LMFDB - Embedded newform 1900.2.c.d.1749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not 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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.d.1749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786i q^{3} +4.82843i q^{7} +2.65685 q^{9} -2.00000 q^{11} +2.24264i q^{13} +4.82843i q^{17} +1.00000 q^{19} +2.82843 q^{21} -6.00000i q^{23} -3.31371i q^{27} -10.4853 q^{29} -1.17157 q^{31} +1.17157i q^{33} +10.2426i q^{37} +1.31371 q^{39} -7.65685 q^{41} -0.828427i q^{43} -0.828427i q^{47} -16.3137 q^{49} +2.82843 q^{51} -5.07107i q^{53} -0.585786i q^{57} -2.34315 q^{59} -2.34315 q^{61} +12.8284i q^{63} +0.585786i q^{67} -3.51472 q^{69} +10.8284 q^{71} +14.4853i q^{73} -9.65685i q^{77} +9.65685 q^{79} +6.02944 q^{81} +9.31371i q^{83} +6.14214i q^{87} -10.4853 q^{89} -10.8284 q^{91} +0.686292i q^{93} +1.75736i q^{97} -5.31371 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} - 8 q^{11} + 4 q^{19} - 8 q^{29} - 16 q^{31} - 40 q^{39} - 8 q^{41} - 20 q^{49} - 32 q^{59} - 32 q^{61} - 48 q^{69} + 32 q^{71} + 16 q^{79} + 92 q^{81} - 8 q^{89} - 32 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^19 - 8 * q^29 - 16 * q^31 - 40 * q^39 - 8 * q^41 - 20 * q^49 - 32 * q^59 - 32 * q^61 - 48 * q^69 + 32 * q^71 + 16 * q^79 + 92 * q^81 - 8 * q^89 - 32 * q^91 + 24 * 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
$3$	− 0.585786i	− 0.338204i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	− 0.585786i	− 0.338204i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.82843i	1.82497i	0.409106	+	0.912487i	$0.365841\pi$
$7$	4.82843i	1.82497i	−0.409106	+	0.912487i	$0.634159\pi$
$8$	0	0
$9$	2.65685	0.885618
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.24264i	0.621997i	0.950410	+	0.310998i	$0.100663\pi$
$13$	2.24264i	0.621997i	−0.950410	+	0.310998i	$0.899337\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843i	1.17107i	0.810649	+	0.585533i	$0.199115\pi$
$17$	4.82843i	1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.31371i	− 0.637723i
$28$	0	0
$29$	−10.4853	−1.94707	−0.973534	−	0.228543i	$-0.926604\pi$
$29$	−10.4853	−1.94707	−0.973534	+	0.228543i	$0.926604\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	1.17157i	0.203945i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.2426i	1.68388i	0.539571	+	0.841940i	$0.318586\pi$
$37$	10.2426i	1.68388i	−0.539571	+	0.841940i	$0.681414\pi$
$38$	0	0
$39$	1.31371	0.210362
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	− 0.828427i	− 0.126334i	−0.998003	−	0.0631670i	$-0.979880\pi$
$43$	− 0.828427i	− 0.126334i	0.998003	−	0.0631670i	$-0.0201201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.828427i	− 0.120839i	−0.998173	−	0.0604193i	$-0.980756\pi$
$47$	− 0.828427i	− 0.120839i	0.998173	−	0.0604193i	$-0.0192438\pi$
$48$	0	0
$49$	−16.3137	−2.33053
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	− 5.07107i	− 0.696565i	−0.937390	−	0.348282i	$-0.886765\pi$
$53$	− 5.07107i	− 0.696565i	0.937390	−	0.348282i	$-0.113235\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.585786i	− 0.0775893i
$58$	0	0
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	−2.34315	−0.300009	−0.150005	−	0.988685i	$-0.547929\pi$
$61$	−2.34315	−0.300009	−0.150005	+	0.988685i	$0.547929\pi$
$62$	0	0
$63$	12.8284i	1.61623i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.585786i	0.0715652i	0.999360	+	0.0357826i	$0.0113924\pi$
$67$	0.585786i	0.0715652i	−0.999360	+	0.0357826i	$0.988608\pi$
$68$	0	0
$69$	−3.51472	−0.423122
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	14.4853i	1.69537i	0.530497	+	0.847687i	$0.322005\pi$
$73$	14.4853i	1.69537i	−0.530497	+	0.847687i	$0.677995\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.65685i	− 1.10050i
$78$	0	0
$79$	9.65685	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$79$	9.65685	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	9.31371i	1.02231i	0.859488	+	0.511156i	$0.170783\pi$
$83$	9.31371i	1.02231i	−0.859488	+	0.511156i	$0.829217\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.14214i	0.658506i
$88$	0	0
$89$	−10.4853	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$89$	−10.4853	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$90$	0	0
$91$	−10.8284	−1.13513
$92$	0	0
