LMFDB - Embedded newform 2352.2.h.i.2255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.1
Root			$1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.i.2255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +O(q^{10})$	$q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +3.31662 q^{11} -4.00000 q^{13} +(1.65831 - 5.50000i) q^{15} -3.31662i q^{17} -7.00000i q^{19} +3.31662 q^{23} -6.00000 q^{25} +(-3.31662 - 4.00000i) q^{27} -6.63325i q^{29} -3.00000i q^{31} +(-5.50000 - 1.65831i) q^{33} -1.00000 q^{37} +(6.63325 + 2.00000i) q^{39} +6.63325i q^{41} +2.00000i q^{43} +(-5.50000 + 8.29156i) q^{45} +9.94987 q^{47} +(-1.65831 + 5.50000i) q^{51} +3.31662i q^{53} +11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} -3.31662 q^{59} +3.00000 q^{61} -13.2665i q^{65} -9.00000i q^{67} +(-5.50000 - 1.65831i) q^{69} +13.2665 q^{71} +7.00000 q^{73} +(9.94987 + 3.00000i) q^{75} -9.00000i q^{79} +(3.50000 + 8.29156i) q^{81} -13.2665 q^{83} +11.0000 q^{85} +(-3.31662 + 11.0000i) q^{87} -3.31662i q^{89} +(-1.50000 + 4.97494i) q^{93} +23.2164 q^{95} +8.00000 q^{97} +(8.29156 + 5.50000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{9}+O(q^{10}) $ 4 * q + 10 * q^9	$ 4 q + 10 q^{9} - 16 q^{13} - 24 q^{25} - 22 q^{33} - 4 q^{37} - 22 q^{45} - 14 q^{57} + 12 q^{61} - 22 q^{69} + 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} + 32 q^{97}+O(q^{100}) $ 4 * q + 10 * q^9 - 16 * q^13 - 24 * q^25 - 22 * q^33 - 4 * q^37 - 22 * q^45 - 14 * q^57 + 12 * q^61 - 22 * q^69 + 28 * q^73 + 14 * q^81 + 44 * q^85 - 6 * q^93 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.65831	−	0.500000i	−0.957427	−	0.288675i
$3$	−1.65831	−	0.500000i	−0.957427	−	0.288675i
$4$		0			0
$5$			3.31662i			1.48324i	0.670820	+	0.741620i	$0.265942\pi$
$5$			3.31662i			1.48324i	−0.670820	+	0.741620i	$0.734058\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.50000	+	1.65831i	0.833333	+	0.552771i
$10$		0			0
$11$	3.31662			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$11$	3.31662			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	1.65831	−	5.50000i	0.428174	−	1.42009i
$16$		0			0
$17$		−	3.31662i		−	0.804400i	−0.915552	−	0.402200i	$-0.868246\pi$
$17$		−	3.31662i		−	0.804400i	0.915552	−	0.402200i	$-0.131754\pi$
$18$		0			0
$19$		−	7.00000i		−	1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$		−	7.00000i		−	1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.31662			0.691564			0.345782	−	0.938315i	$-0.387614\pi$
$23$	3.31662			0.691564			0.345782	+	0.938315i	$0.387614\pi$
$24$		0			0
$25$	−6.00000			−1.20000
$26$		0			0
$27$	−3.31662	−	4.00000i	−0.638285	−	0.769800i
$28$		0			0
$29$		−	6.63325i		−	1.23176i	−0.787839	−	0.615882i	$-0.788800\pi$
$29$		−	6.63325i		−	1.23176i	0.787839	−	0.615882i	$-0.211200\pi$
$30$		0			0
$31$		−	3.00000i		−	0.538816i	−0.963026	−	0.269408i	$-0.913172\pi$
$31$		−	3.00000i		−	0.538816i	0.963026	−	0.269408i	$-0.0868280\pi$
$32$		0			0
$33$	−5.50000	−	1.65831i	−0.957427	−	0.288675i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$	6.63325	+	2.00000i	1.06217	+	0.320256i
$40$		0			0
$41$			6.63325i			1.03594i	0.855399	+	0.517970i	$0.173312\pi$
$41$			6.63325i			1.03594i	−0.855399	+	0.517970i	$0.826688\pi$
$42$		0			0
$43$			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
$45$	−5.50000	+	8.29156i	−0.819892	+	1.23603i
$46$		0			0
$47$	9.94987			1.45134			0.725669	−	0.688044i	$-0.241530\pi$
$47$	9.94987			1.45134			0.725669	+	0.688044i	$0.241530\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−1.65831	+	5.50000i	−0.232210	+	0.770154i
$52$		0			0
$53$			3.31662i			0.455573i	0.973711	+	0.227787i	$0.0731489\pi$
$53$			3.31662i			0.455573i	−0.973711	+	0.227787i	$0.926851\pi$
$54$		0			0
$55$			11.0000i			1.48324i
$56$		0			0
$57$	−3.50000	+	11.6082i	−0.463586	+	1.53754i
$58$		0			0
$59$	−3.31662			−0.431788			−0.215894	−	0.976417i	$-0.569267\pi$
$59$	−3.31662			−0.431788			−0.215894	+	0.976417i	$0.569267\pi$
$60$		0			0
$61$	3.00000			0.384111			0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000			0.384111			0.192055	+	0.981384i	$0.438485\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	13.2665i		−	1.64551i
$66$		0			0
$67$		−	9.00000i		−	1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$		−	9.00000i		−	1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$		0			0
$69$	−5.50000	−	1.65831i	−0.662122	−	0.199637i
$70$		0			0
$71$	13.2665			1.57444			0.787222	−	0.616670i	$-0.211519\pi$
$71$	13.2665			1.57444			0.787222	+	0.616670i	$0.211519\pi$
$72$		0			0
$73$	7.00000			0.819288			0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000			0.819288			0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$	9.94987	+	3.00000i	1.14891	+	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	9.00000i		−	1.01258i	−0.862364	−	0.506290i	$-0.831017\pi$
$79$		−	9.00000i		−	1.01258i	0.862364	−	0.506290i	$-0.168983\pi$
$80$		0			0
$81$	3.50000	+	8.29156i	0.388889	+	0.921285i
$82$		0			0
$83$	−13.2665			−1.45619			−0.728094	−	0.685478i	$-0.759593\pi$
$83$	−13.2665			−1.45619			−0.728094	+	0.685478i	$0.759593\pi$
$84$		0			0
$85$	11.0000			1.19312
$86$		0			0
$87$	−3.31662	+	11.0000i	−0.355580	+	1.17932i
$88$		0			0
$89$		−	3.31662i		−	0.351562i	−0.984429	−	0.175781i	$-0.943755\pi$
