LMFDB - Embedded newform 960.3.j.f.319.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(319,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("960.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.j (of order $2$, degree $1$, 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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 $ x^12 - 6x^11 + 49x^10 - 190x^9 + 792x^8 - 2094x^7 + 5517x^6 - 9954x^5 + 17189x^4 - 19884x^3 + 20996x^2 - 12416*x + 5584
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.6
Root			$0.500000 + 2.65258i$ of defining polynomial
Character	$\chi$	$=$	960.319
Dual form			960.3.j.f.319.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +(4.85927 + 1.17794i) q^{5} +7.06571 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +(4.85927 + 1.17794i) q^{5} +7.06571 q^{7} +3.00000 q^{9} +1.70305i q^{11} +19.3844i q^{13} +(-8.41649 - 2.04025i) q^{15} +13.0069i q^{17} +13.2314i q^{19} -12.2382 q^{21} -5.26392 q^{23} +(22.2249 + 11.4478i) q^{25} -5.19615 q^{27} -33.0071 q^{29} -60.5509i q^{31} -2.94977i q^{33} +(34.3341 + 8.32297i) q^{35} +46.9515i q^{37} -33.5747i q^{39} +65.3233 q^{41} -20.2303 q^{43} +(14.5778 + 3.53382i) q^{45} -40.0636 q^{47} +0.924196 q^{49} -22.5286i q^{51} -11.2001i q^{53} +(-2.00609 + 8.27558i) q^{55} -22.9174i q^{57} +39.8862i q^{59} -32.2030 q^{61} +21.1971 q^{63} +(-22.8336 + 94.1938i) q^{65} -2.76692 q^{67} +9.11738 q^{69} -67.6353i q^{71} +81.8653i q^{73} +(-38.4947 - 19.8282i) q^{75} +12.0333i q^{77} +143.514i q^{79} +9.00000 q^{81} +126.585 q^{83} +(-15.3214 + 63.2041i) q^{85} +57.1699 q^{87} +11.2563 q^{89} +136.964i q^{91} +104.877i q^{93} +(-15.5858 + 64.2948i) q^{95} -49.5443i q^{97} +5.10916i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} + 36 q^{9}+O(q^{10}) $ 12 * q - 4 * q^5 + 36 * q^9	$ 12 q - 4 q^{5} + 36 q^{9} - 24 q^{15} - 80 q^{23} + 28 q^{25} + 40 q^{29} + 144 q^{35} + 136 q^{41} + 224 q^{43} - 12 q^{45} + 208 q^{47} + 212 q^{49} - 192 q^{55} - 40 q^{61} + 96 q^{65} - 352 q^{67} + 192 q^{75} + 108 q^{81} + 64 q^{85} + 48 q^{87} - 8 q^{89} - 176 q^{95}+O(q^{100}) $ 12 * q - 4 * q^5 + 36 * q^9 - 24 * q^15 - 80 * q^23 + 28 * q^25 + 40 * q^29 + 144 * q^35 + 136 * q^41 + 224 * q^43 - 12 * q^45 + 208 * q^47 + 212 * q^49 - 192 * q^55 - 40 * q^61 + 96 * q^65 - 352 * q^67 + 192 * q^75 + 108 * q^81 + 64 * q^85 + 48 * q^87 - 8 * q^89 - 176 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	4.85927	+	1.17794i	0.971853	+	0.235588i
$5$	4.85927	+	1.17794i	0.971853	+	0.235588i
$6$		0			0
$7$	7.06571			1.00939			0.504693	−	0.863299i	$-0.331606\pi$
