LMFDB - Embedded newform 1900.3.e.c.1101.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{-14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 25 $ x^4 + 4*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.4
Root			$1.22474 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.c.1101.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.74166i q^{3} +9.79796 q^{7} -5.00000 q^{9} +4.00000 q^{11} +11.2250i q^{13} +19.5959 q^{17} +(5.00000 - 18.3303i) q^{19} +36.6606i q^{21} +9.79796 q^{23} +14.9666i q^{27} -36.6606i q^{29} +36.6606i q^{31} +14.9666i q^{33} -33.6749i q^{37} -42.0000 q^{39} -36.6606i q^{41} +68.5857 q^{43} -9.79796 q^{47} +47.0000 q^{49} +73.3212i q^{51} -56.1249i q^{53} +(68.5857 + 18.7083i) q^{57} +73.3212i q^{59} +100.000 q^{61} -48.9898 q^{63} -11.2250i q^{67} +36.6606i q^{69} +36.6606i q^{71} +19.5959 q^{73} +39.1918 q^{77} -109.982i q^{79} -101.000 q^{81} +29.3939 q^{83} +137.171 q^{87} +146.642i q^{89} +109.982i q^{91} -137.171 q^{93} -123.475i q^{97} -20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9} + 16 q^{11} + 20 q^{19} - 168 q^{39} + 188 q^{49} + 400 q^{61} - 404 q^{81} - 80 q^{99}+O(q^{100}) $ 4 * q - 20 * q^9 + 16 * q^11 + 20 * q^19 - 168 * q^39 + 188 * q^49 + 400 * q^61 - 404 * q^81 - 80 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			3.74166i			1.24722i	0.781736	+	0.623610i	$0.214334\pi$
$3$			3.74166i			1.24722i	−0.781736	+	0.623610i	$0.785666\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	9.79796			1.39971			0.699854	−	0.714286i	$-0.253248\pi$
$7$	9.79796			1.39971			0.699854	+	0.714286i	$0.253248\pi$
$8$		0			0
$9$	−5.00000			−0.555556
$10$		0			0
$11$	4.00000			0.363636			0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000			0.363636			0.181818	+	0.983332i	$0.441802\pi$
$12$		0			0
$13$			11.2250i			0.863459i	0.902003	+	0.431730i	$0.142097\pi$
$13$			11.2250i			0.863459i	−0.902003	+	0.431730i	$0.857903\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.5959			1.15270			0.576351	−	0.817203i	$-0.304476\pi$
$17$	19.5959			1.15270			0.576351	+	0.817203i	$0.304476\pi$
$18$		0			0
$19$	5.00000	−	18.3303i	0.263158	−	0.964753i
$19$	5.00000	−	18.3303i	0.263158	−	0.964753i
$20$		0			0
$21$			36.6606i			1.74574i
$22$		0			0
$23$	9.79796			0.425998			0.212999	−	0.977052i	$-0.431677\pi$
$23$	9.79796			0.425998			0.212999	+	0.977052i	$0.431677\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			14.9666i			0.554320i
$28$		0			0
$29$		−	36.6606i		−	1.26416i	−0.774904	−	0.632079i	$-0.782202\pi$
$29$		−	36.6606i		−	1.26416i	0.774904	−	0.632079i	$-0.217798\pi$
$30$		0			0
$31$			36.6606i			1.18260i	0.806452	+	0.591300i	$0.201385\pi$
$31$			36.6606i			1.18260i	−0.806452	+	0.591300i	$0.798615\pi$
$32$		0			0
$33$			14.9666i			0.453534i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	33.6749i		−	0.910133i	−0.890457	−	0.455066i	$-0.849616\pi$
$37$		−	33.6749i		−	0.910133i	0.890457	−	0.455066i	$-0.150384\pi$
$38$		0			0
$39$	−42.0000			−1.07692
$40$		0			0
$41$		−	36.6606i		−	0.894161i	−0.894494	−	0.447081i	$-0.852464\pi$
$41$		−	36.6606i		−	0.894161i	0.894494	−	0.447081i	$-0.147536\pi$
$42$		0			0
$43$	68.5857			1.59502			0.797508	−	0.603308i	$-0.206151\pi$
$43$	68.5857			1.59502			0.797508	+	0.603308i	$0.206151\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.79796			−0.208467			−0.104234	−	0.994553i	$-0.533239\pi$
$47$	−9.79796			−0.208467			−0.104234	+	0.994553i	$0.533239\pi$
$48$		0			0
$49$	47.0000			0.959184
$50$		0			0
$51$			73.3212i			1.43767i
$52$		0			0
$53$		−	56.1249i		−	1.05896i	−0.848323	−	0.529480i	$-0.822387\pi$
$53$		−	56.1249i		−	1.05896i	0.848323	−	0.529480i	$-0.177613\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	68.5857	+	18.7083i	1.20326	+	0.328216i
$58$		0			0
$59$			73.3212i			1.24273i	0.783520	+	0.621366i	$0.213422\pi$
$59$			73.3212i			1.24273i	−0.783520	+	0.621366i	$0.786578\pi$
$60$		0			0
$61$	100.000			1.63934			0.819672	−	0.572833i	$-0.194156\pi$
$61$	100.000			1.63934			0.819672	+	0.572833i	$0.194156\pi$
$62$		0			0
$63$	−48.9898			−0.777616
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	11.2250i		−	0.167537i	−0.996485	−	0.0837684i	$-0.973304\pi$
$67$		−	11.2250i		−	0.167537i	0.996485	−	0.0837684i	$-0.0266956\pi$
$68$		0			0
$69$			36.6606i			0.531313i
$70$		0			0
$71$			36.6606i			0.516347i	0.966099	+	0.258173i	$0.0831205\pi$
$71$			36.6606i			0.516347i	−0.966099	+	0.258173i	$0.916879\pi$
$72$		0			0
$73$	19.5959			0.268437			0.134219	−	0.990952i	$-0.457148\pi$
$73$	19.5959			0.268437			0.134219	+	0.990952i	$0.457148\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	39.1918			0.508985
$78$		0			0
$79$		−	109.982i		−	1.39217i	−0.717957	−	0.696087i	$-0.754923\pi$
$79$		−	109.982i		−	1.39217i	0.717957	−	0.696087i	$-0.245077\pi$
$80$		0			0
$81$	−101.000			−1.24691
$82$		0			0
$83$	29.3939			0.354143			0.177072	−	0.984198i	$-0.443338\pi$
$83$	29.3939			0.354143			0.177072	+	0.984198i	$0.443338\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	137.171			1.57668
