LMFDB - Embedded newform 1900.2.l.c.1557.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.l (of order $4$, degree $2$, 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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 184 x^{13} + 588 x^{12} - 1440 x^{11} + 3064 x^{10} - 5344 x^{9} + \cdots + 5800 $ x^16 - 8x^15 + 48x^14 - 184x^13 + 588x^12 - 1440x^11 + 3064x^10 - 5344x^9 + 8028x^8 - 9824x^7 + 9952x^6 - 5472x^5 + 1248x^4 + 6112x^3 - 4224x^2 + 2240*x + 5800
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1557.3
Root			$-0.667207 - 1.54769i$ of defining polynomial
Character	$\chi$	$=$	1900.1557
Dual form			1900.2.l.c.493.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.880486 - 0.880486i) q^{3} +(-1.04930 - 1.04930i) q^{7} -1.44949i q^{9} +O(q^{10})$	$q+(-0.880486 - 0.880486i) q^{3} +(-1.04930 - 1.04930i) q^{7} -1.44949i q^{9} -2.44949 q^{11} +(3.03723 + 3.03723i) q^{13} +(4.66883 + 4.66883i) q^{17} +(4.11084 + 1.44949i) q^{19} +1.84778i q^{21} +(-3.61953 + 3.61953i) q^{23} +(-3.91771 + 3.91771i) q^{27} +8.22167 q^{29} -10.0695i q^{31} +(2.15674 + 2.15674i) q^{33} +(5.19397 - 5.19397i) q^{37} -5.34847i q^{39} -1.84778i q^{41} +(-5.71812 + 5.71812i) q^{43} +(1.04930 + 1.04930i) q^{47} -4.79796i q^{49} -8.22167i q^{51} +(-6.95494 - 6.95494i) q^{53} +(-2.34328 - 4.89579i) q^{57} +10.0695 q^{59} -9.34847 q^{61} +(-1.52094 + 1.52094i) q^{63} +(6.55917 - 6.55917i) q^{67} +6.37389 q^{69} +1.84778i q^{71} +(9.33766 - 9.33766i) q^{73} +(2.57024 + 2.57024i) q^{77} -1.84778 q^{79} +2.55051 q^{81} +(12.4855 - 12.4855i) q^{83} +(-7.23907 - 7.23907i) q^{87} +8.22167 q^{89} -6.37389i q^{91} +(-8.86601 + 8.86601i) q^{93} +(-10.4769 + 10.4769i) q^{97} +3.55051i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{61} + 80 q^{81}+O(q^{100}) $ 16 * q - 32 * q^61 + 80 * q^81

Display 100 coefficients 10 coefficients

Character values

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

$n$	$77$	$401$	$951$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

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
$3$	−0.880486	−	0.880486i	−0.508349	−	0.508349i	0.405671	−	0.914019i	$-0.367038\pi$
$3$	−0.880486	−	0.880486i	−0.508349	−	0.508349i	−0.914019	+	0.405671i	$0.867038\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.04930	−	1.04930i	−0.396596	−	0.396596i	0.480434	−	0.877031i	$-0.340479\pi$
$7$	−1.04930	−	1.04930i	−0.396596	−	0.396596i	−0.877031	+	0.480434i	$0.840479\pi$
$8$		0			0
$9$		−	1.44949i		−	0.483163i
$10$		0			0
$11$	−2.44949			−0.738549			−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949			−0.738549			−0.369274	+	0.929320i	$0.620394\pi$
$12$		0			0
$13$	3.03723	+	3.03723i	0.842375	+	0.842375i	0.989167	−	0.146792i	$-0.0468949\pi$
$13$	3.03723	+	3.03723i	0.842375	+	0.842375i	−0.146792	+	0.989167i	$0.546895\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.66883	+	4.66883i	1.13236	+	1.13236i	0.989784	+	0.142573i	$0.0455375\pi$
$17$	4.66883	+	4.66883i	1.13236	+	1.13236i	0.142573	+	0.989784i	$0.454462\pi$
$18$		0			0
$19$	4.11084	+	1.44949i	0.943091	+	0.332536i
$19$	4.11084	+	1.44949i	0.943091	+	0.332536i
$20$		0			0
$21$			1.84778i			0.403218i
$22$		0			0
$23$	−3.61953	+	3.61953i	−0.754725	+	0.754725i	−0.975357	−	0.220632i	$-0.929188\pi$
$23$	−3.61953	+	3.61953i	−0.754725	+	0.754725i	0.220632	+	0.975357i	$0.429188\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.91771	+	3.91771i	−0.753964	+	0.753964i
$28$		0			0
$29$	8.22167			1.52673			0.763363	−	0.645969i	$-0.223547\pi$
$29$	8.22167			1.52673			0.763363	+	0.645969i	$0.223547\pi$
$30$		0			0
$31$		−	10.0695i		−	1.80853i	−0.426975	−	0.904264i	$-0.640421\pi$
$31$		−	10.0695i		−	1.80853i	0.426975	−	0.904264i	$-0.359579\pi$
$32$		0			0
$33$	2.15674	+	2.15674i	0.375440	+	0.375440i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.19397	−	5.19397i	0.853883	−	0.853883i	−0.136726	−	0.990609i	$-0.543658\pi$
$37$	5.19397	−	5.19397i	0.853883	−	0.853883i	0.990609	+	0.136726i	$0.0436580\pi$
$38$		0			0
$39$		−	5.34847i		−	0.856441i
$40$		0			0
$41$		−	1.84778i		−	0.288575i	−0.989536	−	0.144287i	$-0.953911\pi$
$41$		−	1.84778i		−	0.288575i	0.989536	−	0.144287i	$-0.0460889\pi$
$42$		0			0
$43$	−5.71812	+	5.71812i	−0.872006	+	0.872006i	−0.992691	−	0.120685i	$-0.961491\pi$
$43$	−5.71812	+	5.71812i	−0.872006	+	0.872006i	0.120685	+	0.992691i	$0.461491\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.04930	+	1.04930i	0.153055	+	0.153055i	0.779481	−	0.626426i	$-0.215483\pi$
$47$	1.04930	+	1.04930i	0.153055	+	0.153055i	−0.626426	+	0.779481i	$0.715483\pi$
$48$		0			0
$49$		−	4.79796i		−	0.685423i
$50$		0			0
$51$		−	8.22167i		−	1.15126i
$52$		0			0
$53$	−6.95494	−	6.95494i	−0.955334	−	0.955334i	0.0437100	−	0.999044i	$-0.486082\pi$
$53$	−6.95494	−	6.95494i	−0.955334	−	0.955334i	−0.999044	+	0.0437100i	$0.986082\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.34328	−	4.89579i	−0.310375	−	0.648463i
$58$		0			0
$59$	10.0695			1.31093			0.655466	−	0.755225i	$-0.272472\pi$
$59$	10.0695			1.31093			0.655466	+	0.755225i	$0.272472\pi$
$60$		0			0
$61$	−9.34847			−1.19695			−0.598474	−	0.801142i	$-0.704226\pi$
$61$	−9.34847			−1.19695			−0.598474	+	0.801142i	$0.704226\pi$
$62$		0			0
$63$	−1.52094	+	1.52094i	−0.191621	+	0.191621i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.55917	−	6.55917i	0.801330	−	0.801330i	−0.181973	−	0.983303i	$-0.558248\pi$
$67$	6.55917	−	6.55917i	0.801330	−	0.801330i	0.983303	+	0.181973i	$0.0582485\pi$
$68$		0			0
$69$	6.37389			0.767327
$70$		0			0
$71$			1.84778i			0.219291i	0.993971	+	0.109646i	$0.0349716\pi$
$71$			1.84778i			0.219291i	−0.993971	+	0.109646i	$0.965028\pi$
$72$		0			0
$73$	9.33766	−	9.33766i	1.09289	−	1.09289i	0.0976713	−	0.995219i	$-0.468861\pi$
