LMFDB - Embedded newform 1950.2.e.c.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.e (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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.c.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -5.00000 q^{19} -4.00000 q^{21} +1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +7.00000i q^{37} -5.00000i q^{38} -1.00000 q^{39} +3.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} -9.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -1.00000i q^{52} +9.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -5.00000i q^{57} -3.00000i q^{58} -6.00000 q^{59} +8.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -5.00000i q^{67} -3.00000 q^{71} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} +5.00000 q^{76} -1.00000i q^{78} -11.0000 q^{79} +1.00000 q^{81} +3.00000i q^{82} -6.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -3.00000i q^{87} -6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{93} +9.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} - 10 q^{19} - 8 q^{21} + 2 q^{24} - 2 q^{26} - 6 q^{29} - 8 q^{31} + 2 q^{36} - 2 q^{39} + 6 q^{41} - 18 q^{49} + 2 q^{54} + 8 q^{56} - 12 q^{59} + 16 q^{61} - 2 q^{64} - 6 q^{71} - 14 q^{74} + 10 q^{76} - 22 q^{79} + 2 q^{81} + 8 q^{84} - 4 q^{86} - 12 q^{89} - 8 q^{91} + 18 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 - 10 * q^19 - 8 * q^21 + 2 * q^24 - 2 * q^26 - 6 * q^29 - 8 * q^31 + 2 * q^36 - 2 * q^39 + 6 * q^41 - 18 * q^49 + 2 * q^54 + 8 * q^56 - 12 * q^59 + 16 * q^61 - 2 * q^64 - 6 * q^71 - 14 * q^74 + 10 * q^76 - 22 * q^79 + 2 * q^81 + 8 * q^84 - 4 * q^86 - 12 * q^89 - 8 * q^91 + 18 * q^94 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	− 1.00000i	− 0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	− 5.00000i	− 0.811107i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	− 4.00000i	− 0.617213i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	− 5.00000i	− 0.662266i
$58$	− 3.00000i	− 0.393919i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 4.00000i	− 0.508001i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	1.00000i	0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	− 1.00000i	− 0.113228i
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.00000i	0.331295i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−2.00000	−0.215666
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	9.00000	0.928279
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	1.00000i	0.0962250i
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	4.00000i	0.377964i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	3.00000	0.278543
$117$	− 1.00000i	− 0.0924500i
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.00000i	0.724286i
$123$	3.00000i	0.270501i
$124$	4.00000	0.359211
$125$	0	0
$126$	4.00000	0.356348
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	− 20.0000i	− 1.73422i
$134$	5.00000	0.431934
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	− 3.00000i	− 0.251754i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	− 9.00000i	− 0.742307i
$148$	− 7.00000i	− 0.575396i
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	5.00000i	0.405554i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 11.0000i	− 0.875113i
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	21.0000i	1.62503i	0.582941	+	0.812514i	$0.301902\pi$
$167$	21.0000i	1.62503i	−0.582941	+	0.812514i	$0.698098\pi$
$168$	4.00000i	0.308607i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	5.00000	0.382360
$172$	− 2.00000i	− 0.152499i
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	− 6.00000i	− 0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 4.00000i	− 0.296500i
$183$	8.00000i	0.591377i
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	9.00000i	0.656392i
$189$	4.00000	0.290957
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	18.0000i	1.26648i
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 9.00000i	− 0.618123i
$213$	− 3.00000i	− 0.205557i
$214$	9.00000	0.615227
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 16.0000i	− 1.08615i
$218$	− 11.0000i	− 0.745014i
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	− 7.00000i	− 0.469809i
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	5.00000i	0.331133i
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	6.00000	0.390567
