LMFDB - Embedded newform 3850.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.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:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	3850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{9} -1.00000 q^{11} -2.64575 q^{12} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.64575 q^{17} +4.00000 q^{18} -7.29150 q^{19} +2.64575 q^{21} -1.00000 q^{22} -2.64575 q^{24} +4.00000 q^{26} -2.64575 q^{27} -1.00000 q^{28} -9.29150 q^{29} +6.64575 q^{31} +1.00000 q^{32} +2.64575 q^{33} +4.64575 q^{34} +4.00000 q^{36} +1.00000 q^{37} -7.29150 q^{38} -10.5830 q^{39} +2.64575 q^{42} -2.29150 q^{43} -1.00000 q^{44} +10.6458 q^{47} -2.64575 q^{48} +1.00000 q^{49} -12.2915 q^{51} +4.00000 q^{52} +3.00000 q^{53} -2.64575 q^{54} -1.00000 q^{56} +19.2915 q^{57} -9.29150 q^{58} -7.93725 q^{59} -4.00000 q^{61} +6.64575 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.64575 q^{66} -11.2915 q^{67} +4.64575 q^{68} -6.00000 q^{71} +4.00000 q^{72} +15.2288 q^{73} +1.00000 q^{74} -7.29150 q^{76} +1.00000 q^{77} -10.5830 q^{78} +8.29150 q^{79} -5.00000 q^{81} -6.00000 q^{83} +2.64575 q^{84} -2.29150 q^{86} +24.5830 q^{87} -1.00000 q^{88} +15.2915 q^{89} -4.00000 q^{91} -17.5830 q^{93} +10.6458 q^{94} -2.64575 q^{96} +4.58301 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 8 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 8 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 8 q^{9} - 2 q^{11} + 8 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{18} - 4 q^{19} - 2 q^{22} + 8 q^{26} - 2 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} + 4 q^{34} + 8 q^{36} + 2 q^{37} - 4 q^{38} + 6 q^{43} - 2 q^{44} + 16 q^{47} + 2 q^{49} - 14 q^{51} + 8 q^{52} + 6 q^{53} - 2 q^{56} + 28 q^{57} - 8 q^{58} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{67} + 4 q^{68} - 12 q^{71} + 8 q^{72} + 4 q^{73} + 2 q^{74} - 4 q^{76} + 2 q^{77} + 6 q^{79} - 10 q^{81} - 12 q^{83} + 6 q^{86} + 28 q^{87} - 2 q^{88} + 20 q^{89} - 8 q^{91} - 14 q^{93} + 16 q^{94} - 12 q^{97} + 2 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 8 * q^9 - 2 * q^11 + 8 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^18 - 4 * q^19 - 2 * q^22 + 8 * q^26 - 2 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 + 4 * q^34 + 8 * q^36 + 2 * q^37 - 4 * q^38 + 6 * q^43 - 2 * q^44 + 16 * q^47 + 2 * q^49 - 14 * q^51 + 8 * q^52 + 6 * q^53 - 2 * q^56 + 28 * q^57 - 8 * q^58 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^67 + 4 * q^68 - 12 * q^71 + 8 * q^72 + 4 * q^73 + 2 * q^74 - 4 * q^76 + 2 * q^77 + 6 * q^79 - 10 * q^81 - 12 * q^83 + 6 * q^86 + 28 * q^87 - 2 * q^88 + 20 * q^89 - 8 * q^91 - 14 * q^93 + 16 * q^94 - 12 * q^97 + 2 * q^98 - 8 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.64575	−1.52753	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	−2.64575	−1.52753	−0.763763	+	0.645497i	$0.776650\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.64575	−1.08012
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	4.00000	1.33333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.64575	−0.763763
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.64575	1.12676	0.563380	−	0.826198i	$-0.309501\pi$
$17$	4.64575	1.12676	0.563380	+	0.826198i	$0.309501\pi$
$18$	4.00000	0.942809
$19$	−7.29150	−1.67279	−0.836393	−	0.548131i	$-0.815340\pi$
$19$	−7.29150	−1.67279	−0.836393	+	0.548131i	$0.815340\pi$
$20$	0	0
$21$	2.64575	0.577350
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.64575	−0.540062
$25$	0	0
$26$	4.00000	0.784465
$27$	−2.64575	−0.509175
$28$	−1.00000	−0.188982
$29$	−9.29150	−1.72539	−0.862694	−	0.505726i	$-0.831225\pi$
$29$	−9.29150	−1.72539	−0.862694	+	0.505726i	$0.831225\pi$
$30$	0	0
$31$	6.64575	1.19361	0.596806	−	0.802386i	$-0.296436\pi$
$31$	6.64575	1.19361	0.596806	+	0.802386i	$0.296436\pi$
$32$	1.00000	0.176777
$33$	2.64575	0.460566
$34$	4.64575	0.796740
$35$	0	0
$36$	4.00000	0.666667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−7.29150	−1.18284
$39$	−10.5830	−1.69464
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.64575	0.408248
$43$	−2.29150	−0.349451	−0.174725	−	0.984617i	$-0.555904\pi$
$43$	−2.29150	−0.349451	−0.174725	+	0.984617i	$0.555904\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	10.6458	1.55284	0.776421	−	0.630214i	$-0.217033\pi$
$47$	10.6458	1.55284	0.776421	+	0.630214i	$0.217033\pi$
$48$	−2.64575	−0.381881
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.2915	−1.72115
$52$	4.00000	0.554700
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−2.64575	−0.360041
$55$	0	0
$56$	−1.00000	−0.133631
$57$	19.2915	2.55522
$58$	−9.29150	−1.22003
$59$	−7.93725	−1.03334	−0.516671	−	0.856184i	$-0.672829\pi$
$59$	−7.93725	−1.03334	−0.516671	+	0.856184i	$0.672829\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	6.64575	0.844011
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	2.64575	0.325669
$67$	−11.2915	−1.37948	−0.689738	−	0.724059i	$-0.742274\pi$