$88$		0			0
$89$			146.642i			1.64767i	0.566831	+	0.823834i	$0.308169\pi$
$89$			146.642i			1.64767i	−0.566831	+	0.823834i	$0.691831\pi$
$90$		0			0
$91$			109.982i			1.20859i
$92$		0			0
$93$	−137.171			−1.47496
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	123.475i		−	1.27293i	−0.771304	−	0.636467i	$-0.780395\pi$
$97$		−	123.475i		−	1.27293i	0.771304	−	0.636467i	$-0.219605\pi$
$98$		0			0
$99$	−20.0000			−0.202020
$100$		0			0
$101$	4.00000			0.0396040			0.0198020	−	0.999804i	$-0.493696\pi$
$101$	4.00000			0.0396040			0.0198020	+	0.999804i	$0.493696\pi$
$102$		0			0
$103$		−	78.5748i		−	0.762862i	−0.924397	−	0.381431i	$-0.875431\pi$
$103$		−	78.5748i		−	0.762862i	0.924397	−	0.381431i	$-0.124569\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	101.025i		−	0.944157i	−0.881557	−	0.472078i	$-0.843504\pi$
$107$		−	101.025i		−	0.944157i	0.881557	−	0.472078i	$-0.156496\pi$
$108$		0			0
$109$			73.3212i			0.672672i	0.941742	+	0.336336i	$0.109188\pi$
$109$			73.3212i			0.672672i	−0.941742	+	0.336336i	$0.890812\pi$
$110$		0			0
$111$	126.000			1.13514
$112$		0			0
$113$			11.2250i			0.0993360i	0.998766	+	0.0496680i	$0.0158163\pi$
$113$			11.2250i			0.0993360i	−0.998766	+	0.0496680i	$0.984184\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	56.1249i		−	0.479700i
$118$		0			0
$119$	192.000			1.61345
$120$		0			0
$121$	−105.000			−0.867769
$122$		0			0
$123$	137.171			1.11521
$124$		0			0
$125$		0			0
$126$		0			0
$127$			235.724i			1.85610i	0.372458	+	0.928049i	$0.378515\pi$
$127$			235.724i			1.85610i	−0.372458	+	0.928049i	$0.621485\pi$
$128$		0			0
$129$			256.624i			1.98934i
$130$		0			0
$131$	26.0000			0.198473			0.0992366	−	0.995064i	$-0.468360\pi$
$131$	26.0000			0.198473			0.0992366	+	0.995064i	$0.468360\pi$
$132$		0			0
$133$	48.9898	−	179.600i	0.368344	−	1.35037i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−254.747			−1.85947			−0.929733	−	0.368234i	$-0.879963\pi$
$137$	−254.747			−1.85947			−0.929733	+	0.368234i	$0.879963\pi$
$138$		0			0
$139$	−148.000			−1.06475			−0.532374	−	0.846509i	$-0.678700\pi$
$139$	−148.000			−1.06475			−0.532374	+	0.846509i	$0.678700\pi$
$140$		0			0
$141$		−	36.6606i		−	0.260004i
$142$		0			0
$143$			44.8999i			0.313985i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			175.858i			1.19631i
$148$		0			0
$149$	−52.0000			−0.348993			−0.174497	−	0.984658i	$-0.555830\pi$
$149$	−52.0000			−0.348993			−0.174497	+	0.984658i	$0.555830\pi$
$150$		0			0
$151$			73.3212i			0.485571i	0.970080	+	0.242785i	$0.0780611\pi$
$151$			73.3212i			0.485571i	−0.970080	+	0.242785i	$0.921939\pi$
$152$		0			0
$153$	−97.9796			−0.640389
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−117.576			−0.748889			−0.374444	−	0.927249i	$-0.622167\pi$
$157$	−117.576			−0.748889			−0.374444	+	0.927249i	$0.622167\pi$
$158$		0			0
$159$	210.000			1.32075
$160$		0			0
$161$	96.0000			0.596273
$162$		0			0
$163$	−9.79796			−0.0601102			−0.0300551	−	0.999548i	$-0.509568\pi$
$163$	−9.79796			−0.0601102			−0.0300551	+	0.999548i	$0.509568\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	33.6749i		−	0.201646i	−0.994904	−	0.100823i	$-0.967852\pi$
$167$		−	33.6749i		−	0.201646i	0.994904	−	0.100823i	$-0.0321476\pi$
$168$		0			0
$169$	43.0000			0.254438
$170$		0			0
$171$	−25.0000	+	91.6515i	−0.146199	+	0.535974i
$172$		0			0
$173$			280.624i			1.62211i	0.584973	+	0.811053i	$0.301105\pi$
$173$			280.624i			1.62211i	−0.584973	+	0.811053i	$0.698895\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−274.343			−1.54996
$178$		0			0
$179$		−	146.642i		−	0.819231i	−0.912258	−	0.409616i	$-0.865663\pi$
$179$		−	146.642i		−	0.819231i	0.912258	−	0.409616i	$-0.134337\pi$
$180$		0			0
$181$			256.624i			1.41781i	0.705302	+	0.708907i	$0.250811\pi$
$181$			256.624i			1.41781i	−0.705302	+	0.708907i	$0.749189\pi$
$182$		0			0
$183$			374.166i			2.04462i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	78.3837			0.419164
$188$		0			0
$189$			146.642i			0.775886i
$190$		0			0
$191$	158.000			0.827225			0.413613	−	0.910453i	$-0.364267\pi$
$191$	158.000			0.827225			0.413613	+	0.910453i	$0.364267\pi$
$192$		0			0
$193$			213.274i			1.10505i	0.833497	+	0.552525i	$0.186335\pi$
$193$			213.274i			1.10505i	−0.833497	+	0.552525i	$0.813665\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−176.363			−0.895245			−0.447622	−	0.894223i	$-0.647729\pi$
$197$	−176.363			−0.895245			−0.447622	+	0.894223i	$0.647729\pi$
$198$		0			0
$199$	−302.000			−1.51759			−0.758794	−	0.651331i	$-0.774211\pi$
$199$	−302.000			−1.51759			−0.758794	+	0.651331i	$0.774211\pi$
$200$		0			0
$201$	42.0000			0.208955
$202$		0			0
$203$		−	359.199i		−	1.76945i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−48.9898			−0.236666
$208$		0			0
$209$	20.0000	−	73.3212i	0.0956938	−	0.350819i
$210$		0			0
$211$			183.303i			0.868735i	0.900736	+	0.434367i	$0.143028\pi$
$211$			183.303i			0.868735i	−0.900736	+	0.434367i	$0.856972\pi$
$212$		0			0
$213$	−137.171			−0.643997
$214$		0			0
$215$		0			0
