LMFDB - Embedded newform 1152.3.m.f.991.1

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			991.1
Root			$-1.87459 - 0.697079i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.f.415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.24354 + 5.24354i) q^{5} -5.32796 q^{7} +O(q^{10})$	$q+(-5.24354 + 5.24354i) q^{5} -5.32796 q^{7} +(12.2863 + 12.2863i) q^{11} +(5.73657 + 5.73657i) q^{13} +23.3997 q^{17} +(-11.7492 + 11.7492i) q^{19} -5.80841 q^{23} -29.9894i q^{25} +(18.3914 + 18.3914i) q^{29} -16.9053i q^{31} +(27.9374 - 27.9374i) q^{35} +(-15.3391 + 15.3391i) q^{37} +29.2351i q^{41} +(-33.4099 - 33.4099i) q^{43} +18.2125i q^{47} -20.6128 q^{49} +(-66.9856 + 66.9856i) q^{53} -128.847 q^{55} +(-27.1523 - 27.1523i) q^{59} +(-65.2399 - 65.2399i) q^{61} -60.1599 q^{65} +(37.6951 - 37.6951i) q^{67} -42.6559 q^{71} +106.391i q^{73} +(-65.4607 - 65.4607i) q^{77} -21.2821i q^{79} +(24.1638 - 24.1638i) q^{83} +(-122.697 + 122.697i) q^{85} -52.8029i q^{89} +(-30.5643 - 30.5643i) q^{91} -123.215i q^{95} -21.0222 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5.24354	+	5.24354i	−1.04871	+	1.04871i	−0.0499563	+	0.998751i	$0.515908\pi$
$5$	−5.24354	+	5.24354i	−1.04871	+	1.04871i	−0.998751	+	0.0499563i	$0.984092\pi$
$6$		0			0
$7$	−5.32796			−0.761138			−0.380569	−	0.924753i	$-0.624272\pi$
$7$	−5.32796			−0.761138			−0.380569	+	0.924753i	$0.624272\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.2863	+	12.2863i	1.11693	+	1.11693i	0.992189	+	0.124743i	$0.0398107\pi$
$11$	12.2863	+	12.2863i	1.11693	+	1.11693i	0.124743	+	0.992189i	$0.460189\pi$
$12$		0			0
$13$	5.73657	+	5.73657i	0.441275	+	0.441275i	0.892440	−	0.451165i	$-0.148992\pi$
$13$	5.73657	+	5.73657i	0.441275	+	0.441275i	−0.451165	+	0.892440i	$0.648992\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	23.3997			1.37645			0.688226	−	0.725496i	$-0.258390\pi$
$17$	23.3997			1.37645			0.688226	+	0.725496i	$0.258390\pi$
$18$		0			0
$19$	−11.7492	+	11.7492i	−0.618380	+	0.618380i	−0.945116	−	0.326736i	$-0.894051\pi$
$19$	−11.7492	+	11.7492i	−0.618380	+	0.618380i	0.326736	+	0.945116i	$0.394051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.80841			−0.252540			−0.126270	−	0.991996i	$-0.540301\pi$
$23$	−5.80841			−0.252540			−0.126270	+	0.991996i	$0.540301\pi$
$24$		0			0
$25$		−	29.9894i		−	1.19958i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	18.3914	+	18.3914i	0.634185	+	0.634185i	0.949115	−	0.314930i	$-0.101981\pi$
$29$	18.3914	+	18.3914i	0.634185	+	0.634185i	−0.314930	+	0.949115i	$0.601981\pi$
$30$		0			0
$31$		−	16.9053i		−	0.545332i	−0.962109	−	0.272666i	$-0.912095\pi$
$31$		−	16.9053i		−	0.545332i	0.962109	−	0.272666i	$-0.0879053\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	27.9374	−	27.9374i	0.798211	−	0.798211i
$36$		0			0
$37$	−15.3391	+	15.3391i	−0.414571	+	0.414571i	−0.883327	−	0.468756i	$-0.844702\pi$
$37$	−15.3391	+	15.3391i	−0.414571	+	0.414571i	0.468756	+	0.883327i	$0.344702\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			29.2351i			0.713051i	0.934286	+	0.356526i	$0.116039\pi$
$41$			29.2351i			0.713051i	−0.934286	+	0.356526i	$0.883961\pi$
$42$		0			0
$43$	−33.4099	−	33.4099i	−0.776975	−	0.776975i	0.202340	−	0.979315i	$-0.435145\pi$
$43$	−33.4099	−	33.4099i	−0.776975	−	0.776975i	−0.979315	+	0.202340i	$0.935145\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			18.2125i			0.387500i	0.981051	+	0.193750i	$0.0620650\pi$
$47$			18.2125i			0.387500i	−0.981051	+	0.193750i	$0.937935\pi$
$48$		0			0
$49$	−20.6128			−0.420670
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−66.9856	+	66.9856i	−1.26388	+	1.26388i	−0.314681	+	0.949197i	$0.601898\pi$
$53$	−66.9856	+	66.9856i	−1.26388	+	1.26388i	−0.949197	+	0.314681i	$0.898102\pi$
$54$		0			0
$55$	−128.847			−2.34267
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−27.1523	−	27.1523i	−0.460209	−	0.460209i	0.438515	−	0.898724i	$-0.355505\pi$
$59$	−27.1523	−	27.1523i	−0.460209	−	0.460209i	−0.898724	+	0.438515i	$0.855505\pi$
$60$		0			0
$61$	−65.2399	−	65.2399i	−1.06951	−	1.06951i	−0.997397	−	0.0721103i	$-0.977027\pi$
$61$	−65.2399	−	65.2399i	−1.06951	−	1.06951i	−0.0721103	−	0.997397i	$-0.522973\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−60.1599			−0.925537
$66$		0			0
$67$	37.6951	−	37.6951i	0.562614	−	0.562614i	−0.367435	−	0.930049i	$-0.619764\pi$
$67$	37.6951	−	37.6951i	0.562614	−	0.562614i	0.930049	+	0.367435i	$0.119764\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−42.6559			−0.600788			−0.300394	−	0.953815i	$-0.597118\pi$
$71$	−42.6559			−0.600788			−0.300394	+	0.953815i	$0.597118\pi$
$72$		0			0
$73$			106.391i			1.45742i	0.684825	+	0.728708i	$0.259879\pi$
$73$			106.391i			1.45742i	−0.684825	+	0.728708i	$0.740121\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−65.4607	−	65.4607i	−0.850139	−	0.850139i
$78$		0			0
$79$		−	21.2821i		−	0.269394i	−0.990887	−	0.134697i	$-0.956994\pi$
$79$		−	21.2821i		−	0.269394i	0.990887	−	0.134697i	$-0.0430061\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	24.1638	−	24.1638i	0.291130	−	0.291130i	−0.546396	−	0.837527i	$-0.684001\pi$
$83$	24.1638	−	24.1638i	0.291130	−	0.291130i	0.837527	+	0.546396i	$0.184001\pi$
$84$		0			0
$85$	−122.697	+	122.697i	−1.44350	+	1.44350i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	52.8029i		−	0.593291i	−0.954988	−	0.296645i	$-0.904132\pi$
$89$		−	52.8029i		−	0.593291i	0.954988	−	0.296645i	$-0.0958679\pi$
$90$		0			0