$67$	−11.2915	−1.37948	−0.689738	+	0.724059i	$0.742274\pi$
$68$	4.64575	0.563380
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	4.00000	0.471405
$73$	15.2288	1.78239	0.891196	−	0.453619i	$-0.149867\pi$
$73$	15.2288	1.78239	0.891196	+	0.453619i	$0.149867\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−7.29150	−0.836393
$77$	1.00000	0.113961
$78$	−10.5830	−1.19829
$79$	8.29150	0.932867	0.466433	−	0.884556i	$-0.345539\pi$
$79$	8.29150	0.932867	0.466433	+	0.884556i	$0.345539\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	2.64575	0.288675
$85$	0	0
$86$	−2.29150	−0.247099
$87$	24.5830	2.63557
$88$	−1.00000	−0.106600
$89$	15.2915	1.62090	0.810448	−	0.585811i	$-0.199224\pi$
$89$	15.2915	1.62090	0.810448	+	0.585811i	$0.199224\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−17.5830	−1.82327
$94$	10.6458	1.09803
$95$	0	0
$96$	−2.64575	−0.270031
$97$	4.58301	0.465334	0.232667	−	0.972556i	$-0.425255\pi$
$97$	4.58301	0.465334	0.232667	+	0.972556i	$0.425255\pi$
$98$	1.00000	0.101015
$99$	−4.00000	−0.402015
$100$	0	0
$101$	16.6458	1.65631	0.828157	−	0.560496i	$-0.189390\pi$
$101$	16.6458	1.65631	0.828157	+	0.560496i	$0.189390\pi$
$102$	−12.2915	−1.21704
$103$	−9.35425	−0.921702	−0.460851	−	0.887478i	$-0.652456\pi$
$103$	−9.35425	−0.921702	−0.460851	+	0.887478i	$0.652456\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	3.00000	0.291386
$107$	5.70850	0.551861	0.275931	−	0.961178i	$-0.411014\pi$
$107$	5.70850	0.551861	0.275931	+	0.961178i	$0.411014\pi$
$108$	−2.64575	−0.254588
$109$	−10.5830	−1.01367	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	−10.5830	−1.01367	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	−2.64575	−0.251124
$112$	−1.00000	−0.0944911
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	19.2915	1.80681
$115$	0	0
$116$	−9.29150	−0.862694
$117$	16.0000	1.47920
$118$	−7.93725	−0.730683
$119$	−4.64575	−0.425875
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.00000	−0.362143
$123$	0	0
$124$	6.64575	0.596806
$125$	0	0
$126$	−4.00000	−0.356348
$127$	16.8745	1.49737	0.748685	−	0.662926i	$-0.230685\pi$
$127$	16.8745	1.49737	0.748685	+	0.662926i	$0.230685\pi$
$128$	1.00000	0.0883883
$129$	6.06275	0.533795
$130$	0	0
$131$	21.8745	1.91118	0.955592	−	0.294692i	$-0.0952170\pi$
$131$	21.8745	1.91118	0.955592	+	0.294692i	$0.0952170\pi$
$132$	2.64575	0.230283
$133$	7.29150	0.632253
$134$	−11.2915	−0.975437
$135$	0	0
$136$	4.64575	0.398370
$137$	12.5830	1.07504	0.537519	−	0.843251i	$-0.319362\pi$
$137$	12.5830	1.07504	0.537519	+	0.843251i	$0.319362\pi$
$138$	0	0
$139$	8.58301	0.728001	0.364001	−	0.931399i	$-0.381411\pi$
$139$	8.58301	0.728001	0.364001	+	0.931399i	$0.381411\pi$
$140$	0	0
$141$	−28.1660	−2.37201
$142$	−6.00000	−0.503509
$143$	−4.00000	−0.334497
$144$	4.00000	0.333333
$145$	0	0
$146$	15.2288	1.26034
$147$	−2.64575	−0.218218
$148$	1.00000	0.0821995
$149$	18.5830	1.52238	0.761190	−	0.648529i	$-0.224616\pi$
$149$	18.5830	1.52238	0.761190	+	0.648529i	$0.224616\pi$
$150$	0	0
$151$	14.5830	1.18675	0.593374	−	0.804927i	$-0.297796\pi$
$151$	14.5830	1.18675	0.593374	+	0.804927i	$0.297796\pi$
$152$	−7.29150	−0.591419
$153$	18.5830	1.50235
$154$	1.00000	0.0805823
$155$	0	0
$156$	−10.5830	−0.847319
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	8.29150	0.659637
$159$	−7.93725	−0.629465
$160$	0	0
$161$	0	0
$162$	−5.00000	−0.392837
$163$	13.2915	1.04107	0.520535	−	0.853840i	$-0.325732\pi$
$163$	13.2915	1.04107	0.520535	+	0.853840i	$0.325732\pi$
$164$	0	0
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−9.87451	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$167$	−9.87451	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$168$	2.64575	0.204124
$169$	3.00000	0.230769
$170$	0	0
$171$	−29.1660	−2.23038
$172$	−2.29150	−0.174725
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	24.5830	1.86363
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	21.0000	1.57846
$178$	15.2915	1.14615
$179$	−8.70850	−0.650904	−0.325452	−	0.945559i	$-0.605516\pi$
$179$	−8.70850	−0.650904	−0.325452	+	0.945559i	$0.605516\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	10.5830	0.782318
$184$	0	0
$185$	0	0
$186$	−17.5830	−1.28925
$187$	−4.64575	−0.339731
$188$	10.6458	0.776421
$189$	2.64575	0.192450
$190$	0	0
$191$	−8.70850	−0.630125	−0.315062	−	0.949071i	$-0.602025\pi$
$191$	−8.70850	−0.630125	−0.315062	+	0.949071i	$0.602025\pi$
$192$	−2.64575	−0.190941
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	4.58301	0.329041
$195$	0	0
$196$	1.00000	0.0714286
$197$	9.29150	0.661992	0.330996	−	0.943632i	$-0.392615\pi$
$197$	9.29150	0.661992	0.330996	+	0.943632i	$0.392615\pi$
$198$	−4.00000	−0.284268
$199$	17.2915	1.22576	0.612881	−	0.790175i	$-0.290010\pi$
$199$	17.2915	1.22576	0.612881	+	0.790175i	$0.290010\pi$
$200$	0	0
$201$	29.8745	2.10719
$202$	16.6458	1.17119
$203$	9.29150	0.652136
$204$	−12.2915	−0.860577
$205$	0	0
$206$	−9.35425	−0.651741
$207$	0	0
$208$	4.00000	0.277350
$209$	7.29150	0.504364
$210$	0	0
$211$	2.29150	0.157754	0.0788768	−	0.996884i	$-0.474867\pi$