$73$	9.33766	−	9.33766i	1.09289	−	1.09289i	0.995219	−	0.0976713i	$-0.0311394\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.57024	+	2.57024i	0.292906	+	0.292906i
$78$		0			0
$79$	−1.84778			−0.207891			−0.103946	−	0.994583i	$-0.533147\pi$
$79$	−1.84778			−0.207891			−0.103946	+	0.994583i	$0.533147\pi$
$80$		0			0
$81$	2.55051			0.283390
$82$		0			0
$83$	12.4855	−	12.4855i	1.37047	−	1.37047i	0.510718	−	0.859748i	$-0.329380\pi$
$83$	12.4855	−	12.4855i	1.37047	−	1.37047i	0.859748	−	0.510718i	$-0.170620\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−7.23907	−	7.23907i	−0.776109	−	0.776109i
$88$		0			0
$89$	8.22167			0.871496			0.435748	−	0.900069i	$-0.356484\pi$
$89$	8.22167			0.871496			0.435748	+	0.900069i	$0.356484\pi$
$90$		0			0
$91$		−	6.37389i		−	0.668166i
$92$		0			0
$93$	−8.86601	+	8.86601i	−0.919362	+	0.919362i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.4769	+	10.4769i	−1.06377	+	1.06377i	−0.0659428	+	0.997823i	$0.521005\pi$
$97$	−10.4769	+	10.4769i	−1.06377	+	1.06377i	−0.997823	+	0.0659428i	$0.978995\pi$
$98$		0			0
$99$			3.55051i			0.356840i
$100$		0			0
$101$	13.3485			1.32822			0.664111	−	0.747634i	$-0.268810\pi$
$101$	13.3485			1.32822			0.664111	+	0.747634i	$0.268810\pi$
$102$		0			0
$103$	6.55917	+	6.55917i	0.646294	+	0.646294i	0.952095	−	0.305801i	$-0.0989243\pi$
$103$	6.55917	+	6.55917i	0.646294	+	0.646294i	−0.305801	+	0.952095i	$0.598924\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.11168	+	9.11168i	−0.880859	+	0.880859i	−0.993622	−	0.112763i	$-0.964030\pi$
$107$	−9.11168	+	9.11168i	−0.880859	+	0.880859i	0.112763	+	0.993622i	$0.464030\pi$
$108$		0			0
$109$	10.0695			0.964479			0.482239	−	0.876040i	$-0.339824\pi$
$109$	10.0695			0.964479			0.482239	+	0.876040i	$0.339824\pi$
$110$		0			0
$111$	−9.14643			−0.868141
$112$		0			0
$113$	0.484716	+	0.484716i	0.0455983	+	0.0455983i	0.729538	−	0.683940i	$-0.239735\pi$
$113$	0.484716	+	0.484716i	0.0455983	+	0.0455983i	−0.683940	+	0.729538i	$0.739735\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.40243	−	4.40243i	0.407005	−	0.407005i
$118$		0			0
$119$		−	9.79796i		−	0.898177i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	−1.62694	+	1.62694i	−0.146697	+	0.146697i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.880486	+	0.880486i	−0.0781305	+	0.0781305i	−0.745092	−	0.666962i	$-0.767594\pi$
$127$	−0.880486	+	0.880486i	−0.0781305	+	0.0781305i	0.666962	+	0.745092i	$0.267594\pi$
$128$		0			0
$129$	10.0695			0.886566
$130$		0			0
$131$	−9.79796			−0.856052			−0.428026	−	0.903767i	$-0.640791\pi$
$131$	−9.79796			−0.856052			−0.428026	+	0.903767i	$0.640791\pi$
$132$		0			0
$133$	−2.79254	−	5.83442i	−0.242144	−	0.505909i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.57024	+	2.57024i	0.219590	+	0.219590i	0.808326	−	0.588735i	$-0.200374\pi$
$137$	2.57024	+	2.57024i	0.219590	+	0.219590i	−0.588735	+	0.808326i	$0.700374\pi$
$138$		0			0
$139$		−	10.4495i		−	0.886314i	−0.896444	−	0.443157i	$-0.853858\pi$
$139$		−	10.4495i		−	0.886314i	0.896444	−	0.443157i	$-0.146142\pi$
$140$		0			0
$141$		−	1.84778i		−	0.155611i
$142$		0			0
$143$	−7.43966	−	7.43966i	−0.622135	−	0.622135i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.22453	+	4.22453i	−0.348434	+	0.348434i
$148$		0			0
$149$		−	11.1464i		−	0.913151i	−0.889685	−	0.456575i	$-0.849076\pi$
$149$		−	11.1464i		−	0.913151i	0.889685	−	0.456575i	$-0.150924\pi$
$150$		0			0
$151$		−	8.22167i		−	0.669070i	−0.942383	−	0.334535i	$-0.891421\pi$
$151$		−	8.22167i		−	0.669070i	0.942383	−	0.334535i	$-0.108579\pi$
$152$		0			0
$153$	6.76742	−	6.76742i	0.547113	−	0.547113i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.86601	+	8.86601i	0.707585	+	0.707585i	0.966027	−	0.258442i	$-0.0832091\pi$
$157$	8.86601	+	8.86601i	0.707585	+	0.707585i	−0.258442	+	0.966027i	$0.583209\pi$
$158$		0			0
$159$			12.2474i			0.971286i
$160$		0			0
$161$	7.59592			0.598642
$162$		0			0
$163$	3.61953	−	3.61953i	0.283504	−	0.283504i	−0.551001	−	0.834505i	$-0.685754\pi$
$163$	3.61953	−	3.61953i	0.283504	−	0.283504i	0.834505	+	0.551001i	$0.185754\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.5513	−	16.5513i	1.28078	−	1.28078i	0.340557	−	0.940224i	$-0.389384\pi$
$167$	16.5513	−	16.5513i	1.28078	−	1.28078i	0.940224	−	0.340557i	$-0.110616\pi$
$168$		0			0
$169$			5.44949i			0.419192i
$170$		0			0
$171$	2.10102	−	5.95862i	0.160669	−	0.455667i
$172$		0			0
$173$	10.0811	+	10.0811i	0.766453	+	0.766453i	0.977480	−	0.211027i	$-0.0676809\pi$
$173$	10.0811	+	10.0811i	0.766453	+	0.766453i	−0.211027	+	0.977480i	$0.567681\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.86601	−	8.86601i	−0.666410	−	0.666410i
$178$		0			0
$179$	6.37389			0.476407			0.238204	−	0.971215i	$-0.423441\pi$
$179$	6.37389			0.476407			0.238204	+	0.971215i	$0.423441\pi$
$180$		0			0
$181$		−	8.22167i		−	0.611112i	−0.952174	−	0.305556i	$-0.901158\pi$
$181$		−	8.22167i		−	0.611112i	0.952174	−	0.305556i	$-0.0988424\pi$
$182$		0			0
$183$	8.23119	+	8.23119i	0.608467	+	0.608467i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−11.4362	−	11.4362i	−0.836301	−	0.836301i
$188$		0			0
$189$	8.22167			0.598039
$190$		0			0
$191$	10.8990			0.788622			0.394311	−	0.918977i	$-0.370983\pi$
$191$	10.8990			0.788622			0.394311	+	0.918977i	$0.370983\pi$
$192$		0			0
$193$	−8.71591	−	8.71591i	−0.627385	−	0.627385i	0.320024	−	0.947409i	$-0.396309\pi$
$193$	−8.71591	−	8.71591i	−0.627385	−	0.627385i	−0.947409	+	0.320024i	$0.896309\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.86601	−	8.86601i	−0.631677	−	0.631677i	0.316812	−	0.948489i	$-0.397388\pi$
$197$	−8.86601	−	8.86601i	−0.631677	−	0.631677i	−0.948489	+	0.316812i	$0.897388\pi$
$198$		0			0
$199$			12.6969i			0.900062i	0.893013	+	0.450031i	$0.148587\pi$
$199$			12.6969i			0.900062i	−0.893013	+	0.450031i	$0.851413\pi$