$93$	0.686292i	0.0711651i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.75736i	0.178433i	0.996012	+	0.0892164i	$0.0284363\pi$
$97$	1.75736i	0.178433i	−0.996012	+	0.0892164i	$0.971564\pi$
$98$	0	0
$99$	−5.31371	−0.534048
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	− 15.8995i	− 1.56662i	−0.621629	−	0.783312i	$-0.713529\pi$
$103$	− 15.8995i	− 1.56662i	0.621629	−	0.783312i	$-0.286471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.58579i	− 0.443325i	−0.975123	−	0.221662i	$-0.928852\pi$
$107$	− 4.58579i	− 0.443325i	0.975123	−	0.221662i	$-0.0711483\pi$
$108$	0	0
$109$	8.82843	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	5.07107i	0.477046i	0.971137	+	0.238523i	$0.0766632\pi$
$113$	5.07107i	0.477046i	−0.971137	+	0.238523i	$0.923337\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.95837i	0.550851i
$118$	0	0
$119$	−23.3137	−2.13716
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.48528i	0.404424i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.5563i	1.91282i	0.292033	+	0.956408i	$0.405668\pi$
$127$	21.5563i	1.91282i	−0.292033	+	0.956408i	$0.594332\pi$
$128$	0	0
$129$	−0.485281	−0.0427266
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	4.82843i	0.418678i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.1421i	1.72086i	0.509570	+	0.860429i	$0.329805\pi$
$137$	20.1421i	1.72086i	−0.509570	+	0.860429i	$0.670195\pi$
$138$	0	0
$139$	17.3137	1.46853	0.734265	−	0.678863i	$-0.237527\pi$
$139$	17.3137	1.46853	0.734265	+	0.678863i	$0.237527\pi$
$140$	0	0
$141$	−0.485281	−0.0408681
$142$	0	0
$143$	− 4.48528i	− 0.375078i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.55635i	0.788194i
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	18.1421	1.47639	0.738193	−	0.674590i	$-0.235679\pi$
$151$	18.1421	1.47639	0.738193	+	0.674590i	$0.235679\pi$
$152$	0	0
$153$	12.8284i	1.03712i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.17157i	− 0.253119i	−0.991959	−	0.126560i	$-0.959607\pi$
$157$	− 3.17157i	− 0.253119i	0.991959	−	0.126560i	$-0.0403935\pi$
$158$	0	0
$159$	−2.97056	−0.235581
$160$	0	0
$161$	28.9706	2.28320
$162$	0	0
$163$	− 2.48528i	− 0.194662i	−0.995252	−	0.0973311i	$-0.968969\pi$
$163$	− 2.48528i	− 0.194662i	0.995252	−	0.0973311i	$-0.0310306\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7279i	0.830152i	0.909787	+	0.415076i	$0.136245\pi$
$167$	10.7279i	0.830152i	−0.909787	+	0.415076i	$0.863755\pi$
$168$	0	0
$169$	7.97056	0.613120
$170$	0	0
$171$	2.65685	0.203175
$172$	0	0
$173$	− 3.89949i	− 0.296473i	−0.988952	−	0.148237i	$-0.952640\pi$
$173$	− 3.89949i	− 0.296473i	0.988952	−	0.148237i	$-0.0473597\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.37258i	0.103170i
$178$	0	0
$179$	−21.6569	−1.61871	−0.809355	−	0.587320i	$-0.800183\pi$
$179$	−21.6569	−1.61871	−0.809355	+	0.587320i	$0.800183\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	1.37258i	0.101464i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.65685i	− 0.706179i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	0.585786i	0.0421658i	0.999778	+	0.0210829i	$0.00671140\pi$
$193$	0.585786i	0.0421658i	−0.999778	+	0.0210829i	$0.993289\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.31371i	0.378586i	0.981921	+	0.189293i	$0.0606196\pi$
$197$	5.31371i	0.378586i	−0.981921	+	0.189293i	$0.939380\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	0.343146	0.0242036
$202$	0	0
$203$	− 50.6274i	− 3.55335i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 15.9411i	− 1.10798i
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−11.5147	−0.792706	−0.396353	−	0.918098i	$-0.629724\pi$
$211$	−11.5147	−0.792706	−0.396353	+	0.918098i	$0.629724\pi$
$212$	0	0
$213$	− 6.34315i	− 0.434625i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.65685i	− 0.384012i
$218$	0	0
$219$	8.48528	0.573382
$220$	0	0
$221$	−10.8284	−0.728399
$222$	0	0
$223$	19.2132i	1.28661i	0.765609	+	0.643306i	$0.222438\pi$
$223$	19.2132i	1.28661i	−0.765609	+	0.643306i	$0.777562\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.5858i	− 1.36633i	−0.730266	−	0.683163i	$-0.760604\pi$
$227$	− 20.5858i	− 1.36633i	0.730266	−	0.683163i	$-0.239396\pi$
$228$	0	0
$229$	21.6569	1.43113	0.715563	−	0.698549i	$-0.246170\pi$
$229$	21.6569	1.43113	0.715563	+	0.698549i	$0.246170\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5.65685i	− 0.367452i
$238$	0	0
