LMFDB - Embedded newform 3850.2.a.bh.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:	$1$
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} +2.64575 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} +2.64575 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{9} -1.00000 q^{11} -2.64575 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.645751 q^{17} -4.00000 q^{18} +3.29150 q^{19} -2.64575 q^{21} +1.00000 q^{22} +2.64575 q^{24} +4.00000 q^{26} -2.64575 q^{27} +1.00000 q^{28} +1.29150 q^{29} +1.35425 q^{31} -1.00000 q^{32} +2.64575 q^{33} -0.645751 q^{34} +4.00000 q^{36} -1.00000 q^{37} -3.29150 q^{38} +10.5830 q^{39} +2.64575 q^{42} -8.29150 q^{43} -1.00000 q^{44} -5.35425 q^{47} -2.64575 q^{48} +1.00000 q^{49} -1.70850 q^{51} -4.00000 q^{52} -3.00000 q^{53} +2.64575 q^{54} -1.00000 q^{56} -8.70850 q^{57} -1.29150 q^{58} +7.93725 q^{59} -4.00000 q^{61} -1.35425 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.64575 q^{66} +0.708497 q^{67} +0.645751 q^{68} -6.00000 q^{71} -4.00000 q^{72} +11.2288 q^{73} +1.00000 q^{74} +3.29150 q^{76} -1.00000 q^{77} -10.5830 q^{78} -2.29150 q^{79} -5.00000 q^{81} +6.00000 q^{83} -2.64575 q^{84} +8.29150 q^{86} -3.41699 q^{87} +1.00000 q^{88} +4.70850 q^{89} -4.00000 q^{91} -3.58301 q^{93} +5.35425 q^{94} +2.64575 q^{96} +16.5830 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0.645751	0.156618	0.0783088	−	0.996929i	$-0.475048\pi$
$17$	0.645751	0.156618	0.0783088	+	0.996929i	$0.475048\pi$
$18$	−4.00000	−0.942809
$19$	3.29150	0.755122	0.377561	−	0.925985i	$-0.376763\pi$
$19$	3.29150	0.755122	0.377561	+	0.925985i	$0.376763\pi$
$20$	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
$29$	1.29150	0.239826	0.119913	−	0.992784i	$-0.461738\pi$
$29$	1.29150	0.239826	0.119913	+	0.992784i	$0.461738\pi$
$30$	0	0
$31$	1.35425	0.243230	0.121615	−	0.992577i	$-0.461193\pi$
$31$	1.35425	0.243230	0.121615	+	0.992577i	$0.461193\pi$
$32$	−1.00000	−0.176777
$33$	2.64575	0.460566
$34$	−0.645751	−0.110745
$35$	0	0
$36$	4.00000	0.666667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−3.29150	−0.533952
$39$	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
$43$	−8.29150	−1.26444	−0.632221	−	0.774788i	$-0.717857\pi$
$43$	−8.29150	−1.26444	−0.632221	+	0.774788i	$0.717857\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	−5.35425	−0.780997	−0.390499	−	0.920603i	$-0.627697\pi$
$47$	−5.35425	−0.780997	−0.390499	+	0.920603i	$0.627697\pi$
$48$	−2.64575	−0.381881
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.70850	−0.239237
$52$	−4.00000	−0.554700
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	2.64575	0.360041
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−8.70850	−1.15347
$58$	−1.29150	−0.169583
$59$	7.93725	1.03334	0.516671	−	0.856184i	$-0.327171\pi$
$59$	7.93725	1.03334	0.516671	+	0.856184i	$0.327171\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−1.35425	−0.171990
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.64575	−0.325669
$67$	0.708497	0.0865567	0.0432784	−	0.999063i	$-0.486220\pi$
$67$	0.708497	0.0865567	0.0432784	+	0.999063i	$0.486220\pi$
$68$	0.645751	0.0783088
$69$	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
$73$	11.2288	1.31423	0.657113	−	0.753792i	$-0.271777\pi$
$73$	11.2288	1.31423	0.657113	+	0.753792i	$0.271777\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	3.29150	0.377561
$77$	−1.00000	−0.113961
$78$	−10.5830	−1.19829
$79$	−2.29150	−0.257814	−0.128907	−	0.991657i	$-0.541147\pi$
$79$	−2.29150	−0.257814	−0.128907	+	0.991657i	$0.541147\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−2.64575	−0.288675
$85$	0	0
$86$	8.29150	0.894096
$87$	−3.41699	−0.366340
$88$	1.00000	0.106600
$89$	4.70850	0.499100	0.249550	−	0.968362i	$-0.419717\pi$
$89$	4.70850	0.499100	0.249550	+	0.968362i	$0.419717\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−3.58301	−0.371540
$94$	5.35425	0.552249
$95$	0	0
$96$	2.64575	0.270031
$97$	16.5830	1.68375	0.841875	−	0.539673i	$-0.181452\pi$
$97$	16.5830	1.68375	0.841875	+	0.539673i	$0.181452\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	0	0
$101$	11.3542	1.12979	0.564895	−	0.825163i	$-0.308917\pi$
$101$	11.3542	1.12979	0.564895	+	0.825163i	$0.308917\pi$
$102$	1.70850	0.169166
$103$	14.6458	1.44309	0.721544	−	0.692368i	$-0.243433\pi$
$103$	14.6458	1.44309	0.721544	+	0.692368i	$0.243433\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	3.00000	0.291386
$107$	−16.2915	−1.57496	−0.787480	−	0.616341i	$-0.788614\pi$
$107$	−16.2915	−1.57496	−0.787480	+	0.616341i	$0.788614\pi$
