LMFDB - Embedded newform 3800.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.363328 q^{3} -1.14134 q^{7} -2.86799 q^{9} -2.72666 q^{11} +4.64600 q^{13} +0.858664 q^{17} -1.00000 q^{19} +0.414680 q^{21} +4.41468 q^{23} +2.13201 q^{27} +9.42401 q^{29} -10.2827 q^{31} +0.990671 q^{33} -6.77801 q^{37} -1.68802 q^{39} -7.55602 q^{41} -9.29200 q^{43} +7.00933 q^{47} -5.69735 q^{49} -0.311977 q^{51} +8.64600 q^{53} +0.363328 q^{57} -5.14134 q^{59} +9.45331 q^{61} +3.27334 q^{63} +13.6553 q^{67} -1.60398 q^{69} -5.45331 q^{71} -6.87732 q^{73} +3.11203 q^{77} +17.2920 q^{79} +7.82936 q^{81} +14.2827 q^{83} -3.42401 q^{87} +13.0093 q^{89} -5.30265 q^{91} +3.73599 q^{93} +9.68463 q^{97} +7.82003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} + 11 q^{17} - 3 q^{19} - 3 q^{21} + 9 q^{23} + 19 q^{27} + 3 q^{29} - 14 q^{31} + 24 q^{33} - 14 q^{37} - 5 q^{39} - 10 q^{41} + 10 q^{43} + 4 q^{49}+ \cdots + 36 q^{99}+O(q^{100}) $ 3 * q + q^3 + 5 * q^7 + 4 * q^9 - 4 * q^11 - 5 * q^13 + 11 * q^17 - 3 * q^19 - 3 * q^21 + 9 * q^23 + 19 * q^27 + 3 * q^29 - 14 * q^31 + 24 * q^33 - 14 * q^37 - 5 * q^39 - 10 * q^41 + 10 * q^43 + 4 * q^49 - q^51 + 7 * q^53 - q^57 - 7 * q^59 + 20 * q^61 + 14 * q^63 + q^67 + 33 * q^69 - 8 * q^71 + 13 * q^73 - 16 * q^77 + 14 * q^79 + 15 * q^81 + 26 * q^83 + 15 * q^87 + 18 * q^89 - 37 * q^91 - 14 * q^93 + 6 * q^97 + 36 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.363328	−0.209768	−0.104884	−	0.994484i	$-0.533447\pi$
$3$	−0.363328	−0.209768	−0.104884	+	0.994484i	$0.533447\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.14134	−0.431385	−0.215692	−	0.976461i	$-0.569201\pi$
$7$	−1.14134	−0.431385	−0.215692	+	0.976461i	$0.569201\pi$
$8$	0	0
$9$	−2.86799	−0.955998
$10$	0	0
$11$	−2.72666	−0.822118	−0.411059	−	0.911609i	$-0.634841\pi$
$11$	−2.72666	−0.822118	−0.411059	+	0.911609i	$0.634841\pi$
$12$	0	0
$13$	4.64600	1.28857	0.644284	−	0.764786i	$-0.277155\pi$
$13$	4.64600	1.28857	0.644284	+	0.764786i	$0.277155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.858664	0.208257	0.104128	−	0.994564i	$-0.466795\pi$
$17$	0.858664	0.208257	0.104128	+	0.994564i	$0.466795\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.414680	0.0904905
$22$	0	0
$23$	4.41468	0.920524	0.460262	−	0.887783i	$-0.347755\pi$
$23$	4.41468	0.920524	0.460262	+	0.887783i	$0.347755\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.13201	0.410305
$28$	0	0
$29$	9.42401	1.74999	0.874997	−	0.484128i	$-0.160863\pi$
$29$	9.42401	1.74999	0.874997	+	0.484128i	$0.160863\pi$
$30$	0	0
$31$	−10.2827	−1.84682	−0.923411	−	0.383812i	$-0.874611\pi$
$31$	−10.2827	−1.84682	−0.923411	+	0.383812i	$0.874611\pi$
$32$	0	0
$33$	0.990671	0.172454
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.77801	−1.11430	−0.557149	−	0.830413i	$-0.688105\pi$
$37$	−6.77801	−1.11430	−0.557149	+	0.830413i	$0.688105\pi$
$38$	0	0
$39$	−1.68802	−0.270300
$40$	0	0
$41$	−7.55602	−1.18005	−0.590026	−	0.807384i	$-0.700882\pi$
$41$	−7.55602	−1.18005	−0.590026	+	0.807384i	$0.700882\pi$
$42$	0	0
$43$	−9.29200	−1.41702	−0.708508	−	0.705702i	$-0.750632\pi$
$43$	−9.29200	−1.41702	−0.708508	+	0.705702i	$0.750632\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.00933	1.02242	0.511208	−	0.859457i	$-0.329198\pi$
$47$	7.00933	1.02242	0.511208	+	0.859457i	$0.329198\pi$
$48$	0	0
$49$	−5.69735	−0.813907
$50$	0	0
$51$	−0.311977	−0.0436855
$52$	0	0
$53$	8.64600	1.18762	0.593810	−	0.804605i	$-0.297623\pi$
$53$	8.64600	1.18762	0.593810	+	0.804605i	$0.297623\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.363328	0.0481240
$58$	0	0
$59$	−5.14134	−0.669345	−0.334672	−	0.942335i	$-0.608626\pi$
$59$	−5.14134	−0.669345	−0.334672	+	0.942335i	$0.608626\pi$
$60$	0	0
$61$	9.45331	1.21037	0.605186	−	0.796084i	$-0.293099\pi$
$61$	9.45331	1.21037	0.605186	+	0.796084i	$0.293099\pi$
$62$	0	0
$63$	3.27334	0.412403
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.6553	1.66826	0.834132	−	0.551565i	$-0.185969\pi$
$67$	13.6553	1.66826	0.834132	+	0.551565i	$0.185969\pi$
$68$	0	0
$69$	−1.60398	−0.193096
$70$	0	0
$71$	−5.45331	−0.647189	−0.323595	−	0.946196i	$-0.604891\pi$
$71$	−5.45331	−0.647189	−0.323595	+	0.946196i	$0.604891\pi$
$72$	0	0
$73$	−6.87732	−0.804930	−0.402465	−	0.915435i	$-0.631846\pi$
$73$	−6.87732	−0.804930	−0.402465	+	0.915435i	$0.631846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.11203	0.354649
$78$	0	0
$79$	17.2920	1.94550	0.972751	−	0.231852i	$-0.0744785\pi$
$79$	17.2920	1.94550	0.972751	+	0.231852i	$0.0744785\pi$
$80$	0	0