$239$	−10.3431	−0.669042	−0.334521	−	0.942388i	$-0.608575\pi$
$239$	−10.3431	−0.669042	−0.334521	+	0.942388i	$0.608575\pi$
$240$	0	0
$241$	−30.2843	−1.95078	−0.975391	−	0.220484i	$-0.929236\pi$
$241$	−30.2843	−1.95078	−0.975391	+	0.220484i	$0.929236\pi$
$242$	0	0
$243$	− 13.4731i	− 0.864299i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.24264i	0.142696i
$248$	0	0
$249$	5.45584	0.345750
$250$	0	0
$251$	−13.6569	−0.862013	−0.431006	−	0.902349i	$-0.641841\pi$
$251$	−13.6569	−0.862013	−0.431006	+	0.902349i	$0.641841\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.24264i	0.139892i	0.997551	+	0.0699460i	$0.0222827\pi$
$257$	2.24264i	0.139892i	−0.997551	+	0.0699460i	$0.977717\pi$
$258$	0	0
$259$	−49.4558	−3.07304
$260$	0	0
$261$	−27.8579	−1.72436
$262$	0	0
$263$	3.65685i	0.225491i	0.993624	+	0.112746i	$0.0359645\pi$
$263$	3.65685i	0.225491i	−0.993624	+	0.112746i	$0.964035\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.14214i	0.375893i
$268$	0	0
$269$	22.4853	1.37095	0.685476	−	0.728095i	$-0.259594\pi$
$269$	22.4853	1.37095	0.685476	+	0.728095i	$0.259594\pi$
$270$	0	0
$271$	23.6569	1.43705	0.718526	−	0.695500i	$-0.244817\pi$
$271$	23.6569	1.43705	0.718526	+	0.695500i	$0.244817\pi$
$272$	0	0
$273$	6.34315i	0.383905i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 16.8284i	− 1.01112i	−0.862791	−	0.505561i	$-0.831286\pi$
$277$	− 16.8284i	− 1.01112i	0.862791	−	0.505561i	$-0.168714\pi$
$278$	0	0
$279$	−3.11270	−0.186352
$280$	0	0
$281$	18.9706	1.13169	0.565844	−	0.824512i	$-0.308550\pi$
$281$	18.9706	1.13169	0.565844	+	0.824512i	$0.308550\pi$
$282$	0	0
$283$	− 2.68629i	− 0.159683i	−0.996808	−	0.0798417i	$-0.974559\pi$
$283$	− 2.68629i	− 0.159683i	0.996808	−	0.0798417i	$-0.0254415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 36.9706i	− 2.18230i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	1.02944	0.0603467
$292$	0	0
$293$	− 23.4142i	− 1.36787i	−0.729542	−	0.683936i	$-0.760267\pi$
$293$	− 23.4142i	− 1.36787i	0.729542	−	0.683936i	$-0.239733\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.62742i	0.384562i
$298$	0	0
$299$	13.4558	0.778172
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	− 2.34315i	− 0.134610i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.89949i	− 0.450848i	−0.974261	−	0.225424i	$-0.927623\pi$
$307$	− 7.89949i	− 0.450848i	0.974261	−	0.225424i	$-0.0723767\pi$
$308$	0	0
$309$	−9.31371	−0.529838
$310$	0	0
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	18.0000i	1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$	18.0000i	1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.75736i	0.323366i	0.986843	+	0.161683i	$0.0516921\pi$
$317$	5.75736i	0.323366i	−0.986843	+	0.161683i	$0.948308\pi$
$318$	0	0
$319$	20.9706	1.17413
$320$	0	0
$321$	−2.68629	−0.149934
$322$	0	0
$323$	4.82843i	0.268661i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.17157i	− 0.285989i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	1.17157	0.0643955	0.0321977	−	0.999482i	$-0.489749\pi$
$331$	1.17157	0.0643955	0.0321977	+	0.999482i	$0.489749\pi$
$332$	0	0
$333$	27.2132i	1.49127i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 17.7574i	− 0.967305i	−0.875260	−	0.483652i	$-0.839310\pi$
$337$	− 17.7574i	− 0.967305i	0.875260	−	0.483652i	$-0.160690\pi$
$338$	0	0
$339$	2.97056	0.161339
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	− 44.9706i	− 2.42818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.9706i	− 1.23312i	−0.787306	−	0.616562i	$-0.788525\pi$
$347$	− 22.9706i	− 1.23312i	0.787306	−	0.616562i	$-0.211475\pi$
$348$	0	0
$349$	0.343146	0.0183682	0.00918409	−	0.999958i	$-0.497077\pi$
$349$	0.343146	0.0183682	0.00918409	+	0.999958i	$0.497077\pi$
$350$	0	0
$351$	7.43146	0.396662
$352$	0	0
$353$	− 3.85786i	− 0.205333i	−0.994716	−	0.102667i	$-0.967262\pi$
$353$	− 3.85786i	− 0.205333i	0.994716	−	0.102667i	$-0.0327375\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.6569i	0.722797i
$358$	0	0
$359$	21.3137	1.12489	0.562447	−	0.826833i	$-0.309860\pi$
$359$	21.3137	1.12489	0.562447	+	0.826833i	$0.309860\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.10051i	0.215221i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.1421i	1.46901i	0.678605	+	0.734504i	$0.262585\pi$
$367$	28.1421i	1.46901i	−0.678605	+	0.734504i	$0.737415\pi$
$368$	0	0
$369$	−20.3431	−1.05902