$91$	−30.5643	−	30.5643i	−0.335871	−	0.335871i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	123.215i		−	1.29700i
$96$		0			0
$97$	−21.0222			−0.216724			−0.108362	−	0.994112i	$-0.534560\pi$
$97$	−21.0222			−0.216724			−0.108362	+	0.994112i	$0.534560\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.24960	+	3.24960i	−0.0321743	+	0.0321743i	−0.723011	−	0.690837i	$-0.757242\pi$
$101$	−3.24960	+	3.24960i	−0.0321743	+	0.0321743i	0.690837	+	0.723011i	$0.257242\pi$
$102$		0			0
$103$	105.112			1.02050			0.510252	−	0.860025i	$-0.329552\pi$
$103$	105.112			1.02050			0.510252	+	0.860025i	$0.329552\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−99.6160	−	99.6160i	−0.930991	−	0.930991i	0.0667770	−	0.997768i	$-0.478728\pi$
$107$	−99.6160	−	99.6160i	−0.930991	−	0.930991i	−0.997768	+	0.0667770i	$0.978728\pi$
$108$		0			0
$109$	108.050	+	108.050i	0.991282	+	0.991282i	0.999962	−	0.00868078i	$-0.00276321\pi$
$109$	108.050	+	108.050i	0.991282	+	0.991282i	−0.00868078	+	0.999962i	$0.502763\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	23.2835			0.206048			0.103024	−	0.994679i	$-0.467148\pi$
$113$	23.2835			0.206048			0.103024	+	0.994679i	$0.467148\pi$
$114$		0			0
$115$	30.4566	−	30.4566i	0.264840	−	0.264840i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−124.673			−1.04767
$120$		0			0
$121$			180.904i			1.49508i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	26.1621	+	26.1621i	0.209297	+	0.209297i
$126$		0			0
$127$		−	118.180i		−	0.930550i	−0.885166	−	0.465275i	$-0.845955\pi$
$127$		−	118.180i		−	0.930550i	0.885166	−	0.465275i	$-0.154045\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−69.2067	+	69.2067i	−0.528296	+	0.528296i	−0.920064	−	0.391768i	$-0.871863\pi$
$131$	−69.2067	+	69.2067i	−0.528296	+	0.528296i	0.391768	+	0.920064i	$0.371863\pi$
$132$		0			0
$133$	62.5994	−	62.5994i	0.470672	−	0.470672i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			124.474i			0.908572i	0.890856	+	0.454286i	$0.150106\pi$
$137$			124.474i			0.908572i	−0.890856	+	0.454286i	$0.849894\pi$
$138$		0			0
$139$	−169.014	−	169.014i	−1.21593	−	1.21593i	−0.969046	−	0.246881i	$-0.920594\pi$
$139$	−169.014	−	169.014i	−1.21593	−	1.21593i	−0.246881	−	0.969046i	$-0.579406\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			140.962i			0.985749i
$144$		0			0
$145$	−192.872			−1.33015
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146.988	−	146.988i	0.986495	−	0.986495i	−0.0134145	−	0.999910i	$-0.504270\pi$
$149$	146.988	−	146.988i	0.986495	−	0.986495i	0.999910	+	0.0134145i	$0.00427011\pi$
$150$		0			0
$151$	75.5456			0.500302			0.250151	−	0.968207i	$-0.419520\pi$
$151$	75.5456			0.500302			0.250151	+	0.968207i	$0.419520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	88.6435	+	88.6435i	0.571893	+	0.571893i
$156$		0			0
$157$	81.5356	+	81.5356i	0.519335	+	0.519335i	0.917370	−	0.398035i	$-0.130308\pi$
$157$	81.5356	+	81.5356i	0.519335	+	0.519335i	−0.398035	+	0.917370i	$0.630308\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	30.9470			0.192217
$162$		0			0
$163$	−55.8065	+	55.8065i	−0.342371	+	0.342371i	−0.857258	−	0.514887i	$-0.827834\pi$
$163$	−55.8065	+	55.8065i	−0.342371	+	0.342371i	0.514887	+	0.857258i	$0.327834\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	24.6339			0.147508			0.0737540	−	0.997276i	$-0.476502\pi$
$167$	24.6339			0.147508			0.0737540	+	0.997276i	$0.476502\pi$
$168$		0			0
$169$		−	103.183i		−	0.610553i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.88551	+	4.88551i	0.0282399	+	0.0282399i	0.721086	−	0.692846i	$-0.243643\pi$
$173$	4.88551	+	4.88551i	0.0282399	+	0.0282399i	−0.692846	+	0.721086i	$0.743643\pi$
$174$		0			0
$175$			159.782i			0.913042i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−229.504	+	229.504i	−1.28215	+	1.28215i	−0.342702	+	0.939444i	$0.611342\pi$
$179$	−229.504	+	229.504i	−1.28215	+	1.28215i	−0.939444	+	0.342702i	$0.888658\pi$
$180$		0			0
$181$	−116.607	+	116.607i	−0.644238	+	0.644238i	−0.951595	−	0.307356i	$-0.900556\pi$
$181$	−116.607	+	116.607i	−0.644238	+	0.644238i	0.307356	+	0.951595i	$0.400556\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	160.863i		−	0.869528i
$186$		0			0
$187$	287.495	+	287.495i	1.53740	+	1.53740i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	94.2316i		−	0.493359i	−0.969097	−	0.246680i	$-0.920660\pi$
$191$		−	94.2316i		−	0.493359i	0.969097	−	0.246680i	$-0.0793395\pi$
$192$		0			0
$193$	84.2667			0.436615			0.218308	−	0.975880i	$-0.429946\pi$
$193$	84.2667			0.436615			0.218308	+	0.975880i	$0.429946\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−56.9578	+	56.9578i	−0.289126	+	0.289126i	−0.836734	−	0.547609i	$-0.815538\pi$
$197$	−56.9578	+	56.9578i	−0.289126	+	0.289126i	0.547609	+	0.836734i	$0.315538\pi$
$198$		0			0
$199$	−196.179			−0.985827			−0.492913	−	0.870078i	$-0.664068\pi$
$199$	−196.179			−0.985827			−0.492913	+	0.870078i	$0.664068\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−97.9886	−	97.9886i	−0.482702	−	0.482702i
$204$		0			0
$205$	−153.295	−	153.295i	−0.747782	−	0.747782i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−288.708			−1.38138
$210$		0			0
$211$	−177.340	+	177.340i	−0.840475	+	0.840475i	−0.988921	−	0.148445i	$-0.952573\pi$
$211$	−177.340	+	177.340i	−0.840475	+	0.840475i	0.148445	+	0.988921i	$0.452573\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	350.373			1.62964
$216$		0			0
$217$			90.0707i			0.415072i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	134.234	+	134.234i	0.607394	+	0.607394i
