LMFDB - Embedded newform 1152.3.j.a.161.2

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			161.2
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.a.737.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.52702 - 5.52702i) q^{5} -7.79421i q^{7} +O(q^{10})$	$q+(-5.52702 - 5.52702i) q^{5} -7.79421i q^{7} +(2.48521 + 2.48521i) q^{11} +(-13.3242 - 13.3242i) q^{13} -5.86338i q^{17} +(-18.5712 - 18.5712i) q^{19} +34.3136 q^{23} +36.0959i q^{25} +(21.4872 - 21.4872i) q^{29} -30.6117 q^{31} +(-43.0788 + 43.0788i) q^{35} +(-30.3274 + 30.3274i) q^{37} -3.12709 q^{41} +(9.94981 - 9.94981i) q^{43} -38.4052i q^{47} -11.7497 q^{49} +(61.1826 + 61.1826i) q^{53} -27.4717i q^{55} +(2.98892 + 2.98892i) q^{59} +(-3.88659 - 3.88659i) q^{61} +147.287i q^{65} +(47.0242 + 47.0242i) q^{67} -97.5416 q^{71} +106.904i q^{73} +(19.3703 - 19.3703i) q^{77} -96.8255 q^{79} +(-88.8242 + 88.8242i) q^{83} +(-32.4070 + 32.4070i) q^{85} +54.8651 q^{89} +(-103.852 + 103.852i) q^{91} +205.287i q^{95} -5.00194 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5.52702	−	5.52702i	−1.10540	−	1.10540i	−0.993747	−	0.111658i	$-0.964384\pi$
$5$	−5.52702	−	5.52702i	−1.10540	−	1.10540i	−0.111658	−	0.993747i	$-0.535616\pi$
$6$		0			0
$7$		−	7.79421i		−	1.11346i	−0.830694	−	0.556729i	$-0.812056\pi$
$7$		−	7.79421i		−	1.11346i	0.830694	−	0.556729i	$-0.187944\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.48521	+	2.48521i	0.225929	+	0.225929i	0.810989	−	0.585061i	$-0.198929\pi$
$11$	2.48521	+	2.48521i	0.225929	+	0.225929i	−0.585061	+	0.810989i	$0.698929\pi$
$12$		0			0
$13$	−13.3242	−	13.3242i	−1.02494	−	1.02494i	−0.999681	−	0.0252590i	$-0.991959\pi$
$13$	−13.3242	−	13.3242i	−1.02494	−	1.02494i	−0.0252590	−	0.999681i	$-0.508041\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.86338i		−	0.344905i	−0.985018	−	0.172452i	$-0.944831\pi$
$17$		−	5.86338i		−	0.344905i	0.985018	−	0.172452i	$-0.0551691\pi$
$18$		0			0
$19$	−18.5712	−	18.5712i	−0.977433	−	0.977433i	0.0223177	−	0.999751i	$-0.492895\pi$
$19$	−18.5712	−	18.5712i	−0.977433	−	0.977433i	−0.999751	+	0.0223177i	$0.992895\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	34.3136			1.49190			0.745948	−	0.666004i	$-0.231997\pi$
$23$	34.3136			1.49190			0.745948	+	0.666004i	$0.231997\pi$
$24$		0			0
$25$			36.0959i			1.44384i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	21.4872	−	21.4872i	0.740939	−	0.740939i	−0.231820	−	0.972759i	$-0.574468\pi$
$29$	21.4872	−	21.4872i	0.740939	−	0.740939i	0.972759	+	0.231820i	$0.0744679\pi$
$30$		0			0
$31$	−30.6117			−0.987473			−0.493737	−	0.869612i	$-0.664369\pi$
$31$	−30.6117			−0.987473			−0.493737	+	0.869612i	$0.664369\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−43.0788	+	43.0788i	−1.23082	+	1.23082i
$36$		0			0
$37$	−30.3274	+	30.3274i	−0.819661	+	0.819661i	−0.986059	−	0.166398i	$-0.946786\pi$
$37$	−30.3274	+	30.3274i	−0.819661	+	0.819661i	0.166398	+	0.986059i	$0.446786\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.12709			−0.0762704			−0.0381352	−	0.999273i	$-0.512142\pi$
$41$	−3.12709			−0.0762704			−0.0381352	+	0.999273i	$0.512142\pi$
$42$		0			0
$43$	9.94981	−	9.94981i	0.231391	−	0.231391i	−0.581882	−	0.813273i	$-0.697684\pi$
$43$	9.94981	−	9.94981i	0.231391	−	0.231391i	0.813273	+	0.581882i	$0.197684\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	38.4052i		−	0.817132i	−0.912729	−	0.408566i	$-0.866029\pi$
$47$		−	38.4052i		−	0.817132i	0.912729	−	0.408566i	$-0.133971\pi$
$48$		0			0
$49$	−11.7497			−0.239790
$50$		0			0
$51$		0			0
$52$		0			0
$53$	61.1826	+	61.1826i	1.15439	+	1.15439i	0.985663	+	0.168725i	$0.0539651\pi$
$53$	61.1826	+	61.1826i	1.15439	+	1.15439i	0.168725	+	0.985663i	$0.446035\pi$
$54$		0			0
$55$		−	27.4717i		−	0.499485i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.98892	+	2.98892i	0.0506596	+	0.0506596i	0.731983	−	0.681323i	$-0.238595\pi$
$59$	2.98892	+	2.98892i	0.0506596	+	0.0506596i	−0.681323	+	0.731983i	$0.738595\pi$
$60$		0			0
$61$	−3.88659	−	3.88659i	−0.0637146	−	0.0637146i	0.674531	−	0.738246i	$-0.264346\pi$
$61$	−3.88659	−	3.88659i	−0.0637146	−	0.0637146i	−0.738246	+	0.674531i	$0.764346\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			147.287i			2.26595i
$66$		0			0
$67$	47.0242	+	47.0242i	0.701853	+	0.701853i	0.964808	−	0.262955i	$-0.0846970\pi$
$67$	47.0242	+	47.0242i	0.701853	+	0.701853i	−0.262955	+	0.964808i	$0.584697\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−97.5416			−1.37383			−0.686913	−	0.726740i	$-0.741035\pi$
$71$	−97.5416			−1.37383			−0.686913	+	0.726740i	$0.741035\pi$
$72$		0			0
$73$			106.904i			1.46444i	0.681069	+	0.732219i	$0.261515\pi$
$73$			106.904i			1.46444i	−0.681069	+	0.732219i	$0.738485\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.3703	−	19.3703i	0.251562	−	0.251562i
$78$		0			0
$79$	−96.8255			−1.22564			−0.612820	−	0.790223i	$-0.709965\pi$
$79$	−96.8255			−1.22564			−0.612820	+	0.790223i	$0.709965\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−88.8242	+	88.8242i	−1.07017	+	1.07017i	−0.0728268	+	0.997345i	$0.523202\pi$
$83$	−88.8242	+	88.8242i	−1.07017	+	1.07017i	−0.997345	+	0.0728268i	$0.976798\pi$
$84$		0			0
$85$	−32.4070	+	32.4070i	−0.381259	+	0.381259i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	54.8651			0.616462			0.308231	−	0.951311i	$-0.400263\pi$
$89$	54.8651			0.616462			0.308231	+	0.951311i	$0.400263\pi$
$90$		0			0
$91$	−103.852	+	103.852i	−1.14123	+	1.14123i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			205.287i			2.16092i