$108$	−2.64575	−0.254588
$109$	10.5830	1.01367	0.506834	−	0.862044i	$-0.330816\pi$
$109$	10.5830	1.01367	0.506834	+	0.862044i	$0.330816\pi$
$110$	0	0
$111$	2.64575	0.251124
$112$	1.00000	0.0944911
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	8.70850	0.815626
$115$	0	0
$116$	1.29150	0.119913
$117$	−16.0000	−1.47920
$118$	−7.93725	−0.730683
$119$	0.645751	0.0591959
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.00000	0.362143
$123$	0	0
$124$	1.35425	0.121615
$125$	0	0
$126$	−4.00000	−0.356348
$127$	14.8745	1.31990	0.659950	−	0.751310i	$-0.270578\pi$
$127$	14.8745	1.31990	0.659950	+	0.751310i	$0.270578\pi$
$128$	−1.00000	−0.0883883
$129$	21.9373	1.93147
$130$	0	0
$131$	−9.87451	−0.862740	−0.431370	−	0.902175i	$-0.641970\pi$
$131$	−9.87451	−0.862740	−0.431370	+	0.902175i	$0.641970\pi$
$132$	2.64575	0.230283
$133$	3.29150	0.285409
$134$	−0.708497	−0.0612049
$135$	0	0
$136$	−0.645751	−0.0553727
$137$	8.58301	0.733296	0.366648	−	0.930360i	$-0.380505\pi$
$137$	8.58301	0.733296	0.366648	+	0.930360i	$0.380505\pi$
$138$	0	0
$139$	−12.5830	−1.06728	−0.533638	−	0.845713i	$-0.679176\pi$
$139$	−12.5830	−1.06728	−0.533638	+	0.845713i	$0.679176\pi$
$140$	0	0
$141$	14.1660	1.19299
$142$	6.00000	0.503509
$143$	4.00000	0.334497
$144$	4.00000	0.333333
$145$	0	0
$146$	−11.2288	−0.929299
$147$	−2.64575	−0.218218
$148$	−1.00000	−0.0821995
$149$	−2.58301	−0.211608	−0.105804	−	0.994387i	$-0.533742\pi$
$149$	−2.58301	−0.211608	−0.105804	+	0.994387i	$0.533742\pi$
$150$	0	0
$151$	−6.58301	−0.535717	−0.267859	−	0.963458i	$-0.586316\pi$
$151$	−6.58301	−0.535717	−0.267859	+	0.963458i	$0.586316\pi$
$152$	−3.29150	−0.266976
$153$	2.58301	0.208824
$154$	1.00000	0.0805823
$155$	0	0
$156$	10.5830	0.847319
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	2.29150	0.182302
$159$	7.93725	0.629465
$160$	0	0
$161$	0	0
$162$	5.00000	0.392837
$163$	−2.70850	−0.212146	−0.106073	−	0.994358i	$-0.533828\pi$
$163$	−2.70850	−0.212146	−0.106073	+	0.994358i	$0.533828\pi$
$164$	0	0
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−21.8745	−1.69270	−0.846350	−	0.532627i	$-0.821205\pi$
$167$	−21.8745	−1.69270	−0.846350	+	0.532627i	$0.821205\pi$
$168$	2.64575	0.204124
$169$	3.00000	0.230769
$170$	0	0
$171$	13.1660	1.00683
$172$	−8.29150	−0.632221
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	3.41699	0.259042
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−21.0000	−1.57846
$178$	−4.70850	−0.352917
$179$	−19.2915	−1.44191	−0.720957	−	0.692979i	$-0.756298\pi$
$179$	−19.2915	−1.44191	−0.720957	+	0.692979i	$0.756298\pi$
$180$	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
$186$	3.58301	0.262719
$187$	−0.645751	−0.0472220
$188$	−5.35425	−0.390499
$189$	−2.64575	−0.192450
$190$	0	0
$191$	−19.2915	−1.39588	−0.697942	−	0.716154i	$-0.745901\pi$
$191$	−19.2915	−1.39588	−0.697942	+	0.716154i	$0.745901\pi$
$192$	−2.64575	−0.190941
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−16.5830	−1.19059
$195$	0	0
$196$	1.00000	0.0714286
$197$	1.29150	0.0920158	0.0460079	−	0.998941i	$-0.485350\pi$
$197$	1.29150	0.0920158	0.0460079	+	0.998941i	$0.485350\pi$
$198$	4.00000	0.284268
$199$	6.70850	0.475553	0.237776	−	0.971320i	$-0.423581\pi$
$199$	6.70850	0.475553	0.237776	+	0.971320i	$0.423581\pi$
$200$	0	0
$201$	−1.87451	−0.132218
$202$	−11.3542	−0.798882
$203$	1.29150	0.0906457
$204$	−1.70850	−0.119619
$205$	0	0
$206$	−14.6458	−1.02042
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−3.29150	−0.227678
$210$	0	0
$211$	−8.29150	−0.570811	−0.285405	−	0.958407i	$-0.592128\pi$
$211$	−8.29150	−0.570811	−0.285405	+	0.958407i	$0.592128\pi$
$212$	−3.00000	−0.206041
$213$	15.8745	1.08770
$214$	16.2915	1.11366
$215$	0	0
$216$	2.64575	0.180021
$217$	1.35425	0.0919324
$218$	−10.5830	−0.716772
$219$	−29.7085	−2.00751
$220$	0	0
$221$	−2.58301	−0.173752
$222$	−2.64575	−0.177571
$223$	6.70850	0.449234	0.224617	−	0.974447i	$-0.427887\pi$
$223$	6.70850	0.449234	0.224617	+	0.974447i	$0.427887\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	3.00000	0.199557
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−8.70850	−0.576734
$229$	−6.58301	−0.435017	−0.217509	−	0.976058i	$-0.569793\pi$
$229$	−6.58301	−0.435017	−0.217509	+	0.976058i	$0.569793\pi$
$230$	0	0
$231$	2.64575	0.174078
$232$	−1.29150	−0.0847913
$233$	−14.5830	−0.955364	−0.477682	−	0.878533i	$-0.658523\pi$
$233$	−14.5830	−0.955364	−0.477682	+	0.878533i	$0.658523\pi$
$234$	16.0000	1.04595
$235$	0	0
$236$	7.93725	0.516671
$237$	6.06275	0.393818
$238$	−0.645751	−0.0418578
$239$	18.8745	1.22089	0.610445	−	0.792058i	$-0.290991\pi$