$211$	2.29150	0.157754	0.0788768	+	0.996884i	$0.474867\pi$
$212$	3.00000	0.206041
$213$	15.8745	1.08770
$214$	5.70850	0.390225
$215$	0	0
$216$	−2.64575	−0.180021
$217$	−6.64575	−0.451143
$218$	−10.5830	−0.716772
$219$	−40.2915	−2.72265
$220$	0	0
$221$	18.5830	1.25003
$222$	−2.64575	−0.177571
$223$	−17.2915	−1.15792	−0.578962	−	0.815354i	$-0.696542\pi$
$223$	−17.2915	−1.15792	−0.578962	+	0.815354i	$0.696542\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	3.00000	0.199557
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	19.2915	1.27761
$229$	14.5830	0.963672	0.481836	−	0.876261i	$-0.339970\pi$
$229$	14.5830	0.963672	0.481836	+	0.876261i	$0.339970\pi$
$230$	0	0
$231$	−2.64575	−0.174078
$232$	−9.29150	−0.610017
$233$	−6.58301	−0.431267	−0.215634	−	0.976474i	$-0.569182\pi$
$233$	−6.58301	−0.431267	−0.215634	+	0.976474i	$0.569182\pi$
$234$	16.0000	1.04595
$235$	0	0
$236$	−7.93725	−0.516671
$237$	−21.9373	−1.42498
$238$	−4.64575	−0.301139
$239$	−12.8745	−0.832783	−0.416391	−	0.909185i	$-0.636705\pi$
$239$	−12.8745	−0.832783	−0.416391	+	0.909185i	$0.636705\pi$
$240$	0	0
$241$	19.2288	1.23863	0.619317	−	0.785141i	$-0.287410\pi$
$241$	19.2288	1.23863	0.619317	+	0.785141i	$0.287410\pi$
$242$	1.00000	0.0642824
$243$	21.1660	1.35780
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	−29.1660	−1.85579
$248$	6.64575	0.422006
$249$	15.8745	1.00601
$250$	0	0
$251$	7.93725	0.500995	0.250498	−	0.968117i	$-0.419406\pi$
$251$	7.93725	0.500995	0.250498	+	0.968117i	$0.419406\pi$
$252$	−4.00000	−0.251976
$253$	0	0
$254$	16.8745	1.05880
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.70850	0.168951	0.0844757	−	0.996426i	$-0.473078\pi$
$257$	2.70850	0.168951	0.0844757	+	0.996426i	$0.473078\pi$
$258$	6.06275	0.377450
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−37.1660	−2.30052
$262$	21.8745	1.35141
$263$	−0.291503	−0.0179748	−0.00898741	−	0.999960i	$-0.502861\pi$
$263$	−0.291503	−0.0179748	−0.00898741	+	0.999960i	$0.502861\pi$
$264$	2.64575	0.162835
$265$	0	0
$266$	7.29150	0.447071
$267$	−40.4575	−2.47596
$268$	−11.2915	−0.689738
$269$	−24.5830	−1.49885	−0.749426	−	0.662088i	$-0.769671\pi$
$269$	−24.5830	−1.49885	−0.749426	+	0.662088i	$0.769671\pi$
$270$	0	0
$271$	−18.7085	−1.13646	−0.568230	−	0.822870i	$-0.692372\pi$
$271$	−18.7085	−1.13646	−0.568230	+	0.822870i	$0.692372\pi$
$272$	4.64575	0.281690
$273$	10.5830	0.640513
$274$	12.5830	0.760167
$275$	0	0
$276$	0	0
$277$	25.8745	1.55465	0.777324	−	0.629100i	$-0.216576\pi$
$277$	25.8745	1.55465	0.777324	+	0.629100i	$0.216576\pi$
$278$	8.58301	0.514774
$279$	26.5830	1.59148
$280$	0	0
$281$	−15.2915	−0.912215	−0.456107	−	0.889925i	$-0.650757\pi$
$281$	−15.2915	−0.912215	−0.456107	+	0.889925i	$0.650757\pi$
$282$	−28.1660	−1.67726
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	4.00000	0.235702
$289$	4.58301	0.269589
$290$	0	0
$291$	−12.1255	−0.710809
$292$	15.2288	0.891196
$293$	10.0627	0.587872	0.293936	−	0.955825i	$-0.405035\pi$
$293$	10.0627	0.587872	0.293936	+	0.955825i	$0.405035\pi$
$294$	−2.64575	−0.154303
$295$	0	0
$296$	1.00000	0.0581238
$297$	2.64575	0.153522
$298$	18.5830	1.07648
$299$	0	0
$300$	0	0
$301$	2.29150	0.132080
$302$	14.5830	0.839158
$303$	−44.0405	−2.53006
$304$	−7.29150	−0.418196
$305$	0	0
$306$	18.5830	1.06232
$307$	−26.5830	−1.51717	−0.758586	−	0.651573i	$-0.774110\pi$
$307$	−26.5830	−1.51717	−0.758586	+	0.651573i	$0.774110\pi$
$308$	1.00000	0.0569803
$309$	24.7490	1.40792
$310$	0	0
$311$	14.5203	0.823368	0.411684	−	0.911327i	$-0.364941\pi$
$311$	14.5203	0.823368	0.411684	+	0.911327i	$0.364941\pi$
$312$	−10.5830	−0.599145
$313$	−10.7085	−0.605280	−0.302640	−	0.953105i	$-0.597868\pi$
$313$	−10.7085	−0.605280	−0.302640	+	0.953105i	$0.597868\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	8.29150	0.466433
$317$	15.0000	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$317$	15.0000	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$318$	−7.93725	−0.445099
$319$	9.29150	0.520224
$320$	0	0
$321$	−15.1033	−0.842982
$322$	0	0
$323$	−33.8745	−1.88483
$324$	−5.00000	−0.277778
$325$	0	0
$326$	13.2915	0.736148
$327$	28.0000	1.54840
$328$	0	0
$329$	−10.6458	−0.586919
$330$	0	0
$331$	24.4575	1.34431	0.672153	−	0.740412i	$-0.265370\pi$
$331$	24.4575	1.34431	0.672153	+	0.740412i	$0.265370\pi$
$332$	−6.00000	−0.329293
$333$	4.00000	0.219199
$334$	−9.87451	−0.540309
$335$	0	0
$336$	2.64575	0.144338
$337$	−11.2915	−0.615087	−0.307544	−	0.951534i	$-0.599507\pi$
$337$	−11.2915	−0.615087	−0.307544	+	0.951534i	$0.599507\pi$
$338$	3.00000	0.163178
$339$	−7.93725	−0.431092
$340$	0	0
$341$	−6.64575	−0.359888
$342$	−29.1660	−1.57712
$343$	−1.00000	−0.0539949
$344$	−2.29150	−0.123550
$345$	0	0
$346$	−12.0000	−0.645124
$347$	25.4575	1.36663	0.683315	−	0.730123i	$-0.260537\pi$
$347$	25.4575	1.36663	0.683315	+	0.730123i	$0.260537\pi$
$348$	24.5830	1.31779
$349$	12.6458	0.676912	0.338456	−	0.940982i	$-0.390095\pi$
$349$	12.6458	0.676912	0.338456	+	0.940982i	$0.390095\pi$