$96$		0			0
$97$	−5.00194			−0.0515664			−0.0257832	−	0.999668i	$-0.508208\pi$
$97$	−5.00194			−0.0515664			−0.0257832	+	0.999668i	$0.508208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	37.2481	+	37.2481i	0.368793	+	0.368793i	0.867037	−	0.498244i	$-0.166022\pi$
$101$	37.2481	+	37.2481i	0.368793	+	0.368793i	−0.498244	+	0.867037i	$0.666022\pi$
$102$		0			0
$103$		−	188.028i		−	1.82551i	−0.408505	−	0.912756i	$-0.633950\pi$
$103$		−	188.028i		−	1.82551i	0.408505	−	0.912756i	$-0.366050\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−35.2253	−	35.2253i	−0.329208	−	0.329208i	0.523077	−	0.852285i	$-0.324784\pi$
$107$	−35.2253	−	35.2253i	−0.329208	−	0.329208i	−0.852285	+	0.523077i	$0.824784\pi$
$108$		0			0
$109$	93.0073	+	93.0073i	0.853278	+	0.853278i	0.990535	−	0.137257i	$-0.0438287\pi$
$109$	93.0073	+	93.0073i	0.853278	+	0.853278i	−0.137257	+	0.990535i	$0.543829\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.80833i			0.0337021i	0.999858	+	0.0168510i	$0.00536410\pi$
$113$			3.80833i			0.0337021i	−0.999858	+	0.0168510i	$0.994636\pi$
$114$		0			0
$115$	−189.652	−	189.652i	−1.64915	−	1.64915i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−45.7004			−0.384037
$120$		0			0
$121$		−	108.647i		−	0.897913i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	61.3275	−	61.3275i	0.490620	−	0.490620i
$126$		0			0
$127$	−173.704			−1.36775			−0.683875	−	0.729600i	$-0.739706\pi$
$127$	−173.704			−1.36775			−0.683875	+	0.729600i	$0.739706\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	59.8594	−	59.8594i	0.456942	−	0.456942i	−0.440708	−	0.897650i	$-0.645273\pi$
$131$	59.8594	−	59.8594i	0.456942	−	0.456942i	0.897650	+	0.440708i	$0.145273\pi$
$132$		0			0
$133$	−144.748	+	144.748i	−1.08833	+	1.08833i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.0602			−0.0953299			−0.0476649	−	0.998863i	$-0.515178\pi$
$137$	−13.0602			−0.0953299			−0.0476649	+	0.998863i	$0.515178\pi$
$138$		0			0
$139$	132.497	−	132.497i	0.953219	−	0.953219i	−0.0457344	−	0.998954i	$-0.514563\pi$
$139$	132.497	−	132.497i	0.953219	−	0.953219i	0.998954	+	0.0457344i	$0.0145628\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	66.2271i		−	0.463126i
$144$		0			0
$145$	−237.521			−1.63807
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.4768	−	10.4768i	−0.0703142	−	0.0703142i	0.671075	−	0.741389i	$-0.265833\pi$
$149$	−10.4768	−	10.4768i	−0.0703142	−	0.0703142i	−0.741389	+	0.671075i	$0.765833\pi$
$150$		0			0
$151$			181.207i			1.20005i	0.799982	+	0.600024i	$0.204842\pi$
$151$			181.207i			1.20005i	−0.799982	+	0.600024i	$0.795158\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	169.191	+	169.191i	1.09156	+	1.09156i
$156$		0			0
$157$	−57.9518	−	57.9518i	−0.369120	−	0.369120i	0.498036	−	0.867156i	$-0.334055\pi$
$157$	−57.9518	−	57.9518i	−0.369120	−	0.369120i	−0.867156	+	0.498036i	$0.834055\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	267.447i		−	1.66116i
$162$		0			0
$163$	−87.8044	−	87.8044i	−0.538677	−	0.538677i	0.384463	−	0.923140i	$-0.374387\pi$
$163$	−87.8044	−	87.8044i	−0.538677	−	0.538677i	−0.923140	+	0.384463i	$0.874387\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	86.9131			0.520438			0.260219	−	0.965550i	$-0.416205\pi$
$167$	86.9131			0.520438			0.260219	+	0.965550i	$0.416205\pi$
$168$		0			0
$169$			186.070i			1.10100i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−216.007	+	216.007i	−1.24860	+	1.24860i	−0.292258	+	0.956339i	$0.594407\pi$
$173$	−216.007	+	216.007i	−1.24860	+	1.24860i	−0.956339	+	0.292258i	$0.905593\pi$
$174$		0			0
$175$	281.339			1.60765
$176$		0			0
$177$		0			0
$178$		0			0
$179$	86.1380	−	86.1380i	0.481218	−	0.481218i	−0.424303	−	0.905520i	$-0.639481\pi$
$179$	86.1380	−	86.1380i	0.481218	−	0.481218i	0.905520	+	0.424303i	$0.139481\pi$
$180$		0			0
$181$	−1.84265	+	1.84265i	−0.0101804	+	0.0101804i	−0.712179	−	0.701998i	$-0.752291\pi$
$181$	−1.84265	+	1.84265i	−0.0101804	+	0.0101804i	0.701998	+	0.712179i	$0.252291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	335.241			1.81211
$186$		0			0
$187$	14.5718	−	14.5718i	0.0779238	−	0.0779238i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			82.0328i			0.429491i	0.976670	+	0.214746i	$0.0688922\pi$
$191$			82.0328i			0.429491i	−0.976670	+	0.214746i	$0.931108\pi$
$192$		0			0
$193$	−91.2193			−0.472639			−0.236319	−	0.971675i	$-0.575941\pi$
$193$	−91.2193			−0.472639			−0.236319	+	0.971675i	$0.575941\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.74282	−	8.74282i	−0.0443798	−	0.0443798i	0.684569	−	0.728948i	$-0.259991\pi$
$197$	−8.74282	−	8.74282i	−0.0443798	−	0.0443798i	−0.728948	+	0.684569i	$0.759991\pi$
$198$		0			0
$199$		−	128.824i		−	0.647357i	−0.946167	−	0.323679i	$-0.895080\pi$
$199$		−	128.824i		−	0.647357i	0.946167	−	0.323679i	$-0.104920\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−167.476	−	167.476i	−0.825005	−	0.825005i
$204$		0			0
$205$	17.2835	+	17.2835i	0.0843096	+	0.0843096i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	92.3070i		−	0.441660i
$210$		0			0
$211$	−96.2826	−	96.2826i	−0.456316	−	0.456316i	0.441128	−	0.897444i	$-0.354578\pi$
$211$	−96.2826	−	96.2826i	−0.456316	−	0.456316i	−0.897444	+	0.441128i	$0.854578\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−109.986			−0.511561
$216$		0			0
$217$			238.594i			1.09951i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−78.1250	+	78.1250i	−0.353507	+	0.353507i
$222$		0			0
$223$	−15.9317			−0.0714424			−0.0357212	−	0.999362i	$-0.511373\pi$