$239$	18.8745	1.22089	0.610445	+	0.792058i	$0.290991\pi$
$240$	0	0
$241$	−7.22876	−0.465645	−0.232823	−	0.972519i	$-0.574796\pi$
$241$	−7.22876	−0.465645	−0.232823	+	0.972519i	$0.574796\pi$
$242$	−1.00000	−0.0642824
$243$	21.1660	1.35780
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	−13.1660	−0.837733
$248$	−1.35425	−0.0859949
$249$	−15.8745	−1.00601
$250$	0	0
$251$	−7.93725	−0.500995	−0.250498	−	0.968117i	$-0.580594\pi$
$251$	−7.93725	−0.500995	−0.250498	+	0.968117i	$0.580594\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−14.8745	−0.933310
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.2915	−0.829101	−0.414551	−	0.910026i	$-0.636061\pi$
$257$	−13.2915	−0.829101	−0.414551	+	0.910026i	$0.636061\pi$
$258$	−21.9373	−1.36575
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	5.16601	0.319768
$262$	9.87451	0.610049
$263$	−10.2915	−0.634601	−0.317301	−	0.948325i	$-0.602776\pi$
$263$	−10.2915	−0.634601	−0.317301	+	0.948325i	$0.602776\pi$
$264$	−2.64575	−0.162835
$265$	0	0
$266$	−3.29150	−0.201815
$267$	−12.4575	−0.762387
$268$	0.708497	0.0432784
$269$	−3.41699	−0.208338	−0.104169	−	0.994560i	$-0.533218\pi$
$269$	−3.41699	−0.208338	−0.104169	+	0.994560i	$0.533218\pi$
$270$	0	0
$271$	−29.2915	−1.77933	−0.889666	−	0.456612i	$-0.849063\pi$
$271$	−29.2915	−1.77933	−0.889666	+	0.456612i	$0.849063\pi$
$272$	0.645751	0.0391544
$273$	10.5830	0.640513
$274$	−8.58301	−0.518518
$275$	0	0
$276$	0	0
$277$	5.87451	0.352965	0.176482	−	0.984304i	$-0.443528\pi$
$277$	5.87451	0.352965	0.176482	+	0.984304i	$0.443528\pi$
$278$	12.5830	0.754679
$279$	5.41699	0.324307
$280$	0	0
$281$	−4.70850	−0.280885	−0.140443	−	0.990089i	$-0.544853\pi$
$281$	−4.70850	−0.280885	−0.140443	+	0.990089i	$0.544853\pi$
$282$	−14.1660	−0.843574
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−4.00000	−0.235702
$289$	−16.5830	−0.975471
$290$	0	0
$291$	−43.8745	−2.57197
$292$	11.2288	0.657113
$293$	−25.9373	−1.51527	−0.757635	−	0.652678i	$-0.773645\pi$
$293$	−25.9373	−1.51527	−0.757635	+	0.652678i	$0.773645\pi$
$294$	2.64575	0.154303
$295$	0	0
$296$	1.00000	0.0581238
$297$	2.64575	0.153522
$298$	2.58301	0.149629
$299$	0	0
$300$	0	0
$301$	−8.29150	−0.477914
$302$	6.58301	0.378809
$303$	−30.0405	−1.72578
$304$	3.29150	0.188781
$305$	0	0
$306$	−2.58301	−0.147661
$307$	5.41699	0.309164	0.154582	−	0.987980i	$-0.450597\pi$
$307$	5.41699	0.309164	0.154582	+	0.987980i	$0.450597\pi$
$308$	−1.00000	−0.0569803
$309$	−38.7490	−2.20435
$310$	0	0
$311$	−22.5203	−1.27701	−0.638503	−	0.769619i	$-0.720446\pi$
$311$	−22.5203	−1.27701	−0.638503	+	0.769619i	$0.720446\pi$
$312$	−10.5830	−0.599145
$313$	21.2915	1.20347	0.601733	−	0.798697i	$-0.294477\pi$
$313$	21.2915	1.20347	0.601733	+	0.798697i	$0.294477\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−2.29150	−0.128907
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	−7.93725	−0.445099
$319$	−1.29150	−0.0723103
$320$	0	0
$321$	43.1033	2.40579
$322$	0	0
$323$	2.12549	0.118266
$324$	−5.00000	−0.277778
$325$	0	0
$326$	2.70850	0.150010
$327$	−28.0000	−1.54840
$328$	0	0
$329$	−5.35425	−0.295189
$330$	0	0
$331$	−28.4575	−1.56417	−0.782083	−	0.623174i	$-0.785843\pi$
$331$	−28.4575	−1.56417	−0.782083	+	0.623174i	$0.785843\pi$
$332$	6.00000	0.329293
$333$	−4.00000	−0.219199
$334$	21.8745	1.19692
$335$	0	0
$336$	−2.64575	−0.144338
$337$	0.708497	0.0385943	0.0192972	−	0.999814i	$-0.493857\pi$
$337$	0.708497	0.0385943	0.0192972	+	0.999814i	$0.493857\pi$
$338$	−3.00000	−0.163178
$339$	7.93725	0.431092
$340$	0	0
$341$	−1.35425	−0.0733367
$342$	−13.1660	−0.711936
$343$	1.00000	0.0539949
$344$	8.29150	0.447048
$345$	0	0
$346$	−12.0000	−0.645124
$347$	27.4575	1.47400	0.736998	−	0.675895i	$-0.236243\pi$
$347$	27.4575	1.47400	0.736998	+	0.675895i	$0.236243\pi$
$348$	−3.41699	−0.183170
$349$	7.35425	0.393664	0.196832	−	0.980437i	$-0.436935\pi$
$349$	7.35425	0.393664	0.196832	+	0.980437i	$0.436935\pi$
$350$	0	0
$351$	10.5830	0.564879
$352$	1.00000	0.0533002
$353$	24.4575	1.30174	0.650871	−	0.759188i	$-0.274404\pi$
$353$	24.4575	1.30174	0.650871	+	0.759188i	$0.274404\pi$
$354$	21.0000	1.11614
$355$	0	0
$356$	4.70850	0.249550
$357$	−1.70850	−0.0904233
$358$	19.2915	1.01959
$359$	−31.7490	−1.67565	−0.837824	−	0.545940i	$-0.816173\pi$
$359$	−31.7490	−1.67565	−0.837824	+	0.545940i	$0.816173\pi$
$360$	0	0
$361$	−8.16601	−0.429790
$362$	−2.00000	−0.105118
$363$	−2.64575	−0.138866
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−10.5830	−0.553183
$367$	−22.4575	−1.17227	−0.586136	−	0.810212i	$-0.699352\pi$