$222$		0			0
$223$		−	377.924i		−	1.69473i	−0.531012	−	0.847364i	$-0.678188\pi$
$223$		−	377.924i		−	1.69473i	0.531012	−	0.847364i	$-0.321812\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	103.909	−	103.909i	0.457750	−	0.457750i	−0.440166	−	0.897916i	$-0.645080\pi$
$227$	103.909	−	103.909i	0.457750	−	0.457750i	0.897916	+	0.440166i	$0.145080\pi$
$228$		0			0
$229$	101.055	−	101.055i	0.441290	−	0.441290i	−0.451156	−	0.892445i	$-0.648988\pi$
$229$	101.055	−	101.055i	0.441290	−	0.441290i	0.892445	+	0.451156i	$0.148988\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			287.259i			1.23287i	0.787405	+	0.616436i	$0.211424\pi$
$233$			287.259i			1.23287i	−0.787405	+	0.616436i	$0.788576\pi$
$234$		0			0
$235$	−95.4979	−	95.4979i	−0.406374	−	0.406374i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	150.941i		−	0.631554i	−0.948833	−	0.315777i	$-0.897735\pi$
$239$		−	150.941i		−	0.631554i	0.948833	−	0.315777i	$-0.102265\pi$
$240$		0			0
$241$	37.7817			0.156771			0.0783853	−	0.996923i	$-0.475024\pi$
$241$	37.7817			0.156771			0.0783853	+	0.996923i	$0.475024\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	108.084	−	108.084i	0.441159	−	0.441159i
$246$		0			0
$247$	−134.800			−0.545751
$248$		0			0
$249$		0			0
$250$		0			0
$251$	100.915	+	100.915i	0.402050	+	0.402050i	0.878955	−	0.476905i	$-0.158241\pi$
$251$	100.915	+	100.915i	0.402050	+	0.402050i	−0.476905	+	0.878955i	$0.658241\pi$
$252$		0			0
$253$	−71.3637	−	71.3637i	−0.282070	−	0.282070i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−241.295			−0.938891			−0.469446	−	0.882961i	$-0.655546\pi$
$257$	−241.295			−0.938891			−0.469446	+	0.882961i	$0.655546\pi$
$258$		0			0
$259$	81.7263	−	81.7263i	0.315546	−	0.315546i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	118.747			0.451509			0.225754	−	0.974184i	$-0.427515\pi$
$263$	118.747			0.451509			0.225754	+	0.974184i	$0.427515\pi$
$264$		0			0
$265$		−	702.483i		−	2.65088i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.74853	+	7.74853i	0.0288050	+	0.0288050i	0.721363	−	0.692558i	$-0.243516\pi$
$269$	7.74853	+	7.74853i	0.0288050	+	0.0288050i	−0.692558	+	0.721363i	$0.743516\pi$
$270$		0			0
$271$			131.899i			0.486712i	0.969937	+	0.243356i	$0.0782484\pi$
$271$			131.899i			0.486712i	−0.969937	+	0.243356i	$0.921752\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	368.457	−	368.457i	1.33984	−	1.33984i
$276$		0			0
$277$	202.352	−	202.352i	0.730513	−	0.730513i	−0.240208	−	0.970721i	$-0.577216\pi$
$277$	202.352	−	202.352i	0.730513	−	0.730513i	0.970721	+	0.240208i	$0.0772157\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	68.8493i		−	0.245015i	−0.992468	−	0.122508i	$-0.960906\pi$
$281$		−	68.8493i		−	0.245015i	0.992468	−	0.122508i	$-0.0390936\pi$
$282$		0			0
$283$	−206.773	−	206.773i	−0.730646	−	0.730646i	0.240102	−	0.970748i	$-0.422819\pi$
$283$	−206.773	−	206.773i	−0.730646	−	0.730646i	−0.970748	+	0.240102i	$0.922819\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	155.764i		−	0.542730i
$288$		0			0
$289$	258.545			0.894620
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−361.237	+	361.237i	−1.23289	+	1.23289i	−0.270043	+	0.962848i	$0.587038\pi$
$293$	−361.237	+	361.237i	−1.23289	+	1.23289i	−0.962848	+	0.270043i	$0.912962\pi$
$294$		0			0
$295$	284.749			0.965250
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−33.3204	−	33.3204i	−0.111439	−	0.111439i
$300$		0			0
$301$	178.007	+	178.007i	0.591385	+	0.591385i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	684.176			2.24320
$306$		0			0
$307$	10.9073	−	10.9073i	0.0355286	−	0.0355286i	−0.689119	−	0.724648i	$-0.742002\pi$
$307$	10.9073	−	10.9073i	0.0355286	−	0.0355286i	0.724648	+	0.689119i	$0.242002\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	160.251			0.515278			0.257639	−	0.966241i	$-0.417055\pi$
$311$	160.251			0.515278			0.257639	+	0.966241i	$0.417055\pi$
$312$		0			0
$313$			355.500i			1.13578i	0.823103	+	0.567892i	$0.192241\pi$
$313$			355.500i			1.13578i	−0.823103	+	0.567892i	$0.807759\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	72.5192	+	72.5192i	0.228767	+	0.228767i	0.812178	−	0.583410i	$-0.198282\pi$
$317$	72.5192	+	72.5192i	0.228767	+	0.228767i	−0.583410	+	0.812178i	$0.698282\pi$
$318$		0			0
$319$			451.922i			1.41668i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−274.928	+	274.928i	−0.851170	+	0.851170i
$324$		0			0
$325$	172.036	−	172.036i	0.529343	−	0.529343i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	97.0355i		−	0.294941i
$330$		0			0
$331$	248.096	+	248.096i	0.749536	+	0.749536i	0.974392	−	0.224856i	$-0.0721912\pi$
$331$	248.096	+	248.096i	0.749536	+	0.749536i	−0.224856	+	0.974392i	$0.572191\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			395.312i			1.18003i
$336$		0			0
$337$	−467.271			−1.38656			−0.693280	−	0.720668i	$-0.743835\pi$
$337$	−467.271			−1.38656			−0.693280	+	0.720668i	$0.743835\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	207.703	−	207.703i	0.609098	−	0.609098i
$342$		0			0
$343$	370.894			1.08133
$344$		0			0
$345$		0			0
$346$		0			0
$347$	292.821	+	292.821i	0.843863	+	0.843863i	0.989359	−	0.145496i	$-0.0464776\pi$
$347$	292.821	+	292.821i	0.843863	+	0.843863i	−0.145496	+	0.989359i	$0.546478\pi$
$348$		0			0
$349$	−346.260	−	346.260i	−0.992150	−	0.992150i	0.00781941	−	0.999969i	$-0.497511\pi$
$349$	−346.260	−	346.260i	−0.992150	−	0.992150i	−0.999969	+	0.00781941i	$0.997511\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.01816			−0.0227143			−0.0113572	−	0.999936i	$-0.503615\pi$