$7$	7.06571			1.00939			0.504693	+	0.863299i	$0.331606\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			1.70305i			0.154823i	0.996999	+	0.0774115i	$0.0246655\pi$
$11$			1.70305i			0.154823i	−0.996999	+	0.0774115i	$0.975334\pi$
$12$		0			0
$13$			19.3844i			1.49111i	0.666447	+	0.745553i	$0.267814\pi$
$13$			19.3844i			1.49111i	−0.666447	+	0.745553i	$0.732186\pi$
$14$		0			0
$15$	−8.41649	−	2.04025i	−0.561100	−	0.136017i
$16$		0			0
$17$			13.0069i			0.765113i	0.923932	+	0.382556i	$0.124956\pi$
$17$			13.0069i			0.765113i	−0.923932	+	0.382556i	$0.875044\pi$
$18$		0			0
$19$			13.2314i			0.696389i	0.937422	+	0.348194i	$0.113205\pi$
$19$			13.2314i			0.696389i	−0.937422	+	0.348194i	$0.886795\pi$
$20$		0			0
$21$	−12.2382			−0.582770
$22$		0			0
$23$	−5.26392			−0.228866			−0.114433	−	0.993431i	$-0.536505\pi$
$23$	−5.26392			−0.228866			−0.114433	+	0.993431i	$0.536505\pi$
$24$		0			0
$25$	22.2249	+	11.4478i	0.888997	+	0.457913i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	−33.0071			−1.13817			−0.569087	−	0.822277i	$-0.692703\pi$
$29$	−33.0071			−1.13817			−0.569087	+	0.822277i	$0.692703\pi$
$30$		0			0
$31$		−	60.5509i		−	1.95325i	−0.214941	−	0.976627i	$-0.568956\pi$
$31$		−	60.5509i		−	1.95325i	0.214941	−	0.976627i	$-0.431044\pi$
$32$		0			0
$33$		−	2.94977i		−	0.0893871i
$34$		0			0
$35$	34.3341	+	8.32297i	0.980975	+	0.237799i
$36$		0			0
$37$			46.9515i			1.26896i	0.772940	+	0.634479i	$0.218785\pi$
$37$			46.9515i			1.26896i	−0.772940	+	0.634479i	$0.781215\pi$
$38$		0			0
$39$		−	33.5747i		−	0.860890i
$40$		0			0
$41$	65.3233			1.59325			0.796626	−	0.604472i	$-0.206616\pi$
$41$	65.3233			1.59325			0.796626	+	0.604472i	$0.206616\pi$
$42$		0			0
$43$	−20.2303			−0.470472			−0.235236	−	0.971938i	$-0.575586\pi$
$43$	−20.2303			−0.470472			−0.235236	+	0.971938i	$0.575586\pi$
$44$		0			0
$45$	14.5778	+	3.53382i	0.323951	+	0.0785292i
$46$		0			0
$47$	−40.0636			−0.852417			−0.426209	−	0.904625i	$-0.640151\pi$
$47$	−40.0636			−0.852417			−0.426209	+	0.904625i	$0.640151\pi$
$48$		0			0
$49$	0.924196			0.0188612
$50$		0			0
$51$		−	22.5286i		−	0.441738i
$52$		0			0
$53$		−	11.2001i		−	0.211323i	−0.994402	−	0.105662i	$-0.966304\pi$
$53$		−	11.2001i		−	0.211323i	0.994402	−	0.105662i	$-0.0336960\pi$
$54$		0			0
$55$	−2.00609	+	8.27558i	−0.0364744	+	0.150465i
$56$		0			0
$57$		−	22.9174i		−	0.402060i
$58$		0			0
$59$			39.8862i			0.676038i	0.941139	+	0.338019i	$0.109757\pi$
$59$			39.8862i			0.676038i	−0.941139	+	0.338019i	$0.890243\pi$
$60$		0			0
$61$	−32.2030			−0.527918			−0.263959	−	0.964534i	$-0.585028\pi$
