LMFDB - Embedded newform 3850.2.a.bu.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:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.59774$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +1.00000 q^{11} -3.34596 q^{12} -6.69193 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.19547 q^{17} +8.19547 q^{18} -1.84951 q^{19} -3.34596 q^{21} +1.00000 q^{22} -1.84951 q^{23} -3.34596 q^{24} -6.69193 q^{26} -17.3839 q^{27} +1.00000 q^{28} +6.84242 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.34596 q^{33} +7.19547 q^{34} +8.19547 q^{36} +6.54143 q^{37} -1.84951 q^{38} +22.3909 q^{39} -9.34596 q^{41} -3.34596 q^{42} -0.503544 q^{43} +1.00000 q^{44} -1.84951 q^{46} +1.49646 q^{47} -3.34596 q^{48} +1.00000 q^{49} -24.0758 q^{51} -6.69193 q^{52} -6.84242 q^{53} -17.3839 q^{54} +1.00000 q^{56} +6.18838 q^{57} +6.84242 q^{58} -7.88740 q^{59} +4.50354 q^{61} +6.00000 q^{62} +8.19547 q^{63} +1.00000 q^{64} -3.34596 q^{66} -8.00000 q^{67} +7.19547 q^{68} +6.18838 q^{69} +0.300986 q^{71} +8.19547 q^{72} +10.1884 q^{73} +6.54143 q^{74} -1.84951 q^{76} +1.00000 q^{77} +22.3909 q^{78} -12.2404 q^{79} +33.5793 q^{81} -9.34596 q^{82} +1.30807 q^{83} -3.34596 q^{84} -0.503544 q^{86} -22.8945 q^{87} +1.00000 q^{88} -8.69193 q^{89} -6.69193 q^{91} -1.84951 q^{92} -20.0758 q^{93} +1.49646 q^{94} -3.34596 q^{96} -3.84951 q^{97} +1.00000 q^{98} +8.19547 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} - 2 q^{19} + 3 q^{22} - 2 q^{23} - 12 q^{27} + 3 q^{28} + 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} + 11 q^{36} - 4 q^{37} - 2 q^{38} + 40 q^{39} - 18 q^{41} - 8 q^{43} + 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{54} + 3 q^{56} - 8 q^{57} + 4 q^{58} + 10 q^{59} + 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} - 24 q^{67} + 8 q^{68} - 8 q^{69} + 8 q^{71} + 11 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{76} + 3 q^{77} + 40 q^{78} - 6 q^{79} + 47 q^{81} - 18 q^{82} + 24 q^{83} - 8 q^{86} - 48 q^{87} + 3 q^{88} - 6 q^{89} - 2 q^{92} - 2 q^{94} - 8 q^{97} + 3 q^{98} + 11 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 + 8 * q^17 + 11 * q^18 - 2 * q^19 + 3 * q^22 - 2 * q^23 - 12 * q^27 + 3 * q^28 + 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 + 11 * q^36 - 4 * q^37 - 2 * q^38 + 40 * q^39 - 18 * q^41 - 8 * q^43 + 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^54 + 3 * q^56 - 8 * q^57 + 4 * q^58 + 10 * q^59 + 20 * q^61 + 18 * q^62 + 11 * q^63 + 3 * q^64 - 24 * q^67 + 8 * q^68 - 8 * q^69 + 8 * q^71 + 11 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^76 + 3 * q^77 + 40 * q^78 - 6 * q^79 + 47 * q^81 - 18 * q^82 + 24 * q^83 - 8 * q^86 - 48 * q^87 + 3 * q^88 - 6 * q^89 - 2 * q^92 - 2 * q^94 - 8 * q^97 + 3 * q^98 + 11 * 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
$3$	−3.34596	−1.93179	−0.965896	−	0.258929i	$-0.916630\pi$
$3$	−3.34596	−1.93179	−0.965896	+	0.258929i	$0.916630\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.34596	−1.36598
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	8.19547	2.73182
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.34596	−0.965896
$13$	−6.69193	−1.85601	−0.928003	−	0.372572i	$-0.878476\pi$
$13$	−6.69193	−1.85601	−0.928003	+	0.372572i	$0.878476\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.19547	1.74516	0.872579	−	0.488473i	$-0.162446\pi$
$17$	7.19547	1.74516	0.872579	+	0.488473i	$0.162446\pi$
$18$	8.19547	1.93169
$19$	−1.84951	−0.424306	−0.212153	−	0.977236i	$-0.568048\pi$
$19$	−1.84951	−0.424306	−0.212153	+	0.977236i	$0.568048\pi$
$20$	0	0
$21$	−3.34596	−0.730149
$22$	1.00000	0.213201
$23$	−1.84951	−0.385649	−0.192824	−	0.981233i	$-0.561765\pi$
$23$	−1.84951	−0.385649	−0.192824	+	0.981233i	$0.561765\pi$
$24$	−3.34596	−0.682992
$25$	0	0
$26$	−6.69193	−1.31239
$27$	−17.3839	−3.34552
$28$	1.00000	0.188982
$29$	6.84242	1.27061	0.635303	−	0.772263i	$-0.280875\pi$
$29$	6.84242	1.27061	0.635303	+	0.772263i	$0.280875\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	−3.34596	−0.582457
$34$	7.19547	1.23401
$35$	0	0
$36$	8.19547	1.36591
$37$	6.54143	1.07541	0.537703	−	0.843135i	$-0.319292\pi$
$37$	6.54143	1.07541	0.537703	+	0.843135i	$0.319292\pi$
$38$	−1.84951	−0.300030
$39$	22.3909	3.58542
$40$	0	0
$41$	−9.34596	−1.45959	−0.729797	−	0.683664i	$-0.760385\pi$
$41$	−9.34596	−1.45959	−0.729797	+	0.683664i	$0.760385\pi$
$42$	−3.34596	−0.516293
$43$	−0.503544	−0.0767897	−0.0383949	−	0.999263i	$-0.512224\pi$
$43$	−0.503544	−0.0767897	−0.0383949	+	0.999263i	$0.512224\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.84951	−0.272695
$47$	1.49646	0.218281	0.109140	−	0.994026i	$-0.465190\pi$
$47$	1.49646	0.218281	0.109140	+	0.994026i	$0.465190\pi$
$48$	−3.34596	−0.482948
$49$	1.00000	0.142857
$50$	0	0
$51$	−24.0758	−3.37128
$52$	−6.69193	−0.928003
$53$	−6.84242	−0.939879	−0.469939	−	0.882699i	$-0.655724\pi$
$53$	−6.84242	−0.939879	−0.469939	+	0.882699i	$0.655724\pi$
$54$	−17.3839	−2.36564
$55$	0	0
$56$	1.00000	0.133631
$57$	6.18838	0.819671
$58$	6.84242	0.898454
$59$	−7.88740	−1.02685	−0.513426	−	0.858134i	$-0.671624\pi$
$59$	−7.88740	−1.02685	−0.513426	+	0.858134i	$0.671624\pi$
$60$	0	0
$61$	4.50354	0.576620	0.288310	−	0.957537i	$-0.406907\pi$
$61$	4.50354	0.576620	0.288310	+	0.957537i	$0.406907\pi$
$62$	6.00000	0.762001
$63$	8.19547	1.03253