$370$	0	0
$371$	24.4853	1.27121
$372$	0	0
$373$	9.55635i	0.494809i	0.968912	+	0.247405i	$0.0795776\pi$
$373$	9.55635i	0.494809i	−0.968912	+	0.247405i	$0.920422\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.5147i	− 1.21107i
$378$	0	0
$379$	−22.3431	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$379$	−22.3431	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$380$	0	0
$381$	12.6274	0.646922
$382$	0	0
$383$	2.92893i	0.149661i	0.997196	+	0.0748307i	$0.0238416\pi$
$383$	2.92893i	0.149661i	−0.997196	+	0.0748307i	$0.976158\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.20101i	− 0.111884i
$388$	0	0
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	28.9706	1.46510
$392$	0	0
$393$	3.31371i	0.167154i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.1421i	1.61317i	0.591120	+	0.806584i	$0.298686\pi$
$397$	32.1421i	1.61317i	−0.591120	+	0.806584i	$0.701314\pi$
$398$	0	0
$399$	2.82843	0.141598
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	0	0
$403$	− 2.62742i	− 0.130881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 20.4853i	− 1.01542i
$408$	0	0
$409$	−9.51472	−0.470473	−0.235236	−	0.971938i	$-0.575586\pi$
$409$	−9.51472	−0.470473	−0.235236	+	0.971938i	$0.575586\pi$
$410$	0	0
$411$	11.7990	0.582001
$412$	0	0
$413$	− 11.3137i	− 0.556711i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 10.1421i	− 0.496663i
$418$	0	0
$419$	−9.65685	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$419$	−9.65685	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$420$	0	0
$421$	−8.34315	−0.406620	−0.203310	−	0.979114i	$-0.565170\pi$
$421$	−8.34315	−0.406620	−0.203310	+	0.979114i	$0.565170\pi$
$422$	0	0
$423$	− 2.20101i	− 0.107017i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.3137i	− 0.547509i
$428$	0	0
$429$	−2.62742	−0.126853
$430$	0	0
$431$	33.4558	1.61151	0.805756	−	0.592248i	$-0.201759\pi$
$431$	33.4558	1.61151	0.805756	+	0.592248i	$0.201759\pi$
$432$	0	0
$433$	− 15.2132i	− 0.731100i	−0.930792	−	0.365550i	$-0.880881\pi$
$433$	− 15.2132i	− 0.731100i	0.930792	−	0.365550i	$-0.119119\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	29.6569	1.41544	0.707722	−	0.706491i	$-0.249723\pi$
$439$	29.6569	1.41544	0.707722	+	0.706491i	$0.249723\pi$
$440$	0	0
$441$	−43.3431	−2.06396
$442$	0	0
$443$	34.2843i	1.62889i	0.580237	+	0.814447i	$0.302960\pi$
$443$	34.2843i	1.62889i	−0.580237	+	0.814447i	$0.697040\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.68629i	− 0.221654i
$448$	0	0
$449$	13.5147	0.637799	0.318900	−	0.947789i	$-0.396687\pi$
$449$	13.5147	0.637799	0.318900	+	0.947789i	$0.396687\pi$
$450$	0	0
$451$	15.3137	0.721094
$452$	0	0
$453$	− 10.6274i	− 0.499320i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.9706i	− 0.887405i	−0.896174	−	0.443703i	$-0.853665\pi$
$457$	− 18.9706i	− 0.887405i	0.896174	−	0.443703i	$-0.146335\pi$
$458$	0	0
$459$	16.0000	0.746816
$460$	0	0
$461$	−5.31371	−0.247484	−0.123742	−	0.992314i	$-0.539490\pi$
$461$	−5.31371	−0.247484	−0.123742	+	0.992314i	$0.539490\pi$
$462$	0	0
$463$	− 9.31371i	− 0.432845i	−0.976300	−	0.216422i	$-0.930561\pi$
$463$	− 9.31371i	− 0.432845i	0.976300	−	0.216422i	$-0.0694388\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.31371i	− 0.430987i	−0.976505	−	0.215494i	$-0.930864\pi$
$467$	− 9.31371i	− 0.430987i	0.976505	−	0.215494i	$-0.0691360\pi$
$468$	0	0
$469$	−2.82843	−0.130605
$470$	0	0
$471$	−1.85786	−0.0856059
$472$	0	0
$473$	1.65685i	0.0761822i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 13.4731i	− 0.616890i
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−22.9706	−1.04737
$482$	0	0
$483$	− 16.9706i	− 0.772187i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 24.3848i	− 1.10498i	−0.833520	−	0.552490i	$-0.813678\pi$
$487$	− 24.3848i	− 1.10498i	0.833520	−	0.552490i	$-0.186322\pi$
$488$	0	0
$489$	−1.45584	−0.0658355
$490$	0	0
$491$	35.3137	1.59369	0.796843	−	0.604187i	$-0.206502\pi$
$491$	35.3137	1.59369	0.796843	+	0.604187i	$0.206502\pi$
$492$	0	0
$493$	− 50.6274i	− 2.28014i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	52.2843i	2.34527i
$498$	0	0
$499$	26.9706	1.20737	0.603684	−	0.797224i	$-0.293699\pi$
$499$	26.9706	1.20737	0.603684	+	0.797224i	$0.293699\pi$
$500$	0	0
$501$	6.28427	0.280761
$502$	0	0
$503$	2.97056i	0.132451i	0.997805	+	0.0662254i	$0.0210956\pi$
