LMFDB - Embedded newform 1152.3.j.b.161.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(161,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.9
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.b.737.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900179 + 0.900179i) q^{5} -7.66460i q^{7} +O(q^{10})$	$q+(0.900179 + 0.900179i) q^{5} -7.66460i q^{7} +(4.57563 + 4.57563i) q^{11} +(-10.9824 - 10.9824i) q^{13} -0.0435225i q^{17} +(12.9849 + 12.9849i) q^{19} -18.3139 q^{23} -23.3794i q^{25} +(34.9759 - 34.9759i) q^{29} -38.4592 q^{31} +(6.89951 - 6.89951i) q^{35} +(-11.3885 + 11.3885i) q^{37} -45.6887 q^{41} +(-51.4657 + 51.4657i) q^{43} -56.5712i q^{47} -9.74608 q^{49} +(44.6596 + 44.6596i) q^{53} +8.23778i q^{55} +(-20.7614 - 20.7614i) q^{59} +(2.42706 + 2.42706i) q^{61} -19.7722i q^{65} +(-18.9097 - 18.9097i) q^{67} -114.872 q^{71} -127.648i q^{73} +(35.0704 - 35.0704i) q^{77} -37.0284 q^{79} +(84.4337 - 84.4337i) q^{83} +(0.0391780 - 0.0391780i) q^{85} +136.172 q^{89} +(-84.1756 + 84.1756i) q^{91} +23.3774i q^{95} -173.475 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) $ 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	0.900179	+	0.900179i	0.180036	+	0.180036i	0.791371	−	0.611336i	$-0.209367\pi$
$5$	0.900179	+	0.900179i	0.180036	+	0.180036i	−0.611336	+	0.791371i	$0.709367\pi$
$6$		0			0
$7$		−	7.66460i		−	1.09494i	−0.836824	−	0.547471i	$-0.815591\pi$
$7$		−	7.66460i		−	1.09494i	0.836824	−	0.547471i	$-0.184409\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.57563	+	4.57563i	0.415967	+	0.415967i	0.883811	−	0.467844i	$-0.154969\pi$
$11$	4.57563	+	4.57563i	0.415967	+	0.415967i	−0.467844	+	0.883811i	$0.654969\pi$
$12$		0			0
$13$	−10.9824	−	10.9824i	−0.844799	−	0.844799i	0.144679	−	0.989479i	$-0.453785\pi$
$13$	−10.9824	−	10.9824i	−0.844799	−	0.844799i	−0.989479	+	0.144679i	$0.953785\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.0435225i		−	0.00256015i	−0.999999	−	0.00128007i	$-0.999593\pi$
$17$		−	0.0435225i		−	0.00256015i	0.999999	−	0.00128007i	$-0.000407460\pi$
$18$		0			0
$19$	12.9849	+	12.9849i	0.683414	+	0.683414i	0.960768	−	0.277354i	$-0.0894575\pi$
$19$	12.9849	+	12.9849i	0.683414	+	0.683414i	−0.277354	+	0.960768i	$0.589457\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18.3139			−0.796256			−0.398128	−	0.917330i	$-0.630340\pi$
$23$	−18.3139			−0.796256			−0.398128	+	0.917330i	$0.630340\pi$
$24$		0			0
$25$		−	23.3794i		−	0.935174i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.9759	−	34.9759i	1.20607	−	1.20607i	0.233774	−	0.972291i	$-0.424892\pi$
$29$	34.9759	−	34.9759i	1.20607	−	1.20607i	0.972291	−	0.233774i	$-0.0751077\pi$
$30$		0			0
$31$	−38.4592			−1.24062			−0.620310	−	0.784356i	$-0.712993\pi$
$31$	−38.4592			−1.24062			−0.620310	+	0.784356i	$0.712993\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.89951	−	6.89951i	0.197129	−	0.197129i
$36$		0			0
$37$	−11.3885	+	11.3885i	−0.307797	+	0.307797i	−0.844054	−	0.536258i	$-0.819838\pi$
$37$	−11.3885	+	11.3885i	−0.307797	+	0.307797i	0.536258	+	0.844054i	$0.319838\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−45.6887			−1.11436			−0.557179	−	0.830393i	$-0.688116\pi$
$41$	−45.6887			−1.11436			−0.557179	+	0.830393i	$0.688116\pi$
$42$		0			0
$43$	−51.4657	+	51.4657i	−1.19688	+	1.19688i	−0.221781	+	0.975097i	$0.571187\pi$
$43$	−51.4657	+	51.4657i	−1.19688	+	1.19688i	−0.975097	+	0.221781i	$0.928813\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	56.5712i		−	1.20364i	−0.798631	−	0.601822i	$-0.794442\pi$
$47$		−	56.5712i		−	1.20364i	0.798631	−	0.601822i	$-0.205558\pi$
$48$		0			0
$49$	−9.74608			−0.198900
$50$		0			0
$51$		0			0
$52$		0			0
$53$	44.6596	+	44.6596i	0.842634	+	0.842634i	0.989201	−	0.146567i	$-0.0468223\pi$
$53$	44.6596	+	44.6596i	0.842634	+	0.842634i	−0.146567	+	0.989201i	$0.546822\pi$
$54$		0			0
$55$			8.23778i			0.149778i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−20.7614	−	20.7614i	−0.351887	−	0.351887i	0.508924	−	0.860811i	$-0.330043\pi$
$59$	−20.7614	−	20.7614i	−0.351887	−	0.351887i	−0.860811	+	0.508924i	$0.830043\pi$
$60$		0			0
$61$	2.42706	+	2.42706i	0.0397878	+	0.0397878i	0.726721	−	0.686933i	$-0.241043\pi$
$61$	2.42706	+	2.42706i	0.0397878	+	0.0397878i	−0.686933	+	0.726721i	$0.741043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	19.7722i		−	0.304188i
$66$		0			0
$67$	−18.9097	−	18.9097i	−0.282234	−	0.282234i	0.551766	−	0.833999i	$-0.313954\pi$
$67$	−18.9097	−	18.9097i	−0.282234	−	0.282234i	−0.833999	+	0.551766i	$0.813954\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−114.872			−1.61791			−0.808956	−	0.587869i	$-0.799967\pi$
$71$	−114.872			−1.61791			−0.808956	+	0.587869i	$0.799967\pi$
$72$		0			0
$73$		−	127.648i		−	1.74860i	−0.485388	−	0.874299i	$-0.661322\pi$
$73$		−	127.648i		−	1.74860i	0.485388	−	0.874299i	$-0.338678\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	35.0704	−	35.0704i	0.455460	−	0.455460i
$78$		0			0
$79$	−37.0284			−0.468714			−0.234357	−	0.972151i	$-0.575299\pi$
$79$	−37.0284			−0.468714			−0.234357	+	0.972151i	$0.575299\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	84.4337	−	84.4337i	1.01727	−	1.01727i	0.0174254	−	0.999848i	$-0.494453\pi$
$83$	84.4337	−	84.4337i	1.01727	−	1.01727i	0.999848	−	0.0174254i	$-0.00554695\pi$
$84$		0			0
$85$	0.0391780	−	0.0391780i	0.000460918	−	0.000460918i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	136.172			1.53002			0.765012	−	0.644017i	$-0.222733\pi$
$89$	136.172			1.53002			0.765012	+	0.644017i	$0.222733\pi$