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.34596	−0.411860
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	7.19547	0.872579
$69$	6.18838	0.744994
$70$	0	0
$71$	0.300986	0.0357204	0.0178602	−	0.999840i	$-0.494315\pi$
$71$	0.300986	0.0357204	0.0178602	+	0.999840i	$0.494315\pi$
$72$	8.19547	0.965845
$73$	10.1884	1.19246	0.596230	−	0.802814i	$-0.296665\pi$
$73$	10.1884	1.19246	0.596230	+	0.802814i	$0.296665\pi$
$74$	6.54143	0.760426
$75$	0	0
$76$	−1.84951	−0.212153
$77$	1.00000	0.113961
$78$	22.3909	2.53527
$79$	−12.2404	−1.37716	−0.688579	−	0.725161i	$-0.741765\pi$
$79$	−12.2404	−1.37716	−0.688579	+	0.725161i	$0.741765\pi$
$80$	0	0
$81$	33.5793	3.73104
$82$	−9.34596	−1.03209
$83$	1.30807	0.143580	0.0717899	−	0.997420i	$-0.477129\pi$
$83$	1.30807	0.143580	0.0717899	+	0.997420i	$0.477129\pi$
$84$	−3.34596	−0.365075
$85$	0	0
$86$	−0.503544	−0.0542985
$87$	−22.8945	−2.45455
$88$	1.00000	0.106600
$89$	−8.69193	−0.921342	−0.460671	−	0.887571i	$-0.652391\pi$
$89$	−8.69193	−0.921342	−0.460671	+	0.887571i	$0.652391\pi$
$90$	0	0
$91$	−6.69193	−0.701505
$92$	−1.84951	−0.192824
$93$	−20.0758	−2.08176
$94$	1.49646	0.154348
$95$	0	0
$96$	−3.34596	−0.341496
$97$	−3.84951	−0.390858	−0.195429	−	0.980718i	$-0.562610\pi$
$97$	−3.84951	−0.390858	−0.195429	+	0.980718i	$0.562610\pi$
$98$	1.00000	0.101015
$99$	8.19547	0.823676
$100$	0	0
$101$	6.89448	0.686027	0.343013	−	0.939331i	$-0.388552\pi$
$101$	6.89448	0.686027	0.343013	+	0.939331i	$0.388552\pi$
$102$	−24.0758	−2.38386
$103$	10.5793	1.04241	0.521206	−	0.853431i	$-0.325482\pi$
$103$	10.5793	1.04241	0.521206	+	0.853431i	$0.325482\pi$
$104$	−6.69193	−0.656197
$105$	0	0
$106$	−6.84242	−0.664595
$107$	7.49646	0.724710	0.362355	−	0.932040i	$-0.381973\pi$
$107$	7.49646	0.724710	0.362355	+	0.932040i	$0.381973\pi$
$108$	−17.3839	−1.67276
$109$	1.45857	0.139705	0.0698527	−	0.997557i	$-0.477747\pi$
$109$	1.45857	0.139705	0.0698527	+	0.997557i	$0.477747\pi$
$110$	0	0
$111$	−21.8874	−2.07746
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	6.18838	0.579595
$115$	0	0
$116$	6.84242	0.635303
$117$	−54.8435	−5.07028
$118$	−7.88740	−0.726094
$119$	7.19547	0.659608
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.50354	0.407732
$123$	31.2713	2.81963
$124$	6.00000	0.538816
$125$	0	0
$126$	8.19547	0.730111
$127$	19.7748	1.75473	0.877365	−	0.479824i	$-0.159300\pi$
$127$	19.7748	1.75473	0.877365	+	0.479824i	$0.159300\pi$
$128$	1.00000	0.0883883
$129$	1.68484	0.148342
$130$	0	0
$131$	2.15049	0.187889	0.0939447	−	0.995577i	$-0.470052\pi$
$131$	2.15049	0.187889	0.0939447	+	0.995577i	$0.470052\pi$
$132$	−3.34596	−0.291229
$133$	−1.84951	−0.160373
$134$	−8.00000	−0.691095
$135$	0	0
$136$	7.19547	0.617006
$137$	8.39094	0.716886	0.358443	−	0.933552i	$-0.383308\pi$
$137$	8.39094	0.716886	0.358443	+	0.933552i	$0.383308\pi$
$138$	6.18838	0.526790
$139$	17.9253	1.52040	0.760201	−	0.649687i	$-0.225100\pi$
$139$	17.9253	1.52040	0.760201	+	0.649687i	$0.225100\pi$
$140$	0	0
$141$	−5.00709	−0.421673
$142$	0.300986	0.0252582
$143$	−6.69193	−0.559607
$144$	8.19547	0.682956
$145$	0	0
$146$	10.1884	0.843197
$147$	−3.34596	−0.275970
$148$	6.54143	0.537703
$149$	−0.451479	−0.0369866	−0.0184933	−	0.999829i	$-0.505887\pi$
$149$	−0.451479	−0.0369866	−0.0184933	+	0.999829i	$0.505887\pi$
$150$	0	0
$151$	19.2334	1.56519	0.782594	−	0.622532i	$-0.213896\pi$
$151$	19.2334	1.56519	0.782594	+	0.622532i	$0.213896\pi$
$152$	−1.84951	−0.150015
$153$	58.9703	4.76746
$154$	1.00000	0.0805823
$155$	0	0
$156$	22.3909	1.79271
$157$	9.69901	0.774066	0.387033	−	0.922066i	$-0.373500\pi$
$157$	9.69901	0.774066	0.387033	+	0.922066i	$0.373500\pi$
$158$	−12.2404	−0.973798
$159$	22.8945	1.81565
$160$	0	0
$161$	−1.84951	−0.145762
$162$	33.5793	2.63824
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−9.34596	−0.729797
$165$	0	0
$166$	1.30807	0.101526
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−3.34596	−0.258147
$169$	31.7819	2.44476
$170$	0	0
$171$	−15.1576	−1.15913
$172$	−0.503544	−0.0383949
$173$	15.4965	1.17817	0.589087	−	0.808070i	$-0.299488\pi$
$173$	15.4965	1.17817	0.589087	+	0.808070i	$0.299488\pi$
$174$	−22.8945	−1.73563
$175$	0	0
$176$	1.00000	0.0753778
$177$	26.3909	1.98366
$178$	−8.69193	−0.651487
$179$	−13.8874	−1.03799	−0.518996	−	0.854776i	$-0.673694\pi$
$179$	−13.8874	−1.03799	−0.518996	+	0.854776i	$0.673694\pi$
$180$	0	0
$181$	18.7819	1.39605	0.698023	−	0.716075i	$-0.254063\pi$
$181$	18.7819	1.39605	0.698023	+	0.716075i	$0.254063\pi$
$182$	−6.69193	−0.496039
$183$	−15.0687	−1.11391
$184$	−1.84951	−0.136347
$185$	0	0
$186$	−20.0758	−1.47203
$187$	7.19547	0.526185
$188$	1.49646	0.109140
$189$	−17.3839	−1.26449
$190$	0	0
$191$	−6.39094	−0.462432	−0.231216	−	0.972902i	$-0.574270\pi$
$191$	−6.39094	−0.462432	−0.231216	+	0.972902i	$0.574270\pi$
$192$	−3.34596	−0.241474
$193$	13.1955	0.949831	0.474915	−	0.880031i	$-0.342479\pi$
$193$	13.1955	0.949831	0.474915	+	0.880031i	$0.342479\pi$
$194$	−3.84951	−0.276379
$195$	0	0
$196$	1.00000	0.0714286
$197$	4.69193	0.334286	0.167143	−	0.985933i	$-0.446546\pi$
$197$	4.69193	0.334286	0.167143	+	0.985933i	$0.446546\pi$
$198$	8.19547	0.582427
$199$	−9.49646	−0.673186	−0.336593	−	0.941650i	$-0.609275\pi$
$199$	−9.49646	−0.673186	−0.336593	+	0.941650i	$0.609275\pi$
$200$	0	0
$201$	26.7677	1.88805
$202$	6.89448	0.485094
