LMFDB - Embedded newform 1072.2.g.b.1071.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1072,2,Mod(1071,1072)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1072.1071"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1072 = 2^{4} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1072.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.55996309668$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1071.1
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	1072.1071
Dual form			1072.2.g.b.1071.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{3} -3.00000i q^{5} -0.697224 q^{7} +2.30278 q^{9} -3.00000 q^{11} -3.90833i q^{13} +6.90833i q^{15} -2.60555 q^{17} -1.69722i q^{19} +1.60555 q^{21} +0.394449i q^{23} -4.00000 q^{25} +1.60555 q^{27} -0.394449 q^{29} +1.60555 q^{31} +6.90833 q^{33} +2.09167i q^{35} +5.30278 q^{37} +9.00000i q^{39} +6.90833i q^{41} -8.30278 q^{43} -6.90833i q^{45} +7.30278i q^{47} -6.51388 q^{49} +6.00000 q^{51} +3.00000i q^{53} +9.00000i q^{55} +3.90833i q^{57} +5.21110i q^{59} +3.90833i q^{61} -1.60555 q^{63} -11.7250 q^{65} +(-3.69722 - 7.30278i) q^{67} -0.908327i q^{69} +5.60555i q^{71} -1.39445 q^{73} +9.21110 q^{75} +2.09167 q^{77} -8.30278 q^{79} -10.6056 q^{81} +10.3028i q^{83} +7.81665i q^{85} +0.908327 q^{87} +11.6056 q^{89} +2.72498i q^{91} -3.69722 q^{93} -5.09167 q^{95} +16.8167i q^{97} -6.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 10 q^{7} + 2 q^{9} - 12 q^{11} + 4 q^{17} - 8 q^{21} - 16 q^{25} - 8 q^{27} - 16 q^{29} - 8 q^{31} + 6 q^{33} + 14 q^{37} - 26 q^{43} + 10 q^{49} + 24 q^{51} + 8 q^{63} + 18 q^{65} - 22 q^{67}+ \cdots - 6 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 10 * q^7 + 2 * q^9 - 12 * q^11 + 4 * q^17 - 8 * q^21 - 16 * q^25 - 8 * q^27 - 16 * q^29 - 8 * q^31 + 6 * q^33 + 14 * q^37 - 26 * q^43 + 10 * q^49 + 24 * q^51 + 8 * q^63 + 18 * q^65 - 22 * q^67 - 20 * q^73 + 8 * q^75 + 30 * q^77 - 26 * q^79 - 28 * q^81 - 18 * q^87 + 32 * q^89 - 22 * q^93 - 42 * q^95 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1072\mathbb{Z}\right)^\times$.

$n$	$337$	$671$	$805$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−2.30278			−1.32951			−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278			−1.32951			−0.664754	+	0.747062i	$0.731464\pi$
$4$		0			0
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$	−0.697224			−0.263526			−0.131763	−	0.991281i	$-0.542064\pi$
$7$	−0.697224			−0.263526			−0.131763	+	0.991281i	$0.542064\pi$
$8$		0			0
$9$	2.30278			0.767592
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$		−	3.90833i		−	1.08397i	−0.840387	−	0.541987i	$-0.817672\pi$
$13$		−	3.90833i		−	1.08397i	0.840387	−	0.541987i	$-0.182328\pi$
$14$		0			0
$15$			6.90833i			1.78372i
$16$		0			0
$17$	−2.60555			−0.631939			−0.315970	−	0.948769i	$-0.602330\pi$
$17$	−2.60555			−0.631939			−0.315970	+	0.948769i	$0.602330\pi$
$18$		0			0
$19$		−	1.69722i		−	0.389370i	−0.980866	−	0.194685i	$-0.937632\pi$
$19$		−	1.69722i		−	0.389370i	0.980866	−	0.194685i	$-0.0623685\pi$
$20$		0			0
$21$	1.60555			0.350360
$22$		0			0
$23$			0.394449i			0.0822482i	0.999154	+	0.0411241i	$0.0130939\pi$
$23$			0.394449i			0.0822482i	−0.999154	+	0.0411241i	$0.986906\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	1.60555			0.308988
$28$		0			0
$29$	−0.394449			−0.0732473			−0.0366236	−	0.999329i	$-0.511660\pi$
$29$	−0.394449			−0.0732473			−0.0366236	+	0.999329i	$0.511660\pi$
$30$		0			0
$31$	1.60555			0.288366			0.144183	−	0.989551i	$-0.453945\pi$
$31$	1.60555			0.288366			0.144183	+	0.989551i	$0.453945\pi$
$32$		0			0
$33$	6.90833			1.20259
$34$		0			0
$35$			2.09167i			0.353557i
$36$		0			0
$37$	5.30278			0.871771			0.435885	−	0.900002i	$-0.356435\pi$
$37$	5.30278			0.871771			0.435885	+	0.900002i	$0.356435\pi$
$38$		0			0
$39$			9.00000i			1.44115i
$40$		0			0
$41$			6.90833i			1.07890i	0.842018	+	0.539450i	$0.181368\pi$
$41$			6.90833i			1.07890i	−0.842018	+	0.539450i	$0.818632\pi$
$42$		0			0
$43$	−8.30278			−1.26616			−0.633081	−	0.774086i	$-0.718210\pi$
$43$	−8.30278			−1.26616			−0.633081	+	0.774086i	$0.718210\pi$
$44$		0			0
$45$		−	6.90833i		−	1.02983i
$46$		0			0
$47$			7.30278i			1.06522i	0.846361	+	0.532610i	$0.178789\pi$
$47$			7.30278i			1.06522i	−0.846361	+	0.532610i	$0.821211\pi$
$48$		0			0
$49$	−6.51388			−0.930554
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$			3.00000i			0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$			3.00000i			0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$		0			0
$55$			9.00000i			1.21356i
$56$		0			0
$57$			3.90833i			0.517671i
$58$		0			0
$59$			5.21110i			0.678428i	0.940709	+	0.339214i	$0.110161\pi$
$59$			5.21110i			0.678428i	−0.940709	+	0.339214i	$0.889839\pi$
$60$		0			0
$61$			3.90833i			0.500410i	0.968193	+	0.250205i	$0.0804980\pi$
$61$			3.90833i			0.500410i	−0.968193	+	0.250205i	$0.919502\pi$
$62$		0			0
$63$	−1.60555			−0.202280
$64$		0			0
$65$	−11.7250			−1.45430
$66$		0			0
$67$	−3.69722	−	7.30278i	−0.451688	−	0.892176i
$67$	−3.69722	−	7.30278i	−0.451688	−	0.892176i
$68$		0			0
$69$		−	0.908327i		−	0.109350i
$70$		0			0
$71$			5.60555i			0.665257i	0.943058	+	0.332628i	$0.107935\pi$
$71$			5.60555i			0.665257i	−0.943058	+	0.332628i	$0.892065\pi$