$90$		0			0
$91$	−84.1756	+	84.1756i	−0.925007	+	0.925007i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			23.3774i			0.246078i
$96$		0			0
$97$	−173.475			−1.78840			−0.894202	−	0.447663i	$-0.852256\pi$
$97$	−173.475			−1.78840			−0.894202	+	0.447663i	$0.852256\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−18.4286	−	18.4286i	−0.182462	−	0.182462i	0.609966	−	0.792428i	$-0.291183\pi$
$101$	−18.4286	−	18.4286i	−0.182462	−	0.182462i	−0.792428	+	0.609966i	$0.791183\pi$
$102$		0			0
$103$			88.8974i			0.863082i	0.902093	+	0.431541i	$0.142030\pi$
$103$			88.8974i			0.863082i	−0.902093	+	0.431541i	$0.857970\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−54.3773	−	54.3773i	−0.508199	−	0.508199i	0.405774	−	0.913973i	$-0.367002\pi$
$107$	−54.3773	−	54.3773i	−0.508199	−	0.508199i	−0.913973	+	0.405774i	$0.867002\pi$
$108$		0			0
$109$	−122.019	−	122.019i	−1.11944	−	1.11944i	−0.991823	−	0.127620i	$-0.959266\pi$
$109$	−122.019	−	122.019i	−1.11944	−	1.11944i	−0.127620	−	0.991823i	$-0.540734\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	162.624i		−	1.43915i	−0.694415	−	0.719575i	$-0.744337\pi$
$113$		−	162.624i		−	1.43915i	0.694415	−	0.719575i	$-0.255663\pi$
$114$		0			0
$115$	−16.4858	−	16.4858i	−0.143355	−	0.143355i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.333582			−0.00280321
$120$		0			0
$121$		−	79.1271i		−	0.653943i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	43.5501	−	43.5501i	0.348401	−	0.348401i
$126$		0			0
$127$	−36.2712			−0.285600			−0.142800	−	0.989752i	$-0.545611\pi$
$127$	−36.2712			−0.285600			−0.142800	+	0.989752i	$0.545611\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−59.9003	+	59.9003i	−0.457254	+	0.457254i	−0.897753	−	0.440499i	$-0.854802\pi$
$131$	−59.9003	+	59.9003i	−0.457254	+	0.457254i	0.440499	+	0.897753i	$0.354802\pi$
$132$		0			0
$133$	99.5238	−	99.5238i	0.748299	−	0.748299i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	98.5118			0.719064			0.359532	−	0.933133i	$-0.382936\pi$
$137$	98.5118			0.719064			0.359532	+	0.933133i	$0.382936\pi$
$138$		0			0
$139$	3.39960	−	3.39960i	0.0244575	−	0.0244575i	−0.694772	−	0.719230i	$-0.744495\pi$
$139$	3.39960	−	3.39960i	0.0244575	−	0.0244575i	0.719230	+	0.694772i	$0.244495\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	100.503i		−	0.702817i
$144$		0			0
$145$	62.9691			0.434270
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.03169	+	9.03169i	0.0606154	+	0.0606154i	0.736765	−	0.676149i	$-0.236353\pi$
$149$	9.03169	+	9.03169i	0.0606154	+	0.0606154i	−0.676149	+	0.736765i	$0.736353\pi$
$150$		0			0
$151$			69.0548i			0.457316i	0.973507	+	0.228658i	$0.0734338\pi$
$151$			69.0548i			0.457316i	−0.973507	+	0.228658i	$0.926566\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−34.6202	−	34.6202i	−0.223356	−	0.223356i
$156$		0			0
$157$	−63.7071	−	63.7071i	−0.405778	−	0.405778i	0.474486	−	0.880263i	$-0.342634\pi$
$157$	−63.7071	−	63.7071i	−0.405778	−	0.405778i	−0.880263	+	0.474486i	$0.842634\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			140.369i			0.871855i
$162$		0			0
$163$	−105.629	−	105.629i	−0.648033	−	0.648033i	0.304484	−	0.952517i	$-0.401516\pi$
$163$	−105.629	−	105.629i	−0.648033	−	0.648033i	−0.952517	+	0.304484i	$0.901516\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	255.722			1.53127			0.765634	−	0.643276i	$-0.222425\pi$
$167$	255.722			1.53127			0.765634	+	0.643276i	$0.222425\pi$
$168$		0			0
$169$			72.2259i			0.427372i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−56.9554	+	56.9554i	−0.329222	+	0.329222i	−0.852291	−	0.523069i	$-0.824787\pi$
$173$	−56.9554	+	56.9554i	−0.329222	+	0.329222i	0.523069	+	0.852291i	$0.324787\pi$
$174$		0			0
$175$	−179.193			−1.02396
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−171.658	+	171.658i	−0.958982	+	0.958982i	−0.999191	−	0.0402097i	$-0.987197\pi$
$179$	−171.658	+	171.658i	−0.958982	+	0.958982i	0.0402097	+	0.999191i	$0.487197\pi$
$180$		0			0
$181$	−55.5135	+	55.5135i	−0.306704	+	0.306704i	−0.843630	−	0.536925i	$-0.819586\pi$
$181$	−55.5135	+	55.5135i	−0.306704	+	0.306704i	0.536925	+	0.843630i	$0.319586\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.5034			−0.110829
$186$		0			0
$187$	0.199143	−	0.199143i	0.00106494	−	0.00106494i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	101.947i		−	0.533755i	−0.963730	−	0.266878i	$-0.914008\pi$
$191$		−	101.947i		−	0.533755i	0.963730	−	0.266878i	$-0.0859920\pi$
$192$		0			0
$193$	81.5007			0.422284			0.211142	−	0.977455i	$-0.432282\pi$
$193$	81.5007			0.422284			0.211142	+	0.977455i	$0.432282\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−170.676	−	170.676i	−0.866377	−	0.866377i	0.125692	−	0.992069i	$-0.459885\pi$
$197$	−170.676	−	170.676i	−0.866377	−	0.866377i	−0.992069	+	0.125692i	$0.959885\pi$
$198$		0			0
$199$			187.424i			0.941831i	0.882178	+	0.470916i	$0.156076\pi$
$199$			187.424i			0.941831i	−0.882178	+	0.470916i	$0.843924\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−268.076	−	268.076i	−1.32057	−	1.32057i
$204$		0			0
$205$	−41.1280	−	41.1280i	−0.200624	−	0.200624i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			118.828i			0.568555i
$210$		0			0
$211$	111.675	+	111.675i	0.529265	+	0.529265i	0.920353	−	0.391088i	$-0.127901\pi$
$211$	111.675	+	111.675i	0.529265	+	0.529265i	−0.391088	+	0.920353i	$0.627901\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−92.6567			−0.430962
$216$		0			0
$217$			294.775i			1.35841i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.477981	+	0.477981i	−0.00216281	+	0.00216281i
