LMFDB - Embedded newform 3850.2.a.be.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{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.44949 q^{12} -4.89898 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +4.44949 q^{19} +2.44949 q^{21} +1.00000 q^{22} -6.44949 q^{23} +2.44949 q^{24} +4.89898 q^{26} -1.00000 q^{28} -2.44949 q^{29} -6.89898 q^{31} -1.00000 q^{32} +2.44949 q^{33} -2.00000 q^{34} +3.00000 q^{36} -4.44949 q^{37} -4.44949 q^{38} +12.0000 q^{39} -3.34847 q^{41} -2.44949 q^{42} +2.00000 q^{43} -1.00000 q^{44} +6.44949 q^{46} +6.00000 q^{47} -2.44949 q^{48} +1.00000 q^{49} -4.89898 q^{51} -4.89898 q^{52} -4.44949 q^{53} +1.00000 q^{56} -10.8990 q^{57} +2.44949 q^{58} +4.89898 q^{59} -10.0000 q^{61} +6.89898 q^{62} -3.00000 q^{63} +1.00000 q^{64} -2.44949 q^{66} +7.79796 q^{67} +2.00000 q^{68} +15.7980 q^{69} +3.10102 q^{71} -3.00000 q^{72} +2.89898 q^{73} +4.44949 q^{74} +4.44949 q^{76} +1.00000 q^{77} -12.0000 q^{78} -5.34847 q^{79} -9.00000 q^{81} +3.34847 q^{82} -14.6969 q^{83} +2.44949 q^{84} -2.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -2.89898 q^{89} +4.89898 q^{91} -6.44949 q^{92} +16.8990 q^{93} -6.00000 q^{94} +2.44949 q^{96} -16.4495 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 4 q^{19} + 2 q^{22} - 8 q^{23} - 2 q^{28} - 4 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36} - 4 q^{37} - 4 q^{38} + 24 q^{39} + 8 q^{41} + 4 q^{43} - 2 q^{44} + 8 q^{46} + 12 q^{47} + 2 q^{49} - 4 q^{53} + 2 q^{56} - 12 q^{57} - 20 q^{61} + 4 q^{62} - 6 q^{63} + 2 q^{64} - 4 q^{67} + 4 q^{68} + 12 q^{69} + 16 q^{71} - 6 q^{72} - 4 q^{73} + 4 q^{74} + 4 q^{76} + 2 q^{77} - 24 q^{78} + 4 q^{79} - 18 q^{81} - 8 q^{82} - 4 q^{86} + 12 q^{87} + 2 q^{88} + 4 q^{89} - 8 q^{92} + 24 q^{93} - 12 q^{94} - 28 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 4 * q^19 + 2 * q^22 - 8 * q^23 - 2 * q^28 - 4 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 - 4 * q^37 - 4 * q^38 + 24 * q^39 + 8 * q^41 + 4 * q^43 - 2 * q^44 + 8 * q^46 + 12 * q^47 + 2 * q^49 - 4 * q^53 + 2 * q^56 - 12 * q^57 - 20 * q^61 + 4 * q^62 - 6 * q^63 + 2 * q^64 - 4 * q^67 + 4 * q^68 + 12 * q^69 + 16 * q^71 - 6 * q^72 - 4 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 24 * q^78 + 4 * q^79 - 18 * q^81 - 8 * q^82 - 4 * q^86 + 12 * q^87 + 2 * q^88 + 4 * q^89 - 8 * q^92 + 24 * q^93 - 12 * q^94 - 28 * q^97 - 2 * q^98 - 6 * 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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.44949	1.00000
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.44949	−0.707107
$13$	−4.89898	−1.35873	−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898	−1.35873	−0.679366	+	0.733799i	$0.737745\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−3.00000	−0.707107
$19$	4.44949	1.02078	0.510391	−	0.859942i	$-0.329501\pi$
$19$	4.44949	1.02078	0.510391	+	0.859942i	$0.329501\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	1.00000	0.213201
$23$	−6.44949	−1.34481	−0.672406	−	0.740183i	$-0.734739\pi$
$23$	−6.44949	−1.34481	−0.672406	+	0.740183i	$0.734739\pi$
$24$	2.44949	0.500000
$25$	0	0
$26$	4.89898	0.960769
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−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
$31$	−6.89898	−1.23909	−0.619547	−	0.784960i	$-0.712684\pi$
$31$	−6.89898	−1.23909	−0.619547	+	0.784960i	$0.712684\pi$
$32$	−1.00000	−0.176777
$33$	2.44949	0.426401
$34$	−2.00000	−0.342997
$35$	0	0
$36$	3.00000	0.500000
$37$	−4.44949	−0.731492	−0.365746	−	0.930715i	$-0.619186\pi$
$37$	−4.44949	−0.731492	−0.365746	+	0.930715i	$0.619186\pi$
$38$	−4.44949	−0.721803
$39$	12.0000	1.92154
$40$	0	0
$41$	−3.34847	−0.522943	−0.261472	−	0.965211i	$-0.584208\pi$
$41$	−3.34847	−0.522943	−0.261472	+	0.965211i	$0.584208\pi$
$42$	−2.44949	−0.377964
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	6.44949	0.950925
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−2.44949	−0.353553
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.89898	−0.685994
$52$	−4.89898	−0.679366
$53$	−4.44949	−0.611184	−0.305592	−	0.952162i	$-0.598854\pi$
$53$	−4.44949	−0.611184	−0.305592	+	0.952162i	$0.598854\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	−10.8990	−1.44361
$58$	2.44949	0.321634
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	6.89898	0.876171
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.44949	−0.301511
$67$	7.79796	0.952672	0.476336	−	0.879263i	$-0.341965\pi$
$67$	7.79796	0.952672	0.476336	+	0.879263i	$0.341965\pi$
$68$	2.00000	0.242536
$69$	15.7980	1.90185
$70$	0	0
$71$	3.10102	0.368023	0.184012	−	0.982924i	$-0.441092\pi$
$71$	3.10102	0.368023	0.184012	+	0.982924i	$0.441092\pi$
$72$	−3.00000	−0.353553
$73$	2.89898	0.339300	0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898	0.339300	0.169650	+	0.985504i	$0.445736\pi$
$74$	4.44949	0.517243
$75$	0	0
$76$	4.44949	0.510391
$77$	1.00000	0.113961
$78$	−12.0000	−1.35873
$79$	−5.34847	−0.601750	−0.300875	−	0.953664i	$-0.597279\pi$
$79$	−5.34847	−0.601750	−0.300875	+	0.953664i	$0.597279\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	3.34847	0.369777