$72$		0			0
$73$	−1.39445			−0.163208			−0.0816039	−	0.996665i	$-0.526004\pi$
$73$	−1.39445			−0.163208			−0.0816039	+	0.996665i	$0.526004\pi$
$74$		0			0
$75$	9.21110			1.06361
$76$		0			0
$77$	2.09167			0.238368
$78$		0			0
$79$	−8.30278			−0.934135			−0.467068	−	0.884222i	$-0.654690\pi$
$79$	−8.30278			−0.934135			−0.467068	+	0.884222i	$0.654690\pi$
$80$		0			0
$81$	−10.6056			−1.17839
$82$		0			0
$83$			10.3028i			1.13088i	0.824791	+	0.565438i	$0.191293\pi$
$83$			10.3028i			1.13088i	−0.824791	+	0.565438i	$0.808707\pi$
$84$		0			0
$85$			7.81665i			0.847835i
$86$		0			0
$87$	0.908327			0.0973829
$88$		0			0
$89$	11.6056			1.23019			0.615093	−	0.788455i	$-0.289118\pi$
$89$	11.6056			1.23019			0.615093	+	0.788455i	$0.289118\pi$
$90$		0			0
$91$			2.72498i			0.285656i
$92$		0			0
$93$	−3.69722			−0.383384
$94$		0			0
$95$	−5.09167			−0.522395
$96$		0			0
$97$			16.8167i			1.70747i	0.520706	+	0.853736i	$0.325669\pi$
$97$			16.8167i			1.70747i	−0.520706	+	0.853736i	$0.674331\pi$
$98$		0			0
$99$	−6.90833			−0.694313
$100$		0			0
$101$			12.9083i			1.28443i	0.766526	+	0.642213i	$0.221984\pi$
$101$			12.9083i			1.28443i	−0.766526	+	0.642213i	$0.778016\pi$
$102$		0			0
$103$		−	15.1194i		−	1.48976i	−0.667198	−	0.744881i	$-0.732506\pi$
$103$		−	15.1194i		−	1.48976i	0.667198	−	0.744881i	$-0.267494\pi$
$104$		0			0
$105$		−	4.81665i		−	0.470057i
$106$		0			0
$107$		−	13.4222i		−	1.29757i	−0.760970	−	0.648787i	$-0.775277\pi$
$107$		−	13.4222i		−	1.29757i	0.760970	−	0.648787i	$-0.224723\pi$
$108$		0			0
$109$		−	5.09167i		−	0.487694i	−0.969814	−	0.243847i	$-0.921591\pi$
$109$		−	5.09167i		−	0.487694i	0.969814	−	0.243847i	$-0.0784094\pi$
$110$		0			0
$111$	−12.2111			−1.15903
$112$		0			0
$113$			3.90833i			0.367664i	0.982958	+	0.183832i	$0.0588503\pi$
$113$			3.90833i			0.367664i	−0.982958	+	0.183832i	$0.941150\pi$
$114$		0			0
$115$	1.18335			0.110348
$116$		0			0
$117$		−	9.00000i		−	0.832050i
$118$		0			0
$119$	1.81665			0.166532
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		−	15.9083i		−	1.43441i
$124$		0			0
$125$		−	3.00000i		−	0.268328i
$126$		0			0
$127$		−	10.6972i		−	0.949225i	−0.880195	−	0.474613i	$-0.842588\pi$
$127$		−	10.6972i		−	0.949225i	0.880195	−	0.474613i	$-0.157412\pi$
$128$		0			0
$129$	19.1194			1.68337
$130$		0			0
$131$			8.21110i			0.717407i	0.933451	+	0.358704i	$0.116781\pi$
$131$			8.21110i			0.717407i	−0.933451	+	0.358704i	$0.883219\pi$
$132$		0			0
$133$			1.18335i			0.102609i
$134$		0			0
$135$		−	4.81665i		−	0.414552i
$136$		0			0
$137$		−	4.81665i		−	0.411515i	−0.978603	−	0.205757i	$-0.934034\pi$
$137$		−	4.81665i		−	0.411515i	0.978603	−	0.205757i	$-0.0659657\pi$
$138$		0			0
$139$	−9.42221			−0.799181			−0.399591	−	0.916694i	$-0.630848\pi$
$139$	−9.42221			−0.799181			−0.399591	+	0.916694i	$0.630848\pi$
$140$		0			0
$141$		−	16.8167i		−	1.41622i
$142$		0			0
$143$			11.7250i			0.980492i
$144$		0			0
$145$			1.18335i			0.0982716i
$146$		0			0
$147$	15.0000			1.23718
$148$		0			0
$149$	−7.69722			−0.630581			−0.315291	−	0.948995i	$-0.602102\pi$
$149$	−7.69722			−0.630581			−0.315291	+	0.948995i	$0.602102\pi$
$150$		0			0
$151$			16.8167i			1.36852i	0.729238	+	0.684260i	$0.239875\pi$
$151$			16.8167i			1.36852i	−0.729238	+	0.684260i	$0.760125\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$		−	4.81665i		−	0.386883i
$156$		0			0
$157$	−4.51388			−0.360247			−0.180123	−	0.983644i	$-0.557650\pi$
$157$	−4.51388			−0.360247			−0.180123	+	0.983644i	$0.557650\pi$
$158$		0			0
$159$		−	6.90833i		−	0.547866i
$160$		0			0
$161$		−	0.275019i		−	0.0216746i
$162$		0			0
$163$		−	12.9083i		−	1.01106i	−0.862810	−	0.505529i	$-0.831297\pi$
$163$		−	12.9083i		−	1.01106i	0.862810	−	0.505529i	$-0.168703\pi$
$164$		0			0
$165$		−	20.7250i		−	1.61344i
$166$		0			0
$167$			10.4222i			0.806494i	0.915091	+	0.403247i	$0.132119\pi$
$167$			10.4222i			0.806494i	−0.915091	+	0.403247i	$0.867881\pi$
$168$		0			0
$169$	−2.27502			−0.175001
$170$		0			0
$171$		−	3.90833i		−	0.298877i
$172$		0			0
$173$	12.1194			0.921423			0.460712	−	0.887550i	$-0.347594\pi$
$173$	12.1194			0.921423			0.460712	+	0.887550i	$0.347594\pi$
$174$		0			0
$175$	2.78890			0.210821
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$	19.8167			1.48117			0.740583	−	0.671965i	$-0.234549\pi$
$179$	19.8167			1.48117			0.740583	+	0.671965i	$0.234549\pi$
$180$		0			0
$181$	−8.11943			−0.603512			−0.301756	−	0.953385i	$-0.597573\pi$
$181$	−8.11943			−0.603512			−0.301756	+	0.953385i	$0.597573\pi$
$182$		0			0
$183$		−	9.00000i		−	0.665299i
$184$		0			0
$185$		−	15.9083i		−	1.16960i
$186$		0			0
$187$	7.81665			0.571610
$188$		0			0
$189$	−1.11943			−0.0814265
$190$		0			0
$191$	10.8167			0.782666			0.391333	−	0.920249i	$-0.372014\pi$
$191$	10.8167			0.782666			0.391333	+	0.920249i	$0.372014\pi$
$192$		0			0
$193$	16.7250			1.20389			0.601945	−	0.798537i	$-0.294393\pi$
$193$	16.7250			1.20389			0.601945	+	0.798537i	$0.294393\pi$
$194$		0			0
$195$	27.0000			1.93351
$196$		0			0
$197$			10.8167i			0.770655i	0.922780	+	0.385327i	$0.125911\pi$
$197$			10.8167i			0.770655i	−0.922780	+	0.385327i	$0.874089\pi$
$198$		0			0
$199$		−	1.69722i		−	0.120313i	−0.998189	−	0.0601565i	$-0.980840\pi$