$89$		−	3.31662i		−	0.351562i	0.984429	−	0.175781i	$-0.0562450\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.50000	+	4.97494i	−0.155543	+	0.515877i
$94$		0			0
$95$	23.2164			2.38195
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$	8.29156	+	5.50000i	0.833333	+	0.552771i
$100$		0			0
$101$		−	3.31662i		−	0.330017i	−0.986292	−	0.165008i	$-0.947235\pi$
$101$		−	3.31662i		−	0.330017i	0.986292	−	0.165008i	$-0.0527651\pi$
$102$		0			0
$103$			11.0000i			1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$			11.0000i			1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.31662			−0.320630			−0.160315	−	0.987066i	$-0.551251\pi$
$107$	−3.31662			−0.320630			−0.160315	+	0.987066i	$0.551251\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$	1.65831	+	0.500000i	0.157400	+	0.0474579i
$112$		0			0
$113$		−	6.63325i		−	0.624004i	−0.950082	−	0.312002i	$-0.899000\pi$
$113$		−	6.63325i		−	0.624004i	0.950082	−	0.312002i	$-0.101000\pi$
$114$		0			0
$115$			11.0000i			1.02576i
$116$		0			0
$117$	−10.0000	−	6.63325i	−0.924500	−	0.613244i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	3.31662	−	11.0000i	0.299050	−	0.991837i
$124$		0			0
$125$		−	3.31662i		−	0.296648i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	1.00000	−	3.31662i	0.0880451	−	0.292013i
$130$		0			0
$131$	16.5831			1.44887			0.724437	−	0.689341i	$-0.242100\pi$
$131$	16.5831			1.44887			0.724437	+	0.689341i	$0.242100\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	13.2665	−	11.0000i	1.14180	−	0.946729i
$136$		0			0
$137$		−	16.5831i		−	1.41679i	−0.705815	−	0.708396i	$-0.749419\pi$
$137$		−	16.5831i		−	1.41679i	0.705815	−	0.708396i	$-0.250581\pi$
$138$		0			0
$139$		−	10.0000i		−	0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$		−	10.0000i		−	0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$		0			0
$141$	−16.5000	−	4.97494i	−1.38955	−	0.418965i
$142$		0			0
$143$	−13.2665			−1.10940
$144$		0			0
$145$	22.0000			1.82700
$146$		0			0
$147$		0			0
$148$		0			0
$149$			16.5831i			1.35854i	0.733887	+	0.679271i	$0.237704\pi$
$149$			16.5831i			1.35854i	−0.733887	+	0.679271i	$0.762296\pi$
$150$		0			0
$151$		−	7.00000i		−	0.569652i	−0.958579	−	0.284826i	$-0.908064\pi$
$151$		−	7.00000i		−	0.569652i	0.958579	−	0.284826i	$-0.0919358\pi$
$152$		0			0
$153$	5.50000	−	8.29156i	0.444649	−	0.670333i
$154$		0			0
$155$	9.94987			0.799193
$156$		0			0
$157$	3.00000			0.239426			0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000			0.239426			0.119713	+	0.992809i	$0.461803\pi$
$158$		0			0
$159$	1.65831	−	5.50000i	0.131513	−	0.436178i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			7.00000i			0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$			7.00000i			0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$		0			0
$165$	5.50000	−	18.2414i	0.428174	−	1.42009i
$166$		0			0
$167$	6.63325			0.513296			0.256648	−	0.966505i	$-0.417382\pi$
$167$	6.63325			0.513296			0.256648	+	0.966505i	$0.417382\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	11.6082	−	17.5000i	0.887700	−	1.33826i
$172$		0			0
$173$			9.94987i			0.756475i	0.925709	+	0.378237i	$0.123470\pi$
$173$			9.94987i			0.756475i	−0.925709	+	0.378237i	$0.876530\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.50000	+	1.65831i	0.413405	+	0.124646i
$178$		0			0
$179$	3.31662			0.247896			0.123948	−	0.992289i	$-0.460444\pi$
$179$	3.31662			0.247896			0.123948	+	0.992289i	$0.460444\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$	−4.97494	−	1.50000i	−0.367758	−	0.110883i
$184$		0			0
$185$		−	3.31662i		−	0.243843i
$186$		0			0
$187$		−	11.0000i		−	0.804400i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.94987			0.719948			0.359974	−	0.932962i	$-0.382786\pi$
$191$	9.94987			0.719948			0.359974	+	0.932962i	$0.382786\pi$
$192$		0			0
$193$	−3.00000			−0.215945			−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000			−0.215945			−0.107972	+	0.994154i	$0.534436\pi$
$194$		0			0
$195$	−6.63325	+	22.0000i	−0.475017	+	1.57545i
$196$		0			0
$197$			19.8997i			1.41780i	0.705310	+	0.708899i	$0.250808\pi$
$197$			19.8997i			1.41780i	−0.705310	+	0.708899i	$0.749192\pi$
$198$		0			0
$199$			1.00000i			0.0708881i	0.999372	+	0.0354441i	$0.0112846\pi$
$199$			1.00000i			0.0708881i	−0.999372	+	0.0354441i	$0.988715\pi$
$200$		0			0
$201$	−4.50000	+	14.9248i	−0.317406	+	1.05272i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−22.0000			−1.53655
$206$		0			0
$207$	8.29156	+	5.50000i	0.576303	+	0.382276i
$208$		0			0
$209$		−	23.2164i		−	1.60591i
$210$		0			0
$211$		−	26.0000i		−	1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$		−	26.0000i		−	1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$		0			0
$213$	−22.0000	−	6.63325i	−1.50742	−	0.454503i
$214$		0			0
$215$	−6.63325			−0.452384
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−11.6082	−	3.50000i	−0.784409	−	0.236508i
$220$		0			0
$221$			13.2665i			0.892401i
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	−15.0000	−	9.94987i	−1.00000	−	0.663325i
$226$		0			0
$227$	3.31662			0.220132			0.110066	−	0.993924i	$-0.464894\pi$
$227$	3.31662			0.220132			0.110066	+	0.993924i	$0.464894\pi$
$228$		0			0