$81$	7.82936	0.869929
$82$	0	0
$83$	14.2827	1.56773	0.783863	−	0.620933i	$-0.213246\pi$
$83$	14.2827	1.56773	0.783863	+	0.620933i	$0.213246\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.42401	−0.367092
$88$	0	0
$89$	13.0093	1.37899	0.689493	−	0.724292i	$-0.257833\pi$
$89$	13.0093	1.37899	0.689493	+	0.724292i	$0.257833\pi$
$90$	0	0
$91$	−5.30265	−0.555869
$92$	0	0
$93$	3.73599	0.387404
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.68463	0.983326	0.491663	−	0.870786i	$-0.336389\pi$
$97$	9.68463	0.983326	0.491663	+	0.870786i	$0.336389\pi$
$98$	0	0
$99$	7.82003	0.785943
$100$	0	0
$101$	1.45331	0.144610	0.0723050	−	0.997383i	$-0.476964\pi$
$101$	1.45331	0.144610	0.0723050	+	0.997383i	$0.476964\pi$
$102$	0	0
$103$	−2.23132	−0.219859	−0.109929	−	0.993939i	$-0.535062\pi$
$103$	−2.23132	−0.219859	−0.109929	+	0.993939i	$0.535062\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.6553	−1.32011	−0.660055	−	0.751217i	$-0.729467\pi$
$107$	−13.6553	−1.32011	−0.660055	+	0.751217i	$0.729467\pi$
$108$	0	0
$109$	20.1693	1.93187	0.965935	−	0.258784i	$-0.0833216\pi$
$109$	20.1693	1.93187	0.965935	+	0.258784i	$0.0833216\pi$
$110$	0	0
$111$	2.46264	0.233744
$112$	0	0
$113$	10.0514	0.945552	0.472776	−	0.881183i	$-0.343252\pi$
$113$	10.0514	0.945552	0.472776	+	0.881183i	$0.343252\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.3247	−1.23187
$118$	0	0
$119$	−0.980024	−0.0898387
$120$	0	0
$121$	−3.56534	−0.324122
$122$	0	0
$123$	2.74531	0.247537
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.0700	1.24851	0.624256	−	0.781220i	$-0.285402\pi$
$127$	14.0700	1.24851	0.624256	+	0.781220i	$0.285402\pi$
$128$	0	0
$129$	3.37605	0.297244
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	1.14134	0.0989664
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.85866	0.415104	0.207552	−	0.978224i	$-0.433450\pi$
$137$	4.85866	0.415104	0.207552	+	0.978224i	$0.433450\pi$
$138$	0	0
$139$	−4.17997	−0.354540	−0.177270	−	0.984162i	$-0.556727\pi$
$139$	−4.17997	−0.354540	−0.177270	+	0.984162i	$0.556727\pi$
$140$	0	0
$141$	−2.54669	−0.214470
$142$	0	0
$143$	−12.6680	−1.05936
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.07001	0.170731
$148$	0	0
$149$	1.45331	0.119060	0.0595300	−	0.998227i	$-0.481040\pi$
$149$	1.45331	0.119060	0.0595300	+	0.998227i	$0.481040\pi$
$150$	0	0
$151$	7.27334	0.591896	0.295948	−	0.955204i	$-0.404364\pi$
$151$	7.27334	0.591896	0.295948	+	0.955204i	$0.404364\pi$
$152$	0	0
$153$	−2.46264	−0.199093
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.0093	−1.35749	−0.678746	−	0.734373i	$-0.737476\pi$
$157$	−17.0093	−1.35749	−0.678746	+	0.734373i	$0.737476\pi$
$158$	0	0
$159$	−3.14134	−0.249124
$160$	0	0
$161$	−5.03863	−0.397100
$162$	0	0
$163$	10.1800	0.797357	0.398678	−	0.917091i	$-0.369469\pi$
$163$	10.1800	0.797357	0.398678	+	0.917091i	$0.369469\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.07001	−0.469711	−0.234856	−	0.972030i	$-0.575462\pi$
$167$	−6.07001	−0.469711	−0.234856	+	0.972030i	$0.575462\pi$
$168$	0	0
$169$	8.58532	0.660409
$170$	0	0
$171$	2.86799	0.219321
$172$	0	0
$173$	5.22199	0.397021	0.198510	−	0.980099i	$-0.436390\pi$
$173$	5.22199	0.397021	0.198510	+	0.980099i	$0.436390\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.86799	0.140407
$178$	0	0
$179$	−1.55602	−0.116302	−0.0581510	−	0.998308i	$-0.518520\pi$
$179$	−1.55602	−0.116302	−0.0581510	+	0.998308i	$0.518520\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−3.43466	−0.253897
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.34128	−0.171211
$188$	0	0
$189$	−2.43334	−0.176999
$190$	0	0
$191$	−5.03863	−0.364583	−0.182291	−	0.983245i	$-0.558351\pi$
$191$	−5.03863	−0.364583	−0.182291	+	0.983245i	$0.558351\pi$
$192$	0	0
$193$	9.68463	0.697115	0.348558	−	0.937287i	$-0.386672\pi$
$193$	9.68463	0.697115	0.348558	+	0.937287i	$0.386672\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	19.8867	1.40973	0.704864	−	0.709343i	$-0.251008\pi$
$199$	19.8867	1.40973	0.704864	+	0.709343i	$0.251008\pi$
$200$	0	0
$201$	−4.96137	−0.349948
$202$	0	0
$203$	−10.7560	−0.754920
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−12.6613	−0.880019
$208$	0	0
$209$	2.72666	0.188607
$210$	0	0
$211$	4.31198	0.296849	0.148424	−	0.988924i	$-0.452580\pi$
$211$	4.31198	0.296849	0.148424	+	0.988924i	$0.452580\pi$
$212$	0	0
$213$	1.98134	0.135759
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.7360	0.796691
$218$	0	0
$219$	2.49873	0.168848
$220$	0	0
$221$	3.98935	0.268353
$222$	0	0