$350$	0	0
$351$	−10.5830	−0.564879
$352$	−1.00000	−0.0533002
$353$	28.4575	1.51464	0.757320	−	0.653044i	$-0.226508\pi$
$353$	28.4575	1.51464	0.757320	+	0.653044i	$0.226508\pi$
$354$	21.0000	1.11614
$355$	0	0
$356$	15.2915	0.810448
$357$	12.2915	0.650535
$358$	−8.70850	−0.460258
$359$	31.7490	1.67565	0.837824	−	0.545940i	$-0.183827\pi$
$359$	31.7490	1.67565	0.837824	+	0.545940i	$0.183827\pi$
$360$	0	0
$361$	34.1660	1.79821
$362$	2.00000	0.105118
$363$	−2.64575	−0.138866
$364$	−4.00000	−0.209657
$365$	0	0
$366$	10.5830	0.553183
$367$	−30.4575	−1.58987	−0.794935	−	0.606695i	$-0.792495\pi$
$367$	−30.4575	−1.58987	−0.794935	+	0.606695i	$0.792495\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	−17.5830	−0.911636
$373$	−21.1660	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$373$	−21.1660	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$374$	−4.64575	−0.240226
$375$	0	0
$376$	10.6458	0.549013
$377$	−37.1660	−1.91415
$378$	2.64575	0.136083
$379$	−20.4575	−1.05083	−0.525416	−	0.850846i	$-0.676090\pi$
$379$	−20.4575	−1.05083	−0.525416	+	0.850846i	$0.676090\pi$
$380$	0	0
$381$	−44.6458	−2.28727
$382$	−8.70850	−0.445565
$383$	33.1033	1.69150	0.845749	−	0.533581i	$-0.179154\pi$
$383$	33.1033	1.69150	0.845749	+	0.533581i	$0.179154\pi$
$384$	−2.64575	−0.135015
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−9.16601	−0.465934
$388$	4.58301	0.232667
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	−57.8745	−2.91938
$394$	9.29150	0.468099
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	17.2915	0.866745
$399$	−19.2915	−0.965783
$400$	0	0
$401$	−27.5830	−1.37743	−0.688715	−	0.725032i	$-0.741825\pi$
$401$	−27.5830	−1.37743	−0.688715	+	0.725032i	$0.741825\pi$
$402$	29.8745	1.49001
$403$	26.5830	1.32419
$404$	16.6458	0.828157
$405$	0	0
$406$	9.29150	0.461130
$407$	−1.00000	−0.0495682
$408$	−12.2915	−0.608520
$409$	12.6458	0.625292	0.312646	−	0.949870i	$-0.398785\pi$
$409$	12.6458	0.625292	0.312646	+	0.949870i	$0.398785\pi$
$410$	0	0
$411$	−33.2915	−1.64215
$412$	−9.35425	−0.460851
$413$	7.93725	0.390567
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	−22.7085	−1.11204
$418$	7.29150	0.356639
$419$	5.22876	0.255441	0.127721	−	0.991810i	$-0.459234\pi$
$419$	5.22876	0.255441	0.127721	+	0.991810i	$0.459234\pi$
$420$	0	0
$421$	−4.58301	−0.223362	−0.111681	−	0.993744i	$-0.535623\pi$
$421$	−4.58301	−0.223362	−0.111681	+	0.993744i	$0.535623\pi$
$422$	2.29150	0.111549
$423$	42.5830	2.07046
$424$	3.00000	0.145693
$425$	0	0
$426$	15.8745	0.769122
$427$	4.00000	0.193574
$428$	5.70850	0.275931
$429$	10.5830	0.510952
$430$	0	0
$431$	12.2915	0.592061	0.296030	−	0.955179i	$-0.404337\pi$
$431$	12.2915	0.592061	0.296030	+	0.955179i	$0.404337\pi$
$432$	−2.64575	−0.127294
$433$	6.70850	0.322390	0.161195	−	0.986923i	$-0.448465\pi$
$433$	6.70850	0.322390	0.161195	+	0.986923i	$0.448465\pi$
$434$	−6.64575	−0.319006
$435$	0	0
$436$	−10.5830	−0.506834
$437$	0	0
$438$	−40.2915	−1.92520
$439$	−35.1660	−1.67838	−0.839191	−	0.543837i	$-0.816971\pi$
$439$	−35.1660	−1.67838	−0.839191	+	0.543837i	$0.816971\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	18.5830	0.883903
$443$	17.4170	0.827506	0.413753	−	0.910389i	$-0.364218\pi$
$443$	17.4170	0.827506	0.413753	+	0.910389i	$0.364218\pi$
$444$	−2.64575	−0.125562
$445$	0	0
$446$	−17.2915	−0.818776
$447$	−49.1660	−2.32547
$448$	−1.00000	−0.0472456
$449$	−33.5830	−1.58488	−0.792440	−	0.609950i	$-0.791190\pi$
$449$	−33.5830	−1.58488	−0.792440	+	0.609950i	$0.791190\pi$
$450$	0	0
$451$	0	0
$452$	3.00000	0.141108
$453$	−38.5830	−1.81279
$454$	18.0000	0.844782
$455$	0	0
$456$	19.2915	0.903407
$457$	32.4575	1.51830	0.759149	−	0.650917i	$-0.225615\pi$
$457$	32.4575	1.51830	0.759149	+	0.650917i	$0.225615\pi$
$458$	14.5830	0.681419
$459$	−12.2915	−0.573718
$460$	0	0
$461$	−27.1033	−1.26232	−0.631162	−	0.775651i	$-0.717422\pi$
$461$	−27.1033	−1.26232	−0.631162	+	0.775651i	$0.717422\pi$
$462$	−2.64575	−0.123091
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−9.29150	−0.431347
$465$	0	0
$466$	−6.58301	−0.304952
$467$	10.6458	0.492627	0.246313	−	0.969190i	$-0.420781\pi$
$467$	10.6458	0.492627	0.246313	+	0.969190i	$0.420781\pi$
$468$	16.0000	0.739600
$469$	11.2915	0.521393
$470$	0	0
$471$	−26.4575	−1.21910
$472$	−7.93725	−0.365342
$473$	2.29150	0.105363
$474$	−21.9373	−1.00761
$475$	0	0
$476$	−4.64575	−0.212938
$477$	12.0000	0.549442
$478$	−12.8745	−0.588866
$479$	−22.4575	−1.02611	−0.513055	−	0.858356i	$-0.671486\pi$
$479$	−22.4575	−1.02611	−0.513055	+	0.858356i	$0.671486\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	19.2288	0.875846
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	21.1660	0.960110
$487$	−22.7085	−1.02902	−0.514510	−	0.857484i	$-0.672026\pi$
$487$	−22.7085	−1.02902	−0.514510	+	0.857484i	$0.672026\pi$
$488$	−4.00000	−0.181071
$489$	−35.1660	−1.59026
$490$	0	0
$491$	−11.1255	−0.502086	−0.251043	−	0.967976i	$-0.580774\pi$
$491$	−11.1255	−0.502086	−0.251043	+	0.967976i	$0.580774\pi$
$492$	0	0
$493$	−43.1660	−1.94410