$83$	−14.6969	−1.61320	−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969	−1.61320	−0.806599	+	0.591099i	$0.798694\pi$
$84$	2.44949	0.267261
$85$	0	0
$86$	−2.00000	−0.215666
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	−2.89898	−0.307291	−0.153646	−	0.988126i	$-0.549101\pi$
$89$	−2.89898	−0.307291	−0.153646	+	0.988126i	$0.549101\pi$
$90$	0	0
$91$	4.89898	0.513553
$92$	−6.44949	−0.672406
$93$	16.8990	1.75234
$94$	−6.00000	−0.618853
$95$	0	0
$96$	2.44949	0.250000
$97$	−16.4495	−1.67019	−0.835096	−	0.550104i	$-0.814588\pi$
$97$	−16.4495	−1.67019	−0.835096	+	0.550104i	$0.814588\pi$
$98$	−1.00000	−0.101015
$99$	−3.00000	−0.301511
$100$	0	0
$101$	15.7980	1.57196	0.785978	−	0.618255i	$-0.212160\pi$
$101$	15.7980	1.57196	0.785978	+	0.618255i	$0.212160\pi$
$102$	4.89898	0.485071
$103$	−2.89898	−0.285645	−0.142822	−	0.989748i	$-0.545618\pi$
$103$	−2.89898	−0.285645	−0.142822	+	0.989748i	$0.545618\pi$
$104$	4.89898	0.480384
$105$	0	0
$106$	4.44949	0.432173
$107$	−3.79796	−0.367163	−0.183581	−	0.983005i	$-0.558769\pi$
$107$	−3.79796	−0.367163	−0.183581	+	0.983005i	$0.558769\pi$
$108$	0	0
$109$	−8.24745	−0.789962	−0.394981	−	0.918689i	$-0.629249\pi$
$109$	−8.24745	−0.789962	−0.394981	+	0.918689i	$0.629249\pi$
$110$	0	0
$111$	10.8990	1.03449
$112$	−1.00000	−0.0944911
$113$	9.79796	0.921714	0.460857	−	0.887474i	$-0.347542\pi$
$113$	9.79796	0.921714	0.460857	+	0.887474i	$0.347542\pi$
$114$	10.8990	1.02078
$115$	0	0
$116$	−2.44949	−0.227429
$117$	−14.6969	−1.35873
$118$	−4.89898	−0.450988
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	8.20204	0.739553
$124$	−6.89898	−0.619547
$125$	0	0
$126$	3.00000	0.267261
$127$	0.898979	0.0797715	0.0398858	−	0.999204i	$-0.487301\pi$
$127$	0.898979	0.0797715	0.0398858	+	0.999204i	$0.487301\pi$
$128$	−1.00000	−0.0883883
$129$	−4.89898	−0.431331
$130$	0	0
$131$	16.4495	1.43720	0.718599	−	0.695424i	$-0.244784\pi$
$131$	16.4495	1.43720	0.718599	+	0.695424i	$0.244784\pi$
$132$	2.44949	0.213201
$133$	−4.44949	−0.385820
$134$	−7.79796	−0.673641
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−21.7980	−1.86233	−0.931163	−	0.364604i	$-0.881204\pi$
$137$	−21.7980	−1.86233	−0.931163	+	0.364604i	$0.881204\pi$
$138$	−15.7980	−1.34481
$139$	−1.34847	−0.114376	−0.0571878	−	0.998363i	$-0.518213\pi$
$139$	−1.34847	−0.114376	−0.0571878	+	0.998363i	$0.518213\pi$
$140$	0	0
$141$	−14.6969	−1.23771
$142$	−3.10102	−0.260232
$143$	4.89898	0.409673
$144$	3.00000	0.250000
$145$	0	0
$146$	−2.89898	−0.239921
$147$	−2.44949	−0.202031
$148$	−4.44949	−0.365746
$149$	−13.1464	−1.07700	−0.538499	−	0.842626i	$-0.681008\pi$
$149$	−13.1464	−1.07700	−0.538499	+	0.842626i	$0.681008\pi$
$150$	0	0
$151$	−8.44949	−0.687610	−0.343805	−	0.939041i	$-0.611716\pi$
$151$	−8.44949	−0.687610	−0.343805	+	0.939041i	$0.611716\pi$
$152$	−4.44949	−0.360901
$153$	6.00000	0.485071
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	12.0000	0.960769
$157$	18.6969	1.49218	0.746089	−	0.665846i	$-0.231929\pi$
$157$	18.6969	1.49218	0.746089	+	0.665846i	$0.231929\pi$
$158$	5.34847	0.425501
$159$	10.8990	0.864345
$160$	0	0
$161$	6.44949	0.508291
$162$	9.00000	0.707107
$163$	4.20204	0.329129	0.164565	−	0.986366i	$-0.447378\pi$
$163$	4.20204	0.329129	0.164565	+	0.986366i	$0.447378\pi$
$164$	−3.34847	−0.261472
$165$	0	0
$166$	14.6969	1.14070
$167$	13.7980	1.06772	0.533859	−	0.845573i	$-0.320741\pi$
$167$	13.7980	1.06772	0.533859	+	0.845573i	$0.320741\pi$
$168$	−2.44949	−0.188982
$169$	11.0000	0.846154
$170$	0	0
$171$	13.3485	1.02078
$172$	2.00000	0.152499
$173$	−19.7980	−1.50521	−0.752605	−	0.658472i	$-0.771203\pi$
$173$	−19.7980	−1.50521	−0.752605	+	0.658472i	$0.771203\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−12.0000	−0.901975
$178$	2.89898	0.217288
$179$	10.8990	0.814628	0.407314	−	0.913288i	$-0.366465\pi$
$179$	10.8990	0.814628	0.407314	+	0.913288i	$0.366465\pi$
$180$	0	0
$181$	−23.7980	−1.76889	−0.884444	−	0.466646i	$-0.845462\pi$
$181$	−23.7980	−1.76889	−0.884444	+	0.466646i	$0.845462\pi$
$182$	−4.89898	−0.363137
$183$	24.4949	1.81071
$184$	6.44949	0.475463
$185$	0	0
$186$	−16.8990	−1.23909
$187$	−2.00000	−0.146254
$188$	6.00000	0.437595
$189$	0	0
$190$	0	0
$191$	−14.6969	−1.06343	−0.531717	−	0.846922i	$-0.678453\pi$
$191$	−14.6969	−1.06343	−0.531717	+	0.846922i	$0.678453\pi$
$192$	−2.44949	−0.176777
$193$	18.6969	1.34584	0.672918	−	0.739718i	$-0.265041\pi$
$193$	18.6969	1.34584	0.672918	+	0.739718i	$0.265041\pi$
$194$	16.4495	1.18100
$195$	0	0
$196$	1.00000	0.0714286
$197$	−8.20204	−0.584371	−0.292186	−	0.956362i	$-0.594383\pi$
$197$	−8.20204	−0.584371	−0.292186	+	0.956362i	$0.594383\pi$
$198$	3.00000	0.213201
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−19.1010	−1.34728
$202$	−15.7980	−1.11154
$203$	2.44949	0.171920
$204$	−4.89898	−0.342997
$205$	0	0
$206$	2.89898	0.201981
$207$	−19.3485	−1.34481
$208$	−4.89898	−0.339683
$209$	−4.44949	−0.307778
$210$	0	0
$211$	6.20204	0.426966	0.213483	−	0.976947i	$-0.431519\pi$
$211$	6.20204	0.426966	0.213483	+	0.976947i	$0.431519\pi$
$212$	−4.44949	−0.305592
$213$	−7.59592	−0.520464