$199$		−	1.69722i		−	0.120313i	0.998189	−	0.0601565i	$-0.0191600\pi$
$200$		0			0
$201$	8.51388	+	16.8167i	0.600523	+	1.18616i
$202$		0			0
$203$	0.275019			0.0193026
$204$		0			0
$205$	20.7250			1.44750
$206$		0			0
$207$			0.908327i			0.0631331i
$208$		0			0
$209$			5.09167i			0.352198i
$210$		0			0
$211$		−	26.3305i		−	1.81267i	−0.422561	−	0.906334i	$-0.638869\pi$
$211$		−	26.3305i		−	1.81267i	0.422561	−	0.906334i	$-0.361131\pi$
$212$		0			0
$213$		−	12.9083i		−	0.884464i
$214$		0			0
$215$			24.9083i			1.69873i
$216$		0			0
$217$	−1.11943			−0.0759918
$218$		0			0
$219$	3.21110			0.216986
$220$		0			0
$221$			10.1833i			0.685006i
$222$		0			0
$223$		−	15.6333i		−	1.04688i	−0.852061	−	0.523442i	$-0.824648\pi$
$223$		−	15.6333i		−	1.04688i	0.852061	−	0.523442i	$-0.175352\pi$
$224$		0			0
$225$	−9.21110			−0.614074
$226$		0			0
$227$		−	24.5139i		−	1.62704i	−0.581535	−	0.813522i	$-0.697548\pi$
$227$		−	24.5139i		−	1.62704i	0.581535	−	0.813522i	$-0.302452\pi$
$228$		0			0
$229$			12.9083i			0.853006i	0.904486	+	0.426503i	$0.140255\pi$
$229$			12.9083i			0.853006i	−0.904486	+	0.426503i	$0.859745\pi$
$230$		0			0
$231$	−4.81665			−0.316913
$232$		0			0
$233$		−	27.6333i		−	1.81032i	−0.425073	−	0.905159i	$-0.639752\pi$
$233$		−	27.6333i		−	1.81032i	0.425073	−	0.905159i	$-0.360248\pi$
$234$		0			0
$235$	21.9083			1.42914
$236$		0			0
$237$	19.1194			1.24194
$238$		0			0
$239$	−27.9083			−1.80524			−0.902620	−	0.430439i	$-0.858359\pi$
$239$	−27.9083			−1.80524			−0.902620	+	0.430439i	$0.858359\pi$
$240$		0			0
$241$	−19.7250			−1.27060			−0.635299	−	0.772266i	$-0.719123\pi$
$241$	−19.7250			−1.27060			−0.635299	+	0.772266i	$0.719123\pi$
$242$		0			0
$243$	19.6056			1.25770
$244$		0			0
$245$			19.5416i			1.24847i
$246$		0			0
$247$	−6.63331			−0.422067
$248$		0			0
$249$		−	23.7250i		−	1.50351i
$250$		0			0
$251$	−7.81665			−0.493383			−0.246691	−	0.969094i	$-0.579343\pi$
$251$	−7.81665			−0.493383			−0.246691	+	0.969094i	$0.579343\pi$
$252$		0			0
$253$		−	1.18335i		−	0.0743963i
$254$		0			0
$255$		−	18.0000i		−	1.12720i
$256$		0			0
$257$	5.60555			0.349665			0.174832	−	0.984598i	$-0.444062\pi$
$257$	5.60555			0.349665			0.174832	+	0.984598i	$0.444062\pi$
$258$		0			0
$259$	−3.69722			−0.229734
$260$		0			0
$261$	−0.908327			−0.0562240
$262$		0			0
$263$			3.51388i			0.216675i	0.994114	+	0.108338i	$0.0345527\pi$
$263$			3.51388i			0.216675i	−0.994114	+	0.108338i	$0.965447\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$	−26.7250			−1.63554
$268$		0			0
$269$	−21.1194			−1.28767			−0.643837	−	0.765163i	$-0.722659\pi$
$269$	−21.1194			−1.28767			−0.643837	+	0.765163i	$0.722659\pi$
$270$		0			0
$271$	18.2111			1.10625			0.553123	−	0.833100i	$-0.313436\pi$
$271$	18.2111			1.10625			0.553123	+	0.833100i	$0.313436\pi$
$272$		0			0
$273$		−	6.27502i		−	0.379781i
$274$		0			0
$275$	12.0000			0.723627
$276$		0			0
$277$	−29.0278			−1.74411			−0.872054	−	0.489409i	$-0.837213\pi$
$277$	−29.0278			−1.74411			−0.872054	+	0.489409i	$0.837213\pi$
$278$		0			0
$279$	3.69722			0.221347
$280$		0			0
$281$			14.0917i			0.840639i	0.907376	+	0.420319i	$0.138082\pi$
$281$			14.0917i			0.840639i	−0.907376	+	0.420319i	$0.861918\pi$
$282$		0			0
$283$		−	1.18335i		−	0.0703426i	−0.999381	−	0.0351713i	$-0.988802\pi$
$283$		−	1.18335i		−	0.0703426i	0.999381	−	0.0351713i	$-0.0111977\pi$
$284$		0			0
$285$	11.7250			0.694528
$286$		0			0
$287$		−	4.81665i		−	0.284318i
$288$		0			0
$289$	−10.2111			−0.600653
$290$		0			0
$291$		−	38.7250i		−	2.27010i
$292$		0			0
$293$	−11.6056			−0.678004			−0.339002	−	0.940786i	$-0.610089\pi$
$293$	−11.6056			−0.678004			−0.339002	+	0.940786i	$0.610089\pi$
$294$		0			0
$295$	15.6333			0.910206
$296$		0			0
$297$	−4.81665			−0.279491
$298$		0			0
$299$	1.54163			0.0891550
$300$		0			0
$301$	5.78890			0.333667
$302$		0			0
$303$		−	29.7250i		−	1.70766i
$304$		0			0
$305$	11.7250			0.671370
$306$		0			0
$307$			7.81665i			0.446120i	0.974805	+	0.223060i	$0.0716046\pi$
$307$			7.81665i			0.446120i	−0.974805	+	0.223060i	$0.928395\pi$
$308$		0			0
$309$			34.8167i			1.98065i
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$		−	29.7250i		−	1.68016i	−0.542466	−	0.840078i	$-0.682509\pi$
$313$		−	29.7250i		−	1.68016i	0.542466	−	0.840078i	$-0.317491\pi$
$314$		0			0
$315$			4.81665i			0.271388i
$316$		0			0
$317$	−9.78890			−0.549799			−0.274900	−	0.961473i	$-0.588645\pi$
$317$	−9.78890			−0.549799			−0.274900	+	0.961473i	$0.588645\pi$
$318$		0			0
$319$	1.18335			0.0662547
$320$		0			0
$321$			30.9083i			1.72513i
$322$		0			0
$323$			4.42221i			0.246058i
$324$		0			0
$325$			15.6333i			0.867180i
$326$		0			0
$327$			11.7250i			0.648393i
$328$		0			0
$329$		−	5.09167i		−	0.280713i
$330$		0			0
$331$	−11.0278			−0.606140			−0.303070	−	0.952968i	$-0.598012\pi$
$331$	−11.0278			−0.606140			−0.303070	+	0.952968i	$0.598012\pi$
$332$		0			0
$333$	12.2111			0.669164
$334$		0			0
$335$	−21.9083	+	11.0917i	−1.19698	+	0.606003i
$336$		0			0
$337$			23.0917i			1.25788i	0.777452	+	0.628942i	$0.216512\pi$
$337$			23.0917i			1.25788i	−0.777452	+	0.628942i	$0.783488\pi$
$338$		0			0
$339$		−	9.00000i		−	0.488813i
$340$		0			0
$341$	−4.81665			−0.260836
