LMFDB - Embedded newform 1152.3.m.f.991.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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([2, 1, 0]))

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

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.3
Root			$-0.455024 + 1.94755i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.f.415.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.40572 + 3.40572i) q^{5} +12.1303 q^{7} +O(q^{10})$	$q+(-3.40572 + 3.40572i) q^{5} +12.1303 q^{7} +(9.81086 + 9.81086i) q^{11} +(7.76859 + 7.76859i) q^{13} -9.73087 q^{17} +(-11.2823 + 11.2823i) q^{19} +20.2635 q^{23} +1.80207i q^{25} +(-16.4069 - 16.4069i) q^{29} -26.3542i q^{31} +(-41.3125 + 41.3125i) q^{35} +(23.7263 - 23.7263i) q^{37} +24.7452i q^{41} +(-29.8844 - 29.8844i) q^{43} +31.3325i q^{47} +98.1448 q^{49} +(36.8742 - 36.8742i) q^{53} -66.8262 q^{55} +(-14.1325 - 14.1325i) q^{59} +(42.5199 + 42.5199i) q^{61} -52.9153 q^{65} +(-48.7789 + 48.7789i) q^{67} -7.73935 q^{71} +85.4163i q^{73} +(119.009 + 119.009i) q^{77} +105.294i q^{79} +(-62.1229 + 62.1229i) q^{83} +(33.1407 - 33.1407i) q^{85} +127.172i q^{89} +(94.2355 + 94.2355i) q^{91} -76.8489i q^{95} -147.348 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) $ 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * 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
$5$	−3.40572	+	3.40572i	−0.681145	+	0.681145i	−0.960258	−	0.279113i	$-0.909960\pi$
$5$	−3.40572	+	3.40572i	−0.681145	+	0.681145i	0.279113	+	0.960258i	$0.409960\pi$
$6$		0			0
$7$	12.1303			1.73290			0.866452	−	0.499261i	$-0.166395\pi$
$7$	12.1303			1.73290			0.866452	+	0.499261i	$0.166395\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.81086	+	9.81086i	0.891896	+	0.891896i	0.994702	−	0.102805i	$-0.0327818\pi$
$11$	9.81086	+	9.81086i	0.891896	+	0.891896i	−0.102805	+	0.994702i	$0.532782\pi$
$12$		0			0
$13$	7.76859	+	7.76859i	0.597584	+	0.597584i	0.939669	−	0.342085i	$-0.111133\pi$
$13$	7.76859	+	7.76859i	0.597584	+	0.597584i	−0.342085	+	0.939669i	$0.611133\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.73087			−0.572404			−0.286202	−	0.958169i	$-0.592393\pi$
$17$	−9.73087			−0.572404			−0.286202	+	0.958169i	$0.592393\pi$
$18$		0			0
$19$	−11.2823	+	11.2823i	−0.593806	+	0.593806i	−0.938657	−	0.344851i	$-0.887929\pi$
$19$	−11.2823	+	11.2823i	−0.593806	+	0.593806i	0.344851	+	0.938657i	$0.387929\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.2635			0.881020			0.440510	−	0.897748i	$-0.354798\pi$
$23$	20.2635			0.881020			0.440510	+	0.897748i	$0.354798\pi$
$24$		0			0
$25$			1.80207i			0.0720830i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−16.4069	−	16.4069i	−0.565754	−	0.565754i	0.365182	−	0.930936i	$-0.381007\pi$
$29$	−16.4069	−	16.4069i	−0.565754	−	0.565754i	−0.930936	+	0.365182i	$0.881007\pi$
$30$		0			0
$31$		−	26.3542i		−	0.850134i	−0.905162	−	0.425067i	$-0.860251\pi$
$31$		−	26.3542i		−	0.850134i	0.905162	−	0.425067i	$-0.139749\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−41.3125	+	41.3125i	−1.18036	+	1.18036i
$36$		0			0
$37$	23.7263	−	23.7263i	0.641250	−	0.641250i	−0.309613	−	0.950863i	$-0.600199\pi$
$37$	23.7263	−	23.7263i	0.641250	−	0.641250i	0.950863	+	0.309613i	$0.100199\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			24.7452i			0.603542i	0.953380	+	0.301771i	$0.0975779\pi$
$41$			24.7452i			0.603542i	−0.953380	+	0.301771i	$0.902422\pi$
$42$		0			0
$43$	−29.8844	−	29.8844i	−0.694987	−	0.694987i	0.268338	−	0.963325i	$-0.413526\pi$
$43$	−29.8844	−	29.8844i	−0.694987	−	0.694987i	−0.963325	+	0.268338i	$0.913526\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			31.3325i			0.666648i	0.942812	+	0.333324i	$0.108170\pi$
$47$			31.3325i			0.666648i	−0.942812	+	0.333324i	$0.891830\pi$
$48$		0			0
$49$	98.1448			2.00295
$50$		0			0
$51$		0			0
$52$		0			0
$53$	36.8742	−	36.8742i	0.695739	−	0.695739i	−0.267750	−	0.963489i	$-0.586280\pi$
$53$	36.8742	−	36.8742i	0.695739	−	0.695739i	0.963489	+	0.267750i	$0.0862800\pi$
$54$		0			0
$55$	−66.8262			−1.21502
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.1325	−	14.1325i	−0.239534	−	0.239534i	0.577123	−	0.816657i	$-0.304175\pi$
$59$	−14.1325	−	14.1325i	−0.239534	−	0.239534i	−0.816657	+	0.577123i	$0.804175\pi$
$60$		0			0
$61$	42.5199	+	42.5199i	0.697048	+	0.697048i	0.963773	−	0.266725i	$-0.0859414\pi$
$61$	42.5199	+	42.5199i	0.697048	+	0.697048i	−0.266725	+	0.963773i	$0.585941\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−52.9153			−0.814082
$66$		0			0
$67$	−48.7789	+	48.7789i	−0.728044	+	0.728044i	−0.970230	−	0.242186i	$-0.922136\pi$
$67$	−48.7789	+	48.7789i	−0.728044	+	0.728044i	0.242186	+	0.970230i	$0.422136\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.73935			−0.109005			−0.0545025	−	0.998514i	$-0.517357\pi$
$71$	−7.73935			−0.109005			−0.0545025	+	0.998514i	$0.517357\pi$
$72$		0			0
$73$			85.4163i			1.17009i	0.811002	+	0.585043i	$0.198923\pi$
$73$			85.4163i			1.17009i	−0.811002	+	0.585043i	$0.801077\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	119.009	+	119.009i	1.54557	+	1.54557i
$78$		0			0
$79$			105.294i			1.33283i	0.745581	+	0.666416i	$0.232172\pi$
$79$			105.294i			1.33283i	−0.745581	+	0.666416i	$0.767828\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−62.1229	+	62.1229i	−0.748469	+	0.748469i	−0.974192	−	0.225723i	$-0.927526\pi$
$83$	−62.1229	+	62.1229i	−0.748469	+	0.748469i	0.225723	+	0.974192i	$0.427526\pi$
$84$		0			0