$353$	−8.01816			−0.0227143			−0.0113572	+	0.999936i	$0.503615\pi$
$354$		0			0
$355$	223.668	−	223.668i	0.630051	−	0.630051i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−590.403			−1.64458			−0.822289	−	0.569071i	$-0.807303\pi$
$359$	−590.403			−1.64458			−0.822289	+	0.569071i	$0.807303\pi$
$360$		0			0
$361$			84.9121i			0.235213i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−557.867	−	557.867i	−1.52840	−	1.52840i
$366$		0			0
$367$			397.100i			1.08202i	0.841017	+	0.541008i	$0.181957\pi$
$367$			397.100i			1.08202i	−0.841017	+	0.541008i	$0.818043\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	356.897	−	356.897i	0.961986	−	0.961986i
$372$		0			0
$373$	165.010	−	165.010i	0.442387	−	0.442387i	−0.450427	−	0.892814i	$-0.648728\pi$
$373$	165.010	−	165.010i	0.442387	−	0.442387i	0.892814	+	0.450427i	$0.148728\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			211.007i			0.559700i
$378$		0			0
$379$	206.669	+	206.669i	0.545300	+	0.545300i	0.925078	−	0.379778i	$-0.124000\pi$
$379$	206.669	+	206.669i	0.545300	+	0.545300i	−0.379778	+	0.925078i	$0.624000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			598.414i			1.56244i	0.624257	+	0.781219i	$0.285402\pi$
$383$			598.414i			1.56244i	−0.624257	+	0.781219i	$0.714598\pi$
$384$		0			0
$385$	686.492			1.78310
$386$		0			0
$387$		0			0
$388$		0			0
$389$	186.696	−	186.696i	0.479939	−	0.479939i	−0.425173	−	0.905112i	$-0.639787\pi$
$389$	186.696	−	186.696i	0.479939	−	0.479939i	0.905112	+	0.425173i	$0.139787\pi$
$390$		0			0
$391$	−135.915			−0.347609
$392$		0			0
$393$		0			0
$394$		0			0
$395$	111.594	+	111.594i	0.282515	+	0.282515i
$396$		0			0
$397$	57.3727	+	57.3727i	0.144516	+	0.144516i	0.775663	−	0.631147i	$-0.217416\pi$
$397$	57.3727	+	57.3727i	0.144516	+	0.144516i	−0.631147	+	0.775663i	$0.717416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	466.082			1.16230			0.581149	−	0.813797i	$-0.302603\pi$
$401$	466.082			1.16230			0.581149	+	0.813797i	$0.302603\pi$
$402$		0			0
$403$	96.9784	−	96.9784i	0.240641	−	0.240641i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−376.921			−0.926096
$408$		0			0
$409$		−	597.952i		−	1.46198i	−0.682386	−	0.730992i	$-0.739058\pi$
$409$		−	597.952i		−	1.46198i	0.682386	−	0.730992i	$-0.260942\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	144.667	+	144.667i	0.350282	+	0.350282i
$414$		0			0
$415$			253.408i			0.610621i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.65301	+	4.65301i	−0.0111050	+	0.0111050i	−0.712638	−	0.701532i	$-0.752500\pi$
$419$	−4.65301	+	4.65301i	−0.0111050	+	0.0111050i	0.701532	+	0.712638i	$0.252500\pi$
$420$		0			0
$421$	−34.3754	+	34.3754i	−0.0816519	+	0.0816519i	−0.746753	−	0.665101i	$-0.768388\pi$
$421$	−34.3754	+	34.3754i	−0.0816519	+	0.0816519i	0.665101	+	0.746753i	$0.268388\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	701.742i		−	1.65116i
$426$		0			0
$427$	347.596	+	347.596i	0.814042	+	0.814042i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			423.823i			0.983347i	0.870780	+	0.491674i	$0.163615\pi$
$431$			423.823i			0.983347i	−0.870780	+	0.491674i	$0.836385\pi$
$432$		0			0
$433$	833.377			1.92466			0.962330	−	0.271885i	$-0.0876472\pi$
$433$	833.377			1.92466			0.962330	+	0.271885i	$0.0876472\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	68.2443	−	68.2443i	0.156165	−	0.156165i
$438$		0			0
$439$	32.3193			0.0736203			0.0368102	−	0.999322i	$-0.488280\pi$
$439$	32.3193			0.0736203			0.0368102	+	0.999322i	$0.488280\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	119.527	+	119.527i	0.269813	+	0.269813i	0.829025	−	0.559212i	$-0.188896\pi$
$443$	119.527	+	119.527i	0.269813	+	0.269813i	−0.559212	+	0.829025i	$0.688896\pi$
$444$		0			0
$445$	276.874	+	276.874i	0.622189	+	0.622189i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	182.359			0.406146			0.203073	−	0.979164i	$-0.434907\pi$
$449$	182.359			0.406146			0.203073	+	0.979164i	$0.434907\pi$
$450$		0			0
$451$	−359.190	+	359.190i	−0.796430	+	0.796430i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	320.530			0.704461
$456$		0			0
$457$			272.942i			0.597246i	0.954371	+	0.298623i	$0.0965274\pi$
$457$			272.942i			0.597246i	−0.954371	+	0.298623i	$0.903473\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	188.323	+	188.323i	0.408510	+	0.408510i	0.881219	−	0.472709i	$-0.156724\pi$
$461$	188.323	+	188.323i	0.408510	+	0.408510i	−0.472709	+	0.881219i	$0.656724\pi$
$462$		0			0
$463$		−	116.023i		−	0.250590i	−0.992120	−	0.125295i	$-0.960012\pi$
$463$		−	116.023i		−	0.250590i	0.992120	−	0.125295i	$-0.0399877\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−271.914	+	271.914i	−0.582257	+	0.582257i	−0.935523	−	0.353266i	$-0.885071\pi$
$467$	−271.914	+	271.914i	−0.582257	+	0.582257i	0.353266	+	0.935523i	$0.385071\pi$
$468$		0			0
$469$	−200.838	+	200.838i	−0.428227	+	0.428227i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	820.966i		−	1.73566i
$474$		0			0
$475$	352.352	+	352.352i	0.741793	+	0.741793i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	775.808i		−	1.61964i	−0.586678	−	0.809820i	$-0.699565\pi$
$479$		−	775.808i		−	1.61964i	0.586678	−	0.809820i	$-0.300435\pi$
$480$		0			0
$481$	−175.988			−0.365880
$482$		0			0
$483$		0			0
$484$		0			0
$485$	110.231	−	110.231i	0.227280	−	0.227280i
$486$		0			0
$487$	174.891			0.359118			0.179559	−	0.983747i	$-0.442533\pi$