$367$	−22.4575	−1.17227	−0.586136	+	0.810212i	$0.699352\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	−3.58301	−0.185770
$373$	−21.1660	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$373$	−21.1660	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$374$	0.645751	0.0333910
$375$	0	0
$376$	5.35425	0.276124
$377$	−5.16601	−0.266063
$378$	2.64575	0.136083
$379$	32.4575	1.66723	0.833615	−	0.552346i	$-0.186267\pi$
$379$	32.4575	1.66723	0.833615	+	0.552346i	$0.186267\pi$
$380$	0	0
$381$	−39.3542	−2.01618
$382$	19.2915	0.987039
$383$	25.1033	1.28272	0.641358	−	0.767241i	$-0.278371\pi$
$383$	25.1033	1.28272	0.641358	+	0.767241i	$0.278371\pi$
$384$	2.64575	0.135015
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−33.1660	−1.68592
$388$	16.5830	0.841875
$389$	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
$393$	26.1255	1.31786
$394$	−1.29150	−0.0650650
$395$	0	0
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−6.70850	−0.336267
$399$	−8.70850	−0.435970
$400$	0	0
$401$	−6.41699	−0.320449	−0.160225	−	0.987081i	$-0.551222\pi$
$401$	−6.41699	−0.320449	−0.160225	+	0.987081i	$0.551222\pi$
$402$	1.87451	0.0934920
$403$	−5.41699	−0.269840
$404$	11.3542	0.564895
$405$	0	0
$406$	−1.29150	−0.0640962
$407$	1.00000	0.0495682
$408$	1.70850	0.0845832
$409$	7.35425	0.363644	0.181822	−	0.983331i	$-0.441801\pi$
$409$	7.35425	0.363644	0.181822	+	0.983331i	$0.441801\pi$
$410$	0	0
$411$	−22.7085	−1.12013
$412$	14.6458	0.721544
$413$	7.93725	0.390567
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	33.2915	1.63029
$418$	3.29150	0.160993
$419$	−21.2288	−1.03709	−0.518546	−	0.855050i	$-0.673527\pi$
$419$	−21.2288	−1.03709	−0.518546	+	0.855050i	$0.673527\pi$
$420$	0	0
$421$	16.5830	0.808206	0.404103	−	0.914713i	$-0.367584\pi$
$421$	16.5830	0.808206	0.404103	+	0.914713i	$0.367584\pi$
$422$	8.29150	0.403624
$423$	−21.4170	−1.04133
$424$	3.00000	0.145693
$425$	0	0
$426$	−15.8745	−0.769122
$427$	−4.00000	−0.193574
$428$	−16.2915	−0.787480
$429$	−10.5830	−0.510952
$430$	0	0
$431$	1.70850	0.0822954	0.0411477	−	0.999153i	$-0.486899\pi$
$431$	1.70850	0.0822954	0.0411477	+	0.999153i	$0.486899\pi$
$432$	−2.64575	−0.127294
$433$	−17.2915	−0.830977	−0.415488	−	0.909599i	$-0.636389\pi$
$433$	−17.2915	−0.830977	−0.415488	+	0.909599i	$0.636389\pi$
$434$	−1.35425	−0.0650060
$435$	0	0
$436$	10.5830	0.506834
$437$	0	0
$438$	29.7085	1.41953
$439$	7.16601	0.342015	0.171008	−	0.985270i	$-0.445298\pi$
$439$	7.16601	0.342015	0.171008	+	0.985270i	$0.445298\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	2.58301	0.122861
$443$	−38.5830	−1.83313	−0.916567	−	0.399881i	$-0.869051\pi$
$443$	−38.5830	−1.83313	−0.916567	+	0.399881i	$0.869051\pi$
$444$	2.64575	0.125562
$445$	0	0
$446$	−6.70850	−0.317657
$447$	6.83399	0.323237
$448$	1.00000	0.0472456
$449$	−12.4170	−0.585994	−0.292997	−	0.956113i	$-0.594653\pi$
$449$	−12.4170	−0.585994	−0.292997	+	0.956113i	$0.594653\pi$
$450$	0	0
$451$	0	0
$452$	−3.00000	−0.141108
$453$	17.4170	0.818322
$454$	18.0000	0.844782
$455$	0	0
$456$	8.70850	0.407813
$457$	20.4575	0.956962	0.478481	−	0.878098i	$-0.341188\pi$
$457$	20.4575	0.956962	0.478481	+	0.878098i	$0.341188\pi$
$458$	6.58301	0.307604
$459$	−1.70850	−0.0797458
$460$	0	0
$461$	31.1033	1.44862	0.724312	−	0.689473i	$-0.242158\pi$
$461$	31.1033	1.44862	0.724312	+	0.689473i	$0.242158\pi$
$462$	−2.64575	−0.123091
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	1.29150	0.0599565
$465$	0	0
$466$	14.5830	0.675545
$467$	−5.35425	−0.247765	−0.123883	−	0.992297i	$-0.539535\pi$
$467$	−5.35425	−0.247765	−0.123883	+	0.992297i	$0.539535\pi$
$468$	−16.0000	−0.739600
$469$	0.708497	0.0327154
$470$	0	0
$471$	26.4575	1.21910
$472$	−7.93725	−0.365342
$473$	8.29150	0.381244
$474$	−6.06275	−0.278471
$475$	0	0
$476$	0.645751	0.0295980
$477$	−12.0000	−0.549442
$478$	−18.8745	−0.863300
$479$	30.4575	1.39164	0.695820	−	0.718217i	$-0.255041\pi$
$479$	30.4575	1.39164	0.695820	+	0.718217i	$0.255041\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	7.22876	0.329261
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−21.1660	−0.960110
$487$	33.2915	1.50858	0.754291	−	0.656540i	$-0.227981\pi$
$487$	33.2915	1.50858	0.754291	+	0.656540i	$0.227981\pi$
$488$	4.00000	0.181071
$489$	7.16601	0.324058
$490$	0	0
$491$	−42.8745	−1.93490	−0.967450	−	0.253063i	$-0.918562\pi$
$491$	−42.8745	−1.93490	−0.967450	+	0.253063i	$0.918562\pi$
$492$	0	0
$493$	0.833990	0.0375610
$494$	13.1660	0.592367
$495$	0	0
$496$	1.35425	0.0608076
$497$	−6.00000	−0.269137
$498$	15.8745	0.711354
$499$	−25.8745	−1.15830	−0.579151	−	0.815220i	$-0.696616\pi$