$216$		0			0
$217$			359.199i			1.65530i
$218$		0			0
$219$			73.3212i			0.334800i
$220$		0			0
$221$			219.964i			0.995311i
$222$		0			0
$223$			213.274i			0.956388i	0.878254	+	0.478194i	$0.158708\pi$
$223$			213.274i			0.956388i	−0.878254	+	0.478194i	$0.841292\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	280.624i		−	1.23623i	−0.786088	−	0.618115i	$-0.787897\pi$
$227$		−	280.624i		−	1.23623i	0.786088	−	0.618115i	$-0.212103\pi$
$228$		0			0
$229$	236.000			1.03057			0.515284	−	0.857020i	$-0.327687\pi$
$229$	236.000			1.03057			0.515284	+	0.857020i	$0.327687\pi$
$230$		0			0
$231$			146.642i			0.634816i
$232$		0			0
$233$	−215.555			−0.925129			−0.462565	−	0.886586i	$-0.653071\pi$
$233$	−215.555			−0.925129			−0.462565	+	0.886586i	$0.653071\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	411.514			1.73635
$238$		0			0
$239$	−146.000			−0.610879			−0.305439	−	0.952212i	$-0.598803\pi$
$239$	−146.000			−0.610879			−0.305439	+	0.952212i	$0.598803\pi$
$240$		0			0
$241$			183.303i			0.760593i	0.924865	+	0.380297i	$0.124178\pi$
$241$			183.303i			0.760593i	−0.924865	+	0.380297i	$0.875822\pi$
$242$		0			0
$243$		−	243.208i		−	1.00085i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	205.757	+	56.1249i	0.833025	+	0.227226i
$248$		0			0
$249$			109.982i			0.441694i
$250$		0			0
$251$	−250.000			−0.996016			−0.498008	−	0.867172i	$-0.665935\pi$
$251$	−250.000			−0.996016			−0.498008	+	0.867172i	$0.665935\pi$
$252$		0			0
$253$	39.1918			0.154908
$254$		0			0
$255$		0			0
$256$		0			0
$257$			145.925i			0.567800i	0.958854	+	0.283900i	$0.0916284\pi$
$257$			145.925i			0.567800i	−0.958854	+	0.283900i	$0.908372\pi$
$258$		0			0
$259$		−	329.945i		−	1.27392i
$260$		0			0
$261$			183.303i			0.702310i
$262$		0			0
$263$	88.1816			0.335291			0.167646	−	0.985847i	$-0.446384\pi$
$263$	88.1816			0.335291			0.167646	+	0.985847i	$0.446384\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−548.686			−2.05500
$268$		0			0
$269$			73.3212i			0.272570i	0.990670	+	0.136285i	$0.0435162\pi$
$269$			73.3212i			0.272570i	−0.990670	+	0.136285i	$0.956484\pi$
$270$		0			0
$271$	−68.0000			−0.250923			−0.125461	−	0.992099i	$-0.540041\pi$
$271$	−68.0000			−0.250923			−0.125461	+	0.992099i	$0.540041\pi$
$272$		0			0
$273$	−411.514			−1.50738
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−156.767			−0.565947			−0.282974	−	0.959128i	$-0.591321\pi$
$277$	−156.767			−0.565947			−0.282974	+	0.959128i	$0.591321\pi$
$278$		0			0
$279$		−	183.303i		−	0.657000i
$280$		0			0
$281$			109.982i			0.391394i	0.980664	+	0.195697i	$0.0626970\pi$
$281$			109.982i			0.391394i	−0.980664	+	0.195697i	$0.937303\pi$
$282$		0			0
$283$	480.100			1.69647			0.848233	−	0.529623i	$-0.177667\pi$
$283$	480.100			1.69647			0.848233	+	0.529623i	$0.177667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	359.199i		−	1.25156i
$288$		0			0
$289$	95.0000			0.328720
$290$		0			0
$291$	462.000			1.58763
$292$		0			0
$293$		−	505.124i		−	1.72397i	−0.506932	−	0.861986i	$-0.669221\pi$
$293$		−	505.124i		−	1.72397i	0.506932	−	0.861986i	$-0.330779\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			59.8665i			0.201571i
$298$		0			0
$299$			109.982i			0.367832i
$300$		0			0
$301$	672.000			2.23256
$302$		0			0
$303$			14.9666i			0.0493948i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	101.025i		−	0.329071i	−0.986371	−	0.164535i	$-0.947388\pi$
$307$		−	101.025i		−	0.329071i	0.986371	−	0.164535i	$-0.0526125\pi$
$308$		0			0
$309$	294.000			0.951456
$310$		0			0
$311$	−500.000			−1.60772			−0.803859	−	0.594821i	$-0.797223\pi$
$311$	−500.000			−1.60772			−0.803859	+	0.594821i	$0.797223\pi$
$312$		0			0
$313$	−215.555			−0.688674			−0.344337	−	0.938846i	$-0.611896\pi$
$313$	−215.555			−0.688674			−0.344337	+	0.938846i	$0.611896\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	280.624i		−	0.885250i	−0.896707	−	0.442625i	$-0.854047\pi$
$317$		−	280.624i		−	0.885250i	0.896707	−	0.442625i	$-0.145953\pi$
$318$		0			0
$319$		−	146.642i		−	0.459694i
$320$		0			0
$321$	378.000			1.17757
$322$		0			0
$323$	97.9796	−	359.199i	0.303342	−	1.11207i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−274.343			−0.838969
$328$		0			0
$329$	−96.0000			−0.291793
$330$		0			0
$331$			403.267i			1.21833i	0.793044	+	0.609164i	$0.208495\pi$
$331$			403.267i			1.21833i	−0.793044	+	0.609164i	$0.791505\pi$
$332$		0			0
$333$			168.375i			0.505629i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	101.025i		−	0.299777i	−0.988703	−	0.149888i	$-0.952109\pi$
$337$		−	101.025i		−	0.299777i	0.988703	−	0.149888i	$-0.0478914\pi$
$338$		0			0
$339$	−42.0000			−0.123894
$340$		0			0
$341$			146.642i			0.430036i
$342$		0			0
$343$	−19.5959			−0.0571310
$344$		0			0
$345$		0			0
$346$		0			0
$347$	617.271			1.77888			0.889440	−	0.457052i	$-0.151095\pi$
$347$	617.271			1.77888			0.889440	+	0.457052i	$0.151095\pi$
$348$		0			0
$349$	−230.000			−0.659026			−0.329513	−	0.944151i	$-0.606885\pi$
$349$	−230.000			−0.659026			−0.329513	+	0.944151i	$0.606885\pi$