$487$	174.891			0.359118			0.179559	+	0.983747i	$0.442533\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−348.578	−	348.578i	−0.709934	−	0.709934i	0.256587	−	0.966521i	$-0.417402\pi$
$491$	−348.578	−	348.578i	−0.709934	−	0.709934i	−0.966521	+	0.256587i	$0.917402\pi$
$492$		0			0
$493$	430.352	+	430.352i	0.872926	+	0.872926i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	227.269			0.457282
$498$		0			0
$499$	607.544	−	607.544i	1.21752	−	1.21752i	0.249027	−	0.968496i	$-0.419889\pi$
$499$	607.544	−	607.544i	1.21752	−	1.21752i	0.968496	−	0.249027i	$-0.0801109\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	130.935			0.260309			0.130154	−	0.991494i	$-0.458453\pi$
$503$	130.935			0.260309			0.130154	+	0.991494i	$0.458453\pi$
$504$		0			0
$505$		−	34.0789i		−	0.0674829i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−61.5539	−	61.5539i	−0.120931	−	0.120931i	0.644051	−	0.764982i	$-0.277252\pi$
$509$	−61.5539	−	61.5539i	−0.120931	−	0.120931i	−0.764982	+	0.644051i	$0.777252\pi$
$510$		0			0
$511$		−	566.849i		−	1.10929i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−551.159	+	551.159i	−1.07021	+	1.07021i
$516$		0			0
$517$	−223.763	+	223.763i	−0.432811	+	0.432811i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	32.5929i		−	0.0625584i	−0.999511	−	0.0312792i	$-0.990042\pi$
$521$		−	32.5929i		−	0.0625584i	0.999511	−	0.0312792i	$-0.00995810\pi$
$522$		0			0
$523$	226.407	+	226.407i	0.432900	+	0.432900i	0.889614	−	0.456713i	$-0.150974\pi$
$523$	226.407	+	226.407i	0.432900	+	0.432900i	−0.456713	+	0.889614i	$0.650974\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	395.578i		−	0.750623i
$528$		0			0
$529$	−495.262			−0.936224
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−167.709	+	167.709i	−0.314652	+	0.314652i
$534$		0			0
$535$	1044.68			1.95267
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−253.254	−	253.254i	−0.469859	−	0.469859i
$540$		0			0
$541$	−510.912	−	510.912i	−0.944385	−	0.944385i	0.0541480	−	0.998533i	$-0.482756\pi$
$541$	−510.912	−	510.912i	−0.944385	−	0.944385i	−0.998533	+	0.0541480i	$0.982756\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1133.13			−2.07913
$546$		0			0
$547$	−512.889	+	512.889i	−0.937639	+	0.937639i	−0.998167	−	0.0605271i	$-0.980722\pi$
$547$	−512.889	+	512.889i	−0.937639	+	0.937639i	0.0605271	+	0.998167i	$0.480722\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−432.168			−0.784334
$552$		0			0
$553$			113.390i			0.205046i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	566.691	+	566.691i	1.01740	+	1.01740i	0.999846	+	0.0175529i	$0.00558754\pi$
$557$	566.691	+	566.691i	1.01740	+	1.01740i	0.0175529	+	0.999846i	$0.494412\pi$
$558$		0			0
$559$		−	383.317i		−	0.685720i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	548.653	−	548.653i	0.974517	−	0.974517i	−0.0251665	−	0.999683i	$-0.508012\pi$
$563$	548.653	−	548.653i	0.974517	−	0.974517i	0.999683	+	0.0251665i	$0.00801159\pi$
$564$		0			0
$565$	−122.088	+	122.088i	−0.216085	+	0.216085i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			551.224i			0.968760i	0.874858	+	0.484380i	$0.160955\pi$
$569$			551.224i			0.968760i	−0.874858	+	0.484380i	$0.839045\pi$
$570$		0			0
$571$	−458.387	−	458.387i	−0.802780	−	0.802780i	0.180749	−	0.983529i	$-0.442148\pi$
$571$	−458.387	−	458.387i	−0.802780	−	0.802780i	−0.983529	+	0.180749i	$0.942148\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			174.191i			0.302941i
$576$		0			0
$577$	−718.488			−1.24521			−0.622607	−	0.782535i	$-0.713926\pi$
$577$	−718.488			−1.24521			−0.622607	+	0.782535i	$0.713926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−128.744	+	128.744i	−0.221590	+	0.221590i
$582$		0			0
$583$	−1646.00			−2.82333
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.02450	+	3.02450i	0.00515247	+	0.00515247i	0.709678	−	0.704526i	$-0.248840\pi$
$587$	3.02450	+	3.02450i	0.00515247	+	0.00515247i	−0.704526	+	0.709678i	$0.748840\pi$
$588$		0			0
$589$	198.624	+	198.624i	0.337222	+	0.337222i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−576.193			−0.971657			−0.485829	−	0.874054i	$-0.661482\pi$
$593$	−576.193			−0.971657			−0.485829	+	0.874054i	$0.661482\pi$
$594$		0			0
$595$	653.726	−	653.726i	1.09870	−	1.09870i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1101.40			1.83873			0.919365	−	0.393406i	$-0.128703\pi$
$599$	1101.40			1.83873			0.919365	+	0.393406i	$0.128703\pi$
$600$		0			0
$601$		−	7.11053i		−	0.0118312i	−0.999983	−	0.00591558i	$-0.998117\pi$
$601$		−	7.11053i		−	0.0118312i	0.999983	−	0.00591558i	$-0.00188300\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−948.578	−	948.578i	−1.56790	−	1.56790i
$606$		0			0
$607$			528.384i			0.870485i	0.900313	+	0.435242i	$0.143337\pi$
$607$			528.384i			0.870485i	−0.900313	+	0.435242i	$0.856663\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−104.477	+	104.477i	−0.170994	+	0.170994i
$612$		0			0
$613$	642.364	−	642.364i	1.04790	−	1.04790i	0.0491093	−	0.998793i	$-0.484362\pi$
$613$	642.364	−	642.364i	1.04790	−	1.04790i	0.998793	−	0.0491093i	$-0.0156383\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1068.16i			1.73122i	0.500717	+	0.865611i	$0.333070\pi$
$617$			1068.16i			1.73122i	−0.500717	+	0.865611i	$0.666930\pi$
$618$		0			0
$619$	−691.136	−	691.136i	−1.11654	−	1.11654i	−0.992246	−	0.124290i	$-0.960335\pi$
$619$	−691.136	−	691.136i	−1.11654	−	1.11654i	−0.124290	−	0.992246i	$-0.539665\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			281.332i			0.451576i
$624$		0			0
$625$	475.371			0.760594
