LMFDB - Embedded newform 3850.2.a.bf.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{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
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			$-3.16228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} +3.16228 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} +3.16228 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} -1.00000 q^{11} -3.16228 q^{12} +0.162278 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.837722 q^{17} -7.00000 q^{18} -1.00000 q^{19} +3.16228 q^{21} +1.00000 q^{22} +2.16228 q^{23} +3.16228 q^{24} -0.162278 q^{26} -12.6491 q^{27} -1.00000 q^{28} +3.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} +3.16228 q^{33} +0.837722 q^{34} +7.00000 q^{36} -6.32456 q^{37} +1.00000 q^{38} -0.513167 q^{39} +10.3246 q^{41} -3.16228 q^{42} +0.162278 q^{43} -1.00000 q^{44} -2.16228 q^{46} -4.32456 q^{47} -3.16228 q^{48} +1.00000 q^{49} +2.64911 q^{51} +0.162278 q^{52} -0.837722 q^{53} +12.6491 q^{54} +1.00000 q^{56} +3.16228 q^{57} -3.00000 q^{58} -12.0000 q^{59} +12.3246 q^{61} +7.00000 q^{62} -7.00000 q^{63} +1.00000 q^{64} -3.16228 q^{66} +7.48683 q^{67} -0.837722 q^{68} -6.83772 q^{69} +3.83772 q^{71} -7.00000 q^{72} +4.00000 q^{73} +6.32456 q^{74} -1.00000 q^{76} +1.00000 q^{77} +0.513167 q^{78} -3.16228 q^{79} +19.0000 q^{81} -10.3246 q^{82} +11.6491 q^{83} +3.16228 q^{84} -0.162278 q^{86} -9.48683 q^{87} +1.00000 q^{88} +2.16228 q^{89} -0.162278 q^{91} +2.16228 q^{92} +22.1359 q^{93} +4.32456 q^{94} +3.16228 q^{96} -5.83772 q^{97} -1.00000 q^{98} -7.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 14 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} - 14 q^{18} - 2 q^{19} + 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} + 6 q^{29} - 14 q^{31} - 2 q^{32} + 8 q^{34} + 14 q^{36} + 2 q^{38} - 20 q^{39} + 8 q^{41} - 6 q^{43} - 2 q^{44} + 2 q^{46} + 4 q^{47} + 2 q^{49} - 20 q^{51} - 6 q^{52} - 8 q^{53} + 2 q^{56} - 6 q^{58} - 24 q^{59} + 12 q^{61} + 14 q^{62} - 14 q^{63} + 2 q^{64} - 4 q^{67} - 8 q^{68} - 20 q^{69} + 14 q^{71} - 14 q^{72} + 8 q^{73} - 2 q^{76} + 2 q^{77} + 20 q^{78} + 38 q^{81} - 8 q^{82} - 2 q^{83} + 6 q^{86} + 2 q^{88} - 2 q^{89} + 6 q^{91} - 2 q^{92} - 4 q^{94} - 18 q^{97} - 2 q^{98} - 14 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 - 14 * q^18 - 2 * q^19 + 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 + 6 * q^29 - 14 * q^31 - 2 * q^32 + 8 * q^34 + 14 * q^36 + 2 * q^38 - 20 * q^39 + 8 * q^41 - 6 * q^43 - 2 * q^44 + 2 * q^46 + 4 * q^47 + 2 * q^49 - 20 * q^51 - 6 * q^52 - 8 * q^53 + 2 * q^56 - 6 * q^58 - 24 * q^59 + 12 * q^61 + 14 * q^62 - 14 * q^63 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 20 * q^69 + 14 * q^71 - 14 * q^72 + 8 * q^73 - 2 * q^76 + 2 * q^77 + 20 * q^78 + 38 * q^81 - 8 * q^82 - 2 * q^83 + 6 * q^86 + 2 * q^88 - 2 * q^89 + 6 * q^91 - 2 * q^92 - 4 * q^94 - 18 * q^97 - 2 * q^98 - 14 * 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
$3$	−3.16228	−1.82574	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	−3.16228	−1.82574	−0.912871	+	0.408248i	$0.866140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.16228	1.29099
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	7.00000	2.33333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.16228	−0.912871
$13$	0.162278	0.0450077	0.0225039	−	0.999747i	$-0.492836\pi$
$13$	0.162278	0.0450077	0.0225039	+	0.999747i	$0.492836\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.837722	−0.203178	−0.101589	−	0.994826i	$-0.532393\pi$
$17$	−0.837722	−0.203178	−0.101589	+	0.994826i	$0.532393\pi$
$18$	−7.00000	−1.64992
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	3.16228	0.690066
$22$	1.00000	0.213201
$23$	2.16228	0.450866	0.225433	−	0.974259i	$-0.427620\pi$
$23$	2.16228	0.450866	0.225433	+	0.974259i	$0.427620\pi$
$24$	3.16228	0.645497
$25$	0	0
$26$	−0.162278	−0.0318253
$27$	−12.6491	−2.43432
$28$	−1.00000	−0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	3.16228	0.550482
$34$	0.837722	0.143668
$35$	0	0
$36$	7.00000	1.16667
$37$	−6.32456	−1.03975	−0.519875	−	0.854242i	$-0.674022\pi$
$37$	−6.32456	−1.03975	−0.519875	+	0.854242i	$0.674022\pi$
$38$	1.00000	0.162221
$39$	−0.513167	−0.0821725
$40$	0	0
$41$	10.3246	1.61242	0.806212	−	0.591626i	$-0.201514\pi$
$41$	10.3246	1.61242	0.806212	+	0.591626i	$0.201514\pi$
$42$	−3.16228	−0.487950
$43$	0.162278	0.0247471	0.0123736	−	0.999923i	$-0.496061\pi$
$43$	0.162278	0.0247471	0.0123736	+	0.999923i	$0.496061\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−2.16228	−0.318810
$47$	−4.32456	−0.630801	−0.315401	−	0.948959i	$-0.602139\pi$
$47$	−4.32456	−0.630801	−0.315401	+	0.948959i	$0.602139\pi$
$48$	−3.16228	−0.456435
$49$	1.00000	0.142857
$50$	0	0
$51$	2.64911	0.370950
$52$	0.162278	0.0225039
$53$	−0.837722	−0.115070	−0.0575350	−	0.998343i	$-0.518324\pi$
$53$	−0.837722	−0.115070	−0.0575350	+	0.998343i	$0.518324\pi$
$54$	12.6491	1.72133
$55$	0	0
$56$	1.00000	0.133631
$57$	3.16228	0.418854
$58$	−3.00000	−0.393919
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	12.3246	1.57800	0.788999	−	0.614395i	$-0.210600\pi$
$61$	12.3246	1.57800	0.788999	+	0.614395i	$0.210600\pi$
$62$	7.00000	0.889001
$63$	−7.00000	−0.881917