$350$		0			0
$351$	−168.000			−0.478632
$352$		0			0
$353$	548.686			1.55435			0.777175	−	0.629284i	$-0.216652\pi$
$353$	548.686			1.55435			0.777175	+	0.629284i	$0.216652\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			718.398i			2.01232i
$358$		0			0
$359$	−604.000			−1.68245			−0.841226	−	0.540684i	$-0.818165\pi$
$359$	−604.000			−1.68245			−0.841226	+	0.540684i	$0.818165\pi$
$360$		0			0
$361$	−311.000	−	183.303i	−0.861496	−	0.507765i
$362$		0			0
$363$		−	392.874i		−	1.08230i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−342.929			−0.934410			−0.467205	−	0.884149i	$-0.654739\pi$
$367$	−342.929			−0.934410			−0.467205	+	0.884149i	$0.654739\pi$
$368$		0			0
$369$			183.303i			0.496756i
$370$		0			0
$371$		−	549.909i		−	1.48223i
$372$		0			0
$373$			123.475i			0.331031i	0.986207	+	0.165516i	$0.0529288\pi$
$373$			123.475i			0.331031i	−0.986207	+	0.165516i	$0.947071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	411.514			1.09155
$378$		0			0
$379$		−	329.945i		−	0.870568i	−0.900293	−	0.435284i	$-0.856648\pi$
$379$		−	329.945i		−	0.870568i	0.900293	−	0.435284i	$-0.143352\pi$
$380$		0			0
$381$	−882.000			−2.31496
$382$		0			0
$383$			11.2250i			0.0293080i	0.999893	+	0.0146540i	$0.00466468\pi$
$383$			11.2250i			0.0293080i	−0.999893	+	0.0146540i	$0.995335\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−342.929			−0.886120
$388$		0			0
$389$	−122.000			−0.313625			−0.156812	−	0.987628i	$-0.550122\pi$
$389$	−122.000			−0.313625			−0.156812	+	0.987628i	$0.550122\pi$
$390$		0			0
$391$	192.000			0.491049
$392$		0			0
$393$			97.2831i			0.247540i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	470.302			1.18464			0.592320	−	0.805703i	$-0.298212\pi$
$397$	470.302			1.18464			0.592320	+	0.805703i	$0.298212\pi$
$398$		0			0
$399$	672.000	+	183.303i	1.68421	+	0.459406i
$400$		0			0
$401$		−	219.964i		−	0.548538i	−0.961653	−	0.274269i	$-0.911564\pi$
$401$		−	219.964i		−	0.548538i	0.961653	−	0.274269i	$-0.0884358\pi$
$402$		0			0
$403$	−411.514			−1.02113
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	134.700i		−	0.330957i
$408$		0			0
$409$			329.945i			0.806713i	0.915043	+	0.403356i	$0.132156\pi$
$409$			329.945i			0.806713i	−0.915043	+	0.403356i	$0.867844\pi$
$410$		0			0
$411$		−	953.176i		−	2.31916i
$412$		0			0
$413$			718.398i			1.73946i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	553.765i		−	1.32797i
$418$		0			0
$419$	358.000			0.854415			0.427208	−	0.904154i	$-0.359497\pi$
$419$	358.000			0.854415			0.427208	+	0.904154i	$0.359497\pi$
$420$		0			0
$421$			73.3212i			0.174160i	0.996201	+	0.0870798i	$0.0277535\pi$
$421$			73.3212i			0.174160i	−0.996201	+	0.0870798i	$0.972246\pi$
$422$		0			0
$423$	48.9898			0.115815
$424$		0			0
$425$		0			0
$426$		0			0
$427$	979.796			2.29460
$428$		0			0
$429$	−168.000			−0.391608
$430$		0			0
$431$		−	733.212i		−	1.70119i	−0.525823	−	0.850594i	$-0.676243\pi$
$431$		−	733.212i		−	1.70119i	0.525823	−	0.850594i	$-0.323757\pi$
$432$		0			0
$433$			190.825i			0.440703i	0.975420	+	0.220352i	$0.0707205\pi$
$433$			190.825i			0.440703i	−0.975420	+	0.220352i	$0.929280\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	48.9898	−	179.600i	0.112105	−	0.410983i
$438$		0			0
$439$		−	183.303i		−	0.417547i	−0.977964	−	0.208773i	$-0.933053\pi$
$439$		−	183.303i		−	0.417547i	0.977964	−	0.208773i	$-0.0669471\pi$
$440$		0			0
$441$	−235.000			−0.532880
$442$		0			0
$443$	−264.545			−0.597167			−0.298583	−	0.954384i	$-0.596514\pi$
$443$	−264.545			−0.597167			−0.298583	+	0.954384i	$0.596514\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	194.566i		−	0.435271i
$448$		0			0
$449$			476.588i			1.06144i	0.847546	+	0.530721i	$0.178079\pi$
$449$			476.588i			1.06144i	−0.847546	+	0.530721i	$0.821921\pi$
$450$		0			0
$451$		−	146.642i		−	0.325149i
$452$		0			0
$453$	−274.343			−0.605613
$454$		0			0
$455$		0			0
$456$		0			0
$457$	470.302			1.02911			0.514554	−	0.857458i	$-0.327958\pi$
$457$	470.302			1.02911			0.514554	+	0.857458i	$0.327958\pi$
$458$		0			0
$459$			293.285i			0.638965i
$460$		0			0
$461$	−22.0000			−0.0477223			−0.0238612	−	0.999715i	$-0.507596\pi$
$461$	−22.0000			−0.0477223			−0.0238612	+	0.999715i	$0.507596\pi$
$462$		0			0
$463$	578.080			1.24855			0.624276	−	0.781204i	$-0.285394\pi$
$463$	578.080			1.24855			0.624276	+	0.781204i	$0.285394\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	421.312			0.902168			0.451084	−	0.892482i	$-0.351038\pi$
$467$	421.312			0.902168			0.451084	+	0.892482i	$0.351038\pi$
$468$		0			0
$469$		−	109.982i		−	0.234503i
$470$		0			0
$471$		−	439.927i		−	0.934028i
$472$		0			0
$473$	274.343			0.580006
$474$		0			0
$475$		0			0
$476$		0			0
$477$			280.624i			0.588311i
$478$		0			0
$479$	500.000			1.04384			0.521921	−	0.852994i	$-0.325216\pi$
$479$	500.000			1.04384			0.521921	+	0.852994i	$0.325216\pi$
$480$		0			0
$481$	378.000			0.785863
$482$		0			0
$483$			359.199i			0.743683i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	11.2250i		−	0.0230492i	−0.999934	−	0.0115246i	$-0.996332\pi$