$223$	−15.9317			−0.0714424			−0.0357212	+	0.999362i	$0.511373\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	66.7298	−	66.7298i	0.293964	−	0.293964i	−0.544680	−	0.838644i	$-0.683349\pi$
$227$	66.7298	−	66.7298i	0.293964	−	0.293964i	0.838644	+	0.544680i	$0.183349\pi$
$228$		0			0
$229$	109.514	−	109.514i	0.478229	−	0.478229i	−0.426336	−	0.904565i	$-0.640196\pi$
$229$	109.514	−	109.514i	0.478229	−	0.478229i	0.904565	+	0.426336i	$0.140196\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	85.9062			0.368696			0.184348	−	0.982861i	$-0.440983\pi$
$233$	85.9062			0.368696			0.184348	+	0.982861i	$0.440983\pi$
$234$		0			0
$235$	−212.266	+	212.266i	−0.903261	+	0.903261i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			47.0157i			0.196718i	0.995151	+	0.0983592i	$0.0313594\pi$
$239$			47.0157i			0.196718i	−0.995151	+	0.0983592i	$0.968641\pi$
$240$		0			0
$241$	312.627			1.29721			0.648603	−	0.761127i	$-0.275354\pi$
$241$	312.627			1.29721			0.648603	+	0.761127i	$0.275354\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	64.9408	+	64.9408i	0.265064	+	0.265064i
$246$		0			0
$247$			494.894i			2.00362i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	125.339	+	125.339i	0.499360	+	0.499360i	0.911239	−	0.411878i	$-0.135127\pi$
$251$	125.339	+	125.339i	0.499360	+	0.499360i	−0.411878	+	0.911239i	$0.635127\pi$
$252$		0			0
$253$	85.2767	+	85.2767i	0.337062	+	0.337062i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			341.153i			1.32744i	0.747980	+	0.663722i	$0.231024\pi$
$257$			341.153i			1.32744i	−0.747980	+	0.663722i	$0.768976\pi$
$258$		0			0
$259$	236.378	+	236.378i	0.912658	+	0.912658i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−161.903			−0.615599			−0.307799	−	0.951451i	$-0.599593\pi$
$263$	−161.903			−0.615599			−0.307799	+	0.951451i	$0.599593\pi$
$264$		0			0
$265$		−	676.315i		−	2.55213i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	65.6610	−	65.6610i	0.244093	−	0.244093i	−0.574448	−	0.818541i	$-0.694783\pi$
$269$	65.6610	−	65.6610i	0.244093	−	0.244093i	0.818541	+	0.574448i	$0.194783\pi$
$270$		0			0
$271$	200.090			0.738339			0.369170	−	0.929362i	$-0.379642\pi$
$271$	200.090			0.738339			0.369170	+	0.929362i	$0.379642\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−89.7061	+	89.7061i	−0.326204	+	0.326204i
$276$		0			0
$277$	70.7882	−	70.7882i	0.255553	−	0.255553i	−0.567690	−	0.823243i	$-0.692163\pi$
$277$	70.7882	−	70.7882i	0.255553	−	0.255553i	0.823243	+	0.567690i	$0.192163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−448.021			−1.59438			−0.797190	−	0.603729i	$-0.793681\pi$
$281$	−448.021			−1.59438			−0.797190	+	0.603729i	$0.793681\pi$
$282$		0			0
$283$	187.254	−	187.254i	0.661676	−	0.661676i	−0.294099	−	0.955775i	$-0.595020\pi$
$283$	187.254	−	187.254i	0.661676	−	0.661676i	0.955775	+	0.294099i	$0.0950196\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			24.3732i			0.0849239i
$288$		0			0
$289$	254.621			0.881041
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−200.350	−	200.350i	−0.683788	−	0.683788i	0.277063	−	0.960852i	$-0.410639\pi$
$293$	−200.350	−	200.350i	−0.683788	−	0.683788i	−0.960852	+	0.277063i	$0.910639\pi$
$294$		0			0
$295$		−	33.0396i		−	0.111999i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−457.202	−	457.202i	−1.52910	−	1.52910i
$300$		0			0
$301$	−77.5509	−	77.5509i	−0.257644	−	0.257644i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			42.9625i			0.140861i
$306$		0			0
$307$	118.542	+	118.542i	0.386130	+	0.386130i	0.873305	−	0.487174i	$-0.161972\pi$
$307$	118.542	+	118.542i	0.386130	+	0.386130i	−0.487174	+	0.873305i	$0.661972\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.29491			0.0105946			0.00529729	−	0.999986i	$-0.498314\pi$
$311$	3.29491			0.0105946			0.00529729	+	0.999986i	$0.498314\pi$
$312$		0			0
$313$		−	492.839i		−	1.57457i	−0.616592	−	0.787283i	$-0.711487\pi$
$313$		−	492.839i		−	1.57457i	0.616592	−	0.787283i	$-0.288513\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−118.311	+	118.311i	−0.373222	+	0.373222i	−0.868649	−	0.495427i	$-0.835012\pi$
$317$	−118.311	+	118.311i	−0.373222	+	0.373222i	0.495427	+	0.868649i	$0.335012\pi$
$318$		0			0
$319$	106.801			0.334799
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−108.890	+	108.890i	−0.337121	+	0.337121i
$324$		0			0
$325$	480.950	−	480.950i	1.47985	−	1.47985i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−299.338			−0.909843
$330$		0			0
$331$	52.2818	−	52.2818i	0.157951	−	0.157951i	−0.623707	−	0.781658i	$-0.714374\pi$
$331$	52.2818	−	52.2818i	0.157951	−	0.157951i	0.781658	+	0.623707i	$0.214374\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	519.807i		−	1.55166i
$336$		0			0
$337$	104.504			0.310100			0.155050	−	0.987907i	$-0.450446\pi$
$337$	104.504			0.310100			0.155050	+	0.987907i	$0.450446\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−76.0765	−	76.0765i	−0.223098	−	0.223098i
$342$		0			0
$343$		−	290.337i		−	0.846463i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	312.769	+	312.769i	0.901351	+	0.901351i	0.995553	−	0.0942021i	$-0.0300300\pi$
$347$	312.769	+	312.769i	0.901351	+	0.901351i	−0.0942021	+	0.995553i	$0.530030\pi$
$348$		0			0
$349$	114.172	+	114.172i	0.327141	+	0.327141i	0.851498	−	0.524357i	$-0.175694\pi$
$349$	114.172	+	114.172i	0.327141	+	0.327141i	−0.524357	+	0.851498i	$0.675694\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	312.848i		−	0.886255i	−0.896459	−	0.443127i	$-0.853869\pi$