$342$		0			0
$343$	9.42221			0.508751
$344$		0			0
$345$	−2.72498			−0.146708
$346$		0			0
$347$	−18.6333			−1.00029			−0.500144	−	0.865942i	$-0.666720\pi$
$347$	−18.6333			−1.00029			−0.500144	+	0.865942i	$0.666720\pi$
$348$		0			0
$349$	3.21110			0.171886			0.0859432	−	0.996300i	$-0.472610\pi$
$349$	3.21110			0.171886			0.0859432	+	0.996300i	$0.472610\pi$
$350$		0			0
$351$		−	6.27502i		−	0.334936i
$352$		0			0
$353$		−	6.90833i		−	0.367693i	−0.982955	−	0.183847i	$-0.941145\pi$
$353$		−	6.90833i		−	0.367693i	0.982955	−	0.183847i	$-0.0588550\pi$
$354$		0			0
$355$	16.8167			0.892535
$356$		0			0
$357$	−4.18335			−0.221406
$358$		0			0
$359$			27.3944i			1.44582i	0.690940	+	0.722912i	$0.257197\pi$
$359$			27.3944i			1.44582i	−0.690940	+	0.722912i	$0.742803\pi$
$360$		0			0
$361$	16.1194			0.848391
$362$		0			0
$363$	4.60555			0.241729
$364$		0			0
$365$			4.18335i			0.218966i
$366$		0			0
$367$	22.6056			1.18000			0.590000	−	0.807403i	$-0.299128\pi$
$367$	22.6056			1.18000			0.590000	+	0.807403i	$0.299128\pi$
$368$		0			0
$369$			15.9083i			0.828154i
$370$		0			0
$371$		−	2.09167i		−	0.108594i
$372$		0			0
$373$			23.0917i			1.19564i	0.801630	+	0.597821i	$0.203967\pi$
$373$			23.0917i			1.19564i	−0.801630	+	0.597821i	$0.796033\pi$
$374$		0			0
$375$			6.90833i			0.356744i
$376$		0			0
$377$			1.54163i			0.0793982i
$378$		0			0
$379$	−8.78890			−0.451455			−0.225728	−	0.974190i	$-0.572476\pi$
$379$	−8.78890			−0.451455			−0.225728	+	0.974190i	$0.572476\pi$
$380$		0			0
$381$			24.6333i			1.26200i
$382$		0			0
$383$	−31.5416			−1.61170			−0.805851	−	0.592118i	$-0.798292\pi$
$383$	−31.5416			−1.61170			−0.805851	+	0.592118i	$0.798292\pi$
$384$		0			0
$385$		−	6.27502i		−	0.319805i
$386$		0			0
$387$	−19.1194			−0.971895
$388$		0			0
$389$	−21.3944			−1.08474			−0.542371	−	0.840139i	$-0.682473\pi$
$389$	−21.3944			−1.08474			−0.542371	+	0.840139i	$0.682473\pi$
$390$		0			0
$391$		−	1.02776i		−	0.0519759i
$392$		0			0
$393$		−	18.9083i		−	0.953799i
$394$		0			0
$395$			24.9083i			1.25327i
$396$		0			0
$397$	−28.3305			−1.42187			−0.710934	−	0.703258i	$-0.751728\pi$
$397$	−28.3305			−1.42187			−0.710934	+	0.703258i	$0.751728\pi$
$398$		0			0
$399$		−	2.72498i		−	0.136420i
$400$		0			0
$401$			8.09167i			0.404079i	0.979377	+	0.202039i	$0.0647569\pi$
$401$			8.09167i			0.404079i	−0.979377	+	0.202039i	$0.935243\pi$
$402$		0			0
$403$		−	6.27502i		−	0.312581i
$404$		0			0
$405$			31.8167i			1.58098i
$406$		0			0
$407$	−15.9083			−0.788546
$408$		0			0
$409$		−	18.0000i		−	0.890043i	−0.895520	−	0.445021i	$-0.853196\pi$
$409$		−	18.0000i		−	0.890043i	0.895520	−	0.445021i	$-0.146804\pi$
$410$		0			0
$411$			11.0917i			0.547112i
$412$		0			0
$413$		−	3.63331i		−	0.178783i
$414$		0			0
$415$	30.9083			1.51723
$416$		0			0
$417$	21.6972			1.06252
$418$		0			0
$419$			2.60555i			0.127290i	0.997973	+	0.0636448i	$0.0202725\pi$
$419$			2.60555i			0.127290i	−0.997973	+	0.0636448i	$0.979728\pi$
$420$		0			0
$421$	1.88057			0.0916534			0.0458267	−	0.998949i	$-0.485408\pi$
$421$	1.88057			0.0916534			0.0458267	+	0.998949i	$0.485408\pi$
$422$		0			0
$423$			16.8167i			0.817654i
$424$		0			0
$425$	10.4222			0.505551
$426$		0			0
$427$		−	2.72498i		−	0.131871i
$428$		0			0
$429$		−	27.0000i		−	1.30357i
$430$		0			0
$431$		−	16.0278i		−	0.772030i	−0.922493	−	0.386015i	$-0.873851\pi$
$431$		−	16.0278i		−	0.772030i	0.922493	−	0.386015i	$-0.126149\pi$
$432$		0			0
$433$		−	21.9083i		−	1.05285i	−0.850222	−	0.526424i	$-0.823533\pi$
$433$		−	21.9083i		−	1.05285i	0.850222	−	0.526424i	$-0.176467\pi$
$434$		0			0
$435$		−	2.72498i		−	0.130653i
$436$		0			0
$437$	0.669468			0.0320250
$438$		0			0
$439$			18.5139i			0.883619i	0.897109	+	0.441810i	$0.145663\pi$
$439$			18.5139i			0.883619i	−0.897109	+	0.441810i	$0.854337\pi$
$440$		0			0
$441$	−15.0000			−0.714286
$442$		0			0
$443$	−16.8167			−0.798983			−0.399492	−	0.916737i	$-0.630813\pi$
$443$	−16.8167			−0.798983			−0.399492	+	0.916737i	$0.630813\pi$
$444$		0			0
$445$		−	34.8167i		−	1.65047i
$446$		0			0
$447$	17.7250			0.838363
$448$		0			0
$449$	−29.3305			−1.38419			−0.692097	−	0.721805i	$-0.743313\pi$
$449$	−29.3305			−1.38419			−0.692097	+	0.721805i	$0.743313\pi$
$450$		0			0
$451$		−	20.7250i		−	0.975901i
$452$		0			0
$453$		−	38.7250i		−	1.81946i
$454$		0			0
$455$	8.17494			0.383247
$456$		0			0
$457$	5.42221			0.253640			0.126820	−	0.991926i	$-0.459523\pi$
$457$	5.42221			0.253640			0.126820	+	0.991926i	$0.459523\pi$
$458$		0			0
$459$	−4.18335			−0.195262
$460$		0			0
$461$	−26.3305			−1.22634			−0.613168	−	0.789953i	$-0.710105\pi$
$461$	−26.3305			−1.22634			−0.613168	+	0.789953i	$0.710105\pi$
$462$		0			0
$463$	37.1194			1.72509			0.862543	−	0.505984i	$-0.168870\pi$
$463$	37.1194			1.72509			0.862543	+	0.505984i	$0.168870\pi$
$464$		0			0
$465$			11.0917i			0.514364i
$466$		0			0
$467$			30.3944i			1.40649i	0.710949	+	0.703244i	$0.248266\pi$
$467$			30.3944i			1.40649i	−0.710949	+	0.703244i	$0.751734\pi$
$468$		0			0
$469$	2.57779	+	5.09167i	0.119032	+	0.235112i
$470$		0			0
$471$	10.3944			0.478951
$472$		0			0
$473$	24.9083			1.14529
$474$		0			0
$475$			6.78890i			0.311496i
$476$		0			0
$477$			6.90833i			0.316311i
$478$		0			0