$494$	−29.1660	−1.31224
$495$	0	0
$496$	6.64575	0.298403
$497$	6.00000	0.269137
$498$	15.8745	0.711354
$499$	5.87451	0.262979	0.131490	−	0.991318i	$-0.458024\pi$
$499$	5.87451	0.262979	0.131490	+	0.991318i	$0.458024\pi$
$500$	0	0
$501$	26.1255	1.16720
$502$	7.93725	0.354257
$503$	−28.4575	−1.26886	−0.634429	−	0.772981i	$-0.718765\pi$
$503$	−28.4575	−1.26886	−0.634429	+	0.772981i	$0.718765\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	−7.93725	−0.352506
$508$	16.8745	0.748685
$509$	13.1660	0.583573	0.291787	−	0.956483i	$-0.405750\pi$
$509$	13.1660	0.583573	0.291787	+	0.956483i	$0.405750\pi$
$510$	0	0
$511$	−15.2288	−0.673681
$512$	1.00000	0.0441942
$513$	19.2915	0.851741
$514$	2.70850	0.119467
$515$	0	0
$516$	6.06275	0.266898
$517$	−10.6458	−0.468200
$518$	−1.00000	−0.0439375
$519$	31.7490	1.39363
$520$	0	0
$521$	−30.5830	−1.33987	−0.669933	−	0.742422i	$-0.733677\pi$
$521$	−30.5830	−1.33987	−0.669933	+	0.742422i	$0.733677\pi$
$522$	−37.1660	−1.62671
$523$	30.7085	1.34279	0.671394	−	0.741100i	$-0.265696\pi$
$523$	30.7085	1.34279	0.671394	+	0.741100i	$0.265696\pi$
$524$	21.8745	0.955592
$525$	0	0
$526$	−0.291503	−0.0127101
$527$	30.8745	1.34491
$528$	2.64575	0.115142
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−31.7490	−1.37779
$532$	7.29150	0.316127
$533$	0	0
$534$	−40.4575	−1.75077
$535$	0	0
$536$	−11.2915	−0.487719
$537$	23.0405	0.994272
$538$	−24.5830	−1.05985
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−31.8745	−1.37039	−0.685196	−	0.728359i	$-0.740283\pi$
$541$	−31.8745	−1.37039	−0.685196	+	0.728359i	$0.740283\pi$
$542$	−18.7085	−0.803599
$543$	−5.29150	−0.227080
$544$	4.64575	0.199185
$545$	0	0
$546$	10.5830	0.452911
$547$	−8.87451	−0.379447	−0.189723	−	0.981838i	$-0.560759\pi$
$547$	−8.87451	−0.379447	−0.189723	+	0.981838i	$0.560759\pi$
$548$	12.5830	0.537519
$549$	−16.0000	−0.682863
$550$	0	0
$551$	67.7490	2.88621
$552$	0	0
$553$	−8.29150	−0.352591
$554$	25.8745	1.09930
$555$	0	0
$556$	8.58301	0.364001
$557$	21.8745	0.926853	0.463426	−	0.886135i	$-0.346620\pi$
$557$	21.8745	0.926853	0.463426	+	0.886135i	$0.346620\pi$
$558$	26.5830	1.12535
$559$	−9.16601	−0.387681
$560$	0	0
$561$	12.2915	0.518948
$562$	−15.2915	−0.645033
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	−28.1660	−1.18600
$565$	0	0
$566$	10.0000	0.420331
$567$	5.00000	0.209980
$568$	−6.00000	−0.251754
$569$	−14.7085	−0.616612	−0.308306	−	0.951287i	$-0.599762\pi$
$569$	−14.7085	−0.616612	−0.308306	+	0.951287i	$0.599762\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−4.00000	−0.167248
$573$	23.0405	0.962531
$574$	0	0
$575$	0	0
$576$	4.00000	0.166667
$577$	14.4575	0.601874	0.300937	−	0.953644i	$-0.402701\pi$
$577$	14.4575	0.601874	0.300937	+	0.953644i	$0.402701\pi$
$578$	4.58301	0.190628
$579$	5.29150	0.219907
$580$	0	0
$581$	6.00000	0.248922
$582$	−12.1255	−0.502618
$583$	−3.00000	−0.124247
$584$	15.2288	0.630170
$585$	0	0
$586$	10.0627	0.415688
$587$	9.29150	0.383501	0.191751	−	0.981444i	$-0.438584\pi$
$587$	9.29150	0.383501	0.191751	+	0.981444i	$0.438584\pi$
$588$	−2.64575	−0.109109
$589$	−48.4575	−1.99666
$590$	0	0
$591$	−24.5830	−1.01121
$592$	1.00000	0.0410997
$593$	−13.1660	−0.540663	−0.270332	−	0.962767i	$-0.587133\pi$
$593$	−13.1660	−0.540663	−0.270332	+	0.962767i	$0.587133\pi$
$594$	2.64575	0.108556
$595$	0	0
$596$	18.5830	0.761190
$597$	−45.7490	−1.87238
$598$	0	0
$599$	21.2915	0.869947	0.434974	−	0.900443i	$-0.356758\pi$
$599$	21.2915	0.869947	0.434974	+	0.900443i	$0.356758\pi$
$600$	0	0
$601$	33.9373	1.38433	0.692165	−	0.721740i	$-0.256657\pi$
$601$	33.9373	1.38433	0.692165	+	0.721740i	$0.256657\pi$
$602$	2.29150	0.0933947
$603$	−45.1660	−1.83930
$604$	14.5830	0.593374
$605$	0	0
$606$	−44.0405	−1.78902
$607$	26.4575	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$607$	26.4575	1.07388	0.536939	+	0.843621i	$0.319581\pi$
$608$	−7.29150	−0.295709
$609$	−24.5830	−0.996154
$610$	0	0
$611$	42.5830	1.72272
$612$	18.5830	0.751173
$613$	24.7085	0.997967	0.498983	−	0.866612i	$-0.333707\pi$
$613$	24.7085	0.997967	0.498983	+	0.866612i	$0.333707\pi$
$614$	−26.5830	−1.07280
$615$	0	0
$616$	1.00000	0.0402911
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	24.7490	0.995551
$619$	−14.6458	−0.588662	−0.294331	−	0.955703i	$-0.595097\pi$
$619$	−14.6458	−0.588662	−0.294331	+	0.955703i	$0.595097\pi$
$620$	0	0
$621$	0	0
$622$	14.5203	0.582209
$623$	−15.2915	−0.612641
$624$	−10.5830	−0.423659
$625$	0	0
$626$	−10.7085	−0.427998
$627$	−19.2915	−0.770428
$628$	10.0000	0.399043
$629$	4.64575	0.185238
$630$	0	0
$631$	35.2915	1.40493	0.702466	−	0.711717i	$-0.252082\pi$
$631$	35.2915	1.40493	0.702466	+	0.711717i	$0.252082\pi$
$632$	8.29150	0.329818
$633$	−6.06275	−0.240973
$634$	15.0000	0.595726
$635$	0	0
$636$	−7.93725	−0.314733
$637$	4.00000	0.158486
$638$	9.29150	0.367854
$639$	−24.0000	−0.949425
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	−15.1033	−0.596078
$643$	−17.1033	−0.674487	−0.337243	−	0.941417i	$-0.609495\pi$