$200$		0			0
$201$	−11.5505			−0.814710
$202$		0			0
$203$	−8.62696	−	8.62696i	−0.605494	−	0.605494i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.24648	+	5.24648i	0.364655	+	0.364655i
$208$		0			0
$209$	−10.0695	−	3.55051i	−0.696519	−	0.245594i
$210$		0			0
$211$		−	11.9172i		−	0.820416i	−0.911992	−	0.410208i	$-0.865456\pi$
$211$		−	11.9172i		−	0.820416i	0.911992	−	0.410208i	$-0.134544\pi$
$212$		0			0
$213$	1.62694	−	1.62694i	0.111476	−	0.111476i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−10.5658	+	10.5658i	−0.717255	+	0.717255i
$218$		0			0
$219$	−16.4433			−1.11114
$220$		0			0
$221$			28.3606i			1.90774i
$222$		0			0
$223$	−6.16340	−	6.16340i	−0.412732	−	0.412732i	0.469957	−	0.882689i	$-0.344269\pi$
$223$	−6.16340	−	6.16340i	−0.412732	−	0.412732i	−0.882689	+	0.469957i	$0.844269\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0811	+	10.0811i	−0.669107	+	0.669107i	−0.957509	−	0.288402i	$-0.906876\pi$
$227$	−10.0811	+	10.0811i	−0.669107	+	0.669107i	0.288402	+	0.957509i	$0.406876\pi$
$228$		0			0
$229$		−	14.2474i		−	0.941498i	−0.882267	−	0.470749i	$-0.843984\pi$
$229$		−	14.2474i		−	0.941498i	0.882267	−	0.470749i	$-0.156016\pi$
$230$		0			0
$231$		−	4.52612i		−	0.297797i
$232$		0			0
$233$	0.471647	−	0.471647i	0.0308987	−	0.0308987i	−0.691489	−	0.722387i	$-0.743045\pi$
$233$	0.471647	−	0.471647i	0.0308987	−	0.0308987i	0.722387	+	0.691489i	$0.243045\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.62694	+	1.62694i	0.105681	+	0.105681i
$238$		0			0
$239$			12.0000i			0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$			12.0000i			0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$		0			0
$241$			26.5128i			1.70784i	0.520405	+	0.853920i	$0.325781\pi$
$241$			26.5128i			1.70784i	−0.520405	+	0.853920i	$0.674219\pi$
$242$		0			0
$243$	9.50745	+	9.50745i	0.609903	+	0.609903i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.08311	+	16.8880i	0.514316	+	1.07456i
$248$		0			0
$249$	−21.9867			−1.39335
$250$		0			0
$251$	−22.8990			−1.44537			−0.722685	−	0.691177i	$-0.757092\pi$
$251$	−22.8990			−1.44537			−0.722685	+	0.691177i	$0.757092\pi$
$252$		0			0
$253$	8.86601	−	8.86601i	0.557401	−	0.557401i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.5208	−	17.5208i	1.09292	−	1.09292i	0.0976995	−	0.995216i	$-0.468852\pi$
$257$	17.5208	−	17.5208i	1.09292	−	1.09292i	0.995216	−	0.0976995i	$-0.0311484\pi$
$258$		0			0
$259$	−10.9000			−0.677294
$260$		0			0
$261$		−	11.9172i		−	0.737658i
$262$		0			0
$263$	−12.9572	+	12.9572i	−0.798975	+	0.798975i	−0.982934	−	0.183959i	$-0.941109\pi$
$263$	−12.9572	+	12.9572i	−0.798975	+	0.798975i	0.183959	+	0.982934i	$0.441109\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−7.23907	−	7.23907i	−0.443024	−	0.443024i
$268$		0			0
$269$	21.9867			1.34055			0.670276	−	0.742112i	$-0.266176\pi$
$269$	21.9867			1.34055			0.670276	+	0.742112i	$0.266176\pi$
$270$		0			0
$271$	12.6515			0.768526			0.384263	−	0.923224i	$-0.374456\pi$
$271$	12.6515			0.768526			0.384263	+	0.923224i	$0.374456\pi$
$272$		0			0
$273$	−5.61212	+	5.61212i	−0.339661	+	0.339661i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.1470	+	19.1470i	1.15043	+	1.15043i	0.986467	+	0.163963i	$0.0524277\pi$
$277$	19.1470	+	19.1470i	1.15043	+	1.15043i	0.163963	+	0.986467i	$0.447572\pi$
$278$		0			0
$279$	−14.5956			−0.873814
$280$		0			0
$281$			22.8172i			1.36116i	0.732673	+	0.680581i	$0.238273\pi$
$281$			22.8172i			1.36116i	−0.732673	+	0.680581i	$0.761727\pi$
$282$		0			0
$283$	−3.14789	+	3.14789i	−0.187122	+	0.187122i	−0.794451	−	0.607328i	$-0.792241\pi$
$283$	−3.14789	+	3.14789i	−0.187122	+	0.187122i	0.607328	+	0.794451i	$0.292241\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.93887	+	1.93887i	−0.114448	+	0.114448i
$288$		0			0
$289$			26.5959i			1.56447i
$290$		0			0
$291$	18.4495			1.08153
$292$		0			0
$293$	6.95494	+	6.95494i	0.406312	+	0.406312i	0.880450	−	0.474138i	$-0.157240\pi$
$293$	6.95494	+	6.95494i	0.406312	+	0.406312i	−0.474138	+	0.880450i	$0.657240\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.59640	−	9.59640i	0.556839	−	0.556839i
$298$		0			0
$299$	−21.9867			−1.27152
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	−11.7531	−	11.7531i	−0.675200	−	0.675200i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	16.1556	−	16.1556i	0.922047	−	0.922047i	−0.0751271	−	0.997174i	$-0.523936\pi$
$307$	16.1556	−	16.1556i	0.922047	−	0.922047i	0.997174	+	0.0751271i	$0.0239363\pi$
$308$		0			0
$309$		−	11.5505i		−	0.657086i
$310$		0			0
$311$	5.14643			0.291827			0.145914	−	0.989297i	$-0.453388\pi$
$311$	5.14643			0.291827			0.145914	+	0.989297i	$0.453388\pi$
$312$		0			0
$313$	−16.1051	+	16.1051i	−0.910313	+	0.910313i	−0.996297	−	0.0859838i	$-0.972597\pi$
$313$	−16.1051	+	16.1051i	−0.910313	+	0.910313i	0.0859838	+	0.996297i	$0.472597\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0811	+	10.0811i	−0.566212	+	0.566212i	−0.931065	−	0.364853i	$-0.881119\pi$
$317$	−10.0811	+	10.0811i	−0.566212	+	0.566212i	0.364853	+	0.931065i	$0.381119\pi$
$318$		0			0
$319$	−20.1389			−1.12756
$320$		0			0
$321$	16.0454			0.895567
$322$		0			0
$323$	12.4254	+	25.9602i	0.691366	+	1.44446i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.86601	−	8.86601i	−0.490291	−	0.490291i
$328$		0			0
$329$		−	2.20204i		−	0.121402i
$330$		0			0
$331$		−	12.7478i		−	0.700682i	−0.936622	−	0.350341i	$-0.886066\pi$
$331$		−	12.7478i		−	0.700682i	0.936622	−	0.350341i	$-0.113934\pi$
$332$		0			0
$333$	−7.52860	−	7.52860i	−0.412565	−	0.412565i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.6031	−	13.6031i	0.741006	−	0.741006i	−0.231766	−	0.972772i	$-0.574450\pi$
$337$	13.6031	−	13.6031i	0.741006	−	0.741006i	0.972772	+	0.231766i	$0.0744503\pi$