$353$		−	312.848i		−	0.886255i	0.896459	−	0.443127i	$-0.146131\pi$
$354$		0			0
$355$	539.115	+	539.115i	1.51863	+	1.51863i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	670.421			1.86747			0.933734	−	0.357967i	$-0.116530\pi$
$359$	670.421			1.86747			0.933734	+	0.357967i	$0.116530\pi$
$360$		0			0
$361$			328.781i			0.910751i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	590.861	−	590.861i	1.61880	−	1.61880i
$366$		0			0
$367$	−531.445			−1.44808			−0.724039	−	0.689759i	$-0.757717\pi$
$367$	−531.445			−1.44808			−0.724039	+	0.689759i	$0.757717\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	476.870	−	476.870i	1.28536	−	1.28536i
$372$		0			0
$373$	−25.9291	+	25.9291i	−0.0695151	+	0.0695151i	−0.741010	−	0.671494i	$-0.765653\pi$
$373$	−25.9291	+	25.9291i	−0.0695151	+	0.0695151i	0.671494	+	0.741010i	$0.265653\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−572.601			−1.51884
$378$		0			0
$379$	−109.268	+	109.268i	−0.288307	+	0.288307i	−0.836411	−	0.548103i	$-0.815350\pi$
$379$	−109.268	+	109.268i	−0.288307	+	0.288307i	0.548103	+	0.836411i	$0.315350\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	93.3821i		−	0.243818i	−0.992541	−	0.121909i	$-0.961098\pi$
$383$		−	93.3821i		−	0.243818i	0.992541	−	0.121909i	$-0.0389015\pi$
$384$		0			0
$385$	−214.120			−0.556156
$386$		0			0
$387$		0			0
$388$		0			0
$389$	66.8268	+	66.8268i	0.171791	+	0.171791i	0.787766	−	0.615975i	$-0.211238\pi$
$389$	66.8268	+	66.8268i	0.171791	+	0.171791i	−0.615975	+	0.787766i	$0.711238\pi$
$390$		0			0
$391$		−	201.194i		−	0.514562i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	535.157	+	535.157i	1.35483	+	1.35483i
$396$		0			0
$397$	−329.309	−	329.309i	−0.829494	−	0.829494i	0.157953	−	0.987447i	$-0.449511\pi$
$397$	−329.309	−	329.309i	−0.829494	−	0.829494i	−0.987447	+	0.157953i	$0.949511\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	129.336i		−	0.322534i	−0.986911	−	0.161267i	$-0.948442\pi$
$401$		−	129.336i		−	0.322534i	0.986911	−	0.161267i	$-0.0515581\pi$
$402$		0			0
$403$	407.877	+	407.877i	1.01210	+	1.01210i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−150.740			−0.370369
$408$		0			0
$409$			567.479i			1.38748i	0.720226	+	0.693739i	$0.244038\pi$
$409$			567.479i			1.38748i	−0.720226	+	0.693739i	$0.755962\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	23.2962	−	23.2962i	0.0564074	−	0.0564074i
$414$		0			0
$415$	981.867			2.36594
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.9979	+	11.9979i	−0.0286346	+	0.0286346i	−0.721279	−	0.692645i	$-0.756445\pi$
$419$	−11.9979	+	11.9979i	−0.0286346	+	0.0286346i	0.692645	+	0.721279i	$0.256445\pi$
$420$		0			0
$421$	−301.310	+	301.310i	−0.715700	+	0.715700i	−0.967722	−	0.252022i	$-0.918905\pi$
$421$	−301.310	+	301.310i	−0.715700	+	0.715700i	0.252022	+	0.967722i	$0.418905\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	211.644			0.497987
$426$		0			0
$427$	−30.2929	+	30.2929i	−0.0709436	+	0.0709436i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	700.273i		−	1.62476i	−0.583126	−	0.812382i	$-0.698171\pi$
$431$		−	700.273i		−	1.62476i	0.583126	−	0.812382i	$-0.301829\pi$
$432$		0			0
$433$	−93.2906			−0.215452			−0.107726	−	0.994181i	$-0.534357\pi$
$433$	−93.2906			−0.215452			−0.107726	+	0.994181i	$0.534357\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−637.246	−	637.246i	−1.45823	−	1.45823i
$438$		0			0
$439$			98.5794i			0.224554i	0.993677	+	0.112277i	$0.0358145\pi$
$439$			98.5794i			0.224554i	−0.993677	+	0.112277i	$0.964186\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−574.790	−	574.790i	−1.29749	−	1.29749i	−0.930043	−	0.367450i	$-0.880231\pi$
$443$	−574.790	−	574.790i	−1.29749	−	1.29749i	−0.367450	−	0.930043i	$-0.619769\pi$
$444$		0			0
$445$	−303.241	−	303.241i	−0.681440	−	0.681440i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	79.1258i		−	0.176227i	−0.996110	−	0.0881133i	$-0.971916\pi$
$449$		−	79.1258i		−	0.176227i	0.996110	−	0.0881133i	$-0.0280838\pi$
$450$		0			0
$451$	−7.77148	−	7.77148i	−0.0172317	−	0.0172317i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1147.98			2.52304
$456$		0			0
$457$		−	721.867i		−	1.57958i	−0.613380	−	0.789788i	$-0.710190\pi$
$457$		−	721.867i		−	1.57958i	0.613380	−	0.789788i	$-0.289810\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−57.6748	+	57.6748i	−0.125108	+	0.125108i	−0.766888	−	0.641780i	$-0.778196\pi$
$461$	−57.6748	+	57.6748i	−0.125108	+	0.125108i	0.641780	+	0.766888i	$0.278196\pi$
$462$		0			0
$463$	−469.888			−1.01488			−0.507438	−	0.861688i	$-0.669407\pi$
$463$	−469.888			−1.01488			−0.507438	+	0.861688i	$0.669407\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	284.381	−	284.381i	0.608954	−	0.608954i	−0.333719	−	0.942673i	$-0.608304\pi$
$467$	284.381	−	284.381i	0.608954	−	0.608954i	0.942673	+	0.333719i	$0.108304\pi$
$468$		0			0
$469$	366.516	−	366.516i	0.781485	−	0.781485i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	49.4548			0.104556
$474$		0			0
$475$	670.346	−	670.346i	1.41126	−	1.41126i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	74.7175i		−	0.155986i	−0.996954	−	0.0779932i	$-0.975149\pi$
$479$		−	74.7175i		−	0.155986i	0.996954	−	0.0779932i	$-0.0248512\pi$
$480$		0			0
$481$	808.179			1.68021
$482$		0			0
$483$		0			0
$484$		0			0
$485$	27.6458	+	27.6458i	0.0570017	+	0.0570017i
$486$		0			0
$487$			465.139i			0.955111i	0.878602	+	0.477555i	$0.158477\pi$