$223$	18.5327	1.24104	0.620519	−	0.784191i	$-0.286922\pi$
$223$	18.5327	1.24104	0.620519	+	0.784191i	$0.286922\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.82597	−0.320311	−0.160155	−	0.987092i	$-0.551200\pi$
$227$	−4.82597	−0.320311	−0.160155	+	0.987092i	$0.551200\pi$
$228$	0	0
$229$	23.7360	1.56852	0.784259	−	0.620434i	$-0.213043\pi$
$229$	23.7360	1.56852	0.784259	+	0.620434i	$0.213043\pi$
$230$	0	0
$231$	−1.13069	−0.0743939
$232$	0	0
$233$	14.5653	0.954207	0.477104	−	0.878847i	$-0.341687\pi$
$233$	14.5653	0.954207	0.477104	+	0.878847i	$0.341687\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.28267	−0.408103
$238$	0	0
$239$	2.43334	0.157399	0.0786997	−	0.996898i	$-0.474923\pi$
$239$	2.43334	0.157399	0.0786997	+	0.996898i	$0.474923\pi$
$240$	0	0
$241$	18.1986	1.17228	0.586138	−	0.810211i	$-0.300648\pi$
$241$	18.1986	1.17228	0.586138	+	0.810211i	$0.300648\pi$
$242$	0	0
$243$	−9.24065	−0.592788
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.64600	−0.295618
$248$	0	0
$249$	−5.18930	−0.328858
$250$	0	0
$251$	−15.1120	−0.953863	−0.476931	−	0.878940i	$-0.658251\pi$
$251$	−15.1120	−0.953863	−0.476931	+	0.878940i	$0.658251\pi$
$252$	0	0
$253$	−12.0373	−0.756779
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.2313	−0.762969	−0.381484	−	0.924375i	$-0.624587\pi$
$257$	−12.2313	−0.762969	−0.381484	+	0.924375i	$0.624587\pi$
$258$	0	0
$259$	7.73599	0.480691
$260$	0	0
$261$	−27.0280	−1.67299
$262$	0	0
$263$	−22.5840	−1.39259	−0.696295	−	0.717756i	$-0.745169\pi$
$263$	−22.5840	−1.39259	−0.696295	+	0.717756i	$0.745169\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.72666	−0.289267
$268$	0	0
$269$	−13.1120	−0.799455	−0.399727	−	0.916634i	$-0.630895\pi$
$269$	−13.1120	−0.799455	−0.399727	+	0.916634i	$0.630895\pi$
$270$	0	0
$271$	−20.8960	−1.26934	−0.634670	−	0.772783i	$-0.718864\pi$
$271$	−20.8960	−1.26934	−0.634670	+	0.772783i	$0.718864\pi$
$272$	0	0
$273$	1.92660	0.116603
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.1800	1.21250	0.606248	−	0.795275i	$-0.292674\pi$
$277$	20.1800	1.21250	0.606248	+	0.795275i	$0.292674\pi$
$278$	0	0
$279$	29.4906	1.76556
$280$	0	0
$281$	−22.4626	−1.34001	−0.670004	−	0.742357i	$-0.733708\pi$
$281$	−22.4626	−1.34001	−0.670004	+	0.742357i	$0.733708\pi$
$282$	0	0
$283$	0.829359	0.0493003	0.0246501	−	0.999696i	$-0.492153\pi$
$283$	0.829359	0.0493003	0.0246501	+	0.999696i	$0.492153\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.62395	0.509056
$288$	0	0
$289$	−16.2627	−0.956629
$290$	0	0
$291$	−3.51870	−0.206270
$292$	0	0
$293$	31.2886	1.82790	0.913950	−	0.405827i	$-0.133016\pi$
$293$	31.2886	1.82790	0.913950	+	0.405827i	$0.133016\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.81325	−0.337319
$298$	0	0
$299$	20.5106	1.18616
$300$	0	0
$301$	10.6053	0.611279
$302$	0	0
$303$	−0.528030	−0.0303345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.11929	0.178027	0.0890136	−	0.996030i	$-0.471629\pi$
$307$	3.11929	0.178027	0.0890136	+	0.996030i	$0.471629\pi$
$308$	0	0
$309$	0.810702	0.0461192
$310$	0	0
$311$	−21.4240	−1.21484	−0.607422	−	0.794379i	$-0.707796\pi$
$311$	−21.4240	−1.21484	−0.607422	+	0.794379i	$0.707796\pi$
$312$	0	0
$313$	12.1320	0.685742	0.342871	−	0.939383i	$-0.388601\pi$
$313$	12.1320	0.685742	0.342871	+	0.939383i	$0.388601\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.7487	−0.940701	−0.470350	−	0.882480i	$-0.655872\pi$
$317$	−16.7487	−0.940701	−0.470350	+	0.882480i	$0.655872\pi$
$318$	0	0
$319$	−25.6960	−1.43870
$320$	0	0
$321$	4.96137	0.276916
$322$	0	0
$323$	−0.858664	−0.0477773
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.32808	−0.405244
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−5.97070	−0.328179	−0.164090	−	0.986445i	$-0.552469\pi$
$331$	−5.97070	−0.328179	−0.164090	+	0.986445i	$0.552469\pi$
$332$	0	0
$333$	19.4393	1.06527
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.3340	1.10766	0.553832	−	0.832628i	$-0.313165\pi$
$337$	20.3340	1.10766	0.553832	+	0.832628i	$0.313165\pi$
$338$	0	0
$339$	−3.65194	−0.198346
$340$	0	0
$341$	28.0373	1.51831
$342$	0	0
$343$	14.4919	0.782492
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.8294	−1.11818	−0.559089	−	0.829107i	$-0.688849\pi$
$347$	−20.8294	−1.11818	−0.559089	+	0.829107i	$0.688849\pi$
$348$	0	0
$349$	−20.0187	−1.07157	−0.535787	−	0.844353i	$-0.679985\pi$
$349$	−20.0187	−1.07157	−0.535787	+	0.844353i	$0.679985\pi$
$350$	0	0
$351$	9.90531	0.528706
$352$	0	0
$353$	27.1413	1.44459	0.722294	−	0.691586i	$-0.243088\pi$