$338$		0			0
$339$		−	0.853572i		−	0.0463597i
$340$		0			0
$341$			24.6650i			1.33569i
$342$		0			0
$343$	−12.3795	+	12.3795i	−0.668432	+	0.668432i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.91530	−	9.91530i	−0.532281	−	0.532281i	0.388969	−	0.921251i	$-0.372831\pi$
$347$	−9.91530	−	9.91530i	−0.532281	−	0.532281i	−0.921251	+	0.388969i	$0.872831\pi$
$348$		0			0
$349$			5.79796i			0.310358i	0.987886	+	0.155179i	$0.0495954\pi$
$349$			5.79796i			0.310358i	−0.987886	+	0.155179i	$0.950405\pi$
$350$		0			0
$351$	−23.7980			−1.27024
$352$		0			0
$353$	−8.86601	+	8.86601i	−0.471890	+	0.471890i	−0.902526	−	0.430636i	$-0.858289\pi$
$353$	−8.86601	+	8.86601i	−0.471890	+	0.471890i	0.430636	+	0.902526i	$0.358289\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.62696	+	8.62696i	−0.456587	+	0.456587i
$358$		0			0
$359$			0.247449i			0.0130598i	0.999979	+	0.00652992i	$0.00207855\pi$
$359$			0.247449i			0.0130598i	−0.999979	+	0.00652992i	$0.997921\pi$
$360$		0			0
$361$	14.7980	+	11.9172i	0.778840	+	0.627223i
$362$		0			0
$363$	4.40243	+	4.40243i	0.231068	+	0.231068i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.0977	+	18.0977i	0.944690	+	0.944690i	0.998549	−	0.0538581i	$-0.0171519\pi$
$367$	18.0977	+	18.0977i	0.944690	+	0.944690i	−0.0538581	+	0.998549i	$0.517152\pi$
$368$		0			0
$369$	−2.67834			−0.139429
$370$		0			0
$371$			14.5956i			0.757764i
$372$		0			0
$373$	−5.19397	−	5.19397i	−0.268933	−	0.268933i	0.559737	−	0.828670i	$-0.310902\pi$
$373$	−5.19397	−	5.19397i	−0.268933	−	0.268933i	−0.828670	+	0.559737i	$0.810902\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.9711	+	24.9711i	1.28608	+	1.28608i
$378$		0			0
$379$	−8.22167			−0.422319			−0.211160	−	0.977452i	$-0.567724\pi$
$379$	−8.22167			−0.422319			−0.211160	+	0.977452i	$0.567724\pi$
$380$		0			0
$381$	1.55051			0.0794350
$382$		0			0
$383$	8.14225	+	8.14225i	0.416049	+	0.416049i	0.883840	−	0.467790i	$-0.154950\pi$
$383$	8.14225	+	8.14225i	0.416049	+	0.416049i	−0.467790	+	0.883840i	$0.654950\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.28836	+	8.28836i	0.421321	+	0.421321i
$388$		0			0
$389$			8.20204i			0.415860i	0.978144	+	0.207930i	$0.0666726\pi$
$389$			8.20204i			0.415860i	−0.978144	+	0.207930i	$0.933327\pi$
$390$		0			0
$391$	−33.7980			−1.70924
$392$		0			0
$393$	8.62696	+	8.62696i	0.435173	+	0.435173i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−28.0130	−	28.0130i	−1.40593	−	1.40593i	−0.779383	−	0.626547i	$-0.784468\pi$
$397$	−28.0130	−	28.0130i	−1.40593	−	1.40593i	−0.626547	−	0.779383i	$-0.715532\pi$
$398$		0			0
$399$	−2.67834	+	7.59592i	−0.134085	+	0.380272i
$400$		0			0
$401$			12.7478i			0.636594i	0.947991	+	0.318297i	$0.103111\pi$
$401$			12.7478i			0.636594i	−0.947991	+	0.318297i	$0.896889\pi$
$402$		0			0
$403$	30.5832	−	30.5832i	1.52346	−	1.52346i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.7226	+	12.7226i	−0.630634	+	0.630634i
$408$		0			0
$409$	−18.2911			−0.904438			−0.452219	−	0.891907i	$-0.649367\pi$
$409$	−18.2911			−0.904438			−0.452219	+	0.891907i	$0.649367\pi$
$410$		0			0
$411$		−	4.52612i		−	0.223257i
$412$		0			0
$413$	−10.5658	−	10.5658i	−0.519910	−	0.519910i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.20063	+	9.20063i	−0.450557	+	0.450557i
$418$		0			0
$419$			9.79796i			0.478662i	0.970938	+	0.239331i	$0.0769280\pi$
$419$			9.79796i			0.478662i	−0.970938	+	0.239331i	$0.923072\pi$
$420$		0			0
$421$			18.2911i			0.891455i	0.895169	+	0.445727i	$0.147055\pi$
$421$			18.2911i			0.891455i	−0.895169	+	0.445727i	$0.852945\pi$
$422$		0			0
$423$	1.52094	−	1.52094i	0.0739508	−	0.0739508i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9.80930	+	9.80930i	0.474706	+	0.474706i
$428$		0			0
$429$			13.1010i			0.632523i
$430$		0			0
$431$		−	11.9172i		−	0.574033i	−0.957926	−	0.287016i	$-0.907337\pi$
$431$		−	11.9172i		−	0.574033i	0.957926	−	0.287016i	$-0.0926634\pi$
$432$		0			0
$433$	−6.55917	−	6.55917i	−0.315214	−	0.315214i	0.531712	−	0.846925i	$-0.321549\pi$
$433$	−6.55917	−	6.55917i	−0.315214	−	0.315214i	−0.846925	+	0.531712i	$0.821549\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.1258	+	9.63283i	−0.962747	+	0.460801i
$438$		0			0
$439$	−14.5956			−0.696608			−0.348304	−	0.937382i	$-0.613242\pi$
$439$	−14.5956			−0.696608			−0.348304	+	0.937382i	$0.613242\pi$
$440$		0			0
$441$	−6.95459			−0.331171
$442$		0			0
$443$	19.2530	−	19.2530i	0.914736	−	0.914736i	−0.0819044	−	0.996640i	$-0.526100\pi$
$443$	19.2530	−	19.2530i	0.914736	−	0.914736i	0.996640	+	0.0819044i	$0.0261002\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.81427	+	9.81427i	−0.464199	+	0.464199i
$448$		0			0
$449$	−10.0695			−0.475207			−0.237603	−	0.971362i	$-0.576362\pi$
$449$	−10.0695			−0.475207			−0.237603	+	0.971362i	$0.576362\pi$
$450$		0			0
$451$			4.52612i			0.213126i
$452$		0			0
$453$	−7.23907	+	7.23907i	−0.340121	+	0.340121i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.4362	−	11.4362i	−0.534965	−	0.534965i	0.387081	−	0.922046i	$-0.373483\pi$
$457$	−11.4362	−	11.4362i	−0.534965	−	0.534965i	−0.922046	+	0.387081i	$0.873483\pi$
$458$		0			0
$459$	−36.5823			−1.70751
$460$		0			0
$461$	35.3939			1.64846			0.824229	−	0.566257i	$-0.191609\pi$
$461$	35.3939			1.64846			0.824229	+	0.566257i	$0.191609\pi$
$462$		0			0
$463$	−12.9572	+	12.9572i	−0.602172	+	0.602172i	−0.940888	−	0.338717i	$-0.890007\pi$
$463$	−12.9572	+	12.9572i	−0.602172	+	0.602172i	0.338717	+	0.940888i	$0.390007\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.577648	+	0.577648i	0.0267304	+	0.0267304i	0.720346	−	0.693615i	$-0.243983\pi$
$467$	0.577648	+	0.577648i	0.0267304	+	0.0267304i	−0.693615	+	0.720346i	$0.743983\pi$
$468$		0			0
$469$	−13.7650			−0.635609
$470$		0			0
$471$		−	15.6128i		−	0.719399i
$472$		0			0