$487$			465.139i			0.955111i	−0.878602	+	0.477555i	$0.841523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−165.364	−	165.364i	−0.336791	−	0.336791i	0.518367	−	0.855158i	$-0.326540\pi$
$491$	−165.364	−	165.364i	−0.336791	−	0.336791i	−0.855158	+	0.518367i	$0.826540\pi$
$492$		0			0
$493$	−125.988	−	125.988i	−0.255553	−	0.255553i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			760.260i			1.52970i
$498$		0			0
$499$	−502.645	−	502.645i	−1.00730	−	1.00730i	−0.999973	−	0.00733057i	$-0.997667\pi$
$499$	−502.645	−	502.645i	−1.00730	−	1.00730i	−0.00733057	−	0.999973i	$-0.502333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19.2713			−0.0383127			−0.0191563	−	0.999817i	$-0.506098\pi$
$503$	−19.2713			−0.0383127			−0.0191563	+	0.999817i	$0.506098\pi$
$504$		0			0
$505$		−	411.742i		−	0.815330i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	136.252	−	136.252i	0.267686	−	0.267686i	−0.560481	−	0.828167i	$-0.689384\pi$
$509$	136.252	−	136.252i	0.267686	−	0.267686i	0.828167	+	0.560481i	$0.189384\pi$
$510$		0			0
$511$	833.232			1.63059
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1039.23	+	1039.23i	−2.01793	+	2.01793i
$516$		0			0
$517$	95.4452	−	95.4452i	0.184613	−	0.184613i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−196.906			−0.377939			−0.188969	−	0.981983i	$-0.560515\pi$
$521$	−196.906			−0.377939			−0.188969	+	0.981983i	$0.560515\pi$
$522$		0			0
$523$	−481.313	+	481.313i	−0.920293	+	0.920293i	−0.997050	−	0.0767569i	$-0.975543\pi$
$523$	−481.313	+	481.313i	−0.920293	+	0.920293i	0.0767569	+	0.997050i	$0.475543\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			179.488i			0.340584i
$528$		0			0
$529$	648.424			1.22575
$530$		0			0
$531$		0			0
$532$		0			0
$533$	41.6660	+	41.6660i	0.0781726	+	0.0781726i
$534$		0			0
$535$			389.382i			0.727817i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−29.2005	−	29.2005i	−0.0541753	−	0.0541753i
$540$		0			0
$541$	38.4319	+	38.4319i	0.0710387	+	0.0710387i	0.741733	−	0.670695i	$-0.234004\pi$
$541$	38.4319	+	38.4319i	0.0710387	+	0.0710387i	−0.670695	+	0.741733i	$0.734004\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1028.11i		−	1.88643i
$546$		0			0
$547$	−249.883	−	249.883i	−0.456824	−	0.456824i	0.440788	−	0.897611i	$-0.354699\pi$
$547$	−249.883	−	249.883i	−0.456824	−	0.456824i	−0.897611	+	0.440788i	$0.854699\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−798.089			−1.44844
$552$		0			0
$553$			754.678i			1.36470i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−561.819	+	561.819i	−1.00865	+	1.00865i	−0.00868977	+	0.999962i	$0.502766\pi$
$557$	−561.819	+	561.819i	−1.00865	+	1.00865i	−0.999962	+	0.00868977i	$0.997234\pi$
$558$		0			0
$559$	−265.147			−0.474323
$560$		0			0
$561$		0			0
$562$		0			0
$563$	512.028	−	512.028i	0.909464	−	0.909464i	−0.0867652	−	0.996229i	$-0.527653\pi$
$563$	512.028	−	512.028i	0.909464	−	0.909464i	0.996229	+	0.0867652i	$0.0276530\pi$
$564$		0			0
$565$	21.0487	−	21.0487i	0.0372544	−	0.0372544i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−278.458			−0.489382			−0.244691	−	0.969601i	$-0.578686\pi$
$569$	−278.458			−0.489382			−0.244691	+	0.969601i	$0.578686\pi$
$570$		0			0
$571$	−83.1512	+	83.1512i	−0.145624	+	0.145624i	−0.776160	−	0.630536i	$-0.782835\pi$
$571$	−83.1512	+	83.1512i	−0.145624	+	0.145624i	0.630536	+	0.776160i	$0.282835\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1238.58i			2.15406i
$576$		0			0
$577$	−1079.23			−1.87042			−0.935208	−	0.354098i	$-0.884788\pi$
$577$	−1079.23			−1.87042			−0.935208	+	0.354098i	$0.884788\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	692.315	+	692.315i	1.19159	+	1.19159i
$582$		0			0
$583$			304.104i			0.521619i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−197.819	−	197.819i	−0.337000	−	0.337000i	0.518237	−	0.855237i	$-0.326588\pi$
$587$	−197.819	−	197.819i	−0.337000	−	0.337000i	−0.855237	+	0.518237i	$0.826588\pi$
$588$		0			0
$589$	568.496	+	568.496i	0.965189	+	0.965189i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			313.761i			0.529108i	0.964371	+	0.264554i	$0.0852248\pi$
$593$			313.761i			0.529108i	−0.964371	+	0.264554i	$0.914775\pi$
$594$		0			0
$595$	252.587	+	252.587i	0.424516	+	0.424516i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1076.31			−1.79684			−0.898422	−	0.439133i	$-0.855286\pi$
$599$	−1076.31			−1.79684			−0.898422	+	0.439133i	$0.855286\pi$
$600$		0			0
$601$		−	25.5949i		−	0.0425873i	−0.999773	−	0.0212936i	$-0.993222\pi$
$601$		−	25.5949i		−	0.0425873i	0.999773	−	0.0212936i	$-0.00677848\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−600.497	+	600.497i	−0.992557	+	0.992557i
$606$		0			0
$607$	−576.415			−0.949612			−0.474806	−	0.880090i	$-0.657482\pi$
$607$	−576.415			−0.949612			−0.474806	+	0.880090i	$0.657482\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−511.719	+	511.719i	−0.837511	+	0.837511i
$612$		0			0
$613$	478.265	−	478.265i	0.780205	−	0.780205i	−0.199660	−	0.979865i	$-0.563984\pi$
$613$	478.265	−	478.265i	0.780205	−	0.780205i	0.979865	+	0.199660i	$0.0639839\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1106.10			1.79271			0.896357	−	0.443334i	$-0.146204\pi$
$617$	1106.10			1.79271			0.896357	+	0.443334i	$0.146204\pi$
$618$		0			0
$619$	−592.514	+	592.514i	−0.957212	+	0.957212i	−0.999121	−	0.0419094i	$-0.986656\pi$
$619$	−592.514	+	592.514i	−0.957212	+	0.957212i	0.0419094	+	0.999121i	$0.486656\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	427.630i		−	0.686405i
$624$		0			0
$625$	224.481			0.359170
$626$		0			0
$627$		0			0
$628$		0			0