$353$	27.1413	1.44459	0.722294	+	0.691586i	$0.243088\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.356070	0.0188452
$358$	0	0
$359$	−4.59465	−0.242496	−0.121248	−	0.992622i	$-0.538690\pi$
$359$	−4.59465	−0.242496	−0.121248	+	0.992622i	$0.538690\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.29539	0.0679904
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.1214	1.78112	0.890560	−	0.454865i	$-0.150313\pi$
$367$	34.1214	1.78112	0.890560	+	0.454865i	$0.150313\pi$
$368$	0	0
$369$	21.6706	1.12813
$370$	0	0
$371$	−9.86799	−0.512321
$372$	0	0
$373$	7.81664	0.404730	0.202365	−	0.979310i	$-0.435137\pi$
$373$	7.81664	0.404730	0.202365	+	0.979310i	$0.435137\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	43.7839	2.25499
$378$	0	0
$379$	−23.0573	−1.18437	−0.592187	−	0.805801i	$-0.701735\pi$
$379$	−23.0573	−1.18437	−0.592187	+	0.805801i	$0.701735\pi$
$380$	0	0
$381$	−5.11203	−0.261897
$382$	0	0
$383$	−16.8807	−0.862564	−0.431282	−	0.902217i	$-0.641939\pi$
$383$	−16.8807	−0.862564	−0.431282	+	0.902217i	$0.641939\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.6494	1.35466
$388$	0	0
$389$	−18.3013	−0.927914	−0.463957	−	0.885858i	$-0.653571\pi$
$389$	−18.3013	−0.927914	−0.463957	+	0.885858i	$0.653571\pi$
$390$	0	0
$391$	3.79073	0.191705
$392$	0	0
$393$	1.45331	0.0733099
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.1800	−0.611295	−0.305648	−	0.952145i	$-0.598873\pi$
$397$	−12.1800	−0.611295	−0.305648	+	0.952145i	$0.598873\pi$
$398$	0	0
$399$	−0.414680	−0.0207599
$400$	0	0
$401$	16.5840	0.828166	0.414083	−	0.910239i	$-0.364102\pi$
$401$	16.5840	0.828166	0.414083	+	0.910239i	$0.364102\pi$
$402$	0	0
$403$	−47.7733	−2.37976
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.4813	0.916084
$408$	0	0
$409$	18.5653	0.917997	0.458999	−	0.888437i	$-0.348208\pi$
$409$	18.5653	0.917997	0.458999	+	0.888437i	$0.348208\pi$
$410$	0	0
$411$	−1.76529	−0.0870753
$412$	0	0
$413$	5.86799	0.288745
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.51870	0.0743711
$418$	0	0
$419$	2.84802	0.139135	0.0695674	−	0.997577i	$-0.477838\pi$
$419$	2.84802	0.139135	0.0695674	+	0.997577i	$0.477838\pi$
$420$	0	0
$421$	24.5360	1.19581	0.597907	−	0.801566i	$-0.295999\pi$
$421$	24.5360	1.19581	0.597907	+	0.801566i	$0.295999\pi$
$422$	0	0
$423$	−20.1027	−0.977427
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.7894	−0.522136
$428$	0	0
$429$	4.60266	0.222218
$430$	0	0
$431$	−7.53736	−0.363062	−0.181531	−	0.983385i	$-0.558105\pi$
$431$	−7.53736	−0.363062	−0.181531	+	0.983385i	$0.558105\pi$
$432$	0	0
$433$	0.392633	0.0188687	0.00943437	−	0.999955i	$-0.496997\pi$
$433$	0.392633	0.0188687	0.00943437	+	0.999955i	$0.496997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.41468	−0.211183
$438$	0	0
$439$	5.29200	0.252573	0.126287	−	0.991994i	$-0.459694\pi$
$439$	5.29200	0.252573	0.126287	+	0.991994i	$0.459694\pi$
$440$	0	0
$441$	16.3400	0.778093
$442$	0	0
$443$	22.0187	1.04614	0.523069	−	0.852290i	$-0.324787\pi$
$443$	22.0187	1.04614	0.523069	+	0.852290i	$0.324787\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.528030	−0.0249749
$448$	0	0
$449$	−4.90663	−0.231558	−0.115779	−	0.993275i	$-0.536936\pi$
$449$	−4.90663	−0.231558	−0.115779	+	0.993275i	$0.536936\pi$
$450$	0	0
$451$	20.6027	0.970141
$452$	0	0
$453$	−2.64261	−0.124161
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.9800	−1.26207	−0.631036	−	0.775753i	$-0.717370\pi$
$457$	−26.9800	−1.26207	−0.631036	+	0.775753i	$0.717370\pi$
$458$	0	0
$459$	1.83068	0.0854487
$460$	0	0
$461$	−27.1307	−1.26360	−0.631801	−	0.775131i	$-0.717684\pi$
$461$	−27.1307	−1.26360	−0.631801	+	0.775131i	$0.717684\pi$
$462$	0	0
$463$	16.9253	0.786585	0.393292	−	0.919413i	$-0.371336\pi$
$463$	16.9253	0.786585	0.393292	+	0.919413i	$0.371336\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	42.6213	1.97228	0.986140	−	0.165917i	$-0.0530585\pi$
$467$	42.6213	1.97228	0.986140	+	0.165917i	$0.0530585\pi$
$468$	0	0
$469$	−15.5853	−0.719663
$470$	0	0
$471$	6.17997	0.284758
$472$	0	0
$473$	25.3361	1.16495
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−24.7967	−1.13536
$478$	0	0
$479$	−25.3107	−1.15647	−0.578237	−	0.815869i	$-0.696259\pi$
$479$	−25.3107	−1.15647	−0.578237	+	0.815869i	$0.696259\pi$
$480$	0	0
$481$	−31.4906	−1.43585
$482$	0	0
$483$	1.83068	0.0832987
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.4020	−1.15107	−0.575536	−	0.817776i	$-0.695207\pi$
$487$	−25.4020	−1.15107	−0.575536	+	0.817776i	$0.695207\pi$
$488$	0	0
$489$	−3.69867	−0.167260
$490$	0	0