$222$		0			0
$223$	−135.329			−0.606857			−0.303429	−	0.952854i	$-0.598131\pi$
$223$	−135.329			−0.606857			−0.303429	+	0.952854i	$0.598131\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.74050	−	6.74050i	0.0296938	−	0.0296938i	−0.692104	−	0.721798i	$-0.743316\pi$
$227$	6.74050	−	6.74050i	0.0296938	−	0.0296938i	0.721798	+	0.692104i	$0.243316\pi$
$228$		0			0
$229$	48.9527	−	48.9527i	0.213767	−	0.213767i	−0.592098	−	0.805866i	$-0.701700\pi$
$229$	48.9527	−	48.9527i	0.213767	−	0.213767i	0.805866	+	0.592098i	$0.201700\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	101.494			0.435595			0.217797	−	0.975994i	$-0.430113\pi$
$233$	101.494			0.435595			0.217797	+	0.975994i	$0.430113\pi$
$234$		0			0
$235$	50.9243	−	50.9243i	0.216699	−	0.216699i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			47.7817i			0.199923i	0.994991	+	0.0999617i	$0.0318720\pi$
$239$			47.7817i			0.199923i	−0.994991	+	0.0999617i	$0.968128\pi$
$240$		0			0
$241$	−18.2775			−0.0758402			−0.0379201	−	0.999281i	$-0.512073\pi$
$241$	−18.2775			−0.0758402			−0.0379201	+	0.999281i	$0.512073\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.77322	−	8.77322i	−0.0358090	−	0.0358090i
$246$		0			0
$247$		−	285.210i		−	1.15470i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	300.658	+	300.658i	1.19784	+	1.19784i	0.974812	+	0.223027i	$0.0715938\pi$
$251$	300.658	+	300.658i	1.19784	+	1.19784i	0.223027	+	0.974812i	$0.428406\pi$
$252$		0			0
$253$	−83.7977	−	83.7977i	−0.331216	−	0.331216i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	170.456i		−	0.663254i	−0.943411	−	0.331627i	$-0.892402\pi$
$257$		−	170.456i		−	0.663254i	0.943411	−	0.331627i	$-0.107598\pi$
$258$		0			0
$259$	87.2882	+	87.2882i	0.337020	+	0.337020i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	294.979			1.12159			0.560797	−	0.827953i	$-0.310495\pi$
$263$	294.979			1.12159			0.560797	+	0.827953i	$0.310495\pi$
$264$		0			0
$265$			80.4033i			0.303409i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−129.972	+	129.972i	−0.483165	+	0.483165i	−0.906141	−	0.422976i	$-0.860986\pi$
$269$	−129.972	+	129.972i	−0.483165	+	0.483165i	0.422976	+	0.906141i	$0.360986\pi$
$270$		0			0
$271$	535.196			1.97489			0.987447	−	0.157953i	$-0.0504895\pi$
$271$	535.196			1.97489			0.987447	+	0.157953i	$0.0504895\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	106.975	−	106.975i	0.389001	−	0.389001i
$276$		0			0
$277$	62.0846	−	62.0846i	0.224132	−	0.224132i	−0.586104	−	0.810236i	$-0.699339\pi$
$277$	62.0846	−	62.0846i	0.224132	−	0.224132i	0.810236	+	0.586104i	$0.199339\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	102.299			0.364055			0.182027	−	0.983293i	$-0.441734\pi$
$281$	102.299			0.364055			0.182027	+	0.983293i	$0.441734\pi$
$282$		0			0
$283$	164.906	−	164.906i	0.582705	−	0.582705i	−0.352940	−	0.935646i	$-0.614818\pi$
$283$	164.906	−	164.906i	0.582705	−	0.582705i	0.935646	+	0.352940i	$0.114818\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			350.185i			1.22016i
$288$		0			0
$289$	288.998			0.999993
$290$		0			0
$291$		0			0
$292$		0			0
$293$	203.242	+	203.242i	0.693658	+	0.693658i	0.963035	−	0.269377i	$-0.0868177\pi$
$293$	203.242	+	203.242i	0.693658	+	0.693658i	−0.269377	+	0.963035i	$0.586818\pi$
$294$		0			0
$295$		−	37.3779i		−	0.126705i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	201.130	+	201.130i	0.672677	+	0.672677i
$300$		0			0
$301$	394.464	+	394.464i	1.31051	+	1.31051i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.36957i			0.0143265i
$306$		0			0
$307$	−133.116	−	133.116i	−0.433602	−	0.433602i	0.456250	−	0.889852i	$-0.349192\pi$
$307$	−133.116	−	133.116i	−0.433602	−	0.433602i	−0.889852	+	0.456250i	$0.849192\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−260.983			−0.839175			−0.419587	−	0.907715i	$-0.637825\pi$
$311$	−260.983			−0.839175			−0.419587	+	0.907715i	$0.637825\pi$
$312$		0			0
$313$		−	157.967i		−	0.504686i	−0.967638	−	0.252343i	$-0.918799\pi$
$313$		−	157.967i		−	0.504686i	0.967638	−	0.252343i	$-0.0812011\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−136.209	+	136.209i	−0.429683	+	0.429683i	−0.888520	−	0.458838i	$-0.848266\pi$
$317$	−136.209	+	136.209i	−0.429683	+	0.429683i	0.458838	+	0.888520i	$0.348266\pi$
$318$		0			0
$319$	320.074			1.00337
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.565134	−	0.565134i	0.00174964	−	0.00174964i
$324$		0			0
$325$	−256.761	+	256.761i	−0.790035	+	0.790035i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−433.596			−1.31792
$330$		0			0
$331$	432.414	−	432.414i	1.30639	−	1.30639i	0.382385	−	0.924003i	$-0.375103\pi$
$331$	432.414	−	432.414i	1.30639	−	1.30639i	0.924003	−	0.382385i	$-0.124897\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	34.0442i		−	0.101624i
$336$		0			0
$337$	−383.585			−1.13823			−0.569117	−	0.822256i	$-0.692715\pi$
$337$	−383.585			−1.13823			−0.569117	+	0.822256i	$0.692715\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−175.975	−	175.975i	−0.516057	−	0.516057i
$342$		0			0
$343$		−	300.866i		−	0.877159i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	111.739	+	111.739i	0.322014	+	0.322014i	0.849539	−	0.527526i	$-0.176880\pi$
$347$	111.739	+	111.739i	0.322014	+	0.322014i	−0.527526	+	0.849539i	$0.676880\pi$
$348$		0			0
$349$	431.263	+	431.263i	1.23571	+	1.23571i	0.961738	+	0.273971i	$0.0883374\pi$
$349$	431.263	+	431.263i	1.23571	+	1.23571i	0.273971	+	0.961738i	$0.411663\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			290.324i			0.822448i	0.911534	+	0.411224i	$0.134899\pi$