$499$	−25.8745	−1.15830	−0.579151	+	0.815220i	$0.696616\pi$
$500$	0	0
$501$	57.8745	2.58564
$502$	7.93725	0.354257
$503$	−24.4575	−1.09051	−0.545253	−	0.838271i	$-0.683567\pi$
$503$	−24.4575	−1.09051	−0.545253	+	0.838271i	$0.683567\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	−7.93725	−0.352506
$508$	14.8745	0.659950
$509$	−29.1660	−1.29276	−0.646380	−	0.763015i	$-0.723718\pi$
$509$	−29.1660	−1.29276	−0.646380	+	0.763015i	$0.723718\pi$
$510$	0	0
$511$	11.2288	0.496731
$512$	−1.00000	−0.0441942
$513$	−8.70850	−0.384490
$514$	13.2915	0.586263
$515$	0	0
$516$	21.9373	0.965734
$517$	5.35425	0.235480
$518$	1.00000	0.0439375
$519$	−31.7490	−1.39363
$520$	0	0
$521$	−9.41699	−0.412566	−0.206283	−	0.978492i	$-0.566137\pi$
$521$	−9.41699	−0.412566	−0.206283	+	0.978492i	$0.566137\pi$
$522$	−5.16601	−0.226110
$523$	−41.2915	−1.80555	−0.902776	−	0.430112i	$-0.858474\pi$
$523$	−41.2915	−1.80555	−0.902776	+	0.430112i	$0.858474\pi$
$524$	−9.87451	−0.431370
$525$	0	0
$526$	10.2915	0.448731
$527$	0.874508	0.0380942
$528$	2.64575	0.115142
$529$	−23.0000	−1.00000
$530$	0	0
$531$	31.7490	1.37779
$532$	3.29150	0.142705
$533$	0	0
$534$	12.4575	0.539089
$535$	0	0
$536$	−0.708497	−0.0306024
$537$	51.0405	2.20256
$538$	3.41699	0.147317
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−0.125492	−0.00539533	−0.00269766	−	0.999996i	$-0.500859\pi$
$541$	−0.125492	−0.00539533	−0.00269766	+	0.999996i	$0.500859\pi$
$542$	29.2915	1.25818
$543$	−5.29150	−0.227080
$544$	−0.645751	−0.0276864
$545$	0	0
$546$	−10.5830	−0.452911
$547$	−22.8745	−0.978043	−0.489022	−	0.872272i	$-0.662646\pi$
$547$	−22.8745	−0.978043	−0.489022	+	0.872272i	$0.662646\pi$
$548$	8.58301	0.366648
$549$	−16.0000	−0.682863
$550$	0	0
$551$	4.25098	0.181098
$552$	0	0
$553$	−2.29150	−0.0974446
$554$	−5.87451	−0.249584
$555$	0	0
$556$	−12.5830	−0.533638
$557$	9.87451	0.418396	0.209198	−	0.977873i	$-0.432915\pi$
$557$	9.87451	0.418396	0.209198	+	0.977873i	$0.432915\pi$
$558$	−5.41699	−0.229320
$559$	33.1660	1.40277
$560$	0	0
$561$	1.70850	0.0721328
$562$	4.70850	0.198616
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	14.1660	0.596497
$565$	0	0
$566$	10.0000	0.420331
$567$	−5.00000	−0.209980
$568$	6.00000	0.251754
$569$	−25.2915	−1.06027	−0.530137	−	0.847912i	$-0.677860\pi$
$569$	−25.2915	−1.06027	−0.530137	+	0.847912i	$0.677860\pi$
$570$	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
$573$	51.0405	2.13225
$574$	0	0
$575$	0	0
$576$	4.00000	0.166667
$577$	38.4575	1.60101	0.800504	−	0.599328i	$-0.204565\pi$
$577$	38.4575	1.60101	0.800504	+	0.599328i	$0.204565\pi$
$578$	16.5830	0.689762
$579$	−5.29150	−0.219907
$580$	0	0
$581$	6.00000	0.248922
$582$	43.8745	1.81866
$583$	3.00000	0.124247
$584$	−11.2288	−0.464649
$585$	0	0
$586$	25.9373	1.07146
$587$	1.29150	0.0533060	0.0266530	−	0.999645i	$-0.491515\pi$
$587$	1.29150	0.0533060	0.0266530	+	0.999645i	$0.491515\pi$
$588$	−2.64575	−0.109109
$589$	4.45751	0.183669
$590$	0	0
$591$	−3.41699	−0.140556
$592$	−1.00000	−0.0410997
$593$	−29.1660	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$593$	−29.1660	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$594$	−2.64575	−0.108556
$595$	0	0
$596$	−2.58301	−0.105804
$597$	−17.7490	−0.726419
$598$	0	0
$599$	10.7085	0.437537	0.218769	−	0.975777i	$-0.429796\pi$
$599$	10.7085	0.437537	0.218769	+	0.975777i	$0.429796\pi$
$600$	0	0
$601$	18.0627	0.736795	0.368397	−	0.929668i	$-0.379907\pi$
$601$	18.0627	0.736795	0.368397	+	0.929668i	$0.379907\pi$
$602$	8.29150	0.337936
$603$	2.83399	0.115409
$604$	−6.58301	−0.267859
$605$	0	0
$606$	30.0405	1.22031
$607$	26.4575	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$607$	26.4575	1.07388	0.536939	+	0.843621i	$0.319581\pi$
$608$	−3.29150	−0.133488
$609$	−3.41699	−0.138464
$610$	0	0
$611$	21.4170	0.866439
$612$	2.58301	0.104412
$613$	−35.2915	−1.42541	−0.712705	−	0.701464i	$-0.752530\pi$
$613$	−35.2915	−1.42541	−0.712705	+	0.701464i	$0.752530\pi$
$614$	−5.41699	−0.218612
$615$	0	0
$616$	1.00000	0.0402911
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	38.7490	1.55871
$619$	−9.35425	−0.375979	−0.187989	−	0.982171i	$-0.560197\pi$
$619$	−9.35425	−0.375979	−0.187989	+	0.982171i	$0.560197\pi$
$620$	0	0
$621$	0	0
$622$	22.5203	0.902980
$623$	4.70850	0.188642
$624$	10.5830	0.423659
$625$	0	0
$626$	−21.2915	−0.850980
$627$	8.70850	0.347784
$628$	−10.0000	−0.399043
$629$	−0.645751	−0.0257478
$630$	0	0
$631$	24.7085	0.983630	0.491815	−	0.870700i	$-0.336334\pi$
$631$	24.7085	0.983630	0.491815	+	0.870700i	$0.336334\pi$