$61$	−32.2030			−0.527918			−0.263959	+	0.964534i	$0.585028\pi$
$62$		0			0
$63$	21.1971			0.336462
$64$		0			0
$65$	−22.8336	+	94.1938i	−0.351286	+	1.44914i
$66$		0			0
$67$	−2.76692			−0.0412973			−0.0206486	−	0.999787i	$-0.506573\pi$
$67$	−2.76692			−0.0412973			−0.0206486	+	0.999787i	$0.506573\pi$
$68$		0			0
$69$	9.11738			0.132136
$70$		0			0
$71$		−	67.6353i		−	0.952609i	−0.879280	−	0.476305i	$-0.841976\pi$
$71$		−	67.6353i		−	0.952609i	0.879280	−	0.476305i	$-0.158024\pi$
$72$		0			0
$73$			81.8653i			1.12144i	0.828005	+	0.560721i	$0.189476\pi$
$73$			81.8653i			1.12144i	−0.828005	+	0.560721i	$0.810524\pi$
$74$		0			0
$75$	−38.4947	−	19.8282i	−0.513263	−	0.264376i
$76$		0			0
$77$			12.0333i			0.156276i
$78$		0			0
$79$			143.514i			1.81664i	0.418279	+	0.908319i	$0.362634\pi$
$79$			143.514i			1.81664i	−0.418279	+	0.908319i	$0.637366\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	126.585			1.52512			0.762560	−	0.646918i	$-0.223942\pi$
$83$	126.585			1.52512			0.762560	+	0.646918i	$0.223942\pi$
$84$		0			0
$85$	−15.3214	+	63.2041i	−0.180251	+	0.743577i
$86$		0			0
$87$	57.1699			0.657125
$88$		0			0
$89$	11.2563			0.126475			0.0632374	−	0.997999i	$-0.479857\pi$
$89$	11.2563			0.126475			0.0632374	+	0.997999i	$0.479857\pi$
$90$		0			0
$91$			136.964i			1.50510i
$92$		0			0
$93$			104.877i			1.12771i
$94$		0			0
$95$	−15.5858	+	64.2948i	−0.164061	+	0.676787i
$96$		0			0
$97$		−	49.5443i		−	0.510766i	−0.966840	−	0.255383i	$-0.917798\pi$
$97$		−	49.5443i		−	0.510766i	0.966840	−	0.255383i	$-0.0822015\pi$
$98$		0			0
$99$			5.10916i			0.0516076i
$100$		0			0
$101$	−57.1327			−0.565671			−0.282835	−	0.959169i	$-0.591275\pi$
$101$	−57.1327			−0.565671			−0.282835	+	0.959169i	$0.591275\pi$
$102$		0			0
$103$	−63.1664			−0.613266			−0.306633	−	0.951828i	$-0.599202\pi$
$103$	−63.1664			−0.613266			−0.306633	+	0.951828i	$0.599202\pi$
$104$		0			0
$105$	−59.4685	−	14.4158i	−0.566366	−	0.137293i
$106$		0			0
$107$	−59.1537			−0.552838			−0.276419	−	0.961037i	$-0.589148\pi$
$107$	−59.1537			−0.552838			−0.276419	+	0.961037i	$0.589148\pi$
$108$		0			0
$109$	159.657			1.46474			0.732370	−	0.680906i	$-0.238414\pi$
$109$	159.657			1.46474			0.732370	+	0.680906i	$0.238414\pi$
$110$		0			0
$111$		−	81.3223i		−	0.732634i
$112$		0			0
$113$			145.680i			1.28920i	0.764519	+	0.644601i	$0.222976\pi$
$113$			145.680i			1.28920i	−0.764519	+	0.644601i	$0.777024\pi$
$114$		0			0
$115$	−25.5788	−	6.20058i	−0.222424	−	0.0539181i
$116$		0			0
$117$			58.1531i			0.497035i
$118$		0			0
$119$			91.9031i			0.772295i
$120$		0			0