$203$	6.84242	0.480244
$204$	−24.0758	−1.68564
$205$	0	0
$206$	10.5793	0.737096
$207$	−15.1576	−1.05352
$208$	−6.69193	−0.464002
$209$	−1.84951	−0.127933
$210$	0	0
$211$	−2.39094	−0.164599	−0.0822996	−	0.996608i	$-0.526226\pi$
$211$	−2.39094	−0.164599	−0.0822996	+	0.996608i	$0.526226\pi$
$212$	−6.84242	−0.469939
$213$	−1.00709	−0.0690045
$214$	7.49646	0.512447
$215$	0	0
$216$	−17.3839	−1.18282
$217$	6.00000	0.407307
$218$	1.45857	0.0987866
$219$	−34.0900	−2.30359
$220$	0	0
$221$	−48.1516	−3.23902
$222$	−21.8874	−1.46899
$223$	−0.188383	−0.0126150	−0.00630752	−	0.999980i	$-0.502008\pi$
$223$	−0.188383	−0.0126150	−0.00630752	+	0.999980i	$0.502008\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	6.99291	0.464136	0.232068	−	0.972700i	$-0.425451\pi$
$227$	6.99291	0.464136	0.232068	+	0.972700i	$0.425451\pi$
$228$	6.18838	0.409836
$229$	−9.19547	−0.607654	−0.303827	−	0.952727i	$-0.598264\pi$
$229$	−9.19547	−0.607654	−0.303827	+	0.952727i	$0.598264\pi$
$230$	0	0
$231$	−3.34596	−0.220148
$232$	6.84242	0.449227
$233$	0.992912	0.0650479	0.0325239	−	0.999471i	$-0.489645\pi$
$233$	0.992912	0.0650479	0.0325239	+	0.999471i	$0.489645\pi$
$234$	−54.8435	−3.58523
$235$	0	0
$236$	−7.88740	−0.513426
$237$	40.9561	2.66038
$238$	7.19547	0.466413
$239$	−1.14341	−0.0739607	−0.0369804	−	0.999316i	$-0.511774\pi$
$239$	−1.14341	−0.0739607	−0.0369804	+	0.999316i	$0.511774\pi$
$240$	0	0
$241$	8.33888	0.537154	0.268577	−	0.963258i	$-0.413447\pi$
$241$	8.33888	0.537154	0.268577	+	0.963258i	$0.413447\pi$
$242$	1.00000	0.0642824
$243$	−60.2036	−3.86206
$244$	4.50354	0.288310
$245$	0	0
$246$	31.2713	1.99378
$247$	12.3768	0.787515
$248$	6.00000	0.381000
$249$	−4.37677	−0.277366
$250$	0	0
$251$	23.8874	1.50776	0.753880	−	0.657013i	$-0.228180\pi$
$251$	23.8874	1.50776	0.753880	+	0.657013i	$0.228180\pi$
$252$	8.19547	0.516266
$253$	−1.84951	−0.116278
$254$	19.7748	1.24078
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.6243	1.97267	0.986335	−	0.164753i	$-0.0526827\pi$
$257$	31.6243	1.97267	0.986335	+	0.164753i	$0.0526827\pi$
$258$	1.68484	0.104893
$259$	6.54143	0.406465
$260$	0	0
$261$	56.0768	3.47107
$262$	2.15049	0.132858
$263$	−13.3839	−0.825284	−0.412642	−	0.910893i	$-0.635394\pi$
$263$	−13.3839	−0.825284	−0.412642	+	0.910893i	$0.635394\pi$
$264$	−3.34596	−0.205930
$265$	0	0
$266$	−1.84951	−0.113401
$267$	29.0829	1.77984
$268$	−8.00000	−0.488678
$269$	5.79744	0.353476	0.176738	−	0.984258i	$-0.443445\pi$
$269$	5.79744	0.353476	0.176738	+	0.984258i	$0.443445\pi$
$270$	0	0
$271$	−20.0758	−1.21952	−0.609758	−	0.792587i	$-0.708734\pi$
$271$	−20.0758	−1.21952	−0.609758	+	0.792587i	$0.708734\pi$
$272$	7.19547	0.436289
$273$	22.3909	1.35516
$274$	8.39094	0.506915
$275$	0	0
$276$	6.18838	0.372497
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	17.9253	1.07509
$279$	49.1728	2.94390
$280$	0	0
$281$	−6.30099	−0.375885	−0.187943	−	0.982180i	$-0.560182\pi$
$281$	−6.30099	−0.375885	−0.187943	+	0.982180i	$0.560182\pi$
$282$	−5.00709	−0.298168
$283$	−22.3909	−1.33100	−0.665502	−	0.746396i	$-0.731782\pi$
$283$	−22.3909	−1.33100	−0.665502	+	0.746396i	$0.731782\pi$
$284$	0.300986	0.0178602
$285$	0	0
$286$	−6.69193	−0.395702
$287$	−9.34596	−0.551675
$288$	8.19547	0.482923
$289$	34.7748	2.04558
$290$	0	0
$291$	12.8803	0.755057
$292$	10.1884	0.596230
$293$	10.6919	0.624629	0.312315	−	0.949979i	$-0.398896\pi$
$293$	10.6919	0.624629	0.312315	+	0.949979i	$0.398896\pi$
$294$	−3.34596	−0.195141
$295$	0	0
$296$	6.54143	0.380213
$297$	−17.3839	−1.00871
$298$	−0.451479	−0.0261535
$299$	12.3768	0.715767
$300$	0	0
$301$	−0.503544	−0.0290238
$302$	19.2334	1.10676
$303$	−23.0687	−1.32526
$304$	−1.84951	−0.106077
$305$	0	0
$306$	58.9703	3.37111
$307$	9.08287	0.518387	0.259193	−	0.965825i	$-0.416543\pi$
$307$	9.08287	0.518387	0.259193	+	0.965825i	$0.416543\pi$
$308$	1.00000	0.0569803
$309$	−35.3980	−2.01372
$310$	0	0
$311$	−11.3839	−0.645519	−0.322760	−	0.946481i	$-0.604611\pi$
$311$	−11.3839	−0.645519	−0.322760	+	0.946481i	$0.604611\pi$
$312$	22.3909	1.26764
$313$	23.6243	1.33532	0.667662	−	0.744464i	$-0.267295\pi$
$313$	23.6243	1.33532	0.667662	+	0.744464i	$0.267295\pi$
$314$	9.69901	0.547347
$315$	0	0
$316$	−12.2404	−0.688579
$317$	9.53435	0.535502	0.267751	−	0.963488i	$-0.413720\pi$
$317$	9.53435	0.535502	0.267751	+	0.963488i	$0.413720\pi$
$318$	22.8945	1.28386
$319$	6.84242	0.383102
$320$	0	0
$321$	−25.0829	−1.39999
$322$	−1.84951	−0.103069
$323$	−13.3081	−0.740481
$324$	33.5793	1.86552
$325$	0	0
$326$	8.00000	0.443079
$327$	−4.88031	−0.269882
$328$	−9.34596	−0.516044
$329$	1.49646	0.0825023
$330$	0	0
$331$	0.503544	0.0276773	0.0138386	−	0.999904i	$-0.495595\pi$
$331$	0.503544	0.0276773	0.0138386	+	0.999904i	$0.495595\pi$
$332$	1.30807	0.0717899
$333$	53.6101	2.93782
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−3.34596	−0.182537
$337$	−12.3909	−0.674978	−0.337489	−	0.941330i	$-0.609578\pi$
$337$	−12.3909	−0.674978	−0.337489	+	0.941330i	$0.609578\pi$
$338$	31.7819	1.72871
$339$	−46.8435	−2.54419
$340$	0	0
$341$	6.00000	0.324918
$342$	−15.1576	−0.819628
$343$	1.00000	0.0539949
$344$	−0.503544	−0.0271493
$345$	0	0
$346$	15.4965	0.833095
$347$	−20.8803	−1.12091	−0.560457	−	0.828184i	$-0.689374\pi$
$347$	−20.8803	−1.12091	−0.560457	+	0.828184i	$0.689374\pi$
$348$	−22.8945	−1.22727