$473$	14.0065	−	14.0065i	0.644019	−	0.644019i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−10.0811	+	10.0811i	−0.461582	+	0.461582i
$478$		0			0
$479$		−	5.14643i		−	0.235146i	−0.993064	−	0.117573i	$-0.962489\pi$
$479$		−	5.14643i		−	0.235146i	0.993064	−	0.117573i	$-0.0375115\pi$
$480$		0			0
$481$	31.5505			1.43858
$482$		0			0
$483$	−6.68810	−	6.68810i	−0.304319	−	0.304319i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	11.6642	−	11.6642i	0.528555	−	0.528555i	−0.391586	−	0.920141i	$-0.628074\pi$
$487$	11.6642	−	11.6642i	0.528555	−	0.528555i	0.920141	+	0.391586i	$0.128074\pi$
$488$		0			0
$489$	−6.37389			−0.288237
$490$		0			0
$491$	40.2929			1.81839			0.909196	−	0.416369i	$-0.136698\pi$
$491$	40.2929			1.81839			0.909196	+	0.416369i	$0.136698\pi$
$492$		0			0
$493$	38.3856	+	38.3856i	1.72880	+	1.72880i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.93887	−	1.93887i	0.0869700	−	0.0869700i
$498$		0			0
$499$			10.9444i			0.489938i	0.969531	+	0.244969i	$0.0787778\pi$
$499$			10.9444i			0.489938i	−0.969531	+	0.244969i	$0.921222\pi$
$500$		0			0
$501$	−29.1464			−1.30217
$502$		0			0
$503$	10.8586	−	10.8586i	0.484161	−	0.484161i	−0.422297	−	0.906458i	$-0.638776\pi$
$503$	10.8586	−	10.8586i	0.484161	−	0.484161i	0.906458	+	0.422297i	$0.138776\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.79820	−	4.79820i	0.213095	−	0.213095i
$508$		0			0
$509$	−3.69556			−0.163803			−0.0819014	−	0.996640i	$-0.526099\pi$
$509$	−3.69556			−0.163803			−0.0819014	+	0.996640i	$0.526099\pi$
$510$		0			0
$511$	−19.5959			−0.866872
$512$		0			0
$513$	−21.7838	+	10.4264i	−0.961776	+	0.460336i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.57024	−	2.57024i	−0.113039	−	0.113039i
$518$		0			0
$519$		−	17.7526i		−	0.779251i
$520$		0			0
$521$			44.8039i			1.96290i	0.191729	+	0.981448i	$0.438590\pi$
$521$			44.8039i			1.96290i	−0.191729	+	0.981448i	$0.561410\pi$
$522$		0			0
$523$	−18.3123	−	18.3123i	−0.800741	−	0.800741i	0.182470	−	0.983211i	$-0.441591\pi$
$523$	−18.3123	−	18.3123i	−0.800741	−	0.800741i	−0.983211	+	0.182470i	$0.941591\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	47.0125	−	47.0125i	2.04790	−	2.04790i
$528$		0			0
$529$		−	3.20204i		−	0.139219i
$530$		0			0
$531$		−	14.5956i		−	0.633394i
$532$		0			0
$533$	5.61212	−	5.61212i	0.243088	−	0.243088i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.61212	−	5.61212i	−0.242181	−	0.242181i
$538$		0			0
$539$			11.7526i			0.506218i
$540$		0			0
$541$	−29.3485			−1.26179			−0.630895	−	0.775869i	$-0.717312\pi$
$541$	−29.3485			−1.26179			−0.630895	+	0.775869i	$0.717312\pi$
$542$		0			0
$543$	−7.23907	+	7.23907i	−0.310658	+	0.310658i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.46356	+	2.46356i	−0.105334	+	0.105334i	−0.757810	−	0.652475i	$-0.773731\pi$
$547$	−2.46356	+	2.46356i	−0.105334	+	0.105334i	0.652475	+	0.757810i	$0.273731\pi$
$548$		0			0
$549$			13.5505i			0.578322i
$550$		0			0
$551$	33.7980	+	11.9172i	1.43984	+	0.507691i
$552$		0			0
$553$	1.93887	+	1.93887i	0.0824490	+	0.0824490i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0484	−	17.0484i	−0.722363	−	0.722363i	0.246723	−	0.969086i	$-0.420646\pi$
$557$	−17.0484	−	17.0484i	−0.722363	−	0.722363i	−0.969086	+	0.246723i	$0.920646\pi$
$558$		0			0
$559$	−34.7345			−1.46911
$560$		0			0
$561$			20.1389i			0.850265i
$562$		0			0
$563$	−30.4612	−	30.4612i	−1.28379	−	1.28379i	−0.938495	−	0.345292i	$-0.887780\pi$
$563$	−30.4612	−	30.4612i	−1.28379	−	1.28379i	−0.345292	−	0.938495i	$-0.612220\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.67624	−	2.67624i	−0.112391	−	0.112391i
$568$		0			0
$569$	−22.8172			−0.956549			−0.478274	−	0.878211i	$-0.658738\pi$
$569$	−22.8172			−0.956549			−0.478274	+	0.878211i	$0.658738\pi$
$570$		0			0
$571$	−4.24745			−0.177750			−0.0888751	−	0.996043i	$-0.528327\pi$
$571$	−4.24745			−0.177750			−0.0888751	+	0.996043i	$0.528327\pi$
$572$		0			0
$573$	−9.59640	−	9.59640i	−0.400895	−	0.400895i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.80930	−	9.80930i	−0.408367	−	0.408367i	0.472802	−	0.881169i	$-0.343243\pi$
$577$	−9.80930	−	9.80930i	−0.408367	−	0.408367i	−0.881169	+	0.472802i	$0.843243\pi$
$578$		0			0
$579$			15.3485i			0.637861i
$580$		0			0
$581$	−26.2020			−1.08704
$582$		0			0
$583$	17.0361	+	17.0361i	0.705561	+	0.705561i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.6827	+	16.6827i	0.688570	+	0.688570i	0.961916	−	0.273346i	$-0.0881304\pi$
$587$	16.6827	+	16.6827i	0.688570	+	0.688570i	−0.273346	+	0.961916i	$0.588130\pi$
$588$		0			0
$589$	14.5956	−	41.3939i	0.601400	−	1.70560i
$590$		0			0
$591$			15.6128i			0.642224i
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	11.1795	−	11.1795i	0.457545	−	0.457545i
$598$		0			0
$599$	38.4300			1.57021			0.785104	−	0.619364i	$-0.212609\pi$
$599$	38.4300			1.57021			0.785104	+	0.619364i	$0.212609\pi$
$600$		0			0
$601$		−	30.2084i		−	1.23222i	−0.787658	−	0.616112i	$-0.788707\pi$
$601$		−	30.2084i		−	1.23222i	0.787658	−	0.616112i	$-0.211293\pi$
$602$		0			0
$603$	−9.50745	−	9.50745i	−0.387173	−	0.387173i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.7520	+	25.7520i	−1.04524	+	1.04524i	−0.0463129	+	0.998927i	$0.514747\pi$
$607$	−25.7520	+	25.7520i	−1.04524	+	1.04524i	−0.998927	+	0.0463129i	$0.985253\pi$
$608$		0			0
$609$			15.1918i			0.615604i
$610$		0			0
$611$			6.37389i			0.257860i
$612$		0			0
$613$	27.0697	−	27.0697i	1.09333	−	1.09333i	0.0981636	−	0.995170i	$-0.468703\pi$
$613$	27.0697	−	27.0697i	1.09333	−	1.09333i	0.995170	−	0.0981636i	$-0.0312969\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.04189	−	3.04189i	−0.122462	−	0.122462i	0.643220	−	0.765682i	$-0.277598\pi$
$617$	−3.04189	−	3.04189i	−0.122462	−	0.122462i	−0.765682	+	0.643220i	$0.777598\pi$