$121$	118.100			0.976030
$122$		0			0
$123$	−113.143			−0.919865
$124$		0			0
$125$	94.5119	+	81.8077i	0.756096	+	0.654461i
$126$		0			0
$127$	236.822			1.86474			0.932370	−	0.361506i	$-0.117737\pi$
$127$	236.822			1.86474			0.932370	+	0.361506i	$0.117737\pi$
$128$		0			0
$129$	35.0399			0.271627
$130$		0			0
$131$		−	106.346i		−	0.811802i	−0.913917	−	0.405901i	$-0.866958\pi$
$131$		−	106.346i		−	0.811802i	0.913917	−	0.405901i	$-0.133042\pi$
$132$		0			0
$133$			93.4891i			0.702925i
$134$		0			0
$135$	−25.2495	−	6.12075i	−0.187033	−	0.0453389i
$136$		0			0
$137$			138.800i			1.01314i	0.862199	+	0.506570i	$0.169087\pi$
$137$			138.800i			1.01314i	−0.862199	+	0.506570i	$0.830913\pi$
$138$		0			0
$139$			24.3036i			0.174846i	0.996171	+	0.0874229i	$0.0278631\pi$
$139$			24.3036i			0.174846i	−0.996171	+	0.0874229i	$0.972137\pi$
$140$		0			0
$141$	69.3922			0.492143
$142$		0			0
$143$	−33.0126			−0.230857
$144$		0			0
$145$	−160.390	−	38.8803i	−1.10614	−	0.268140i
$146$		0			0
$147$	−1.60076			−0.0108895
$148$		0			0
$149$	178.349			1.19698			0.598488	−	0.801132i	$-0.295768\pi$
$149$	178.349			1.19698			0.598488	+	0.801132i	$0.295768\pi$
$150$		0			0
$151$			170.810i			1.13119i	0.824682	+	0.565596i	$0.191354\pi$
$151$			170.810i			1.13119i	−0.824682	+	0.565596i	$0.808646\pi$
$152$		0			0
$153$			39.0208i			0.255038i
$154$		0			0
$155$	71.3252	−	294.233i	0.460163	−	1.89828i
$156$		0			0
$157$		−	83.1986i		−	0.529927i	−0.964258	−	0.264964i	$-0.914640\pi$
$157$		−	83.1986i		−	0.529927i	0.964258	−	0.264964i	$-0.0853599\pi$
$158$		0			0
$159$			19.3992i			0.122007i
$160$		0			0
$161$	−37.1933			−0.231014
$162$		0			0
$163$	81.2870			0.498693			0.249346	−	0.968414i	$-0.419784\pi$
$163$	81.2870			0.498693			0.249346	+	0.968414i	$0.419784\pi$
$164$		0			0
$165$	3.47465	−	14.3337i	0.0210585	−	0.0868711i
$166$		0			0
$167$	−314.667			−1.88423			−0.942116	−	0.335288i	$-0.891166\pi$
$167$	−314.667			−1.88423			−0.942116	+	0.335288i	$0.891166\pi$
$168$		0			0
$169$	−206.754			−1.22339
$170$		0			0
$171$			39.6942i			0.232130i
$172$		0			0
$173$		−	143.143i		−	0.827419i	−0.910409	−	0.413709i	$-0.864233\pi$
$173$		−	143.143i		−	0.827419i	0.910409	−	0.413709i	$-0.135767\pi$
$174$		0			0
$175$	157.035	+	80.8870i	0.897341	+	0.462212i
$176$		0			0
$177$		−	69.0850i		−	0.390310i
$178$		0			0
$179$		−	229.059i		−	1.27966i	−0.768517	−	0.639829i	$-0.779005\pi$
$179$		−	229.059i		−	1.27966i	0.768517	−	0.639829i	$-0.220995\pi$
$180$		0			0
$181$	−155.190			−0.857401			−0.428700	−	0.903447i	$-0.641028\pi$
$181$	−155.190			−0.857401			−0.428700	+	0.903447i	$0.641028\pi$