$643$	−17.1033	−0.674487	−0.337243	+	0.941417i	$0.609495\pi$
$644$	0	0
$645$	0	0
$646$	−33.8745	−1.33277
$647$	7.93725	0.312046	0.156023	−	0.987753i	$-0.450133\pi$
$647$	7.93725	0.312046	0.156023	+	0.987753i	$0.450133\pi$
$648$	−5.00000	−0.196419
$649$	7.93725	0.311564
$650$	0	0
$651$	17.5830	0.689132
$652$	13.2915	0.520535
$653$	−28.7490	−1.12504	−0.562518	−	0.826785i	$-0.690167\pi$
$653$	−28.7490	−1.12504	−0.562518	+	0.826785i	$0.690167\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	0	0
$657$	60.9150	2.37652
$658$	−10.6458	−0.415015
$659$	−30.5830	−1.19134	−0.595672	−	0.803228i	$-0.703114\pi$
$659$	−30.5830	−1.19134	−0.595672	+	0.803228i	$0.703114\pi$
$660$	0	0
$661$	−25.2915	−0.983725	−0.491863	−	0.870673i	$-0.663684\pi$
$661$	−25.2915	−0.983725	−0.491863	+	0.870673i	$0.663684\pi$
$662$	24.4575	0.950568
$663$	−49.1660	−1.90945
$664$	−6.00000	−0.232845
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	−9.87451	−0.382056
$669$	45.7490	1.76876
$670$	0	0
$671$	4.00000	0.154418
$672$	2.64575	0.102062
$673$	−42.4575	−1.63662	−0.818308	−	0.574779i	$-0.805088\pi$
$673$	−42.4575	−1.63662	−0.818308	+	0.574779i	$0.805088\pi$
$674$	−11.2915	−0.434932
$675$	0	0
$676$	3.00000	0.115385
$677$	−3.47974	−0.133737	−0.0668687	−	0.997762i	$-0.521301\pi$
$677$	−3.47974	−0.133737	−0.0668687	+	0.997762i	$0.521301\pi$
$678$	−7.93725	−0.304828
$679$	−4.58301	−0.175880
$680$	0	0
$681$	−47.6235	−1.82494
$682$	−6.64575	−0.254479
$683$	−9.29150	−0.355529	−0.177765	−	0.984073i	$-0.556887\pi$
$683$	−9.29150	−0.355529	−0.177765	+	0.984073i	$0.556887\pi$
$684$	−29.1660	−1.11519
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−38.5830	−1.47203
$688$	−2.29150	−0.0873627
$689$	12.0000	0.457164
$690$	0	0
$691$	−29.3542	−1.11669	−0.558344	−	0.829609i	$-0.688563\pi$
$691$	−29.3542	−1.11669	−0.558344	+	0.829609i	$0.688563\pi$
$692$	−12.0000	−0.456172
$693$	4.00000	0.151947
$694$	25.4575	0.966354
$695$	0	0
$696$	24.5830	0.931816
$697$	0	0
$698$	12.6458	0.478649
$699$	17.4170	0.658771
$700$	0	0
$701$	−1.16601	−0.0440396	−0.0220198	−	0.999758i	$-0.507010\pi$
$701$	−1.16601	−0.0440396	−0.0220198	+	0.999758i	$0.507010\pi$
$702$	−10.5830	−0.399430
$703$	−7.29150	−0.275004
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	28.4575	1.07101
$707$	−16.6458	−0.626028
$708$	21.0000	0.789228
$709$	−32.1660	−1.20802	−0.604010	−	0.796977i	$-0.706431\pi$
$709$	−32.1660	−1.20802	−0.604010	+	0.796977i	$0.706431\pi$
$710$	0	0
$711$	33.1660	1.24382
$712$	15.2915	0.573073
$713$	0	0
$714$	12.2915	0.459998
$715$	0	0
$716$	−8.70850	−0.325452
$717$	34.0627	1.27210
$718$	31.7490	1.18486
$719$	−39.8745	−1.48707	−0.743534	−	0.668698i	$-0.766852\pi$
$719$	−39.8745	−1.48707	−0.743534	+	0.668698i	$0.766852\pi$
$720$	0	0
$721$	9.35425	0.348370
$722$	34.1660	1.27153
$723$	−50.8745	−1.89204
$724$	2.00000	0.0743294
$725$	0	0
$726$	−2.64575	−0.0981930
$727$	−25.2288	−0.935683	−0.467841	−	0.883812i	$-0.654968\pi$
$727$	−25.2288	−0.935683	−0.467841	+	0.883812i	$0.654968\pi$
$728$	−4.00000	−0.148250
$729$	−41.0000	−1.51852
$730$	0	0
$731$	−10.6458	−0.393747
$732$	10.5830	0.391159
$733$	−7.22876	−0.267000	−0.133500	−	0.991049i	$-0.542622\pi$
$733$	−7.22876	−0.267000	−0.133500	+	0.991049i	$0.542622\pi$
$734$	−30.4575	−1.12421
$735$	0	0
$736$	0	0
$737$	11.2915	0.415928
$738$	0	0
$739$	13.7085	0.504275	0.252138	−	0.967691i	$-0.418866\pi$
$739$	13.7085	0.504275	0.252138	+	0.967691i	$0.418866\pi$
$740$	0	0
$741$	77.1660	2.83476
$742$	−3.00000	−0.110133
$743$	5.12549	0.188036	0.0940180	−	0.995570i	$-0.470029\pi$
$743$	5.12549	0.188036	0.0940180	+	0.995570i	$0.470029\pi$
$744$	−17.5830	−0.644624
$745$	0	0
$746$	−21.1660	−0.774943
$747$	−24.0000	−0.878114
$748$	−4.64575	−0.169865
$749$	−5.70850	−0.208584
$750$	0	0
$751$	−4.58301	−0.167236	−0.0836181	−	0.996498i	$-0.526648\pi$
$751$	−4.58301	−0.167236	−0.0836181	+	0.996498i	$0.526648\pi$
$752$	10.6458	0.388211
$753$	−21.0000	−0.765283
$754$	−37.1660	−1.35351
$755$	0	0
$756$	2.64575	0.0962250
$757$	−36.1660	−1.31448	−0.657238	−	0.753683i	$-0.728275\pi$
$757$	−36.1660	−1.31448	−0.657238	+	0.753683i	$0.728275\pi$
$758$	−20.4575	−0.743050
$759$	0	0
$760$	0	0
$761$	13.9373	0.505225	0.252613	−	0.967568i	$-0.418710\pi$
$761$	13.9373	0.505225	0.252613	+	0.967568i	$0.418710\pi$
$762$	−44.6458	−1.61734
$763$	10.5830	0.383131
$764$	−8.70850	−0.315062
$765$	0	0
$766$	33.1033	1.19607
$767$	−31.7490	−1.14639
$768$	−2.64575	−0.0954703
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	−7.16601	−0.258078
$772$	−2.00000	−0.0719816
$773$	3.87451	0.139356	0.0696782	−	0.997570i	$-0.477803\pi$
$773$	3.87451	0.139356	0.0696782	+	0.997570i	$0.477803\pi$
$774$	−9.16601	−0.329465
$775$	0	0
$776$	4.58301	0.164520
$777$	2.64575	0.0949158
$778$	3.00000	0.107555
$779$	0	0
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	24.5830	0.878525
$784$	1.00000	0.0357143
$785$	0	0
$786$	−57.8745	−2.06432
$787$	−38.5830	−1.37534	−0.687668	−	0.726025i	$-0.741365\pi$
$787$	−38.5830	−1.37534	−0.687668	+	0.726025i	$0.741365\pi$