$85$	33.1407	−	33.1407i	0.389890	−	0.389890i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			127.172i			1.42890i	0.699685	+	0.714451i	$0.253324\pi$
$89$			127.172i			1.42890i	−0.699685	+	0.714451i	$0.746676\pi$
$90$		0			0
$91$	94.2355	+	94.2355i	1.03555	+	1.03555i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	76.8489i		−	0.808936i
$96$		0			0
$97$	−147.348			−1.51905			−0.759525	−	0.650478i	$-0.774569\pi$
$97$	−147.348			−1.51905			−0.759525	+	0.650478i	$0.774569\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.7690	−	12.7690i	0.126426	−	0.126426i	−0.641063	−	0.767489i	$-0.721506\pi$
$101$	12.7690	−	12.7690i	0.126426	−	0.126426i	0.767489	+	0.641063i	$0.221506\pi$
$102$		0			0
$103$	−17.7621			−0.172448			−0.0862240	−	0.996276i	$-0.527480\pi$
$103$	−17.7621			−0.172448			−0.0862240	+	0.996276i	$0.527480\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.8889	−	15.8889i	−0.148494	−	0.148494i	0.628951	−	0.777445i	$-0.283485\pi$
$107$	−15.8889	−	15.8889i	−0.148494	−	0.148494i	−0.777445	+	0.628951i	$0.783485\pi$
$108$		0			0
$109$	79.3257	+	79.3257i	0.727758	+	0.727758i	0.970173	−	0.242414i	$-0.0779394\pi$
$109$	79.3257	+	79.3257i	0.727758	+	0.727758i	−0.242414	+	0.970173i	$0.577939\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	167.538			1.48263			0.741317	−	0.671155i	$-0.234201\pi$
$113$	167.538			1.48263			0.741317	+	0.671155i	$0.234201\pi$
$114$		0			0
$115$	−69.0118	+	69.0118i	−0.600102	+	0.600102i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−118.039			−0.991921
$120$		0			0
$121$			71.5059i			0.590958i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−91.2805	−	91.2805i	−0.730244	−	0.730244i
$126$		0			0
$127$			198.247i			1.56100i	0.625156	+	0.780500i	$0.285035\pi$
$127$			198.247i			1.56100i	−0.625156	+	0.780500i	$0.714965\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	134.339	−	134.339i	1.02549	−	1.02549i	0.0258197	−	0.999667i	$-0.491780\pi$
$131$	134.339	−	134.339i	1.02549	−	1.02549i	0.999667	−	0.0258197i	$-0.00821957\pi$
$132$		0			0
$133$	−136.858	+	136.858i	−1.02901	+	1.02901i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	255.937i		−	1.86816i	−0.357069	−	0.934078i	$-0.616224\pi$
$137$		−	255.937i		−	1.86816i	0.357069	−	0.934078i	$-0.383776\pi$
$138$		0			0
$139$	21.7231	+	21.7231i	0.156281	+	0.156281i	0.780917	−	0.624635i	$-0.214752\pi$
$139$	21.7231	+	21.7231i	0.156281	+	0.156281i	−0.624635	+	0.780917i	$0.714752\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			152.433i			1.06597i
$144$		0			0
$145$	111.755			0.770722
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−34.2444	+	34.2444i	−0.229828	+	0.229828i	−0.812621	−	0.582793i	$-0.801960\pi$
$149$	−34.2444	+	34.2444i	−0.229828	+	0.229828i	0.582793	+	0.812621i	$0.301960\pi$
$150$		0			0
$151$	−14.4645			−0.0957913			−0.0478956	−	0.998852i	$-0.515251\pi$
$151$	−14.4645			−0.0957913			−0.0478956	+	0.998852i	$0.515251\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	89.7550	+	89.7550i	0.579064	+	0.579064i
$156$		0			0
$157$	−31.4652	−	31.4652i	−0.200415	−	0.200415i	0.599763	−	0.800178i	$-0.295262\pi$
$157$	−31.4652	−	31.4652i	−0.200415	−	0.200415i	−0.800178	+	0.599763i	$0.795262\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	245.802			1.52672
$162$		0			0
$163$	−31.4002	+	31.4002i	−0.192640	+	0.192640i	−0.796836	−	0.604196i	$-0.793494\pi$
$163$	−31.4002	+	31.4002i	−0.192640	+	0.192640i	0.604196	+	0.796836i	$0.293494\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−36.4796			−0.218441			−0.109220	−	0.994018i	$-0.534835\pi$
$167$	−36.4796			−0.218441			−0.109220	+	0.994018i	$0.534835\pi$
$168$		0			0
$169$		−	48.2981i		−	0.285788i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	97.6419	+	97.6419i	0.564404	+	0.564404i	0.930555	−	0.366151i	$-0.119325\pi$
$173$	97.6419	+	97.6419i	0.564404	+	0.564404i	−0.366151	+	0.930555i	$0.619325\pi$
$174$		0			0
$175$			21.8598i			0.124913i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	89.7427	−	89.7427i	0.501356	−	0.501356i	−0.410503	−	0.911859i	$-0.634647\pi$
$179$	89.7427	−	89.7427i	0.501356	−	0.501356i	0.911859	+	0.410503i	$0.134647\pi$
$180$		0			0
$181$	115.497	−	115.497i	0.638108	−	0.638108i	−0.311981	−	0.950088i	$-0.600992\pi$
$181$	115.497	−	115.497i	0.638108	−	0.638108i	0.950088	+	0.311981i	$0.100992\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			161.610i			0.873569i
$186$		0			0
$187$	−95.4682	−	95.4682i	−0.510525	−	0.510525i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	62.6278i		−	0.327894i	−0.986469	−	0.163947i	$-0.947577\pi$
$191$		−	62.6278i		−	0.327894i	0.986469	−	0.163947i	$-0.0524227\pi$
$192$		0			0
$193$	223.342			1.15721			0.578607	−	0.815607i	$-0.303597\pi$
$193$	223.342			1.15721			0.578607	+	0.815607i	$0.303597\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	29.0959	−	29.0959i	0.147695	−	0.147695i	−0.629393	−	0.777087i	$-0.716696\pi$
$197$	29.0959	−	29.0959i	0.147695	−	0.147695i	0.777087	+	0.629393i	$0.216696\pi$
$198$		0			0
$199$	11.6967			0.0587776			0.0293888	−	0.999568i	$-0.490644\pi$
$199$	11.6967			0.0587776			0.0293888	+	0.999568i	$0.490644\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−199.021	−	199.021i	−0.980398	−	0.980398i
$204$		0			0
$205$	−84.2755	−	84.2755i	−0.411100	−	0.411100i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−221.378			−1.05923
$210$		0			0
$211$	0.215765	−	0.215765i	0.00102258	−	0.00102258i	−0.706595	−	0.707618i	$-0.749770\pi$
$211$	0.215765	−	0.215765i	0.00102258	−	0.00102258i	0.707618	+	0.706595i	$0.249770\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	203.556			0.946773