$349$	−22.1884	−1.18772	−0.593858	−	0.804570i	$-0.702396\pi$
$349$	−22.1884	−1.18772	−0.593858	+	0.804570i	$0.702396\pi$
$350$	0	0
$351$	116.331	6.20931
$352$	1.00000	0.0533002
$353$	31.6243	1.68319	0.841596	−	0.540108i	$-0.181617\pi$
$353$	31.6243	1.68319	0.841596	+	0.540108i	$0.181617\pi$
$354$	26.3909	1.40266
$355$	0	0
$356$	−8.69193	−0.460671
$357$	−24.0758	−1.27423
$358$	−13.8874	−0.733972
$359$	30.5273	1.61117	0.805584	−	0.592482i	$-0.201852\pi$
$359$	30.5273	1.61117	0.805584	+	0.592482i	$0.201852\pi$
$360$	0	0
$361$	−15.5793	−0.819964
$362$	18.7819	0.987154
$363$	−3.34596	−0.175618
$364$	−6.69193	−0.350752
$365$	0	0
$366$	−15.0687	−0.787653
$367$	10.5793	0.552236	0.276118	−	0.961124i	$-0.410952\pi$
$367$	10.5793	0.552236	0.276118	+	0.961124i	$0.410952\pi$
$368$	−1.84951	−0.0964122
$369$	−76.5946	−3.98735
$370$	0	0
$371$	−6.84242	−0.355241
$372$	−20.0758	−1.04088
$373$	−36.4667	−1.88818	−0.944088	−	0.329695i	$-0.893054\pi$
$373$	−36.4667	−1.88818	−0.944088	+	0.329695i	$0.893054\pi$
$374$	7.19547	0.372069
$375$	0	0
$376$	1.49646	0.0771738
$377$	−45.7890	−2.35825
$378$	−17.3839	−0.894129
$379$	14.9929	0.770134	0.385067	−	0.922889i	$-0.374178\pi$
$379$	14.9929	0.770134	0.385067	+	0.922889i	$0.374178\pi$
$380$	0	0
$381$	−66.1657	−3.38977
$382$	−6.39094	−0.326989
$383$	−35.1813	−1.79768	−0.898840	−	0.438277i	$-0.855589\pi$
$383$	−35.1813	−1.79768	−0.898840	+	0.438277i	$0.855589\pi$
$384$	−3.34596	−0.170748
$385$	0	0
$386$	13.1955	0.671632
$387$	−4.12678	−0.209776
$388$	−3.84951	−0.195429
$389$	2.30099	0.116665	0.0583323	−	0.998297i	$-0.481422\pi$
$389$	2.30099	0.116665	0.0583323	+	0.998297i	$0.481422\pi$
$390$	0	0
$391$	−13.3081	−0.673018
$392$	1.00000	0.0505076
$393$	−7.19547	−0.362963
$394$	4.69193	0.236376
$395$	0	0
$396$	8.19547	0.411838
$397$	−18.0758	−0.907197	−0.453599	−	0.891206i	$-0.649860\pi$
$397$	−18.0758	−0.907197	−0.453599	+	0.891206i	$0.649860\pi$
$398$	−9.49646	−0.476014
$399$	6.18838	0.309807
$400$	0	0
$401$	22.5793	1.12756	0.563779	−	0.825926i	$-0.309347\pi$
$401$	22.5793	1.12756	0.563779	+	0.825926i	$0.309347\pi$
$402$	26.7677	1.33505
$403$	−40.1516	−2.00009
$404$	6.89448	0.343013
$405$	0	0
$406$	6.84242	0.339584
$407$	6.54143	0.324247
$408$	−24.0758	−1.19193
$409$	1.04498	0.0516708	0.0258354	−	0.999666i	$-0.491775\pi$
$409$	1.04498	0.0516708	0.0258354	+	0.999666i	$0.491775\pi$
$410$	0	0
$411$	−28.0758	−1.38488
$412$	10.5793	0.521206
$413$	−7.88740	−0.388113
$414$	−15.1576	−0.744954
$415$	0	0
$416$	−6.69193	−0.328099
$417$	−59.9774	−2.93710
$418$	−1.84951	−0.0904623
$419$	5.49646	0.268519	0.134260	−	0.990946i	$-0.457134\pi$
$419$	5.49646	0.268519	0.134260	+	0.990946i	$0.457134\pi$
$420$	0	0
$421$	2.60197	0.126812	0.0634062	−	0.997988i	$-0.479804\pi$
$421$	2.60197	0.126812	0.0634062	+	0.997988i	$0.479804\pi$
$422$	−2.39094	−0.116389
$423$	12.2642	0.596304
$424$	−6.84242	−0.332297
$425$	0	0
$426$	−1.00709	−0.0487936
$427$	4.50354	0.217942
$428$	7.49646	0.362355
$429$	22.3909	1.08104
$430$	0	0
$431$	19.2334	0.926438	0.463219	−	0.886244i	$-0.346694\pi$
$431$	19.2334	0.926438	0.463219	+	0.886244i	$0.346694\pi$
$432$	−17.3839	−0.836381
$433$	8.93237	0.429263	0.214631	−	0.976695i	$-0.431145\pi$
$433$	8.93237	0.429263	0.214631	+	0.976695i	$0.431145\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	1.45857	0.0698527
$437$	3.42068	0.163633
$438$	−34.0900	−1.62888
$439$	−0.300986	−0.0143653	−0.00718264	−	0.999974i	$-0.502286\pi$
$439$	−0.300986	−0.0143653	−0.00718264	+	0.999974i	$0.502286\pi$
$440$	0	0
$441$	8.19547	0.390260
$442$	−48.1516	−2.29034
$443$	−30.7677	−1.46182	−0.730909	−	0.682475i	$-0.760904\pi$
$443$	−30.7677	−1.46182	−0.730909	+	0.682475i	$0.760904\pi$
$444$	−21.8874	−1.03873
$445$	0	0
$446$	−0.188383	−0.00892018
$447$	1.51063	0.0714504
$448$	1.00000	0.0472456
$449$	−1.19547	−0.0564177	−0.0282089	−	0.999602i	$-0.508980\pi$
$449$	−1.19547	−0.0564177	−0.0282089	+	0.999602i	$0.508980\pi$
$450$	0	0
$451$	−9.34596	−0.440084
$452$	14.0000	0.658505
$453$	−64.3541	−3.02362
$454$	6.99291	0.328194
$455$	0	0
$456$	6.18838	0.289798
$457$	25.7748	1.20569	0.602847	−	0.797857i	$-0.294033\pi$
$457$	25.7748	1.20569	0.602847	+	0.797857i	$0.294033\pi$
$458$	−9.19547	−0.429676
$459$	−125.085	−5.83847
$460$	0	0
$461$	11.5722	0.538973	0.269486	−	0.963004i	$-0.413146\pi$
$461$	11.5722	0.538973	0.269486	+	0.963004i	$0.413146\pi$
$462$	−3.34596	−0.155668
$463$	32.9182	1.52984	0.764919	−	0.644126i	$-0.222779\pi$
$463$	32.9182	1.52984	0.764919	+	0.644126i	$0.222779\pi$
$464$	6.84242	0.317651
$465$	0	0
$466$	0.992912	0.0459958
$467$	14.6399	0.677452	0.338726	−	0.940885i	$-0.390004\pi$
$467$	14.6399	0.677452	0.338726	+	0.940885i	$0.390004\pi$
$468$	−54.8435	−2.53514
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−32.4525	−1.49533
$472$	−7.88740	−0.363047
$473$	−0.503544	−0.0231530
$474$	40.9561	1.88118
$475$	0	0
$476$	7.19547	0.329804
$477$	−56.0768	−2.56758
$478$	−1.14341	−0.0522981
$479$	10.3909	0.474774	0.237387	−	0.971415i	$-0.423709\pi$
$479$	10.3909	0.474774	0.237387	+	0.971415i	$0.423709\pi$
$480$	0	0
$481$	−43.7748	−1.99596
$482$	8.33888	0.379825
$483$	6.18838	0.281581
$484$	1.00000	0.0454545
$485$	0	0
$486$	−60.2036	−2.73089
$487$	−39.3091	−1.78127	−0.890634	−	0.454722i	$-0.849739\pi$
$487$	−39.3091	−1.78127	−0.890634	+	0.454722i	$0.849739\pi$
$488$	4.50354	0.203866