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.16228	−0.389249
$67$	7.48683	0.914662	0.457331	−	0.889296i	$-0.348805\pi$
$67$	7.48683	0.914662	0.457331	+	0.889296i	$0.348805\pi$
$68$	−0.837722	−0.101589
$69$	−6.83772	−0.823165
$70$	0	0
$71$	3.83772	0.455454	0.227727	−	0.973725i	$-0.426871\pi$
$71$	3.83772	0.455454	0.227727	+	0.973725i	$0.426871\pi$
$72$	−7.00000	−0.824958
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	6.32456	0.735215
$75$	0	0
$76$	−1.00000	−0.114708
$77$	1.00000	0.113961
$78$	0.513167	0.0581047
$79$	−3.16228	−0.355784	−0.177892	−	0.984050i	$-0.556928\pi$
$79$	−3.16228	−0.355784	−0.177892	+	0.984050i	$0.556928\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	−10.3246	−1.14016
$83$	11.6491	1.27866	0.639328	−	0.768934i	$-0.279213\pi$
$83$	11.6491	1.27866	0.639328	+	0.768934i	$0.279213\pi$
$84$	3.16228	0.345033
$85$	0	0
$86$	−0.162278	−0.0174989
$87$	−9.48683	−1.01710
$88$	1.00000	0.106600
$89$	2.16228	0.229201	0.114600	−	0.993412i	$-0.463441\pi$
$89$	2.16228	0.229201	0.114600	+	0.993412i	$0.463441\pi$
$90$	0	0
$91$	−0.162278	−0.0170113
$92$	2.16228	0.225433
$93$	22.1359	2.29539
$94$	4.32456	0.446044
$95$	0	0
$96$	3.16228	0.322749
$97$	−5.83772	−0.592731	−0.296365	−	0.955075i	$-0.595775\pi$
$97$	−5.83772	−0.592731	−0.296365	+	0.955075i	$0.595775\pi$
$98$	−1.00000	−0.101015
$99$	−7.00000	−0.703526
$100$	0	0
$101$	−18.4868	−1.83951	−0.919754	−	0.392495i	$-0.871612\pi$
$101$	−18.4868	−1.83951	−0.919754	+	0.392495i	$0.871612\pi$
$102$	−2.64911	−0.262301
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−0.162278	−0.0159126
$105$	0	0
$106$	0.837722	0.0813668
$107$	−0.486833	−0.0470639	−0.0235320	−	0.999723i	$-0.507491\pi$
$107$	−0.486833	−0.0470639	−0.0235320	+	0.999723i	$0.507491\pi$
$108$	−12.6491	−1.21716
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	−1.00000	−0.0944911
$113$	10.3246	0.971252	0.485626	−	0.874167i	$-0.338592\pi$
$113$	10.3246	0.971252	0.485626	+	0.874167i	$0.338592\pi$
$114$	−3.16228	−0.296174
$115$	0	0
$116$	3.00000	0.278543
$117$	1.13594	0.105018
$118$	12.0000	1.10469
$119$	0.837722	0.0767939
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.3246	−1.11581
$123$	−32.6491	−2.94387
$124$	−7.00000	−0.628619
$125$	0	0
$126$	7.00000	0.623610
$127$	14.3246	1.27110	0.635549	−	0.772060i	$-0.280774\pi$
$127$	14.3246	1.27110	0.635549	+	0.772060i	$0.280774\pi$
$128$	−1.00000	−0.0883883
$129$	−0.513167	−0.0451818
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	3.16228	0.275241
$133$	1.00000	0.0867110
$134$	−7.48683	−0.646764
$135$	0	0
$136$	0.837722	0.0718341
$137$	7.32456	0.625779	0.312889	−	0.949790i	$-0.398703\pi$
$137$	7.32456	0.625779	0.312889	+	0.949790i	$0.398703\pi$
$138$	6.83772	0.582066
$139$	−17.3246	−1.46945	−0.734725	−	0.678365i	$-0.762689\pi$
$139$	−17.3246	−1.46945	−0.734725	+	0.678365i	$0.762689\pi$
$140$	0	0
$141$	13.6754	1.15168
$142$	−3.83772	−0.322055
$143$	−0.162278	−0.0135703
$144$	7.00000	0.583333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	−3.16228	−0.260820
$148$	−6.32456	−0.519875
$149$	−1.67544	−0.137258	−0.0686289	−	0.997642i	$-0.521862\pi$
$149$	−1.67544	−0.137258	−0.0686289	+	0.997642i	$0.521862\pi$
$150$	0	0
$151$	19.1623	1.55940	0.779702	−	0.626151i	$-0.215371\pi$
$151$	19.1623	1.55940	0.779702	+	0.626151i	$0.215371\pi$
$152$	1.00000	0.0811107
$153$	−5.86406	−0.474081
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−0.513167	−0.0410862
$157$	−1.16228	−0.0927599	−0.0463799	−	0.998924i	$-0.514768\pi$
$157$	−1.16228	−0.0927599	−0.0463799	+	0.998924i	$0.514768\pi$
$158$	3.16228	0.251577
$159$	2.64911	0.210088
$160$	0	0
$161$	−2.16228	−0.170411
$162$	−19.0000	−1.49278
$163$	22.9737	1.79944	0.899718	−	0.436471i	$-0.143772\pi$
$163$	22.9737	1.79944	0.899718	+	0.436471i	$0.143772\pi$
$164$	10.3246	0.806212
$165$	0	0
$166$	−11.6491	−0.904146
$167$	3.48683	0.269819	0.134910	−	0.990858i	$-0.456926\pi$
$167$	3.48683	0.269819	0.134910	+	0.990858i	$0.456926\pi$
$168$	−3.16228	−0.243975
$169$	−12.9737	−0.997974
$170$	0	0
$171$	−7.00000	−0.535303
$172$	0.162278	0.0123736
$173$	−18.4868	−1.40553	−0.702764	−	0.711423i	$-0.748051\pi$
$173$	−18.4868	−1.40553	−0.702764	+	0.711423i	$0.748051\pi$
$174$	9.48683	0.719195
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	37.9473	2.85230
$178$	−2.16228	−0.162070
$179$	−18.1359	−1.35554	−0.677772	−	0.735272i	$-0.737054\pi$
$179$	−18.1359	−1.35554	−0.677772	+	0.735272i	$0.737054\pi$
$180$	0	0
$181$	6.32456	0.470100	0.235050	−	0.971983i	$-0.424475\pi$
$181$	6.32456	0.470100	0.235050	+	0.971983i	$0.424475\pi$
$182$	0.162278	0.0120288
$183$	−38.9737	−2.88102
$184$	−2.16228	−0.159405
$185$	0	0
$186$	−22.1359	−1.62309
$187$	0.837722	0.0612603
$188$	−4.32456	−0.315401
$189$	12.6491	0.920087
$190$	0	0
$191$	16.8114	1.21643	0.608215	−	0.793773i	$-0.291886\pi$
$191$	16.8114	1.21643	0.608215	+	0.793773i	$0.291886\pi$
$192$	−3.16228	−0.228218
$193$	5.67544	0.408527	0.204264	−	0.978916i	$-0.434520\pi$
$193$	5.67544	0.408527	0.204264	+	0.978916i	$0.434520\pi$
$194$	5.83772	0.419124
$195$	0	0
$196$	1.00000	0.0714286
$197$	−11.6491	−0.829965	−0.414982	−	0.909829i	$-0.636212\pi$
$197$	−11.6491	−0.829965	−0.414982	+	0.909829i	$0.636212\pi$
$198$	7.00000	0.497468
$199$	−24.2982	−1.72246	−0.861228	−	0.508219i	$-0.830304\pi$