$353$			290.324i			0.822448i	−0.911534	+	0.411224i	$0.865101\pi$
$354$		0			0
$355$	−103.405	−	103.405i	−0.291282	−	0.291282i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−81.9699			−0.228329			−0.114164	−	0.993462i	$-0.536419\pi$
$359$	−81.9699			−0.228329			−0.114164	+	0.993462i	$0.536419\pi$
$360$		0			0
$361$		−	23.7867i		−	0.0658912i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	114.906	−	114.906i	0.314810	−	0.314810i
$366$		0			0
$367$	112.713			0.307121			0.153560	−	0.988139i	$-0.450926\pi$
$367$	112.713			0.307121			0.153560	+	0.988139i	$0.450926\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	342.298	−	342.298i	0.922636	−	0.922636i
$372$		0			0
$373$	32.7139	−	32.7139i	0.0877049	−	0.0877049i	−0.661893	−	0.749598i	$-0.730247\pi$
$373$	32.7139	−	32.7139i	0.0877049	−	0.0877049i	0.749598	+	0.661893i	$0.230247\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−768.238			−2.03777
$378$		0			0
$379$	−413.066	+	413.066i	−1.08988	+	1.08988i	−0.0943432	+	0.995540i	$0.530075\pi$
$379$	−413.066	+	413.066i	−1.08988	+	1.08988i	−0.995540	+	0.0943432i	$0.969925\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			616.895i			1.61069i	0.592805	+	0.805346i	$0.298020\pi$
$383$			616.895i			1.61069i	−0.592805	+	0.805346i	$0.701980\pi$
$384$		0			0
$385$	63.1393			0.163998
$386$		0			0
$387$		0			0
$388$		0			0
$389$	414.902	+	414.902i	1.06659	+	1.06659i	0.997619	+	0.0689668i	$0.0219703\pi$
$389$	414.902	+	414.902i	1.06659	+	1.06659i	0.0689668	+	0.997619i	$0.478030\pi$
$390$		0			0
$391$			0.797066i			0.00203853i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−33.3322	−	33.3322i	−0.0843854	−	0.0843854i
$396$		0			0
$397$	252.200	+	252.200i	0.635266	+	0.635266i	0.949384	−	0.314118i	$-0.101709\pi$
$397$	252.200	+	252.200i	0.635266	+	0.635266i	−0.314118	+	0.949384i	$0.601709\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			161.463i			0.402651i	0.979524	+	0.201325i	$0.0645249\pi$
$401$			161.463i			0.402651i	−0.979524	+	0.201325i	$0.935475\pi$
$402$		0			0
$403$	422.375	+	422.375i	1.04808	+	1.04808i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−104.219			−0.256067
$408$		0			0
$409$			156.176i			0.381847i	0.981605	+	0.190924i	$0.0611483\pi$
$409$			156.176i			0.381847i	−0.981605	+	0.190924i	$0.938852\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−159.127	+	159.127i	−0.385296	+	0.385296i
$414$		0			0
$415$	152.011			0.366291
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−530.843	+	530.843i	−1.26693	+	1.26693i	−0.319260	+	0.947667i	$0.603434\pi$
$419$	−530.843	+	530.843i	−1.26693	+	1.26693i	−0.947667	+	0.319260i	$0.896566\pi$
$420$		0			0
$421$	−198.004	+	198.004i	−0.470318	+	0.470318i	−0.902017	−	0.431700i	$-0.857914\pi$
$421$	−198.004	+	198.004i	−0.470318	+	0.470318i	0.431700	+	0.902017i	$0.357914\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.01753			−0.00239418
$426$		0			0
$427$	18.6024	−	18.6024i	0.0435654	−	0.0435654i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	418.584i		−	0.971192i	−0.874183	−	0.485596i	$-0.838603\pi$
$431$		−	418.584i		−	0.971192i	0.874183	−	0.485596i	$-0.161397\pi$
$432$		0			0
$433$	747.383			1.72606			0.863029	−	0.505154i	$-0.168564\pi$
$433$	747.383			1.72606			0.863029	+	0.505154i	$0.168564\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−237.803	−	237.803i	−0.544172	−	0.544172i
$438$		0			0
$439$		−	304.573i		−	0.693789i	−0.937904	−	0.346894i	$-0.887236\pi$
$439$		−	304.573i		−	0.693789i	0.937904	−	0.346894i	$-0.112764\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	479.719	+	479.719i	1.08289	+	1.08289i	0.996239	+	0.0866493i	$0.0276159\pi$
$443$	479.719	+	479.719i	1.08289	+	1.08289i	0.0866493	+	0.996239i	$0.472384\pi$
$444$		0			0
$445$	122.579	+	122.579i	0.275459	+	0.275459i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	485.831i		−	1.08203i	−0.841013	−	0.541015i	$-0.818040\pi$
$449$		−	485.831i		−	1.08203i	0.841013	−	0.541015i	$-0.181960\pi$
$450$		0			0
$451$	−209.055	−	209.055i	−0.463536	−	0.463536i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−151.546			−0.333069
$456$		0			0
$457$			413.632i			0.905102i	0.891738	+	0.452551i	$0.149486\pi$
$457$			413.632i			0.905102i	−0.891738	+	0.452551i	$0.850514\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−54.2195	+	54.2195i	−0.117613	+	0.117613i	−0.763464	−	0.645851i	$-0.776503\pi$
$461$	−54.2195	+	54.2195i	−0.117613	+	0.117613i	0.645851	+	0.763464i	$0.276503\pi$
$462$		0			0
$463$	135.028			0.291637			0.145819	−	0.989311i	$-0.453418\pi$
$463$	135.028			0.291637			0.145819	+	0.989311i	$0.453418\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	583.642	−	583.642i	1.24977	−	1.24977i	0.293947	−	0.955822i	$-0.405031\pi$
$467$	583.642	−	583.642i	1.24977	−	1.24977i	0.955822	−	0.293947i	$-0.0949689\pi$
$468$		0			0
$469$	−144.935	+	144.935i	−0.309030	+	0.309030i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−470.977			−0.995722
$474$		0			0
$475$	303.578	−	303.578i	0.639111	−	0.639111i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			51.0721i			0.106622i	0.998578	+	0.0533112i	$0.0169775\pi$
$479$			51.0721i			0.106622i	−0.998578	+	0.0533112i	$0.983022\pi$
$480$		0			0
$481$	250.146			0.520053
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−156.159	−	156.159i	−0.321977	−	0.321977i
$486$		0			0
$487$		−	490.285i		−	1.00674i	−0.864070	−	0.503372i	$-0.832092\pi$