$229$	29.0000			1.91637			0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000			1.91637			0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.5831i			1.08640i	0.839604	+	0.543198i	$0.182787\pi$
$233$			16.5831i			1.08640i	−0.839604	+	0.543198i	$0.817213\pi$
$234$		0			0
$235$			33.0000i			2.15268i
$236$		0			0
$237$	−4.50000	+	14.9248i	−0.292306	+	0.969471i
$238$		0			0
$239$	−6.63325			−0.429069			−0.214535	−	0.976716i	$-0.568823\pi$
$239$	−6.63325			−0.429069			−0.214535	+	0.976716i	$0.568823\pi$
$240$		0			0
$241$	19.0000			1.22390			0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000			1.22390			0.611949	+	0.790897i	$0.290386\pi$
$242$		0			0
$243$	−1.65831	−	15.5000i	−0.106381	−	0.994325i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			28.0000i			1.78160i
$248$		0			0
$249$	22.0000	+	6.63325i	1.39419	+	0.420365i
$250$		0			0
$251$	−19.8997			−1.25606			−0.628031	−	0.778189i	$-0.716139\pi$
$251$	−19.8997			−1.25606			−0.628031	+	0.778189i	$0.716139\pi$
$252$		0			0
$253$	11.0000			0.691564
$254$		0			0
$255$	−18.2414	−	5.50000i	−1.14232	−	0.344423i
$256$		0			0
$257$		−	23.2164i		−	1.44820i	−0.689696	−	0.724099i	$-0.742256\pi$
$257$		−	23.2164i		−	1.44820i	0.689696	−	0.724099i	$-0.257744\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	11.0000	−	16.5831i	0.680883	−	1.02647i
$262$		0			0
$263$	−3.31662			−0.204512			−0.102256	−	0.994758i	$-0.532606\pi$
$263$	−3.31662			−0.204512			−0.102256	+	0.994758i	$0.532606\pi$
$264$		0			0
$265$	−11.0000			−0.675725
$266$		0			0
$267$	−1.65831	+	5.50000i	−0.101487	+	0.336595i
$268$		0			0
$269$			16.5831i			1.01109i	0.862800	+	0.505545i	$0.168709\pi$
$269$			16.5831i			1.01109i	−0.862800	+	0.505545i	$0.831291\pi$
$270$		0			0
$271$		−	17.0000i		−	1.03268i	−0.856385	−	0.516338i	$-0.827295\pi$
$271$		−	17.0000i		−	1.03268i	0.856385	−	0.516338i	$-0.172705\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.8997			−1.20000
$276$		0			0
$277$	−13.0000			−0.781094			−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000			−0.781094			−0.390547	+	0.920583i	$0.627714\pi$
$278$		0			0
$279$	4.97494	−	7.50000i	0.297842	−	0.449013i
$280$		0			0
$281$			6.63325i			0.395706i	0.980232	+	0.197853i	$0.0633969\pi$
$281$			6.63325i			0.395706i	−0.980232	+	0.197853i	$0.936603\pi$
$282$		0			0
$283$		−	15.0000i		−	0.891657i	−0.895118	−	0.445829i	$-0.852909\pi$
$283$		−	15.0000i		−	0.891657i	0.895118	−	0.445829i	$-0.147091\pi$
$284$		0			0
$285$	−38.5000	−	11.6082i	−2.28054	−	0.687610i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.00000			0.352941
$290$		0			0
$291$	−13.2665	−	4.00000i	−0.777696	−	0.234484i
$292$		0			0
$293$			13.2665i			0.775037i	0.921862	+	0.387519i	$0.126668\pi$
$293$			13.2665i			0.775037i	−0.921862	+	0.387519i	$0.873332\pi$
$294$		0			0
$295$		−	11.0000i		−	0.640445i
$296$		0			0
$297$	−11.0000	−	13.2665i	−0.638285	−	0.769800i
$298$		0			0
$299$	−13.2665			−0.767221
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1.65831	+	5.50000i	−0.0952676	+	0.315967i
$304$		0			0
$305$			9.94987i			0.569728i
$306$		0			0
$307$			6.00000i			0.342438i	0.985233	+	0.171219i	$0.0547706\pi$
$307$			6.00000i			0.342438i	−0.985233	+	0.171219i	$0.945229\pi$
$308$		0			0
$309$	5.50000	−	18.2414i	0.312884	−	1.03772i
$310$		0			0
$311$	9.94987			0.564206			0.282103	−	0.959384i	$-0.408968\pi$
$311$	9.94987			0.564206			0.282103	+	0.959384i	$0.408968\pi$
$312$		0			0
$313$	−9.00000			−0.508710			−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000			−0.508710			−0.254355	+	0.967111i	$0.581863\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	29.8496i		−	1.67652i	−0.545269	−	0.838261i	$-0.683573\pi$
$317$		−	29.8496i		−	1.67652i	0.545269	−	0.838261i	$-0.316427\pi$
$318$		0			0
$319$		−	22.0000i		−	1.23176i
$320$		0			0
$321$	5.50000	+	1.65831i	0.306980	+	0.0925580i
$322$		0			0
$323$	−23.2164			−1.29179
$324$		0			0
$325$	24.0000			1.33128
$326$		0			0
$327$	−8.29156	−	2.50000i	−0.458524	−	0.138250i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	17.0000i		−	0.934405i	−0.884150	−	0.467202i	$-0.845262\pi$
$331$		−	17.0000i		−	0.934405i	0.884150	−	0.467202i	$-0.154738\pi$
$332$		0			0
$333$	−2.50000	−	1.65831i	−0.136999	−	0.0908750i
$334$		0			0
$335$	29.8496			1.63086
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$		0			0
$339$	−3.31662	+	11.0000i	−0.180134	+	0.597438i
$340$		0			0
$341$		−	9.94987i		−	0.538816i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	5.50000	−	18.2414i	0.296110	−	0.982086i
$346$		0			0
$347$	−29.8496			−1.60241			−0.801206	−	0.598389i	$-0.795808\pi$
$347$	−29.8496			−1.60241			−0.801206	+	0.598389i	$0.795808\pi$
$348$		0			0
$349$	−8.00000			−0.428230			−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000			−0.428230			−0.214115	+	0.976808i	$0.568687\pi$
$350$		0			0
$351$	13.2665	+	16.0000i	0.708113	+	0.854017i
$352$		0			0
$353$			36.4829i			1.94179i	0.239511	+	0.970894i	$0.423013\pi$
$353$			36.4829i			1.94179i	−0.239511	+	0.970894i	$0.576987\pi$
$354$		0			0
$355$			44.0000i			2.33528i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	36.4829			1.92549			0.962746	−	0.270407i	$-0.0871582\pi$
$359$	36.4829			1.92549			0.962746	+	0.270407i	$0.0871582\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$			23.2164i			1.21520i