$214$	3.79796	0.259623
$215$	0	0
$216$	0	0
$217$	6.89898	0.468333
$218$	8.24745	0.558588
$219$	−7.10102	−0.479842
$220$	0	0
$221$	−9.79796	−0.659082
$222$	−10.8990	−0.731492
$223$	24.6969	1.65383	0.826915	−	0.562327i	$-0.190094\pi$
$223$	24.6969	1.65383	0.826915	+	0.562327i	$0.190094\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.79796	−0.651751
$227$	26.6969	1.77194	0.885969	−	0.463744i	$-0.153494\pi$
$227$	26.6969	1.77194	0.885969	+	0.463744i	$0.153494\pi$
$228$	−10.8990	−0.721803
$229$	21.7980	1.44045	0.720225	−	0.693741i	$-0.244039\pi$
$229$	21.7980	1.44045	0.720225	+	0.693741i	$0.244039\pi$
$230$	0	0
$231$	−2.44949	−0.161165
$232$	2.44949	0.160817
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	14.6969	0.960769
$235$	0	0
$236$	4.89898	0.318896
$237$	13.1010	0.851003
$238$	2.00000	0.129641
$239$	23.1464	1.49722	0.748609	−	0.663012i	$-0.230722\pi$
$239$	23.1464	1.49722	0.748609	+	0.663012i	$0.230722\pi$
$240$	0	0
$241$	19.3485	1.24634	0.623172	−	0.782085i	$-0.285844\pi$
$241$	19.3485	1.24634	0.623172	+	0.782085i	$0.285844\pi$
$242$	−1.00000	−0.0642824
$243$	22.0454	1.41421
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−8.20204	−0.522943
$247$	−21.7980	−1.38697
$248$	6.89898	0.438086
$249$	36.0000	2.28141
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−3.00000	−0.188982
$253$	6.44949	0.405476
$254$	−0.898979	−0.0564070
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.2474	1.38776	0.693879	−	0.720092i	$-0.255900\pi$
$257$	22.2474	1.38776	0.693879	+	0.720092i	$0.255900\pi$
$258$	4.89898	0.304997
$259$	4.44949	0.276478
$260$	0	0
$261$	−7.34847	−0.454859
$262$	−16.4495	−1.01625
$263$	11.1010	0.684518	0.342259	−	0.939606i	$-0.388808\pi$
$263$	11.1010	0.684518	0.342259	+	0.939606i	$0.388808\pi$
$264$	−2.44949	−0.150756
$265$	0	0
$266$	4.44949	0.272816
$267$	7.10102	0.434575
$268$	7.79796	0.476336
$269$	9.79796	0.597392	0.298696	−	0.954348i	$-0.403448\pi$
$269$	9.79796	0.597392	0.298696	+	0.954348i	$0.403448\pi$
$270$	0	0
$271$	−6.69694	−0.406810	−0.203405	−	0.979095i	$-0.565201\pi$
$271$	−6.69694	−0.406810	−0.203405	+	0.979095i	$0.565201\pi$
$272$	2.00000	0.121268
$273$	−12.0000	−0.726273
$274$	21.7980	1.31686
$275$	0	0
$276$	15.7980	0.950925
$277$	−9.10102	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$277$	−9.10102	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$278$	1.34847	0.0808758
$279$	−20.6969	−1.23909
$280$	0	0
$281$	14.8990	0.888799	0.444399	−	0.895829i	$-0.353417\pi$
$281$	14.8990	0.888799	0.444399	+	0.895829i	$0.353417\pi$
$282$	14.6969	0.875190
$283$	20.4949	1.21830	0.609148	−	0.793057i	$-0.291512\pi$
$283$	20.4949	1.21830	0.609148	+	0.793057i	$0.291512\pi$
$284$	3.10102	0.184012
$285$	0	0
$286$	−4.89898	−0.289683
$287$	3.34847	0.197654
$288$	−3.00000	−0.176777
$289$	−13.0000	−0.764706
$290$	0	0
$291$	40.2929	2.36201
$292$	2.89898	0.169650
$293$	20.4949	1.19732	0.598662	−	0.801001i	$-0.295699\pi$
$293$	20.4949	1.19732	0.598662	+	0.801001i	$0.295699\pi$
$294$	2.44949	0.142857
$295$	0	0
$296$	4.44949	0.258621
$297$	0	0
$298$	13.1464	0.761552
$299$	31.5959	1.82724
$300$	0	0
$301$	−2.00000	−0.115278
$302$	8.44949	0.486213
$303$	−38.6969	−2.22308
$304$	4.44949	0.255196
$305$	0	0
$306$	−6.00000	−0.342997
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	1.00000	0.0569803
$309$	7.10102	0.403963
$310$	0	0
$311$	0.696938	0.0395198	0.0197599	−	0.999805i	$-0.493710\pi$
$311$	0.696938	0.0395198	0.0197599	+	0.999805i	$0.493710\pi$
$312$	−12.0000	−0.679366
$313$	10.6515	0.602060	0.301030	−	0.953615i	$-0.402670\pi$
$313$	10.6515	0.602060	0.301030	+	0.953615i	$0.402670\pi$
$314$	−18.6969	−1.05513
$315$	0	0
$316$	−5.34847	−0.300875
$317$	27.1464	1.52470	0.762348	−	0.647168i	$-0.224047\pi$
$317$	27.1464	1.52470	0.762348	+	0.647168i	$0.224047\pi$
$318$	−10.8990	−0.611184
$319$	2.44949	0.137145
$320$	0	0
$321$	9.30306	0.519246
$322$	−6.44949	−0.359416
$323$	8.89898	0.495152
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−4.20204	−0.232730
$327$	20.2020	1.11718
$328$	3.34847	0.184888
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−28.6969	−1.57733	−0.788663	−	0.614825i	$-0.789226\pi$
$331$	−28.6969	−1.57733	−0.788663	+	0.614825i	$0.789226\pi$
$332$	−14.6969	−0.806599
$333$	−13.3485	−0.731492
$334$	−13.7980	−0.754991
$335$	0	0
$336$	2.44949	0.133631
$337$	23.7980	1.29636	0.648179	−	0.761488i	$-0.275531\pi$
$337$	23.7980	1.29636	0.648179	+	0.761488i	$0.275531\pi$
$338$	−11.0000	−0.598321
$339$	−24.0000	−1.30350
$340$	0	0
$341$	6.89898	0.373601
$342$	−13.3485	−0.721803
$343$	−1.00000	−0.0539949
$344$	−2.00000	−0.107833
$345$	0	0
$346$	19.7980	1.06434
$347$	0.202041	0.0108461	0.00542307	−	0.999985i	$-0.498274\pi$
$347$	0.202041	0.0108461	0.00542307	+	0.999985i	$0.498274\pi$
$348$	6.00000	0.321634
$349$	16.6969	0.893767	0.446883	−	0.894592i	$-0.352534\pi$
$349$	16.6969	0.893767	0.446883	+	0.894592i	$0.352534\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	33.8434	1.80130	0.900650	−	0.434545i	$-0.143091\pi$
$353$	33.8434	1.80130	0.900650	+	0.434545i	$0.143091\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−2.89898	−0.153646