$216$		0			0
$217$		−	319.684i		−	1.47320i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−75.5951	−	75.5951i	−0.342059	−	0.342059i
$222$		0			0
$223$		−	371.347i		−	1.66523i	−0.553850	−	0.832617i	$-0.686842\pi$
$223$		−	371.347i		−	1.66523i	0.553850	−	0.832617i	$-0.313158\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−209.823	+	209.823i	−0.924330	+	0.924330i	−0.997332	−	0.0730018i	$-0.976742\pi$
$227$	−209.823	+	209.823i	−0.924330	+	0.924330i	0.0730018	+	0.997332i	$0.476742\pi$
$228$		0			0
$229$	−152.751	+	152.751i	−0.667037	+	0.667037i	−0.957029	−	0.289992i	$-0.906347\pi$
$229$	−152.751	+	152.751i	−0.667037	+	0.667037i	0.289992	+	0.957029i	$0.406347\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	272.899i		−	1.17124i	−0.810586	−	0.585619i	$-0.800851\pi$
$233$		−	272.899i		−	1.17124i	0.810586	−	0.585619i	$-0.199149\pi$
$234$		0			0
$235$	−106.710	−	106.710i	−0.454084	−	0.454084i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			104.650i			0.437866i	0.975740	+	0.218933i	$0.0702576\pi$
$239$			104.650i			0.437866i	−0.975740	+	0.218933i	$0.929742\pi$
$240$		0			0
$241$	148.875			0.617737			0.308869	−	0.951105i	$-0.400050\pi$
$241$	148.875			0.617737			0.308869	+	0.951105i	$0.400050\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−334.254	+	334.254i	−1.36430	+	1.36430i
$246$		0			0
$247$	−175.295			−0.709698
$248$		0			0
$249$		0			0
$250$		0			0
$251$	143.712	+	143.712i	0.572558	+	0.572558i	0.932843	−	0.360284i	$-0.117320\pi$
$251$	143.712	+	143.712i	0.572558	+	0.572558i	−0.360284	+	0.932843i	$0.617320\pi$
$252$		0			0
$253$	198.802	+	198.802i	0.785778	+	0.785778i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−134.023			−0.521489			−0.260745	−	0.965408i	$-0.583968\pi$
$257$	−134.023			−0.521489			−0.260745	+	0.965408i	$0.583968\pi$
$258$		0			0
$259$	287.807	−	287.807i	1.11122	−	1.11122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−290.386			−1.10413			−0.552066	−	0.833801i	$-0.686160\pi$
$263$	−290.386			−1.10413			−0.552066	+	0.833801i	$0.686160\pi$
$264$		0			0
$265$			251.166i			0.947798i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−74.2628	−	74.2628i	−0.276070	−	0.276070i	0.555468	−	0.831538i	$-0.312539\pi$
$269$	−74.2628	−	74.2628i	−0.276070	−	0.276070i	−0.831538	+	0.555468i	$0.812539\pi$
$270$		0			0
$271$			70.8329i			0.261376i	0.991424	+	0.130688i	$0.0417186\pi$
$271$			70.8329i			0.261376i	−0.991424	+	0.130688i	$0.958281\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−17.6799	+	17.6799i	−0.0642906	+	0.0642906i
$276$		0			0
$277$	96.6953	−	96.6953i	0.349081	−	0.349081i	−0.510686	−	0.859767i	$-0.670609\pi$
$277$	96.6953	−	96.6953i	0.349081	−	0.349081i	0.859767	+	0.510686i	$0.170609\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			138.151i			0.491640i	0.969316	+	0.245820i	$0.0790572\pi$
$281$			138.151i			0.491640i	−0.969316	+	0.245820i	$0.920943\pi$
$282$		0			0
$283$	295.011	+	295.011i	1.04244	+	1.04244i	0.999059	+	0.0433821i	$0.0138133\pi$
$283$	295.011	+	295.011i	1.04244	+	1.04244i	0.0433821	+	0.999059i	$0.486187\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			300.168i			1.04588i
$288$		0			0
$289$	−194.310			−0.672353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−33.4759	+	33.4759i	−0.114252	+	0.114252i	−0.761922	−	0.647669i	$-0.775744\pi$
$293$	−33.4759	+	33.4759i	−0.114252	+	0.114252i	0.647669	+	0.761922i	$0.275744\pi$
$294$		0			0
$295$	96.2630			0.326315
$296$		0			0
$297$		0			0
$298$		0			0
$299$	157.418	+	157.418i	0.526483	+	0.526483i
$300$		0			0
$301$	−362.508	−	362.508i	−1.20434	−	1.20434i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−289.622			−0.949582
$306$		0			0
$307$	92.6638	−	92.6638i	0.301836	−	0.301836i	−0.539896	−	0.841732i	$-0.681536\pi$
$307$	92.6638	−	92.6638i	0.301836	−	0.301836i	0.841732	+	0.539896i	$0.181536\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.5610			0.0596817			0.0298408	−	0.999555i	$-0.490500\pi$
$311$	18.5610			0.0596817			0.0298408	+	0.999555i	$0.490500\pi$
$312$		0			0
$313$		−	55.1534i		−	0.176209i	−0.996111	−	0.0881045i	$-0.971919\pi$
$313$		−	55.1534i		−	0.176209i	0.996111	−	0.0881045i	$-0.0280809\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	62.2977	+	62.2977i	0.196523	+	0.196523i	0.798507	−	0.601985i	$-0.205623\pi$
$317$	62.2977	+	62.2977i	0.196523	+	0.196523i	−0.601985	+	0.798507i	$0.705623\pi$
$318$		0			0
$319$		−	321.931i		−	1.00919i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	109.787	−	109.787i	0.339897	−	0.339897i
$324$		0			0
$325$	−13.9996	+	13.9996i	−0.0430756	+	0.0430756i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			380.073i			1.15524i
$330$		0			0
$331$	−373.767	−	373.767i	−1.12921	−	1.12921i	−0.990307	−	0.138899i	$-0.955644\pi$
$331$	−373.767	−	373.767i	−1.12921	−	1.12921i	−0.138899	−	0.990307i	$-0.544356\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	332.255i		−	0.991807i
$336$		0			0
$337$	−519.936			−1.54284			−0.771419	−	0.636328i	$-0.780453\pi$
$337$	−519.936			−1.54284			−0.771419	+	0.636328i	$0.780453\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	258.557	−	258.557i	0.758231	−	0.758231i
$342$		0			0
$343$	596.142			1.73802
$344$		0			0
$345$		0			0
$346$		0			0
$347$	122.160	+	122.160i	0.352045	+	0.352045i	0.860870	−	0.508825i	$-0.169920\pi$
$347$	122.160	+	122.160i	0.352045	+	0.352045i	−0.508825	+	0.860870i	$0.669920\pi$
$348$		0			0
$349$	279.483	+	279.483i	0.800810	+	0.800810i	0.983222	−	0.182412i	$-0.0583906\pi$