$182$		0			0
$183$	55.7773			0.304794
$184$		0			0
$185$	−55.3060	+	228.150i	−0.298951	+	1.23324i
$186$		0			0
$187$	−22.1515			−0.118457
$188$		0			0
$189$	−36.7145			−0.194257
$190$		0			0
$191$		−	299.037i		−	1.56564i	−0.622249	−	0.782819i	$-0.713781\pi$
$191$		−	299.037i		−	1.56564i	0.622249	−	0.782819i	$-0.286219\pi$
$192$		0			0
$193$			112.150i			0.581090i	0.956861	+	0.290545i	$0.0938366\pi$
$193$			112.150i			0.581090i	−0.956861	+	0.290545i	$0.906163\pi$
$194$		0			0
$195$	39.5489	−	163.148i	0.202815	−	0.836659i
$196$		0			0
$197$		−	348.881i		−	1.77097i	−0.464668	−	0.885485i	$-0.653826\pi$
$197$		−	348.881i		−	1.77097i	0.464668	−	0.885485i	$-0.346174\pi$
$198$		0			0
$199$			217.793i			1.09444i	0.836990	+	0.547218i	$0.184313\pi$
$199$			217.793i			1.09444i	−0.836990	+	0.547218i	$0.815687\pi$
$200$		0			0
$201$	4.79244			0.0238430
$202$		0			0
$203$	−233.218			−1.14886
$204$		0			0
$205$	317.423	+	76.9469i	1.54841	+	0.375351i
$206$		0			0
$207$	−15.7918			−0.0762887
$208$		0			0
$209$	−22.5337			−0.107817
$210$		0			0
$211$			318.221i			1.50816i	0.656785	+	0.754078i	$0.271916\pi$
$211$			318.221i			1.50816i	−0.656785	+	0.754078i	$0.728084\pi$
$212$		0			0
$213$			117.148i			0.549989i
$214$		0			0
$215$	−98.3043	−	23.8300i	−0.457229	−	0.110837i
$216$		0			0
$217$		−	427.835i		−	1.97159i
$218$		0			0
$219$		−	141.795i		−	0.647465i
$220$		0			0
$221$	−252.131			−1.14086
$222$		0			0
$223$	185.718			0.832816			0.416408	−	0.909178i	$-0.363289\pi$
$223$	185.718			0.832816			0.416408	+	0.909178i	$0.363289\pi$
$224$		0			0
$225$	66.6748	+	34.3435i	0.296332	+	0.152638i
$226$		0			0
$227$	286.576			1.26245			0.631225	−	0.775600i	$-0.282552\pi$
$227$	286.576			1.26245			0.631225	+	0.775600i	$0.282552\pi$
$228$		0			0
$229$	−109.601			−0.478607			−0.239304	−	0.970945i	$-0.576919\pi$
$229$	−109.601			−0.478607			−0.239304	+	0.970945i	$0.576919\pi$
$230$		0			0
$231$		−	20.8422i		−	0.0902261i
$232$		0			0
$233$		−	404.676i		−	1.73681i	−0.495857	−	0.868404i	$-0.665146\pi$
$233$		−	404.676i		−	1.73681i	0.495857	−	0.868404i	$-0.334854\pi$
$234$		0			0
$235$	−194.680	−	47.1925i	−0.828424	−	0.200819i
$236$		0			0
$237$		−	248.574i		−	1.04884i
$238$		0			0
$239$		−	101.788i		−	0.425890i	−0.977064	−	0.212945i	$-0.931694\pi$
$239$		−	101.788i		−	0.425890i	0.977064	−	0.212945i	$-0.0683056\pi$
$240$		0			0
$241$	111.710			0.463526			0.231763	−	0.972772i	$-0.425551\pi$
$241$	111.710			0.463526			0.231763	+	0.972772i	$0.425551\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	4.49092	+	1.08865i	0.0183303	+	0.00444346i