$357$	4.89898	0.259281
$358$	−10.8990	−0.576029
$359$	−5.34847	−0.282281	−0.141141	−	0.989990i	$-0.545077\pi$
$359$	−5.34847	−0.282281	−0.141141	+	0.989990i	$0.545077\pi$
$360$	0	0
$361$	0.797959	0.0419978
$362$	23.7980	1.25079
$363$	−2.44949	−0.128565
$364$	4.89898	0.256776
$365$	0	0
$366$	−24.4949	−1.28037
$367$	0.696938	0.0363799	0.0181899	−	0.999835i	$-0.494210\pi$
$367$	0.696938	0.0363799	0.0181899	+	0.999835i	$0.494210\pi$
$368$	−6.44949	−0.336203
$369$	−10.0454	−0.522943
$370$	0	0
$371$	4.44949	0.231006
$372$	16.8990	0.876171
$373$	−31.7980	−1.64644	−0.823218	−	0.567725i	$-0.807824\pi$
$373$	−31.7980	−1.64644	−0.823218	+	0.567725i	$0.807824\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000	0.618031
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−2.20204	−0.112814
$382$	14.6969	0.751961
$383$	−34.4949	−1.76261	−0.881303	−	0.472551i	$-0.843333\pi$
$383$	−34.4949	−1.76261	−0.881303	+	0.472551i	$0.843333\pi$
$384$	2.44949	0.125000
$385$	0	0
$386$	−18.6969	−0.951649
$387$	6.00000	0.304997
$388$	−16.4495	−0.835096
$389$	11.7980	0.598180	0.299090	−	0.954225i	$-0.403317\pi$
$389$	11.7980	0.598180	0.299090	+	0.954225i	$0.403317\pi$
$390$	0	0
$391$	−12.8990	−0.652329
$392$	−1.00000	−0.0505076
$393$	−40.2929	−2.03251
$394$	8.20204	0.413213
$395$	0	0
$396$	−3.00000	−0.150756
$397$	6.69694	0.336110	0.168055	−	0.985778i	$-0.446251\pi$
$397$	6.69694	0.336110	0.168055	+	0.985778i	$0.446251\pi$
$398$	16.0000	0.802008
$399$	10.8990	0.545631
$400$	0	0
$401$	−23.5959	−1.17832	−0.589162	−	0.808015i	$-0.700542\pi$
$401$	−23.5959	−1.17832	−0.589162	+	0.808015i	$0.700542\pi$
$402$	19.1010	0.952672
$403$	33.7980	1.68360
$404$	15.7980	0.785978
$405$	0	0
$406$	−2.44949	−0.121566
$407$	4.44949	0.220553
$408$	4.89898	0.242536
$409$	−25.1464	−1.24341	−0.621705	−	0.783251i	$-0.713560\pi$
$409$	−25.1464	−1.24341	−0.621705	+	0.783251i	$0.713560\pi$
$410$	0	0
$411$	53.3939	2.63373
$412$	−2.89898	−0.142822
$413$	−4.89898	−0.241063
$414$	19.3485	0.950925
$415$	0	0
$416$	4.89898	0.240192
$417$	3.30306	0.161752
$418$	4.44949	0.217632
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	8.69694	0.423863	0.211931	−	0.977285i	$-0.432025\pi$
$421$	8.69694	0.423863	0.211931	+	0.977285i	$0.432025\pi$
$422$	−6.20204	−0.301911
$423$	18.0000	0.875190
$424$	4.44949	0.216086
$425$	0	0
$426$	7.59592	0.368023
$427$	10.0000	0.483934
$428$	−3.79796	−0.183581
$429$	−12.0000	−0.579365
$430$	0	0
$431$	24.9444	1.20153	0.600764	−	0.799426i	$-0.294863\pi$
$431$	24.9444	1.20153	0.600764	+	0.799426i	$0.294863\pi$
$432$	0	0
$433$	33.3485	1.60263	0.801313	−	0.598246i	$-0.204135\pi$
$433$	33.3485	1.60263	0.801313	+	0.598246i	$0.204135\pi$
$434$	−6.89898	−0.331162
$435$	0	0
$436$	−8.24745	−0.394981
$437$	−28.6969	−1.37276
$438$	7.10102	0.339300
$439$	−5.30306	−0.253101	−0.126551	−	0.991960i	$-0.540391\pi$
$439$	−5.30306	−0.253101	−0.126551	+	0.991960i	$0.540391\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	9.79796	0.466041
$443$	−0.202041	−0.00959926	−0.00479963	−	0.999988i	$-0.501528\pi$
$443$	−0.202041	−0.00959926	−0.00479963	+	0.999988i	$0.501528\pi$
$444$	10.8990	0.517243
$445$	0	0
$446$	−24.6969	−1.16943
$447$	32.2020	1.52310
$448$	−1.00000	−0.0472456
$449$	−37.3939	−1.76473	−0.882363	−	0.470569i	$-0.844049\pi$
$449$	−37.3939	−1.76473	−0.882363	+	0.470569i	$0.844049\pi$
$450$	0	0
$451$	3.34847	0.157673
$452$	9.79796	0.460857
$453$	20.6969	0.972427
$454$	−26.6969	−1.25295
$455$	0	0
$456$	10.8990	0.510391
$457$	−13.5959	−0.635990	−0.317995	−	0.948092i	$-0.603010\pi$
$457$	−13.5959	−0.635990	−0.317995	+	0.948092i	$0.603010\pi$
$458$	−21.7980	−1.01855
$459$	0	0
$460$	0	0
$461$	−0.696938	−0.0324597	−0.0162298	−	0.999868i	$-0.505166\pi$
$461$	−0.696938	−0.0324597	−0.0162298	+	0.999868i	$0.505166\pi$
$462$	2.44949	0.113961
$463$	−14.4495	−0.671525	−0.335762	−	0.941947i	$-0.608994\pi$
$463$	−14.4495	−0.671525	−0.335762	+	0.941947i	$0.608994\pi$
$464$	−2.44949	−0.113715
$465$	0	0
$466$	6.00000	0.277945
$467$	17.5505	0.812141	0.406070	−	0.913842i	$-0.366899\pi$
$467$	17.5505	0.812141	0.406070	+	0.913842i	$0.366899\pi$
$468$	−14.6969	−0.679366
$469$	−7.79796	−0.360076
$470$	0	0
$471$	−45.7980	−2.11026
$472$	−4.89898	−0.225494
$473$	−2.00000	−0.0919601
$474$	−13.1010	−0.601750
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−13.3485	−0.611184
$478$	−23.1464	−1.05869
$479$	19.5959	0.895360	0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959	0.895360	0.447680	+	0.894194i	$0.352250\pi$
$480$	0	0
$481$	21.7980	0.993901
$482$	−19.3485	−0.881299
$483$	−15.7980	−0.718832
$484$	1.00000	0.0454545
$485$	0	0
$486$	−22.0454	−1.00000
$487$	−26.9444	−1.22097	−0.610483	−	0.792029i	$-0.709025\pi$
$487$	−26.9444	−1.22097	−0.610483	+	0.792029i	$0.709025\pi$
$488$	10.0000	0.452679
$489$	−10.2929	−0.465459
$490$	0	0
$491$	39.5959	1.78694	0.893469	−	0.449124i	$-0.148264\pi$
$491$	39.5959	1.78694	0.893469	+	0.449124i	$0.148264\pi$
$492$	8.20204	0.369777
$493$	−4.89898	−0.220639
$494$	21.7980	0.980737
$495$	0	0
$496$	−6.89898	−0.309773
$497$	−3.10102	−0.139100