$349$	279.483	+	279.483i	0.800810	+	0.800810i	−0.182412	+	0.983222i	$0.558391\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	212.266			0.601320			0.300660	−	0.953731i	$-0.402793\pi$
$353$	212.266			0.601320			0.300660	+	0.953731i	$0.402793\pi$
$354$		0			0
$355$	26.3581	−	26.3581i	0.0742482	−	0.0742482i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	435.033			1.21179			0.605895	−	0.795545i	$-0.292815\pi$
$359$	435.033			1.21179			0.605895	+	0.795545i	$0.292815\pi$
$360$		0			0
$361$			106.419i			0.294789i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−290.905	−	290.905i	−0.796999	−	0.796999i
$366$		0			0
$367$			125.535i			0.342058i	0.985266	+	0.171029i	$0.0547091\pi$
$367$			125.535i			0.342058i	−0.985266	+	0.171029i	$0.945291\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	447.295	−	447.295i	1.20565	−	1.20565i
$372$		0			0
$373$	302.389	−	302.389i	0.810694	−	0.810694i	−0.174044	−	0.984738i	$-0.555683\pi$
$373$	302.389	−	302.389i	0.810694	−	0.810694i	0.984738	+	0.174044i	$0.0556835\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	254.917i		−	0.676171i
$378$		0			0
$379$	−189.784	−	189.784i	−0.500751	−	0.500751i	0.410921	−	0.911671i	$-0.365207\pi$
$379$	−189.784	−	189.784i	−0.500751	−	0.500751i	−0.911671	+	0.410921i	$0.865207\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	639.916i		−	1.67080i	−0.549644	−	0.835399i	$-0.685237\pi$
$383$		−	639.916i		−	1.67080i	0.549644	−	0.835399i	$-0.314763\pi$
$384$		0			0
$385$	−810.623			−2.10551
$386$		0			0
$387$		0			0
$388$		0			0
$389$	499.333	−	499.333i	1.28363	−	1.28363i	0.345046	−	0.938586i	$-0.387863\pi$
$389$	499.333	−	499.333i	1.28363	−	1.28363i	0.938586	−	0.345046i	$-0.112137\pi$
$390$		0			0
$391$	−197.181			−0.504300
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−358.601	−	358.601i	−0.907851	−	0.907851i
$396$		0			0
$397$	−492.518	−	492.518i	−1.24060	−	1.24060i	−0.959753	−	0.280846i	$-0.909385\pi$
$397$	−492.518	−	492.518i	−1.24060	−	1.24060i	−0.280846	−	0.959753i	$-0.590615\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−705.045			−1.75822			−0.879109	−	0.476621i	$-0.841862\pi$
$401$	−705.045			−1.75822			−0.879109	+	0.476621i	$0.841862\pi$
$402$		0			0
$403$	204.735	−	204.735i	0.508026	−	0.508026i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	465.550			1.14386
$408$		0			0
$409$		−	279.815i		−	0.684144i	−0.939674	−	0.342072i	$-0.888871\pi$
$409$		−	279.815i		−	0.684144i	0.939674	−	0.342072i	$-0.111129\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−171.432	−	171.432i	−0.415090	−	0.415090i
$414$		0			0
$415$		−	423.147i		−	1.01963i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−573.583	+	573.583i	−1.36893	+	1.36893i	−0.506968	+	0.861965i	$0.669234\pi$
$419$	−573.583	+	573.583i	−1.36893	+	1.36893i	−0.861965	+	0.506968i	$0.830766\pi$
$420$		0			0
$421$	213.341	−	213.341i	0.506749	−	0.506749i	−0.406778	−	0.913527i	$-0.633348\pi$
$421$	213.341	−	213.341i	0.506749	−	0.506749i	0.913527	+	0.406778i	$0.133348\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	17.5358i		−	0.0412606i
$426$		0			0
$427$	515.781	+	515.781i	1.20792	+	1.20792i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			166.900i			0.387239i	0.981077	+	0.193619i	$0.0620227\pi$
$431$			166.900i			0.387239i	−0.981077	+	0.193619i	$0.937977\pi$
$432$		0			0
$433$	233.153			0.538459			0.269230	−	0.963076i	$-0.413231\pi$
$433$	233.153			0.538459			0.269230	+	0.963076i	$0.413231\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−228.619	+	228.619i	−0.523155	+	0.523155i
$438$		0			0
$439$	440.480			1.00337			0.501686	−	0.865050i	$-0.332713\pi$
$439$	440.480			1.00337			0.501686	+	0.865050i	$0.332713\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−312.524	−	312.524i	−0.705473	−	0.705473i	0.260107	−	0.965580i	$-0.416242\pi$
$443$	−312.524	−	312.524i	−0.705473	−	0.705473i	−0.965580	+	0.260107i	$0.916242\pi$
$444$		0			0
$445$	−433.114	−	433.114i	−0.973290	−	0.973290i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	734.338			1.63550			0.817748	−	0.575576i	$-0.195222\pi$
$449$	734.338			1.63550			0.817748	+	0.575576i	$0.195222\pi$
$450$		0			0
$451$	−242.772	+	242.772i	−0.538297	+	0.538297i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−641.880			−1.41073
$456$		0			0
$457$		−	692.749i		−	1.51586i	−0.652335	−	0.757931i	$-0.726211\pi$
$457$		−	692.749i		−	1.51586i	0.652335	−	0.757931i	$-0.273789\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	298.447	+	298.447i	0.647391	+	0.647391i	0.952362	−	0.304971i	$-0.0986467\pi$
$461$	298.447	+	298.447i	0.647391	+	0.647391i	−0.304971	+	0.952362i	$0.598647\pi$
$462$		0			0
$463$		−	281.830i		−	0.608705i	−0.952560	−	0.304352i	$-0.901560\pi$
$463$		−	281.830i		−	0.608705i	0.952560	−	0.304352i	$-0.0984400\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	198.116	−	198.116i	0.424232	−	0.424232i	−0.462426	−	0.886658i	$-0.653021\pi$
$467$	198.116	−	198.116i	0.424232	−	0.424232i	0.886658	+	0.462426i	$0.153021\pi$
$468$		0			0
$469$	−591.704	+	591.704i	−1.26163	+	1.26163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	586.384i		−	1.23971i
$474$		0			0
$475$	−20.3316	−	20.3316i	−0.0428033	−	0.0428033i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	917.713i		−	1.91589i	−0.286945	−	0.957947i	$-0.592640\pi$
$479$		−	917.713i		−	1.91589i	0.286945	−	0.957947i	$-0.407360\pi$
$480$		0			0
$481$	368.639			0.766401
$482$		0			0
$483$		0			0
$484$		0			0
$485$	501.826	−	501.826i	1.03469	−	1.03469i