$237$	− 11.0000i	− 0.714527i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	−8.00000	−0.512148
$245$	0	0
$246$	−3.00000	−0.191273
$247$	− 5.00000i	− 0.318142i
$248$	4.00000i	0.254000i
$249$	6.00000	0.380235
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	− 2.00000i	− 0.124515i
$259$	−28.0000	−1.73984
$260$	0	0
$261$	3.00000	0.185695
$262$	3.00000i	0.185341i
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	20.0000	1.22628
$267$	− 6.00000i	− 0.367194i
$268$	5.00000i	0.305424i
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	− 8.00000i	− 0.479808i
$279$	4.00000	0.239474
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	9.00000i	0.535942i
$283$	32.0000i	1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$	32.0000i	1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	8.00000	0.468968
$292$	4.00000i	0.234082i
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	12.0000i	0.695141i
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 16.0000i	− 0.920697i
$303$	18.0000i	1.03407i
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	1.00000i	0.0566139i
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	11.0000	0.618798
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	− 9.00000i	− 0.504695i
$319$	0	0
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	− 11.0000i	− 0.608301i
$328$	− 3.00000i	− 0.165647i
$329$	36.0000	1.98474
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	6.00000i	0.329293i
$333$	− 7.00000i	− 0.383598i
$334$	−21.0000	−1.14907
$335$	0	0
$336$	−4.00000	−0.218218
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	5.00000i	0.270369i
$343$	− 8.00000i	− 0.431959i
$344$	2.00000	0.107833
$345$	0	0
$346$	−21.0000	−1.12897
$347$	9.00000i	0.483145i	0.970383	+	0.241573i	$0.0776632\pi$
$347$	9.00000i	0.483145i	−0.970383	+	0.241573i	$0.922337\pi$
$348$	3.00000i	0.160817i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	− 16.0000i	− 0.840941i
$363$	− 11.0000i	− 0.577350i
$364$	4.00000	0.209657
$365$	0	0
$366$	−8.00000	−0.418167
$367$	19.0000i	0.991792i	0.868382	+	0.495896i	$0.165160\pi$
$367$	19.0000i	0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$	0	0
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−36.0000	−1.86903
$372$	4.00000i	0.207390i
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	0	0
$376$	−9.00000	−0.464140
$377$	− 3.00000i	− 0.154508i
$378$	4.00000i	0.205738i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	18.0000i	0.920960i
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−20.0000	−1.01797
$387$	− 2.00000i	− 0.101666i
$388$	8.00000i	0.406138i
$389$	−27.0000	−1.36895	−0.684477	−	0.729034i	$-0.739969\pi$
$389$	−27.0000	−1.36895	−0.684477	+	0.729034i	$0.739969\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	3.00000i	0.151330i
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	31.0000i	1.55585i	0.628360	+	0.777923i	$0.283727\pi$
$397$	31.0000i	1.55585i	−0.628360	+	0.777923i	$0.716273\pi$
$398$	7.00000i	0.350878i
$399$	20.0000	1.00125
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	5.00000i	0.249377i
$403$	− 4.00000i	− 0.199254i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	16.0000i	0.788263i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	27.0000	1.31904	0.659518	−	0.751689i	$-0.270760\pi$
$419$	27.0000	1.31904	0.659518	+	0.751689i	$0.270760\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	− 4.00000i	− 0.194717i
$423$	9.00000i	0.437595i
$424$	9.00000	0.437079
$425$	0	0
$426$	3.00000	0.145350
$427$	32.0000i	1.54859i
$428$	9.00000i	0.435031i
$429$	0	0
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	11.0000	0.526804
$437$	0	0
$438$	4.00000i	0.191127i
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	−14.0000	−0.662919
$447$	12.0000i	0.567581i
$448$	− 4.00000i	− 0.188982i
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	0	0
$452$	− 18.0000i	− 0.846649i
$453$	− 16.0000i	− 0.751746i
$454$	6.00000	0.281594
$455$	0	0
$456$	−5.00000	−0.234146
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	7.00000i	0.327089i
$459$	0	0
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−24.0000	−1.11178
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	1.00000i	0.0462250i
$469$	20.0000	0.923514
$470$	0	0
$471$	−10.0000	−0.460776
$472$	6.00000i	0.276172i
$473$	0	0
$474$	11.0000	0.505247
$475$	0	0
$476$	0	0
$477$	− 9.00000i	− 0.412082i
$478$	24.0000i	1.09773i