$498$	−36.0000	−1.61320
$499$	32.2929	1.44563	0.722813	−	0.691043i	$-0.242849\pi$
$499$	32.2929	1.44563	0.722813	+	0.691043i	$0.242849\pi$
$500$	0	0
$501$	−33.7980	−1.50998
$502$	−20.0000	−0.892644
$503$	−9.79796	−0.436869	−0.218435	−	0.975852i	$-0.570095\pi$
$503$	−9.79796	−0.436869	−0.218435	+	0.975852i	$0.570095\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−6.44949	−0.286715
$507$	−26.9444	−1.19664
$508$	0.898979	0.0398858
$509$	13.5959	0.602628	0.301314	−	0.953525i	$-0.402575\pi$
$509$	13.5959	0.602628	0.301314	+	0.953525i	$0.402575\pi$
$510$	0	0
$511$	−2.89898	−0.128243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.2474	−0.981293
$515$	0	0
$516$	−4.89898	−0.215666
$517$	−6.00000	−0.263880
$518$	−4.44949	−0.195499
$519$	48.4949	2.12869
$520$	0	0
$521$	−23.3939	−1.02490	−0.512452	−	0.858716i	$-0.671263\pi$
$521$	−23.3939	−1.02490	−0.512452	+	0.858716i	$0.671263\pi$
$522$	7.34847	0.321634
$523$	−39.5959	−1.73141	−0.865704	−	0.500556i	$-0.833129\pi$
$523$	−39.5959	−1.73141	−0.865704	+	0.500556i	$0.833129\pi$
$524$	16.4495	0.718599
$525$	0	0
$526$	−11.1010	−0.484027
$527$	−13.7980	−0.601049
$528$	2.44949	0.106600
$529$	18.5959	0.808518
$530$	0	0
$531$	14.6969	0.637793
$532$	−4.44949	−0.192910
$533$	16.4041	0.710540
$534$	−7.10102	−0.307291
$535$	0	0
$536$	−7.79796	−0.336821
$537$	−26.6969	−1.15206
$538$	−9.79796	−0.422420
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	7.34847	0.315935	0.157968	−	0.987444i	$-0.449506\pi$
$541$	7.34847	0.315935	0.157968	+	0.987444i	$0.449506\pi$
$542$	6.69694	0.287658
$543$	58.2929	2.50159
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	12.0000	0.513553
$547$	37.3939	1.59885	0.799423	−	0.600768i	$-0.205138\pi$
$547$	37.3939	1.59885	0.799423	+	0.600768i	$0.205138\pi$
$548$	−21.7980	−0.931163
$549$	−30.0000	−1.28037
$550$	0	0
$551$	−10.8990	−0.464312
$552$	−15.7980	−0.672406
$553$	5.34847	0.227440
$554$	9.10102	0.386665
$555$	0	0
$556$	−1.34847	−0.0571878
$557$	21.1010	0.894079	0.447039	−	0.894514i	$-0.352478\pi$
$557$	21.1010	0.894079	0.447039	+	0.894514i	$0.352478\pi$
$558$	20.6969	0.876171
$559$	−9.79796	−0.414410
$560$	0	0
$561$	4.89898	0.206835
$562$	−14.8990	−0.628476
$563$	33.7980	1.42441	0.712207	−	0.701969i	$-0.247696\pi$
$563$	33.7980	1.42441	0.712207	+	0.701969i	$0.247696\pi$
$564$	−14.6969	−0.618853
$565$	0	0
$566$	−20.4949	−0.861465
$567$	9.00000	0.377964
$568$	−3.10102	−0.130116
$569$	−13.5959	−0.569971	−0.284985	−	0.958532i	$-0.591989\pi$
$569$	−13.5959	−0.569971	−0.284985	+	0.958532i	$0.591989\pi$
$570$	0	0
$571$	34.6969	1.45202	0.726011	−	0.687683i	$-0.241372\pi$
$571$	34.6969	1.45202	0.726011	+	0.687683i	$0.241372\pi$
$572$	4.89898	0.204837
$573$	36.0000	1.50392
$574$	−3.34847	−0.139762
$575$	0	0
$576$	3.00000	0.125000
$577$	−17.3485	−0.722226	−0.361113	−	0.932522i	$-0.617603\pi$
$577$	−17.3485	−0.722226	−0.361113	+	0.932522i	$0.617603\pi$
$578$	13.0000	0.540729
$579$	−45.7980	−1.90330
$580$	0	0
$581$	14.6969	0.609732
$582$	−40.2929	−1.67019
$583$	4.44949	0.184279
$584$	−2.89898	−0.119961
$585$	0	0
$586$	−20.4949	−0.846636
$587$	9.55051	0.394192	0.197096	−	0.980384i	$-0.436849\pi$
$587$	9.55051	0.394192	0.197096	+	0.980384i	$0.436849\pi$
$588$	−2.44949	−0.101015
$589$	−30.6969	−1.26485
$590$	0	0
$591$	20.0908	0.826426
$592$	−4.44949	−0.182873
$593$	−5.10102	−0.209474	−0.104737	−	0.994500i	$-0.533400\pi$
$593$	−5.10102	−0.209474	−0.104737	+	0.994500i	$0.533400\pi$
$594$	0	0
$595$	0	0
$596$	−13.1464	−0.538499
$597$	39.1918	1.60402
$598$	−31.5959	−1.29205
$599$	22.6969	0.927372	0.463686	−	0.886000i	$-0.346527\pi$
$599$	22.6969	0.927372	0.463686	+	0.886000i	$0.346527\pi$
$600$	0	0
$601$	−44.2474	−1.80489	−0.902446	−	0.430804i	$-0.858230\pi$
$601$	−44.2474	−1.80489	−0.902446	+	0.430804i	$0.858230\pi$
$602$	2.00000	0.0815139
$603$	23.3939	0.952672
$604$	−8.44949	−0.343805
$605$	0	0
$606$	38.6969	1.57196
$607$	−39.7980	−1.61535	−0.807675	−	0.589628i	$-0.799274\pi$
$607$	−39.7980	−1.61535	−0.807675	+	0.589628i	$0.799274\pi$
$608$	−4.44949	−0.180451
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−29.3939	−1.18915
$612$	6.00000	0.242536
$613$	−32.2929	−1.30430	−0.652148	−	0.758092i	$-0.726132\pi$
$613$	−32.2929	−1.30430	−0.652148	+	0.758092i	$0.726132\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	32.4949	1.30820	0.654098	−	0.756410i	$-0.273049\pi$
$617$	32.4949	1.30820	0.654098	+	0.756410i	$0.273049\pi$
$618$	−7.10102	−0.285645
$619$	−2.20204	−0.0885075	−0.0442538	−	0.999020i	$-0.514091\pi$
$619$	−2.20204	−0.0885075	−0.0442538	+	0.999020i	$0.514091\pi$
$620$	0	0
$621$	0	0
$622$	−0.696938	−0.0279447
$623$	2.89898	0.116145
$624$	12.0000	0.480384
$625$	0	0
$626$	−10.6515	−0.425721
$627$	10.8990	0.435263
$628$	18.6969	0.746089
$629$	−8.89898	−0.354826
$630$	0	0
$631$	26.2020	1.04309	0.521543	−	0.853225i	$-0.325356\pi$
$631$	26.2020	1.04309	0.521543	+	0.853225i	$0.325356\pi$
$632$	5.34847	0.212751
$633$	−15.1918	−0.603821
$634$	−27.1464	−1.07812
$635$	0	0
$636$	10.8990	0.432173
$637$	−4.89898	−0.194105
$638$	−2.44949	−0.0969762
$639$	9.30306	0.368023