$366$		0			0
$367$		−	1.00000i		−	0.0521996i	−0.999659	−	0.0260998i	$-0.991691\pi$
$367$		−	1.00000i		−	0.0521996i	0.999659	−	0.0260998i	$-0.00830876\pi$
$368$		0			0
$369$	−11.0000	+	16.5831i	−0.572637	+	0.863283i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.0000			−0.673114			−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000			−0.673114			−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$	−1.65831	+	5.50000i	−0.0856349	+	0.284019i
$376$		0			0
$377$			26.5330i			1.36652i
$378$		0			0
$379$			24.0000i			1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$			24.0000i			1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$		0			0
$381$	1.00000	−	3.31662i	0.0512316	−	0.169916i
$382$		0			0
$383$	−16.5831			−0.847358			−0.423679	−	0.905812i	$-0.639262\pi$
$383$	−16.5831			−0.847358			−0.423679	+	0.905812i	$0.639262\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.31662	+	5.00000i	−0.168594	+	0.254164i
$388$		0			0
$389$		−	3.31662i		−	0.168160i	−0.996459	−	0.0840798i	$-0.973205\pi$
$389$		−	3.31662i		−	0.168160i	0.996459	−	0.0840798i	$-0.0267951\pi$
$390$		0			0
$391$		−	11.0000i		−	0.556294i
$392$		0			0
$393$	−27.5000	−	8.29156i	−1.38719	−	0.418254i
$394$		0			0
$395$	29.8496			1.50190
$396$		0			0
$397$	13.0000			0.652451			0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000			0.652451			0.326226	+	0.945292i	$0.394223\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.5831i			0.828122i	0.910249	+	0.414061i	$0.135890\pi$
$401$			16.5831i			0.828122i	−0.910249	+	0.414061i	$0.864110\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$	−27.5000	+	11.6082i	−1.36649	+	0.576815i
$406$		0			0
$407$	−3.31662			−0.164399
$408$		0			0
$409$	−15.0000			−0.741702			−0.370851	−	0.928692i	$-0.620934\pi$
$409$	−15.0000			−0.741702			−0.370851	+	0.928692i	$0.620934\pi$
$410$		0			0
$411$	−8.29156	+	27.5000i	−0.408993	+	1.35647i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	44.0000i		−	2.15988i
$416$		0			0
$417$	−5.00000	+	16.5831i	−0.244851	+	0.812079i
$418$		0			0
$419$	26.5330			1.29622			0.648111	−	0.761546i	$-0.275559\pi$
$419$	26.5330			1.29622			0.648111	+	0.761546i	$0.275559\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	24.8747	+	16.5000i	1.20945	+	0.802257i
$424$		0			0
$425$			19.8997i			0.965280i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	22.0000	+	6.63325i	1.06217	+	0.320256i
$430$		0			0
$431$	−23.2164			−1.11829			−0.559147	−	0.829069i	$-0.688871\pi$
$431$	−23.2164			−1.11829			−0.559147	+	0.829069i	$0.688871\pi$
$432$		0			0
$433$	−20.0000			−0.961139			−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000			−0.961139			−0.480569	+	0.876957i	$0.659570\pi$
$434$		0			0
$435$	−36.4829	−	11.0000i	−1.74922	−	0.527410i
$436$		0			0
$437$		−	23.2164i		−	1.11059i
$438$		0			0
$439$			9.00000i			0.429547i	0.976664	+	0.214773i	$0.0689013\pi$
$439$			9.00000i			0.429547i	−0.976664	+	0.214773i	$0.931099\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.8496			−1.41820			−0.709099	−	0.705109i	$-0.750898\pi$
$443$	−29.8496			−1.41820			−0.709099	+	0.705109i	$0.750898\pi$
$444$		0			0
$445$	11.0000			0.521450
$446$		0			0
$447$	8.29156	−	27.5000i	0.392177	−	1.30071i
$448$		0			0
$449$		−	19.8997i		−	0.939127i	−0.882899	−	0.469564i	$-0.844411\pi$
$449$		−	19.8997i		−	0.939127i	0.882899	−	0.469564i	$-0.155589\pi$
$450$		0			0
$451$			22.0000i			1.03594i
$452$		0			0
$453$	−3.50000	+	11.6082i	−0.164444	+	0.545400i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.00000			0.140334			0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000			0.140334			0.0701670	+	0.997535i	$0.477647\pi$
$458$		0			0
$459$	−13.2665	+	11.0000i	−0.619227	+	0.513436i
$460$		0			0
$461$		−	33.1662i		−	1.54471i	−0.635193	−	0.772353i	$-0.719080\pi$
$461$		−	33.1662i		−	1.54471i	0.635193	−	0.772353i	$-0.280920\pi$
$462$		0			0
$463$			38.0000i			1.76601i	0.469364	+	0.883005i	$0.344483\pi$
$463$			38.0000i			1.76601i	−0.469364	+	0.883005i	$0.655517\pi$
$464$		0			0
$465$	−16.5000	−	4.97494i	−0.765169	−	0.230707i
$466$		0			0
$467$	3.31662			0.153475			0.0767375	−	0.997051i	$-0.475550\pi$
$467$	3.31662			0.153475			0.0767375	+	0.997051i	$0.475550\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.97494	−	1.50000i	−0.229233	−	0.0691164i
$472$		0			0
$473$			6.63325i			0.304997i
$474$		0			0
$475$			42.0000i			1.92709i
$476$		0			0
$477$	−5.50000	+	8.29156i	−0.251828	+	0.379645i
$478$		0			0
$479$	−16.5831			−0.757702			−0.378851	−	0.925458i	$-0.623681\pi$
$479$	−16.5831			−0.757702			−0.378851	+	0.925458i	$0.623681\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$			26.5330i			1.20480i
$486$		0			0
$487$		−	17.0000i		−	0.770344i	−0.922845	−	0.385172i	$-0.874142\pi$
$487$		−	17.0000i		−	0.770344i	0.922845	−	0.385172i	$-0.125858\pi$
$488$		0			0
$489$	3.50000	−	11.6082i	0.158275	−	0.524940i
$490$		0			0
$491$	−6.63325			−0.299354			−0.149677	−	0.988735i	$-0.547823\pi$
$491$	−6.63325			−0.299354			−0.149677	+	0.988735i	$0.547823\pi$
$492$		0			0
$493$	−22.0000			−0.990830
$494$		0			0
$495$	−18.2414	+	27.5000i	−0.819892	+	1.23603i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			35.0000i			1.56682i	0.621508	+	0.783408i	$0.286520\pi$