$246$		0			0
$247$	−256.482			−1.03839
$248$		0			0
$249$	−219.251			−0.880528
$250$		0			0
$251$			168.763i			0.672363i	0.941797	+	0.336182i	$0.109136\pi$
$251$			168.763i			0.672363i	−0.941797	+	0.336182i	$0.890864\pi$
$252$		0			0
$253$		−	8.96473i		−	0.0354337i
$254$		0			0
$255$	26.5374	−	109.473i	0.104068	−	0.429305i
$256$		0			0
$257$		−	320.743i		−	1.24803i	−0.781414	−	0.624013i	$-0.785501\pi$
$257$		−	320.743i		−	1.24803i	0.781414	−	0.624013i	$-0.214499\pi$
$258$		0			0
$259$			331.745i			1.28087i
$260$		0			0
$261$	−99.0212			−0.379392
$262$		0			0
$263$	−58.2208			−0.221372			−0.110686	−	0.993855i	$-0.535305\pi$
$263$	−58.2208			−0.221372			−0.110686	+	0.993855i	$0.535305\pi$
$264$		0			0
$265$	13.1931	−	54.4244i	0.0497851	−	0.205375i
$266$		0			0
$267$	−19.4964			−0.0730203
$268$		0			0
$269$	−259.743			−0.965586			−0.482793	−	0.875734i	$-0.660378\pi$
$269$	−259.743			−0.965586			−0.482793	+	0.875734i	$0.660378\pi$
$270$		0			0
$271$			62.7802i			0.231661i	0.993269	+	0.115831i	$0.0369530\pi$
$271$			62.7802i			0.231661i	−0.993269	+	0.115831i	$0.963047\pi$
$272$		0			0
$273$		−	237.229i		−	0.868971i
$274$		0			0
$275$	−19.4963	+	37.8502i	−0.0708955	+	0.137637i
$276$		0			0
$277$		−	419.214i		−	1.51341i	−0.653757	−	0.756705i	$-0.726808\pi$
$277$		−	419.214i		−	1.51341i	0.653757	−	0.756705i	$-0.273192\pi$
$278$		0			0
$279$		−	181.653i		−	0.651085i
$280$		0			0
$281$	429.741			1.52933			0.764664	−	0.644430i	$-0.222905\pi$
$281$	429.741			1.52933			0.764664	+	0.644430i	$0.222905\pi$
$282$		0			0
$283$	207.152			0.731988			0.365994	−	0.930617i	$-0.380729\pi$
$283$	207.152			0.731988			0.365994	+	0.930617i	$0.380729\pi$
$284$		0			0
$285$	26.9953	−	111.362i	0.0947205	−	0.390743i
$286$		0			0
$287$	461.556			1.60821
$288$		0			0
$289$	119.820			0.414602
$290$		0			0
$291$			85.8132i			0.294891i
$292$		0			0
$293$			157.986i			0.539200i	0.962972	+	0.269600i	$0.0868914\pi$
$293$			157.986i			0.539200i	−0.962972	+	0.269600i	$0.913109\pi$
$294$		0			0
$295$	−46.9835	+	193.818i	−0.159266	+	0.657009i
$296$		0			0
$297$		−	8.84932i		−	0.0297957i
$298$		0			0
$299$		−	102.038i		−	0.341263i
$300$		0			0
$301$	−142.941			−0.474888
$302$		0			0
$303$	98.9568			0.326590
$304$		0			0
$305$	−156.483	−	37.9332i	−0.513059	−	0.124371i
$306$		0			0
$307$	16.9945			0.0553568			0.0276784	−	0.999617i	$-0.491189\pi$
$307$	16.9945			0.0553568			0.0276784	+	0.999617i	$0.491189\pi$
$308$		0			0
$309$	109.407			0.354069
$310$		0			0
$311$		−	430.263i		−	1.38348i	−0.722146	−	0.691741i	$-0.756844\pi$