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	8.00000i	0.364390i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	− 8.00000i	− 0.362143i
$489$	16.0000	0.723545
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 3.00000i	− 0.135250i
$493$	0	0
$494$	5.00000	0.224961
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 12.0000i	− 0.538274i
$498$	6.00000i	0.268866i
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	9.00000i	0.401690i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	− 7.00000i	− 0.310575i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000i	0.0441942i
$513$	5.00000i	0.220755i
$514$	−12.0000	−0.529297
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	− 28.0000i	− 1.23025i
$519$	−21.0000	−0.921798
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	3.00000i	0.131306i
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	6.00000	0.260378
$532$	20.0000i	0.867110i
$533$	3.00000i	0.129944i
$534$	6.00000	0.259645
$535$	0	0
$536$	−5.00000	−0.215967
$537$	− 12.0000i	− 0.517838i
$538$	21.0000i	0.905374i
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	− 28.0000i	− 1.20270i
$543$	− 16.0000i	− 0.686626i
$544$	0	0
$545$	0	0
$546$	4.00000	0.171184
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	3.00000i	0.128154i
$549$	−8.00000	−0.341432
$550$	0	0
$551$	15.0000	0.639021
$552$	0	0
$553$	− 44.0000i	− 1.87107i
$554$	14.0000	0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	4.00000i	0.169334i
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	− 15.0000i	− 0.632737i
$563$	− 21.0000i	− 0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$	− 21.0000i	− 0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$	−9.00000	−0.378968
$565$	0	0
$566$	−32.0000	−1.34506
$567$	4.00000i	0.167984i
$568$	3.00000i	0.125877i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	17.0000i	0.707107i
$579$	−20.0000	−0.831172
$580$	0	0
$581$	24.0000	0.995688
$582$	8.00000i	0.331611i
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−24.0000	−0.991431
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	9.00000i	0.371154i
$589$	20.0000	0.824086
$590$	0	0
$591$	24.0000	0.987228
$592$	7.00000i	0.287698i
$593$	3.00000i	0.123195i	0.998101	+	0.0615976i	$0.0196196\pi$
$593$	3.00000i	0.123195i	−0.998101	+	0.0615976i	$0.980380\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	7.00000i	0.286491i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	− 8.00000i	− 0.326056i
$603$	5.00000i	0.203616i
$604$	16.0000	0.651031
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 23.0000i	− 0.933541i	−0.884378	−	0.466771i	$-0.845417\pi$
$607$	− 23.0000i	− 0.933541i	0.884378	−	0.466771i	$-0.154583\pi$
$608$	− 5.00000i	− 0.202777i
$609$	12.0000	0.486265
$610$	0	0
$611$	9.00000	0.364101
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	0	0
$617$	− 33.0000i	− 1.32853i	−0.747497	−	0.664265i	$-0.768745\pi$
$617$	− 33.0000i	− 1.32853i	0.747497	−	0.664265i	$-0.231255\pi$
$618$	16.0000i	0.643614i
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	− 24.0000i	− 0.961540i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	11.0000i	0.437557i
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	9.00000	0.356873
$637$	− 9.00000i	− 0.356593i
$638$	0	0
$639$	3.00000	0.118678
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	9.00000i	0.355202i
$643$	23.0000i	0.907031i	0.891248	+	0.453516i	$0.149830\pi$
$643$	23.0000i	0.907031i	−0.891248	+	0.453516i	$0.850170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	16.0000i	0.626608i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	11.0000	0.430134
$655$	0	0
$656$	3.00000	0.117130
$657$	4.00000i	0.156055i
$658$	36.0000i	1.40343i
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	7.00000	0.271244
$667$	0	0
$668$	− 21.0000i	− 0.812514i
$669$	−14.0000	−0.541271
$670$	0	0
$671$	0	0
$672$	− 4.00000i	− 0.154303i
$673$	29.0000i	1.11787i	0.829212	+	0.558934i	$0.188789\pi$
$673$	29.0000i	1.11787i	−0.829212	+	0.558934i	$0.811211\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	− 18.0000i	− 0.691286i
$679$	32.0000	1.22805
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	8.00000	0.305441
$687$	7.00000i	0.267067i
$688$	2.00000i	0.0762493i
$689$	−9.00000	−0.342873
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	− 21.0000i	− 0.798300i
$693$	0	0
$694$	−9.00000	−0.341635
$695$	0	0
$696$	−3.00000	−0.113715
$697$	0	0
$698$	− 2.00000i	− 0.0757011i