$632$	2.29150	0.0911511
$633$	21.9373	0.871928
$634$	15.0000	0.595726
$635$	0	0
$636$	7.93725	0.314733
$637$	−4.00000	−0.158486
$638$	1.29150	0.0511311
$639$	−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	$0.481130\pi$
$642$	−43.1033	−1.70115
$643$	−41.1033	−1.62095	−0.810477	−	0.585770i	$-0.800792\pi$
$643$	−41.1033	−1.62095	−0.810477	+	0.585770i	$0.800792\pi$
$644$	0	0
$645$	0	0
$646$	−2.12549	−0.0836264
$647$	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
$651$	−3.58301	−0.140429
$652$	−2.70850	−0.106073
$653$	−34.7490	−1.35983	−0.679917	−	0.733289i	$-0.737984\pi$
$653$	−34.7490	−1.35983	−0.679917	+	0.733289i	$0.737984\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	0	0
$657$	44.9150	1.75230
$658$	5.35425	0.208730
$659$	−9.41699	−0.366834	−0.183417	−	0.983035i	$-0.558716\pi$
$659$	−9.41699	−0.366834	−0.183417	+	0.983035i	$0.558716\pi$
$660$	0	0
$661$	−14.7085	−0.572094	−0.286047	−	0.958216i	$-0.592341\pi$
$661$	−14.7085	−0.572094	−0.286047	+	0.958216i	$0.592341\pi$
$662$	28.4575	1.10603
$663$	6.83399	0.265410
$664$	−6.00000	−0.232845
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	−21.8745	−0.846350
$669$	−17.7490	−0.686217
$670$	0	0
$671$	4.00000	0.154418
$672$	2.64575	0.102062
$673$	−10.4575	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$673$	−10.4575	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$674$	−0.708497	−0.0272903
$675$	0	0
$676$	3.00000	0.115385
$677$	40.5203	1.55732	0.778660	−	0.627446i	$-0.215900\pi$
$677$	40.5203	1.55732	0.778660	+	0.627446i	$0.215900\pi$
$678$	−7.93725	−0.304828
$679$	16.5830	0.636397
$680$	0	0
$681$	47.6235	1.82494
$682$	1.35425	0.0518569
$683$	−1.29150	−0.0494180	−0.0247090	−	0.999695i	$-0.507866\pi$
$683$	−1.29150	−0.0494180	−0.0247090	+	0.999695i	$0.507866\pi$
$684$	13.1660	0.503415
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	17.4170	0.664500
$688$	−8.29150	−0.316111
$689$	12.0000	0.457164
$690$	0	0
$691$	−34.6458	−1.31799	−0.658993	−	0.752149i	$-0.729017\pi$
$691$	−34.6458	−1.31799	−0.658993	+	0.752149i	$0.729017\pi$
$692$	12.0000	0.456172
$693$	−4.00000	−0.151947
$694$	−27.4575	−1.04227
$695$	0	0
$696$	3.41699	0.129521
$697$	0	0
$698$	−7.35425	−0.278362
$699$	38.5830	1.45934
$700$	0	0
$701$	41.1660	1.55482	0.777409	−	0.628995i	$-0.216533\pi$
$701$	41.1660	1.55482	0.777409	+	0.628995i	$0.216533\pi$
$702$	−10.5830	−0.399430
$703$	−3.29150	−0.124141
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−24.4575	−0.920471
$707$	11.3542	0.427020
$708$	−21.0000	−0.789228
$709$	10.1660	0.381793	0.190896	−	0.981610i	$-0.438861\pi$
$709$	10.1660	0.381793	0.190896	+	0.981610i	$0.438861\pi$
$710$	0	0
$711$	−9.16601	−0.343752
$712$	−4.70850	−0.176458
$713$	0	0
$714$	1.70850	0.0639389
$715$	0	0
$716$	−19.2915	−0.720957
$717$	−49.9373	−1.86494
$718$	31.7490	1.18486
$719$	−8.12549	−0.303030	−0.151515	−	0.988455i	$-0.548415\pi$
$719$	−8.12549	−0.303030	−0.151515	+	0.988455i	$0.548415\pi$
$720$	0	0
$721$	14.6458	0.545436
$722$	8.16601	0.303907
$723$	19.1255	0.711285
$724$	2.00000	0.0743294
$725$	0	0
$726$	2.64575	0.0981930
$727$	−1.22876	−0.0455721	−0.0227860	−	0.999740i	$-0.507254\pi$
$727$	−1.22876	−0.0455721	−0.0227860	+	0.999740i	$0.507254\pi$
$728$	4.00000	0.148250
$729$	−41.0000	−1.51852
$730$	0	0
$731$	−5.35425	−0.198034
$732$	10.5830	0.391159
$733$	−19.2288	−0.710230	−0.355115	−	0.934823i	$-0.615558\pi$
$733$	−19.2288	−0.710230	−0.355115	+	0.934823i	$0.615558\pi$
$734$	22.4575	0.828922
$735$	0	0
$736$	0	0
$737$	−0.708497	−0.0260978
$738$	0	0
$739$	24.2915	0.893577	0.446789	−	0.894640i	$-0.352568\pi$
$739$	24.2915	0.893577	0.446789	+	0.894640i	$0.352568\pi$
$740$	0	0
$741$	34.8340	1.27966
$742$	3.00000	0.110133
$743$	−36.8745	−1.35279	−0.676397	−	0.736537i	$-0.736460\pi$
$743$	−36.8745	−1.35279	−0.676397	+	0.736537i	$0.736460\pi$
$744$	3.58301	0.131359
$745$	0	0
$746$	21.1660	0.774943
$747$	24.0000	0.878114
$748$	−0.645751	−0.0236110
$749$	−16.2915	−0.595279
$750$	0	0
$751$	16.5830	0.605122	0.302561	−	0.953130i	$-0.402158\pi$
$751$	16.5830	0.605122	0.302561	+	0.953130i	$0.402158\pi$
$752$	−5.35425	−0.195249
$753$	21.0000	0.765283
$754$	5.16601	0.188135
$755$	0	0
$756$	−2.64575	−0.0962250
$757$	−6.16601	−0.224107	−0.112054	−	0.993702i	$-0.535743\pi$
$757$	−6.16601	−0.224107	−0.112054	+	0.993702i	$0.535743\pi$
$758$	−32.4575	−1.17891
$759$	0	0
$760$	0	0
$761$	−1.93725	−0.0702254	−0.0351127	−	0.999383i	$-0.511179\pi$
$761$	−1.93725	−0.0702254	−0.0351127	+	0.999383i	$0.511179\pi$
$762$	39.3542	1.42565
$763$	10.5830	0.383131
$764$	−19.2915	−0.697942
$765$	0	0