$491$	20.6240	0.930746	0.465373	−	0.885115i	$-0.345920\pi$
$491$	20.6240	0.930746	0.465373	+	0.885115i	$0.345920\pi$
$492$	0	0
$493$	8.09206	0.364448
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.22406	0.279187
$498$	0	0
$499$	17.3693	0.777555	0.388778	−	0.921332i	$-0.372897\pi$
$499$	17.3693	0.777555	0.388778	+	0.921332i	$0.372897\pi$
$500$	0	0
$501$	2.20541	0.0985303
$502$	0	0
$503$	14.9987	0.668758	0.334379	−	0.942439i	$-0.391473\pi$
$503$	14.9987	0.668758	0.334379	+	0.942439i	$0.391473\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.11929	−0.138533
$508$	0	0
$509$	−19.9160	−0.882759	−0.441380	−	0.897320i	$-0.645511\pi$
$509$	−19.9160	−0.882759	−0.441380	+	0.897320i	$0.645511\pi$
$510$	0	0
$511$	7.84934	0.347234
$512$	0	0
$513$	−2.13201	−0.0941304
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.1120	−0.840546
$518$	0	0
$519$	−1.89730	−0.0832821
$520$	0	0
$521$	−14.8294	−0.649686	−0.324843	−	0.945768i	$-0.605311\pi$
$521$	−14.8294	−0.649686	−0.324843	+	0.945768i	$0.605311\pi$
$522$	0	0
$523$	−11.9966	−0.524575	−0.262288	−	0.964990i	$-0.584477\pi$
$523$	−11.9966	−0.524575	−0.262288	+	0.964990i	$0.584477\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.82936	−0.384613
$528$	0	0
$529$	−3.51060	−0.152635
$530$	0	0
$531$	14.7453	0.639892
$532$	0	0
$533$	−35.1053	−1.52058
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.565344	0.0243964
$538$	0	0
$539$	15.5347	0.669128
$540$	0	0
$541$	−2.01866	−0.0867889	−0.0433944	−	0.999058i	$-0.513817\pi$
$541$	−2.01866	−0.0867889	−0.0433944	+	0.999058i	$0.513817\pi$
$542$	0	0
$543$	−7.99322	−0.343022
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.6940	0.799296	0.399648	−	0.916669i	$-0.369132\pi$
$547$	18.6940	0.799296	0.399648	+	0.916669i	$0.369132\pi$
$548$	0	0
$549$	−27.1120	−1.15711
$550$	0	0
$551$	−9.42401	−0.401476
$552$	0	0
$553$	−19.7360	−0.839259
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.0866	−0.469754	−0.234877	−	0.972025i	$-0.575469\pi$
$557$	−11.0866	−0.469754	−0.234877	+	0.972025i	$0.575469\pi$
$558$	0	0
$559$	−43.1706	−1.82592
$560$	0	0
$561$	0.850654	0.0359146
$562$	0	0
$563$	−2.69396	−0.113537	−0.0567685	−	0.998387i	$-0.518080\pi$
$563$	−2.69396	−0.113537	−0.0567685	+	0.998387i	$0.518080\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8.93593	−0.375274
$568$	0	0
$569$	29.4134	1.23307	0.616536	−	0.787327i	$-0.288535\pi$
$569$	29.4134	1.23307	0.616536	+	0.787327i	$0.288535\pi$
$570$	0	0
$571$	−33.0466	−1.38296	−0.691479	−	0.722396i	$-0.743041\pi$
$571$	−33.0466	−1.38296	−0.691479	+	0.722396i	$0.743041\pi$
$572$	0	0
$573$	1.83068	0.0764777
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.396022	0.0164866	0.00824331	−	0.999966i	$-0.497376\pi$
$577$	0.396022	0.0164866	0.00824331	+	0.999966i	$0.497376\pi$
$578$	0	0
$579$	−3.51870	−0.146232
$580$	0	0
$581$	−16.3013	−0.676293
$582$	0	0
$583$	−23.5747	−0.976363
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0773	0.581031	0.290515	−	0.956870i	$-0.406173\pi$
$587$	14.0773	0.581031	0.290515	+	0.956870i	$0.406173\pi$
$588$	0	0
$589$	10.2827	0.423690
$590$	0	0
$591$	−0.726656	−0.0298907
$592$	0	0
$593$	−5.43466	−0.223175	−0.111587	−	0.993755i	$-0.535593\pi$
$593$	−5.43466	−0.223175	−0.111587	+	0.993755i	$0.535593\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.22538	−0.295715
$598$	0	0
$599$	11.6333	0.475323	0.237662	−	0.971348i	$-0.423619\pi$
$599$	11.6333	0.475323	0.237662	+	0.971348i	$0.423619\pi$
$600$	0	0
$601$	8.12136	0.331277	0.165639	−	0.986187i	$-0.447031\pi$
$601$	8.12136	0.331277	0.165639	+	0.986187i	$0.447031\pi$
$602$	0	0
$603$	−39.1634	−1.59486
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.5327	−0.427507	−0.213754	−	0.976888i	$-0.568569\pi$
$607$	−10.5327	−0.427507	−0.213754	+	0.976888i	$0.568569\pi$
$608$	0	0
$609$	3.90794	0.158358
$610$	0	0
$611$	32.5653	1.31745
$612$	0	0
$613$	−47.3293	−1.91161	−0.955807	−	0.293996i	$-0.905015\pi$
$613$	−47.3293	−1.91161	−0.955807	+	0.293996i	$0.905015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.3693	1.34340	0.671698	−	0.740825i	$-0.265565\pi$
$617$	33.3693	1.34340	0.671698	+	0.740825i	$0.265565\pi$
$618$	0	0
$619$	−33.8760	−1.36159	−0.680796	−	0.732473i	$-0.738366\pi$
$619$	−33.8760	−1.36159	−0.680796	+	0.732473i	$0.738366\pi$
$620$	0	0
$621$	9.41213	0.377696
$622$	0	0
$623$	−14.8480	−0.594873
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.990671	−0.0395636
$628$	0	0
$629$	−5.82003	−0.232060
$630$	0	0
$631$	27.8573	1.10898	0.554492	−	0.832189i	$-0.312913\pi$