$499$			35.0000i			1.56682i	−0.621508	+	0.783408i	$0.713480\pi$
$500$		0			0
$501$	−11.0000	−	3.31662i	−0.491444	−	0.148176i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	11.0000			0.489494
$506$		0			0
$507$	−4.97494	−	1.50000i	−0.220945	−	0.0666173i
$508$		0			0
$509$			16.5831i			0.735034i	0.930017	+	0.367517i	$0.119792\pi$
$509$			16.5831i			0.735034i	−0.930017	+	0.367517i	$0.880208\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−28.0000	+	23.2164i	−1.23623	+	1.02503i
$514$		0			0
$515$	−36.4829			−1.60763
$516$		0			0
$517$	33.0000			1.45134
$518$		0			0
$519$	4.97494	−	16.5000i	0.218376	−	0.724270i
$520$		0			0
$521$		−	23.2164i		−	1.01713i	−0.861024	−	0.508564i	$-0.830177\pi$
$521$		−	23.2164i		−	1.01713i	0.861024	−	0.508564i	$-0.169823\pi$
$522$		0			0
$523$		−	19.0000i		−	0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$		−	19.0000i		−	0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.94987			−0.433423
$528$		0			0
$529$	−12.0000			−0.521739
$530$		0			0
$531$	−8.29156	−	5.50000i	−0.359823	−	0.238680i
$532$		0			0
$533$		−	26.5330i		−	1.14927i
$534$		0			0
$535$		−	11.0000i		−	0.475571i
$536$		0			0
$537$	−5.50000	−	1.65831i	−0.237343	−	0.0715615i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	37.0000			1.59075			0.795377	−	0.606115i	$-0.207273\pi$
$541$	37.0000			1.59075			0.795377	+	0.606115i	$0.207273\pi$
$542$		0			0
$543$	−29.8496	−	9.00000i	−1.28097	−	0.386227i
$544$		0			0
$545$			16.5831i			0.710343i
$546$		0			0
$547$			40.0000i			1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$			40.0000i			1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$		0			0
$549$	7.50000	+	4.97494i	0.320092	+	0.212325i
$550$		0			0
$551$	−46.4327			−1.97810
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.65831	+	5.50000i	−0.0703914	+	0.233462i
$556$		0			0
$557$		−	3.31662i		−	0.140530i	−0.997528	−	0.0702650i	$-0.977616\pi$
$557$		−	3.31662i		−	0.140530i	0.997528	−	0.0702650i	$-0.0223845\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$	−5.50000	+	18.2414i	−0.232210	+	0.770154i
$562$		0			0
$563$	9.94987			0.419337			0.209669	−	0.977773i	$-0.432761\pi$
$563$	9.94987			0.419337			0.209669	+	0.977773i	$0.432761\pi$
$564$		0			0
$565$	22.0000			0.925547
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	29.8496i		−	1.25136i	−0.780079	−	0.625681i	$-0.784821\pi$
$569$		−	29.8496i		−	1.25136i	0.780079	−	0.625681i	$-0.215179\pi$
$570$		0			0
$571$			5.00000i			0.209243i	0.994512	+	0.104622i	$0.0333632\pi$
$571$			5.00000i			0.209243i	−0.994512	+	0.104622i	$0.966637\pi$
$572$		0			0
$573$	−16.5000	−	4.97494i	−0.689297	−	0.207831i
$574$		0			0
$575$	−19.8997			−0.829877
$576$		0			0
$577$	35.0000			1.45707			0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000			1.45707			0.728535	+	0.685009i	$0.240202\pi$
$578$		0			0
$579$	4.97494	+	1.50000i	0.206751	+	0.0623379i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			11.0000i			0.455573i
$584$		0			0
$585$	22.0000	−	33.1662i	0.909588	−	1.37126i
$586$		0			0
$587$	−26.5330			−1.09513			−0.547567	−	0.836762i	$-0.684446\pi$
$587$	−26.5330			−1.09513			−0.547567	+	0.836762i	$0.684446\pi$
$588$		0			0
$589$	−21.0000			−0.865290
$590$		0			0
$591$	9.94987	−	33.0000i	0.409283	−	1.35744i
$592$		0			0
$593$			3.31662i			0.136197i	0.997679	+	0.0680987i	$0.0216933\pi$
$593$			3.31662i			0.136197i	−0.997679	+	0.0680987i	$0.978307\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.500000	−	1.65831i	0.0204636	−	0.0678702i
$598$		0			0
$599$	9.94987			0.406541			0.203270	−	0.979123i	$-0.434843\pi$
$599$	9.94987			0.406541			0.203270	+	0.979123i	$0.434843\pi$
$600$		0			0
$601$	−16.0000			−0.652654			−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000			−0.652654			−0.326327	+	0.945257i	$0.605811\pi$
$602$		0			0
$603$	14.9248	−	22.5000i	0.607785	−	0.916271i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	7.00000i		−	0.284121i	−0.989858	−	0.142061i	$-0.954627\pi$
$607$		−	7.00000i		−	0.284121i	0.989858	−	0.142061i	$-0.0453728\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−39.7995			−1.61012
$612$		0			0
$613$	1.00000			0.0403896			0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000			0.0403896			0.0201948	+	0.999796i	$0.493571\pi$
$614$		0			0
$615$	36.4829	+	11.0000i	1.47113	+	0.443563i
$616$		0			0
$617$			6.63325i			0.267045i	0.991046	+	0.133522i	$0.0426288\pi$
$617$			6.63325i			0.267045i	−0.991046	+	0.133522i	$0.957371\pi$
$618$		0			0
$619$		−	25.0000i		−	1.00483i	−0.864625	−	0.502417i	$-0.832444\pi$
$619$		−	25.0000i		−	1.00483i	0.864625	−	0.502417i	$-0.167556\pi$
$620$		0			0
$621$	−11.0000	−	13.2665i	−0.441415	−	0.532366i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−11.6082	+	38.5000i	−0.463586	+	1.53754i
$628$		0			0
$629$			3.31662i			0.132242i
$630$		0			0
$631$		−	18.0000i		−	0.716569i	−0.933613	−	0.358284i	$-0.883362\pi$
$631$		−	18.0000i		−	0.716569i	0.933613	−	0.358284i	$-0.116638\pi$
$632$		0			0
$633$	−13.0000	+	43.1161i	−0.516704	+	1.71371i
$634$		0			0
$635$	−6.63325			−0.263232
$636$		0			0
$637$		0			0
$638$		0			0
$639$	33.1662	+	22.0000i	1.31204	+	0.870307i
$640$		0			0
$641$		−	36.4829i		−	1.44099i	−0.693462	−	0.720493i	$-0.743915\pi$
$641$		−	36.4829i		−	1.44099i	0.693462	−	0.720493i	$-0.256085\pi$