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−358.931	+	358.931i	−0.570637	+	0.570637i
$630$		0			0
$631$	486.622			0.771191			0.385596	−	0.922668i	$-0.373996\pi$
$631$	486.622			0.771191			0.385596	+	0.922668i	$0.373996\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	619.681	+	619.681i	0.975875	+	0.975875i
$636$		0			0
$637$	−118.247	−	118.247i	−0.185631	−	0.185631i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	691.017			1.07803			0.539015	−	0.842296i	$-0.318797\pi$
$641$	691.017			1.07803			0.539015	+	0.842296i	$0.318797\pi$
$642$		0			0
$643$	−652.605	+	652.605i	−1.01494	+	1.01494i	−0.0150512	+	0.999887i	$0.504791\pi$
$643$	−652.605	+	652.605i	−1.01494	+	1.01494i	−0.999887	+	0.0150512i	$0.995209\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1156.72			−1.78782			−0.893911	−	0.448245i	$-0.852049\pi$
$647$	−1156.72			−1.78782			−0.893911	+	0.448245i	$0.852049\pi$
$648$		0			0
$649$		−	667.201i		−	1.02804i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−209.105	−	209.105i	−0.320222	−	0.320222i	0.528630	−	0.848852i	$-0.322706\pi$
$653$	−209.105	−	209.105i	−0.320222	−	0.320222i	−0.848852	+	0.528630i	$0.822706\pi$
$654$		0			0
$655$		−	725.776i		−	1.10806i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	533.902	−	533.902i	0.810170	−	0.810170i	−0.174489	−	0.984659i	$-0.555827\pi$
$659$	533.902	−	533.902i	0.810170	−	0.810170i	0.984659	+	0.174489i	$0.0558274\pi$
$660$		0			0
$661$	−283.120	+	283.120i	−0.428320	+	0.428320i	−0.888056	−	0.459736i	$-0.847944\pi$
$661$	−283.120	+	283.120i	−0.428320	+	0.428320i	0.459736	+	0.888056i	$0.347944\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			656.484i			0.987195i
$666$		0			0
$667$	−106.825	−	106.825i	−0.160157	−	0.160157i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1603.11i		−	2.38913i
$672$		0			0
$673$	−397.854			−0.591164			−0.295582	−	0.955317i	$-0.595514\pi$
$673$	−397.854			−0.591164			−0.295582	+	0.955317i	$0.595514\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	289.959	−	289.959i	0.428299	−	0.428299i	−0.459749	−	0.888049i	$-0.652061\pi$
$677$	289.959	−	289.959i	0.428299	−	0.428299i	0.888049	+	0.459749i	$0.152061\pi$
$678$		0			0
$679$	112.005			0.164956
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−150.197	−	150.197i	−0.219908	−	0.219908i	0.588551	−	0.808460i	$-0.299698\pi$
$683$	−150.197	−	150.197i	−0.219908	−	0.219908i	−0.808460	+	0.588551i	$0.799698\pi$
$684$		0			0
$685$	−652.686	−	652.686i	−0.952826	−	0.952826i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−768.535			−1.11544
$690$		0			0
$691$	−791.212	+	791.212i	−1.14502	+	1.14502i	−0.157506	+	0.987518i	$0.550345\pi$
$691$	−791.212	+	791.212i	−1.14502	+	1.14502i	−0.987518	+	0.157506i	$0.949655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1772.46			2.55030
$696$		0			0
$697$			684.092i			0.981481i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	900.201	+	900.201i	1.28417	+	1.28417i	0.938274	+	0.345893i	$0.112424\pi$
$701$	900.201	+	900.201i	1.28417	+	1.28417i	0.345893	+	0.938274i	$0.387576\pi$
$702$		0			0
$703$		−	360.445i		−	0.512724i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.3138	−	17.3138i	0.0244891	−	0.0244891i
$708$		0			0
$709$	−128.490	+	128.490i	−0.181227	+	0.181227i	−0.791891	−	0.610663i	$-0.790903\pi$
$709$	−128.490	+	128.490i	−0.181227	+	0.181227i	0.610663	+	0.791891i	$0.290903\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			98.1928i			0.137718i
$714$		0			0
$715$	−739.140	−	739.140i	−1.03376	−	1.03376i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1246.14i			1.73315i	0.499045	+	0.866576i	$0.333684\pi$
$719$			1246.14i			1.73315i	−0.499045	+	0.866576i	$0.666316\pi$
$720$		0			0
$721$	−560.033			−0.776745
$722$		0			0
$723$		0			0
$724$		0			0
$725$	551.546	−	551.546i	0.760753	−	0.760753i
$726$		0			0
$727$	−1130.07			−1.55443			−0.777216	−	0.629234i	$-0.783369\pi$
$727$	−1130.07			−1.55443			−0.777216	+	0.629234i	$0.783369\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−781.782	−	781.782i	−1.06947	−	1.06947i
$732$		0			0
$733$	708.087	+	708.087i	0.966012	+	0.966012i	0.999441	−	0.0334292i	$-0.0106428\pi$
$733$	708.087	+	708.087i	0.966012	+	0.966012i	−0.0334292	+	0.999441i	$0.510643\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	926.264			1.25680
$738$		0			0
$739$	32.7516	−	32.7516i	0.0443188	−	0.0443188i	−0.684600	−	0.728919i	$-0.740023\pi$
$739$	32.7516	−	32.7516i	0.0443188	−	0.0443188i	0.728919	+	0.684600i	$0.240023\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−708.128			−0.953066			−0.476533	−	0.879157i	$-0.658107\pi$
$743$	−708.128			−0.953066			−0.476533	+	0.879157i	$0.658107\pi$
$744$		0			0
$745$			1541.47i			2.06909i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	530.751	+	530.751i	0.708612	+	0.708612i
$750$		0			0
$751$			1242.37i			1.65429i	0.561990	+	0.827144i	$0.310036\pi$
$751$			1242.37i			1.65429i	−0.561990	+	0.827144i	$0.689964\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−396.127	+	396.127i	−0.524671	+	0.524671i
$756$		0			0
$757$	311.304	−	311.304i	0.411233	−	0.411233i	−0.470935	−	0.882168i	$-0.656083\pi$
$757$	311.304	−	311.304i	0.411233	−	0.411233i	0.882168	+	0.470935i	$0.156083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	179.137i		−	0.235397i	−0.993049	−	0.117699i	$-0.962448\pi$
$761$		−	179.137i		−	0.235397i	0.993049	−	0.117699i	$-0.0375517\pi$
$762$		0			0
$763$	−575.685	−	575.685i	−0.754502	−	0.754502i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	311.523i		−	0.406158i