$489$	−26.7677	−1.21048
$490$	0	0
$491$	23.7748	1.07294	0.536471	−	0.843919i	$-0.319757\pi$
$491$	23.7748	1.07294	0.536471	+	0.843919i	$0.319757\pi$
$492$	31.2713	1.40982
$493$	49.2344	2.21741
$494$	12.3768	0.556857
$495$	0	0
$496$	6.00000	0.269408
$497$	0.300986	0.0135011
$498$	−4.37677	−0.196128
$499$	−6.11260	−0.273638	−0.136819	−	0.990596i	$-0.543688\pi$
$499$	−6.11260	−0.273638	−0.136819	+	0.990596i	$0.543688\pi$
$500$	0	0
$501$	26.7677	1.19589
$502$	23.8874	1.06615
$503$	13.1586	0.586715	0.293358	−	0.956003i	$-0.405227\pi$
$503$	13.1586	0.586715	0.293358	+	0.956003i	$0.405227\pi$
$504$	8.19547	0.365055
$505$	0	0
$506$	−1.84951	−0.0822206
$507$	−106.341	−4.72277
$508$	19.7748	0.877365
$509$	34.1799	1.51500	0.757499	−	0.652836i	$-0.226421\pi$
$509$	34.1799	1.51500	0.757499	+	0.652836i	$0.226421\pi$
$510$	0	0
$511$	10.1884	0.450708
$512$	1.00000	0.0441942
$513$	32.1516	1.41953
$514$	31.6243	1.39489
$515$	0	0
$516$	1.68484	0.0741709
$517$	1.49646	0.0658141
$518$	6.54143	0.287414
$519$	−51.8506	−2.27599
$520$	0	0
$521$	−26.6778	−1.16877	−0.584387	−	0.811475i	$-0.698665\pi$
$521$	−26.6778	−1.16877	−0.584387	+	0.811475i	$0.698665\pi$
$522$	56.0768	2.45442
$523$	−25.4880	−1.11451	−0.557256	−	0.830341i	$-0.688146\pi$
$523$	−25.4880	−1.11451	−0.557256	+	0.830341i	$0.688146\pi$
$524$	2.15049	0.0939447
$525$	0	0
$526$	−13.3839	−0.583564
$527$	43.1728	1.88064
$528$	−3.34596	−0.145614
$529$	−19.5793	−0.851275
$530$	0	0
$531$	−64.6409	−2.80518
$532$	−1.84951	−0.0801863
$533$	62.5425	2.70902
$534$	29.0829	1.25854
$535$	0	0
$536$	−8.00000	−0.345547
$537$	46.4667	2.00519
$538$	5.79744	0.249945
$539$	1.00000	0.0430730
$540$	0	0
$541$	4.45148	0.191384	0.0956920	−	0.995411i	$-0.469494\pi$
$541$	4.45148	0.191384	0.0956920	+	0.995411i	$0.469494\pi$
$542$	−20.0758	−0.862329
$543$	−62.8435	−2.69687
$544$	7.19547	0.308503
$545$	0	0
$546$	22.3909	0.958244
$547$	1.38385	0.0591693	0.0295846	−	0.999562i	$-0.490582\pi$
$547$	1.38385	0.0591693	0.0295846	+	0.999562i	$0.490582\pi$
$548$	8.39094	0.358443
$549$	36.9087	1.57522
$550$	0	0
$551$	−12.6551	−0.539126
$552$	6.18838	0.263395
$553$	−12.2404	−0.520517
$554$	−22.0000	−0.934690
$555$	0	0
$556$	17.9253	0.760201
$557$	6.70610	0.284147	0.142073	−	0.989856i	$-0.454623\pi$
$557$	6.70610	0.284147	0.142073	+	0.989856i	$0.454623\pi$
$558$	49.1728	2.08165
$559$	3.36968	0.142522
$560$	0	0
$561$	−24.0758	−1.01648
$562$	−6.30099	−0.265791
$563$	−3.59488	−0.151506	−0.0757532	−	0.997127i	$-0.524136\pi$
$563$	−3.59488	−0.151506	−0.0757532	+	0.997127i	$0.524136\pi$
$564$	−5.00709	−0.210836
$565$	0	0
$566$	−22.3909	−0.941161
$567$	33.5793	1.41020
$568$	0.300986	0.0126291
$569$	31.7606	1.33147	0.665737	−	0.746186i	$-0.268117\pi$
$569$	31.7606	1.33147	0.665737	+	0.746186i	$0.268117\pi$
$570$	0	0
$571$	−5.68484	−0.237903	−0.118952	−	0.992900i	$-0.537953\pi$
$571$	−5.68484	−0.237903	−0.118952	+	0.992900i	$0.537953\pi$
$572$	−6.69193	−0.279804
$573$	21.3839	0.893323
$574$	−9.34596	−0.390093
$575$	0	0
$576$	8.19547	0.341478
$577$	−13.5343	−0.563442	−0.281721	−	0.959496i	$-0.590905\pi$
$577$	−13.5343	−0.563442	−0.281721	+	0.959496i	$0.590905\pi$
$578$	34.7748	1.44644
$579$	−44.1516	−1.83488
$580$	0	0
$581$	1.30807	0.0542680
$582$	12.8803	0.533906
$583$	−6.84242	−0.283384
$584$	10.1884	0.421598
$585$	0	0
$586$	10.6919	0.441679
$587$	0.0520650	0.00214895	0.00107448	−	0.999999i	$-0.499658\pi$
$587$	0.0520650	0.00214895	0.00107448	+	0.999999i	$0.499658\pi$
$588$	−3.34596	−0.137985
$589$	−11.0970	−0.457246
$590$	0	0
$591$	−15.6990	−0.645771
$592$	6.54143	0.268851
$593$	−31.9774	−1.31315	−0.656576	−	0.754260i	$-0.727996\pi$
$593$	−31.9774	−1.31315	−0.656576	+	0.754260i	$0.727996\pi$
$594$	−17.3839	−0.713268
$595$	0	0
$596$	−0.451479	−0.0184933
$597$	31.7748	1.30046
$598$	12.3768	0.506124
$599$	21.0829	0.861423	0.430711	−	0.902490i	$-0.358263\pi$
$599$	21.0829	0.861423	0.430711	+	0.902490i	$0.358263\pi$
$600$	0	0
$601$	−21.6469	−0.882997	−0.441499	−	0.897262i	$-0.645553\pi$
$601$	−21.6469	−0.882997	−0.441499	+	0.897262i	$0.645553\pi$
$602$	−0.503544	−0.0205229
$603$	−65.5638	−2.66996
$604$	19.2334	0.782594
$605$	0	0
$606$	−23.0687	−0.937102
$607$	−21.6622	−0.879241	−0.439621	−	0.898184i	$-0.644887\pi$
$607$	−21.6622	−0.879241	−0.439621	+	0.898184i	$0.644887\pi$
$608$	−1.84951	−0.0750074
$609$	−22.8945	−0.927731
$610$	0	0
$611$	−10.0142	−0.405130
$612$	58.9703	2.38373
$613$	−8.31516	−0.335846	−0.167923	−	0.985800i	$-0.553706\pi$
$613$	−8.31516	−0.335846	−0.167923	+	0.985800i	$0.553706\pi$
$614$	9.08287	0.366555
$615$	0	0
$616$	1.00000	0.0402911
$617$	−12.4667	−0.501891	−0.250946	−	0.968001i	$-0.580742\pi$
$617$	−12.4667	−0.501891	−0.250946	+	0.968001i	$0.580742\pi$
$618$	−35.3980	−1.42392
$619$	5.27125	0.211869	0.105935	−	0.994373i	$-0.466217\pi$
$619$	5.27125	0.211869	0.105935	+	0.994373i	$0.466217\pi$
$620$	0	0
$621$	32.1516	1.29020
$622$	−11.3839	−0.456451
$623$	−8.69193	−0.348235
$624$	22.3909	0.896355
$625$	0	0
$626$	23.6243	0.944217
$627$	6.18838	0.247140
$628$	9.69901	0.387033
$629$	47.0687	1.87675
$630$	0	0
$631$	−17.1586	−0.683075	−0.341537	−	0.939868i	$-0.610948\pi$
$631$	−17.1586	−0.683075	−0.341537	+	0.939868i	$0.610948\pi$
$632$	−12.2404	−0.486899
$633$	8.00000	0.317971
$634$	9.53435	0.378657
$635$	0	0
$636$	22.8945	0.907825
$637$	−6.69193	−0.265144