$788$	9.29150	0.330996
$789$	0.771243	0.0274570
$790$	0	0
$791$	−3.00000	−0.106668
$792$	−4.00000	−0.142134
$793$	−16.0000	−0.568177
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	17.2915	0.612881
$797$	33.8745	1.19990	0.599948	−	0.800039i	$-0.295188\pi$
$797$	33.8745	1.19990	0.599948	+	0.800039i	$0.295188\pi$
$798$	−19.2915	−0.682912
$799$	49.4575	1.74968
$800$	0	0
$801$	61.1660	2.16119
$802$	−27.5830	−0.973990
$803$	−15.2288	−0.537411
$804$	29.8745	1.05359
$805$	0	0
$806$	26.5830	0.936346
$807$	65.0405	2.28953
$808$	16.6458	0.585595
$809$	−39.8745	−1.40191	−0.700957	−	0.713204i	$-0.747243\pi$
$809$	−39.8745	−1.40191	−0.700957	+	0.713204i	$0.747243\pi$
$810$	0	0
$811$	−41.1660	−1.44553	−0.722767	−	0.691092i	$-0.757130\pi$
$811$	−41.1660	−1.44553	−0.722767	+	0.691092i	$0.757130\pi$
$812$	9.29150	0.326068
$813$	49.4980	1.73597
$814$	−1.00000	−0.0350500
$815$	0	0
$816$	−12.2915	−0.430289
$817$	16.7085	0.584556
$818$	12.6458	0.442148
$819$	−16.0000	−0.559085
$820$	0	0
$821$	−39.2915	−1.37128	−0.685641	−	0.727939i	$-0.740478\pi$
$821$	−39.2915	−1.37128	−0.685641	+	0.727939i	$0.740478\pi$
$822$	−33.2915	−1.16117
$823$	−22.1255	−0.771246	−0.385623	−	0.922656i	$-0.626013\pi$
$823$	−22.1255	−0.771246	−0.385623	+	0.922656i	$0.626013\pi$
$824$	−9.35425	−0.325871
$825$	0	0
$826$	7.93725	0.276172
$827$	−24.8745	−0.864971	−0.432486	−	0.901641i	$-0.642363\pi$
$827$	−24.8745	−0.864971	−0.432486	+	0.901641i	$0.642363\pi$
$828$	0	0
$829$	5.29150	0.183781	0.0918907	−	0.995769i	$-0.470709\pi$
$829$	5.29150	0.183781	0.0918907	+	0.995769i	$0.470709\pi$
$830$	0	0
$831$	−68.4575	−2.37476
$832$	4.00000	0.138675
$833$	4.64575	0.160966
$834$	−22.7085	−0.786331
$835$	0	0
$836$	7.29150	0.252182
$837$	−17.5830	−0.607758
$838$	5.22876	0.180624
$839$	21.2915	0.735064	0.367532	−	0.930011i	$-0.380203\pi$
$839$	21.2915	0.735064	0.367532	+	0.930011i	$0.380203\pi$
$840$	0	0
$841$	57.3320	1.97697
$842$	−4.58301	−0.157941
$843$	40.4575	1.39343
$844$	2.29150	0.0788768
$845$	0	0
$846$	42.5830	1.46403
$847$	−1.00000	−0.0343604
$848$	3.00000	0.103020
$849$	−26.4575	−0.908019
$850$	0	0
$851$	0	0
$852$	15.8745	0.543852
$853$	−38.9778	−1.33457	−0.667287	−	0.744801i	$-0.732544\pi$
$853$	−38.9778	−1.33457	−0.667287	+	0.744801i	$0.732544\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	5.70850	0.195112
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	10.5830	0.361298
$859$	−9.22876	−0.314881	−0.157441	−	0.987528i	$-0.550324\pi$
$859$	−9.22876	−0.314881	−0.157441	+	0.987528i	$0.550324\pi$
$860$	0	0
$861$	0	0
$862$	12.2915	0.418650
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	−2.64575	−0.0900103
$865$	0	0
$866$	6.70850	0.227964
$867$	−12.1255	−0.411803
$868$	−6.64575	−0.225571
$869$	−8.29150	−0.281270
$870$	0	0
$871$	−45.1660	−1.53039
$872$	−10.5830	−0.358386
$873$	18.3320	0.620445
$874$	0	0
$875$	0	0
$876$	−40.2915	−1.36132
$877$	35.7490	1.20716	0.603579	−	0.797303i	$-0.293741\pi$
$877$	35.7490	1.20716	0.603579	+	0.797303i	$0.293741\pi$
$878$	−35.1660	−1.18680
$879$	−26.6235	−0.897989
$880$	0	0
$881$	40.4575	1.36305	0.681524	−	0.731796i	$-0.261317\pi$
$881$	40.4575	1.36305	0.681524	+	0.731796i	$0.261317\pi$
$882$	4.00000	0.134687
$883$	2.45751	0.0827019	0.0413510	−	0.999145i	$-0.486834\pi$
$883$	2.45751	0.0827019	0.0413510	+	0.999145i	$0.486834\pi$
$884$	18.5830	0.625014
$885$	0	0
$886$	17.4170	0.585135
$887$	−55.7490	−1.87187	−0.935934	−	0.352174i	$-0.885442\pi$
$887$	−55.7490	−1.87187	−0.935934	+	0.352174i	$0.885442\pi$
$888$	−2.64575	−0.0887856
$889$	−16.8745	−0.565953
$890$	0	0
$891$	5.00000	0.167506
$892$	−17.2915	−0.578962
$893$	−77.6235	−2.59757
$894$	−49.1660	−1.64436
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−33.5830	−1.12068
$899$	−61.7490	−2.05944
$900$	0	0
$901$	13.9373	0.464317
$902$	0	0
$903$	−6.06275	−0.201756
$904$	3.00000	0.0997785
$905$	0	0
$906$	−38.5830	−1.28183
$907$	5.74902	0.190893	0.0954465	−	0.995435i	$-0.469572\pi$
$907$	5.74902	0.190893	0.0954465	+	0.995435i	$0.469572\pi$
$908$	18.0000	0.597351
$909$	66.5830	2.20842
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	19.2915	0.638805
$913$	6.00000	0.198571
$914$	32.4575	1.07360
$915$	0	0
$916$	14.5830	0.481836
$917$	−21.8745	−0.722360
$918$	−12.2915	−0.405680
$919$	20.8745	0.688586	0.344293	−	0.938862i	$-0.388119\pi$
$919$	20.8745	0.688586	0.344293	+	0.938862i	$0.388119\pi$
$920$	0	0
$921$	70.3320	2.31752
$922$	−27.1033	−0.892598
$923$	−24.0000	−0.789970
$924$	−2.64575	−0.0870388
$925$	0	0
$926$	−14.0000	−0.460069
$927$	−37.4170	−1.22894
$928$	−9.29150	−0.305009
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	−7.29150	−0.238969
$932$	−6.58301	−0.215634
$933$	−38.4170	−1.25772
$934$	10.6458	0.348340
$935$	0	0
$936$	16.0000	0.522976
$937$	−28.5203	−0.931716	−0.465858	−	0.884859i	$-0.654254\pi$
$937$	−28.5203	−0.931716	−0.465858	+	0.884859i	$0.654254\pi$
$938$	11.2915	0.368681
$939$	28.3320	0.924581
$940$	0	0
$941$	22.0627	0.719225	0.359612	−	0.933102i	$-0.382909\pi$