$486$		0			0
$487$	−426.183			−0.875119			−0.437559	−	0.899190i	$-0.644157\pi$
$487$	−426.183			−0.875119			−0.437559	+	0.899190i	$0.644157\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−266.299	−	266.299i	−0.542361	−	0.542361i	0.381859	−	0.924220i	$-0.375284\pi$
$491$	−266.299	−	266.299i	−0.542361	−	0.542361i	−0.924220	+	0.381859i	$0.875284\pi$
$492$		0			0
$493$	159.653	+	159.653i	0.323840	+	0.323840i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−93.8809			−0.188895
$498$		0			0
$499$	264.104	−	264.104i	0.529266	−	0.529266i	−0.391088	−	0.920353i	$-0.627901\pi$
$499$	264.104	−	264.104i	0.529266	−	0.529266i	0.920353	+	0.391088i	$0.127901\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	574.766			1.14268			0.571338	−	0.820715i	$-0.306425\pi$
$503$	574.766			1.14268			0.571338	+	0.820715i	$0.306425\pi$
$504$		0			0
$505$			86.9756i			0.172229i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	170.592	+	170.592i	0.335152	+	0.335152i	0.854539	−	0.519387i	$-0.173840\pi$
$509$	170.592	+	170.592i	0.335152	+	0.335152i	−0.519387	+	0.854539i	$0.673840\pi$
$510$		0			0
$511$			1036.13i			2.02765i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	60.4930	−	60.4930i	0.117462	−	0.117462i
$516$		0			0
$517$	−307.398	+	307.398i	−0.594581	+	0.594581i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			37.1210i			0.0712496i	0.999365	+	0.0356248i	$0.0113421\pi$
$521$			37.1210i			0.0712496i	−0.999365	+	0.0356248i	$0.988658\pi$
$522$		0			0
$523$	199.555	+	199.555i	0.381558	+	0.381558i	0.871663	−	0.490105i	$-0.163042\pi$
$523$	199.555	+	199.555i	0.381558	+	0.381558i	−0.490105	+	0.871663i	$0.663042\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			256.449i			0.486620i
$528$		0			0
$529$	−118.392			−0.223804
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−192.236	+	192.236i	−0.360667	+	0.360667i
$534$		0			0
$535$	108.226			0.202292
$536$		0			0
$537$		0			0
$538$		0			0
$539$	962.884	+	962.884i	1.78643	+	1.78643i
$540$		0			0
$541$	−278.121	−	278.121i	−0.514086	−	0.514086i	0.401690	−	0.915776i	$-0.368423\pi$
$541$	−278.121	−	278.121i	−0.514086	−	0.514086i	−0.915776	+	0.401690i	$0.868423\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−540.323			−0.991418
$546$		0			0
$547$	−724.938	+	724.938i	−1.32530	+	1.32530i	−0.415876	+	0.909421i	$0.636525\pi$
$547$	−724.938	+	724.938i	−1.32530	+	1.32530i	−0.909421	+	0.415876i	$0.863475\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	370.215			0.671897
$552$		0			0
$553$			1277.25i			2.30967i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	268.298	+	268.298i	0.481685	+	0.481685i	0.905669	−	0.423985i	$-0.139369\pi$
$557$	268.298	+	268.298i	0.481685	+	0.481685i	−0.423985	+	0.905669i	$0.639369\pi$
$558$		0			0
$559$		−	464.320i		−	0.830625i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−78.4662	+	78.4662i	−0.139372	+	0.139372i	−0.773350	−	0.633979i	$-0.781421\pi$
$563$	−78.4662	+	78.4662i	−0.139372	+	0.139372i	0.633979	+	0.773350i	$0.281421\pi$
$564$		0			0
$565$	−570.587	+	570.587i	−1.00989	+	1.00989i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			801.999i			1.40949i	0.709461	+	0.704744i	$0.248938\pi$
$569$			801.999i			1.40949i	−0.709461	+	0.704744i	$0.751062\pi$
$570$		0			0
$571$	−79.9964	−	79.9964i	−0.140099	−	0.140099i	0.633579	−	0.773678i	$-0.281585\pi$
$571$	−79.9964	−	79.9964i	−0.140099	−	0.140099i	−0.773678	+	0.633579i	$0.781585\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			36.5163i			0.0635066i
$576$		0			0
$577$	−237.186			−0.411068			−0.205534	−	0.978650i	$-0.565893\pi$
$577$	−237.186			−0.411068			−0.205534	+	0.978650i	$0.565893\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−753.571	+	753.571i	−1.29702	+	1.29702i
$582$		0			0
$583$	723.534			1.24105
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−267.958	−	267.958i	−0.456487	−	0.456487i	0.441014	−	0.897500i	$-0.354619\pi$
$587$	−267.958	−	267.958i	−0.456487	−	0.456487i	−0.897500	+	0.441014i	$0.854619\pi$
$588$		0			0
$589$	297.336	+	297.336i	0.504815	+	0.504815i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	607.086			1.02375			0.511877	−	0.859059i	$-0.328950\pi$
$593$	607.086			1.02375			0.511877	+	0.859059i	$0.328950\pi$
$594$		0			0
$595$	402.007	−	402.007i	0.675642	−	0.675642i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−575.392			−0.960587			−0.480294	−	0.877108i	$-0.659470\pi$
$599$	−575.392			−0.960587			−0.480294	+	0.877108i	$0.659470\pi$
$600$		0			0
$601$		−	310.094i		−	0.515963i	−0.966150	−	0.257981i	$-0.916943\pi$
$601$		−	310.094i		−	0.515963i	0.966150	−	0.257981i	$-0.0830573\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−243.529	−	243.529i	−0.402528	−	0.402528i
$606$		0			0
$607$		−	556.510i		−	0.916820i	−0.888741	−	0.458410i	$-0.848419\pi$
$607$		−	556.510i		−	0.916820i	0.888741	−	0.458410i	$-0.151581\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−243.409	+	243.409i	−0.398378	+	0.398378i
$612$		0			0
$613$	326.241	−	326.241i	0.532204	−	0.532204i	−0.389024	−	0.921228i	$-0.627188\pi$
$613$	326.241	−	326.241i	0.532204	−	0.532204i	0.921228	+	0.389024i	$0.127188\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			502.068i			0.813725i	0.913490	+	0.406862i	$0.133377\pi$
$617$			502.068i			0.813725i	−0.913490	+	0.406862i	$0.866623\pi$
$618$		0			0
$619$	−304.429	−	304.429i	−0.491808	−	0.491808i	0.417067	−	0.908876i	$-0.363058\pi$
$619$	−304.429	−	304.429i	−0.491808	−	0.491808i	−0.908876	+	0.417067i	$0.863058\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1542.64i			2.47615i