$311$		−	430.263i		−	1.38348i	0.722146	−	0.691741i	$-0.243156\pi$
$312$		0			0
$313$			340.367i			1.08743i	0.839268	+	0.543717i	$0.182984\pi$
$313$			340.367i			1.08743i	−0.839268	+	0.543717i	$0.817016\pi$
$314$		0			0
$315$	103.002	+	24.9689i	0.326992	+	0.0792664i
$316$		0			0
$317$		−	347.597i		−	1.09652i	−0.836308	−	0.548260i	$-0.815291\pi$
$317$		−	347.597i		−	1.09652i	0.836308	−	0.548260i	$-0.184709\pi$
$318$		0			0
$319$		−	56.2128i		−	0.176216i
$320$		0			0
$321$	102.457			0.319181
$322$		0			0
$323$	−172.100			−0.532816
$324$		0			0
$325$	−221.909	+	430.816i	−0.682797	+	1.32559i
$326$		0			0
$327$	−276.534			−0.845668
$328$		0			0
$329$	−283.078			−0.860418
$330$		0			0
$331$			521.268i			1.57483i	0.616424	+	0.787414i	$0.288581\pi$
$331$			521.268i			1.57483i	−0.616424	+	0.787414i	$0.711419\pi$
$332$		0			0
$333$			140.854i			0.422986i
$334$		0			0
$335$	−13.4452	−	3.25926i	−0.0401349	−	0.00972914i
$336$		0			0
$337$		−	124.784i		−	0.370278i	−0.982712	−	0.185139i	$-0.940726\pi$
$337$		−	124.784i		−	0.370278i	0.982712	−	0.185139i	$-0.0592735\pi$
$338$		0			0
$339$		−	252.325i		−	0.744321i
$340$		0			0
$341$	103.121			0.302409
$342$		0			0
$343$	−339.689			−0.990348
$344$		0			0
$345$	44.3038	+	10.7397i	0.128417	+	0.0311296i
$346$		0			0
$347$	−133.616			−0.385060			−0.192530	−	0.981291i	$-0.561669\pi$
$347$	−133.616			−0.385060			−0.192530	+	0.981291i	$0.561669\pi$
$348$		0			0
$349$	−265.623			−0.761098			−0.380549	−	0.924761i	$-0.624265\pi$
$349$	−265.623			−0.761098			−0.380549	+	0.924761i	$0.624265\pi$
$350$		0			0
$351$		−	100.724i		−	0.286963i
$352$		0			0
$353$		−	196.286i		−	0.556050i	−0.960574	−	0.278025i	$-0.910320\pi$
$353$		−	196.286i		−	0.556050i	0.960574	−	0.278025i	$-0.0896798\pi$
$354$		0			0
$355$	79.6702	−	328.658i	0.224423	−	0.925796i
$356$		0			0
$357$		−	159.181i		−	0.445885i
$358$		0			0
$359$			139.938i			0.389800i	0.980823	+	0.194900i	$0.0624383\pi$
$359$			139.938i			0.389800i	−0.980823	+	0.194900i	$0.937562\pi$
$360$		0			0
$361$	185.930			0.515043
$362$		0			0
$363$	−204.555			−0.563511
$364$		0			0
$365$	−96.4323	+	397.805i	−0.264198	+	1.08988i
$366$		0			0
$367$	−6.22040			−0.0169493			−0.00847465	−	0.999964i	$-0.502698\pi$
$367$	−6.22040			−0.0169493			−0.00847465	+	0.999964i	$0.502698\pi$
$368$		0			0
$369$	195.970			0.531084
$370$		0			0
$371$		−	79.1368i		−	0.213307i
$372$		0			0
$373$		−	687.926i		−	1.84431i	−0.386825	−	0.922153i	$-0.626428\pi$
$373$		−	687.926i		−	1.84431i	0.386825	−	0.922153i	$-0.373572\pi$
$374$		0			0
$375$	−163.699	−	141.695i	−0.436532	−	0.377853i
$376$		0			0