$479$			23.2111i			1.06054i	0.847828	+	0.530271i	$0.177910\pi$
$479$			23.2111i			1.06054i	−0.847828	+	0.530271i	$0.822090\pi$
$480$		0			0
$481$		−	20.7250i		−	0.944978i
$482$		0			0
$483$			0.633308i			0.0288165i
$484$		0			0
$485$	50.4500			2.29081
$486$		0			0
$487$	15.4222			0.698847			0.349423	−	0.936965i	$-0.386377\pi$
$487$	15.4222			0.698847			0.349423	+	0.936965i	$0.386377\pi$
$488$		0			0
$489$			29.7250i			1.34421i
$490$		0			0
$491$			7.42221i			0.334959i	0.985876	+	0.167480i	$0.0535629\pi$
$491$			7.42221i			0.334959i	−0.985876	+	0.167480i	$0.946437\pi$
$492$		0			0
$493$	1.02776			0.0462878
$494$		0			0
$495$			20.7250i			0.931519i
$496$		0			0
$497$		−	3.90833i		−	0.175312i
$498$		0			0
$499$	1.60555			0.0718743			0.0359372	−	0.999354i	$-0.488558\pi$
$499$	1.60555			0.0718743			0.0359372	+	0.999354i	$0.488558\pi$
$500$		0			0
$501$		−	24.0000i		−	1.07224i
$502$		0			0
$503$	38.4500			1.71440			0.857200	−	0.514984i	$-0.172202\pi$
$503$	38.4500			1.71440			0.857200	+	0.514984i	$0.172202\pi$
$504$		0			0
$505$	38.7250			1.72324
$506$		0			0
$507$	5.23886			0.232666
$508$		0			0
$509$	20.6056			0.913325			0.456663	−	0.889640i	$-0.349045\pi$
$509$	20.6056			0.913325			0.456663	+	0.889640i	$0.349045\pi$
$510$		0			0
$511$	0.972244			0.0430095
$512$		0			0
$513$		−	2.72498i		−	0.120311i
$514$		0			0
$515$	−45.3583			−1.99872
$516$		0			0
$517$		−	21.9083i		−	0.963527i
$518$		0			0
$519$	−27.9083			−1.22504
$520$		0			0
$521$		−	7.81665i		−	0.342454i	−0.985232	−	0.171227i	$-0.945227\pi$
$521$		−	7.81665i		−	0.342454i	0.985232	−	0.171227i	$-0.0547731\pi$
$522$		0			0
$523$			25.3028i			1.10641i	0.833044	+	0.553207i	$0.186596\pi$
$523$			25.3028i			1.10641i	−0.833044	+	0.553207i	$0.813404\pi$
$524$		0			0
$525$	−6.42221			−0.280288
$526$		0			0
$527$	−4.18335			−0.182229
$528$		0			0
$529$	22.8444			0.993235
$530$		0			0
$531$			12.0000i			0.520756i
$532$		0			0
$533$	27.0000			1.16950
$534$		0			0
$535$	−40.2666			−1.74088
$536$		0			0
$537$	−45.6333			−1.96922
$538$		0			0
$539$	19.5416			0.841718
$540$		0			0
$541$		−	18.0000i		−	0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$		−	18.0000i		−	0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$		0			0
$543$	18.6972			0.802375
$544$		0			0
$545$	−15.2750			−0.654310
$546$		0			0
$547$	−43.3305			−1.85268			−0.926340	−	0.376689i	$-0.877063\pi$
$547$	−43.3305			−1.85268			−0.926340	+	0.376689i	$0.877063\pi$
$548$		0			0
$549$			9.00000i			0.384111i
$550$		0			0
$551$			0.669468i			0.0285203i
$552$		0			0
$553$	5.78890			0.246169
$554$		0			0
$555$			36.6333i			1.55500i
$556$		0			0
$557$	21.7527			0.921693			0.460847	−	0.887480i	$-0.347546\pi$
$557$	21.7527			0.921693			0.460847	+	0.887480i	$0.347546\pi$
$558$		0			0
$559$			32.4500i			1.37249i
$560$		0			0
$561$	−18.0000			−0.759961
$562$		0			0
$563$	3.63331			0.153126			0.0765628	−	0.997065i	$-0.475605\pi$
$563$	3.63331			0.153126			0.0765628	+	0.997065i	$0.475605\pi$
$564$		0			0
$565$	11.7250			0.493274
$566$		0			0
$567$	7.39445			0.310538
$568$		0			0
$569$	19.0278			0.797685			0.398843	−	0.917019i	$-0.369412\pi$
$569$	19.0278			0.797685			0.398843	+	0.917019i	$0.369412\pi$
$570$		0			0
$571$			28.5416i			1.19443i	0.802081	+	0.597215i	$0.203726\pi$
$571$			28.5416i			1.19443i	−0.802081	+	0.597215i	$0.796274\pi$
$572$		0			0
$573$	−24.9083			−1.04056
$574$		0			0
$575$		−	1.57779i		−	0.0657986i
$576$		0			0
$577$			16.8167i			0.700086i	0.936734	+	0.350043i	$0.113833\pi$
$577$			16.8167i			0.700086i	−0.936734	+	0.350043i	$0.886167\pi$
$578$		0			0
$579$	−38.5139			−1.60058
$580$		0			0
$581$		−	7.18335i		−	0.298015i
$582$		0			0
$583$		−	9.00000i		−	0.372742i
$584$		0			0
$585$	−27.0000			−1.11631
$586$		0			0
$587$	9.90833			0.408960			0.204480	−	0.978871i	$-0.434450\pi$
$587$	9.90833			0.408960			0.204480	+	0.978871i	$0.434450\pi$
$588$		0			0
$589$		−	2.72498i		−	0.112281i
$590$		0			0
$591$		−	24.9083i		−	1.02459i
$592$		0			0
$593$		−	21.6333i		−	0.888373i	−0.895934	−	0.444187i	$-0.853493\pi$
$593$		−	21.6333i		−	0.888373i	0.895934	−	0.444187i	$-0.146507\pi$
$594$		0			0
$595$		−	5.44996i		−	0.223427i
$596$		0			0
$597$			3.90833i			0.159957i
$598$		0			0
$599$	−35.4500			−1.44845			−0.724223	−	0.689566i	$-0.757801\pi$
$599$	−35.4500			−1.44845			−0.724223	+	0.689566i	$0.757801\pi$
$600$		0			0
$601$	3.21110			0.130984			0.0654918	−	0.997853i	$-0.479138\pi$
$601$	3.21110			0.130984			0.0654918	+	0.997853i	$0.479138\pi$
$602$		0			0
$603$	−8.51388	−	16.8167i	−0.346712	−	0.684827i
$604$		0			0
$605$			6.00000i			0.243935i
$606$		0			0
$607$			12.9083i			0.523933i	0.965077	+	0.261966i	$0.0843710\pi$
$607$			12.9083i			0.523933i	−0.965077	+	0.261966i	$0.915629\pi$
$608$		0			0
$609$	−0.633308			−0.0256629
$610$		0			0
$611$	28.5416			1.15467
$612$		0			0
$613$	19.8444			0.801508			0.400754	−	0.916186i	$-0.368748\pi$
$613$	19.8444			0.801508			0.400754	+	0.916186i	$0.368748\pi$
$614$		0			0
$615$	−47.7250			−1.92446
$616$		0			0
$617$	−12.5139			−0.503790			−0.251895	−	0.967755i	$-0.581054\pi$
$617$	−12.5139			−0.503790			−0.251895	+	0.967755i	$0.581054\pi$
$618$		0			0
$619$			2.36669i			0.0951254i	0.998868	+	0.0475627i	$0.0151454\pi$