$638$	6.84242	0.270894
$639$	2.46672	0.0975820
$640$	0	0
$641$	24.7677	0.978266	0.489133	−	0.872209i	$-0.337313\pi$
$641$	24.7677	0.978266	0.489133	+	0.872209i	$0.337313\pi$
$642$	−25.0829	−0.989942
$643$	−1.03080	−0.0406509	−0.0203254	−	0.999793i	$-0.506470\pi$
$643$	−1.03080	−0.0406509	−0.0203254	+	0.999793i	$0.506470\pi$
$644$	−1.84951	−0.0728808
$645$	0	0
$646$	−13.3081	−0.523599
$647$	−12.7904	−0.502841	−0.251420	−	0.967878i	$-0.580898\pi$
$647$	−12.7904	−0.502841	−0.251420	+	0.967878i	$0.580898\pi$
$648$	33.5793	1.31912
$649$	−7.88740	−0.309607
$650$	0	0
$651$	−20.0758	−0.786832
$652$	8.00000	0.313304
$653$	22.3162	0.873301	0.436651	−	0.899631i	$-0.356165\pi$
$653$	22.3162	0.873301	0.436651	+	0.899631i	$0.356165\pi$
$654$	−4.88031	−0.190835
$655$	0	0
$656$	−9.34596	−0.364899
$657$	83.4986	3.25759
$658$	1.49646	0.0583379
$659$	22.6919	0.883952	0.441976	−	0.897027i	$-0.354278\pi$
$659$	22.6919	0.883952	0.441976	+	0.897027i	$0.354278\pi$
$660$	0	0
$661$	43.5638	1.69443	0.847217	−	0.531247i	$-0.178276\pi$
$661$	43.5638	1.69443	0.847217	+	0.531247i	$0.178276\pi$
$662$	0.503544	0.0195708
$663$	161.113	6.25712
$664$	1.30807	0.0507631
$665$	0	0
$666$	53.6101	2.07735
$667$	−12.6551	−0.490008
$668$	−8.00000	−0.309529
$669$	0.630322	0.0243697
$670$	0	0
$671$	4.50354	0.173857
$672$	−3.34596	−0.129073
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−12.3909	−0.477281
$675$	0	0
$676$	31.7819	1.22238
$677$	16.0984	0.618713	0.309356	−	0.950946i	$-0.399886\pi$
$677$	16.0984	0.618713	0.309356	+	0.950946i	$0.399886\pi$
$678$	−46.8435	−1.79901
$679$	−3.84951	−0.147731
$680$	0	0
$681$	−23.3980	−0.896614
$682$	6.00000	0.229752
$683$	−29.0829	−1.11282	−0.556412	−	0.830906i	$-0.687823\pi$
$683$	−29.0829	−1.11282	−0.556412	+	0.830906i	$0.687823\pi$
$684$	−15.1576	−0.579565
$685$	0	0
$686$	1.00000	0.0381802
$687$	30.7677	1.17386
$688$	−0.503544	−0.0191974
$689$	45.7890	1.74442
$690$	0	0
$691$	7.58641	0.288601	0.144300	−	0.989534i	$-0.453907\pi$
$691$	7.58641	0.288601	0.144300	+	0.989534i	$0.453907\pi$
$692$	15.4965	0.589087
$693$	8.19547	0.311320
$694$	−20.8803	−0.792606
$695$	0	0
$696$	−22.8945	−0.867813
$697$	−67.2486	−2.54722
$698$	−22.1884	−0.839843
$699$	−3.32225	−0.125659
$700$	0	0
$701$	4.07471	0.153900	0.0769499	−	0.997035i	$-0.475482\pi$
$701$	4.07471	0.153900	0.0769499	+	0.997035i	$0.475482\pi$
$702$	116.331	4.39065
$703$	−12.0984	−0.456301
$704$	1.00000	0.0376889
$705$	0	0
$706$	31.6243	1.19020
$707$	6.89448	0.259294
$708$	26.3909	0.991832
$709$	−46.4525	−1.74456	−0.872281	−	0.489005i	$-0.837360\pi$
$709$	−46.4525	−1.74456	−0.872281	+	0.489005i	$0.837360\pi$
$710$	0	0
$711$	−100.316	−3.76215
$712$	−8.69193	−0.325744
$713$	−11.0970	−0.415588
$714$	−24.0758	−0.901013
$715$	0	0
$716$	−13.8874	−0.518996
$717$	3.82579	0.142877
$718$	30.5273	1.13927
$719$	−17.8732	−0.666559	−0.333279	−	0.942828i	$-0.608155\pi$
$719$	−17.8732	−0.666559	−0.333279	+	0.942828i	$0.608155\pi$
$720$	0	0
$721$	10.5793	0.393995
$722$	−15.5793	−0.579802
$723$	−27.9016	−1.03767
$724$	18.7819	0.698023
$725$	0	0
$726$	−3.34596	−0.124180
$727$	11.8874	0.440879	0.220440	−	0.975401i	$-0.429251\pi$
$727$	11.8874	0.440879	0.220440	+	0.975401i	$0.429251\pi$
$728$	−6.69193	−0.248019
$729$	100.701	3.72967
$730$	0	0
$731$	−3.62323	−0.134010
$732$	−15.0687	−0.556955
$733$	31.6764	1.16999	0.584997	−	0.811036i	$-0.301096\pi$
$733$	31.6764	1.16999	0.584997	+	0.811036i	$0.301096\pi$
$734$	10.5793	0.390490
$735$	0	0
$736$	−1.84951	−0.0681737
$737$	−8.00000	−0.294684
$738$	−76.5946	−2.81948
$739$	28.5567	1.05047	0.525237	−	0.850956i	$-0.323977\pi$
$739$	28.5567	1.05047	0.525237	+	0.850956i	$0.323977\pi$
$740$	0	0
$741$	−41.4122	−1.52132
$742$	−6.84242	−0.251193
$743$	−13.9858	−0.513090	−0.256545	−	0.966532i	$-0.582584\pi$
$743$	−13.9858	−0.513090	−0.256545	+	0.966532i	$0.582584\pi$
$744$	−20.0758	−0.736014
$745$	0	0
$746$	−36.4667	−1.33514
$747$	10.7203	0.392234
$748$	7.19547	0.263092
$749$	7.49646	0.273915
$750$	0	0
$751$	18.9171	0.690296	0.345148	−	0.938548i	$-0.387829\pi$
$751$	18.9171	0.690296	0.345148	+	0.938548i	$0.387829\pi$
$752$	1.49646	0.0545701
$753$	−79.9264	−2.91268
$754$	−45.7890	−1.66754
$755$	0	0
$756$	−17.3839	−0.632245
$757$	−10.9182	−0.396829	−0.198414	−	0.980118i	$-0.563579\pi$
$757$	−10.9182	−0.396829	−0.198414	+	0.980118i	$0.563579\pi$
$758$	14.9929	0.544567
$759$	6.18838	0.224624
$760$	0	0
$761$	−30.3531	−1.10030	−0.550149	−	0.835067i	$-0.685429\pi$
$761$	−30.3531	−1.10030	−0.550149	+	0.835067i	$0.685429\pi$
$762$	−66.1657	−2.39693
$763$	1.45857	0.0528036
$764$	−6.39094	−0.231216
$765$	0	0
$766$	−35.1813	−1.27115
$767$	52.7819	1.90584
$768$	−3.34596	−0.120737
$769$	−19.7369	−0.711731	−0.355865	−	0.934537i	$-0.615814\pi$
$769$	−19.7369	−0.711731	−0.355865	+	0.934537i	$0.615814\pi$
$770$	0	0
$771$	−105.814	−3.81079
$772$	13.1955	0.474915
$773$	24.3909	0.877281	0.438641	−	0.898663i	$-0.355460\pi$
$773$	24.3909	0.877281	0.438641	+	0.898663i	$0.355460\pi$
$774$	−4.12678	−0.148334
$775$	0	0
$776$	−3.84951	−0.138189
$777$	−21.8874	−0.785206
$778$	2.30099	0.0824943
$779$	17.2854	0.619315
$780$	0	0
$781$	0.300986	0.0107701
$782$	−13.3081	−0.475896
$783$	−118.948	−4.25084
$784$	1.00000	0.0357143
$785$	0	0
$786$	−7.19547	−0.256654
$787$	−33.7890	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$787$	−33.7890	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$788$	4.69193	0.167143