$199$	−24.2982	−1.72246	−0.861228	+	0.508219i	$0.830304\pi$
$200$	0	0
$201$	−23.6754	−1.66994
$202$	18.4868	1.30073
$203$	−3.00000	−0.210559
$204$	2.64911	0.185475
$205$	0	0
$206$	−13.0000	−0.905753
$207$	15.1359	1.05202
$208$	0.162278	0.0112519
$209$	1.00000	0.0691714
$210$	0	0
$211$	4.64911	0.320058	0.160029	−	0.987112i	$-0.448841\pi$
$211$	4.64911	0.320058	0.160029	+	0.987112i	$0.448841\pi$
$212$	−0.837722	−0.0575350
$213$	−12.1359	−0.831541
$214$	0.486833	0.0332792
$215$	0	0
$216$	12.6491	0.860663
$217$	7.00000	0.475191
$218$	−11.0000	−0.745014
$219$	−12.6491	−0.854748
$220$	0	0
$221$	−0.135944	−0.00914456
$222$	−20.0000	−1.34231
$223$	−15.6754	−1.04971	−0.524853	−	0.851193i	$-0.675880\pi$
$223$	−15.6754	−1.04971	−0.524853	+	0.851193i	$0.675880\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−10.3246	−0.686779
$227$	−13.3246	−0.884382	−0.442191	−	0.896921i	$-0.645799\pi$
$227$	−13.3246	−0.884382	−0.442191	+	0.896921i	$0.645799\pi$
$228$	3.16228	0.209427
$229$	22.6491	1.49670	0.748348	−	0.663307i	$-0.230847\pi$
$229$	22.6491	1.49670	0.748348	+	0.663307i	$0.230847\pi$
$230$	0	0
$231$	−3.16228	−0.208063
$232$	−3.00000	−0.196960
$233$	−23.1623	−1.51741	−0.758706	−	0.651434i	$-0.774168\pi$
$233$	−23.1623	−1.51741	−0.758706	+	0.651434i	$0.774168\pi$
$234$	−1.13594	−0.0742590
$235$	0	0
$236$	−12.0000	−0.781133
$237$	10.0000	0.649570
$238$	−0.837722	−0.0543015
$239$	−12.9737	−0.839197	−0.419598	−	0.907710i	$-0.637829\pi$
$239$	−12.9737	−0.839197	−0.419598	+	0.907710i	$0.637829\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−1.00000	−0.0642824
$243$	−22.1359	−1.42002
$244$	12.3246	0.788999
$245$	0	0
$246$	32.6491	2.08163
$247$	−0.162278	−0.0103255
$248$	7.00000	0.444500
$249$	−36.8377	−2.33450
$250$	0	0
$251$	−8.51317	−0.537346	−0.268673	−	0.963231i	$-0.586585\pi$
$251$	−8.51317	−0.537346	−0.268673	+	0.963231i	$0.586585\pi$
$252$	−7.00000	−0.440959
$253$	−2.16228	−0.135941
$254$	−14.3246	−0.898803
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.1623	−0.883419	−0.441709	−	0.897158i	$-0.645628\pi$
$257$	−14.1623	−0.883419	−0.441709	+	0.897158i	$0.645628\pi$
$258$	0.513167	0.0319484
$259$	6.32456	0.392989
$260$	0	0
$261$	21.0000	1.29987
$262$	9.00000	0.556022
$263$	−23.1623	−1.42825	−0.714124	−	0.700020i	$-0.753175\pi$
$263$	−23.1623	−1.42825	−0.714124	+	0.700020i	$0.753175\pi$
$264$	−3.16228	−0.194625
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	−6.83772	−0.418462
$268$	7.48683	0.457331
$269$	5.16228	0.314750	0.157375	−	0.987539i	$-0.449697\pi$
$269$	5.16228	0.314750	0.157375	+	0.987539i	$0.449697\pi$
$270$	0	0
$271$	−22.1359	−1.34466	−0.672331	−	0.740250i	$-0.734707\pi$
$271$	−22.1359	−1.34466	−0.672331	+	0.740250i	$0.734707\pi$
$272$	−0.837722	−0.0507944
$273$	0.513167	0.0310583
$274$	−7.32456	−0.442493
$275$	0	0
$276$	−6.83772	−0.411583
$277$	−25.2982	−1.52002	−0.760011	−	0.649910i	$-0.774807\pi$
$277$	−25.2982	−1.52002	−0.760011	+	0.649910i	$0.774807\pi$
$278$	17.3246	1.03906
$279$	−49.0000	−2.93355
$280$	0	0
$281$	−12.1359	−0.723970	−0.361985	−	0.932184i	$-0.617901\pi$
$281$	−12.1359	−0.723970	−0.361985	+	0.932184i	$0.617901\pi$
$282$	−13.6754	−0.814361
$283$	−0.324555	−0.0192928	−0.00964641	−	0.999953i	$-0.503071\pi$
$283$	−0.324555	−0.0192928	−0.00964641	+	0.999953i	$0.503071\pi$
$284$	3.83772	0.227727
$285$	0	0
$286$	0.162278	0.00959568
$287$	−10.3246	−0.609439
$288$	−7.00000	−0.412479
$289$	−16.2982	−0.958719
$290$	0	0
$291$	18.4605	1.08217
$292$	4.00000	0.234082
$293$	4.32456	0.252643	0.126322	−	0.991989i	$-0.459683\pi$
$293$	4.32456	0.252643	0.126322	+	0.991989i	$0.459683\pi$
$294$	3.16228	0.184428
$295$	0	0
$296$	6.32456	0.367607
$297$	12.6491	0.733976
$298$	1.67544	0.0970559
$299$	0.350889	0.0202925
$300$	0	0
$301$	−0.162278	−0.00935353
$302$	−19.1623	−1.10267
$303$	58.4605	3.35847
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	5.86406	0.335226
$307$	−25.2982	−1.44385	−0.721923	−	0.691974i	$-0.756741\pi$
$307$	−25.2982	−1.44385	−0.721923	+	0.691974i	$0.756741\pi$
$308$	1.00000	0.0569803
$309$	−41.1096	−2.33864
$310$	0	0
$311$	−9.97367	−0.565555	−0.282777	−	0.959186i	$-0.591256\pi$
$311$	−9.97367	−0.565555	−0.282777	+	0.959186i	$0.591256\pi$
$312$	0.513167	0.0290524
$313$	14.3246	0.809672	0.404836	−	0.914389i	$-0.367329\pi$
$313$	14.3246	0.809672	0.404836	+	0.914389i	$0.367329\pi$
$314$	1.16228	0.0655911
$315$	0	0
$316$	−3.16228	−0.177892
$317$	15.4868	0.869827	0.434914	−	0.900472i	$-0.356779\pi$
$317$	15.4868	0.869827	0.434914	+	0.900472i	$0.356779\pi$
$318$	−2.64911	−0.148555
$319$	−3.00000	−0.167968
$320$	0	0
$321$	1.53950	0.0859266
$322$	2.16228	0.120499
$323$	0.837722	0.0466121
$324$	19.0000	1.05556
$325$	0	0
$326$	−22.9737	−1.27239
$327$	−34.7851	−1.92362
$328$	−10.3246	−0.570078
$329$	4.32456	0.238420
$330$	0	0
$331$	6.32456	0.347629	0.173814	−	0.984778i	$-0.444391\pi$
$331$	6.32456	0.347629	0.173814	+	0.984778i	$0.444391\pi$
$332$	11.6491	0.639328
$333$	−44.2719	−2.42608
$334$	−3.48683	−0.190791
$335$	0	0
$336$	3.16228	0.172516
$337$	1.48683	0.0809930	0.0404965	−	0.999180i	$-0.487106\pi$
$337$	1.48683	0.0809930	0.0404965	+	0.999180i	$0.487106\pi$
$338$	12.9737	0.705674
$339$	−32.6491	−1.77326
$340$	0	0
$341$	7.00000	0.379071