$619$			2.36669i			0.0951254i	−0.998868	+	0.0475627i	$0.984855\pi$
$620$		0			0
$621$			0.633308i			0.0254138i
$622$		0			0
$623$	−8.09167			−0.324186
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		−	11.7250i		−	0.468251i
$628$		0			0
$629$	−13.8167			−0.550906
$630$		0			0
$631$	−22.8806			−0.910861			−0.455431	−	0.890271i	$-0.650515\pi$
$631$	−22.8806			−0.910861			−0.455431	+	0.890271i	$0.650515\pi$
$632$		0			0
$633$			60.6333i			2.40996i
$634$		0			0
$635$	−32.0917			−1.27352
$636$		0			0
$637$			25.4584i			1.00870i
$638$		0			0
$639$			12.9083i			0.510646i
$640$		0			0
$641$		−	3.00000i		−	0.118493i	−0.998243	−	0.0592464i	$-0.981130\pi$
$641$		−	3.00000i		−	0.118493i	0.998243	−	0.0592464i	$-0.0188698\pi$
$642$		0			0
$643$		−	11.8806i		−	0.468524i	−0.972174	−	0.234262i	$-0.924733\pi$
$643$		−	11.8806i		−	0.468524i	0.972174	−	0.234262i	$-0.0752674\pi$
$644$		0			0
$645$		−	57.3583i		−	2.25848i
$646$		0			0
$647$	−19.8167			−0.779073			−0.389537	−	0.921011i	$-0.627365\pi$
$647$	−19.8167			−0.779073			−0.389537	+	0.921011i	$0.627365\pi$
$648$		0			0
$649$		−	15.6333i		−	0.613661i
$650$		0			0
$651$	2.57779			0.101032
$652$		0			0
$653$		−	45.6333i		−	1.78577i	−0.450285	−	0.892885i	$-0.648678\pi$
$653$		−	45.6333i		−	1.78577i	0.450285	−	0.892885i	$-0.351322\pi$
$654$		0			0
$655$	24.6333			0.962503
$656$		0			0
$657$	−3.21110			−0.125277
$658$		0			0
$659$			35.0555i			1.36557i	0.730620	+	0.682784i	$0.239231\pi$
$659$			35.0555i			1.36557i	−0.730620	+	0.682784i	$0.760769\pi$
$660$		0			0
$661$			24.6333i			0.958125i	0.877781	+	0.479062i	$0.159023\pi$
$661$			24.6333i			0.958125i	−0.877781	+	0.479062i	$0.840977\pi$
$662$		0			0
$663$		−	23.4500i		−	0.910721i
$664$		0			0
$665$	3.55004			0.137665
$666$		0			0
$667$		−	0.155590i		−	0.00602446i
$668$		0			0
$669$			36.0000i			1.39184i
$670$		0			0
$671$		−	11.7250i		−	0.452638i
$672$		0			0
$673$		−	12.9083i		−	0.497579i	−0.968558	−	0.248790i	$-0.919967\pi$
$673$		−	12.9083i		−	0.497579i	0.968558	−	0.248790i	$-0.0800328\pi$
$674$		0			0
$675$	−6.42221			−0.247191
$676$		0			0
$677$		−	28.8167i		−	1.10751i	−0.832678	−	0.553757i	$-0.813194\pi$
$677$		−	28.8167i		−	1.10751i	0.832678	−	0.553757i	$-0.186806\pi$
$678$		0			0
$679$		−	11.7250i		−	0.449963i
$680$		0			0
$681$			56.4500i			2.16317i
$682$		0			0
$683$	−3.90833			−0.149548			−0.0747740	−	0.997201i	$-0.523824\pi$
$683$	−3.90833			−0.149548			−0.0747740	+	0.997201i	$0.523824\pi$
$684$		0			0
$685$	−14.4500			−0.552105
$686$		0			0
$687$		−	29.7250i		−	1.13408i
$688$		0			0
$689$	11.7250			0.446686
$690$		0			0
$691$		−	29.2111i		−	1.11124i	−0.831435	−	0.555621i	$-0.812480\pi$
$691$		−	29.2111i		−	1.11124i	0.831435	−	0.555621i	$-0.187520\pi$
$692$		0			0
$693$	4.81665			0.182970
$694$		0			0
$695$			28.2666i			1.07221i
$696$		0			0
$697$		−	18.0000i		−	0.681799i
$698$		0			0
$699$			63.6333i			2.40683i
$700$		0			0
$701$			3.63331i			0.137228i	0.997643	+	0.0686141i	$0.0218577\pi$
$701$			3.63331i			0.137228i	−0.997643	+	0.0686141i	$0.978142\pi$
$702$		0			0
$703$		−	9.00000i		−	0.339441i
$704$		0			0
$705$	−50.4500			−1.90006
$706$		0			0
$707$		−	9.00000i		−	0.338480i
$708$		0			0
$709$	4.60555			0.172965			0.0864826	−	0.996253i	$-0.472437\pi$
$709$	4.60555			0.172965			0.0864826	+	0.996253i	$0.472437\pi$
$710$		0			0
$711$	−19.1194			−0.717035
$712$		0			0
$713$			0.633308i			0.0237176i
$714$		0			0
$715$	35.1749			1.31547
$716$		0			0
$717$	64.2666			2.40008
$718$		0			0
$719$			50.2111i			1.87256i	0.351257	+	0.936279i	$0.385754\pi$
$719$			50.2111i			1.87256i	−0.351257	+	0.936279i	$0.614246\pi$
$720$		0			0
$721$			10.5416i			0.392591i
$722$		0			0
$723$	45.4222			1.68927
$724$		0			0
$725$	1.57779			0.0585978
$726$		0			0
$727$	12.4222			0.460714			0.230357	−	0.973106i	$-0.426011\pi$
$727$	12.4222			0.460714			0.230357	+	0.973106i	$0.426011\pi$
$728$		0			0
$729$	−13.3305			−0.493723
$730$		0			0
$731$	21.6333			0.800137
$732$		0			0
$733$		−	10.1833i		−	0.376130i	−0.982157	−	0.188065i	$-0.939778\pi$
$733$		−	10.1833i		−	0.376130i	0.982157	−	0.188065i	$-0.0602216\pi$
$734$		0			0
$735$		−	45.0000i		−	1.65985i
$736$		0			0
$737$	11.0917	+	21.9083i	0.408567	+	0.807004i
$738$		0			0
$739$	−38.9361			−1.43229			−0.716143	−	0.697953i	$-0.754094\pi$
$739$	−38.9361			−1.43229			−0.716143	+	0.697953i	$0.754094\pi$
$740$		0			0
$741$	15.2750			0.561142
$742$		0			0
$743$			1.42221i			0.0521756i	0.999660	+	0.0260878i	$0.00830495\pi$
$743$			1.42221i			0.0521756i	−0.999660	+	0.0260878i	$0.991695\pi$
$744$		0			0
$745$			23.0917i			0.846013i
$746$		0			0
$747$			23.7250i			0.868052i
$748$		0			0
$749$			9.35829i			0.341944i
$750$		0			0
$751$		−	42.6333i		−	1.55571i	−0.628443	−	0.777856i	$-0.716307\pi$
$751$		−	42.6333i		−	1.55571i	0.628443	−	0.777856i	$-0.283693\pi$
$752$		0			0
$753$	18.0000			0.655956
$754$		0			0
$755$	50.4500			1.83606
$756$		0			0
$757$			27.0000i			0.981332i	0.871348	+	0.490666i	$0.163246\pi$
$757$			27.0000i			0.981332i	−0.871348	+	0.490666i	$0.836754\pi$
$758$		0			0
$759$			2.72498i			0.0989105i
$760$		0			0
$761$	−6.78890			−0.246097			−0.123049	−	0.992401i	$-0.539267\pi$
$761$	−6.78890			−0.246097			−0.123049	+	0.992401i	$0.539267\pi$