$789$	44.7819	1.59428
$790$	0	0
$791$	14.0000	0.497783
$792$	8.19547	0.291213
$793$	−30.1374	−1.07021
$794$	−18.0758	−0.641485
$795$	0	0
$796$	−9.49646	−0.336593
$797$	−36.8435	−1.30506	−0.652532	−	0.757761i	$-0.726293\pi$
$797$	−36.8435	−1.30506	−0.652532	+	0.757761i	$0.726293\pi$
$798$	6.18838	0.219066
$799$	10.7677	0.380934
$800$	0	0
$801$	−71.2344	−2.51694
$802$	22.5793	0.797304
$803$	10.1884	0.359540
$804$	26.7677	0.944024
$805$	0	0
$806$	−40.1516	−1.41428
$807$	−19.3980	−0.682843
$808$	6.89448	0.242547
$809$	37.6990	1.32543	0.662713	−	0.748873i	$-0.269405\pi$
$809$	37.6990	1.32543	0.662713	+	0.748873i	$0.269405\pi$
$810$	0	0
$811$	38.3304	1.34596	0.672981	−	0.739660i	$-0.265013\pi$
$811$	38.3304	1.34596	0.672981	+	0.739660i	$0.265013\pi$
$812$	6.84242	0.240122
$813$	67.1728	2.35585
$814$	6.54143	0.229277
$815$	0	0
$816$	−24.0758	−0.842821
$817$	0.931308	0.0325823
$818$	1.04498	0.0365368
$819$	−54.8435	−1.91639
$820$	0	0
$821$	−18.1363	−0.632962	−0.316481	−	0.948599i	$-0.602501\pi$
$821$	−18.1363	−0.632962	−0.316481	+	0.948599i	$0.602501\pi$
$822$	−28.0758	−0.979255
$823$	7.86368	0.274111	0.137055	−	0.990563i	$-0.456236\pi$
$823$	7.86368	0.274111	0.137055	+	0.990563i	$0.456236\pi$
$824$	10.5793	0.368548
$825$	0	0
$826$	−7.88740	−0.274438
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−15.1576	−0.526762
$829$	2.80453	0.0974053	0.0487027	−	0.998813i	$-0.484491\pi$
$829$	2.80453	0.0974053	0.0487027	+	0.998813i	$0.484491\pi$
$830$	0	0
$831$	73.6112	2.55354
$832$	−6.69193	−0.232001
$833$	7.19547	0.249308
$834$	−59.9774	−2.07685
$835$	0	0
$836$	−1.84951	−0.0639665
$837$	−104.303	−3.60524
$838$	5.49646	0.189872
$839$	36.9929	1.27714	0.638569	−	0.769565i	$-0.279527\pi$
$839$	36.9929	1.27714	0.638569	+	0.769565i	$0.279527\pi$
$840$	0	0
$841$	17.8187	0.614438
$842$	2.60197	0.0896699
$843$	21.0829	0.726133
$844$	−2.39094	−0.0822996
$845$	0	0
$846$	12.2642	0.421651
$847$	1.00000	0.0343604
$848$	−6.84242	−0.234970
$849$	74.9193	2.57122
$850$	0	0
$851$	−12.0984	−0.414729
$852$	−1.00709	−0.0345023
$853$	49.2571	1.68653	0.843265	−	0.537498i	$-0.180630\pi$
$853$	49.2571	1.68653	0.843265	+	0.537498i	$0.180630\pi$
$854$	4.50354	0.154108
$855$	0	0
$856$	7.49646	0.256224
$857$	20.3541	0.695283	0.347642	−	0.937627i	$-0.386983\pi$
$857$	20.3541	0.695283	0.347642	+	0.937627i	$0.386983\pi$
$858$	22.3909	0.764414
$859$	−34.2783	−1.16956	−0.584781	−	0.811191i	$-0.698820\pi$
$859$	−34.2783	−1.16956	−0.584781	+	0.811191i	$0.698820\pi$
$860$	0	0
$861$	31.2713	1.06572
$862$	19.2334	0.655091
$863$	12.4373	0.423371	0.211685	−	0.977338i	$-0.432105\pi$
$863$	12.4373	0.423371	0.211685	+	0.977338i	$0.432105\pi$
$864$	−17.3839	−0.591411
$865$	0	0
$866$	8.93237	0.303534
$867$	−116.355	−3.95163
$868$	6.00000	0.203653
$869$	−12.2404	−0.415229
$870$	0	0
$871$	53.5354	1.81398
$872$	1.45857	0.0493933
$873$	−31.5485	−1.06776
$874$	3.42068	0.115706
$875$	0	0
$876$	−34.0900	−1.15179
$877$	−34.4809	−1.16434	−0.582169	−	0.813068i	$-0.697796\pi$
$877$	−34.4809	−1.16434	−0.582169	+	0.813068i	$0.697796\pi$
$878$	−0.300986	−0.0101578
$879$	−35.7748	−1.20665
$880$	0	0
$881$	27.3839	0.922585	0.461293	−	0.887248i	$-0.347386\pi$
$881$	27.3839	0.922585	0.461293	+	0.887248i	$0.347386\pi$
$882$	8.19547	0.275956
$883$	−22.0900	−0.743386	−0.371693	−	0.928356i	$-0.621223\pi$
$883$	−22.0900	−0.743386	−0.371693	+	0.928356i	$0.621223\pi$
$884$	−48.1516	−1.61951
$885$	0	0
$886$	−30.7677	−1.03366
$887$	−25.7890	−0.865909	−0.432954	−	0.901416i	$-0.642529\pi$
$887$	−25.7890	−0.865909	−0.432954	+	0.901416i	$0.642529\pi$
$888$	−21.8874	−0.734493
$889$	19.7748	0.663225
$890$	0	0
$891$	33.5793	1.12495
$892$	−0.188383	−0.00630752
$893$	−2.76771	−0.0926178
$894$	1.51063	0.0505231
$895$	0	0
$896$	1.00000	0.0334077
$897$	−41.4122	−1.38271
$898$	−1.19547	−0.0398934
$899$	41.0545	1.36924
$900$	0	0
$901$	−49.2344	−1.64024
$902$	−9.34596	−0.311187
$903$	1.68484	0.0560679
$904$	14.0000	0.465633
$905$	0	0
$906$	−64.3541	−2.13802
$907$	45.4596	1.50946	0.754731	−	0.656034i	$-0.227767\pi$
$907$	45.4596	1.50946	0.754731	+	0.656034i	$0.227767\pi$
$908$	6.99291	0.232068
$909$	56.5035	1.87410
$910$	0	0
$911$	−15.8506	−0.525153	−0.262576	−	0.964911i	$-0.584572\pi$
$911$	−15.8506	−0.525153	−0.262576	+	0.964911i	$0.584572\pi$
$912$	6.18838	0.204918
$913$	1.30807	0.0432909
$914$	25.7748	0.852554
$915$	0	0
$916$	−9.19547	−0.303827
$917$	2.15049	0.0710155
$918$	−125.085	−4.12842
$919$	18.6314	0.614593	0.307296	−	0.951614i	$-0.400576\pi$
$919$	18.6314	0.614593	0.307296	+	0.951614i	$0.400576\pi$
$920$	0	0
$921$	−30.3909	−1.00142
$922$	11.5722	0.381111
$923$	−2.01418	−0.0662974
$924$	−3.34596	−0.110074
$925$	0	0
$926$	32.9182	1.08176
$927$	86.7025	2.84768
$928$	6.84242	0.224613
$929$	−29.2486	−0.959616	−0.479808	−	0.877374i	$-0.659294\pi$
$929$	−29.2486	−0.959616	−0.479808	+	0.877374i	$0.659294\pi$
$930$	0	0
$931$	−1.84951	−0.0606151
$932$	0.992912	0.0325239
$933$	38.0900	1.24701
$934$	14.6399	0.479031
$935$	0	0
$936$	−54.8435	−1.79262
$937$	46.4441	1.51726	0.758631	−	0.651521i	$-0.225869\pi$
$937$	46.4441	1.51726	0.758631	+	0.651521i	$0.225869\pi$
$938$	−8.00000	−0.261209
$939$	−79.0460	−2.57957
$940$	0	0
$941$	−39.2713	−1.28021	−0.640103	−	0.768289i	$-0.721108\pi$