$342$	7.00000	0.378517
$343$	−1.00000	−0.0539949
$344$	−0.162278	−0.00874943
$345$	0	0
$346$	18.4868	0.993858
$347$	−30.9737	−1.66275	−0.831377	−	0.555709i	$-0.812447\pi$
$347$	−30.9737	−1.66275	−0.831377	+	0.555709i	$0.812447\pi$
$348$	−9.48683	−0.508548
$349$	−18.1623	−0.972204	−0.486102	−	0.873902i	$-0.661582\pi$
$349$	−18.1623	−0.972204	−0.486102	+	0.873902i	$0.661582\pi$
$350$	0	0
$351$	−2.05267	−0.109563
$352$	1.00000	0.0533002
$353$	−11.5132	−0.612784	−0.306392	−	0.951905i	$-0.599122\pi$
$353$	−11.5132	−0.612784	−0.306392	+	0.951905i	$0.599122\pi$
$354$	−37.9473	−2.01688
$355$	0	0
$356$	2.16228	0.114600
$357$	−2.64911	−0.140206
$358$	18.1359	0.958514
$359$	−8.64911	−0.456483	−0.228241	−	0.973605i	$-0.573298\pi$
$359$	−8.64911	−0.456483	−0.228241	+	0.973605i	$0.573298\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−6.32456	−0.332411
$363$	−3.16228	−0.165977
$364$	−0.162278	−0.00850566
$365$	0	0
$366$	38.9737	2.03719
$367$	23.3246	1.21753	0.608766	−	0.793350i	$-0.291665\pi$
$367$	23.3246	1.21753	0.608766	+	0.793350i	$0.291665\pi$
$368$	2.16228	0.112717
$369$	72.2719	3.76232
$370$	0	0
$371$	0.837722	0.0434924
$372$	22.1359	1.14770
$373$	26.3246	1.36303	0.681517	−	0.731802i	$-0.261321\pi$
$373$	26.3246	1.36303	0.681517	+	0.731802i	$0.261321\pi$
$374$	−0.837722	−0.0433176
$375$	0	0
$376$	4.32456	0.223022
$377$	0.486833	0.0250732
$378$	−12.6491	−0.650600
$379$	9.81139	0.503977	0.251989	−	0.967730i	$-0.418915\pi$
$379$	9.81139	0.503977	0.251989	+	0.967730i	$0.418915\pi$
$380$	0	0
$381$	−45.2982	−2.32070
$382$	−16.8114	−0.860145
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	3.16228	0.161374
$385$	0	0
$386$	−5.67544	−0.288873
$387$	1.13594	0.0577433
$388$	−5.83772	−0.296365
$389$	−6.83772	−0.346686	−0.173343	−	0.984861i	$-0.555457\pi$
$389$	−6.83772	−0.346686	−0.173343	+	0.984861i	$0.555457\pi$
$390$	0	0
$391$	−1.81139	−0.0916058
$392$	−1.00000	−0.0505076
$393$	28.4605	1.43564
$394$	11.6491	0.586874
$395$	0	0
$396$	−7.00000	−0.351763
$397$	20.3246	1.02006	0.510030	−	0.860157i	$-0.329634\pi$
$397$	20.3246	1.02006	0.510030	+	0.860157i	$0.329634\pi$
$398$	24.2982	1.21796
$399$	−3.16228	−0.158312
$400$	0	0
$401$	15.9737	0.797687	0.398843	−	0.917019i	$-0.369412\pi$
$401$	15.9737	0.797687	0.398843	+	0.917019i	$0.369412\pi$
$402$	23.6754	1.18082
$403$	−1.13594	−0.0565854
$404$	−18.4868	−0.919754
$405$	0	0
$406$	3.00000	0.148888
$407$	6.32456	0.313497
$408$	−2.64911	−0.131151
$409$	24.4605	1.20949	0.604747	−	0.796418i	$-0.293274\pi$
$409$	24.4605	1.20949	0.604747	+	0.796418i	$0.293274\pi$
$410$	0	0
$411$	−23.1623	−1.14251
$412$	13.0000	0.640464
$413$	12.0000	0.590481
$414$	−15.1359	−0.743891
$415$	0	0
$416$	−0.162278	−0.00795632
$417$	54.7851	2.68284
$418$	−1.00000	−0.0489116
$419$	−33.6228	−1.64258	−0.821290	−	0.570511i	$-0.806745\pi$
$419$	−33.6228	−1.64258	−0.821290	+	0.570511i	$0.806745\pi$
$420$	0	0
$421$	−34.1359	−1.66368	−0.831842	−	0.555012i	$-0.812713\pi$
$421$	−34.1359	−1.66368	−0.831842	+	0.555012i	$0.812713\pi$
$422$	−4.64911	−0.226315
$423$	−30.2719	−1.47187
$424$	0.837722	0.0406834
$425$	0	0
$426$	12.1359	0.587988
$427$	−12.3246	−0.596427
$428$	−0.486833	−0.0235320
$429$	0.513167	0.0247759
$430$	0	0
$431$	27.6228	1.33054	0.665271	−	0.746602i	$-0.268316\pi$
$431$	27.6228	1.33054	0.665271	+	0.746602i	$0.268316\pi$
$432$	−12.6491	−0.608581
$433$	30.1623	1.44951	0.724753	−	0.689008i	$-0.241954\pi$
$433$	30.1623	1.44951	0.724753	+	0.689008i	$0.241954\pi$
$434$	−7.00000	−0.336011
$435$	0	0
$436$	11.0000	0.526804
$437$	−2.16228	−0.103436
$438$	12.6491	0.604398
$439$	−5.67544	−0.270874	−0.135437	−	0.990786i	$-0.543244\pi$
$439$	−5.67544	−0.270874	−0.135437	+	0.990786i	$0.543244\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0.135944	0.00646618
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	20.0000	0.949158
$445$	0	0
$446$	15.6754	0.742254
$447$	5.29822	0.250597
$448$	−1.00000	−0.0472456
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−10.3246	−0.486164
$452$	10.3246	0.485626
$453$	−60.5964	−2.84707
$454$	13.3246	0.625352
$455$	0	0
$456$	−3.16228	−0.148087
$457$	−14.8377	−0.694079	−0.347040	−	0.937850i	$-0.612813\pi$
$457$	−14.8377	−0.694079	−0.347040	+	0.937850i	$0.612813\pi$
$458$	−22.6491	−1.05832
$459$	10.5964	0.494600
$460$	0	0
$461$	−13.6754	−0.636929	−0.318464	−	0.947935i	$-0.603167\pi$
$461$	−13.6754	−0.636929	−0.318464	+	0.947935i	$0.603167\pi$
$462$	3.16228	0.147122
$463$	6.16228	0.286385	0.143193	−	0.989695i	$-0.454263\pi$
$463$	6.16228	0.286385	0.143193	+	0.989695i	$0.454263\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	23.1623	1.07297
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	1.13594	0.0525090
$469$	−7.48683	−0.345710
$470$	0	0
$471$	3.67544	0.169356
$472$	12.0000	0.552345
$473$	−0.162278	−0.00746153
$474$	−10.0000	−0.459315
$475$	0	0
$476$	0.837722	0.0383969
$477$	−5.86406	−0.268497
$478$	12.9737	0.593402
$479$	12.9737	0.592782	0.296391	−	0.955067i	$-0.404217\pi$
$479$	12.9737	0.592782	0.296391	+	0.955067i	$0.404217\pi$
$480$	0	0
$481$	−1.02633	−0.0467968
$482$	−2.00000	−0.0910975
$483$	6.83772	0.311127
$484$	1.00000	0.0454545
$485$	0	0
$486$	22.1359	1.00411
$487$	−35.1359	−1.59216	−0.796081	−	0.605190i	$-0.793097\pi$