$629$	177.821	+	177.821i	0.282705	+	0.282705i
$630$		0			0
$631$		−	845.724i		−	1.34029i	−0.742229	−	0.670146i	$-0.766231\pi$
$631$		−	845.724i		−	1.34029i	0.742229	−	0.670146i	$-0.233769\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	960.067	+	960.067i	1.51192	+	1.51192i
$636$		0			0
$637$	156.555	+	156.555i	0.245770	+	0.245770i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1167.54i			1.82143i	0.413035	+	0.910715i	$0.364469\pi$
$641$			1167.54i			1.82143i	−0.413035	+	0.910715i	$0.635531\pi$
$642$		0			0
$643$	−296.182	−	296.182i	−0.460626	−	0.460626i	0.438235	−	0.898861i	$-0.355604\pi$
$643$	−296.182	−	296.182i	−0.460626	−	0.460626i	−0.898861	+	0.438235i	$0.855604\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	113.278			0.175082			0.0875412	−	0.996161i	$-0.472099\pi$
$647$	113.278			0.175082			0.0875412	+	0.996161i	$0.472099\pi$
$648$		0			0
$649$			14.8562i			0.0228909i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.6005	+	24.6005i	−0.0376730	+	0.0376730i	−0.725692	−	0.688019i	$-0.758480\pi$
$653$	−24.6005	+	24.6005i	−0.0376730	+	0.0376730i	0.688019	+	0.725692i	$0.258480\pi$
$654$		0			0
$655$	−661.689			−1.01021
$656$		0			0
$657$		0			0
$658$		0			0
$659$	430.149	−	430.149i	0.652730	−	0.652730i	−0.300919	−	0.953650i	$-0.597293\pi$
$659$	430.149	−	430.149i	0.652730	−	0.652730i	0.953650	+	0.300919i	$0.0972935\pi$
$660$		0			0
$661$	−755.093	+	755.093i	−1.14235	+	1.14235i	−0.154330	+	0.988019i	$0.549322\pi$
$661$	−755.093	+	755.093i	−1.14235	+	1.14235i	−0.988019	+	0.154330i	$0.950678\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1600.05			2.40609
$666$		0			0
$667$	737.305	−	737.305i	1.10540	−	1.10540i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	19.3180i		−	0.0287899i
$672$		0			0
$673$	−148.466			−0.220604			−0.110302	−	0.993898i	$-0.535182\pi$
$673$	−148.466			−0.220604			−0.110302	+	0.993898i	$0.535182\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−422.759	−	422.759i	−0.624459	−	0.624459i	0.322209	−	0.946668i	$-0.395575\pi$
$677$	−422.759	−	422.759i	−0.624459	−	0.624459i	−0.946668	+	0.322209i	$0.895575\pi$
$678$		0			0
$679$			38.9862i			0.0574171i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−473.364	−	473.364i	−0.693066	−	0.693066i	0.269839	−	0.962905i	$-0.413029\pi$
$683$	−473.364	−	473.364i	−0.693066	−	0.693066i	−0.962905	+	0.269839i	$0.913029\pi$
$684$		0			0
$685$	72.1840	+	72.1840i	0.105378	+	0.105378i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1630.42i		−	2.36636i
$690$		0			0
$691$	−286.969	−	286.969i	−0.415295	−	0.415295i	0.468283	−	0.883579i	$-0.344873\pi$
$691$	−286.969	−	286.969i	−0.415295	−	0.415295i	−0.883579	+	0.468283i	$0.844873\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1464.63			−2.10739
$696$		0			0
$697$			18.3353i			0.0263060i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	490.554	−	490.554i	0.699791	−	0.699791i	−0.264574	−	0.964365i	$-0.585231\pi$
$701$	490.554	−	490.554i	0.699791	−	0.699791i	0.964365	+	0.264574i	$0.0852314\pi$
$702$		0			0
$703$	1126.44			1.60233
$704$		0			0
$705$		0			0
$706$		0			0
$707$	290.319	−	290.319i	0.410635	−	0.410635i
$708$		0			0
$709$	−471.995	+	471.995i	−0.665719	+	0.665719i	−0.956722	−	0.291003i	$-0.906011\pi$
$709$	−471.995	+	471.995i	−0.665719	+	0.665719i	0.291003	+	0.956722i	$0.406011\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1050.40			−1.47321
$714$		0			0
$715$	−366.038	+	366.038i	−0.511942	+	0.511942i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1035.64i			1.44040i	0.693769	+	0.720198i	$0.255949\pi$
$719$			1035.64i			1.44040i	−0.693769	+	0.720198i	$0.744051\pi$
$720$		0			0
$721$	−1465.53			−2.03263
$722$		0			0
$723$		0			0
$724$		0			0
$725$	775.602	+	775.602i	1.06980	+	1.06980i
$726$		0			0
$727$			172.311i			0.237016i	0.992953	+	0.118508i	$0.0378112\pi$
$727$			172.311i			0.237016i	−0.992953	+	0.118508i	$0.962189\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−58.3395	−	58.3395i	−0.0798078	−	0.0798078i
$732$		0			0
$733$	729.510	+	729.510i	0.995239	+	0.995239i	0.999989	−	0.00474987i	$-0.00151193\pi$
$733$	729.510	+	729.510i	0.995239	+	0.995239i	−0.00474987	+	0.999989i	$0.501512\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			233.730i			0.317137i
$738$		0			0
$739$	−183.822	−	183.822i	−0.248745	−	0.248745i	0.571711	−	0.820455i	$-0.306280\pi$
$739$	−183.822	−	183.822i	−0.248745	−	0.248745i	−0.820455	+	0.571711i	$0.806280\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1276.92			1.71860			0.859300	−	0.511471i	$-0.170899\pi$
$743$	1276.92			1.71860			0.859300	+	0.511471i	$0.170899\pi$
$744$		0			0
$745$			115.811i			0.155451i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−274.553	+	274.553i	−0.366560	+	0.366560i
$750$		0			0
$751$	−841.133			−1.12002			−0.560009	−	0.828487i	$-0.689202\pi$
$751$	−841.133			−1.12002			−0.560009	+	0.828487i	$0.689202\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1001.54	−	1001.54i	1.32654	−	1.32654i
$756$		0			0
$757$	−695.488	+	695.488i	−0.918743	+	0.918743i	−0.996938	−	0.0781952i	$-0.975084\pi$
$757$	−695.488	+	695.488i	−0.918743	+	0.918743i	0.0781952	+	0.996938i	$0.475084\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−449.501			−0.590671			−0.295335	−	0.955394i	$-0.595431\pi$
$761$	−449.501			−0.590671			−0.295335	+	0.955394i	$0.595431\pi$
$762$		0			0
$763$	724.918	−	724.918i	0.950090	−	0.950090i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	79.6500i		−	0.103846i
$768$		0			0