$631$	27.8573	1.10898	0.554492	+	0.832189i	$0.312913\pi$
$632$	0	0
$633$	−1.56666	−0.0622693
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−26.4699	−1.04878
$638$	0	0
$639$	15.6401	0.618711
$640$	0	0
$641$	−4.74531	−0.187429	−0.0937143	−	0.995599i	$-0.529874\pi$
$641$	−4.74531	−0.187429	−0.0937143	+	0.995599i	$0.529874\pi$
$642$	0	0
$643$	6.28267	0.247764	0.123882	−	0.992297i	$-0.460466\pi$
$643$	6.28267	0.247764	0.123882	+	0.992297i	$0.460466\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.1507	−0.634948	−0.317474	−	0.948267i	$-0.602835\pi$
$647$	−16.1507	−0.634948	−0.317474	+	0.948267i	$0.602835\pi$
$648$	0	0
$649$	14.0187	0.550280
$650$	0	0
$651$	−4.26401	−0.167120
$652$	0	0
$653$	−29.8760	−1.16914	−0.584569	−	0.811344i	$-0.698736\pi$
$653$	−29.8760	−1.16914	−0.584569	+	0.811344i	$0.698736\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	19.7241	0.769511
$658$	0	0
$659$	−17.4427	−0.679470	−0.339735	−	0.940521i	$-0.610337\pi$
$659$	−17.4427	−0.679470	−0.339735	+	0.940521i	$0.610337\pi$
$660$	0	0
$661$	−2.87732	−0.111915	−0.0559574	−	0.998433i	$-0.517821\pi$
$661$	−2.87732	−0.111915	−0.0559574	+	0.998433i	$0.517821\pi$
$662$	0	0
$663$	−1.44944	−0.0562918
$664$	0	0
$665$	0	0
$666$	0	0
$667$	41.6040	1.61091
$668$	0	0
$669$	−6.73344	−0.260330
$670$	0	0
$671$	−25.7759	−0.995069
$672$	0	0
$673$	−10.9393	−0.421680	−0.210840	−	0.977521i	$-0.567620\pi$
$673$	−10.9393	−0.421680	−0.210840	+	0.977521i	$0.567620\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.0900	−0.733688	−0.366844	−	0.930283i	$-0.619562\pi$
$677$	−19.0900	−0.733688	−0.366844	+	0.930283i	$0.619562\pi$
$678$	0	0
$679$	−11.0534	−0.424191
$680$	0	0
$681$	1.75341	0.0671909
$682$	0	0
$683$	−25.7687	−0.986011	−0.493006	−	0.870026i	$-0.664102\pi$
$683$	−25.7687	−0.986011	−0.493006	+	0.870026i	$0.664102\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.62395	−0.329024
$688$	0	0
$689$	40.1693	1.53033
$690$	0	0
$691$	−27.2334	−1.03601	−0.518004	−	0.855378i	$-0.673325\pi$
$691$	−27.2334	−1.03601	−0.518004	+	0.855378i	$0.673325\pi$
$692$	0	0
$693$	−8.92528	−0.339043
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.48808	−0.245753
$698$	0	0
$699$	−5.29200	−0.200162
$700$	0	0
$701$	49.6960	1.87699	0.938497	−	0.345288i	$-0.112219\pi$
$701$	49.6960	1.87699	0.938497	+	0.345288i	$0.112219\pi$
$702$	0	0
$703$	6.77801	0.255637
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.65872	−0.0623825
$708$	0	0
$709$	5.64006	0.211817	0.105908	−	0.994376i	$-0.466225\pi$
$709$	5.64006	0.211817	0.105908	+	0.994376i	$0.466225\pi$
$710$	0	0
$711$	−49.5933	−1.85990
$712$	0	0
$713$	−45.3947	−1.70005
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.884100	−0.0330173
$718$	0	0
$719$	−33.4240	−1.24651	−0.623253	−	0.782021i	$-0.714189\pi$
$719$	−33.4240	−1.24651	−0.623253	+	0.782021i	$0.714189\pi$
$720$	0	0
$721$	2.54669	0.0948436
$722$	0	0
$723$	−6.61208	−0.245906
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.3306	0.531494	0.265747	−	0.964043i	$-0.414381\pi$
$727$	14.3306	0.531494	0.265747	+	0.964043i	$0.414381\pi$
$728$	0	0
$729$	−20.1307	−0.745581
$730$	0	0
$731$	−7.97871	−0.295103
$732$	0	0
$733$	40.8853	1.51013	0.755067	−	0.655648i	$-0.227604\pi$
$733$	40.8853	1.51013	0.755067	+	0.655648i	$0.227604\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.2334	−1.37151
$738$	0	0
$739$	−28.8294	−1.06051	−0.530253	−	0.847840i	$-0.677903\pi$
$739$	−28.8294	−1.06051	−0.530253	+	0.847840i	$0.677903\pi$
$740$	0	0
$741$	1.68802	0.0620111
$742$	0	0
$743$	8.05135	0.295375	0.147688	−	0.989034i	$-0.452817\pi$
$743$	8.05135	0.295375	0.147688	+	0.989034i	$0.452817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−40.9626	−1.49874
$748$	0	0
$749$	15.5853	0.569475
$750$	0	0
$751$	47.3107	1.72639	0.863195	−	0.504870i	$-0.168460\pi$
$751$	47.3107	1.72639	0.863195	+	0.504870i	$0.168460\pi$
$752$	0	0
$753$	5.49063	0.200090
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.20541	−0.225539	−0.112770	−	0.993621i	$-0.535972\pi$
$757$	−6.20541	−0.225539	−0.112770	+	0.993621i	$0.535972\pi$
$758$	0	0
$759$	4.37350	0.158748
$760$	0	0
$761$	43.8280	1.58877	0.794383	−	0.607418i	$-0.207795\pi$
$761$	43.8280	1.58877	0.794383	+	0.607418i	$0.207795\pi$
$762$	0	0
$763$	−23.0200	−0.833379
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.8867	−0.862497
$768$	0	0
$769$	−3.22538	−0.116310	−0.0581551	−	0.998308i	$-0.518522\pi$
$769$	−3.22538	−0.116310	−0.0581551	+	0.998308i	$0.518522\pi$
$770$	0	0
$771$	4.44398	0.160046
$772$	0	0