$624$		0			0
$625$	576.701			0.922721
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−230.877	+	230.877i	−0.367054	+	0.367054i
$630$		0			0
$631$	8.60592			0.0136385			0.00681927	−	0.999977i	$-0.497829\pi$
$631$	8.60592			0.0136385			0.00681927	+	0.999977i	$0.497829\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−675.174	−	675.174i	−1.06327	−	1.06327i
$636$		0			0
$637$	762.446	+	762.446i	1.19693	+	1.19693i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	445.780			0.695445			0.347722	−	0.937598i	$-0.386955\pi$
$641$	445.780			0.695445			0.347722	+	0.937598i	$0.386955\pi$
$642$		0			0
$643$	118.001	−	118.001i	0.183517	−	0.183517i	−0.609369	−	0.792886i	$-0.708577\pi$
$643$	118.001	−	118.001i	0.183517	−	0.183517i	0.792886	+	0.609369i	$0.208577\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1081.35			−1.67132			−0.835662	−	0.549243i	$-0.814916\pi$
$647$	−1081.35			−1.67132			−0.835662	+	0.549243i	$0.814916\pi$
$648$		0			0
$649$		−	277.305i		−	0.427280i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−586.227	−	586.227i	−0.897744	−	0.897744i	0.0974927	−	0.995236i	$-0.468918\pi$
$653$	−586.227	−	586.227i	−0.897744	−	0.897744i	−0.995236	+	0.0974927i	$0.968918\pi$
$654$		0			0
$655$			915.041i			1.39701i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	469.999	−	469.999i	0.713201	−	0.713201i	−0.254003	−	0.967204i	$-0.581747\pi$
$659$	469.999	−	469.999i	0.713201	−	0.713201i	0.967204	+	0.254003i	$0.0817472\pi$
$660$		0			0
$661$	884.745	−	884.745i	1.33849	−	1.33849i	0.440976	−	0.897519i	$-0.354632\pi$
$661$	884.745	−	884.745i	1.33849	−	1.33849i	0.897519	−	0.440976i	$-0.145368\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	932.202i		−	1.40181i
$666$		0			0
$667$	−332.460	−	332.460i	−0.498441	−	0.498441i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			834.314i			1.24339i
$672$		0			0
$673$	684.329			1.01683			0.508417	−	0.861111i	$-0.330231\pi$
$673$	684.329			1.01683			0.508417	+	0.861111i	$0.330231\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−383.762	+	383.762i	−0.566857	+	0.566857i	−0.931246	−	0.364390i	$-0.881278\pi$
$677$	−383.762	+	383.762i	−0.566857	+	0.566857i	0.364390	+	0.931246i	$0.381278\pi$
$678$		0			0
$679$	−1787.38			−2.63237
$680$		0			0
$681$		0			0
$682$		0			0
$683$	903.626	+	903.626i	1.32302	+	1.32302i	0.911315	+	0.411709i	$0.135068\pi$
$683$	903.626	+	903.626i	1.32302	+	1.32302i	0.411709	+	0.911315i	$0.364932\pi$
$684$		0			0
$685$	871.652	+	871.652i	1.27248	+	1.27248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	572.920			0.831524
$690$		0			0
$691$	63.6870	−	63.6870i	0.0921665	−	0.0921665i	−0.659520	−	0.751687i	$-0.729241\pi$
$691$	63.6870	−	63.6870i	0.0921665	−	0.0921665i	0.751687	+	0.659520i	$0.229241\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−147.966			−0.212901
$696$		0			0
$697$		−	240.793i		−	0.345470i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	218.312	+	218.312i	0.311430	+	0.311430i	0.845463	−	0.534033i	$-0.179324\pi$
$701$	218.312	+	218.312i	0.311430	+	0.311430i	−0.534033	+	0.845463i	$0.679324\pi$
$702$		0			0
$703$			535.374i			0.761557i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	154.893	−	154.893i	0.219084	−	0.219084i
$708$		0			0
$709$	822.199	−	822.199i	1.15966	−	1.15966i	0.175112	−	0.984548i	$-0.443971\pi$
$709$	822.199	−	822.199i	1.15966	−	1.15966i	0.984548	−	0.175112i	$-0.0560288\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	534.026i		−	0.748985i
$714$		0			0
$715$	−519.145	−	519.145i	−0.726077	−	0.726077i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	340.913i		−	0.474149i	−0.971491	−	0.237074i	$-0.923811\pi$
$719$		−	340.913i		−	0.474149i	0.971491	−	0.237074i	$-0.0761885\pi$
$720$		0			0
$721$	−215.461			−0.298836
$722$		0			0
$723$		0			0
$724$		0			0
$725$	29.5664	−	29.5664i	0.0407813	−	0.0407813i
$726$		0			0
$727$	−803.090			−1.10466			−0.552331	−	0.833625i	$-0.686262\pi$
$727$	−803.090			−1.10466			−0.552331	+	0.833625i	$0.686262\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	290.802	+	290.802i	0.397813	+	0.397813i
$732$		0			0
$733$	−481.592	−	481.592i	−0.657015	−	0.657015i	0.297658	−	0.954673i	$-0.403795\pi$
$733$	−481.592	−	481.592i	−0.657015	−	0.657015i	−0.954673	+	0.297658i	$0.903795\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−957.127			−1.29868
$738$		0			0
$739$	−173.622	+	173.622i	−0.234941	+	0.234941i	−0.814752	−	0.579810i	$-0.803127\pi$
$739$	−173.622	+	173.622i	−0.234941	+	0.234941i	0.579810	+	0.814752i	$0.303127\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1316.22			−1.77149			−0.885744	−	0.464173i	$-0.846351\pi$
$743$	−1316.22			−1.77149			−0.885744	+	0.464173i	$0.846351\pi$
$744$		0			0
$745$		−	233.254i		−	0.313093i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−192.737	−	192.737i	−0.257326	−	0.257326i
$750$		0			0
$751$		−	322.977i		−	0.430062i	−0.976607	−	0.215031i	$-0.931015\pi$
$751$		−	322.977i		−	0.430062i	0.976607	−	0.215031i	$-0.0689853\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	49.2621	−	49.2621i	0.0652478	−	0.0652478i
$756$		0			0
$757$	80.2744	−	80.2744i	0.106043	−	0.106043i	−0.652095	−	0.758138i	$-0.726110\pi$
$757$	80.2744	−	80.2744i	0.106043	−	0.106043i	0.758138	+	0.652095i	$0.226110\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			596.664i			0.784053i	0.919954	+	0.392027i	$0.128226\pi$
$761$			596.664i			0.784053i	−0.919954	+	0.392027i	$0.871774\pi$
$762$		0			0