$503$	2.97056i	0.132451i	−0.997805	+	0.0662254i	$0.978904\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.66905i	− 0.207360i
$508$	0	0
$509$	33.7990	1.49811	0.749057	−	0.662506i	$-0.230507\pi$
$509$	33.7990	1.49811	0.749057	+	0.662506i	$0.230507\pi$
$510$	0	0
$511$	−69.9411	−3.09401
$512$	0	0
$513$	− 3.31371i	− 0.146304i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.65685i	0.0728684i
$518$	0	0
$519$	−2.28427	−0.100268
$520$	0	0
$521$	28.3431	1.24174	0.620868	−	0.783915i	$-0.286780\pi$
$521$	28.3431	1.24174	0.620868	+	0.783915i	$0.286780\pi$
$522$	0	0
$523$	− 6.72792i	− 0.294191i	−0.989122	−	0.147096i	$-0.953007\pi$
$523$	− 6.72792i	− 0.294191i	0.989122	−	0.147096i	$-0.0469925\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.65685i	− 0.246416i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−6.22540	−0.270159
$532$	0	0
$533$	− 17.1716i	− 0.743783i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.6863i	0.547454i
$538$	0	0
$539$	32.6274	1.40536
$540$	0	0
$541$	11.3137	0.486414	0.243207	−	0.969974i	$-0.421801\pi$
$541$	11.3137	0.486414	0.243207	+	0.969974i	$0.421801\pi$
$542$	0	0
$543$	10.5442i	0.452493i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 36.3848i	− 1.55570i	−0.628450	−	0.777850i	$-0.716310\pi$
$547$	− 36.3848i	− 1.55570i	0.628450	−	0.777850i	$-0.283690\pi$
$548$	0	0
$549$	−6.22540	−0.265693
$550$	0	0
$551$	−10.4853	−0.446688
$552$	0	0
$553$	46.6274i	1.98280i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 41.7990i	− 1.77108i	−0.464563	−	0.885540i	$-0.653789\pi$
$557$	− 41.7990i	− 1.77108i	0.464563	−	0.885540i	$-0.346211\pi$
$558$	0	0
$559$	1.85786	0.0785793
$560$	0	0
$561$	−5.65685	−0.238833
$562$	0	0
$563$	− 11.4142i	− 0.481052i	−0.970643	−	0.240526i	$-0.922680\pi$
$563$	− 11.4142i	− 0.481052i	0.970643	−	0.240526i	$-0.0773199\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	29.1127i	1.22262i
$568$	0	0
$569$	−0.828427	−0.0347295	−0.0173647	−	0.999849i	$-0.505528\pi$
$569$	−0.828427	−0.0347295	−0.0173647	+	0.999849i	$0.505528\pi$
$570$	0	0
$571$	−14.6863	−0.614602	−0.307301	−	0.951612i	$-0.599426\pi$
$571$	−14.6863	−0.614602	−0.307301	+	0.951612i	$0.599426\pi$
$572$	0	0
$573$	− 2.34315i	− 0.0978863i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	0	0
$579$	0.343146	0.0142607
$580$	0	0
$581$	−44.9706	−1.86569
$582$	0	0
$583$	10.1421i	0.420044i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.9706i	0.617901i	0.951078	+	0.308951i	$0.0999778\pi$
$587$	14.9706i	0.617901i	−0.951078	+	0.308951i	$0.900022\pi$
$588$	0	0
$589$	−1.17157	−0.0482738
$590$	0	0
$591$	3.11270	0.128039
$592$	0	0
$593$	8.62742i	0.354286i	0.984185	+	0.177143i	$0.0566854\pi$
$593$	8.62742i	0.354286i	−0.984185	+	0.177143i	$0.943315\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.05887i	− 0.247973i
$598$	0	0
$599$	4.68629	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$599$	4.68629	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	1.55635i	0.0633794i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5.07107i	− 0.205828i	−0.994690	−	0.102914i	$-0.967183\pi$
$607$	− 5.07107i	− 0.205828i	0.994690	−	0.102914i	$-0.0328167\pi$
$608$	0	0
$609$	−29.6569	−1.20176
$610$	0	0
$611$	1.85786	0.0751611
$612$	0	0
$613$	17.5147i	0.707413i	0.935356	+	0.353706i	$0.115079\pi$
$613$	17.5147i	0.707413i	−0.935356	+	0.353706i	$0.884921\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.20101i	0.249643i	0.992179	+	0.124822i	$0.0398359\pi$
$617$	6.20101i	0.249643i	−0.992179	+	0.124822i	$0.960164\pi$
$618$	0	0
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	0	0
$621$	−19.8823	−0.797847
$622$	0	0
$623$	− 50.6274i	− 2.02834i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.17157i	0.0467881i
$628$	0	0
$629$	−49.4558	−1.97193
$630$	0	0
$631$	−3.65685	−0.145577	−0.0727885	−	0.997347i	$-0.523190\pi$
$631$	−3.65685	−0.145577	−0.0727885	+	0.997347i	$0.523190\pi$
$632$	0	0
$633$	6.74517i	0.268096i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 36.5858i	− 1.44958i
$638$	0	0
$639$	28.7696	1.13811
$640$	0	0
$641$	−21.3137	−0.841841	−0.420920	−	0.907098i	$-0.638293\pi$
$641$	−21.3137	−0.841841	−0.420920	+	0.907098i	$0.638293\pi$
$642$	0	0
$643$	24.6274i	0.971211i	0.874178	+	0.485605i	$0.161401\pi$
$643$	24.6274i	0.971211i	−0.874178	+	0.485605i	$0.838599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6.68629i	− 0.262865i	−0.991325	−	0.131433i	$-0.958042\pi$