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−9.30306	−0.367163
$643$	−10.9444	−0.431604	−0.215802	−	0.976437i	$-0.569237\pi$
$643$	−10.9444	−0.431604	−0.215802	+	0.976437i	$0.569237\pi$
$644$	6.44949	0.254145
$645$	0	0
$646$	−8.89898	−0.350126
$647$	50.0908	1.96927	0.984637	−	0.174616i	$-0.0558685\pi$
$647$	50.0908	1.96927	0.984637	+	0.174616i	$0.0558685\pi$
$648$	9.00000	0.353553
$649$	−4.89898	−0.192302
$650$	0	0
$651$	−16.8990	−0.662323
$652$	4.20204	0.164565
$653$	−26.6515	−1.04295	−0.521477	−	0.853265i	$-0.674619\pi$
$653$	−26.6515	−1.04295	−0.521477	+	0.853265i	$0.674619\pi$
$654$	−20.2020	−0.789962
$655$	0	0
$656$	−3.34847	−0.130736
$657$	8.69694	0.339300
$658$	6.00000	0.233904
$659$	−3.10102	−0.120799	−0.0603993	−	0.998174i	$-0.519237\pi$
$659$	−3.10102	−0.120799	−0.0603993	+	0.998174i	$0.519237\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	28.6969	1.11534
$663$	24.0000	0.932083
$664$	14.6969	0.570352
$665$	0	0
$666$	13.3485	0.517243
$667$	15.7980	0.611699
$668$	13.7980	0.533859
$669$	−60.4949	−2.33887
$670$	0	0
$671$	10.0000	0.386046
$672$	−2.44949	−0.0944911
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−23.7980	−0.916663
$675$	0	0
$676$	11.0000	0.423077
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	24.0000	0.921714
$679$	16.4495	0.631273
$680$	0	0
$681$	−65.3939	−2.50590
$682$	−6.89898	−0.264176
$683$	0.696938	0.0266676	0.0133338	−	0.999911i	$-0.495756\pi$
$683$	0.696938	0.0266676	0.0133338	+	0.999911i	$0.495756\pi$
$684$	13.3485	0.510391
$685$	0	0
$686$	1.00000	0.0381802
$687$	−53.3939	−2.03710
$688$	2.00000	0.0762493
$689$	21.7980	0.830436
$690$	0	0
$691$	−5.30306	−0.201738	−0.100869	−	0.994900i	$-0.532162\pi$
$691$	−5.30306	−0.201738	−0.100869	+	0.994900i	$0.532162\pi$
$692$	−19.7980	−0.752605
$693$	3.00000	0.113961
$694$	−0.202041	−0.00766937
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−6.69694	−0.253665
$698$	−16.6969	−0.631988
$699$	14.6969	0.555889
$700$	0	0
$701$	30.9444	1.16875	0.584377	−	0.811483i	$-0.301339\pi$
$701$	30.9444	1.16875	0.584377	+	0.811483i	$0.301339\pi$
$702$	0	0
$703$	−19.7980	−0.746694
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−33.8434	−1.27371
$707$	−15.7980	−0.594143
$708$	−12.0000	−0.450988
$709$	3.30306	0.124049	0.0620245	−	0.998075i	$-0.480244\pi$
$709$	3.30306	0.124049	0.0620245	+	0.998075i	$0.480244\pi$
$710$	0	0
$711$	−16.0454	−0.601750
$712$	2.89898	0.108644
$713$	44.4949	1.66635
$714$	−4.89898	−0.183340
$715$	0	0
$716$	10.8990	0.407314
$717$	−56.6969	−2.11739
$718$	5.34847	0.199603
$719$	14.2020	0.529647	0.264823	−	0.964297i	$-0.414686\pi$
$719$	14.2020	0.529647	0.264823	+	0.964297i	$0.414686\pi$
$720$	0	0
$721$	2.89898	0.107964
$722$	−0.797959	−0.0296970
$723$	−47.3939	−1.76260
$724$	−23.7980	−0.884444
$725$	0	0
$726$	2.44949	0.0909091
$727$	11.3939	0.422576	0.211288	−	0.977424i	$-0.432234\pi$
$727$	11.3939	0.422576	0.211288	+	0.977424i	$0.432234\pi$
$728$	−4.89898	−0.181568
$729$	−27.0000	−1.00000
$730$	0	0
$731$	4.00000	0.147945
$732$	24.4949	0.905357
$733$	−15.7980	−0.583511	−0.291755	−	0.956493i	$-0.594239\pi$
$733$	−15.7980	−0.583511	−0.291755	+	0.956493i	$0.594239\pi$
$734$	−0.696938	−0.0257245
$735$	0	0
$736$	6.44949	0.237731
$737$	−7.79796	−0.287242
$738$	10.0454	0.369777
$739$	5.79796	0.213281	0.106641	−	0.994298i	$-0.465991\pi$
$739$	5.79796	0.213281	0.106641	+	0.994298i	$0.465991\pi$
$740$	0	0
$741$	53.3939	1.96147
$742$	−4.44949	−0.163346
$743$	−14.6969	−0.539178	−0.269589	−	0.962975i	$-0.586888\pi$
$743$	−14.6969	−0.539178	−0.269589	+	0.962975i	$0.586888\pi$
$744$	−16.8990	−0.619547
$745$	0	0
$746$	31.7980	1.16421
$747$	−44.0908	−1.61320
$748$	−2.00000	−0.0731272
$749$	3.79796	0.138774
$750$	0	0
$751$	22.6969	0.828223	0.414112	−	0.910226i	$-0.364092\pi$
$751$	22.6969	0.828223	0.414112	+	0.910226i	$0.364092\pi$
$752$	6.00000	0.218797
$753$	−48.9898	−1.78529
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	−51.1464	−1.85895	−0.929474	−	0.368887i	$-0.879739\pi$
$757$	−51.1464	−1.85895	−0.929474	+	0.368887i	$0.879739\pi$
$758$	28.0000	1.01701
$759$	−15.7980	−0.573430
$760$	0	0
$761$	39.8434	1.44432	0.722160	−	0.691726i	$-0.243149\pi$
$761$	39.8434	1.44432	0.722160	+	0.691726i	$0.243149\pi$
$762$	2.20204	0.0797715
$763$	8.24745	0.298578
$764$	−14.6969	−0.531717
$765$	0	0
$766$	34.4949	1.24635
$767$	−24.0000	−0.866590
$768$	−2.44949	−0.0883883
$769$	20.2474	0.730142	0.365071	−	0.930980i	$-0.381045\pi$
$769$	20.2474	0.730142	0.365071	+	0.930980i	$0.381045\pi$
$770$	0	0
$771$	−54.4949	−1.96259
$772$	18.6969	0.672918
$773$	−5.39388	−0.194004	−0.0970021	−	0.995284i	$-0.530925\pi$
$773$	−5.39388	−0.194004	−0.0970021	+	0.995284i	$0.530925\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	16.4495	0.590502
$777$	−10.8990	−0.390999
$778$	−11.7980	−0.422977
$779$	−14.8990	−0.533811
$780$	0	0
$781$	−3.10102	−0.110963
$782$	12.8990	0.461267
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	40.2929	1.43720
$787$	9.79796	0.349260	0.174630	−	0.984634i	$-0.444127\pi$
$787$	9.79796	0.349260	0.174630	+	0.984634i	$0.444127\pi$
$788$	−8.20204	−0.292186
$789$	−27.1918	−0.968055