$766$	−25.1033	−0.907018
$767$	−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
$771$	35.1660	1.26647
$772$	2.00000	0.0719816
$773$	27.8745	1.00258	0.501288	−	0.865280i	$-0.332860\pi$
$773$	27.8745	1.00258	0.501288	+	0.865280i	$0.332860\pi$
$774$	33.1660	1.19213
$775$	0	0
$776$	−16.5830	−0.595295
$777$	2.64575	0.0949158
$778$	−3.00000	−0.107555
$779$	0	0
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	−3.41699	−0.122113
$784$	1.00000	0.0357143
$785$	0	0
$786$	−26.1255	−0.931865
$787$	17.4170	0.620849	0.310424	−	0.950598i	$-0.399529\pi$
$787$	17.4170	0.620849	0.310424	+	0.950598i	$0.399529\pi$
$788$	1.29150	0.0460079
$789$	27.2288	0.969369
$790$	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
$796$	6.70850	0.237776
$797$	−2.12549	−0.0752888	−0.0376444	−	0.999291i	$-0.511985\pi$
$797$	−2.12549	−0.0752888	−0.0376444	+	0.999291i	$0.511985\pi$
$798$	8.70850	0.308277
$799$	−3.45751	−0.122318
$800$	0	0
$801$	18.8340	0.665466
$802$	6.41699	0.226592
$803$	−11.2288	−0.396254
$804$	−1.87451	−0.0661088
$805$	0	0
$806$	5.41699	0.190806
$807$	9.04052	0.318241
$808$	−11.3542	−0.399441
$809$	−8.12549	−0.285677	−0.142839	−	0.989746i	$-0.545623\pi$
$809$	−8.12549	−0.285677	−0.142839	+	0.989746i	$0.545623\pi$
$810$	0	0
$811$	1.16601	0.0409442	0.0204721	−	0.999790i	$-0.493483\pi$
$811$	1.16601	0.0409442	0.0204721	+	0.999790i	$0.493483\pi$
$812$	1.29150	0.0453229
$813$	77.4980	2.71797
$814$	−1.00000	−0.0350500
$815$	0	0
$816$	−1.70850	−0.0598094
$817$	−27.2915	−0.954809
$818$	−7.35425	−0.257135
$819$	−16.0000	−0.559085
$820$	0	0
$821$	−28.7085	−1.00193	−0.500967	−	0.865467i	$-0.667022\pi$
$821$	−28.7085	−1.00193	−0.500967	+	0.865467i	$0.667022\pi$
$822$	22.7085	0.792050
$823$	53.8745	1.87795	0.938974	−	0.343989i	$-0.111778\pi$
$823$	53.8745	1.87795	0.938974	+	0.343989i	$0.111778\pi$
$824$	−14.6458	−0.510209
$825$	0	0
$826$	−7.93725	−0.276172
$827$	−6.87451	−0.239050	−0.119525	−	0.992831i	$-0.538137\pi$
$827$	−6.87451	−0.239050	−0.119525	+	0.992831i	$0.538137\pi$
$828$	0	0
$829$	−5.29150	−0.183781	−0.0918907	−	0.995769i	$-0.529291\pi$
$829$	−5.29150	−0.183781	−0.0918907	+	0.995769i	$0.529291\pi$
$830$	0	0
$831$	−15.5425	−0.539163
$832$	−4.00000	−0.138675
$833$	0.645751	0.0223740
$834$	−33.2915	−1.15279
$835$	0	0
$836$	−3.29150	−0.113839
$837$	−3.58301	−0.123847
$838$	21.2288	0.733335
$839$	10.7085	0.369698	0.184849	−	0.982767i	$-0.440820\pi$
$839$	10.7085	0.369698	0.184849	+	0.982767i	$0.440820\pi$
$840$	0	0
$841$	−27.3320	−0.942483
$842$	−16.5830	−0.571488
$843$	12.4575	0.429060
$844$	−8.29150	−0.285405
$845$	0	0
$846$	21.4170	0.736331
$847$	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
$853$	−50.9778	−1.74545	−0.872723	−	0.488216i	$-0.837648\pi$
$853$	−50.9778	−1.74545	−0.872723	+	0.488216i	$0.837648\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	16.2915	0.556832
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	10.5830	0.361298
$859$	17.2288	0.587838	0.293919	−	0.955830i	$-0.405040\pi$
$859$	17.2288	0.587838	0.293919	+	0.955830i	$0.405040\pi$
$860$	0	0
$861$	0	0
$862$	−1.70850	−0.0581917
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	2.64575	0.0900103
$865$	0	0
$866$	17.2915	0.587589
$867$	43.8745	1.49006
$868$	1.35425	0.0459662
$869$	2.29150	0.0777339
$870$	0	0
$871$	−2.83399	−0.0960261
$872$	−10.5830	−0.358386
$873$	66.3320	2.24500
$874$	0	0
$875$	0	0
$876$	−29.7085	−1.00376
$877$	27.7490	0.937018	0.468509	−	0.883459i	$-0.344791\pi$
$877$	27.7490	0.937018	0.468509	+	0.883459i	$0.344791\pi$
$878$	−7.16601	−0.241841
$879$	68.6235	2.31461
$880$	0	0
$881$	−12.4575	−0.419704	−0.209852	−	0.977733i	$-0.567298\pi$
$881$	−12.4575	−0.419704	−0.209852	+	0.977733i	$0.567298\pi$
$882$	−4.00000	−0.134687
$883$	50.4575	1.69803	0.849015	−	0.528368i	$-0.177196\pi$
$883$	50.4575	1.69803	0.849015	+	0.528368i	$0.177196\pi$
$884$	−2.58301	−0.0868759
$885$	0	0
$886$	38.5830	1.29622
$887$	−7.74902	−0.260186	−0.130093	−	0.991502i	$-0.541528\pi$
$887$	−7.74902	−0.260186	−0.130093	+	0.991502i	$0.541528\pi$
$888$	−2.64575	−0.0887856
$889$	14.8745	0.498875
$890$	0	0
$891$	5.00000	0.167506
$892$	6.70850	0.224617
$893$	−17.6235	−0.589749
$894$	−6.83399	−0.228563
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	12.4170	0.414360
$899$	1.74902	0.0583329
$900$	0	0
$901$	−1.93725	−0.0645393
$902$	0	0
$903$	21.9373	0.730026
$904$	3.00000	0.0997785
$905$	0	0
$906$	−17.4170	−0.578641
$907$	57.7490	1.91752	0.958762	−	0.284209i	$-0.0917310\pi$
$907$	57.7490	1.91752	0.958762	+	0.284209i	$0.0917310\pi$
$908$	−18.0000	−0.597351