$768$		0			0
$769$	−967.409			−1.25801			−0.629005	−	0.777402i	$-0.716537\pi$
$769$	−967.409			−1.25801			−0.629005	+	0.777402i	$0.716537\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−96.7342	+	96.7342i	−0.125141	+	0.125141i	−0.766904	−	0.641762i	$-0.778204\pi$
$773$	−96.7342	+	96.7342i	−0.125141	+	0.125141i	0.641762	+	0.766904i	$0.278204\pi$
$774$		0			0
$775$	−506.979			−0.654166
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−343.489	−	343.489i	−0.440936	−	0.440936i
$780$		0			0
$781$	−524.082	−	524.082i	−0.671039	−	0.671039i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−855.070			−1.08926
$786$		0			0
$787$	381.038	−	381.038i	0.484166	−	0.484166i	−0.422293	−	0.906459i	$-0.638775\pi$
$787$	381.038	−	381.038i	0.484166	−	0.484166i	0.906459	+	0.422293i	$0.138775\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−124.054			−0.156831
$792$		0			0
$793$		−	748.507i		−	0.943893i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−371.148	−	371.148i	−0.465681	−	0.465681i	0.434831	−	0.900512i	$-0.356808\pi$
$797$	−371.148	−	371.148i	−0.465681	−	0.465681i	−0.900512	+	0.434831i	$0.856808\pi$
$798$		0			0
$799$			426.167i			0.533375i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1307.15	+	1307.15i	−1.62783	+	1.62783i
$804$		0			0
$805$	−162.272	+	162.272i	−0.201580	+	0.201580i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			309.566i			0.382653i	0.981526	+	0.191326i	$0.0612789\pi$
$809$			309.566i			0.382653i	−0.981526	+	0.191326i	$0.938721\pi$
$810$		0			0
$811$	27.2916	+	27.2916i	0.0336517	+	0.0336517i	0.723732	−	0.690081i	$-0.242425\pi$
$811$	27.2916	+	27.2916i	0.0336517	+	0.0336517i	−0.690081	+	0.723732i	$0.742425\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	585.247i		−	0.718095i
$816$		0			0
$817$	785.081			0.960931
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−879.903	+	879.903i	−1.07175	+	1.07175i	−0.0745264	+	0.997219i	$0.523745\pi$
$821$	−879.903	+	879.903i	−1.07175	+	1.07175i	−0.997219	+	0.0745264i	$0.976255\pi$
$822$		0			0
$823$	68.6842			0.0834559			0.0417280	−	0.999129i	$-0.486714\pi$
$823$	68.6842			0.0834559			0.0417280	+	0.999129i	$0.486714\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	942.097	+	942.097i	1.13917	+	1.13917i	0.988599	+	0.150575i	$0.0481126\pi$
$827$	942.097	+	942.097i	1.13917	+	1.13917i	0.150575	+	0.988599i	$0.451887\pi$
$828$		0			0
$829$	568.532	+	568.532i	0.685805	+	0.685805i	0.961302	−	0.275497i	$-0.0888424\pi$
$829$	568.532	+	568.532i	0.685805	+	0.685805i	−0.275497	+	0.961302i	$0.588842\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−482.333			−0.579031
$834$		0			0
$835$	−129.169	+	129.169i	−0.154693	+	0.154693i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1346.87			1.60533			0.802666	−	0.596429i	$-0.203414\pi$
$839$	1346.87			1.60533			0.802666	+	0.596429i	$0.203414\pi$
$840$		0			0
$841$		−	164.515i		−	0.195618i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	541.046	+	541.046i	0.640291	+	0.640291i
$846$		0			0
$847$		−	963.851i		−	1.13796i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	89.0960	−	89.0960i	0.104696	−	0.104696i
$852$		0			0
$853$	−74.4816	+	74.4816i	−0.0873172	+	0.0873172i	−0.749416	−	0.662099i	$-0.769666\pi$
$853$	−74.4816	+	74.4816i	−0.0873172	+	0.0873172i	0.662099	+	0.749416i	$0.269666\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	53.7221i		−	0.0626862i	−0.999509	−	0.0313431i	$-0.990022\pi$
$857$		−	53.7221i		−	0.0626862i	0.999509	−	0.0313431i	$-0.00997845\pi$
$858$		0			0
$859$	537.704	+	537.704i	0.625965	+	0.625965i	0.947050	−	0.321085i	$-0.104048\pi$
$859$	537.704	+	537.704i	0.625965	+	0.625965i	−0.321085	+	0.947050i	$0.604048\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1390.97i			1.61178i	0.592064	+	0.805891i	$0.298313\pi$
$863$			1390.97i			1.61178i	−0.592064	+	0.805891i	$0.701687\pi$
$864$		0			0
$865$	−51.2347			−0.0592309
$866$		0			0
$867$		0			0
$868$		0			0
$869$	261.477	−	261.477i	0.300895	−	0.300895i
$870$		0			0
$871$	432.482			0.496535
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−139.391	−	139.391i	−0.159303	−	0.159303i
$876$		0			0
$877$	−940.115	−	940.115i	−1.07197	−	1.07197i	−0.997201	−	0.0747652i	$-0.976179\pi$
$877$	−940.115	−	940.115i	−1.07197	−	1.07197i	−0.0747652	−	0.997201i	$-0.523821\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	140.985			0.160029			0.0800143	−	0.996794i	$-0.474503\pi$
$881$	140.985			0.160029			0.0800143	+	0.996794i	$0.474503\pi$
$882$		0			0
$883$	482.231	−	482.231i	0.546127	−	0.546127i	−0.379191	−	0.925318i	$-0.623798\pi$
$883$	482.231	−	482.231i	0.546127	−	0.546127i	0.925318	+	0.379191i	$0.123798\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−266.180			−0.300091			−0.150045	−	0.988679i	$-0.547942\pi$
$887$	−266.180			−0.300091			−0.150045	+	0.988679i	$0.547942\pi$
$888$		0			0
$889$			629.658i			0.708277i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−213.982	−	213.982i	−0.239622	−	0.239622i
$894$		0			0
$895$		−	2406.83i		−	2.68919i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	310.911	−	310.911i	0.345841	−	0.345841i
$900$		0			0
$901$	−1567.44	+	1567.44i	−1.73967	+	1.73967i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1222.87i		−	1.35124i
$906$		0			0
$907$	303.117	+	303.117i	0.334197	+	0.334197i	0.854178	−	0.519981i	$-0.174061\pi$