$487$	−35.1359	−1.59216	−0.796081	+	0.605190i	$0.793097\pi$
$488$	−12.3246	−0.557906
$489$	−72.6491	−3.28531
$490$	0	0
$491$	−33.1359	−1.49540	−0.747702	−	0.664034i	$-0.768843\pi$
$491$	−33.1359	−1.49540	−0.747702	+	0.664034i	$0.768843\pi$
$492$	−32.6491	−1.47194
$493$	−2.51317	−0.113187
$494$	0.162278	0.00730122
$495$	0	0
$496$	−7.00000	−0.314309
$497$	−3.83772	−0.172145
$498$	36.8377	1.65074
$499$	0.324555	0.0145291	0.00726455	−	0.999974i	$-0.497688\pi$
$499$	0.324555	0.0145291	0.00726455	+	0.999974i	$0.497688\pi$
$500$	0	0
$501$	−11.0263	−0.492620
$502$	8.51317	0.379961
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	7.00000	0.311805
$505$	0	0
$506$	2.16228	0.0961250
$507$	41.0263	1.82204
$508$	14.3246	0.635549
$509$	−8.51317	−0.377339	−0.188670	−	0.982041i	$-0.560418\pi$
$509$	−8.51317	−0.377339	−0.188670	+	0.982041i	$0.560418\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	12.6491	0.558472
$514$	14.1623	0.624671
$515$	0	0
$516$	−0.513167	−0.0225909
$517$	4.32456	0.190194
$518$	−6.32456	−0.277885
$519$	58.4605	2.56613
$520$	0	0
$521$	−10.8114	−0.473656	−0.236828	−	0.971552i	$-0.576108\pi$
$521$	−10.8114	−0.473656	−0.236828	+	0.971552i	$0.576108\pi$
$522$	−21.0000	−0.919145
$523$	−31.6491	−1.38392	−0.691959	−	0.721936i	$-0.743252\pi$
$523$	−31.6491	−1.38392	−0.691959	+	0.721936i	$0.743252\pi$
$524$	−9.00000	−0.393167
$525$	0	0
$526$	23.1623	1.00992
$527$	5.86406	0.255442
$528$	3.16228	0.137620
$529$	−18.3246	−0.796720
$530$	0	0
$531$	−84.0000	−3.64529
$532$	1.00000	0.0433555
$533$	1.67544	0.0725716
$534$	6.83772	0.295897
$535$	0	0
$536$	−7.48683	−0.323382
$537$	57.3509	2.47487
$538$	−5.16228	−0.222562
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−3.64911	−0.156888	−0.0784438	−	0.996919i	$-0.524995\pi$
$541$	−3.64911	−0.156888	−0.0784438	+	0.996919i	$0.524995\pi$
$542$	22.1359	0.950820
$543$	−20.0000	−0.858282
$544$	0.837722	0.0359170
$545$	0	0
$546$	−0.513167	−0.0219615
$547$	−23.8377	−1.01923	−0.509614	−	0.860403i	$-0.670212\pi$
$547$	−23.8377	−1.01923	−0.509614	+	0.860403i	$0.670212\pi$
$548$	7.32456	0.312889
$549$	86.2719	3.68199
$550$	0	0
$551$	−3.00000	−0.127804
$552$	6.83772	0.291033
$553$	3.16228	0.134474
$554$	25.2982	1.07482
$555$	0	0
$556$	−17.3246	−0.734725
$557$	−4.67544	−0.198105	−0.0990525	−	0.995082i	$-0.531581\pi$
$557$	−4.67544	−0.198105	−0.0990525	+	0.995082i	$0.531581\pi$
$558$	49.0000	2.07434
$559$	0.0263340	0.00111381
$560$	0	0
$561$	−2.64911	−0.111846
$562$	12.1359	0.511924
$563$	24.9737	1.05251	0.526257	−	0.850325i	$-0.323595\pi$
$563$	24.9737	1.05251	0.526257	+	0.850325i	$0.323595\pi$
$564$	13.6754	0.575840
$565$	0	0
$566$	0.324555	0.0136421
$567$	−19.0000	−0.797925
$568$	−3.83772	−0.161027
$569$	29.1623	1.22255	0.611273	−	0.791420i	$-0.290658\pi$
$569$	29.1623	1.22255	0.611273	+	0.791420i	$0.290658\pi$
$570$	0	0
$571$	−32.8114	−1.37311	−0.686557	−	0.727076i	$-0.740879\pi$
$571$	−32.8114	−1.37311	−0.686557	+	0.727076i	$0.740879\pi$
$572$	−0.162278	−0.00678517
$573$	−53.1623	−2.22089
$574$	10.3246	0.430939
$575$	0	0
$576$	7.00000	0.291667
$577$	−17.6228	−0.733646	−0.366823	−	0.930291i	$-0.619554\pi$
$577$	−17.6228	−0.733646	−0.366823	+	0.930291i	$0.619554\pi$
$578$	16.2982	0.677917
$579$	−17.9473	−0.745866
$580$	0	0
$581$	−11.6491	−0.483287
$582$	−18.4605	−0.765212
$583$	0.837722	0.0346949
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−4.32456	−0.178646
$587$	−30.9737	−1.27842	−0.639210	−	0.769032i	$-0.720739\pi$
$587$	−30.9737	−1.27842	−0.639210	+	0.769032i	$0.720739\pi$
$588$	−3.16228	−0.130410
$589$	7.00000	0.288430
$590$	0	0
$591$	36.8377	1.51530
$592$	−6.32456	−0.259938
$593$	32.7851	1.34632	0.673160	−	0.739496i	$-0.264936\pi$
$593$	32.7851	1.34632	0.673160	+	0.739496i	$0.264936\pi$
$594$	−12.6491	−0.518999
$595$	0	0
$596$	−1.67544	−0.0686289
$597$	76.8377	3.14476
$598$	−0.350889	−0.0143489
$599$	42.9737	1.75586	0.877928	−	0.478792i	$-0.158925\pi$
$599$	42.9737	1.75586	0.877928	+	0.478792i	$0.158925\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0.162278	0.00661394
$603$	52.4078	2.13421
$604$	19.1623	0.779702
$605$	0	0
$606$	−58.4605	−2.37480
$607$	−39.9473	−1.62141	−0.810706	−	0.585453i	$-0.800917\pi$
$607$	−39.9473	−1.62141	−0.810706	+	0.585453i	$0.800917\pi$
$608$	1.00000	0.0405554
$609$	9.48683	0.384426
$610$	0	0
$611$	−0.701779	−0.0283909
$612$	−5.86406	−0.237040
$613$	−16.6491	−0.672451	−0.336226	−	0.941781i	$-0.609150\pi$
$613$	−16.6491	−0.672451	−0.336226	+	0.941781i	$0.609150\pi$
$614$	25.2982	1.02095
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	5.64911	0.227425	0.113712	−	0.993514i	$-0.463726\pi$
$617$	5.64911	0.227425	0.113712	+	0.993514i	$0.463726\pi$
$618$	41.1096	1.65367
$619$	33.8114	1.35899	0.679497	−	0.733678i	$-0.262198\pi$
$619$	33.8114	1.35899	0.679497	+	0.733678i	$0.262198\pi$
$620$	0	0
$621$	−27.3509	−1.09755
$622$	9.97367	0.399908
$623$	−2.16228	−0.0866298
$624$	−0.513167	−0.0205431
$625$	0	0
$626$	−14.3246	−0.572524
$627$	−3.16228	−0.126289
$628$	−1.16228	−0.0463799
$629$	5.29822	0.211254
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	3.16228	0.125789
$633$	−14.7018	−0.584343
$634$	−15.4868	−0.615061
$635$	0	0
$636$	2.64911	0.105044
$637$	0.162278	0.00642967
$638$	3.00000	0.118771
$639$	26.8641	1.06273