$642$		0			0
$643$		−	2.00000i		−	0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$		−	2.00000i		−	0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$		0			0
$645$	11.0000	+	3.31662i	0.433125	+	0.130592i
$646$		0			0
$647$	−3.31662			−0.130390			−0.0651950	−	0.997873i	$-0.520767\pi$
$647$	−3.31662			−0.130390			−0.0651950	+	0.997873i	$0.520767\pi$
$648$		0			0
$649$	−11.0000			−0.431788
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	3.31662i		−	0.129790i	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$		−	3.31662i		−	0.129790i	0.997892	−	0.0648948i	$-0.0206712\pi$
$654$		0			0
$655$			55.0000i			2.14903i
$656$		0			0
$657$	17.5000	+	11.6082i	0.682740	+	0.452878i
$658$		0			0
$659$	−26.5330			−1.03358			−0.516789	−	0.856113i	$-0.672873\pi$
$659$	−26.5330			−1.03358			−0.516789	+	0.856113i	$0.672873\pi$
$660$		0			0
$661$	39.0000			1.51692			0.758462	−	0.651717i	$-0.225951\pi$
$661$	39.0000			1.51692			0.758462	+	0.651717i	$0.225951\pi$
$662$		0			0
$663$	6.63325	−	22.0000i	0.257614	−	0.854409i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	22.0000i		−	0.851843i
$668$		0			0
$669$	−7.00000	+	23.2164i	−0.270636	+	0.897597i
$670$		0			0
$671$	9.94987			0.384111
$672$		0			0
$673$	40.0000			1.54189			0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000			1.54189			0.770943	+	0.636904i	$0.219785\pi$
$674$		0			0
$675$	19.8997	+	24.0000i	0.765942	+	0.923760i
$676$		0			0
$677$		−	9.94987i		−	0.382405i	−0.981551	−	0.191202i	$-0.938761\pi$
$677$		−	9.94987i		−	0.382405i	0.981551	−	0.191202i	$-0.0612387\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.50000	−	1.65831i	−0.210760	−	0.0635467i
$682$		0			0
$683$	−23.2164			−0.888350			−0.444175	−	0.895940i	$-0.646503\pi$
$683$	−23.2164			−0.888350			−0.444175	+	0.895940i	$0.646503\pi$
$684$		0			0
$685$	55.0000			2.10144
$686$		0			0
$687$	−48.0911	−	14.5000i	−1.83479	−	0.553210i
$688$		0			0
$689$		−	13.2665i		−	0.505413i
$690$		0			0
$691$		−	33.0000i		−	1.25538i	−0.778464	−	0.627690i	$-0.784001\pi$
$691$		−	33.0000i		−	1.25538i	0.778464	−	0.627690i	$-0.215999\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.1662			1.25807
$696$		0			0
$697$	22.0000			0.833309
$698$		0			0
$699$	8.29156	−	27.5000i	0.313616	−	1.04015i
$700$		0			0
$701$		−	13.2665i		−	0.501069i	−0.968108	−	0.250534i	$-0.919394\pi$
$701$		−	13.2665i		−	0.501069i	0.968108	−	0.250534i	$-0.0806063\pi$
$702$		0			0
$703$			7.00000i			0.264010i
$704$		0			0
$705$	16.5000	−	54.7243i	0.621426	−	2.06104i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−33.0000			−1.23934			−0.619671	−	0.784862i	$-0.712734\pi$
$709$	−33.0000			−1.23934			−0.619671	+	0.784862i	$0.712734\pi$
$710$		0			0
$711$	14.9248	−	22.5000i	0.559724	−	0.843816i
$712$		0			0
$713$		−	9.94987i		−	0.372626i
$714$		0			0
$715$		−	44.0000i		−	1.64551i
$716$		0			0
$717$	11.0000	+	3.31662i	0.410803	+	0.123862i
$718$		0			0
$719$	−23.2164			−0.865825			−0.432912	−	0.901436i	$-0.642514\pi$
$719$	−23.2164			−0.865825			−0.432912	+	0.901436i	$0.642514\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−31.5079	−	9.50000i	−1.17179	−	0.353309i
$724$		0			0
$725$			39.7995i			1.47812i
$726$		0			0
$727$		−	10.0000i		−	0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$		−	10.0000i		−	0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$		0			0
$729$	−5.00000	+	26.5330i	−0.185185	+	0.982704i
$730$		0			0
$731$	6.63325			0.245340
$732$		0			0
$733$	7.00000			0.258551			0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000			0.258551			0.129275	+	0.991609i	$0.458735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	29.8496i		−	1.09952i
$738$		0			0
$739$		−	31.0000i		−	1.14035i	−0.821522	−	0.570177i	$-0.806875\pi$
$739$		−	31.0000i		−	1.14035i	0.821522	−	0.570177i	$-0.193125\pi$
$740$		0			0
$741$	14.0000	−	46.4327i	0.514303	−	1.70575i
$742$		0			0
$743$	13.2665			0.486701			0.243350	−	0.969938i	$-0.421754\pi$
$743$	13.2665			0.486701			0.243350	+	0.969938i	$0.421754\pi$
$744$		0			0
$745$	−55.0000			−2.01504
$746$		0			0
$747$	−33.1662	−	22.0000i	−1.21349	−	0.804938i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			5.00000i			0.182453i	0.995830	+	0.0912263i	$0.0290787\pi$
$751$			5.00000i			0.182453i	−0.995830	+	0.0912263i	$0.970921\pi$
$752$		0			0
$753$	33.0000	+	9.94987i	1.20259	+	0.362594i
$754$		0			0
$755$	23.2164			0.844930
$756$		0			0
$757$	−48.0000			−1.74459			−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000			−1.74459			−0.872295	+	0.488980i	$0.837369\pi$
$758$		0			0
$759$	−18.2414	−	5.50000i	−0.662122	−	0.199637i
$760$		0			0
$761$		−	23.2164i		−	0.841593i	−0.907155	−	0.420796i	$-0.861751\pi$
$761$		−	23.2164i		−	0.841593i	0.907155	−	0.420796i	$-0.138249\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	27.5000	+	18.2414i	0.994265	+	0.659521i
$766$		0			0
$767$	13.2665			0.479026
$768$		0			0
$769$	−16.0000			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$770$		0			0
$771$	−11.6082	+	38.5000i	−0.418059	+	1.38654i
$772$		0			0
$773$		−	16.5831i		−	0.596454i	−0.954495	−	0.298227i	$-0.903605\pi$
$773$		−	16.5831i		−	0.596454i	0.954495	−	0.298227i	$-0.0963952\pi$
$774$		0			0