$647$	− 6.68629i	− 0.262865i	0.991325	−	0.131433i	$-0.0419577\pi$
$648$	0	0
$649$	4.68629	0.183953
$650$	0	0
$651$	−3.31371	−0.129874
$652$	0	0
$653$	13.7990i	0.539996i	0.962861	+	0.269998i	$0.0870231\pi$
$653$	13.7990i	0.539996i	−0.962861	+	0.269998i	$0.912977\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	38.4853i	1.50145i
$658$	0	0
$659$	34.6274	1.34889	0.674446	−	0.738324i	$-0.264382\pi$
$659$	34.6274	1.34889	0.674446	+	0.738324i	$0.264382\pi$
$660$	0	0
$661$	−12.6274	−0.491150	−0.245575	−	0.969378i	$-0.578977\pi$
$661$	−12.6274	−0.491150	−0.245575	+	0.969378i	$0.578977\pi$
$662$	0	0
$663$	6.34315i	0.246347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	62.9117i	2.43595i
$668$	0	0
$669$	11.2548	0.435137
$670$	0	0
$671$	4.68629	0.180912
$672$	0	0
$673$	10.2426i	0.394825i	0.980321	+	0.197412i	$0.0632538\pi$
$673$	10.2426i	0.394825i	−0.980321	+	0.197412i	$0.936746\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25.5563i	− 0.982210i	−0.871100	−	0.491105i	$-0.836593\pi$
$677$	− 25.5563i	− 0.982210i	0.871100	−	0.491105i	$-0.163407\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	−12.0589	−0.462097
$682$	0	0
$683$	− 38.7279i	− 1.48188i	−0.671570	−	0.740941i	$-0.734380\pi$
$683$	− 38.7279i	− 1.48188i	0.671570	−	0.740941i	$-0.265620\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.6863i	− 0.484012i
$688$	0	0
$689$	11.3726	0.433261
$690$	0	0
$691$	−15.6569	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$691$	−15.6569	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$692$	0	0
$693$	− 25.6569i	− 0.974623i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 36.9706i	− 1.40036i
$698$	0	0
$699$	5.85786	0.221565
$700$	0	0
$701$	49.6569	1.87551	0.937757	−	0.347293i	$-0.112899\pi$
$701$	49.6569	1.87551	0.937757	+	0.347293i	$0.112899\pi$
$702$	0	0
$703$	10.2426i	0.386309i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.3137i	0.726367i
$708$	0	0
$709$	−18.9706	−0.712454	−0.356227	−	0.934399i	$-0.615937\pi$
$709$	−18.9706	−0.712454	−0.356227	+	0.934399i	$0.615937\pi$
$710$	0	0
$711$	25.6569	0.962207
$712$	0	0
$713$	7.02944i	0.263254i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.05887i	0.226273i
$718$	0	0
$719$	−38.2843	−1.42776	−0.713881	−	0.700267i	$-0.753064\pi$
$719$	−38.2843	−1.42776	−0.713881	+	0.700267i	$0.753064\pi$
$720$	0	0
$721$	76.7696	2.85905
$722$	0	0
$723$	17.7401i	0.659762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.5147i	1.09464i	0.836923	+	0.547320i	$0.184352\pi$
$727$	29.5147i	1.09464i	−0.836923	+	0.547320i	$0.815648\pi$
$728$	0	0
$729$	10.1960	0.377628
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	39.6569i	1.46476i	0.680896	+	0.732380i	$0.261590\pi$
$733$	39.6569i	1.46476i	−0.680896	+	0.732380i	$0.738410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.17157i	− 0.0431554i
$738$	0	0
$739$	−34.6274	−1.27379	−0.636895	−	0.770951i	$-0.719782\pi$
$739$	−34.6274	−1.27379	−0.636895	+	0.770951i	$0.719782\pi$
$740$	0	0
$741$	1.31371	0.0482603
$742$	0	0
$743$	14.0416i	0.515137i	0.966260	+	0.257569i	$0.0829214\pi$
$743$	14.0416i	0.515137i	−0.966260	+	0.257569i	$0.917079\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.7452i	0.905378i
$748$	0	0
$749$	22.1421	0.809056
$750$	0	0
$751$	14.1421	0.516054	0.258027	−	0.966138i	$-0.416928\pi$
$751$	14.1421	0.516054	0.258027	+	0.966138i	$0.416928\pi$
$752$	0	0
$753$	8.00000i	0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 19.6569i	− 0.714441i	−0.934020	−	0.357220i	$-0.883725\pi$
$757$	− 19.6569i	− 0.714441i	0.934020	−	0.357220i	$-0.116275\pi$
$758$	0	0
$759$	7.02944	0.255152
$760$	0	0
$761$	30.6274	1.11024	0.555121	−	0.831769i	$-0.312672\pi$
$761$	30.6274	1.11024	0.555121	+	0.831769i	$0.312672\pi$
$762$	0	0
$763$	42.6274i	1.54322i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.25483i	− 0.189741i
$768$	0	0
$769$	−38.6274	−1.39294	−0.696470	−	0.717586i	$-0.745247\pi$
$769$	−38.6274	−1.39294	−0.696470	+	0.717586i	$0.745247\pi$
$770$	0	0
$771$	1.31371	0.0473121
$772$	0	0
$773$	− 0.100505i	− 0.00361492i	−0.999998	−	0.00180746i	$-0.999425\pi$
$773$	− 0.100505i	− 0.00361492i	0.999998	−	0.00180746i	$-0.000575332\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	28.9706i	1.03931i
$778$	0	0
$779$	−7.65685	−0.274335