$640$	0	0
$641$	46.9473	1.85431	0.927154	−	0.374680i	$-0.122248\pi$
$641$	46.9473	1.85431	0.927154	+	0.374680i	$0.122248\pi$
$642$	−1.53950	−0.0607593
$643$	48.7851	1.92389	0.961947	−	0.273235i	$-0.0880936\pi$
$643$	48.7851	1.92389	0.961947	+	0.273235i	$0.0880936\pi$
$644$	−2.16228	−0.0852057
$645$	0	0
$646$	−0.837722	−0.0329597
$647$	24.9737	0.981816	0.490908	−	0.871211i	$-0.336665\pi$
$647$	24.9737	0.981816	0.490908	+	0.871211i	$0.336665\pi$
$648$	−19.0000	−0.746390
$649$	12.0000	0.471041
$650$	0	0
$651$	−22.1359	−0.867576
$652$	22.9737	0.899718
$653$	−36.8377	−1.44157	−0.720786	−	0.693158i	$-0.756219\pi$
$653$	−36.8377	−1.44157	−0.720786	+	0.693158i	$0.756219\pi$
$654$	34.7851	1.36020
$655$	0	0
$656$	10.3246	0.403106
$657$	28.0000	1.09238
$658$	−4.32456	−0.168589
$659$	25.4605	0.991800	0.495900	−	0.868380i	$-0.334838\pi$
$659$	25.4605	0.991800	0.495900	+	0.868380i	$0.334838\pi$
$660$	0	0
$661$	−3.16228	−0.122998	−0.0614992	−	0.998107i	$-0.519588\pi$
$661$	−3.16228	−0.122998	−0.0614992	+	0.998107i	$0.519588\pi$
$662$	−6.32456	−0.245811
$663$	0.429891	0.0166956
$664$	−11.6491	−0.452073
$665$	0	0
$666$	44.2719	1.71550
$667$	6.48683	0.251171
$668$	3.48683	0.134910
$669$	49.5701	1.91649
$670$	0	0
$671$	−12.3246	−0.475784
$672$	−3.16228	−0.121988
$673$	24.6491	0.950153	0.475077	−	0.879944i	$-0.342420\pi$
$673$	24.6491	0.950153	0.475077	+	0.879944i	$0.342420\pi$
$674$	−1.48683	−0.0572707
$675$	0	0
$676$	−12.9737	−0.498987
$677$	−21.1359	−0.812320	−0.406160	−	0.913802i	$-0.633132\pi$
$677$	−21.1359	−0.812320	−0.406160	+	0.913802i	$0.633132\pi$
$678$	32.6491	1.25388
$679$	5.83772	0.224031
$680$	0	0
$681$	42.1359	1.61465
$682$	−7.00000	−0.268044
$683$	−33.4868	−1.28134	−0.640669	−	0.767817i	$-0.721343\pi$
$683$	−33.4868	−1.28134	−0.640669	+	0.767817i	$0.721343\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	1.00000	0.0381802
$687$	−71.6228	−2.73258
$688$	0.162278	0.00618678
$689$	−0.135944	−0.00517904
$690$	0	0
$691$	42.3246	1.61010	0.805051	−	0.593206i	$-0.202138\pi$
$691$	42.3246	1.61010	0.805051	+	0.593206i	$0.202138\pi$
$692$	−18.4868	−0.702764
$693$	7.00000	0.265908
$694$	30.9737	1.17574
$695$	0	0
$696$	9.48683	0.359597
$697$	−8.64911	−0.327608
$698$	18.1623	0.687452
$699$	73.2456	2.77040
$700$	0	0
$701$	−17.6491	−0.666598	−0.333299	−	0.942821i	$-0.608162\pi$
$701$	−17.6491	−0.666598	−0.333299	+	0.942821i	$0.608162\pi$
$702$	2.05267	0.0774730
$703$	6.32456	0.238535
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	11.5132	0.433304
$707$	18.4868	0.695269
$708$	37.9473	1.42615
$709$	1.16228	0.0436503	0.0218251	−	0.999762i	$-0.493052\pi$
$709$	1.16228	0.0436503	0.0218251	+	0.999762i	$0.493052\pi$
$710$	0	0
$711$	−22.1359	−0.830163
$712$	−2.16228	−0.0810348
$713$	−15.1359	−0.566846
$714$	2.64911	0.0991405
$715$	0	0
$716$	−18.1359	−0.677772
$717$	41.0263	1.53216
$718$	8.64911	0.322782
$719$	12.9737	0.483836	0.241918	−	0.970297i	$-0.422223\pi$
$719$	12.9737	0.483836	0.241918	+	0.970297i	$0.422223\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	18.0000	0.669891
$723$	−6.32456	−0.235213
$724$	6.32456	0.235050
$725$	0	0
$726$	3.16228	0.117363
$727$	−7.64911	−0.283690	−0.141845	−	0.989889i	$-0.545303\pi$
$727$	−7.64911	−0.283690	−0.141845	+	0.989889i	$0.545303\pi$
$728$	0.162278	0.00601441
$729$	13.0000	0.481481
$730$	0	0
$731$	−0.135944	−0.00502806
$732$	−38.9737	−1.44051
$733$	6.16228	0.227609	0.113804	−	0.993503i	$-0.463696\pi$
$733$	6.16228	0.227609	0.113804	+	0.993503i	$0.463696\pi$
$734$	−23.3246	−0.860925
$735$	0	0
$736$	−2.16228	−0.0797026
$737$	−7.48683	−0.275781
$738$	−72.2719	−2.66036
$739$	−43.6228	−1.60469	−0.802345	−	0.596861i	$-0.796414\pi$
$739$	−43.6228	−1.60469	−0.802345	+	0.596861i	$0.796414\pi$
$740$	0	0
$741$	0.513167	0.0188517
$742$	−0.837722	−0.0307538
$743$	−17.1623	−0.629623	−0.314811	−	0.949154i	$-0.601941\pi$
$743$	−17.1623	−0.629623	−0.314811	+	0.949154i	$0.601941\pi$
$744$	−22.1359	−0.811543
$745$	0	0
$746$	−26.3246	−0.963810
$747$	81.5438	2.98353
$748$	0.837722	0.0306302
$749$	0.486833	0.0177885
$750$	0	0
$751$	32.4868	1.18546	0.592731	−	0.805401i	$-0.298050\pi$
$751$	32.4868	1.18546	0.592731	+	0.805401i	$0.298050\pi$
$752$	−4.32456	−0.157700
$753$	26.9210	0.981056
$754$	−0.486833	−0.0177294
$755$	0	0
$756$	12.6491	0.460044
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−9.81139	−0.356366
$759$	6.83772	0.248194
$760$	0	0
$761$	−16.4605	−0.596693	−0.298346	−	0.954458i	$-0.596435\pi$
$761$	−16.4605	−0.596693	−0.298346	+	0.954458i	$0.596435\pi$
$762$	45.2982	1.64098
$763$	−11.0000	−0.398227
$764$	16.8114	0.608215
$765$	0	0
$766$	15.0000	0.541972
$767$	−1.94733	−0.0703141
$768$	−3.16228	−0.114109
$769$	2.83772	0.102331	0.0511654	−	0.998690i	$-0.483706\pi$
$769$	2.83772	0.102331	0.0511654	+	0.998690i	$0.483706\pi$
$770$	0	0
$771$	44.7851	1.61289
$772$	5.67544	0.204264
$773$	−53.2982	−1.91700	−0.958502	−	0.285086i	$-0.907978\pi$
$773$	−53.2982	−1.91700	−0.958502	+	0.285086i	$0.907978\pi$
$774$	−1.13594	−0.0408307
$775$	0	0
$776$	5.83772	0.209562
$777$	−20.0000	−0.717496
$778$	6.83772	0.245144
$779$	−10.3246	−0.369916
$780$	0	0
$781$	−3.83772	−0.137325
$782$	1.81139	0.0647751
$783$	−37.9473	−1.35613
$784$	1.00000	0.0357143
$785$	0	0
$786$	−28.4605	−1.01515