$775$			18.0000i			0.646579i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.4327			1.66363
$780$		0			0
$781$	44.0000			1.57444
$782$		0			0
$783$	−26.5330	+	22.0000i	−0.948212	+	0.786216i
$784$		0			0
$785$			9.94987i			0.355126i
$786$		0			0
$787$		−	45.0000i		−	1.60408i	−0.597272	−	0.802038i	$-0.703749\pi$
$787$		−	45.0000i		−	1.60408i	0.597272	−	0.802038i	$-0.296251\pi$
$788$		0			0
$789$	5.50000	+	1.65831i	0.195805	+	0.0590375i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.0000			−0.426132
$794$		0			0
$795$	18.2414	+	5.50000i	0.646957	+	0.195065i
$796$		0			0
$797$			6.63325i			0.234962i	0.993075	+	0.117481i	$0.0374819\pi$
$797$			6.63325i			0.234962i	−0.993075	+	0.117481i	$0.962518\pi$
$798$		0			0
$799$		−	33.0000i		−	1.16746i
$800$		0			0
$801$	5.50000	−	8.29156i	0.194333	−	0.292968i
$802$		0			0
$803$	23.2164			0.819288
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.29156	−	27.5000i	0.291877	−	0.968046i
$808$		0			0
$809$			3.31662i			0.116606i	0.998299	+	0.0583032i	$0.0185690\pi$
$809$			3.31662i			0.116606i	−0.998299	+	0.0583032i	$0.981431\pi$
$810$		0			0
$811$		−	30.0000i		−	1.05344i	−0.850038	−	0.526721i	$-0.823421\pi$
$811$		−	30.0000i		−	1.05344i	0.850038	−	0.526721i	$-0.176579\pi$
$812$		0			0
$813$	−8.50000	+	28.1913i	−0.298108	+	0.988712i
$814$		0			0
$815$	−23.2164			−0.813234
$816$		0			0
$817$	14.0000			0.489798
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	36.4829i		−	1.27326i	−0.771169	−	0.636631i	$-0.780328\pi$
$821$		−	36.4829i		−	1.27326i	0.771169	−	0.636631i	$-0.219672\pi$
$822$		0			0
$823$			15.0000i			0.522867i	0.965221	+	0.261434i	$0.0841952\pi$
$823$			15.0000i			0.522867i	−0.965221	+	0.261434i	$0.915805\pi$
$824$		0			0
$825$	33.0000	+	9.94987i	1.14891	+	0.346410i
$826$		0			0
$827$	39.7995			1.38396			0.691982	−	0.721915i	$-0.256738\pi$
$827$	39.7995			1.38396			0.691982	+	0.721915i	$0.256738\pi$
$828$		0			0
$829$	29.0000			1.00721			0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000			1.00721			0.503606	+	0.863934i	$0.332006\pi$
$830$		0			0
$831$	21.5581	+	6.50000i	0.747841	+	0.225483i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			22.0000i			0.761341i
$836$		0			0
$837$	−12.0000	+	9.94987i	−0.414781	+	0.343918i
$838$		0			0
$839$	13.2665			0.458010			0.229005	−	0.973425i	$-0.426453\pi$
$839$	13.2665			0.458010			0.229005	+	0.973425i	$0.426453\pi$
$840$		0			0
$841$	−15.0000			−0.517241
$842$		0			0
$843$	3.31662	−	11.0000i	0.114231	−	0.378860i
$844$		0			0
$845$			9.94987i			0.342286i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−7.50000	+	24.8747i	−0.257399	+	0.853697i
$850$		0			0
$851$	−3.31662			−0.113692
$852$		0			0
$853$	−20.0000			−0.684787			−0.342393	−	0.939557i	$-0.611238\pi$
$853$	−20.0000			−0.684787			−0.342393	+	0.939557i	$0.611238\pi$
$854$		0			0
$855$	58.0409	+	38.5000i	1.98496	+	1.31667i
$856$		0			0
$857$			16.5831i			0.566469i	0.959051	+	0.283234i	$0.0914074\pi$
$857$			16.5831i			0.566469i	−0.959051	+	0.283234i	$0.908593\pi$
$858$		0			0
$859$			17.0000i			0.580033i	0.957022	+	0.290016i	$0.0936607\pi$
$859$			17.0000i			0.580033i	−0.957022	+	0.290016i	$0.906339\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.4829			−1.24189			−0.620946	−	0.783854i	$-0.713251\pi$
$863$	−36.4829			−1.24189			−0.620946	+	0.783854i	$0.713251\pi$
$864$		0			0
$865$	−33.0000			−1.12203
$866$		0			0
$867$	−9.94987	−	3.00000i	−0.337915	−	0.101885i
$868$		0			0
$869$		−	29.8496i		−	1.01258i
$870$		0			0
$871$			36.0000i			1.21981i
$872$		0			0
$873$	20.0000	+	13.2665i	0.676897	+	0.449003i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.0000			−0.776655			−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000			−0.776655			−0.388327	+	0.921521i	$0.626947\pi$
$878$		0			0
$879$	6.63325	−	22.0000i	0.223734	−	0.742042i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$	−5.50000	+	18.2414i	−0.184880	+	0.613179i
$886$		0			0
$887$	23.2164			0.779530			0.389765	−	0.920914i	$-0.372556\pi$
$887$	23.2164			0.779530			0.389765	+	0.920914i	$0.372556\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	11.6082	+	27.5000i	0.388889	+	0.921285i
$892$		0			0
$893$		−	69.6491i		−	2.33072i
$894$		0			0
$895$			11.0000i			0.367689i
$896$		0			0
$897$	22.0000	+	6.63325i	0.734559	+	0.221478i
$898$		0			0
$899$	−19.8997			−0.663694
$900$		0			0
$901$	11.0000			0.366463
$902$		0			0
$903$		0			0
$904$		0			0
$905$			59.6992i			1.98447i
$906$		0			0
$907$		−	53.0000i		−	1.75984i	−0.475125	−	0.879918i	$-0.657597\pi$
$907$		−	53.0000i		−	1.75984i	0.475125	−	0.879918i	$-0.342403\pi$
$908$		0			0
$909$	5.50000	−	8.29156i	0.182423	−	0.275014i
$910$		0			0
$911$	−26.5330			−0.879077			−0.439539	−	0.898224i	$-0.644858\pi$
$911$	−26.5330			−0.879077			−0.439539	+	0.898224i	$0.644858\pi$
$912$		0			0
$913$	−44.0000			−1.45619
$914$		0			0
$915$	4.97494	−	16.5000i	0.164466	−	0.545473i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	23.0000i		−	0.758700i	−0.925253	−	0.379350i	$-0.876148\pi$
$919$		−	23.0000i		−	0.758700i	0.925253	−	0.379350i	$-0.123852\pi$
$920$		0			0
$921$	3.00000	−	9.94987i	0.0988534	−	0.327860i
$922$		0			0
$923$	−53.0660			−1.74669
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	−18.2414	+	27.5000i	−0.599127	+	0.903219i
$928$		0			0