$763$	962.246	+	962.246i	1.26113	+	1.26113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	219.580i		−	0.286284i
$768$		0			0
$769$	1515.31			1.97050			0.985249	−	0.171129i	$-0.0547416\pi$
$769$	1515.31			1.97050			0.985249	+	0.171129i	$0.0547416\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	607.901	−	607.901i	0.786418	−	0.786418i	−0.194487	−	0.980905i	$-0.562304\pi$
$773$	607.901	−	607.901i	0.786418	−	0.786418i	0.980905	+	0.194487i	$0.0623042\pi$
$774$		0			0
$775$	47.4922			0.0612802
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−279.184	−	279.184i	−0.358387	−	0.358387i
$780$		0			0
$781$	−75.9297	−	75.9297i	−0.0972211	−	0.0972211i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	214.324			0.273024
$786$		0			0
$787$	−356.009	+	356.009i	−0.452362	+	0.452362i	−0.896138	−	0.443776i	$-0.853639\pi$
$787$	−356.009	+	356.009i	−0.452362	+	0.452362i	0.443776	+	0.896138i	$0.353639\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2032.29			2.56926
$792$		0			0
$793$			660.640i			0.833089i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	971.380	+	971.380i	1.21880	+	1.21880i	0.968054	+	0.250742i	$0.0806745\pi$
$797$	971.380	+	971.380i	1.21880	+	1.21880i	0.250742	+	0.968054i	$0.419326\pi$
$798$		0			0
$799$		−	304.892i		−	0.381592i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−838.008	+	838.008i	−1.04360	+	1.04360i
$804$		0			0
$805$	−837.135	+	837.135i	−1.03992	+	1.03992i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	678.276i		−	0.838412i	−0.907891	−	0.419206i	$-0.862308\pi$
$809$		−	678.276i		−	0.838412i	0.907891	−	0.419206i	$-0.137692\pi$
$810$		0			0
$811$	−204.625	−	204.625i	−0.252312	−	0.252312i	0.569606	−	0.821918i	$-0.307096\pi$
$811$	−204.625	−	204.625i	−0.252312	−	0.252312i	−0.821918	+	0.569606i	$0.807096\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	213.881i		−	0.262431i
$816$		0			0
$817$	674.331			0.825375
$818$		0			0
$819$		0			0
$820$		0			0
$821$	326.524	−	326.524i	0.397715	−	0.397715i	−0.479711	−	0.877426i	$-0.659259\pi$
$821$	326.524	−	326.524i	0.397715	−	0.397715i	0.877426	+	0.479711i	$0.159259\pi$
$822$		0			0
$823$	804.270			0.977241			0.488621	−	0.872496i	$-0.337500\pi$
$823$	804.270			0.977241			0.488621	+	0.872496i	$0.337500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	848.530	+	848.530i	1.02603	+	1.02603i	0.999652	+	0.0263821i	$0.00839864\pi$
$827$	848.530	+	848.530i	1.02603	+	1.02603i	0.0263821	+	0.999652i	$0.491601\pi$
$828$		0			0
$829$	−49.5139	−	49.5139i	−0.0597273	−	0.0597273i	0.676612	−	0.736340i	$-0.263447\pi$
$829$	−49.5139	−	49.5139i	−0.0597273	−	0.0597273i	−0.736340	+	0.676612i	$0.763447\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−955.034			−1.14650
$834$		0			0
$835$	124.240	−	124.240i	0.148790	−	0.148790i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−866.213			−1.03244			−0.516218	−	0.856457i	$-0.672660\pi$
$839$	−866.213			−1.03244			−0.516218	+	0.856457i	$0.672660\pi$
$840$		0			0
$841$		−	302.629i		−	0.359844i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	164.490	+	164.490i	0.194663	+	0.194663i
$846$		0			0
$847$			867.390i			1.02407i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	480.776	−	480.776i	0.564954	−	0.564954i
$852$		0			0
$853$	−313.947	+	313.947i	−0.368050	+	0.368050i	−0.866766	−	0.498715i	$-0.833805\pi$
$853$	−313.947	+	313.947i	−0.368050	+	0.368050i	0.498715	+	0.866766i	$0.333805\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			473.297i			0.552272i	0.961119	+	0.276136i	$0.0890540\pi$
$857$			473.297i			0.552272i	−0.961119	+	0.276136i	$0.910946\pi$
$858$		0			0
$859$	595.383	+	595.383i	0.693112	+	0.693112i	0.962915	−	0.269803i	$-0.0869586\pi$
$859$	595.383	+	595.383i	0.693112	+	0.693112i	−0.269803	+	0.962915i	$0.586959\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			742.134i			0.859947i	0.902842	+	0.429973i	$0.141477\pi$
$863$			742.134i			0.859947i	−0.902842	+	0.429973i	$0.858523\pi$
$864$		0			0
$865$	−665.083			−0.768882
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1033.02	+	1033.02i	−1.18875	+	1.18875i
$870$		0			0
$871$	−757.887			−0.870134
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1107.26	−	1107.26i	−1.26544	−	1.26544i
$876$		0			0
$877$	791.224	+	791.224i	0.902194	+	0.902194i	0.995626	−	0.0934320i	$-0.0297838\pi$
$877$	791.224	+	791.224i	0.902194	+	0.902194i	−0.0934320	+	0.995626i	$0.529784\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1524.92			1.73090			0.865450	−	0.500995i	$-0.167033\pi$
$881$	1524.92			1.73090			0.865450	+	0.500995i	$0.167033\pi$
$882$		0			0
$883$	314.328	−	314.328i	0.355978	−	0.355978i	−0.506350	−	0.862328i	$-0.669006\pi$
$883$	314.328	−	314.328i	0.355978	−	0.355978i	0.862328	+	0.506350i	$0.169006\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1520.80			1.71454			0.857271	−	0.514866i	$-0.172158\pi$
$887$	1520.80			1.71454			0.857271	+	0.514866i	$0.172158\pi$
$888$		0			0
$889$			2404.80i			2.70506i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−353.503	−	353.503i	−0.395860	−	0.395860i
$894$		0			0
$895$			611.278i			0.682992i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−432.389	+	432.389i	−0.480967	+	0.480967i
$900$		0			0
$901$	−358.818	+	358.818i	−0.398244	+	0.398244i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			786.705i			0.869288i
$906$		0			0
$907$	216.886	+	216.886i	0.239125	+	0.239125i	0.816488	−	0.577363i	$-0.195918\pi$
$907$	216.886	+	216.886i	0.239125	+	0.239125i	−0.577363	+	0.816488i	$0.695918\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			799.632i			0.877752i	0.898548	+	0.438876i	$0.144623\pi$