$787$	−53.6228	−1.91145	−0.955723	−	0.294269i	$-0.904924\pi$
$787$	−53.6228	−1.91145	−0.955723	+	0.294269i	$0.904924\pi$
$788$	−11.6491	−0.414982
$789$	73.2456	2.60761
$790$	0	0
$791$	−10.3246	−0.367099
$792$	7.00000	0.248734
$793$	2.00000	0.0710221
$794$	−20.3246	−0.721291
$795$	0	0
$796$	−24.2982	−0.861228
$797$	19.8114	0.701755	0.350878	−	0.936421i	$-0.385883\pi$
$797$	19.8114	0.701755	0.350878	+	0.936421i	$0.385883\pi$
$798$	3.16228	0.111943
$799$	3.62278	0.128165
$800$	0	0
$801$	15.1359	0.534802
$802$	−15.9737	−0.564050
$803$	−4.00000	−0.141157
$804$	−23.6754	−0.834969
$805$	0	0
$806$	1.13594	0.0400119
$807$	−16.3246	−0.574652
$808$	18.4868	0.650365
$809$	−9.48683	−0.333539	−0.166770	−	0.985996i	$-0.553334\pi$
$809$	−9.48683	−0.333539	−0.166770	+	0.985996i	$0.553334\pi$
$810$	0	0
$811$	−3.02633	−0.106269	−0.0531345	−	0.998587i	$-0.516921\pi$
$811$	−3.02633	−0.106269	−0.0531345	+	0.998587i	$0.516921\pi$
$812$	−3.00000	−0.105279
$813$	70.0000	2.45501
$814$	−6.32456	−0.221676
$815$	0	0
$816$	2.64911	0.0927374
$817$	−0.162278	−0.00567738
$818$	−24.4605	−0.855241
$819$	−1.13594	−0.0396931
$820$	0	0
$821$	−8.02633	−0.280121	−0.140060	−	0.990143i	$-0.544730\pi$
$821$	−8.02633	−0.280121	−0.140060	+	0.990143i	$0.544730\pi$
$822$	23.1623	0.807877
$823$	48.6491	1.69580	0.847901	−	0.530155i	$-0.177866\pi$
$823$	48.6491	1.69580	0.847901	+	0.530155i	$0.177866\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−46.1096	−1.60339	−0.801694	−	0.597735i	$-0.796068\pi$
$827$	−46.1096	−1.60339	−0.801694	+	0.597735i	$0.796068\pi$
$828$	15.1359	0.526010
$829$	17.3509	0.602621	0.301311	−	0.953526i	$-0.402576\pi$
$829$	17.3509	0.602621	0.301311	+	0.953526i	$0.402576\pi$
$830$	0	0
$831$	80.0000	2.77517
$832$	0.162278	0.00562597
$833$	−0.837722	−0.0290254
$834$	−54.7851	−1.89705
$835$	0	0
$836$	1.00000	0.0345857
$837$	88.5438	3.06052
$838$	33.6228	1.16148
$839$	11.0263	0.380671	0.190336	−	0.981719i	$-0.439042\pi$
$839$	11.0263	0.380671	0.190336	+	0.981719i	$0.439042\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	34.1359	1.17640
$843$	38.3772	1.32178
$844$	4.64911	0.160029
$845$	0	0
$846$	30.2719	1.04077
$847$	−1.00000	−0.0343604
$848$	−0.837722	−0.0287675
$849$	1.02633	0.0352237
$850$	0	0
$851$	−13.6754	−0.468788
$852$	−12.1359	−0.415771
$853$	13.6228	0.466435	0.233217	−	0.972425i	$-0.425075\pi$
$853$	13.6228	0.466435	0.233217	+	0.972425i	$0.425075\pi$
$854$	12.3246	0.421738
$855$	0	0
$856$	0.486833	0.0166396
$857$	−16.4605	−0.562280	−0.281140	−	0.959667i	$-0.590713\pi$
$857$	−16.4605	−0.562280	−0.281140	+	0.959667i	$0.590713\pi$
$858$	−0.513167	−0.0175192
$859$	−20.4605	−0.698103	−0.349052	−	0.937104i	$-0.613496\pi$
$859$	−20.4605	−0.698103	−0.349052	+	0.937104i	$0.613496\pi$
$860$	0	0
$861$	32.6491	1.11268
$862$	−27.6228	−0.940836
$863$	16.8114	0.572266	0.286133	−	0.958190i	$-0.407630\pi$
$863$	16.8114	0.572266	0.286133	+	0.958190i	$0.407630\pi$
$864$	12.6491	0.430331
$865$	0	0
$866$	−30.1623	−1.02496
$867$	51.5395	1.75037
$868$	7.00000	0.237595
$869$	3.16228	0.107273
$870$	0	0
$871$	1.21495	0.0411669
$872$	−11.0000	−0.372507
$873$	−40.8641	−1.38304
$874$	2.16228	0.0731401
$875$	0	0
$876$	−12.6491	−0.427374
$877$	7.97367	0.269252	0.134626	−	0.990897i	$-0.457017\pi$
$877$	7.97367	0.269252	0.134626	+	0.990897i	$0.457017\pi$
$878$	5.67544	0.191537
$879$	−13.6754	−0.461261
$880$	0	0
$881$	−40.8114	−1.37497	−0.687485	−	0.726198i	$-0.741285\pi$
$881$	−40.8114	−1.37497	−0.687485	+	0.726198i	$0.741285\pi$
$882$	−7.00000	−0.235702
$883$	29.6754	0.998658	0.499329	−	0.866412i	$-0.333580\pi$
$883$	29.6754	0.998658	0.499329	+	0.866412i	$0.333580\pi$
$884$	−0.135944	−0.00457228
$885$	0	0
$886$	−24.0000	−0.806296
$887$	43.9473	1.47561	0.737803	−	0.675016i	$-0.235863\pi$
$887$	43.9473	1.47561	0.737803	+	0.675016i	$0.235863\pi$
$888$	−20.0000	−0.671156
$889$	−14.3246	−0.480430
$890$	0	0
$891$	−19.0000	−0.636524
$892$	−15.6754	−0.524853
$893$	4.32456	0.144716
$894$	−5.29822	−0.177199
$895$	0	0
$896$	1.00000	0.0334077
$897$	−1.10961	−0.0370488
$898$	3.00000	0.100111
$899$	−21.0000	−0.700389
$900$	0	0
$901$	0.701779	0.0233796
$902$	10.3246	0.343770
$903$	0.513167	0.0170771
$904$	−10.3246	−0.343390
$905$	0	0
$906$	60.5964	2.01318
$907$	21.1623	0.702682	0.351341	−	0.936248i	$-0.385726\pi$
$907$	21.1623	0.702682	0.351341	+	0.936248i	$0.385726\pi$
$908$	−13.3246	−0.442191
$909$	−129.408	−4.29219
$910$	0	0
$911$	59.2982	1.96464	0.982319	−	0.187216i	$-0.0599463\pi$
$911$	59.2982	1.96464	0.982319	+	0.187216i	$0.0599463\pi$
$912$	3.16228	0.104713
$913$	−11.6491	−0.385529
$914$	14.8377	0.490788
$915$	0	0
$916$	22.6491	0.748348
$917$	9.00000	0.297206
$918$	−10.5964	−0.349735
$919$	54.5964	1.80097	0.900485	−	0.434887i	$-0.143212\pi$
$919$	54.5964	1.80097	0.900485	+	0.434887i	$0.143212\pi$
$920$	0	0
$921$	80.0000	2.63609
$922$	13.6754	0.450377
$923$	0.622777	0.0204989
$924$	−3.16228	−0.104031
$925$	0	0
$926$	−6.16228	−0.202505
$927$	91.0000	2.98883
$928$	−3.00000	−0.0984798
$929$	−43.4605	−1.42589	−0.712946	−	0.701219i	$-0.752640\pi$
$929$	−43.4605	−1.42589	−0.712946	+	0.701219i	$0.752640\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−23.1623	−0.758706
$933$	31.5395	1.03256
$934$	30.0000	0.981630
$935$	0	0
$936$	−1.13594	−0.0371295