$762$		0			0
$763$			3.55004i			0.128520i
$764$		0			0
$765$			18.0000i			0.650791i
$766$		0			0
$767$	20.3667			0.735399
$768$		0			0
$769$			32.4500i			1.17018i	0.810970	+	0.585088i	$0.198940\pi$
$769$			32.4500i			1.17018i	−0.810970	+	0.585088i	$0.801060\pi$
$770$		0			0
$771$	−12.9083			−0.464882
$772$		0			0
$773$	−25.6611			−0.922964			−0.461482	−	0.887149i	$-0.652682\pi$
$773$	−25.6611			−0.922964			−0.461482	+	0.887149i	$0.652682\pi$
$774$		0			0
$775$	−6.42221			−0.230692
$776$		0			0
$777$	8.51388			0.305434
$778$		0			0
$779$	11.7250			0.420091
$780$		0			0
$781$		−	16.8167i		−	0.601747i
$782$		0			0
$783$	−0.633308			−0.0226326
$784$		0			0
$785$			13.5416i			0.483322i
$786$		0			0
$787$	−21.8444			−0.778669			−0.389335	−	0.921096i	$-0.627295\pi$
$787$	−21.8444			−0.778669			−0.389335	+	0.921096i	$0.627295\pi$
$788$		0			0
$789$		−	8.09167i		−	0.288071i
$790$		0			0
$791$		−	2.72498i		−	0.0968892i
$792$		0			0
$793$	15.2750			0.542432
$794$		0			0
$795$	−20.7250			−0.735039
$796$		0			0
$797$	−26.4861			−0.938187			−0.469093	−	0.883149i	$-0.655419\pi$
$797$	−26.4861			−0.938187			−0.469093	+	0.883149i	$0.655419\pi$
$798$		0			0
$799$		−	19.0278i		−	0.673154i
$800$		0			0
$801$	26.7250			0.944281
$802$		0			0
$803$	4.18335			0.147627
$804$		0			0
$805$	−0.825058			−0.0290795
$806$		0			0
$807$	48.6333			1.71197
$808$		0			0
$809$		−	44.7250i		−	1.57245i	−0.617943	−	0.786223i	$-0.712034\pi$
$809$		−	44.7250i		−	1.57245i	0.617943	−	0.786223i	$-0.287966\pi$
$810$		0			0
$811$	7.60555			0.267067			0.133534	−	0.991044i	$-0.457368\pi$
$811$	7.60555			0.267067			0.133534	+	0.991044i	$0.457368\pi$
$812$		0			0
$813$	−41.9361			−1.47076
$814$		0			0
$815$	−38.7250			−1.35648
$816$		0			0
$817$			14.0917i			0.493005i
$818$		0			0
$819$			6.27502i			0.219267i
$820$		0			0
$821$	−29.6056			−1.03324			−0.516620	−	0.856215i	$-0.672810\pi$
$821$	−29.6056			−1.03324			−0.516620	+	0.856215i	$0.672810\pi$
$822$		0			0
$823$		−	42.1194i		−	1.46819i	−0.679046	−	0.734096i	$-0.737606\pi$
$823$		−	42.1194i		−	1.46819i	0.679046	−	0.734096i	$-0.262394\pi$
$824$		0			0
$825$	−27.6333			−0.962068
$826$		0			0
$827$			50.6056i			1.75973i	0.475226	+	0.879864i	$0.342366\pi$
$827$			50.6056i			1.75973i	−0.475226	+	0.879864i	$0.657634\pi$
$828$		0			0
$829$	−33.8444			−1.17546			−0.587732	−	0.809055i	$-0.699979\pi$
$829$	−33.8444			−1.17546			−0.587732	+	0.809055i	$0.699979\pi$
$830$		0			0
$831$	66.8444			2.31881
$832$		0			0
$833$	16.9722			0.588053
$834$		0			0
$835$	31.2666			1.08203
$836$		0			0
$837$	2.57779			0.0891016
$838$		0			0
$839$			12.2389i			0.422532i	0.977429	+	0.211266i	$0.0677587\pi$
$839$			12.2389i			0.422532i	−0.977429	+	0.211266i	$0.932241\pi$
$840$		0			0
$841$	−28.8444			−0.994635
$842$		0			0
$843$		−	32.4500i		−	1.11764i
$844$		0			0
$845$			6.82506i			0.234789i
$846$		0			0
$847$	1.39445			0.0479138
$848$		0			0
$849$			2.72498i			0.0935211i
$850$		0			0
$851$			2.09167i			0.0717016i
$852$		0			0
$853$	−11.6972			−0.400505			−0.200253	−	0.979744i	$-0.564176\pi$
$853$	−11.6972			−0.400505			−0.200253	+	0.979744i	$0.564176\pi$
$854$		0			0
$855$	−11.7250			−0.400986
$856$		0			0
$857$			48.6333i			1.66128i	0.556808	+	0.830641i	$0.312026\pi$
$857$			48.6333i			1.66128i	−0.556808	+	0.830641i	$0.687974\pi$
$858$		0			0
$859$			5.09167i			0.173726i	0.996220	+	0.0868628i	$0.0276842\pi$
$859$			5.09167i			0.173726i	−0.996220	+	0.0868628i	$0.972316\pi$
$860$		0			0
$861$			11.0917i			0.378003i
$862$		0			0
$863$		−	28.9361i		−	0.984996i	−0.870314	−	0.492498i	$-0.836084\pi$
$863$		−	28.9361i		−	0.984996i	0.870314	−	0.492498i	$-0.163916\pi$
$864$		0			0
$865$		−	36.3583i		−	1.23622i
$866$		0			0
$867$	23.5139			0.798573
$868$		0			0
$869$	24.9083			0.844957
$870$		0			0
$871$	−28.5416	+	14.4500i	−0.967096	+	0.489618i
$872$		0			0
$873$			38.7250i			1.31064i
$874$		0			0
$875$			2.09167i			0.0707115i
$876$		0			0
$877$	14.0278			0.473684			0.236842	−	0.971548i	$-0.423888\pi$
$877$	14.0278			0.473684			0.236842	+	0.971548i	$0.423888\pi$
$878$		0			0
$879$	26.7250			0.901411
$880$		0			0
$881$	27.2389			0.917700			0.458850	−	0.888514i	$-0.348261\pi$
$881$	27.2389			0.917700			0.458850	+	0.888514i	$0.348261\pi$
$882$		0			0
$883$	−23.6611			−0.796258			−0.398129	−	0.917329i	$-0.630340\pi$
$883$	−23.6611			−0.796258			−0.398129	+	0.917329i	$0.630340\pi$
$884$		0			0
$885$	−36.0000			−1.21013
$886$		0			0
$887$		−	27.2389i		−	0.914591i	−0.889315	−	0.457296i	$-0.848818\pi$
$887$		−	27.2389i		−	0.914591i	0.889315	−	0.457296i	$-0.151182\pi$
$888$		0			0
$889$			7.45837i			0.250146i
$890$		0			0
$891$	31.8167			1.06590
$892$		0			0
$893$	12.3944			0.414764
$894$		0			0
$895$		−	59.4500i		−	1.98719i
$896$		0			0
$897$	−3.55004			−0.118532
$898$		0			0
$899$	−0.633308			−0.0211220
$900$		0			0
$901$		−	7.81665i		−	0.260410i
$902$		0			0
$903$	−13.3305			−0.443612
$904$		0			0
$905$			24.3583i			0.809697i
$906$		0			0
$907$			46.5416i			1.54539i	0.634778	+	0.772695i	$0.281092\pi$
$907$			46.5416i			1.54539i	−0.634778	+	0.772695i	$0.718908\pi$
$908$		0			0
$909$			29.7250i			0.985915i
$910$		0			0
$911$			35.3305i			1.17055i	0.810834	+	0.585276i	$0.199014\pi$