$909$	45.4170	1.50639
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−8.70850	−0.288367
$913$	−6.00000	−0.198571
$914$	−20.4575	−0.676674
$915$	0	0
$916$	−6.58301	−0.217509
$917$	−9.87451	−0.326085
$918$	1.70850	0.0563888
$919$	−10.8745	−0.358717	−0.179358	−	0.983784i	$-0.557402\pi$
$919$	−10.8745	−0.358717	−0.179358	+	0.983784i	$0.557402\pi$
$920$	0	0
$921$	−14.3320	−0.472256
$922$	−31.1033	−1.02433
$923$	24.0000	0.789970
$924$	2.64575	0.0870388
$925$	0	0
$926$	−14.0000	−0.460069
$927$	58.5830	1.92412
$928$	−1.29150	−0.0423957
$929$	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
$931$	3.29150	0.107875
$932$	−14.5830	−0.477682
$933$	59.5830	1.95066
$934$	5.35425	0.175196
$935$	0	0
$936$	16.0000	0.522976
$937$	−8.52026	−0.278345	−0.139172	−	0.990268i	$-0.544444\pi$
$937$	−8.52026	−0.278345	−0.139172	+	0.990268i	$0.544444\pi$
$938$	−0.708497	−0.0231333
$939$	−56.3320	−1.83833
$940$	0	0
$941$	37.9373	1.23672	0.618360	−	0.785895i	$-0.287798\pi$
$941$	37.9373	1.23672	0.618360	+	0.785895i	$0.287798\pi$
$942$	−26.4575	−0.862032
$943$	0	0
$944$	7.93725	0.258336
$945$	0	0
$946$	−8.29150	−0.269580
$947$	−39.0405	−1.26865	−0.634323	−	0.773068i	$-0.718721\pi$
$947$	−39.0405	−1.26865	−0.634323	+	0.773068i	$0.718721\pi$
$948$	6.06275	0.196909
$949$	−44.9150	−1.45800
$950$	0	0
$951$	39.6863	1.28692
$952$	−0.645751	−0.0209289
$953$	53.6235	1.73704	0.868518	−	0.495657i	$-0.165073\pi$
$953$	53.6235	1.73704	0.868518	+	0.495657i	$0.165073\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	18.8745	0.610445
$957$	3.41699	0.110456
$958$	−30.4575	−0.984038
$959$	8.58301	0.277160
$960$	0	0
$961$	−29.1660	−0.940839
$962$	−4.00000	−0.128965
$963$	−65.1660	−2.09995
$964$	−7.22876	−0.232823
$965$	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
$969$	−5.62352	−0.180654
$970$	0	0
$971$	39.8745	1.27963	0.639817	−	0.768527i	$-0.279010\pi$
$971$	39.8745	1.27963	0.639817	+	0.768527i	$0.279010\pi$
$972$	21.1660	0.678900
$973$	−12.5830	−0.403393
$974$	−33.2915	−1.06673
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−13.7490	−0.439870	−0.219935	−	0.975515i	$-0.570585\pi$
$977$	−13.7490	−0.439870	−0.219935	+	0.975515i	$0.570585\pi$
$978$	−7.16601	−0.229144
$979$	−4.70850	−0.150484
$980$	0	0
$981$	42.3320	1.35156
$982$	42.8745	1.36818
$983$	−46.5203	−1.48377	−0.741883	−	0.670529i	$-0.766067\pi$
$983$	−46.5203	−1.48377	−0.741883	+	0.670529i	$0.766067\pi$
$984$	0	0
$985$	0	0
$986$	−0.833990	−0.0265596
$987$	14.1660	0.450909
$988$	−13.1660	−0.418867
$989$	0	0
$990$	0	0
$991$	−3.16601	−0.100572	−0.0502858	−	0.998735i	$-0.516013\pi$
$991$	−3.16601	−0.100572	−0.0502858	+	0.998735i	$0.516013\pi$
$992$	−1.35425	−0.0429974
$993$	75.2915	2.38930
$994$	6.00000	0.190308
$995$	0	0
$996$	−15.8745	−0.503003
$997$	−24.3948	−0.772590	−0.386295	−	0.922375i	$-0.626245\pi$
$997$	−24.3948	−0.772590	−0.386295	+	0.922375i	$0.626245\pi$
$998$	25.8745	0.819043
$999$	2.64575	0.0837079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bh.1.1	✓	2
5.2	odd	4		3850.2.c.v.1849.2		4
5.3	odd	4		3850.2.c.v.1849.3		4
5.4	even	2		3850.2.a.bo.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3850.2.a.bh.1.1	✓	2	1.1	even	1	trivial
3850.2.a.bo.1.2	yes	2	5.4	even	2
3850.2.c.v.1849.2		4	5.2	odd	4
3850.2.c.v.1849.3		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} +2.64575 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.64575 q^{3} +1.00000 q^{4} +2.64575 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{9} -1.00000 q^{11} -2.64575 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.645751 q^{17} -4.00000 q^{18} +3.29150 q^{19} -2.64575 q^{21} +1.00000 q^{22} +2.64575 q^{24} +4.00000 q^{26} -2.64575 q^{27} +1.00000 q^{28} +1.29150 q^{29} +1.35425 q^{31} -1.00000 q^{32} +2.64575 q^{33} -0.645751 q^{34} +4.00000 q^{36} -1.00000 q^{37} -3.29150 q^{38} +10.5830 q^{39} +2.64575 q^{42} -8.29150 q^{43} -1.00000 q^{44} -5.35425 q^{47} -2.64575 q^{48} +1.00000 q^{49} -1.70850 q^{51} -4.00000 q^{52} -3.00000 q^{53} +2.64575 q^{54} -1.00000 q^{56} -8.70850 q^{57} -1.29150 q^{58} +7.93725 q^{59} -4.00000 q^{61} -1.35425 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.64575 q^{66} +0.708497 q^{67} +0.645751 q^{68} -6.00000 q^{71} -4.00000 q^{72} +11.2288 q^{73} +1.00000 q^{74} +3.29150 q^{76} -1.00000 q^{77} -10.5830 q^{78} -2.29150 q^{79} -5.00000 q^{81} +6.00000 q^{83} -2.64575 q^{84} +8.29150 q^{86} -3.41699 q^{87} +1.00000 q^{88} +4.70850 q^{89} -4.00000 q^{91} -3.58301 q^{93} +5.35425 q^{94} +2.64575 q^{96} +16.5830 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more