$911$			799.632i			0.877752i	−0.898548	+	0.438876i	$0.855377\pi$
$912$		0			0
$913$	−1218.96			−1.33511
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1629.57	−	1629.57i	1.77707	−	1.77707i
$918$		0			0
$919$	−640.590			−0.697051			−0.348525	−	0.937299i	$-0.613317\pi$
$919$	−640.590			−0.697051			−0.348525	+	0.937299i	$0.613317\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−60.1238	−	60.1238i	−0.0651396	−	0.0651396i
$924$		0			0
$925$	42.7565	+	42.7565i	0.0462232	+	0.0462232i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−118.633			−0.127699			−0.0638496	−	0.997960i	$-0.520338\pi$
$929$	−118.633			−0.127699			−0.0638496	+	0.997960i	$0.520338\pi$
$930$		0			0
$931$	−1107.30	+	1107.30i	−1.18937	+	1.18937i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	650.277			0.695483
$936$		0			0
$937$		−	731.334i		−	0.780506i	−0.920708	−	0.390253i	$-0.872388\pi$
$937$		−	731.334i		−	0.780506i	0.920708	−	0.390253i	$-0.127612\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	980.281	+	980.281i	1.04174	+	1.04174i	0.999090	+	0.0426536i	$0.0135812\pi$
$941$	980.281	+	980.281i	1.04174	+	1.04174i	0.0426536	+	0.999090i	$0.486419\pi$
$942$		0			0
$943$			501.424i			0.531733i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−240.008	+	240.008i	−0.253441	+	0.253441i	−0.822380	−	0.568939i	$-0.807354\pi$
$947$	−240.008	+	240.008i	−0.253441	+	0.253441i	0.568939	+	0.822380i	$0.307354\pi$
$948$		0			0
$949$	−663.564	+	663.564i	−0.699225	+	0.699225i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	780.049i		−	0.818519i	−0.912418	−	0.409259i	$-0.865787\pi$
$953$		−	780.049i		−	0.818519i	0.912418	−	0.409259i	$-0.134213\pi$
$954$		0			0
$955$	213.293	+	213.293i	0.223344	+	0.223344i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	3104.60i		−	3.23733i
$960$		0			0
$961$	266.459			0.277272
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−760.642	+	760.642i	−0.788230	+	0.788230i
$966$		0			0
$967$	1783.10			1.84395			0.921975	−	0.387249i	$-0.126575\pi$
$967$	1783.10			1.84395			0.921975	+	0.387249i	$0.126575\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−159.340	−	159.340i	−0.164099	−	0.164099i	0.620281	−	0.784380i	$-0.287019\pi$
$971$	−159.340	−	159.340i	−0.164099	−	0.164099i	−0.784380	+	0.620281i	$0.787019\pi$
$972$		0			0
$973$	263.509	+	263.509i	0.270821	+	0.270821i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	970.922			0.993779			0.496889	−	0.867814i	$-0.334475\pi$
$977$	970.922			0.993779			0.496889	+	0.867814i	$0.334475\pi$
$978$		0			0
$979$	−1247.67	+	1247.67i	−1.27443	+	1.27443i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1266.90			−1.28881			−0.644406	−	0.764684i	$-0.722895\pi$
$983$	−1266.90			−1.28881			−0.644406	+	0.764684i	$0.722895\pi$
$984$		0			0
$985$			198.185i			0.201203i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−605.562	−	605.562i	−0.612297	−	0.612297i
$990$		0			0
$991$			222.422i			0.224442i	0.993683	+	0.112221i	$0.0357964\pi$
$991$			222.422i			0.224442i	−0.993683	+	0.112221i	$0.964204\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−39.8359	+	39.8359i	−0.0400360	+	0.0400360i
$996$		0			0
$997$	441.746	−	441.746i	0.443075	−	0.443075i	−0.449969	−	0.893044i	$-0.648565\pi$
$997$	441.746	−	441.746i	0.443075	−	0.443075i	0.893044	+	0.449969i	$0.148565\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.f.991.3		16
3.2	odd	2		384.3.l.a.223.7		16
4.3	odd	2		1152.3.m.c.991.3		16
8.3	odd	2		576.3.m.c.559.6		16
8.5	even	2		144.3.m.c.19.4		16
12.11	even	2		384.3.l.b.223.3		16
16.3	odd	4		144.3.m.c.91.4		16
16.5	even	4		1152.3.m.c.415.3		16
16.11	odd	4	inner	1152.3.m.f.415.3		16
16.13	even	4		576.3.m.c.271.6		16
24.5	odd	2		48.3.l.a.19.5	✓	16
24.11	even	2		192.3.l.a.175.6		16
48.5	odd	4		384.3.l.b.31.3		16
48.11	even	4		384.3.l.a.31.7		16
48.29	odd	4		192.3.l.a.79.6		16
48.35	even	4		48.3.l.a.43.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.5	✓	16	24.5	odd	2
48.3.l.a.43.5	yes	16	48.35	even	4
144.3.m.c.19.4		16	8.5	even	2
144.3.m.c.91.4		16	16.3	odd	4
192.3.l.a.79.6		16	48.29	odd	4
192.3.l.a.175.6		16	24.11	even	2
384.3.l.a.31.7		16	48.11	even	4
384.3.l.a.223.7		16	3.2	odd	2
384.3.l.b.31.3		16	48.5	odd	4
384.3.l.b.223.3		16	12.11	even	2
576.3.m.c.271.6		16	16.13	even	4
576.3.m.c.559.6		16	8.3	odd	2
1152.3.m.c.415.3		16	16.5	even	4
1152.3.m.c.991.3		16	4.3	odd	2
1152.3.m.f.415.3		16	16.11	odd	4	inner
1152.3.m.f.991.3		16	1.1	even	1	trivial

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.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.40572 + 3.40572i) q^{5} +12.1303 q^{7} +O(q^{10})\)	\(q+(-3.40572 + 3.40572i) q^{5} +12.1303 q^{7} +(9.81086 + 9.81086i) q^{11} +(7.76859 + 7.76859i) q^{13} -9.73087 q^{17} +(-11.2823 + 11.2823i) q^{19} +20.2635 q^{23} +1.80207i q^{25} +(-16.4069 - 16.4069i) q^{29} -26.3542i q^{31} +(-41.3125 + 41.3125i) q^{35} +(23.7263 - 23.7263i) q^{37} +24.7452i q^{41} +(-29.8844 - 29.8844i) q^{43} +31.3325i q^{47} +98.1448 q^{49} +(36.8742 - 36.8742i) q^{53} -66.8262 q^{55} +(-14.1325 - 14.1325i) q^{59} +(42.5199 + 42.5199i) q^{61} -52.9153 q^{65} +(-48.7789 + 48.7789i) q^{67} -7.73935 q^{71} +85.4163i q^{73} +(119.009 + 119.009i) q^{77} +105.294i q^{79} +(-62.1229 + 62.1229i) q^{83} +(33.1407 - 33.1407i) q^{85} +127.172i q^{89} +(94.2355 + 94.2355i) q^{91} -76.8489i q^{95} -147.348 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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