$618$		0			0
$619$			30.0454i			1.20763i	0.797126	+	0.603813i	$0.206353\pi$
$619$			30.0454i			1.20763i	−0.797126	+	0.603813i	$0.793647\pi$
$620$		0			0
$621$		−	28.3606i		−	1.13807i
$622$		0			0
$623$	−8.62696	−	8.62696i	−0.345632	−	0.345632i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	5.73984	+	11.9922i	0.229227	+	0.478922i
$628$		0			0
$629$	48.4995			1.93380
$630$		0			0
$631$	−22.4495			−0.893700			−0.446850	−	0.894609i	$-0.647454\pi$
$631$	−22.4495			−0.893700			−0.446850	+	0.894609i	$0.647454\pi$
$632$		0			0
$633$	−10.4930	+	10.4930i	−0.417057	+	0.417057i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.5725	−	14.5725i	0.577383	−	0.577383i
$638$		0			0
$639$	2.67834			0.105953
$640$		0			0
$641$			46.6517i			1.84263i	0.388815	+	0.921316i	$0.372884\pi$
$641$			46.6517i			1.84263i	−0.388815	+	0.921316i	$0.627116\pi$
$642$		0			0
$643$	−12.0139	+	12.0139i	−0.473782	+	0.473782i	−0.903136	−	0.429354i	$-0.858741\pi$
$643$	−12.0139	+	12.0139i	−0.473782	+	0.473782i	0.429354	+	0.903136i	$0.358741\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.28836	+	8.28836i	0.325849	+	0.325849i	0.851006	−	0.525157i	$-0.175993\pi$
$647$	8.28836	+	8.28836i	0.325849	+	0.325849i	−0.525157	+	0.851006i	$0.675993\pi$
$648$		0			0
$649$	−24.6650			−0.968187
$650$		0			0
$651$	18.6061			0.729231
$652$		0			0
$653$	−8.39436	+	8.39436i	−0.328497	+	0.328497i	−0.852015	−	0.523518i	$-0.824619\pi$
$653$	−8.39436	+	8.39436i	−0.328497	+	0.328497i	0.523518	+	0.852015i	$0.324619\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−13.5348	−	13.5348i	−0.528044	−	0.528044i
$658$		0			0
$659$	18.2911			0.712521			0.356261	−	0.934387i	$-0.384052\pi$
$659$	18.2911			0.712521			0.356261	+	0.934387i	$0.384052\pi$
$660$		0			0
$661$		−	31.0389i		−	1.20727i	−0.797259	−	0.603637i	$-0.793718\pi$
$661$		−	31.0389i		−	1.20727i	0.797259	−	0.603637i	$-0.206282\pi$
$662$		0			0
$663$	24.9711	−	24.9711i	0.969797	−	0.969797i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−29.7586	+	29.7586i	−1.15226	+	1.15226i
$668$		0			0
$669$			10.8536i			0.419623i
$670$		0			0
$671$	22.8990			0.884005
$672$		0			0
$673$	−6.55917	−	6.55917i	−0.252837	−	0.252837i	0.569296	−	0.822133i	$-0.307216\pi$
$673$	−6.55917	−	6.55917i	−0.252837	−	0.252837i	−0.822133	+	0.569296i	$0.807216\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.67202	+	1.67202i	−0.0642611	+	0.0642611i	−0.738507	−	0.674246i	$-0.764469\pi$
$677$	−1.67202	+	1.67202i	−0.0642611	+	0.0642611i	0.674246	+	0.738507i	$0.264469\pi$
$678$		0			0
$679$	21.9867			0.843772
$680$		0			0
$681$	17.7526			0.680279
$682$		0			0
$683$	2.64146	+	2.64146i	0.101073	+	0.101073i	0.755835	−	0.654762i	$-0.227231\pi$
$683$	2.64146	+	2.64146i	0.101073	+	0.101073i	−0.654762	+	0.755835i	$0.727231\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−12.5447	+	12.5447i	−0.478609	+	0.478609i
$688$		0			0
$689$		−	42.2474i		−	1.60950i
$690$		0			0
$691$	−5.55051			−0.211151			−0.105576	−	0.994411i	$-0.533669\pi$
$691$	−5.55051			−0.211151			−0.105576	+	0.994411i	$0.533669\pi$
$692$		0			0
$693$	3.72553	−	3.72553i	0.141521	−	0.141521i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	8.62696	−	8.62696i	0.326770	−	0.326770i
$698$		0			0
$699$	−0.830558			−0.0314146
$700$		0			0
$701$	20.9444			0.791059			0.395529	−	0.918453i	$-0.370561\pi$
$701$	20.9444			0.791059			0.395529	+	0.918453i	$0.370561\pi$
$702$		0			0
$703$	28.8802	−	13.8229i	1.08924	−	0.521342i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.0065	−	14.0065i	−0.526768	−	0.526768i
$708$		0			0
$709$		−	4.00000i		−	0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$		−	4.00000i		−	0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$		0			0
$711$			2.67834i			0.100445i
$712$		0			0
$713$	36.4467	+	36.4467i	1.36494	+	1.36494i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.5658	−	10.5658i	0.394588	−	0.394588i
$718$		0			0
$719$		−	24.2474i		−	0.904277i	−0.891948	−	0.452139i	$-0.850661\pi$
$719$		−	24.2474i		−	0.904277i	0.891948	−	0.452139i	$-0.149339\pi$
$720$		0			0
$721$		−	13.7650i		−	0.512636i
$722$		0			0
$723$	23.3441	−	23.3441i	0.868178	−	0.868178i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.577648	−	0.577648i	−0.0214238	−	0.0214238i	0.696314	−	0.717738i	$-0.254822\pi$
$727$	−0.577648	−	0.577648i	−0.0214238	−	0.0214238i	−0.717738	+	0.696314i	$0.754822\pi$
$728$		0			0
$729$		−	24.3939i		−	0.903477i
$730$		0			0
$731$	−53.3939			−1.97484
$732$		0			0
$733$	8.86601	−	8.86601i	0.327474	−	0.327474i	−0.524152	−	0.851625i	$-0.675617\pi$
$733$	8.86601	−	8.86601i	0.327474	−	0.327474i	0.851625	+	0.524152i	$0.175617\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0666	+	16.0666i	−0.591821	+	0.591821i
$738$		0			0
$739$		−	36.2929i		−	1.33505i	−0.744585	−	0.667527i	$-0.767353\pi$
$739$		−	36.2929i		−	1.33505i	0.744585	−	0.667527i	$-0.232647\pi$
$740$		0			0
$741$	7.75255	−	21.9867i	0.284797	−	0.807701i
$742$		0			0
$743$	5.98551	+	5.98551i	0.219587	+	0.219587i	0.808324	−	0.588737i	$-0.200375\pi$
$743$	5.98551	+	5.98551i	0.219587	+	0.219587i	−0.588737	+	0.808324i	$0.700375\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−18.0977	−	18.0977i	−0.662159	−	0.662159i
$748$		0			0
$749$	19.1217			0.698691
$750$		0			0
$751$		−	37.4128i		−	1.36521i	−0.730786	−	0.682606i	$-0.760846\pi$
$751$		−	37.4128i		−	1.36521i	0.730786	−	0.682606i	$-0.239154\pi$
$752$		0			0
$753$	20.1622	+	20.1622i	0.734752	+	0.734752i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−11.9079	−	11.9079i	−0.432800	−	0.432800i	0.456780	−	0.889580i	$-0.349003\pi$
$757$	−11.9079	−	11.9079i	−0.432800	−	0.432800i	−0.889580	+	0.456780i	$0.849003\pi$
$758$		0			0
$759$	−15.6128			−0.566708
$760$		0			0
$761$	1.34847			0.0488820			0.0244410	−	0.999701i	$-0.492219\pi$