$929$			23.2164i			0.761705i	0.924636	+	0.380852i	$0.124369\pi$
$929$			23.2164i			0.761705i	−0.924636	+	0.380852i	$0.875631\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−16.5000	−	4.97494i	−0.540186	−	0.162872i
$934$		0			0
$935$	36.4829			1.19312
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	14.9248	+	4.50000i	0.487053	+	0.146852i
$940$		0			0
$941$		−	9.94987i		−	0.324357i	−0.986761	−	0.162178i	$-0.948148\pi$
$941$		−	9.94987i		−	0.324357i	0.986761	−	0.162178i	$-0.0518520\pi$
$942$		0			0
$943$			22.0000i			0.716419i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.94987			−0.323327			−0.161664	−	0.986846i	$-0.551686\pi$
$947$	−9.94987			−0.323327			−0.161664	+	0.986846i	$0.551686\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		0			0
$951$	−14.9248	+	49.5000i	−0.483970	+	1.60515i
$952$		0			0
$953$		−	19.8997i		−	0.644616i	−0.946635	−	0.322308i	$-0.895541\pi$
$953$		−	19.8997i		−	0.644616i	0.946635	−	0.322308i	$-0.104459\pi$
$954$		0			0
$955$			33.0000i			1.06785i
$956$		0			0
$957$	−11.0000	+	36.4829i	−0.355580	+	1.17932i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.0000			0.709677
$962$		0			0
$963$	−8.29156	−	5.50000i	−0.267192	−	0.177235i
$964$		0			0
$965$		−	9.94987i		−	0.320298i
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		0			0
$969$	38.5000	+	11.6082i	1.23680	+	0.372909i
$970$		0			0
$971$	16.5831			0.532178			0.266089	−	0.963948i	$-0.414269\pi$
$971$	16.5831			0.532178			0.266089	+	0.963948i	$0.414269\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−39.7995	−	12.0000i	−1.27460	−	0.384308i
$976$		0			0
$977$		−	3.31662i		−	0.106108i	−0.998592	−	0.0530541i	$-0.983104\pi$
$977$		−	3.31662i		−	0.106108i	0.998592	−	0.0530541i	$-0.0168956\pi$
$978$		0			0
$979$		−	11.0000i		−	0.351562i
$980$		0			0
$981$	12.5000	+	8.29156i	0.399094	+	0.264729i
$982$		0			0
$983$	56.3826			1.79833			0.899163	−	0.437614i	$-0.144176\pi$
$983$	56.3826			1.79833			0.899163	+	0.437614i	$0.144176\pi$
$984$		0			0
$985$	−66.0000			−2.10293
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.63325i			0.210925i
$990$		0			0
$991$			43.0000i			1.36594i	0.730446	+	0.682970i	$0.239312\pi$
$991$			43.0000i			1.36594i	−0.730446	+	0.682970i	$0.760688\pi$
$992$		0			0
$993$	−8.50000	+	28.1913i	−0.269739	+	0.894624i
$994$		0			0
$995$	−3.31662			−0.105144
$996$		0			0
$997$	−5.00000			−0.158352			−0.0791758	−	0.996861i	$-0.525229\pi$
$997$	−5.00000			−0.158352			−0.0791758	+	0.996861i	$0.525229\pi$
$998$		0			0
$999$	3.31662	+	4.00000i	0.104933	+	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.i.2255.1		4
3.2	odd	2	inner	2352.2.h.i.2255.3		4
4.3	odd	2	inner	2352.2.h.i.2255.4		4
7.2	even	3		336.2.bj.f.95.4	yes	8
7.4	even	3		336.2.bj.f.191.3	yes	8
7.6	odd	2		2352.2.h.j.2255.4		4
12.11	even	2	inner	2352.2.h.i.2255.2		4
21.2	odd	6		336.2.bj.f.95.2	yes	8
21.11	odd	6		336.2.bj.f.191.1	yes	8
21.20	even	2		2352.2.h.j.2255.2		4
28.11	odd	6		336.2.bj.f.191.2	yes	8
28.23	odd	6		336.2.bj.f.95.1	✓	8
28.27	even	2		2352.2.h.j.2255.1		4
84.11	even	6		336.2.bj.f.191.4	yes	8
84.23	even	6		336.2.bj.f.95.3	yes	8
84.83	odd	2		2352.2.h.j.2255.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bj.f.95.1	✓	8	28.23	odd	6
336.2.bj.f.95.2	yes	8	21.2	odd	6
336.2.bj.f.95.3	yes	8	84.23	even	6
336.2.bj.f.95.4	yes	8	7.2	even	3
336.2.bj.f.191.1	yes	8	21.11	odd	6
336.2.bj.f.191.2	yes	8	28.11	odd	6
336.2.bj.f.191.3	yes	8	7.4	even	3
336.2.bj.f.191.4	yes	8	84.11	even	6
2352.2.h.i.2255.1		4	1.1	even	1	trivial
2352.2.h.i.2255.2		4	12.11	even	2	inner
2352.2.h.i.2255.3		4	3.2	odd	2	inner
2352.2.h.i.2255.4		4	4.3	odd	2	inner
2352.2.h.j.2255.1		4	28.27	even	2
2352.2.h.j.2255.2		4	21.20	even	2
2352.2.h.j.2255.3		4	84.83	odd	2
2352.2.h.j.2255.4		4	7.6	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +O(q^{10})\)	\(q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +3.31662 q^{11} -4.00000 q^{13} +(1.65831 - 5.50000i) q^{15} -3.31662i q^{17} -7.00000i q^{19} +3.31662 q^{23} -6.00000 q^{25} +(-3.31662 - 4.00000i) q^{27} -6.63325i q^{29} -3.00000i q^{31} +(-5.50000 - 1.65831i) q^{33} -1.00000 q^{37} +(6.63325 + 2.00000i) q^{39} +6.63325i q^{41} +2.00000i q^{43} +(-5.50000 + 8.29156i) q^{45} +9.94987 q^{47} +(-1.65831 + 5.50000i) q^{51} +3.31662i q^{53} +11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} -3.31662 q^{59} +3.00000 q^{61} -13.2665i q^{65} -9.00000i q^{67} +(-5.50000 - 1.65831i) q^{69} +13.2665 q^{71} +7.00000 q^{73} +(9.94987 + 3.00000i) q^{75} -9.00000i q^{79} +(3.50000 + 8.29156i) q^{81} -13.2665 q^{83} +11.0000 q^{85} +(-3.31662 + 11.0000i) q^{87} -3.31662i q^{89} +(-1.50000 + 4.97494i) q^{93} +23.2164 q^{95} +8.00000 q^{97} +(8.29156 + 5.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{9}+O(q^{10}) \) 4 * q + 10 * q^9	\( 4 q + 10 q^{9} - 16 q^{13} - 24 q^{25} - 22 q^{33} - 4 q^{37} - 22 q^{45} - 14 q^{57} + 12 q^{61} - 22 q^{69} + 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} + 32 q^{97}+O(q^{100}) \) 4 * q + 10 * q^9 - 16 * q^13 - 24 * q^25 - 22 * q^33 - 4 * q^37 - 22 * q^45 - 14 * q^57 + 12 * q^61 - 22 * q^69 + 28 * q^73 + 14 * q^81 + 44 * q^85 - 6 * q^93 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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