$769$	−1445.97			−1.88033			−0.940163	−	0.340725i	$-0.889328\pi$
$769$	−1445.97			−1.88033			−0.940163	+	0.340725i	$0.889328\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−186.543	−	186.543i	−0.241323	−	0.241323i	0.576074	−	0.817397i	$-0.304584\pi$
$773$	−186.543	−	186.543i	−0.241323	−	0.241323i	−0.817397	+	0.576074i	$0.804584\pi$
$774$		0			0
$775$		−	1104.96i		−	1.42575i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	58.0738	+	58.0738i	0.0745492	+	0.0745492i
$780$		0			0
$781$	−242.412	−	242.412i	−0.310386	−	0.310386i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			640.602i			0.816053i
$786$		0			0
$787$	−753.754	−	753.754i	−0.957756	−	0.957756i	0.0413871	−	0.999143i	$-0.486822\pi$
$787$	−753.754	−	753.754i	−0.957756	−	0.957756i	−0.999143	+	0.0413871i	$0.986822\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	29.6829			0.0375258
$792$		0			0
$793$			103.572i			0.130607i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	531.870	−	531.870i	0.667340	−	0.667340i	−0.289759	−	0.957100i	$-0.593575\pi$
$797$	531.870	−	531.870i	0.667340	−	0.667340i	0.957100	+	0.289759i	$0.0935752\pi$
$798$		0			0
$799$	−225.184			−0.281833
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−265.679	+	265.679i	−0.330858	+	0.330858i
$804$		0			0
$805$	−1478.19	+	1478.19i	−1.83626	+	1.83626i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	540.279			0.667836			0.333918	−	0.942602i	$-0.391629\pi$
$809$	540.279			0.667836			0.333918	+	0.942602i	$0.391629\pi$
$810$		0			0
$811$	−702.871	+	702.871i	−0.866672	+	0.866672i	−0.992102	−	0.125430i	$-0.959969\pi$
$811$	−702.871	+	702.871i	−0.866672	+	0.866672i	0.125430	+	0.992102i	$0.459969\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			970.594i			1.19091i
$816$		0			0
$817$	−369.560			−0.452338
$818$		0			0
$819$		0			0
$820$		0			0
$821$	286.031	+	286.031i	0.348393	+	0.348393i	0.859511	−	0.511117i	$-0.170768\pi$
$821$	286.031	+	286.031i	0.348393	+	0.348393i	−0.511117	+	0.859511i	$0.670768\pi$
$822$		0			0
$823$		−	215.600i		−	0.261969i	−0.991384	−	0.130984i	$-0.958186\pi$
$823$		−	215.600i		−	0.261969i	0.991384	−	0.130984i	$-0.0418138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	375.162	+	375.162i	0.453642	+	0.453642i	0.896561	−	0.442920i	$-0.146057\pi$
$827$	375.162	+	375.162i	0.453642	+	0.453642i	−0.442920	+	0.896561i	$0.646057\pi$
$828$		0			0
$829$	−22.5151	−	22.5151i	−0.0271594	−	0.0271594i	0.693397	−	0.720556i	$-0.256113\pi$
$829$	−22.5151	−	22.5151i	−0.0271594	−	0.0271594i	−0.720556	+	0.693397i	$0.756113\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			68.8929i			0.0827046i
$834$		0			0
$835$	−480.370	−	480.370i	−0.575294	−	0.575294i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1139.61			1.35830			0.679151	−	0.733999i	$-0.262348\pi$
$839$	1139.61			1.35830			0.679151	+	0.733999i	$0.262348\pi$
$840$		0			0
$841$		−	82.4023i		−	0.0979814i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1028.41	−	1028.41i	1.21705	−	1.21705i
$846$		0			0
$847$	−846.821			−0.999788
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1040.64	+	1040.64i	−1.22285	+	1.22285i
$852$		0			0
$853$	405.464	−	405.464i	0.475339	−	0.475339i	−0.428298	−	0.903637i	$-0.640887\pi$
$853$	405.464	−	405.464i	0.475339	−	0.475339i	0.903637	+	0.428298i	$0.140887\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1006.21			1.17411			0.587056	−	0.809546i	$-0.300287\pi$
$857$	1006.21			1.17411			0.587056	+	0.809546i	$0.300287\pi$
$858$		0			0
$859$	−626.283	+	626.283i	−0.729083	+	0.729083i	−0.970437	−	0.241354i	$-0.922409\pi$
$859$	−626.283	+	626.283i	−0.729083	+	0.729083i	0.241354	+	0.970437i	$0.422409\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			926.795i			1.07392i	0.843607	+	0.536961i	$0.180428\pi$
$863$			926.795i			1.07392i	−0.843607	+	0.536961i	$0.819572\pi$
$864$		0			0
$865$	2387.76			2.76041
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−240.632	−	240.632i	−0.276907	−	0.276907i
$870$		0			0
$871$		−	1253.12i		−	1.43872i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−478.000	−	478.000i	−0.546285	−	0.546285i
$876$		0			0
$877$	558.537	+	558.537i	0.636872	+	0.636872i	0.949783	−	0.312911i	$-0.101304\pi$
$877$	558.537	+	558.537i	0.636872	+	0.636872i	−0.312911	+	0.949783i	$0.601304\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	969.827i		−	1.10083i	−0.834893	−	0.550413i	$-0.814470\pi$
$881$		−	969.827i		−	1.10083i	0.834893	−	0.550413i	$-0.185530\pi$
$882$		0			0
$883$	−549.585	−	549.585i	−0.622407	−	0.622407i	0.323739	−	0.946146i	$-0.395060\pi$
$883$	−549.585	−	549.585i	−0.622407	−	0.622407i	−0.946146	+	0.323739i	$0.895060\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1552.40			−1.75017			−0.875086	−	0.483968i	$-0.839195\pi$
$887$	−1552.40			−1.75017			−0.875086	+	0.483968i	$0.839195\pi$
$888$		0			0
$889$			1353.89i			1.52293i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−713.232	+	713.232i	−0.798692	+	0.798692i
$894$		0			0
$895$	−952.173			−1.06388
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−657.760	+	657.760i	−0.731657	+	0.731657i
$900$		0			0
$901$	358.737	−	358.737i	0.398154	−	0.398154i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.3687			0.0225069
$906$		0			0
$907$	451.019	−	451.019i	0.497265	−	0.497265i	−0.413321	−	0.910586i	$-0.635631\pi$
$907$	451.019	−	451.019i	0.497265	−	0.497265i	0.910586	+	0.413321i	$0.135631\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1490.19i		−	1.63578i	−0.575378	−	0.817888i	$-0.695145\pi$