$773$	−6.16470	−0.221729	−0.110864	−	0.993836i	$-0.535362\pi$
$773$	−6.16470	−0.221729	−0.110864	+	0.993836i	$0.535362\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.81070	−0.100833
$778$	0	0
$779$	7.55602	0.270722
$780$	0	0
$781$	14.8693	0.532066
$782$	0	0
$783$	20.0921	0.718031
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.1527	−1.25306	−0.626530	−	0.779397i	$-0.715525\pi$
$787$	−35.1527	−1.25306	−0.626530	+	0.779397i	$0.715525\pi$
$788$	0	0
$789$	8.20541	0.292120
$790$	0	0
$791$	−11.4720	−0.407896
$792$	0	0
$793$	43.9201	1.55965
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.3420	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$797$	30.3420	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$798$	0	0
$799$	6.01866	0.212925
$800$	0	0
$801$	−37.3107	−1.31831
$802$	0	0
$803$	18.7521	0.661747
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.76397	0.167700
$808$	0	0
$809$	−4.96137	−0.174432	−0.0872162	−	0.996189i	$-0.527797\pi$
$809$	−4.96137	−0.174432	−0.0872162	+	0.996189i	$0.527797\pi$
$810$	0	0
$811$	29.8094	1.04675	0.523375	−	0.852103i	$-0.324673\pi$
$811$	29.8094	1.04675	0.523375	+	0.852103i	$0.324673\pi$
$812$	0	0
$813$	7.59210	0.266267
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.29200	0.325086
$818$	0	0
$819$	15.2080	0.531409
$820$	0	0
$821$	−11.1520	−0.389207	−0.194603	−	0.980882i	$-0.562342\pi$
$821$	−11.1520	−0.389207	−0.194603	+	0.980882i	$0.562342\pi$
$822$	0	0
$823$	17.1413	0.597509	0.298755	−	0.954330i	$-0.403429\pi$
$823$	17.1413	0.597509	0.298755	+	0.954330i	$0.403429\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.8633	−0.864581	−0.432291	−	0.901734i	$-0.642295\pi$
$827$	−24.8633	−0.864581	−0.432291	+	0.901734i	$0.642295\pi$
$828$	0	0
$829$	−18.7746	−0.652069	−0.326035	−	0.945358i	$-0.605713\pi$
$829$	−18.7746	−0.652069	−0.326035	+	0.945358i	$0.605713\pi$
$830$	0	0
$831$	−7.33195	−0.254343
$832$	0	0
$833$	−4.89211	−0.169502
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−21.9227	−0.757761
$838$	0	0
$839$	29.0280	1.00216	0.501079	−	0.865402i	$-0.332937\pi$
$839$	29.0280	1.00216	0.501079	+	0.865402i	$0.332937\pi$
$840$	0	0
$841$	59.8119	2.06248
$842$	0	0
$843$	8.16131	0.281091
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.06926	0.139821
$848$	0	0
$849$	−0.301330	−0.0103416
$850$	0	0
$851$	−29.9227	−1.02574
$852$	0	0
$853$	−40.7894	−1.39660	−0.698301	−	0.715804i	$-0.746060\pi$
$853$	−40.7894	−1.39660	−0.698301	+	0.715804i	$0.746060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.6354	1.11480	0.557401	−	0.830243i	$-0.311799\pi$
$857$	32.6354	1.11480	0.557401	+	0.830243i	$0.311799\pi$
$858$	0	0
$859$	33.4906	1.14269	0.571343	−	0.820712i	$-0.306423\pi$
$859$	33.4906	1.14269	0.571343	+	0.820712i	$0.306423\pi$
$860$	0	0
$861$	−3.13333	−0.106783
$862$	0	0
$863$	−5.13795	−0.174898	−0.0874489	−	0.996169i	$-0.527871\pi$
$863$	−5.13795	−0.174898	−0.0874489	+	0.996169i	$0.527871\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.90870	0.200670
$868$	0	0
$869$	−47.1493	−1.59943
$870$	0	0
$871$	63.4427	2.14967
$872$	0	0
$873$	−27.7755	−0.940057
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.0220	0.541026	0.270513	−	0.962716i	$-0.412807\pi$
$877$	16.0220	0.541026	0.270513	+	0.962716i	$0.412807\pi$
$878$	0	0
$879$	−11.3680	−0.383434
$880$	0	0
$881$	11.5306	0.388475	0.194238	−	0.980955i	$-0.437777\pi$
$881$	11.5306	0.388475	0.194238	+	0.980955i	$0.437777\pi$
$882$	0	0
$883$	39.5161	1.32982	0.664911	−	0.746923i	$-0.268470\pi$
$883$	39.5161	1.32982	0.664911	+	0.746923i	$0.268470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.8153	0.698910	0.349455	−	0.936953i	$-0.386367\pi$
$887$	20.8153	0.698910	0.349455	+	0.936953i	$0.386367\pi$
$888$	0	0
$889$	−16.0586	−0.538588
$890$	0	0
$891$	−21.3480	−0.715184
$892$	0	0
$893$	−7.00933	−0.234558
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.45208	−0.248818
$898$	0	0
$899$	−96.9040	−3.23193
$900$	0	0
$901$	7.42401	0.247330
$902$	0	0
$903$	−3.85320	−0.128227
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.0339	−1.06367	−0.531835	−	0.846848i	$-0.678497\pi$
$907$	−32.0339	−1.06367	−0.531835	+	0.846848i	$0.678497\pi$
$908$	0	0
$909$	−4.16809	−0.138247
$910$	0	0
$911$	−2.12136	−0.0702838	−0.0351419	−	0.999382i	$-0.511188\pi$
$911$	−2.12136	−0.0702838	−0.0351419	+	0.999382i	$0.511188\pi$
$912$	0	0
$913$	−38.9439	−1.28886
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.56534	0.150761
$918$	0	0
$919$	−26.4333	−0.871955	−0.435978	−	0.899957i	$-0.643597\pi$