$487$		−	11.2250i		−	0.0230492i	0.999934	−	0.0115246i	$-0.00366848\pi$
$488$		0			0
$489$		−	36.6606i		−	0.0749706i
$490$		0			0
$491$	230.000			0.468432			0.234216	−	0.972185i	$-0.424748\pi$
$491$	230.000			0.468432			0.234216	+	0.972185i	$0.424748\pi$
$492$		0			0
$493$		−	718.398i		−	1.45720i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			359.199i			0.722735i
$498$		0			0
$499$	620.000			1.24248			0.621242	−	0.783618i	$-0.286628\pi$
$499$	620.000			1.24248			0.621242	+	0.783618i	$0.286628\pi$
$500$		0			0
$501$	126.000			0.251497
$502$		0			0
$503$	656.463			1.30510			0.652548	−	0.757747i	$-0.273700\pi$
$503$	656.463			1.30510			0.652548	+	0.757747i	$0.273700\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			160.891i			0.317340i
$508$		0			0
$509$		−	769.873i		−	1.51252i	−0.654271	−	0.756260i	$-0.727024\pi$
$509$		−	769.873i		−	1.51252i	0.654271	−	0.756260i	$-0.272976\pi$
$510$		0			0
$511$	192.000			0.375734
$512$		0			0
$513$	274.343	+	74.8331i	0.534781	+	0.145874i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−39.1918			−0.0758063
$518$		0			0
$519$	−1050.00			−2.02312
$520$		0			0
$521$		−	659.891i		−	1.26659i	−0.773912	−	0.633293i	$-0.781703\pi$
$521$		−	659.891i		−	1.26659i	0.773912	−	0.633293i	$-0.218297\pi$
$522$		0			0
$523$			841.873i			1.60970i	0.593479	+	0.804850i	$0.297754\pi$
$523$			841.873i			1.60970i	−0.593479	+	0.804850i	$0.702246\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			718.398i			1.36318i
$528$		0			0
$529$	−433.000			−0.818526
$530$		0			0
$531$		−	366.606i		−	0.690407i
$532$		0			0
$533$	411.514			0.772072
$534$		0			0
$535$		0			0
$536$		0			0
$537$	548.686			1.02176
$538$		0			0
$539$	188.000			0.348794
$540$		0			0
$541$	−620.000			−1.14603			−0.573013	−	0.819546i	$-0.694226\pi$
$541$	−620.000			−1.14603			−0.573013	+	0.819546i	$0.694226\pi$
$542$		0			0
$543$	−960.200			−1.76832
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	931.673i		−	1.70324i	−0.524159	−	0.851620i	$-0.675620\pi$
$547$		−	931.673i		−	1.70324i	0.524159	−	0.851620i	$-0.324380\pi$
$548$		0			0
$549$	−500.000			−0.910747
$550$		0			0
$551$	−672.000	−	183.303i	−1.21960	−	0.332673i
$552$		0			0
$553$		−	1077.60i		−	1.94864i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−548.686			−0.985073			−0.492537	−	0.870292i	$-0.663930\pi$
$557$	−548.686			−0.985073			−0.492537	+	0.870292i	$0.663930\pi$
$558$		0			0
$559$			769.873i			1.37723i
$560$		0			0
$561$			293.285i			0.522789i
$562$		0			0
$563$		−	325.524i		−	0.578196i	−0.957300	−	0.289098i	$-0.906645\pi$
$563$		−	325.524i		−	0.578196i	0.957300	−	0.289098i	$-0.0933553\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−989.594			−1.74532
$568$		0			0
$569$			36.6606i			0.0644299i	0.999481	+	0.0322149i	$0.0102561\pi$
$569$			36.6606i			0.0644299i	−0.999481	+	0.0322149i	$0.989744\pi$
$570$		0			0
$571$	−332.000			−0.581436			−0.290718	−	0.956809i	$-0.593894\pi$
$571$	−332.000			−0.581436			−0.290718	+	0.956809i	$0.593894\pi$
$572$		0			0
$573$			591.182i			1.03173i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−979.796			−1.69809			−0.849043	−	0.528323i	$-0.822821\pi$
$577$	−979.796			−1.69809			−0.849043	+	0.528323i	$0.822821\pi$
$578$		0			0
$579$	−798.000			−1.37824
$580$		0			0
$581$	288.000			0.495697
$582$		0			0
$583$		−	224.499i		−	0.385076i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	852.422			1.45217			0.726084	−	0.687606i	$-0.241338\pi$
$587$	852.422			1.45217			0.726084	+	0.687606i	$0.241338\pi$
$588$		0			0
$589$	672.000	+	183.303i	1.14092	+	0.311211i
$590$		0			0
$591$		−	659.891i		−	1.11657i
$592$		0			0
$593$	489.898			0.826135			0.413067	−	0.910700i	$-0.364457\pi$
$593$	489.898			0.826135			0.413067	+	0.910700i	$0.364457\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1129.98i		−	1.89276i
$598$		0			0
$599$		−	916.515i		−	1.53008i	−0.643986	−	0.765038i	$-0.722720\pi$
$599$		−	916.515i		−	1.53008i	0.643986	−	0.765038i	$-0.277280\pi$
$600$		0			0
$601$			843.194i			1.40298i	0.712677	+	0.701492i	$0.247483\pi$
$601$			843.194i			1.40298i	−0.712677	+	0.701492i	$0.752517\pi$
$602$		0			0
$603$			56.1249i			0.0930761i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			976.573i			1.60885i	0.594054	+	0.804426i	$0.297527\pi$
$607$			976.573i			1.60885i	−0.594054	+	0.804426i	$0.702473\pi$
$608$		0			0
$609$	1344.00			2.20690
$610$		0			0
$611$		−	109.982i		−	0.180003i
$612$		0			0
$613$	1077.78			1.75820			0.879099	−	0.476639i	$-0.158145\pi$
$613$	1077.78			1.75820			0.879099	+	0.476639i	$0.158145\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−685.857			−1.11160			−0.555800	−	0.831316i	$-0.687588\pi$
$617$	−685.857			−1.11160			−0.555800	+	0.831316i	$0.687588\pi$
$618$		0			0
$619$	−676.000			−1.09208			−0.546042	−	0.837758i	$-0.683866\pi$
$619$	−676.000			−1.09208			−0.546042	+	0.837758i	$0.683866\pi$
$620$		0			0
$621$			146.642i			0.236139i
$622$		0			0
$623$			1436.80i			2.30625i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	274.343	+	74.8331i	0.437548	+	0.119351i