$699$	−24.0000	−0.907763
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	1.00000i	0.0377426i
$703$	− 35.0000i	− 1.32005i
$704$	0	0
$705$	0	0
$706$	3.00000	0.112906
$707$	72.0000i	2.70784i
$708$	6.00000i	0.225494i
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	24.0000i	0.896296i
$718$	3.00000i	0.111959i
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	6.00000i	0.223297i
$723$	8.00000i	0.297523i
$724$	16.0000	0.594635
$725$	0	0
$726$	11.0000	0.408248
$727$	52.0000i	1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$	52.0000i	1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$	4.00000i	0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 8.00000i	− 0.295689i
$733$	5.00000i	0.184679i	0.995728	+	0.0923396i	$0.0294345\pi$
$733$	5.00000i	0.184679i	−0.995728	+	0.0923396i	$0.970565\pi$
$734$	−19.0000	−0.701303
$735$	0	0
$736$	0	0
$737$	0	0
$738$	− 3.00000i	− 0.110432i
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	5.00000	0.183680
$742$	− 36.0000i	− 1.32160i
$743$	− 39.0000i	− 1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$	− 39.0000i	− 1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	6.00000i	0.219529i
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	− 9.00000i	− 0.328196i
$753$	9.00000i	0.327978i
$754$	3.00000	0.109254
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 32.0000i	− 1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$	− 32.0000i	− 1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	− 7.00000i	− 0.253583i
$763$	− 44.0000i	− 1.59291i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−15.0000	−0.541972
$767$	− 6.00000i	− 0.216647i
$768$	1.00000i	0.0360844i
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	− 20.0000i	− 0.719816i
$773$	36.0000i	1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$	36.0000i	1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	−8.00000	−0.287183
$777$	− 28.0000i	− 1.00449i
$778$	− 27.0000i	− 0.967997i
$779$	−15.0000	−0.537431
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.00000i	0.107211i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−3.00000	−0.107006
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	24.0000i	0.854965i
$789$	6.00000	0.213606
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	8.00000i	0.284088i
$794$	−31.0000	−1.10015
$795$	0	0
$796$	−7.00000	−0.248108
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	20.0000i	0.707992i
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	− 30.0000i	− 1.05934i
$803$	0	0
$804$	−5.00000	−0.176336
$805$	0	0
$806$	4.00000	0.140894
$807$	21.0000i	0.739235i
$808$	− 18.0000i	− 0.633238i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	12.0000i	0.421117i
$813$	− 28.0000i	− 0.982003i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.0000i	− 0.349856i
$818$	4.00000i	0.139857i
$819$	4.00000	0.139771
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	3.00000i	0.104637i
$823$	23.0000i	0.801730i	0.916137	+	0.400865i	$0.131290\pi$
$823$	23.0000i	0.801730i	−0.916137	+	0.400865i	$0.868710\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	24.0000	0.835067
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	− 1.00000i	− 0.0346688i
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	27.0000i	0.932700i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 22.0000i	− 0.758170i
$843$	− 15.0000i	− 0.516627i
$844$	4.00000	0.137686
$845$	0	0
$846$	−9.00000	−0.309426
$847$	− 44.0000i	− 1.51186i
$848$	9.00000i	0.309061i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	0	0
$852$	3.00000i	0.102778i
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	−9.00000	−0.307614
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	− 33.0000i	− 1.12398i
$863$	− 27.0000i	− 0.919091i	−0.888154	−	0.459545i	$-0.848012\pi$
$863$	− 27.0000i	− 0.919091i	0.888154	−	0.459545i	$-0.151988\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−11.0000	−0.373795
$867$	17.0000i	0.577350i
$868$	16.0000i	0.543075i
$869$	0	0
$870$	0	0
$871$	5.00000	0.169419
$872$	11.0000i	0.372507i
$873$	8.00000i	0.270759i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 35.0000i	− 1.18187i	−0.806721	−	0.590933i	$-0.798760\pi$
$877$	− 35.0000i	− 1.18187i	0.806721	−	0.590933i	$-0.201240\pi$
$878$	− 29.0000i	− 0.978703i
$879$	−24.0000	−0.809500
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	9.00000i	0.303046i