$919$	−26.4333	−0.871955	−0.435978	+	0.899957i	$0.643597\pi$
$920$	0	0
$921$	−1.13333	−0.0373444
$922$	0	0
$923$	−25.3361	−0.833948
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.39941	0.210184
$928$	0	0
$929$	17.7907	0.583695	0.291847	−	0.956465i	$-0.405730\pi$
$929$	17.7907	0.583695	0.291847	+	0.956465i	$0.405730\pi$
$930$	0	0
$931$	5.69735	0.186723
$932$	0	0
$933$	7.78395	0.254835
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.46396	0.243837	0.121918	−	0.992540i	$-0.461095\pi$
$937$	7.46396	0.243837	0.121918	+	0.992540i	$0.461095\pi$
$938$	0	0
$939$	−4.40790	−0.143846
$940$	0	0
$941$	−2.67869	−0.0873229	−0.0436615	−	0.999046i	$-0.513902\pi$
$941$	−2.67869	−0.0873229	−0.0436615	+	0.999046i	$0.513902\pi$
$942$	0	0
$943$	−33.3574	−1.08627
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.5840	−1.38379	−0.691897	−	0.721996i	$-0.743225\pi$
$947$	−42.5840	−1.38379	−0.691897	+	0.721996i	$0.743225\pi$
$948$	0	0
$949$	−31.9520	−1.03721
$950$	0	0
$951$	6.08528	0.197329
$952$	0	0
$953$	−60.0046	−1.94374	−0.971870	−	0.235517i	$-0.924322\pi$
$953$	−60.0046	−1.94374	−0.971870	+	0.235517i	$0.924322\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.33609	0.301793
$958$	0	0
$959$	−5.54537	−0.179069
$960$	0	0
$961$	74.7333	2.41075
$962$	0	0
$963$	39.1634	1.26202
$964$	0	0
$965$	0	0
$966$	0	0
$967$	61.1307	1.96583	0.982915	−	0.184059i	$-0.0589237\pi$
$967$	61.1307	1.96583	0.982915	+	0.184059i	$0.0589237\pi$
$968$	0	0
$969$	0.311977	0.0100221
$970$	0	0
$971$	54.6213	1.75288	0.876441	−	0.481510i	$-0.159911\pi$
$971$	54.6213	1.75288	0.876441	+	0.481510i	$0.159911\pi$
$972$	0	0
$973$	4.77075	0.152943
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.3060	0.617655	0.308827	−	0.951118i	$-0.400063\pi$
$977$	19.3060	0.617655	0.308827	+	0.951118i	$0.400063\pi$
$978$	0	0
$979$	−35.4720	−1.13369
$980$	0	0
$981$	−57.8455	−1.84686
$982$	0	0
$983$	−25.1820	−0.803182	−0.401591	−	0.915819i	$-0.631543\pi$
$983$	−25.1820	−0.803182	−0.401591	+	0.915819i	$0.631543\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.90663	0.0925189
$988$	0	0
$989$	−41.0212	−1.30440
$990$	0	0
$991$	−5.82003	−0.184879	−0.0924397	−	0.995718i	$-0.529467\pi$
$991$	−5.82003	−0.184879	−0.0924397	+	0.995718i	$0.529467\pi$
$992$	0	0
$993$	2.16932	0.0688414
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.97201	−0.284147	−0.142073	−	0.989856i	$-0.545377\pi$
$997$	−8.97201	−0.284147	−0.142073	+	0.989856i	$0.545377\pi$
$998$	0	0
$999$	−14.4508	−0.457202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.y.1.2		3
4.3	odd	2		7600.2.a.bo.1.2		3
5.2	odd	4		3800.2.d.k.3649.4		6
5.3	odd	4		3800.2.d.k.3649.3		6
5.4	even	2		760.2.a.h.1.2	✓	3
15.14	odd	2		6840.2.a.bj.1.3		3
20.19	odd	2		1520.2.a.r.1.2		3
40.19	odd	2		6080.2.a.bs.1.2		3
40.29	even	2		6080.2.a.bw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.2	✓	3	5.4	even	2
1520.2.a.r.1.2		3	20.19	odd	2
3800.2.a.y.1.2		3	1.1	even	1	trivial
3800.2.d.k.3649.3		6	5.3	odd	4
3800.2.d.k.3649.4		6	5.2	odd	4
6080.2.a.bs.1.2		3	40.19	odd	2
6080.2.a.bw.1.2		3	40.29	even	2
6840.2.a.bj.1.3		3	15.14	odd	2
7600.2.a.bo.1.2		3	4.3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q-0.363328 q^{3} -1.14134 q^{7} -2.86799 q^{9} -2.72666 q^{11} +4.64600 q^{13} +0.858664 q^{17} -1.00000 q^{19} +0.414680 q^{21} +4.41468 q^{23} +2.13201 q^{27} +9.42401 q^{29} -10.2827 q^{31} +0.990671 q^{33} -6.77801 q^{37} -1.68802 q^{39} -7.55602 q^{41} -9.29200 q^{43} +7.00933 q^{47} -5.69735 q^{49} -0.311977 q^{51} +8.64600 q^{53} +0.363328 q^{57} -5.14134 q^{59} +9.45331 q^{61} +3.27334 q^{63} +13.6553 q^{67} -1.60398 q^{69} -5.45331 q^{71} -6.87732 q^{73} +3.11203 q^{77} +17.2920 q^{79} +7.82936 q^{81} +14.2827 q^{83} -3.42401 q^{87} +13.0093 q^{89} -5.30265 q^{91} +3.73599 q^{93} +9.68463 q^{97} +7.82003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} + 11 q^{17} - 3 q^{19} - 3 q^{21} + 9 q^{23} + 19 q^{27} + 3 q^{29} - 14 q^{31} + 24 q^{33} - 14 q^{37} - 5 q^{39} - 10 q^{41} + 10 q^{43} + 4 q^{49}+ \cdots + 36 q^{99}+O(q^{100}) \) 3 * q + q^3 + 5 * q^7 + 4 * q^9 - 4 * q^11 - 5 * q^13 + 11 * q^17 - 3 * q^19 - 3 * q^21 + 9 * q^23 + 19 * q^27 + 3 * q^29 - 14 * q^31 + 24 * q^33 - 14 * q^37 - 5 * q^39 - 10 * q^41 + 10 * q^43 + 4 * q^49 - q^51 + 7 * q^53 - q^57 - 7 * q^59 + 20 * q^61 + 14 * q^63 + q^67 + 33 * q^69 - 8 * q^71 + 13 * q^73 - 16 * q^77 + 14 * q^79 + 15 * q^81 + 26 * q^83 + 15 * q^87 + 18 * q^89 - 37 * q^91 - 14 * q^93 + 6 * q^97 + 36 * q^99

Introduction

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