$628$		0			0
$629$		−	659.891i		−	1.04911i
$630$		0			0
$631$	124.000			0.196513			0.0982567	−	0.995161i	$-0.468673\pi$
$631$	124.000			0.196513			0.0982567	+	0.995161i	$0.468673\pi$
$632$		0			0
$633$	−685.857			−1.08350
$634$		0			0
$635$		0			0
$636$		0			0
$637$			527.574i			0.828216i
$638$		0			0
$639$		−	183.303i		−	0.286859i
$640$		0			0
$641$		−	623.230i		−	0.972278i	−0.873881	−	0.486139i	$-0.838405\pi$
$641$		−	623.230i		−	0.972278i	0.873881	−	0.486139i	$-0.161595\pi$
$642$		0			0
$643$	−244.949			−0.380947			−0.190474	−	0.981692i	$-0.561002\pi$
$643$	−244.949			−0.380947			−0.190474	+	0.981692i	$0.561002\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−342.929			−0.530029			−0.265014	−	0.964244i	$-0.585377\pi$
$647$	−342.929			−0.530029			−0.265014	+	0.964244i	$0.585377\pi$
$648$		0			0
$649$			293.285i			0.451903i
$650$		0			0
$651$	−1344.00			−2.06452
$652$		0			0
$653$	−1038.58			−1.59048			−0.795240	−	0.606295i	$-0.792655\pi$
$653$	−1038.58			−1.59048			−0.795240	+	0.606295i	$0.792655\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−97.9796			−0.149132
$658$		0			0
$659$		−	879.855i		−	1.33514i	−0.744549	−	0.667568i	$-0.767335\pi$
$659$		−	879.855i		−	1.33514i	0.744549	−	0.667568i	$-0.232665\pi$
$660$		0			0
$661$		−	586.570i		−	0.887397i	−0.896176	−	0.443699i	$-0.853666\pi$
$661$		−	586.570i		−	0.887397i	0.896176	−	0.443699i	$-0.146334\pi$
$662$		0			0
$663$	−823.029			−1.24137
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	359.199i		−	0.538529i
$668$		0			0
$669$	−798.000			−1.19283
$670$		0			0
$671$	400.000			0.596125
$672$		0			0
$673$		−	1156.17i		−	1.71794i	−0.512028	−	0.858969i	$-0.671106\pi$
$673$		−	1156.17i		−	1.71794i	0.512028	−	0.858969i	$-0.328894\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			325.524i			0.480833i	0.970670	+	0.240417i	$0.0772841\pi$
$677$			325.524i			0.480833i	−0.970670	+	0.240417i	$0.922716\pi$
$678$		0			0
$679$		−	1209.80i		−	1.78174i
$680$		0			0
$681$	1050.00			1.54185
$682$		0			0
$683$		−	1133.72i		−	1.65992i	−0.557826	−	0.829958i	$-0.688364\pi$
$683$		−	1133.72i		−	1.65992i	0.557826	−	0.829958i	$-0.311636\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			883.031i			1.28534i
$688$		0			0
$689$	630.000			0.914369
$690$		0			0
$691$	−572.000			−0.827786			−0.413893	−	0.910326i	$-0.635831\pi$
$691$	−572.000			−0.827786			−0.413893	+	0.910326i	$0.635831\pi$
$692$		0			0
$693$	−195.959			−0.282769
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	718.398i		−	1.03070i
$698$		0			0
$699$		−	806.533i		−	1.15384i
$700$		0			0
$701$	820.000			1.16976			0.584879	−	0.811121i	$-0.301142\pi$
$701$	820.000			1.16976			0.584879	+	0.811121i	$0.301142\pi$
$702$		0			0
$703$	−617.271	−	168.375i	−0.878053	−	0.239509i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	39.1918			0.0554340
$708$		0			0
$709$	−278.000			−0.392102			−0.196051	−	0.980594i	$-0.562812\pi$
$709$	−278.000			−0.392102			−0.196051	+	0.980594i	$0.562812\pi$
$710$		0			0
$711$			549.909i			0.773430i
$712$		0			0
$713$			359.199i			0.503786i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	546.282i		−	0.761900i
$718$		0			0
$719$	−604.000			−0.840056			−0.420028	−	0.907511i	$-0.637980\pi$
$719$	−604.000			−0.840056			−0.420028	+	0.907511i	$0.637980\pi$
$720$		0			0
$721$		−	769.873i		−	1.06778i
$722$		0			0
$723$	−685.857			−0.948627
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−617.271			−0.849067			−0.424533	−	0.905412i	$-0.639562\pi$
$727$	−617.271			−0.849067			−0.424533	+	0.905412i	$0.639562\pi$
$728$		0			0
$729$	1.00000			0.00137174
$730$		0			0
$731$	1344.00			1.83858
$732$		0			0
$733$	137.171			0.187137			0.0935685	−	0.995613i	$-0.470173\pi$
$733$	137.171			0.187137			0.0935685	+	0.995613i	$0.470173\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	44.8999i		−	0.0609225i
$738$		0			0
$739$	298.000			0.403248			0.201624	−	0.979463i	$-0.435378\pi$
$739$	298.000			0.403248			0.201624	+	0.979463i	$0.435378\pi$
$740$		0			0
$741$	−210.000	+	769.873i	−0.283401	+	1.03896i
$742$		0			0
$743$		−	1313.32i		−	1.76759i	−0.467871	−	0.883797i	$-0.654979\pi$
$743$		−	1313.32i		−	1.76759i	0.467871	−	0.883797i	$-0.345021\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−146.969			−0.196746
$748$		0			0
$749$		−	989.836i		−	1.32154i
$750$		0			0
$751$		−	659.891i		−	0.878683i	−0.898320	−	0.439341i	$-0.855212\pi$
$751$		−	659.891i		−	0.878683i	0.898320	−	0.439341i	$-0.144788\pi$
$752$		0			0
$753$		−	935.414i		−	1.24225i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1058.18			−1.39786			−0.698930	−	0.715190i	$-0.746340\pi$
$757$	−1058.18			−1.39786			−0.698930	+	0.715190i	$0.746340\pi$
$758$		0			0
$759$			146.642i			0.193205i
$760$		0			0
$761$	−788.000			−1.03548			−0.517740	−	0.855538i	$-0.673226\pi$
$761$	−788.000			−1.03548			−0.517740	+	0.855538i	$0.673226\pi$
$762$		0			0
$763$			718.398i			0.941544i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−823.029			−1.07305