$487$		−	490.285i		−	1.00674i	0.864070	−	0.503372i	$-0.167908\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	219.757	+	219.757i	0.447570	+	0.447570i	0.894546	−	0.446976i	$-0.147499\pi$
$491$	219.757	+	219.757i	0.447570	+	0.447570i	−0.446976	+	0.894546i	$0.647499\pi$
$492$		0			0
$493$	−1.52224	−	1.52224i	−0.00308770	−	0.00308770i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			880.446i			1.77152i
$498$		0			0
$499$	435.155	+	435.155i	0.872054	+	0.872054i	0.992696	−	0.120642i	$-0.0384955\pi$
$499$	435.155	+	435.155i	0.872054	+	0.872054i	−0.120642	+	0.992696i	$0.538495\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−46.8803			−0.0932014			−0.0466007	−	0.998914i	$-0.514839\pi$
$503$	−46.8803			−0.0932014			−0.0466007	+	0.998914i	$0.514839\pi$
$504$		0			0
$505$		−	33.1781i		−	0.0656993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−166.731	+	166.731i	−0.327566	+	0.327566i	−0.851660	−	0.524094i	$-0.824404\pi$
$509$	−166.731	+	166.731i	−0.327566	+	0.327566i	0.524094	+	0.851660i	$0.324404\pi$
$510$		0			0
$511$	−978.368			−1.91461
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−80.0236	+	80.0236i	−0.155386	+	0.155386i
$516$		0			0
$517$	258.849	−	258.849i	0.500676	−	0.500676i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	351.071			0.673840			0.336920	−	0.941533i	$-0.390615\pi$
$521$	351.071			0.673840			0.336920	+	0.941533i	$0.390615\pi$
$522$		0			0
$523$	214.780	−	214.780i	0.410670	−	0.410670i	−0.471302	−	0.881972i	$-0.656216\pi$
$523$	214.780	−	214.780i	0.410670	−	0.410670i	0.881972	+	0.471302i	$0.156216\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.67384i			0.00317617i
$528$		0			0
$529$	−193.601			−0.365976
$530$		0			0
$531$		0			0
$532$		0			0
$533$	501.771	+	501.771i	0.941409	+	0.941409i
$534$		0			0
$535$		−	97.8986i		−	0.182988i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−44.5945	−	44.5945i	−0.0827356	−	0.0827356i
$540$		0			0
$541$	−579.728	−	579.728i	−1.07159	−	1.07159i	−0.997232	−	0.0743548i	$-0.976310\pi$
$541$	−579.728	−	579.728i	−1.07159	−	1.07159i	−0.0743548	−	0.997232i	$-0.523690\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	219.679i		−	0.403080i
$546$		0			0
$547$	100.142	+	100.142i	0.183075	+	0.183075i	0.792694	−	0.609619i	$-0.208678\pi$
$547$	100.142	+	100.142i	0.183075	+	0.183075i	−0.609619	+	0.792694i	$0.708678\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	908.314			1.64848
$552$		0			0
$553$			283.808i			0.513215i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	644.771	−	644.771i	1.15758	−	1.15758i	0.172584	−	0.984995i	$-0.444788\pi$
$557$	644.771	−	644.771i	1.15758	−	1.15758i	0.984995	−	0.172584i	$-0.0552116\pi$
$558$		0			0
$559$	1130.43			2.02224
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−418.555	+	418.555i	−0.743436	+	0.743436i	−0.973238	−	0.229801i	$-0.926192\pi$
$563$	−418.555	+	418.555i	−0.743436	+	0.743436i	0.229801	+	0.973238i	$0.426192\pi$
$564$		0			0
$565$	146.391	−	146.391i	0.259098	−	0.259098i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	652.285			1.14637			0.573186	−	0.819426i	$-0.305707\pi$
$569$	652.285			1.14637			0.573186	+	0.819426i	$0.305707\pi$
$570$		0			0
$571$	13.6933	−	13.6933i	0.0239813	−	0.0239813i	−0.695014	−	0.718996i	$-0.744602\pi$
$571$	13.6933	−	13.6933i	0.0239813	−	0.0239813i	0.718996	+	0.695014i	$0.244602\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			428.167i			0.744638i
$576$		0			0
$577$	−269.997			−0.467932			−0.233966	−	0.972245i	$-0.575170\pi$
$577$	−269.997			−0.467932			−0.233966	+	0.972245i	$0.575170\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−647.150	−	647.150i	−1.11386	−	1.11386i
$582$		0			0
$583$			408.692i			0.701015i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−784.885	−	784.885i	−1.33711	−	1.33711i	−0.898844	−	0.438269i	$-0.855592\pi$
$587$	−784.885	−	784.885i	−1.33711	−	1.33711i	−0.438269	−	0.898844i	$-0.644408\pi$
$588$		0			0
$589$	−499.388	−	499.388i	−0.847857	−	0.847857i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	451.885i		−	0.762032i	−0.924569	−	0.381016i	$-0.875574\pi$
$593$		−	451.885i		−	0.762032i	0.924569	−	0.381016i	$-0.124426\pi$
$594$		0			0
$595$	−0.300284	−	0.300284i	−0.000504679	−	0.000504679i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−664.313			−1.10904			−0.554518	−	0.832171i	$-0.687098\pi$
$599$	−664.313			−1.10904			−0.554518	+	0.832171i	$0.687098\pi$
$600$		0			0
$601$		−	496.693i		−	0.826444i	−0.910630	−	0.413222i	$-0.864403\pi$
$601$		−	496.693i		−	0.826444i	0.910630	−	0.413222i	$-0.135597\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	71.2286	−	71.2286i	0.117733	−	0.117733i
$606$		0			0
$607$	−595.964			−0.981819			−0.490910	−	0.871210i	$-0.663335\pi$
$607$	−595.964			−0.981819			−0.490910	+	0.871210i	$0.663335\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−621.287	+	621.287i	−1.01684	+	1.01684i
$612$		0			0
$613$	−228.109	+	228.109i	−0.372119	+	0.372119i	−0.868249	−	0.496129i	$-0.834754\pi$
$613$	−228.109	+	228.109i	−0.372119	+	0.372119i	0.496129	+	0.868249i	$0.334754\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	199.680			0.323631			0.161815	−	0.986821i	$-0.448265\pi$
$617$	199.680			0.323631			0.161815	+	0.986821i	$0.448265\pi$
$618$		0			0
$619$	−8.72024	+	8.72024i	−0.0140876	+	0.0140876i	−0.714116	−	0.700028i	$-0.753171\pi$
$619$	−8.72024	+	8.72024i	−0.0140876	+	0.0140876i	0.700028	+	0.714116i	$0.253171\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1043.70i		−	1.67529i
$624$		0			0
$625$	−506.078			−0.809725