$761$	1.34847			0.0488820			0.0244410	+	0.999701i	$0.492219\pi$
$762$		0			0
$763$	−10.5658	−	10.5658i	−0.382509	−	0.382509i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	30.5832	+	30.5832i	1.10430	+	1.10430i
$768$		0			0
$769$			33.3485i			1.20258i	0.799032	+	0.601288i	$0.205346\pi$
$769$			33.3485i			1.20258i	−0.799032	+	0.601288i	$0.794654\pi$
$770$		0			0
$771$	−30.8536			−1.11116
$772$		0			0
$773$	−18.7081	−	18.7081i	−0.672883	−	0.672883i	0.285497	−	0.958380i	$-0.407841\pi$
$773$	−18.7081	−	18.7081i	−0.672883	−	0.672883i	−0.958380	+	0.285497i	$0.907841\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.59730	+	9.59730i	0.344301	+	0.344301i
$778$		0			0
$779$	2.67834	−	7.59592i	0.0959614	−	0.272152i
$780$		0			0
$781$		−	4.52612i		−	0.161957i
$782$		0			0
$783$	−32.2102	+	32.2102i	−1.15110	+	1.15110i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7.74648	−	7.74648i	0.276132	−	0.276132i	−0.555431	−	0.831563i	$-0.687447\pi$
$787$	7.74648	−	7.74648i	0.276132	−	0.276132i	0.831563	+	0.555431i	$0.187447\pi$
$788$		0			0
$789$	22.8172			0.812315
$790$		0			0
$791$		−	1.01722i		−	0.0361682i
$792$		0			0
$793$	−28.3934	−	28.3934i	−1.00828	−	1.00828i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.67202	+	1.67202i	−0.0592261	+	0.0592261i	−0.736099	−	0.676873i	$-0.763334\pi$
$797$	−1.67202	+	1.67202i	−0.0592261	+	0.0592261i	0.676873	+	0.736099i	$0.263334\pi$
$798$		0			0
$799$			9.79796i			0.346627i
$800$		0			0
$801$		−	11.9172i		−	0.421075i
$802$		0			0
$803$	−22.8725	+	22.8725i	−0.807153	+	0.807153i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.3590	−	19.3590i	−0.681468	−	0.681468i
$808$		0			0
$809$			53.3939i			1.87723i	0.344968	+	0.938614i	$0.387890\pi$
$809$			53.3939i			1.87723i	−0.344968	+	0.938614i	$0.612110\pi$
$810$		0			0
$811$			5.54334i			0.194653i	0.995253	+	0.0973264i	$0.0310291\pi$
$811$			5.54334i			0.194653i	−0.995253	+	0.0973264i	$0.968971\pi$
$812$		0			0
$813$	−11.1395	−	11.1395i	−0.390679	−	0.390679i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−31.7946	+	15.2179i	−1.11235	+	0.532407i
$818$		0			0
$819$	−9.23889			−0.322833
$820$		0			0
$821$	−32.2020			−1.12386			−0.561929	−	0.827185i	$-0.689941\pi$
$821$	−32.2020			−1.12386			−0.561929	+	0.827185i	$0.689941\pi$
$822$		0			0
$823$	−30.6892	+	30.6892i	−1.06976	+	1.06976i	−0.0723816	+	0.997377i	$0.523060\pi$
$823$	−30.6892	+	30.6892i	−1.06976	+	1.06976i	−0.997377	+	0.0723816i	$0.976940\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21.8342	−	21.8342i	0.759251	−	0.759251i	−0.216935	−	0.976186i	$-0.569606\pi$
$827$	21.8342	−	21.8342i	0.759251	−	0.759251i	0.976186	+	0.216935i	$0.0696059\pi$
$828$		0			0
$829$	26.5128			0.920828			0.460414	−	0.887704i	$-0.347701\pi$
$829$	26.5128			0.920828			0.460414	+	0.887704i	$0.347701\pi$
$830$		0			0
$831$		−	33.7173i		−	1.16964i
$832$		0			0
$833$	22.4008	−	22.4008i	0.776143	−	0.776143i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	39.4492	+	39.4492i	1.36356	+	1.36356i
$838$		0			0
$839$	42.9561			1.48301			0.741505	−	0.670947i	$-0.234112\pi$
$839$	42.9561			1.48301			0.741505	+	0.670947i	$0.234112\pi$
$840$		0			0
$841$	38.5959			1.33089
$842$		0			0
$843$	20.0903	−	20.0903i	0.691945	−	0.691945i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.24648	+	5.24648i	0.180271	+	0.180271i
$848$		0			0
$849$	5.54334			0.190247
$850$		0			0
$851$			37.5995i			1.28889i
$852$		0			0
$853$	−24.9711	+	24.9711i	−0.854994	+	0.854994i	−0.990743	−	0.135750i	$-0.956656\pi$
$853$	−24.9711	+	24.9711i	−0.854994	+	0.854994i	0.135750	+	0.990743i	$0.456656\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.5819	−	15.5819i	0.532268	−	0.532268i	−0.388979	−	0.921247i	$-0.627172\pi$
$857$	15.5819	−	15.5819i	0.532268	−	0.532268i	0.921247	+	0.388979i	$0.127172\pi$
$858$		0			0
$859$			32.0000i			1.09183i	0.837842	+	0.545913i	$0.183817\pi$
$859$			32.0000i			1.09183i	−0.837842	+	0.545913i	$0.816183\pi$
$860$		0			0
$861$	3.41429			0.116359
$862$		0			0
$863$	30.4612	+	30.4612i	1.03691	+	1.03691i	0.999292	+	0.0376196i	$0.0119775\pi$
$863$	30.4612	+	30.4612i	1.03691	+	1.03691i	0.0376196	+	0.999292i	$0.488022\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.4173	−	23.4173i	0.795294	−	0.795294i
$868$		0			0
$869$	4.52612			0.153538
$870$		0			0
$871$	39.8434			1.35004
$872$		0			0
$873$	15.1861	+	15.1861i	0.513973	+	0.513973i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.71591	−	8.71591i	0.294315	−	0.294315i	−0.544467	−	0.838782i	$-0.683268\pi$
$877$	8.71591	−	8.71591i	0.294315	−	0.294315i	0.838782	+	0.544467i	$0.183268\pi$
$878$		0			0
$879$		−	12.2474i		−	0.413096i
$880$		0			0
$881$	−23.1464			−0.779823			−0.389911	−	0.920852i	$-0.627494\pi$
$881$	−23.1464			−0.779823			−0.389911	+	0.920852i	$0.627494\pi$
$882$		0			0
$883$	−11.3302	+	11.3302i	−0.381293	+	0.381293i	−0.871568	−	0.490275i	$-0.836896\pi$
$883$	−11.3302	+	11.3302i	−0.381293	+	0.381293i	0.490275	+	0.871568i	$0.336896\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.7386	+	17.7386i	−0.595605	+	0.595605i	−0.939140	−	0.343535i	$-0.888376\pi$
$887$	−17.7386	+	17.7386i	−0.595605	+	0.595605i	0.343535	+	0.939140i	$0.388376\pi$
$888$		0			0
$889$	1.84778			0.0619725
$890$		0			0
$891$	−6.24745			−0.209297
$892$		0			0
$893$	2.79254	+	5.83442i	0.0934487	+	0.195242i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	19.3590	+	19.3590i	0.646377	+	0.646377i
$898$		0			0
$899$		−	82.7878i		−	2.76113i
$900$		0			0
$901$		−	64.9428i		−	2.16356i
$902$		0			0
$903$	−10.5658	−	10.5658i	−0.351609	−	0.351609i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.6642	+	11.6642i	−0.387303	+	0.387303i	−0.873724	−	0.486421i	$-0.838302\pi$
$907$	−11.6642	+	11.6642i	−0.387303	+	0.387303i	0.486421	+	0.873724i	$0.338302\pi$