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	7.00000i	0.234905i
$889$	−28.0000	−0.939090
$890$	0	0
$891$	0	0
$892$	− 14.0000i	− 0.468755i
$893$	45.0000i	1.50587i
$894$	−12.0000	−0.401340
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	− 21.0000i	− 0.700779i
$899$	12.0000	0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	18.0000	0.598671
$905$	0	0
$906$	16.0000	0.531564
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	6.00000i	0.199117i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	− 5.00000i	− 0.165567i
$913$	0	0
$914$	8.00000	0.264616
$915$	0	0
$916$	−7.00000	−0.231287
$917$	12.0000i	0.396275i
$918$	0	0
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	24.0000i	0.790398i
$923$	− 3.00000i	− 0.0987462i
$924$	0	0
$925$	0	0
$926$	−26.0000	−0.854413
$927$	16.0000i	0.525509i
$928$	− 3.00000i	− 0.0984798i
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	45.0000	1.47482
$932$	− 24.0000i	− 0.786146i
$933$	− 18.0000i	− 0.589294i
$934$	−33.0000	−1.07979
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	20.0000i	0.653023i
$939$	1.00000	0.0326338
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	− 10.0000i	− 0.325818i
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	11.0000i	0.357263i
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	21.0000i	0.678479i
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 7.00000i	− 0.225689i
$963$	9.00000i	0.290021i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	0	0
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 32.0000i	− 1.02587i
$974$	−34.0000	−1.08943
$975$	0	0
$976$	8.00000	0.256074
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	16.0000i	0.511624i
$979$	0	0
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	3.00000	0.0956365
$985$	0	0
$986$	0	0
$987$	36.0000i	1.14589i
$988$	5.00000i	0.159071i
$989$	0	0
$990$	0	0
$991$	41.0000	1.30241	0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000	1.30241	0.651204	+	0.758903i	$0.274264\pi$
$992$	− 4.00000i	− 0.127000i
$993$	20.0000i	0.634681i
$994$	12.0000	0.380617
$995$	0	0
$996$	−6.00000	−0.190117
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 11.0000i	− 0.348199i
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.c.1249.2		2
3.2	odd	2		5850.2.e.s.5149.1		2
5.2	odd	4		1950.2.a.f.1.1	✓	1
5.3	odd	4		1950.2.a.t.1.1	yes	1
5.4	even	2	inner	1950.2.e.c.1249.1		2
15.2	even	4		5850.2.a.be.1.1		1
15.8	even	4		5850.2.a.y.1.1		1
15.14	odd	2		5850.2.e.s.5149.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.f.1.1	✓	1	5.2	odd	4
1950.2.a.t.1.1	yes	1	5.3	odd	4
1950.2.e.c.1249.1		2	5.4	even	2	inner
1950.2.e.c.1249.2		2	1.1	even	1	trivial
5850.2.a.y.1.1		1	15.8	even	4
5850.2.a.be.1.1		1	15.2	even	4
5850.2.e.s.5149.1		2	3.2	odd	2
5850.2.e.s.5149.2		2	15.14	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -5.00000 q^{19} -4.00000 q^{21} +1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +7.00000i q^{37} -5.00000i q^{38} -1.00000 q^{39} +3.00000 q^{41} -4.00000i q^{42} +2.00000i q^{43} -9.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -1.00000i q^{52} +9.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -5.00000i q^{57} -3.00000i q^{58} -6.00000 q^{59} +8.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -5.00000i q^{67} -3.00000 q^{71} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} +5.00000 q^{76} -1.00000i q^{78} -11.0000 q^{79} +1.00000 q^{81} +3.00000i q^{82} -6.00000i q^{83} +4.00000 q^{84} -2.00000 q^{86} -3.00000i q^{87} -6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{93} +9.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} - 10 q^{19} - 8 q^{21} + 2 q^{24} - 2 q^{26} - 6 q^{29} - 8 q^{31} + 2 q^{36} - 2 q^{39} + 6 q^{41} - 18 q^{49} + 2 q^{54} + 8 q^{56} - 12 q^{59} + 16 q^{61} - 2 q^{64} - 6 q^{71} - 14 q^{74} + 10 q^{76} - 22 q^{79} + 2 q^{81} + 8 q^{84} - 4 q^{86} - 12 q^{89} - 8 q^{91} + 18 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 - 10 * q^19 - 8 * q^21 + 2 * q^24 - 2 * q^26 - 6 * q^29 - 8 * q^31 + 2 * q^36 - 2 * q^39 + 6 * q^41 - 18 * q^49 + 2 * q^54 + 8 * q^56 - 12 * q^59 + 16 * q^61 - 2 * q^64 - 6 * q^71 - 14 * q^74 + 10 * q^76 - 22 * q^79 + 2 * q^81 + 8 * q^84 - 4 * q^86 - 12 * q^89 - 8 * q^91 + 18 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

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Groups

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