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.495655	+	0.495655i	0.000788005	+	0.000788005i
$630$		0			0
$631$			558.798i			0.885575i	0.896627	+	0.442788i	$0.146010\pi$
$631$			558.798i			0.885575i	−0.896627	+	0.442788i	$0.853990\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−32.6506	−	32.6506i	−0.0514182	−	0.0514182i
$636$		0			0
$637$	107.035	+	107.035i	0.168030	+	0.168030i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			549.401i			0.857100i	0.903518	+	0.428550i	$0.140975\pi$
$641$			549.401i			0.857100i	−0.903518	+	0.428550i	$0.859025\pi$
$642$		0			0
$643$	−615.401	−	615.401i	−0.957078	−	0.957078i	0.0420379	−	0.999116i	$-0.486615\pi$
$643$	−615.401	−	615.401i	−0.957078	−	0.957078i	−0.999116	+	0.0420379i	$0.986615\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1040.44			1.60810			0.804051	−	0.594560i	$-0.202674\pi$
$647$	1040.44			1.60810			0.804051	+	0.594560i	$0.202674\pi$
$648$		0			0
$649$		−	189.993i		−	0.292747i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−359.946	+	359.946i	−0.551218	+	0.551218i	−0.926792	−	0.375574i	$-0.877446\pi$
$653$	−359.946	+	359.946i	−0.551218	+	0.551218i	0.375574	+	0.926792i	$0.377446\pi$
$654$		0			0
$655$	−107.842			−0.164644
$656$		0			0
$657$		0			0
$658$		0			0
$659$	675.277	−	675.277i	1.02470	−	1.02470i	0.0250117	−	0.999687i	$-0.492038\pi$
$659$	675.277	−	675.277i	1.02470	−	1.02470i	0.999687	−	0.0250117i	$-0.00796231\pi$
$660$		0			0
$661$	118.754	−	118.754i	0.179659	−	0.179659i	−0.611548	−	0.791207i	$-0.709453\pi$
$661$	118.754	−	118.754i	0.179659	−	0.179659i	0.791207	+	0.611548i	$0.209453\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	179.178			0.269441
$666$		0			0
$667$	−640.545	+	640.545i	−0.960337	+	0.960337i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.2106i			0.0331008i
$672$		0			0
$673$	−226.637			−0.336757			−0.168378	−	0.985722i	$-0.553853\pi$
$673$	−226.637			−0.336757			−0.168378	+	0.985722i	$0.553853\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	529.591	+	529.591i	0.782261	+	0.782261i	0.980212	−	0.197951i	$-0.0634287\pi$
$677$	529.591	+	529.591i	0.782261	+	0.782261i	−0.197951	+	0.980212i	$0.563429\pi$
$678$		0			0
$679$			1329.62i			1.95820i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−582.521	−	582.521i	−0.852885	−	0.852885i	0.137602	−	0.990488i	$-0.456060\pi$
$683$	−582.521	−	582.521i	−0.852885	−	0.852885i	−0.990488	+	0.137602i	$0.956060\pi$
$684$		0			0
$685$	88.6783	+	88.6783i	0.129457	+	0.129457i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	980.938i		−	1.42371i
$690$		0			0
$691$	−670.134	−	670.134i	−0.969803	−	0.969803i	0.0297546	−	0.999557i	$-0.490527\pi$
$691$	−670.134	−	670.134i	−0.969803	−	0.969803i	−0.999557	+	0.0297546i	$0.990527\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.12050			0.00880647
$696$		0			0
$697$			1.98848i			0.00285292i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−291.428	+	291.428i	−0.415732	+	0.415732i	−0.883730	−	0.467998i	$-0.844976\pi$
$701$	−291.428	+	291.428i	−0.415732	+	0.415732i	0.467998	+	0.883730i	$0.344976\pi$
$702$		0			0
$703$	−295.756			−0.420705
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−141.248	+	141.248i	−0.199785	+	0.199785i
$708$		0			0
$709$	459.988	−	459.988i	0.648784	−	0.648784i	−0.303915	−	0.952699i	$-0.598294\pi$
$709$	459.988	−	459.988i	0.648784	−	0.648784i	0.952699	+	0.303915i	$0.0982940\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	704.338			0.987852
$714$		0			0
$715$	90.4706	−	90.4706i	0.126532	−	0.126532i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	285.232i		−	0.396707i	−0.980131	−	0.198353i	$-0.936441\pi$
$719$		−	285.232i		−	0.396707i	0.980131	−	0.198353i	$-0.0635594\pi$
$720$		0			0
$721$	681.363			0.945025
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−817.714	−	817.714i	−1.12788	−	1.12788i
$726$		0			0
$727$		−	114.350i		−	0.157290i	−0.996903	−	0.0786451i	$-0.974941\pi$
$727$		−	114.350i		−	0.157290i	0.996903	−	0.0786451i	$-0.0250594\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.23992	+	2.23992i	0.00306418	+	0.00306418i
$732$		0			0
$733$	158.553	+	158.553i	0.216307	+	0.216307i	0.806940	−	0.590633i	$-0.201122\pi$
$733$	158.553	+	158.553i	0.216307	+	0.216307i	−0.590633	+	0.806940i	$0.701122\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	173.047i		−	0.234800i
$738$		0			0
$739$	−381.275	−	381.275i	−0.515933	−	0.515933i	0.400405	−	0.916338i	$-0.368869\pi$
$739$	−381.275	−	381.275i	−0.515933	−	0.515933i	−0.916338	+	0.400405i	$0.868869\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−445.947			−0.600198			−0.300099	−	0.953908i	$-0.597020\pi$
$743$	−445.947			−0.600198			−0.300099	+	0.953908i	$0.597020\pi$
$744$		0			0
$745$			16.2603i			0.0218259i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−416.780	+	416.780i	−0.556449	+	0.556449i
$750$		0			0
$751$	−774.061			−1.03071			−0.515353	−	0.856978i	$-0.672339\pi$
$751$	−774.061			−1.03071			−0.515353	+	0.856978i	$0.672339\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−62.1617	+	62.1617i	−0.0823333	+	0.0823333i
$756$		0			0
$757$	431.310	−	431.310i	0.569763	−	0.569763i	−0.362299	−	0.932062i	$-0.618008\pi$
$757$	431.310	−	431.310i	0.569763	−	0.569763i	0.932062	+	0.362299i	$0.118008\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1493.05			1.96195			0.980977	−	0.194123i	$-0.0621861\pi$
$761$	1493.05			1.96195			0.980977	+	0.194123i	$0.0621861\pi$
$762$		0			0
$763$	−935.229	+	935.229i	−1.22573	+	1.22573i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			456.019i			0.594548i