$780$	0	0
$781$	−21.6569	−0.774943
$782$	0	0
$783$	34.7452i	1.24169i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.8406i	1.49146i	0.666250	+	0.745729i	$0.267898\pi$
$787$	41.8406i	1.49146i	−0.666250	+	0.745729i	$0.732102\pi$
$788$	0	0
$789$	2.14214	0.0762620
$790$	0	0
$791$	−24.4853	−0.870596
$792$	0	0
$793$	− 5.25483i	− 0.186605i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.8995i	− 0.421502i	−0.977540	−	0.210751i	$-0.932409\pi$
$797$	− 11.8995i	− 0.421502i	0.977540	−	0.210751i	$-0.0675909\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−27.8579	−0.984309
$802$	0	0
$803$	− 28.9706i	− 1.02235i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.1716i	− 0.463661i
$808$	0	0
$809$	39.2548	1.38013	0.690063	−	0.723749i	$-0.257583\pi$
$809$	39.2548	1.38013	0.690063	+	0.723749i	$0.257583\pi$
$810$	0	0
$811$	19.7990	0.695237	0.347618	−	0.937636i	$-0.386991\pi$
$811$	19.7990	0.695237	0.347618	+	0.937636i	$0.386991\pi$
$812$	0	0
$813$	− 13.8579i	− 0.486017i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.828427i	− 0.0289830i
$818$	0	0
$819$	−28.7696	−1.00529
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 15.1716i	− 0.528848i	−0.964407	−	0.264424i	$-0.914818\pi$
$823$	− 15.1716i	− 0.528848i	0.964407	−	0.264424i	$-0.0851818\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.0711i	1.01090i	0.862856	+	0.505450i	$0.168674\pi$
$827$	29.0711i	1.01090i	−0.862856	+	0.505450i	$0.831326\pi$
$828$	0	0
$829$	40.4264	1.40407	0.702034	−	0.712144i	$-0.252276\pi$
$829$	40.4264	1.40407	0.702034	+	0.712144i	$0.252276\pi$
$830$	0	0
$831$	−9.85786	−0.341966
$832$	0	0
$833$	− 78.7696i	− 2.72920i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.88225i	0.134190i
$838$	0	0
$839$	−43.5980	−1.50517	−0.752585	−	0.658495i	$-0.771193\pi$
$839$	−43.5980	−1.50517	−0.752585	+	0.658495i	$0.771193\pi$
$840$	0	0
$841$	80.9411	2.79107
$842$	0	0
$843$	− 11.1127i	− 0.382742i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 33.7990i	− 1.16135i
$848$	0	0
$849$	−1.57359	−0.0540056
$850$	0	0
$851$	61.4558	2.10668
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.2132i	0.929585i	0.885420	+	0.464793i	$0.153871\pi$
$857$	27.2132i	0.929585i	−0.885420	+	0.464793i	$0.846129\pi$
$858$	0	0
$859$	24.2843	0.828569	0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843	0.828569	0.414284	+	0.910148i	$0.364032\pi$
$860$	0	0
$861$	−21.6569	−0.738064
$862$	0	0
$863$	− 3.89949i	− 0.132740i	−0.997795	−	0.0663702i	$-0.978858\pi$
$863$	− 3.89949i	− 0.132740i	0.997795	−	0.0663702i	$-0.0211418\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.69848i	0.125607i
$868$	0	0
$869$	−19.3137	−0.655173
$870$	0	0
$871$	−1.31371	−0.0445133
$872$	0	0
$873$	4.66905i	0.158023i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.0711i	1.25180i	0.779903	+	0.625901i	$0.215268\pi$
$877$	37.0711i	1.25180i	−0.779903	+	0.625901i	$0.784732\pi$
$878$	0	0
$879$	−13.7157	−0.462620
$880$	0	0
$881$	44.2843	1.49198	0.745988	−	0.665960i	$-0.231978\pi$
$881$	44.2843	1.49198	0.745988	+	0.665960i	$0.231978\pi$
$882$	0	0
$883$	− 39.1716i	− 1.31823i	−0.752043	−	0.659114i	$-0.770931\pi$
$883$	− 39.1716i	− 1.31823i	0.752043	−	0.659114i	$-0.229069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.0711i	1.64765i	0.566848	+	0.823823i	$0.308163\pi$
$887$	49.0711i	1.64765i	−0.566848	+	0.823823i	$0.691837\pi$
$888$	0	0
$889$	−104.083	−3.49084
$890$	0	0
$891$	−12.0589	−0.403987
$892$	0	0
$893$	− 0.828427i	− 0.0277223i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.88225i	− 0.263181i
$898$	0	0
$899$	12.2843	0.409703
$900$	0	0
$901$	24.4853	0.815723
$902$	0	0
$903$	− 2.34315i	− 0.0779750i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.3848i	0.676865i	0.940991	+	0.338433i	$0.109897\pi$
$907$	20.3848i	0.676865i	−0.940991	+	0.338433i	$0.890103\pi$
$908$	0	0
$909$	10.6274	0.352489
$910$	0	0
$911$	4.20101	0.139186	0.0695928	−	0.997575i	$-0.477830\pi$
$911$	4.20101	0.139186	0.0695928	+	0.997575i	$0.477830\pi$
$912$	0	0
$913$	− 18.6274i	− 0.616478i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 27.3137i	− 0.901978i
$918$	0	0
$919$	−35.3137	−1.16489	−0.582446	−	0.812869i	$-0.697904\pi$
$919$	−35.3137	−1.16489	−0.582446	+	0.812869i	$0.697904\pi$
$920$	0	0
$921$	−4.62742	−0.152479
$922$	0	0
$923$	24.2843i	0.799327i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 42.2426i	− 1.38743i