$911$			35.3305i			1.17055i	−0.810834	+	0.585276i	$0.800986\pi$
$912$		0			0
$913$		−	30.9083i		−	1.02292i
$914$		0			0
$915$	−27.0000			−0.892592
$916$		0			0
$917$		−	5.72498i		−	0.189056i
$918$		0			0
$919$	−15.8444			−0.522659			−0.261329	−	0.965250i	$-0.584161\pi$
$919$	−15.8444			−0.522659			−0.261329	+	0.965250i	$0.584161\pi$
$920$		0			0
$921$		−	18.0000i		−	0.593120i
$922$		0			0
$923$	21.9083			0.721121
$924$		0			0
$925$	−21.2111			−0.697417
$926$		0			0
$927$		−	34.8167i		−	1.14353i
$928$		0			0
$929$		−	50.4500i		−	1.65521i	−0.561311	−	0.827605i	$-0.689703\pi$
$929$		−	50.4500i		−	1.65521i	0.561311	−	0.827605i	$-0.310297\pi$
$930$		0			0
$931$			11.0555i			0.362330i
$932$		0			0
$933$	55.2666			1.80935
$934$		0			0
$935$		−	23.4500i		−	0.766896i
$936$		0			0
$937$			6.63331i			0.216701i	0.994113	+	0.108350i	$0.0345568\pi$
$937$			6.63331i			0.216701i	−0.994113	+	0.108350i	$0.965443\pi$
$938$		0			0
$939$			68.4500i			2.23378i
$940$		0			0
$941$			19.1833i			0.625359i	0.949859	+	0.312680i	$0.101227\pi$
$941$			19.1833i			0.625359i	−0.949859	+	0.312680i	$0.898773\pi$
$942$		0			0
$943$	−2.72498			−0.0887376
$944$		0			0
$945$			3.35829i			0.109245i
$946$		0			0
$947$		−	38.3305i		−	1.24557i	−0.782391	−	0.622787i	$-0.786000\pi$
$947$		−	38.3305i		−	1.24557i	0.782391	−	0.622787i	$-0.214000\pi$
$948$		0			0
$949$			5.44996i			0.176913i
$950$		0			0
$951$	22.5416			0.730963
$952$		0			0
$953$	−5.60555			−0.181582			−0.0907908	−	0.995870i	$-0.528939\pi$
$953$	−5.60555			−0.181582			−0.0907908	+	0.995870i	$0.528939\pi$
$954$		0			0
$955$		−	32.4500i		−	1.05006i
$956$		0			0
$957$	−2.72498			−0.0880861
$958$		0			0
$959$			3.35829i			0.108445i
$960$		0			0
$961$	−28.4222			−0.916845
$962$		0			0
$963$		−	30.9083i		−	0.996007i
$964$		0			0
$965$		−	50.1749i		−	1.61519i
$966$		0			0
$967$		−	30.3944i		−	0.977420i	−0.872446	−	0.488710i	$-0.837468\pi$
$967$		−	30.3944i		−	0.977420i	0.872446	−	0.488710i	$-0.162532\pi$
$968$		0			0
$969$		−	10.1833i		−	0.327136i
$970$		0			0
$971$		−	20.2111i		−	0.648605i	−0.945953	−	0.324303i	$-0.894870\pi$
$971$		−	20.2111i		−	0.648605i	0.945953	−	0.324303i	$-0.105130\pi$
$972$		0			0
$973$	6.56939			0.210605
$974$		0			0
$975$		−	36.0000i		−	1.15292i
$976$		0			0
$977$	−17.6056			−0.563251			−0.281626	−	0.959524i	$-0.590874\pi$
$977$	−17.6056			−0.563251			−0.281626	+	0.959524i	$0.590874\pi$
$978$		0			0
$979$	−34.8167			−1.11275
$980$		0			0
$981$		−	11.7250i		−	0.374350i
$982$		0			0
$983$	52.8167			1.68459			0.842295	−	0.539017i	$-0.181204\pi$
$983$	52.8167			1.68459			0.842295	+	0.539017i	$0.181204\pi$
$984$		0			0
$985$	32.4500			1.03394
$986$		0			0
$987$			11.7250i			0.373210i
$988$		0			0
$989$		−	3.27502i		−	0.104140i
$990$		0			0
$991$	55.7527			1.77104			0.885522	−	0.464597i	$-0.153801\pi$
$991$	55.7527			1.77104			0.885522	+	0.464597i	$0.153801\pi$
$992$		0			0
$993$	25.3944			0.805868
$994$		0			0
$995$	−5.09167			−0.161417
$996$		0			0
$997$	17.6611			0.559332			0.279666	−	0.960097i	$-0.409776\pi$
$997$	17.6611			0.559332			0.279666	+	0.960097i	$0.409776\pi$
$998$		0			0
$999$	8.51388			0.269367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1072.2.g.b.1071.1	✓	4
4.3	odd	2		1072.2.g.d.1071.3	yes	4
67.66	odd	2		1072.2.g.d.1071.4	yes	4
268.267	even	2	inner	1072.2.g.b.1071.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1072.2.g.b.1071.1	✓	4	1.1	even	1	trivial
1072.2.g.b.1071.2	yes	4	268.267	even	2	inner
1072.2.g.d.1071.3	yes	4	4.3	odd	2
1072.2.g.d.1071.4	yes	4	67.66	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1072 = 2^{4} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1072.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} -3.00000i q^{5} -0.697224 q^{7} +2.30278 q^{9} -3.00000 q^{11} -3.90833i q^{13} +6.90833i q^{15} -2.60555 q^{17} -1.69722i q^{19} +1.60555 q^{21} +0.394449i q^{23} -4.00000 q^{25} +1.60555 q^{27} -0.394449 q^{29} +1.60555 q^{31} +6.90833 q^{33} +2.09167i q^{35} +5.30278 q^{37} +9.00000i q^{39} +6.90833i q^{41} -8.30278 q^{43} -6.90833i q^{45} +7.30278i q^{47} -6.51388 q^{49} +6.00000 q^{51} +3.00000i q^{53} +9.00000i q^{55} +3.90833i q^{57} +5.21110i q^{59} +3.90833i q^{61} -1.60555 q^{63} -11.7250 q^{65} +(-3.69722 - 7.30278i) q^{67} -0.908327i q^{69} +5.60555i q^{71} -1.39445 q^{73} +9.21110 q^{75} +2.09167 q^{77} -8.30278 q^{79} -10.6056 q^{81} +10.3028i q^{83} +7.81665i q^{85} +0.908327 q^{87} +11.6056 q^{89} +2.72498i q^{91} -3.69722 q^{93} -5.09167 q^{95} +16.8167i q^{97} -6.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 10 q^{7} + 2 q^{9} - 12 q^{11} + 4 q^{17} - 8 q^{21} - 16 q^{25} - 8 q^{27} - 16 q^{29} - 8 q^{31} + 6 q^{33} + 14 q^{37} - 26 q^{43} + 10 q^{49} + 24 q^{51} + 8 q^{63} + 18 q^{65} - 22 q^{67}+ \cdots - 6 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 10 * q^7 + 2 * q^9 - 12 * q^11 + 4 * q^17 - 8 * q^21 - 16 * q^25 - 8 * q^27 - 16 * q^29 - 8 * q^31 + 6 * q^33 + 14 * q^37 - 26 * q^43 + 10 * q^49 + 24 * q^51 + 8 * q^63 + 18 * q^65 - 22 * q^67 - 20 * q^73 + 8 * q^75 + 30 * q^77 - 26 * q^79 - 28 * q^81 - 18 * q^87 + 32 * q^89 - 22 * q^93 - 42 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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