$768$		0			0
$769$	−823.023			−1.07025			−0.535126	−	0.844772i	$-0.679736\pi$
$769$	−823.023			−1.07025			−0.535126	+	0.844772i	$0.679736\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−322.532	−	322.532i	−0.417247	−	0.417247i	0.467007	−	0.884254i	$-0.345332\pi$
$773$	−322.532	−	322.532i	−0.417247	−	0.417247i	−0.884254	+	0.467007i	$0.845332\pi$
$774$		0			0
$775$			899.152i			1.16020i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−593.261	−	593.261i	−0.761567	−	0.761567i
$780$		0			0
$781$	−525.611	−	525.611i	−0.672998	−	0.672998i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	114.696i		−	0.146109i
$786$		0			0
$787$	470.300	+	470.300i	0.597586	+	0.597586i	0.939670	−	0.342083i	$-0.111133\pi$
$787$	470.300	+	470.300i	0.597586	+	0.597586i	−0.342083	+	0.939670i	$0.611133\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1246.45			−1.57579
$792$		0			0
$793$		−	53.3098i		−	0.0672255i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	775.967	−	775.967i	0.973610	−	0.973610i	−0.0260505	−	0.999661i	$-0.508293\pi$
$797$	775.967	−	775.967i	0.973610	−	0.973610i	0.999661	+	0.0260505i	$0.00829308\pi$
$798$		0			0
$799$	−2.46212			−0.00308150
$800$		0			0
$801$		0			0
$802$		0			0
$803$	584.069	−	584.069i	0.727359	−	0.727359i
$804$		0			0
$805$	−126.357	+	126.357i	−0.156965	+	0.156965i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−812.614			−1.00447			−0.502234	−	0.864732i	$-0.667488\pi$
$809$	−812.614			−1.00447			−0.502234	+	0.864732i	$0.667488\pi$
$810$		0			0
$811$	229.002	−	229.002i	0.282369	−	0.282369i	−0.551684	−	0.834053i	$-0.686015\pi$
$811$	229.002	−	229.002i	0.282369	−	0.282369i	0.834053	+	0.551684i	$0.186015\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	190.171i		−	0.233338i
$816$		0			0
$817$	−1336.55			−1.63592
$818$		0			0
$819$		0			0
$820$		0			0
$821$	781.504	+	781.504i	0.951893	+	0.951893i	0.998895	−	0.0470014i	$-0.0149665\pi$
$821$	781.504	+	781.504i	0.951893	+	0.951893i	−0.0470014	+	0.998895i	$0.514967\pi$
$822$		0			0
$823$		−	757.548i		−	0.920472i	−0.887797	−	0.460236i	$-0.847765\pi$
$823$		−	757.548i		−	0.920472i	0.887797	−	0.460236i	$-0.152235\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−168.079	−	168.079i	−0.203240	−	0.203240i	0.598147	−	0.801387i	$-0.295904\pi$
$827$	−168.079	−	168.079i	−0.203240	−	0.203240i	−0.801387	+	0.598147i	$0.795904\pi$
$828$		0			0
$829$	−1044.13	−	1044.13i	−1.25950	−	1.25950i	−0.951331	−	0.308171i	$-0.900283\pi$
$829$	−1044.13	−	1044.13i	−1.25950	−	1.25950i	−0.308171	−	0.951331i	$-0.599717\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.424174i			0.000509212i
$834$		0			0
$835$	230.195	+	230.195i	0.275683	+	0.275683i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1052.57			1.25456			0.627278	−	0.778796i	$-0.284169\pi$
$839$	1052.57			1.25456			0.627278	+	0.778796i	$0.284169\pi$
$840$		0			0
$841$		−	1605.63i		−	1.90919i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−65.0162	+	65.0162i	−0.0769423	+	0.0769423i
$846$		0			0
$847$	−606.478			−0.716030
$848$		0			0
$849$		0			0
$850$		0			0
$851$	208.567	−	208.567i	0.245085	−	0.245085i
$852$		0			0
$853$	101.196	−	101.196i	0.118636	−	0.118636i	−0.645296	−	0.763932i	$-0.723266\pi$
$853$	101.196	−	101.196i	0.118636	−	0.118636i	0.763932	+	0.645296i	$0.223266\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	280.731			0.327574			0.163787	−	0.986496i	$-0.447629\pi$
$857$	280.731			0.327574			0.163787	+	0.986496i	$0.447629\pi$
$858$		0			0
$859$	−743.635	+	743.635i	−0.865699	+	0.865699i	−0.991993	−	0.126294i	$-0.959692\pi$
$859$	−743.635	+	743.635i	−0.865699	+	0.865699i	0.126294	+	0.991993i	$0.459692\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			98.7005i			0.114369i	0.998364	+	0.0571845i	$0.0182123\pi$
$863$			98.7005i			0.114369i	−0.998364	+	0.0571845i	$0.981788\pi$
$864$		0			0
$865$	−102.540			−0.118543
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−169.429	−	169.429i	−0.194970	−	0.194970i
$870$		0			0
$871$			415.347i			0.476862i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−333.794	−	333.794i	−0.381479	−	0.381479i
$876$		0			0
$877$	−552.841	−	552.841i	−0.630378	−	0.630378i	0.317785	−	0.948163i	$-0.397061\pi$
$877$	−552.841	−	552.841i	−0.630378	−	0.630378i	−0.948163	+	0.317785i	$0.897061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	316.248i		−	0.358965i	−0.983761	−	0.179482i	$-0.942558\pi$
$881$		−	316.248i		−	0.358965i	0.983761	−	0.179482i	$-0.0574423\pi$
$882$		0			0
$883$	−98.6469	−	98.6469i	−0.111718	−	0.111718i	0.649038	−	0.760756i	$-0.275172\pi$
$883$	−98.6469	−	98.6469i	−0.111718	−	0.111718i	−0.760756	+	0.649038i	$0.775172\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1426.43			−1.60815			−0.804074	−	0.594530i	$-0.797338\pi$
$887$	−1426.43			−1.60815			−0.804074	+	0.594530i	$0.797338\pi$
$888$		0			0
$889$			278.004i			0.312715i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	734.570	−	734.570i	0.822586	−	0.822586i
$894$		0			0
$895$	−309.045			−0.345302
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1345.15	+	1345.15i	−1.49627	+	1.49627i
$900$		0			0
$901$	1.94370	−	1.94370i	0.00215727	−	0.00215727i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−99.9442			−0.110436
$906$		0			0
$907$	238.634	−	238.634i	0.263103	−	0.263103i	−0.563211	−	0.826313i	$-0.690434\pi$
$907$	238.634	−	238.634i	0.263103	−	0.263103i	0.826313	+	0.563211i	$0.190434\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1309.38i		−	1.43730i	−0.695374	−	0.718648i	$-0.744761\pi$