$907$	303.117	+	303.117i	0.334197	+	0.334197i	−0.519981	+	0.854178i	$0.674061\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			296.228i			0.325168i	0.986695	+	0.162584i	$0.0519829\pi$
$911$			296.228i			0.325168i	−0.986695	+	0.162584i	$0.948017\pi$
$912$		0			0
$913$	593.765			0.650346
$914$		0			0
$915$		0			0
$916$		0			0
$917$	368.731	−	368.731i	0.402106	−	0.402106i
$918$		0			0
$919$	−228.052			−0.248153			−0.124076	−	0.992273i	$-0.539597\pi$
$919$	−228.052			−0.248153			−0.124076	+	0.992273i	$0.539597\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−244.699	−	244.699i	−0.265113	−	0.265113i
$924$		0			0
$925$	460.011	+	460.011i	0.497309	+	0.497309i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	574.026			0.617897			0.308948	−	0.951079i	$-0.400023\pi$
$929$	574.026			0.617897			0.308948	+	0.951079i	$0.400023\pi$
$930$		0			0
$931$	242.184	−	242.184i	0.260133	−	0.260133i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3014.98			−3.22457
$936$		0			0
$937$			1098.22i			1.17206i	0.810291	+	0.586028i	$0.199309\pi$
$937$			1098.22i			1.17206i	−0.810291	+	0.586028i	$0.800691\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−857.669	−	857.669i	−0.911444	−	0.911444i	0.0849418	−	0.996386i	$-0.472930\pi$
$941$	−857.669	−	857.669i	−0.911444	−	0.911444i	−0.996386	+	0.0849418i	$0.972930\pi$
$942$		0			0
$943$		−	169.810i		−	0.180074i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1041.67	−	1041.67i	1.09997	−	1.09997i	0.105556	−	0.994413i	$-0.466338\pi$
$947$	1041.67	−	1041.67i	1.09997	−	1.09997i	0.994413	−	0.105556i	$-0.0336622\pi$
$948$		0			0
$949$	−610.322	+	610.322i	−0.643121	+	0.643121i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			910.089i			0.954973i	0.878639	+	0.477486i	$0.158452\pi$
$953$			910.089i			0.954973i	−0.878639	+	0.477486i	$0.841548\pi$
$954$		0			0
$955$	494.107	+	494.107i	0.517389	+	0.517389i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	663.195i		−	0.691548i
$960$		0			0
$961$	675.212			0.702614
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−441.856	+	441.856i	−0.457882	+	0.457882i
$966$		0			0
$967$	−695.071			−0.718791			−0.359396	−	0.933185i	$-0.617017\pi$
$967$	−695.071			−0.718791			−0.359396	+	0.933185i	$0.617017\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1208.40	−	1208.40i	−1.24449	−	1.24449i	−0.958120	−	0.286366i	$-0.907553\pi$
$971$	−1208.40	−	1208.40i	−1.24449	−	1.24449i	−0.286366	−	0.958120i	$-0.592447\pi$
$972$		0			0
$973$	900.500	+	900.500i	0.925488	+	0.925488i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−141.036			−0.144356			−0.0721780	−	0.997392i	$-0.522995\pi$
$977$	−141.036			−0.144356			−0.0721780	+	0.997392i	$0.522995\pi$
$978$		0			0
$979$	648.750	−	648.750i	0.662666	−	0.662666i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1692.71			−1.72199			−0.860994	−	0.508616i	$-0.830157\pi$
$983$	−1692.71			−1.72199			−0.860994	+	0.508616i	$0.830157\pi$
$984$		0			0
$985$		−	597.320i		−	0.606417i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	194.059	+	194.059i	0.196217	+	0.196217i
$990$		0			0
$991$			1532.62i			1.54654i	0.634079	+	0.773268i	$0.281379\pi$
$991$			1532.62i			1.54654i	−0.634079	+	0.773268i	$0.718621\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1028.67	−	1028.67i	1.03384	−	1.03384i
$996$		0			0
$997$	−1131.91	+	1131.91i	−1.13532	+	1.13532i	−0.146039	+	0.989279i	$0.546652\pi$
$997$	−1131.91	+	1131.91i	−1.13532	+	1.13532i	−0.989279	+	0.146039i	$0.953348\pi$
$998$		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.991.1		16
3.2	odd	2		384.3.l.a.223.4		16
4.3	odd	2		1152.3.m.c.991.1		16
8.3	odd	2		576.3.m.c.559.8		16
8.5	even	2		144.3.m.c.19.2		16
12.11	even	2		384.3.l.b.223.8		16
16.3	odd	4		144.3.m.c.91.2		16
16.5	even	4		1152.3.m.c.415.1		16
16.11	odd	4	inner	1152.3.m.f.415.1		16
16.13	even	4		576.3.m.c.271.8		16
24.5	odd	2		48.3.l.a.19.7	✓	16
24.11	even	2		192.3.l.a.175.1		16
48.5	odd	4		384.3.l.b.31.8		16
48.11	even	4		384.3.l.a.31.4		16
48.29	odd	4		192.3.l.a.79.1		16
48.35	even	4		48.3.l.a.43.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.7	✓	16	24.5	odd	2
48.3.l.a.43.7	yes	16	48.35	even	4
144.3.m.c.19.2		16	8.5	even	2
144.3.m.c.91.2		16	16.3	odd	4
192.3.l.a.79.1		16	48.29	odd	4
192.3.l.a.175.1		16	24.11	even	2
384.3.l.a.31.4		16	48.11	even	4
384.3.l.a.223.4		16	3.2	odd	2
384.3.l.b.31.8		16	48.5	odd	4
384.3.l.b.223.8		16	12.11	even	2
576.3.m.c.271.8		16	16.13	even	4
576.3.m.c.559.8		16	8.3	odd	2
1152.3.m.c.415.1		16	16.5	even	4
1152.3.m.c.991.1		16	4.3	odd	2
1152.3.m.f.415.1		16	16.11	odd	4	inner
1152.3.m.f.991.1		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)

\(f(q)\)	\(=\)	\(q+(-5.24354 + 5.24354i) q^{5} -5.32796 q^{7} +O(q^{10})\)	\(q+(-5.24354 + 5.24354i) q^{5} -5.32796 q^{7} +(12.2863 + 12.2863i) q^{11} +(5.73657 + 5.73657i) q^{13} +23.3997 q^{17} +(-11.7492 + 11.7492i) q^{19} -5.80841 q^{23} -29.9894i q^{25} +(18.3914 + 18.3914i) q^{29} -16.9053i q^{31} +(27.9374 - 27.9374i) q^{35} +(-15.3391 + 15.3391i) q^{37} +29.2351i q^{41} +(-33.4099 - 33.4099i) q^{43} +18.2125i q^{47} -20.6128 q^{49} +(-66.9856 + 66.9856i) q^{53} -128.847 q^{55} +(-27.1523 - 27.1523i) q^{59} +(-65.2399 - 65.2399i) q^{61} -60.1599 q^{65} +(37.6951 - 37.6951i) q^{67} -42.6559 q^{71} +106.391i q^{73} +(-65.4607 - 65.4607i) q^{77} -21.2821i q^{79} +(24.1638 - 24.1638i) q^{83} +(-122.697 + 122.697i) q^{85} -52.8029i q^{89} +(-30.5643 - 30.5643i) q^{91} -123.215i q^{95} -21.0222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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