$937$	31.4868	1.02863	0.514315	−	0.857601i	$-0.328046\pi$
$937$	31.4868	1.02863	0.514315	+	0.857601i	$0.328046\pi$
$938$	7.48683	0.244454
$939$	−45.2982	−1.47825
$940$	0	0
$941$	−40.3246	−1.31454	−0.657271	−	0.753654i	$-0.728289\pi$
$941$	−40.3246	−1.31454	−0.657271	+	0.753654i	$0.728289\pi$
$942$	−3.67544	−0.119752
$943$	22.3246	0.726988
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0.162278	0.00527610
$947$	−17.1623	−0.557699	−0.278850	−	0.960335i	$-0.589953\pi$
$947$	−17.1623	−0.557699	−0.278850	+	0.960335i	$0.589953\pi$
$948$	10.0000	0.324785
$949$	0.649111	0.0210710
$950$	0	0
$951$	−48.9737	−1.58808
$952$	−0.837722	−0.0271507
$953$	−40.4605	−1.31064	−0.655322	−	0.755350i	$-0.727467\pi$
$953$	−40.4605	−1.31064	−0.655322	+	0.755350i	$0.727467\pi$
$954$	5.86406	0.189856
$955$	0	0
$956$	−12.9737	−0.419598
$957$	9.48683	0.306666
$958$	−12.9737	−0.419160
$959$	−7.32456	−0.236522
$960$	0	0
$961$	18.0000	0.580645
$962$	1.02633	0.0330903
$963$	−3.40783	−0.109816
$964$	2.00000	0.0644157
$965$	0	0
$966$	−6.83772	−0.220000
$967$	−6.46050	−0.207756	−0.103878	−	0.994590i	$-0.533125\pi$
$967$	−6.46050	−0.207756	−0.103878	+	0.994590i	$0.533125\pi$
$968$	−1.00000	−0.0321412
$969$	−2.64911	−0.0851017
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	−22.1359	−0.710011
$973$	17.3246	0.555400
$974$	35.1359	1.12583
$975$	0	0
$976$	12.3246	0.394499
$977$	−53.2982	−1.70516	−0.852581	−	0.522596i	$-0.824964\pi$
$977$	−53.2982	−1.70516	−0.852581	+	0.522596i	$0.824964\pi$
$978$	72.6491	2.32306
$979$	−2.16228	−0.0691067
$980$	0	0
$981$	77.0000	2.45842
$982$	33.1359	1.05741
$983$	−8.02633	−0.256000	−0.128000	−	0.991774i	$-0.540856\pi$
$983$	−8.02633	−0.256000	−0.128000	+	0.991774i	$0.540856\pi$
$984$	32.6491	1.04082
$985$	0	0
$986$	2.51317	0.0800355
$987$	−13.6754	−0.435294
$988$	−0.162278	−0.00516274
$989$	0.350889	0.0111576
$990$	0	0
$991$	33.9473	1.07837	0.539186	−	0.842187i	$-0.318732\pi$
$991$	33.9473	1.07837	0.539186	+	0.842187i	$0.318732\pi$
$992$	7.00000	0.222250
$993$	−20.0000	−0.634681
$994$	3.83772	0.121725
$995$	0	0
$996$	−36.8377	−1.16725
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−0.324555	−0.0102736
$999$	80.0000	2.53109

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3850.2.a.bf.1.1	✓	2	1.1	even	1	trivial
3850.2.a.bq.1.2	yes	2	5.4	even	2
3850.2.c.t.1849.2		4	5.2	odd	4
3850.2.c.t.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} +3.16228 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} +3.16228 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} -1.00000 q^{11} -3.16228 q^{12} +0.162278 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.837722 q^{17} -7.00000 q^{18} -1.00000 q^{19} +3.16228 q^{21} +1.00000 q^{22} +2.16228 q^{23} +3.16228 q^{24} -0.162278 q^{26} -12.6491 q^{27} -1.00000 q^{28} +3.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} +3.16228 q^{33} +0.837722 q^{34} +7.00000 q^{36} -6.32456 q^{37} +1.00000 q^{38} -0.513167 q^{39} +10.3246 q^{41} -3.16228 q^{42} +0.162278 q^{43} -1.00000 q^{44} -2.16228 q^{46} -4.32456 q^{47} -3.16228 q^{48} +1.00000 q^{49} +2.64911 q^{51} +0.162278 q^{52} -0.837722 q^{53} +12.6491 q^{54} +1.00000 q^{56} +3.16228 q^{57} -3.00000 q^{58} -12.0000 q^{59} +12.3246 q^{61} +7.00000 q^{62} -7.00000 q^{63} +1.00000 q^{64} -3.16228 q^{66} +7.48683 q^{67} -0.837722 q^{68} -6.83772 q^{69} +3.83772 q^{71} -7.00000 q^{72} +4.00000 q^{73} +6.32456 q^{74} -1.00000 q^{76} +1.00000 q^{77} +0.513167 q^{78} -3.16228 q^{79} +19.0000 q^{81} -10.3246 q^{82} +11.6491 q^{83} +3.16228 q^{84} -0.162278 q^{86} -9.48683 q^{87} +1.00000 q^{88} +2.16228 q^{89} -0.162278 q^{91} +2.16228 q^{92} +22.1359 q^{93} +4.32456 q^{94} +3.16228 q^{96} -5.83772 q^{97} -1.00000 q^{98} -7.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 14 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} - 14 q^{18} - 2 q^{19} + 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} + 6 q^{29} - 14 q^{31} - 2 q^{32} + 8 q^{34} + 14 q^{36} + 2 q^{38} - 20 q^{39} + 8 q^{41} - 6 q^{43} - 2 q^{44} + 2 q^{46} + 4 q^{47} + 2 q^{49} - 20 q^{51} - 6 q^{52} - 8 q^{53} + 2 q^{56} - 6 q^{58} - 24 q^{59} + 12 q^{61} + 14 q^{62} - 14 q^{63} + 2 q^{64} - 4 q^{67} - 8 q^{68} - 20 q^{69} + 14 q^{71} - 14 q^{72} + 8 q^{73} - 2 q^{76} + 2 q^{77} + 20 q^{78} + 38 q^{81} - 8 q^{82} - 2 q^{83} + 6 q^{86} + 2 q^{88} - 2 q^{89} + 6 q^{91} - 2 q^{92} - 4 q^{94} - 18 q^{97} - 2 q^{98} - 14 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 - 14 * q^18 - 2 * q^19 + 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 + 6 * q^29 - 14 * q^31 - 2 * q^32 + 8 * q^34 + 14 * q^36 + 2 * q^38 - 20 * q^39 + 8 * q^41 - 6 * q^43 - 2 * q^44 + 2 * q^46 + 4 * q^47 + 2 * q^49 - 20 * q^51 - 6 * q^52 - 8 * q^53 + 2 * q^56 - 6 * q^58 - 24 * q^59 + 12 * q^61 + 14 * q^62 - 14 * q^63 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 20 * q^69 + 14 * q^71 - 14 * q^72 + 8 * q^73 - 2 * q^76 + 2 * q^77 + 20 * q^78 + 38 * q^81 - 8 * q^82 - 2 * q^83 + 6 * q^86 + 2 * q^88 - 2 * q^89 + 6 * q^91 - 2 * q^92 - 4 * q^94 - 18 * q^97 - 2 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

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Database

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