$911$		−	1309.38i		−	1.43730i	0.695374	−	0.718648i	$-0.255239\pi$
$912$		0			0
$913$	772.676			0.846304
$914$		0			0
$915$		0			0
$916$		0			0
$917$	459.112	+	459.112i	0.500667	+	0.500667i
$918$		0			0
$919$		−	511.000i		−	0.556039i	−0.960575	−	0.278019i	$-0.910322\pi$
$919$		−	511.000i		−	0.556039i	0.960575	−	0.278019i	$-0.0896779\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1261.57	+	1261.57i	1.36681	+	1.36681i
$924$		0			0
$925$	266.255	+	266.255i	0.287844	+	0.287844i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	386.394i		−	0.415924i	−0.978137	−	0.207962i	$-0.933317\pi$
$929$		−	386.394i		−	0.415924i	0.978137	−	0.207962i	$-0.0666831\pi$
$930$		0			0
$931$	−126.551	−	126.551i	−0.135931	−	0.135931i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.358529			0.000383453
$936$		0			0
$937$		−	927.440i		−	0.989797i	−0.868951	−	0.494898i	$-0.835205\pi$
$937$		−	927.440i		−	0.989797i	0.868951	−	0.494898i	$-0.164795\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	645.746	−	645.746i	0.686234	−	0.686234i	−0.275164	−	0.961397i	$-0.588732\pi$
$941$	645.746	−	645.746i	0.686234	−	0.686234i	0.961397	+	0.275164i	$0.0887320\pi$
$942$		0			0
$943$	836.737			0.887314
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1017.28	−	1017.28i	1.07421	−	1.07421i	0.0771972	−	0.997016i	$-0.475403\pi$
$947$	1017.28	−	1017.28i	1.07421	−	1.07421i	0.997016	−	0.0771972i	$-0.0245971\pi$
$948$		0			0
$949$	−1401.88	+	1401.88i	−1.47721	+	1.47721i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−214.062			−0.224619			−0.112309	−	0.993673i	$-0.535825\pi$
$953$	−214.062			−0.224619			−0.112309	+	0.993673i	$0.535825\pi$
$954$		0			0
$955$	91.7708	−	91.7708i	0.0960951	−	0.0960951i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	755.054i		−	0.787334i
$960$		0			0
$961$	518.114			0.539140
$962$		0			0
$963$		0			0
$964$		0			0
$965$	73.3653	+	73.3653i	0.0760262	+	0.0760262i
$966$		0			0
$967$		−	440.861i		−	0.455906i	−0.973672	−	0.227953i	$-0.926797\pi$
$967$		−	440.861i		−	0.455906i	0.973672	−	0.227953i	$-0.0732033\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−343.159	−	343.159i	−0.353407	−	0.353407i	0.507968	−	0.861376i	$-0.330397\pi$
$971$	−343.159	−	343.159i	−0.353407	−	0.353407i	−0.861376	+	0.507968i	$0.830397\pi$
$972$		0			0
$973$	−26.0566	−	26.0566i	−0.0267796	−	0.0267796i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			388.323i			0.397465i	0.980054	+	0.198733i	$0.0636825\pi$
$977$			388.323i			0.397465i	−0.980054	+	0.198733i	$0.936317\pi$
$978$		0			0
$979$	623.073	+	623.073i	0.636439	+	0.636439i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1344.22			1.36746			0.683731	−	0.729734i	$-0.260356\pi$
$983$	1344.22			1.36746			0.683731	+	0.729734i	$0.260356\pi$
$984$		0			0
$985$		−	307.279i		−	0.311958i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	942.538	−	942.538i	0.953021	−	0.953021i
$990$		0			0
$991$	−1058.38			−1.06799			−0.533997	−	0.845486i	$-0.679311\pi$
$991$	−1058.38			−1.06799			−0.533997	+	0.845486i	$0.679311\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−168.716	+	168.716i	−0.169563	+	0.169563i
$996$		0			0
$997$	554.656	−	554.656i	0.556325	−	0.556325i	−0.371934	−	0.928259i	$-0.621305\pi$
$997$	554.656	−	554.656i	0.556325	−	0.556325i	0.928259	+	0.371934i	$0.121305\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.b.161.9		32
3.2	odd	2	inner	1152.3.j.b.161.8		32
4.3	odd	2		1152.3.j.a.161.9		32
8.3	odd	2		144.3.j.a.125.6	yes	32
8.5	even	2		576.3.j.a.17.8		32
12.11	even	2		1152.3.j.a.161.8		32
16.3	odd	4		144.3.j.a.53.11	yes	32
16.5	even	4	inner	1152.3.j.b.737.8		32
16.11	odd	4		1152.3.j.a.737.8		32
16.13	even	4		576.3.j.a.305.9		32
24.5	odd	2		576.3.j.a.17.9		32
24.11	even	2		144.3.j.a.125.11	yes	32
48.5	odd	4	inner	1152.3.j.b.737.9		32
48.11	even	4		1152.3.j.a.737.9		32
48.29	odd	4		576.3.j.a.305.8		32
48.35	even	4		144.3.j.a.53.6	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.6	✓	32	48.35	even	4
144.3.j.a.53.11	yes	32	16.3	odd	4
144.3.j.a.125.6	yes	32	8.3	odd	2
144.3.j.a.125.11	yes	32	24.11	even	2
576.3.j.a.17.8		32	8.5	even	2
576.3.j.a.17.9		32	24.5	odd	2
576.3.j.a.305.8		32	48.29	odd	4
576.3.j.a.305.9		32	16.13	even	4
1152.3.j.a.161.8		32	12.11	even	2
1152.3.j.a.161.9		32	4.3	odd	2
1152.3.j.a.737.8		32	16.11	odd	4
1152.3.j.a.737.9		32	48.11	even	4
1152.3.j.b.161.8		32	3.2	odd	2	inner
1152.3.j.b.161.9		32	1.1	even	1	trivial
1152.3.j.b.737.8		32	16.5	even	4	inner
1152.3.j.b.737.9		32	48.5	odd	4	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.900179 + 0.900179i) q^{5} -7.66460i q^{7} +O(q^{10})\)	\(q+(0.900179 + 0.900179i) q^{5} -7.66460i q^{7} +(4.57563 + 4.57563i) q^{11} +(-10.9824 - 10.9824i) q^{13} -0.0435225i q^{17} +(12.9849 + 12.9849i) q^{19} -18.3139 q^{23} -23.3794i q^{25} +(34.9759 - 34.9759i) q^{29} -38.4592 q^{31} +(6.89951 - 6.89951i) q^{35} +(-11.3885 + 11.3885i) q^{37} -45.6887 q^{41} +(-51.4657 + 51.4657i) q^{43} -56.5712i q^{47} -9.74608 q^{49} +(44.6596 + 44.6596i) q^{53} +8.23778i q^{55} +(-20.7614 - 20.7614i) q^{59} +(2.42706 + 2.42706i) q^{61} -19.7722i q^{65} +(-18.9097 - 18.9097i) q^{67} -114.872 q^{71} -127.648i q^{73} +(35.0704 - 35.0704i) q^{77} -37.0284 q^{79} +(84.4337 - 84.4337i) q^{83} +(0.0391780 - 0.0391780i) q^{85} +136.172 q^{89} +(-84.1756 + 84.1756i) q^{91} +23.3774i q^{95} -173.475 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) \) 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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