$911$		−	1490.19i		−	1.63578i	0.575378	−	0.817888i	$-0.304855\pi$
$912$		0			0
$913$	−441.494			−0.483565
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−466.557	−	466.557i	−0.508786	−	0.508786i
$918$		0			0
$919$			949.287i			1.03296i	0.856300	+	0.516478i	$0.172757\pi$
$919$			949.287i			1.03296i	−0.856300	+	0.516478i	$0.827243\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1299.67	+	1299.67i	1.40809	+	1.40809i
$924$		0			0
$925$	−1094.70	−	1094.70i	−1.18346	−	1.18346i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1078.41i			1.16083i	0.814322	+	0.580413i	$0.197109\pi$
$929$			1078.41i			1.16083i	−0.814322	+	0.580413i	$0.802891\pi$
$930$		0			0
$931$	218.206	+	218.206i	0.234378	+	0.234378i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−161.077			−0.172275
$936$		0			0
$937$			1158.42i			1.23631i	0.786056	+	0.618155i	$0.212120\pi$
$937$			1158.42i			1.23631i	−0.786056	+	0.618155i	$0.787880\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1027.40	+	1027.40i	−1.09181	+	1.09181i	−0.0964770	+	0.995335i	$0.530757\pi$
$941$	−1027.40	+	1027.40i	−1.09181	+	1.09181i	−0.995335	+	0.0964770i	$0.969243\pi$
$942$		0			0
$943$	−107.302			−0.113788
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−890.208	+	890.208i	−0.940030	+	0.940030i	−0.998301	−	0.0582710i	$-0.981441\pi$
$947$	−890.208	+	890.208i	−0.940030	+	0.940030i	0.0582710	+	0.998301i	$0.481441\pi$
$948$		0			0
$949$	1424.41	−	1424.41i	1.50096	−	1.50096i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−566.070			−0.593987			−0.296994	−	0.954879i	$-0.595984\pi$
$953$	−566.070			−0.593987			−0.296994	+	0.954879i	$0.595984\pi$
$954$		0			0
$955$	453.397	−	453.397i	0.474761	−	0.474761i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			101.794i			0.106146i
$960$		0			0
$961$	−23.9259			−0.0248968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	504.171	+	504.171i	0.522457	+	0.522457i
$966$		0			0
$967$		−	461.918i		−	0.477682i	−0.971059	−	0.238841i	$-0.923233\pi$
$967$		−	461.918i		−	0.477682i	0.971059	−	0.238841i	$-0.0767674\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−703.059	−	703.059i	−0.724057	−	0.724057i	0.245372	−	0.969429i	$-0.421090\pi$
$971$	−703.059	−	703.059i	−0.724057	−	0.724057i	−0.969429	+	0.245372i	$0.921090\pi$
$972$		0			0
$973$	−1032.71	−	1032.71i	−1.06137	−	1.06137i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1020.60i			1.04462i	0.852755	+	0.522311i	$0.174930\pi$
$977$			1020.60i			1.04462i	−0.852755	+	0.522311i	$0.825070\pi$
$978$		0			0
$979$	136.352	+	136.352i	0.139276	+	0.139276i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−837.639			−0.852125			−0.426063	−	0.904694i	$-0.640100\pi$
$983$	−837.639			−0.852125			−0.426063	+	0.904694i	$0.640100\pi$
$984$		0			0
$985$			96.6435i			0.0981152i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	341.414	−	341.414i	0.345211	−	0.345211i
$990$		0			0
$991$	273.535			0.276019			0.138009	−	0.990431i	$-0.455930\pi$
$991$	273.535			0.276019			0.138009	+	0.990431i	$0.455930\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−712.013	+	712.013i	−0.715591	+	0.715591i
$996$		0			0
$997$	−77.9043	+	77.9043i	−0.0781387	+	0.0781387i	−0.745096	−	0.666957i	$-0.767596\pi$
$997$	−77.9043	+	77.9043i	−0.0781387	+	0.0781387i	0.666957	+	0.745096i	$0.267596\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.a.161.2		32
3.2	odd	2	inner	1152.3.j.a.161.15		32
4.3	odd	2		1152.3.j.b.161.2		32
8.3	odd	2		576.3.j.a.17.15		32
8.5	even	2		144.3.j.a.125.2	yes	32
12.11	even	2		1152.3.j.b.161.15		32
16.3	odd	4		576.3.j.a.305.2		32
16.5	even	4	inner	1152.3.j.a.737.15		32
16.11	odd	4		1152.3.j.b.737.15		32
16.13	even	4		144.3.j.a.53.15	yes	32
24.5	odd	2		144.3.j.a.125.15	yes	32
24.11	even	2		576.3.j.a.17.2		32
48.5	odd	4	inner	1152.3.j.a.737.2		32
48.11	even	4		1152.3.j.b.737.2		32
48.29	odd	4		144.3.j.a.53.2	✓	32
48.35	even	4		576.3.j.a.305.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.2	✓	32	48.29	odd	4
144.3.j.a.53.15	yes	32	16.13	even	4
144.3.j.a.125.2	yes	32	8.5	even	2
144.3.j.a.125.15	yes	32	24.5	odd	2
576.3.j.a.17.2		32	24.11	even	2
576.3.j.a.17.15		32	8.3	odd	2
576.3.j.a.305.2		32	16.3	odd	4
576.3.j.a.305.15		32	48.35	even	4
1152.3.j.a.161.2		32	1.1	even	1	trivial
1152.3.j.a.161.15		32	3.2	odd	2	inner
1152.3.j.a.737.2		32	48.5	odd	4	inner
1152.3.j.a.737.15		32	16.5	even	4	inner
1152.3.j.b.161.2		32	4.3	odd	2
1152.3.j.b.161.15		32	12.11	even	2
1152.3.j.b.737.2		32	48.11	even	4
1152.3.j.b.737.15		32	16.11	odd	4

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)

\(f(q)\)	\(=\)	\(q+(-5.52702 - 5.52702i) q^{5} -7.79421i q^{7} +O(q^{10})\)	\(q+(-5.52702 - 5.52702i) q^{5} -7.79421i q^{7} +(2.48521 + 2.48521i) q^{11} +(-13.3242 - 13.3242i) q^{13} -5.86338i q^{17} +(-18.5712 - 18.5712i) q^{19} +34.3136 q^{23} +36.0959i q^{25} +(21.4872 - 21.4872i) q^{29} -30.6117 q^{31} +(-43.0788 + 43.0788i) q^{35} +(-30.3274 + 30.3274i) q^{37} -3.12709 q^{41} +(9.94981 - 9.94981i) q^{43} -38.4052i q^{47} -11.7497 q^{49} +(61.1826 + 61.1826i) q^{53} -27.4717i q^{55} +(2.98892 + 2.98892i) q^{59} +(-3.88659 - 3.88659i) q^{61} +147.287i q^{65} +(47.0242 + 47.0242i) q^{67} -97.5416 q^{71} +106.904i q^{73} +(19.3703 - 19.3703i) q^{77} -96.8255 q^{79} +(-88.8242 + 88.8242i) q^{83} +(-32.4070 + 32.4070i) q^{85} +54.8651 q^{89} +(-103.852 + 103.852i) q^{91} +205.287i q^{95} -5.00194 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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