.a.br.1.1		2	1.1	even	1	trivial
4050.2.a.bu.1.2		2	5.4	even	2
4050.2.a.by.1.1		2	3.2	odd	2
4050.2.c.w.649.2		4	15.8	even	4
4050.2.c.w.649.3		4	15.2	even	4
4050.2.c.y.649.1		4	5.2	odd	4
4050.2.c.y.649.4		4	5.3	odd	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4050.2.a.br.1.1

Level	$4050$
Weight	$2$
Character	4050.1
Self dual	yes
Analytic conductor	$32.339$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$