$941$	−39.2713	−1.28021	−0.640103	+	0.768289i	$0.721108\pi$
$942$	−32.4525	−1.05736
$943$	17.2854	0.562891
$944$	−7.88740	−0.256713
$945$	0	0
$946$	−0.503544	−0.0163716
$947$	−54.2415	−1.76261	−0.881306	−	0.472546i	$-0.843335\pi$
$947$	−54.2415	−1.76261	−0.881306	+	0.472546i	$0.843335\pi$
$948$	40.9561	1.33019
$949$	−68.1799	−2.21321
$950$	0	0
$951$	−31.9016	−1.03448
$952$	7.19547	0.233207
$953$	−59.3612	−1.92290	−0.961449	−	0.274983i	$-0.911328\pi$
$953$	−59.3612	−1.92290	−0.961449	+	0.274983i	$0.911328\pi$
$954$	−56.0768	−1.81555
$955$	0	0
$956$	−1.14341	−0.0369804
$957$	−22.8945	−0.740074
$958$	10.3909	0.335716
$959$	8.39094	0.270958
$960$	0	0
$961$	5.00000	0.161290
$962$	−43.7748	−1.41136
$963$	61.4370	1.97978
$964$	8.33888	0.268577
$965$	0	0
$966$	6.18838	0.199108
$967$	18.7677	0.603529	0.301764	−	0.953383i	$-0.402424\pi$
$967$	18.7677	0.603529	0.301764	+	0.953383i	$0.402424\pi$
$968$	1.00000	0.0321412
$969$	44.5283	1.43046
$970$	0	0
$971$	−59.4370	−1.90742	−0.953712	−	0.300722i	$-0.902772\pi$
$971$	−59.4370	−1.90742	−0.953712	+	0.300722i	$0.902772\pi$
$972$	−60.2036	−1.93103
$973$	17.9253	0.574658
$974$	−39.3091	−1.25955
$975$	0	0
$976$	4.50354	0.144155
$977$	32.7677	1.04833	0.524166	−	0.851616i	$-0.324377\pi$
$977$	32.7677	1.04833	0.524166	+	0.851616i	$0.324377\pi$
$978$	−26.7677	−0.855937
$979$	−8.69193	−0.277795
$980$	0	0
$981$	11.9536	0.381650
$982$	23.7748	0.758684
$983$	−13.4207	−0.428053	−0.214027	−	0.976828i	$-0.568658\pi$
$983$	−13.4207	−0.428053	−0.214027	+	0.976828i	$0.568658\pi$
$984$	31.2713	0.996891
$985$	0	0
$986$	49.2344	1.56794
$987$	−5.00709	−0.159377
$988$	12.3768	0.393757
$989$	0.931308	0.0296139
$990$	0	0
$991$	−38.1657	−1.21237	−0.606187	−	0.795322i	$-0.707302\pi$
$991$	−38.1657	−1.21237	−0.606187	+	0.795322i	$0.707302\pi$
$992$	6.00000	0.190500
$993$	−1.68484	−0.0534668
$994$	0.300986	0.00954669
$995$	0	0
$996$	−4.37677	−0.138683
$997$	33.6622	1.06609	0.533046	−	0.846086i	$-0.321047\pi$
$997$	33.6622	1.06609	0.533046	+	0.846086i	$0.321047\pi$
$998$	−6.11260	−0.193491
$999$	−113.715	−3.59779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bu.1.1		3
5.2	odd	4		3850.2.c.z.1849.6		6
5.3	odd	4		3850.2.c.z.1849.1		6
5.4	even	2		770.2.a.l.1.3	✓	3
15.14	odd	2		6930.2.a.cl.1.3		3
20.19	odd	2		6160.2.a.bi.1.1		3
35.34	odd	2		5390.2.a.bz.1.1		3
55.54	odd	2		8470.2.a.cl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.3	✓	3	5.4	even	2
3850.2.a.bu.1.1		3	1.1	even	1	trivial
3850.2.c.z.1849.1		6	5.3	odd	4
3850.2.c.z.1849.6		6	5.2	odd	4
5390.2.a.bz.1.1		3	35.34	odd	2
6160.2.a.bi.1.1		3	20.19	odd	2
6930.2.a.cl.1.3		3	15.14	odd	2
8470.2.a.cl.1.3		3	55.54	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +1.00000 q^{11} -3.34596 q^{12} -6.69193 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.19547 q^{17} +8.19547 q^{18} -1.84951 q^{19} -3.34596 q^{21} +1.00000 q^{22} -1.84951 q^{23} -3.34596 q^{24} -6.69193 q^{26} -17.3839 q^{27} +1.00000 q^{28} +6.84242 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.34596 q^{33} +7.19547 q^{34} +8.19547 q^{36} +6.54143 q^{37} -1.84951 q^{38} +22.3909 q^{39} -9.34596 q^{41} -3.34596 q^{42} -0.503544 q^{43} +1.00000 q^{44} -1.84951 q^{46} +1.49646 q^{47} -3.34596 q^{48} +1.00000 q^{49} -24.0758 q^{51} -6.69193 q^{52} -6.84242 q^{53} -17.3839 q^{54} +1.00000 q^{56} +6.18838 q^{57} +6.84242 q^{58} -7.88740 q^{59} +4.50354 q^{61} +6.00000 q^{62} +8.19547 q^{63} +1.00000 q^{64} -3.34596 q^{66} -8.00000 q^{67} +7.19547 q^{68} +6.18838 q^{69} +0.300986 q^{71} +8.19547 q^{72} +10.1884 q^{73} +6.54143 q^{74} -1.84951 q^{76} +1.00000 q^{77} +22.3909 q^{78} -12.2404 q^{79} +33.5793 q^{81} -9.34596 q^{82} +1.30807 q^{83} -3.34596 q^{84} -0.503544 q^{86} -22.8945 q^{87} +1.00000 q^{88} -8.69193 q^{89} -6.69193 q^{91} -1.84951 q^{92} -20.0758 q^{93} +1.49646 q^{94} -3.34596 q^{96} -3.84951 q^{97} +1.00000 q^{98} +8.19547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} - 2 q^{19} + 3 q^{22} - 2 q^{23} - 12 q^{27} + 3 q^{28} + 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} + 11 q^{36} - 4 q^{37} - 2 q^{38} + 40 q^{39} - 18 q^{41} - 8 q^{43} + 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{54} + 3 q^{56} - 8 q^{57} + 4 q^{58} + 10 q^{59} + 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} - 24 q^{67} + 8 q^{68} - 8 q^{69} + 8 q^{71} + 11 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{76} + 3 q^{77} + 40 q^{78} - 6 q^{79} + 47 q^{81} - 18 q^{82} + 24 q^{83} - 8 q^{86} - 48 q^{87} + 3 q^{88} - 6 q^{89} - 2 q^{92} - 2 q^{94} - 8 q^{97} + 3 q^{98} + 11 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 + 8 * q^17 + 11 * q^18 - 2 * q^19 + 3 * q^22 - 2 * q^23 - 12 * q^27 + 3 * q^28 + 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 + 11 * q^36 - 4 * q^37 - 2 * q^38 + 40 * q^39 - 18 * q^41 - 8 * q^43 + 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^54 + 3 * q^56 - 8 * q^57 + 4 * q^58 + 10 * q^59 + 20 * q^61 + 18 * q^62 + 11 * q^63 + 3 * q^64 - 24 * q^67 + 8 * q^68 - 8 * q^69 + 8 * q^71 + 11 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^76 + 3 * q^77 + 40 * q^78 - 6 * q^79 + 47 * q^81 - 18 * q^82 + 24 * q^83 - 8 * q^86 - 48 * q^87 + 3 * q^88 - 6 * q^89 - 2 * q^92 - 2 * q^94 - 8 * q^97 + 3 * q^98 + 11 * q^99

Introduction

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