$941$	22.0627	0.719225	0.359612	+	0.933102i	$0.382909\pi$
$942$	−26.4575	−0.862032
$943$	0	0
$944$	−7.93725	−0.258336
$945$	0	0
$946$	2.29150	0.0745032
$947$	−35.0405	−1.13866	−0.569332	−	0.822108i	$-0.692798\pi$
$947$	−35.0405	−1.13866	−0.569332	+	0.822108i	$0.692798\pi$
$948$	−21.9373	−0.712489
$949$	60.9150	1.97739
$950$	0	0
$951$	−39.6863	−1.28692
$952$	−4.64575	−0.150570
$953$	41.6235	1.34832	0.674159	−	0.738586i	$-0.264506\pi$
$953$	41.6235	1.34832	0.674159	+	0.738586i	$0.264506\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−12.8745	−0.416391
$957$	−24.5830	−0.794656
$958$	−22.4575	−0.725569
$959$	−12.5830	−0.406326
$960$	0	0
$961$	13.1660	0.424710
$962$	4.00000	0.128965
$963$	22.8340	0.735815
$964$	19.2288	0.619317
$965$	0	0
$966$	0	0
$967$	10.5830	0.340327	0.170163	−	0.985416i	$-0.445570\pi$
$967$	10.5830	0.340327	0.170163	+	0.985416i	$0.445570\pi$
$968$	1.00000	0.0321412
$969$	89.6235	2.87912
$970$	0	0
$971$	8.12549	0.260759	0.130380	−	0.991464i	$-0.458380\pi$
$971$	8.12549	0.260759	0.130380	+	0.991464i	$0.458380\pi$
$972$	21.1660	0.678900
$973$	−8.58301	−0.275159
$974$	−22.7085	−0.727627
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−49.7490	−1.59161	−0.795806	−	0.605552i	$-0.792953\pi$
$977$	−49.7490	−1.59161	−0.795806	+	0.605552i	$0.792953\pi$
$978$	−35.1660	−1.12449
$979$	−15.2915	−0.488719
$980$	0	0
$981$	−42.3320	−1.35156
$982$	−11.1255	−0.355029
$983$	9.47974	0.302357	0.151178	−	0.988506i	$-0.451693\pi$
$983$	9.47974	0.302357	0.151178	+	0.988506i	$0.451693\pi$
$984$	0	0
$985$	0	0
$986$	−43.1660	−1.37469
$987$	28.1660	0.896534
$988$	−29.1660	−0.927894
$989$	0	0
$990$	0	0
$991$	39.1660	1.24415	0.622075	−	0.782958i	$-0.286290\pi$
$991$	39.1660	1.24415	0.622075	+	0.782958i	$0.286290\pi$
$992$	6.64575	0.211003
$993$	−64.7085	−2.05346
$994$	6.00000	0.190308
$995$	0	0
$996$	15.8745	0.503003
$997$	−44.3948	−1.40600	−0.702998	−	0.711192i	$-0.748156\pi$
$997$	−44.3948	−1.40600	−0.702998	+	0.711192i	$0.748156\pi$
$998$	5.87451	0.185954
$999$	−2.64575	−0.0837079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bo.1.1	yes	2
5.2	odd	4		3850.2.c.v.1849.4		4
5.3	odd	4		3850.2.c.v.1849.1		4
5.4	even	2		3850.2.a.bh.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3850.2.a.bh.1.2	✓	2	5.4	even	2
3850.2.a.bo.1.1	yes	2	1.1	even	1	trivial
3850.2.c.v.1849.1		4	5.3	odd	4
3850.2.c.v.1849.4		4	5.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} -2.64575 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.00000 q^{9} -1.00000 q^{11} -2.64575 q^{12} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.64575 q^{17} +4.00000 q^{18} -7.29150 q^{19} +2.64575 q^{21} -1.00000 q^{22} -2.64575 q^{24} +4.00000 q^{26} -2.64575 q^{27} -1.00000 q^{28} -9.29150 q^{29} +6.64575 q^{31} +1.00000 q^{32} +2.64575 q^{33} +4.64575 q^{34} +4.00000 q^{36} +1.00000 q^{37} -7.29150 q^{38} -10.5830 q^{39} +2.64575 q^{42} -2.29150 q^{43} -1.00000 q^{44} +10.6458 q^{47} -2.64575 q^{48} +1.00000 q^{49} -12.2915 q^{51} +4.00000 q^{52} +3.00000 q^{53} -2.64575 q^{54} -1.00000 q^{56} +19.2915 q^{57} -9.29150 q^{58} -7.93725 q^{59} -4.00000 q^{61} +6.64575 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.64575 q^{66} -11.2915 q^{67} +4.64575 q^{68} -6.00000 q^{71} +4.00000 q^{72} +15.2288 q^{73} +1.00000 q^{74} -7.29150 q^{76} +1.00000 q^{77} -10.5830 q^{78} +8.29150 q^{79} -5.00000 q^{81} -6.00000 q^{83} +2.64575 q^{84} -2.29150 q^{86} +24.5830 q^{87} -1.00000 q^{88} +15.2915 q^{89} -4.00000 q^{91} -17.5830 q^{93} +10.6458 q^{94} -2.64575 q^{96} +4.58301 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 8 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 8 q^{9} - 2 q^{11} + 8 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{18} - 4 q^{19} - 2 q^{22} + 8 q^{26} - 2 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} + 4 q^{34} + 8 q^{36} + 2 q^{37} - 4 q^{38} + 6 q^{43} - 2 q^{44} + 16 q^{47} + 2 q^{49} - 14 q^{51} + 8 q^{52} + 6 q^{53} - 2 q^{56} + 28 q^{57} - 8 q^{58} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{67} + 4 q^{68} - 12 q^{71} + 8 q^{72} + 4 q^{73} + 2 q^{74} - 4 q^{76} + 2 q^{77} + 6 q^{79} - 10 q^{81} - 12 q^{83} + 6 q^{86} + 28 q^{87} - 2 q^{88} + 20 q^{89} - 8 q^{91} - 14 q^{93} + 16 q^{94} - 12 q^{97} + 2 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 8 * q^9 - 2 * q^11 + 8 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^18 - 4 * q^19 - 2 * q^22 + 8 * q^26 - 2 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 + 4 * q^34 + 8 * q^36 + 2 * q^37 - 4 * q^38 + 6 * q^43 - 2 * q^44 + 16 * q^47 + 2 * q^49 - 14 * q^51 + 8 * q^52 + 6 * q^53 - 2 * q^56 + 28 * q^57 - 8 * q^58 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^67 + 4 * q^68 - 12 * q^71 + 8 * q^72 + 4 * q^73 + 2 * q^74 - 4 * q^76 + 2 * q^77 + 6 * q^79 - 10 * q^81 - 12 * q^83 + 6 * q^86 + 28 * q^87 - 2 * q^88 + 20 * q^89 - 8 * q^91 - 14 * q^93 + 16 * q^94 - 12 * q^97 + 2 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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