$908$		0			0
$909$		−	19.3485i		−	0.641748i
$910$		0			0
$911$			24.6650i			0.817189i	0.912716	+	0.408594i	$0.133981\pi$
$911$			24.6650i			0.817189i	−0.912716	+	0.408594i	$0.866019\pi$
$912$		0			0
$913$	−30.5832	+	30.5832i	−1.01216	+	1.01216i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.2810	+	10.2810i	0.339507	+	0.339507i
$918$		0			0
$919$			8.40408i			0.277225i	0.990347	+	0.138613i	$0.0442643\pi$
$919$			8.40408i			0.277225i	−0.990347	+	0.138613i	$0.955736\pi$
$920$		0			0
$921$	−28.4495			−0.937443
$922$		0			0
$923$	−5.61212	+	5.61212i	−0.184725	+	0.184725i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	9.50745	−	9.50745i	0.312266	−	0.312266i
$928$		0			0
$929$			2.20204i			0.0722466i	0.999347	+	0.0361233i	$0.0115009\pi$
$929$			2.20204i			0.0722466i	−0.999347	+	0.0361233i	$0.988499\pi$
$930$		0			0
$931$	6.95459	−	19.7236i	0.227928	−	0.646416i
$932$		0			0
$933$	−4.53136	−	4.53136i	−0.148350	−	0.148350i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	25.9144	+	25.9144i	0.846586	+	0.846586i	0.989705	−	0.143120i	$-0.0457133\pi$
$937$	25.9144	+	25.9144i	0.846586	+	0.846586i	−0.143120	+	0.989705i	$0.545713\pi$
$938$		0			0
$939$	28.3606			0.925513
$940$		0			0
$941$			12.7478i			0.415566i	0.978175	+	0.207783i	$0.0666248\pi$
$941$			12.7478i			0.415566i	−0.978175	+	0.207783i	$0.933375\pi$
$942$		0			0
$943$	6.68810	+	6.68810i	0.217794	+	0.217794i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.61953	−	3.61953i	−0.117619	−	0.117619i	0.645847	−	0.763466i	$-0.276504\pi$
$947$	−3.61953	−	3.61953i	−0.117619	−	0.117619i	−0.763466	+	0.645847i	$0.776504\pi$
$948$		0			0
$949$	56.7212			1.84125
$950$		0			0
$951$	17.7526			0.575666
$952$		0			0
$953$	−15.5819	−	15.5819i	−0.504747	−	0.504747i	0.408162	−	0.912909i	$-0.366170\pi$
$953$	−15.5819	−	15.5819i	−0.504747	−	0.504747i	−0.912909	+	0.408162i	$0.866170\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	17.7320	+	17.7320i	0.573195	+	0.573195i
$958$		0			0
$959$		−	5.39388i		−	0.174177i
$960$		0			0
$961$	−70.3939			−2.27077
$962$		0			0
$963$	13.2073	+	13.2073i	0.425599	+	0.425599i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−12.4855	−	12.4855i	−0.401508	−	0.401508i	0.477256	−	0.878764i	$-0.341631\pi$
$967$	−12.4855	−	12.4855i	−0.401508	−	0.401508i	−0.878764	+	0.477256i	$0.841631\pi$
$968$		0			0
$969$	11.9172	−	33.7980i	0.382837	−	1.08575i
$970$		0			0
$971$		−	18.2911i		−	0.586990i	−0.955961	−	0.293495i	$-0.905182\pi$
$971$		−	18.2911i		−	0.586990i	0.955961	−	0.293495i	$-0.0948184\pi$
$972$		0			0
$973$	−10.9646	+	10.9646i	−0.351509	+	0.351509i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.2433	+	30.2433i	−0.967570	+	0.967570i	−0.999490	−	0.0319203i	$-0.989838\pi$
$977$	−30.2433	+	30.2433i	−0.967570	+	0.967570i	0.0319203	+	0.999490i	$0.489838\pi$
$978$		0			0
$979$	−20.1389			−0.643642
$980$		0			0
$981$		−	14.5956i		−	0.466001i
$982$		0			0
$983$	8.14225	+	8.14225i	0.259697	+	0.259697i	0.824931	−	0.565234i	$-0.191214\pi$
$983$	8.14225	+	8.14225i	0.259697	+	0.259697i	−0.565234	+	0.824931i	$0.691214\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.93887	+	1.93887i	−0.0617148	+	0.0617148i
$988$		0			0
$989$		−	41.3939i		−	1.31625i
$990$		0			0
$991$		−	32.0561i		−	1.01830i	−0.860679	−	0.509148i	$-0.829960\pi$
$991$		−	32.0561i		−	1.01830i	0.860679	−	0.509148i	$-0.170040\pi$
$992$		0			0
$993$	−11.2242	+	11.2242i	−0.356191	+	0.356191i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.57024	+	2.57024i	0.0814003	+	0.0814003i	0.746635	−	0.665234i	$-0.231668\pi$
$997$	2.57024	+	2.57024i	0.0814003	+	0.0814003i	−0.665234	+	0.746635i	$0.731668\pi$
$998$		0			0
$999$			40.6969i			1.28759i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.l.c.1557.3	yes	16
5.2	odd	4	inner	1900.2.l.c.493.4	yes	16
5.3	odd	4	inner	1900.2.l.c.493.5	yes	16
5.4	even	2	inner	1900.2.l.c.1557.6	yes	16
19.18	odd	2	inner	1900.2.l.c.1557.5	yes	16
95.18	even	4	inner	1900.2.l.c.493.3	✓	16
95.37	even	4	inner	1900.2.l.c.493.6	yes	16
95.94	odd	2	inner	1900.2.l.c.1557.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.l.c.493.3	✓	16	95.18	even	4	inner
1900.2.l.c.493.4	yes	16	5.2	odd	4	inner
1900.2.l.c.493.5	yes	16	5.3	odd	4	inner
1900.2.l.c.493.6	yes	16	95.37	even	4	inner
1900.2.l.c.1557.3	yes	16	1.1	even	1	trivial
1900.2.l.c.1557.4	yes	16	95.94	odd	2	inner
1900.2.l.c.1557.5	yes	16	19.18	odd	2	inner
1900.2.l.c.1557.6	yes	16	5.4	even	2	inner

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.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.880486 - 0.880486i) q^{3} +(-1.04930 - 1.04930i) q^{7} -1.44949i q^{9} +O(q^{10})\)	\(q+(-0.880486 - 0.880486i) q^{3} +(-1.04930 - 1.04930i) q^{7} -1.44949i q^{9} -2.44949 q^{11} +(3.03723 + 3.03723i) q^{13} +(4.66883 + 4.66883i) q^{17} +(4.11084 + 1.44949i) q^{19} +1.84778i q^{21} +(-3.61953 + 3.61953i) q^{23} +(-3.91771 + 3.91771i) q^{27} +8.22167 q^{29} -10.0695i q^{31} +(2.15674 + 2.15674i) q^{33} +(5.19397 - 5.19397i) q^{37} -5.34847i q^{39} -1.84778i q^{41} +(-5.71812 + 5.71812i) q^{43} +(1.04930 + 1.04930i) q^{47} -4.79796i q^{49} -8.22167i q^{51} +(-6.95494 - 6.95494i) q^{53} +(-2.34328 - 4.89579i) q^{57} +10.0695 q^{59} -9.34847 q^{61} +(-1.52094 + 1.52094i) q^{63} +(6.55917 - 6.55917i) q^{67} +6.37389 q^{69} +1.84778i q^{71} +(9.33766 - 9.33766i) q^{73} +(2.57024 + 2.57024i) q^{77} -1.84778 q^{79} +2.55051 q^{81} +(12.4855 - 12.4855i) q^{83} +(-7.23907 - 7.23907i) q^{87} +8.22167 q^{89} -6.37389i q^{91} +(-8.86601 + 8.86601i) q^{93} +(-10.4769 + 10.4769i) q^{97} +3.55051i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{61} + 80 q^{81}+O(q^{100}) \) 16 * q - 32 * q^61 + 80 * q^81

Introduction

L-functions

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