$928$	0	0
$929$	−6.68629	−0.219370	−0.109685	−	0.993966i	$-0.534984\pi$
$929$	−6.68629	−0.219370	−0.109685	+	0.993966i	$0.534984\pi$
$930$	0	0
$931$	−16.3137	−0.534660
$932$	0	0
$933$	8.20101i	0.268489i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.1421i	1.31139i	0.755027	+	0.655693i	$0.227624\pi$
$937$	40.1421i	1.31139i	−0.755027	+	0.655693i	$0.772376\pi$
$938$	0	0
$939$	10.5442	0.344096
$940$	0	0
$941$	−30.2843	−0.987239	−0.493620	−	0.869678i	$-0.664326\pi$
$941$	−30.2843	−0.987239	−0.493620	+	0.869678i	$0.664326\pi$
$942$	0	0
$943$	45.9411i	1.49605i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.34315i	0.271116i	0.990769	+	0.135558i	$0.0432827\pi$
$947$	8.34315i	0.271116i	−0.990769	+	0.135558i	$0.956717\pi$
$948$	0	0
$949$	−32.4853	−1.05452
$950$	0	0
$951$	3.37258	0.109363
$952$	0	0
$953$	5.75736i	0.186499i	0.995643	+	0.0932496i	$0.0297254\pi$
$953$	5.75736i	0.186499i	−0.995643	+	0.0932496i	$0.970275\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.2843i	− 0.397094i
$958$	0	0
$959$	−97.2548	−3.14052
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	− 12.1838i	− 0.392616i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 20.6274i	− 0.663333i	−0.943397	−	0.331667i	$-0.892389\pi$
$967$	− 20.6274i	− 0.663333i	0.943397	−	0.331667i	$-0.107611\pi$
$968$	0	0
$969$	2.82843	0.0908622
$970$	0	0
$971$	−35.1127	−1.12682	−0.563410	−	0.826177i	$-0.690511\pi$
$971$	−35.1127	−1.12682	−0.563410	+	0.826177i	$0.690511\pi$
$972$	0	0
$973$	83.5980i	2.68003i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.27208i	− 0.296640i	−0.988939	−	0.148320i	$-0.952613\pi$
$977$	− 9.27208i	− 0.296640i	0.988939	−	0.148320i	$-0.0473866\pi$
$978$	0	0
$979$	20.9706	0.670222
$980$	0	0
$981$	23.4558	0.748887
$982$	0	0
$983$	− 23.4142i	− 0.746797i	−0.927671	−	0.373399i	$-0.878192\pi$
$983$	− 23.4142i	− 0.746797i	0.927671	−	0.373399i	$-0.121808\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.34315i	− 0.0745832i
$988$	0	0
$989$	−4.97056	−0.158055
$990$	0	0
$991$	−51.1127	−1.62365	−0.811824	−	0.583902i	$-0.801525\pi$
$991$	−51.1127	−1.62365	−0.811824	+	0.583902i	$0.801525\pi$
$992$	0	0
$993$	− 0.686292i	− 0.0217788i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.5147i	0.428015i	0.976832	+	0.214008i	$0.0686518\pi$
$997$	13.5147i	0.428015i	−0.976832	+	0.214008i	$0.931348\pi$
$998$	0	0
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.d.1749.2		4
5.2	odd	4		380.2.a.c.1.2	✓	2
5.3	odd	4		1900.2.a.e.1.1		2
5.4	even	2	inner	1900.2.c.d.1749.3		4
15.2	even	4		3420.2.a.g.1.1		2
20.3	even	4		7600.2.a.u.1.2		2
20.7	even	4		1520.2.a.o.1.1		2
40.27	even	4		6080.2.a.y.1.2		2
40.37	odd	4		6080.2.a.bl.1.1		2
95.37	even	4		7220.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.c.1.2	✓	2	5.2	odd	4
1520.2.a.o.1.1		2	20.7	even	4
1900.2.a.e.1.1		2	5.3	odd	4
1900.2.c.d.1749.2		4	1.1	even	1	trivial
1900.2.c.d.1749.3		4	5.4	even	2	inner
3420.2.a.g.1.1		2	15.2	even	4
6080.2.a.y.1.2		2	40.27	even	4
6080.2.a.bl.1.1		2	40.37	odd	4
7220.2.a.m.1.1		2	95.37	even	4
7600.2.a.u.1.2		2	20.3	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.585786i q^{3} +4.82843i q^{7} +2.65685 q^{9} -2.00000 q^{11} +2.24264i q^{13} +4.82843i q^{17} +1.00000 q^{19} +2.82843 q^{21} -6.00000i q^{23} -3.31371i q^{27} -10.4853 q^{29} -1.17157 q^{31} +1.17157i q^{33} +10.2426i q^{37} +1.31371 q^{39} -7.65685 q^{41} -0.828427i q^{43} -0.828427i q^{47} -16.3137 q^{49} +2.82843 q^{51} -5.07107i q^{53} -0.585786i q^{57} -2.34315 q^{59} -2.34315 q^{61} +12.8284i q^{63} +0.585786i q^{67} -3.51472 q^{69} +10.8284 q^{71} +14.4853i q^{73} -9.65685i q^{77} +9.65685 q^{79} +6.02944 q^{81} +9.31371i q^{83} +6.14214i q^{87} -10.4853 q^{89} -10.8284 q^{91} +0.686292i q^{93} +1.75736i q^{97} -5.31371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} - 8 q^{11} + 4 q^{19} - 8 q^{29} - 16 q^{31} - 40 q^{39} - 8 q^{41} - 20 q^{49} - 32 q^{59} - 32 q^{61} - 48 q^{69} + 32 q^{71} + 16 q^{79} + 92 q^{81} - 8 q^{89} - 32 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^19 - 8 * q^29 - 16 * q^31 - 40 * q^39 - 8 * q^41 - 20 * q^49 - 32 * q^59 - 32 * q^61 - 48 * q^69 + 32 * q^71 + 16 * q^79 + 92 * q^81 - 8 * q^89 - 32 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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