$790$	0	0
$791$	−9.79796	−0.348375
$792$	3.00000	0.106600
$793$	48.9898	1.73968
$794$	−6.69694	−0.237665
$795$	0	0
$796$	−16.0000	−0.567105
$797$	50.2929	1.78146	0.890732	−	0.454529i	$-0.150192\pi$
$797$	50.2929	1.78146	0.890732	+	0.454529i	$0.150192\pi$
$798$	−10.8990	−0.385820
$799$	12.0000	0.424529
$800$	0	0
$801$	−8.69694	−0.307291
$802$	23.5959	0.833201
$803$	−2.89898	−0.102303
$804$	−19.1010	−0.673641
$805$	0	0
$806$	−33.7980	−1.19048
$807$	−24.0000	−0.844840
$808$	−15.7980	−0.555770
$809$	24.6969	0.868298	0.434149	−	0.900841i	$-0.357049\pi$
$809$	24.6969	0.868298	0.434149	+	0.900841i	$0.357049\pi$
$810$	0	0
$811$	−19.5505	−0.686511	−0.343256	−	0.939242i	$-0.611530\pi$
$811$	−19.5505	−0.686511	−0.343256	+	0.939242i	$0.611530\pi$
$812$	2.44949	0.0859602
$813$	16.4041	0.575316
$814$	−4.44949	−0.155955
$815$	0	0
$816$	−4.89898	−0.171499
$817$	8.89898	0.311336
$818$	25.1464	0.879224
$819$	14.6969	0.513553
$820$	0	0
$821$	17.5505	0.612517	0.306259	−	0.951948i	$-0.400923\pi$
$821$	17.5505	0.612517	0.306259	+	0.951948i	$0.400923\pi$
$822$	−53.3939	−1.86233
$823$	−38.5403	−1.34343	−0.671715	−	0.740809i	$-0.734442\pi$
$823$	−38.5403	−1.34343	−0.671715	+	0.740809i	$0.734442\pi$
$824$	2.89898	0.100991
$825$	0	0
$826$	4.89898	0.170457
$827$	21.7980	0.757989	0.378995	−	0.925399i	$-0.376270\pi$
$827$	21.7980	0.757989	0.378995	+	0.925399i	$0.376270\pi$
$828$	−19.3485	−0.672406
$829$	−18.2020	−0.632183	−0.316092	−	0.948729i	$-0.602371\pi$
$829$	−18.2020	−0.632183	−0.316092	+	0.948729i	$0.602371\pi$
$830$	0	0
$831$	22.2929	0.773331
$832$	−4.89898	−0.169842
$833$	2.00000	0.0692959
$834$	−3.30306	−0.114376
$835$	0	0
$836$	−4.44949	−0.153889
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−44.6969	−1.54311	−0.771555	−	0.636163i	$-0.780521\pi$
$839$	−44.6969	−1.54311	−0.771555	+	0.636163i	$0.780521\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	−8.69694	−0.299716
$843$	−36.4949	−1.25695
$844$	6.20204	0.213483
$845$	0	0
$846$	−18.0000	−0.618853
$847$	−1.00000	−0.0343604
$848$	−4.44949	−0.152796
$849$	−50.2020	−1.72293
$850$	0	0
$851$	28.6969	0.983718
$852$	−7.59592	−0.260232
$853$	−4.20204	−0.143875	−0.0719376	−	0.997409i	$-0.522918\pi$
$853$	−4.20204	−0.143875	−0.0719376	+	0.997409i	$0.522918\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	3.79796	0.129812
$857$	−48.6969	−1.66346	−0.831728	−	0.555184i	$-0.812648\pi$
$857$	−48.6969	−1.66346	−0.831728	+	0.555184i	$0.812648\pi$
$858$	12.0000	0.409673
$859$	−10.2020	−0.348089	−0.174045	−	0.984738i	$-0.555684\pi$
$859$	−10.2020	−0.348089	−0.174045	+	0.984738i	$0.555684\pi$
$860$	0	0
$861$	−8.20204	−0.279525
$862$	−24.9444	−0.849609
$863$	−7.34847	−0.250145	−0.125072	−	0.992148i	$-0.539916\pi$
$863$	−7.34847	−0.250145	−0.125072	+	0.992148i	$0.539916\pi$
$864$	0	0
$865$	0	0
$866$	−33.3485	−1.13323
$867$	31.8434	1.08146
$868$	6.89898	0.234167
$869$	5.34847	0.181434
$870$	0	0
$871$	−38.2020	−1.29443
$872$	8.24745	0.279294
$873$	−49.3485	−1.67019
$874$	28.6969	0.970688
$875$	0	0
$876$	−7.10102	−0.239921
$877$	−39.3939	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$877$	−39.3939	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$878$	5.30306	0.178970
$879$	−50.2020	−1.69327
$880$	0	0
$881$	−23.3939	−0.788160	−0.394080	−	0.919076i	$-0.628937\pi$
$881$	−23.3939	−0.788160	−0.394080	+	0.919076i	$0.628937\pi$
$882$	−3.00000	−0.101015
$883$	−14.8990	−0.501391	−0.250695	−	0.968066i	$-0.580659\pi$
$883$	−14.8990	−0.501391	−0.250695	+	0.968066i	$0.580659\pi$
$884$	−9.79796	−0.329541
$885$	0	0
$886$	0.202041	0.00678770
$887$	52.9898	1.77922	0.889612	−	0.456718i	$-0.150975\pi$
$887$	52.9898	1.77922	0.889612	+	0.456718i	$0.150975\pi$
$888$	−10.8990	−0.365746
$889$	−0.898979	−0.0301508
$890$	0	0
$891$	9.00000	0.301511
$892$	24.6969	0.826915
$893$	26.6969	0.893379
$894$	−32.2020	−1.07700
$895$	0	0
$896$	1.00000	0.0334077
$897$	−77.3939	−2.58411
$898$	37.3939	1.24785
$899$	16.8990	0.563613
$900$	0	0
$901$	−8.89898	−0.296468
$902$	−3.34847	−0.111492
$903$	4.89898	0.163028
$904$	−9.79796	−0.325875
$905$	0	0
$906$	−20.6969	−0.687610
$907$	21.1010	0.700648	0.350324	−	0.936629i	$-0.386071\pi$
$907$	21.1010	0.700648	0.350324	+	0.936629i	$0.386071\pi$
$908$	26.6969	0.885969
$909$	47.3939	1.57196
$910$	0	0
$911$	−41.3939	−1.37144	−0.685720	−	0.727865i	$-0.740513\pi$
$911$	−41.3939	−1.37144	−0.685720	+	0.727865i	$0.740513\pi$
$912$	−10.8990	−0.360901
$913$	14.6969	0.486398
$914$	13.5959	0.449713
$915$	0	0
$916$	21.7980	0.720225
$917$	−16.4495	−0.543210
$918$	0	0
$919$	36.9444	1.21868	0.609341	−	0.792908i	$-0.291434\pi$
$919$	36.9444	1.21868	0.609341	+	0.792908i	$0.291434\pi$
$920$	0	0
$921$	−78.3837	−2.58283
$922$	0.696938	0.0229524
$923$	−15.1918	−0.500045
$924$	−2.44949	−0.0805823
$925$	0	0
$926$	14.4495	0.474840
$927$	−8.69694	−0.285645
$928$	2.44949	0.0804084
$929$	−12.2020	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$929$	−12.2020	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$930$	0	0
$931$	4.44949	0.145826
$932$	−6.00000	−0.196537
$933$	−1.70714	−0.0558894
$934$	−17.5505	−0.574270
$935$	0	0