$768$		0			0
$769$	548.000			0.712614			0.356307	−	0.934369i	$-0.384036\pi$
$769$	548.000			0.712614			0.356307	+	0.934369i	$0.384036\pi$
$770$		0			0
$771$	−546.000			−0.708171
$772$		0			0
$773$			819.423i			1.06006i	0.847980	+	0.530028i	$0.177819\pi$
$773$			819.423i			1.06006i	−0.847980	+	0.530028i	$0.822181\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1234.54			1.58886
$778$		0			0
$779$	−672.000	−	183.303i	−0.862644	−	0.235306i
$780$		0			0
$781$			146.642i			0.187762i
$782$		0			0
$783$	548.686			0.700748
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	931.673i		−	1.18383i	−0.806001	−	0.591914i	$-0.798372\pi$
$787$		−	931.673i		−	1.18383i	0.806001	−	0.591914i	$-0.201628\pi$
$788$		0			0
$789$			329.945i			0.418182i
$790$		0			0
$791$			109.982i			0.139041i
$792$		0			0
$793$			1122.50i			1.41551i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			774.523i			0.971798i	0.874015	+	0.485899i	$0.161508\pi$
$797$			774.523i			0.971798i	−0.874015	+	0.485899i	$0.838492\pi$
$798$		0			0
$799$	−192.000			−0.240300
$800$		0			0
$801$		−	733.212i		−	0.915371i
$802$		0			0
$803$	78.3837			0.0976135
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−274.343			−0.339954
$808$		0			0
$809$	−638.000			−0.788628			−0.394314	−	0.918976i	$-0.629018\pi$
$809$	−638.000			−0.788628			−0.394314	+	0.918976i	$0.629018\pi$
$810$		0			0
$811$		−	73.3212i		−	0.0904084i	−0.998978	−	0.0452042i	$-0.985606\pi$
$811$		−	73.3212i		−	0.0904084i	0.998978	−	0.0452042i	$-0.0143938\pi$
$812$		0			0
$813$		−	254.433i		−	0.312955i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	342.929	−	1257.20i	0.419741	−	1.53880i
$818$		0			0
$819$		−	549.909i		−	0.671440i
$820$		0			0
$821$	−1318.00			−1.60536			−0.802680	−	0.596410i	$-0.796593\pi$
$821$	−1318.00			−1.60536			−0.802680	+	0.596410i	$0.796593\pi$
$822$		0			0
$823$	−88.1816			−0.107147			−0.0535733	−	0.998564i	$-0.517061\pi$
$823$	−88.1816			−0.107147			−0.0535733	+	0.998564i	$0.517061\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1133.72i			1.37089i	0.728127	+	0.685443i	$0.240391\pi$
$827$			1133.72i			1.37089i	−0.728127	+	0.685443i	$0.759609\pi$
$828$		0			0
$829$			1466.42i			1.76891i	0.466628	+	0.884454i	$0.345469\pi$
$829$			1466.42i			1.76891i	−0.466628	+	0.884454i	$0.654531\pi$
$830$		0			0
$831$		−	586.570i		−	0.705860i
$832$		0			0
$833$	921.008			1.10565
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−548.686			−0.655538
$838$		0			0
$839$		−	623.230i		−	0.742825i	−0.928468	−	0.371413i	$-0.878874\pi$
$839$		−	623.230i		−	0.742825i	0.928468	−	0.371413i	$-0.121126\pi$
$840$		0			0
$841$	−503.000			−0.598098
$842$		0			0
$843$	−411.514			−0.488155
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1028.79			−1.21462
$848$		0			0
$849$			1796.37i			2.11587i
$850$		0			0
$851$		−	329.945i		−	0.387715i
$852$		0			0
$853$	921.008			1.07973			0.539864	−	0.841752i	$-0.318476\pi$
$853$	921.008			1.07973			0.539864	+	0.841752i	$0.318476\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1133.72i			1.32290i	0.749991	+	0.661448i	$0.230058\pi$
$857$			1133.72i			1.32290i	−0.749991	+	0.661448i	$0.769942\pi$
$858$		0			0
$859$	−746.000			−0.868452			−0.434226	−	0.900804i	$-0.642978\pi$
$859$	−746.000			−0.868452			−0.434226	+	0.900804i	$0.642978\pi$
$860$		0			0
$861$	1344.00			1.56098
$862$		0			0
$863$		−	235.724i		−	0.273145i	−0.990630	−	0.136573i	$-0.956391\pi$
$863$		−	235.724i		−	0.273145i	0.990630	−	0.136573i	$-0.0436087\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			355.457i			0.409986i
$868$		0			0
$869$		−	439.927i		−	0.506245i
$870$		0			0
$871$	126.000			0.144661
$872$		0			0
$873$			617.373i			0.707186i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			145.925i			0.166391i	0.996533	+	0.0831953i	$0.0265125\pi$
$877$			145.925i			0.166391i	−0.996533	+	0.0831953i	$0.973487\pi$
$878$		0			0
$879$	1890.00			2.15017
$880$		0			0
$881$	−1028.00			−1.16686			−0.583428	−	0.812165i	$-0.698289\pi$
$881$	−1028.00			−1.16686			−0.583428	+	0.812165i	$0.698289\pi$
$882$		0			0
$883$	930.806			1.05414			0.527070	−	0.849822i	$-0.323290\pi$
$883$	930.806			1.05414			0.527070	+	0.849822i	$0.323290\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			168.375i			0.189825i	0.995486	+	0.0949124i	$0.0302571\pi$
$887$			168.375i			0.189825i	−0.995486	+	0.0949124i	$0.969743\pi$
$888$		0			0
$889$			2309.62i			2.59800i
$890$		0			0
$891$	−404.000			−0.453423
$892$		0			0
$893$	−48.9898	+	179.600i	−0.0548598	+	0.201119i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−411.514			−0.458767
$898$		0			0
$899$	1344.00			1.49499
$900$		0			0
$901$		−	1099.82i		−	1.22066i
$902$		0			0
$903$			2514.39i			2.78449i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			347.974i			0.383654i	0.981429	+	0.191827i	$0.0614412\pi$
$907$			347.974i			0.383654i	−0.981429	+	0.191827i	$0.938559\pi$
$908$		0			0
$909$	−20.0000			−0.0220022
$910$		0			0
$911$			1246.46i			1.36823i	0.729373	+	0.684117i	$0.239812\pi$