$936$	14.6969	0.480384
$937$	4.69694	0.153442	0.0767211	−	0.997053i	$-0.475555\pi$
$937$	4.69694	0.153442	0.0767211	+	0.997053i	$0.475555\pi$
$938$	7.79796	0.254612
$939$	−26.0908	−0.851442
$940$	0	0
$941$	51.3939	1.67539	0.837696	−	0.546136i	$-0.183902\pi$
$941$	51.3939	1.67539	0.837696	+	0.546136i	$0.183902\pi$
$942$	45.7980	1.49218
$943$	21.5959	0.703260
$944$	4.89898	0.159448
$945$	0	0
$946$	2.00000	0.0650256
$947$	−52.2929	−1.69929	−0.849645	−	0.527355i	$-0.823184\pi$
$947$	−52.2929	−1.69929	−0.849645	+	0.527355i	$0.823184\pi$
$948$	13.1010	0.425501
$949$	−14.2020	−0.461018
$950$	0	0
$951$	−66.4949	−2.15624
$952$	2.00000	0.0648204
$953$	−39.1010	−1.26661	−0.633303	−	0.773904i	$-0.718301\pi$
$953$	−39.1010	−1.26661	−0.633303	+	0.773904i	$0.718301\pi$
$954$	13.3485	0.432173
$955$	0	0
$956$	23.1464	0.748609
$957$	−6.00000	−0.193952
$958$	−19.5959	−0.633115
$959$	21.7980	0.703893
$960$	0	0
$961$	16.5959	0.535352
$962$	−21.7980	−0.702794
$963$	−11.3939	−0.367163
$964$	19.3485	0.623172
$965$	0	0
$966$	15.7980	0.508291
$967$	36.8990	1.18659	0.593296	−	0.804985i	$-0.297827\pi$
$967$	36.8990	1.18659	0.593296	+	0.804985i	$0.297827\pi$
$968$	−1.00000	−0.0321412
$969$	−21.7980	−0.700251
$970$	0	0
$971$	−8.89898	−0.285582	−0.142791	−	0.989753i	$-0.545608\pi$
$971$	−8.89898	−0.285582	−0.142791	+	0.989753i	$0.545608\pi$
$972$	22.0454	0.707107
$973$	1.34847	0.0432299
$974$	26.9444	0.863354
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−40.9898	−1.31138	−0.655690	−	0.755030i	$-0.727622\pi$
$977$	−40.9898	−1.31138	−0.655690	+	0.755030i	$0.727622\pi$
$978$	10.2929	0.329129
$979$	2.89898	0.0926518
$980$	0	0
$981$	−24.7423	−0.789962
$982$	−39.5959	−1.26356
$983$	−14.8990	−0.475204	−0.237602	−	0.971363i	$-0.576361\pi$
$983$	−14.8990	−0.475204	−0.237602	+	0.971363i	$0.576361\pi$
$984$	−8.20204	−0.261472
$985$	0	0
$986$	4.89898	0.156015
$987$	14.6969	0.467809
$988$	−21.7980	−0.693485
$989$	−12.8990	−0.410164
$990$	0	0
$991$	−24.4949	−0.778106	−0.389053	−	0.921215i	$-0.627198\pi$
$991$	−24.4949	−0.778106	−0.389053	+	0.921215i	$0.627198\pi$
$992$	6.89898	0.219043
$993$	70.2929	2.23068
$994$	3.10102	0.0983584
$995$	0	0
$996$	36.0000	1.14070
$997$	23.7980	0.753689	0.376844	−	0.926277i	$-0.377009\pi$
$997$	23.7980	0.753689	0.376844	+	0.926277i	$0.377009\pi$
$998$	−32.2929	−1.02221
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.be.1.1		2
5.2	odd	4		770.2.c.d.309.2	✓	4
5.3	odd	4		770.2.c.d.309.3	yes	4
5.4	even	2		3850.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.c.d.309.2	✓	4	5.2	odd	4
770.2.c.d.309.3	yes	4	5.3	odd	4
3850.2.a.be.1.1		2	1.1	even	1	trivial
3850.2.a.bp.1.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.44949 q^{12} -4.89898 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +4.44949 q^{19} +2.44949 q^{21} +1.00000 q^{22} -6.44949 q^{23} +2.44949 q^{24} +4.89898 q^{26} -1.00000 q^{28} -2.44949 q^{29} -6.89898 q^{31} -1.00000 q^{32} +2.44949 q^{33} -2.00000 q^{34} +3.00000 q^{36} -4.44949 q^{37} -4.44949 q^{38} +12.0000 q^{39} -3.34847 q^{41} -2.44949 q^{42} +2.00000 q^{43} -1.00000 q^{44} +6.44949 q^{46} +6.00000 q^{47} -2.44949 q^{48} +1.00000 q^{49} -4.89898 q^{51} -4.89898 q^{52} -4.44949 q^{53} +1.00000 q^{56} -10.8990 q^{57} +2.44949 q^{58} +4.89898 q^{59} -10.0000 q^{61} +6.89898 q^{62} -3.00000 q^{63} +1.00000 q^{64} -2.44949 q^{66} +7.79796 q^{67} +2.00000 q^{68} +15.7980 q^{69} +3.10102 q^{71} -3.00000 q^{72} +2.89898 q^{73} +4.44949 q^{74} +4.44949 q^{76} +1.00000 q^{77} -12.0000 q^{78} -5.34847 q^{79} -9.00000 q^{81} +3.34847 q^{82} -14.6969 q^{83} +2.44949 q^{84} -2.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -2.89898 q^{89} +4.89898 q^{91} -6.44949 q^{92} +16.8990 q^{93} -6.00000 q^{94} +2.44949 q^{96} -16.4495 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 4 q^{19} + 2 q^{22} - 8 q^{23} - 2 q^{28} - 4 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36} - 4 q^{37} - 4 q^{38} + 24 q^{39} + 8 q^{41} + 4 q^{43} - 2 q^{44} + 8 q^{46} + 12 q^{47} + 2 q^{49} - 4 q^{53} + 2 q^{56} - 12 q^{57} - 20 q^{61} + 4 q^{62} - 6 q^{63} + 2 q^{64} - 4 q^{67} + 4 q^{68} + 12 q^{69} + 16 q^{71} - 6 q^{72} - 4 q^{73} + 4 q^{74} + 4 q^{76} + 2 q^{77} - 24 q^{78} + 4 q^{79} - 18 q^{81} - 8 q^{82} - 4 q^{86} + 12 q^{87} + 2 q^{88} + 4 q^{89} - 8 q^{92} + 24 q^{93} - 12 q^{94} - 28 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 4 * q^19 + 2 * q^22 - 8 * q^23 - 2 * q^28 - 4 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 - 4 * q^37 - 4 * q^38 + 24 * q^39 + 8 * q^41 + 4 * q^43 - 2 * q^44 + 8 * q^46 + 12 * q^47 + 2 * q^49 - 4 * q^53 + 2 * q^56 - 12 * q^57 - 20 * q^61 + 4 * q^62 - 6 * q^63 + 2 * q^64 - 4 * q^67 + 4 * q^68 + 12 * q^69 + 16 * q^71 - 6 * q^72 - 4 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 24 * q^78 + 4 * q^79 - 18 * q^81 - 8 * q^82 - 4 * q^86 + 12 * q^87 + 2 * q^88 + 4 * q^89 - 8 * q^92 + 24 * q^93 - 12 * q^94 - 28 * q^97 - 2 * q^98 - 6 * q^99

Introduction

L-functions

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Database

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