$911$			1246.46i			1.36823i	−0.729373	+	0.684117i	$0.760188\pi$
$912$		0			0
$913$	117.576			0.128779
$914$		0			0
$915$		0			0
$916$		0			0
$917$	254.747			0.277805
$918$		0			0
$919$	1150.00			1.25136			0.625680	−	0.780080i	$-0.284822\pi$
$919$	1150.00			1.25136			0.625680	+	0.780080i	$0.284822\pi$
$920$		0			0
$921$	378.000			0.410423
$922$		0			0
$923$	−411.514			−0.445844
$924$		0			0
$925$		0			0
$926$		0			0
$927$			392.874i			0.423812i
$928$		0			0
$929$	−1454.00			−1.56512			−0.782562	−	0.622573i	$-0.786087\pi$
$929$	−1454.00			−1.56512			−0.782562	+	0.622573i	$0.786087\pi$
$930$		0			0
$931$	235.000	−	861.524i	0.252417	−	0.925375i
$932$		0			0
$933$		−	1870.83i		−	2.00518i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	823.029			0.878366			0.439183	−	0.898398i	$-0.355268\pi$
$937$	823.029			0.878366			0.439183	+	0.898398i	$0.355268\pi$
$938$		0			0
$939$		−	806.533i		−	0.858928i
$940$		0			0
$941$			329.945i			0.350633i	0.984512	+	0.175316i	$0.0560948\pi$
$941$			329.945i			0.350633i	−0.984512	+	0.175316i	$0.943905\pi$
$942$		0			0
$943$		−	359.199i		−	0.380911i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	754.443			0.796666			0.398333	−	0.917241i	$-0.369589\pi$
$947$	754.443			0.796666			0.398333	+	0.917241i	$0.369589\pi$
$948$		0			0
$949$			219.964i			0.231785i
$950$		0			0
$951$	1050.00			1.10410
$952$		0			0
$953$			931.673i			0.977621i	0.872390	+	0.488810i	$0.162569\pi$
$953$			931.673i			0.977621i	−0.872390	+	0.488810i	$0.837431\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	548.686			0.573339
$958$		0			0
$959$	−2496.00			−2.60271
$960$		0			0
$961$	−383.000			−0.398543
$962$		0			0
$963$			505.124i			0.524531i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−891.614			−0.922042			−0.461021	−	0.887389i	$-0.652517\pi$
$967$	−891.614			−0.922042			−0.461021	+	0.887389i	$0.652517\pi$
$968$		0			0
$969$	1344.00	+	366.606i	1.38700	+	0.378334i
$970$		0			0
$971$		−	73.3212i		−	0.0755110i	−0.999287	−	0.0377555i	$-0.987979\pi$
$971$		−	73.3212i		−	0.0755110i	0.999287	−	0.0377555i	$-0.0120208\pi$
$972$		0			0
$973$	−1450.10			−1.49034
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	280.624i		−	0.287231i	−0.989634	−	0.143615i	$-0.954127\pi$
$977$		−	280.624i		−	0.287231i	0.989634	−	0.143615i	$-0.0458728\pi$
$978$		0			0
$979$			586.570i			0.599152i
$980$		0			0
$981$		−	366.606i		−	0.373706i
$982$		0			0
$983$			460.224i			0.468183i	0.972215	+	0.234091i	$0.0752115\pi$
$983$			460.224i			0.468183i	−0.972215	+	0.234091i	$0.924788\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	359.199i		−	0.363930i
$988$		0			0
$989$	672.000			0.679474
$990$		0			0
$991$			73.3212i			0.0739871i	0.999316	+	0.0369935i	$0.0117781\pi$
$991$			73.3212i			0.0739871i	−0.999316	+	0.0369935i	$0.988222\pi$
$992$		0			0
$993$	−1508.89			−1.51952
$994$		0			0
$995$		0			0
$996$		0			0
$997$	529.090			0.530682			0.265341	−	0.964155i	$-0.414515\pi$
$997$	529.090			0.530682			0.265341	+	0.964155i	$0.414515\pi$
$998$		0			0
$999$	504.000			0.504505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.c.1101.4		4
5.2	odd	4		380.3.g.a.189.4	yes	4
5.3	odd	4		380.3.g.a.189.1	✓	4
5.4	even	2	inner	1900.3.e.c.1101.1		4
15.2	even	4		3420.3.h.c.2089.4		4
15.8	even	4		3420.3.h.c.2089.1		4
19.18	odd	2	inner	1900.3.e.c.1101.2		4
95.18	even	4		380.3.g.a.189.3	yes	4
95.37	even	4		380.3.g.a.189.2	yes	4
95.94	odd	2	inner	1900.3.e.c.1101.3		4
285.113	odd	4		3420.3.h.c.2089.2		4
285.227	odd	4		3420.3.h.c.2089.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.a.189.1	✓	4	5.3	odd	4
380.3.g.a.189.2	yes	4	95.37	even	4
380.3.g.a.189.3	yes	4	95.18	even	4
380.3.g.a.189.4	yes	4	5.2	odd	4
1900.3.e.c.1101.1		4	5.4	even	2	inner
1900.3.e.c.1101.2		4	19.18	odd	2	inner
1900.3.e.c.1101.3		4	95.94	odd	2	inner
1900.3.e.c.1101.4		4	1.1	even	1	trivial
3420.3.h.c.2089.1		4	15.8	even	4
3420.3.h.c.2089.2		4	285.113	odd	4
3420.3.h.c.2089.3		4	285.227	odd	4
3420.3.h.c.2089.4		4	15.2	even	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.74166i q^{3} +9.79796 q^{7} -5.00000 q^{9} +4.00000 q^{11} +11.2250i q^{13} +19.5959 q^{17} +(5.00000 - 18.3303i) q^{19} +36.6606i q^{21} +9.79796 q^{23} +14.9666i q^{27} -36.6606i q^{29} +36.6606i q^{31} +14.9666i q^{33} -33.6749i q^{37} -42.0000 q^{39} -36.6606i q^{41} +68.5857 q^{43} -9.79796 q^{47} +47.0000 q^{49} +73.3212i q^{51} -56.1249i q^{53} +(68.5857 + 18.7083i) q^{57} +73.3212i q^{59} +100.000 q^{61} -48.9898 q^{63} -11.2250i q^{67} +36.6606i q^{69} +36.6606i q^{71} +19.5959 q^{73} +39.1918 q^{77} -109.982i q^{79} -101.000 q^{81} +29.3939 q^{83} +137.171 q^{87} +146.642i q^{89} +109.982i q^{91} -137.171 q^{93} -123.475i q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9} + 16 q^{11} + 20 q^{19} - 168 q^{39} + 188 q^{49} + 400 q^{61} - 404 q^{81} - 80 q^{99}+O(q^{100}) \) 4 * q - 20 * q^9 + 16 * q^11 + 20 * q^19 - 168 * q^39 + 188 * q^49 + 400 * q^61 - 404 * q^81 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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