$377$		−	639.821i		−	1.69714i
$378$		0			0
$379$			264.922i			0.699004i	0.936936	+	0.349502i	$0.113649\pi$
$379$			264.922i			0.699004i	−0.936936	+	0.349502i	$0.886351\pi$
$380$		0			0
$381$	−410.188			−1.07661
$382$		0			0
$383$	335.299			0.875454			0.437727	−	0.899108i	$-0.355784\pi$
$383$	335.299			0.875454			0.437727	+	0.899108i	$0.355784\pi$
$384$		0			0
$385$	−14.1744	+	58.4728i	−0.0368168	+	0.151877i
$386$		0			0
$387$	−60.6909			−0.156824
$388$		0			0
$389$	355.544			0.913996			0.456998	−	0.889468i	$-0.348925\pi$
$389$	355.544			0.913996			0.456998	+	0.889468i	$0.348925\pi$
$390$		0			0
$391$		−	68.4674i		−	0.175108i
$392$		0			0
$393$			184.197i			0.468694i
$394$		0			0
$395$	−169.051	+	697.374i	−0.427977	+	1.76550i
$396$		0			0
$397$		−	238.927i		−	0.601832i	−0.953651	−	0.300916i	$-0.902708\pi$
$397$		−	238.927i		−	0.601832i	0.953651	−	0.300916i	$-0.0972924\pi$
$398$		0			0
$399$		−	161.928i		−	0.405834i
$400$		0			0
$401$	44.9170			0.112013			0.0560063	−	0.998430i	$-0.482163\pi$
$401$	44.9170			0.112013			0.0560063	+	0.998430i	$0.482163\pi$
$402$		0			0
$403$	1173.74			2.91251
$404$		0			0
$405$	43.7334	+	10.6014i	0.107984	+	0.0261764i
$406$		0			0
$407$	−79.9608			−0.196464
$408$		0			0
$409$	484.137			1.18371			0.591854	−	0.806045i	$-0.298396\pi$
$409$	484.137			1.18371			0.591854	+	0.806045i	$0.298396\pi$
$410$		0			0
$411$		−	240.409i		−	0.584937i
$412$		0			0
$413$			281.824i			0.682383i
$414$		0			0
$415$	615.110	+	149.109i	1.48219	+	0.359299i
$416$		0			0
$417$		−	42.0950i		−	0.100947i
$418$		0			0
$419$			374.895i			0.894739i	0.894349	+	0.447369i	$0.147639\pi$
$419$			374.895i			0.894739i	−0.894349	+	0.447369i	$0.852361\pi$
$420$		0			0
$421$	−16.8787			−0.0400919			−0.0200459	−	0.999799i	$-0.506381\pi$
$421$	−16.8787			−0.0400919			−0.0200459	+	0.999799i	$0.506381\pi$
$422$		0			0
$423$	−120.191			−0.284139
$424$		0			0
$425$	−148.901	+	289.078i	−0.350355	+	0.680183i
$426$		0			0
$427$	−227.537			−0.532874
$428$		0			0
$429$	57.1795			0.133285
$430$		0			0
$431$		−	29.0390i		−	0.0673759i	−0.999432	−	0.0336879i	$-0.989275\pi$
$431$		−	29.0390i		−	0.0673759i	0.999432	−	0.0336879i	$-0.0107252\pi$
$432$		0			0
$433$			128.629i			0.297064i	0.988908	+	0.148532i	$0.0474548\pi$
$433$			128.629i			0.297064i	−0.988908	+	0.148532i	$0.952545\pi$
$434$		0			0
$435$	277.804	+	67.3426i	0.638629	+	0.154811i
$436$		0			0
$437$		−	69.6490i		−	0.159380i
$438$		0			0
$439$			67.3591i			0.153438i	0.997053	+	0.0767188i	$0.0244444\pi$
$439$			67.3591i			0.153438i	−0.997053	+	0.0767188i	$0.975556\pi$
$440$		0			0
$441$	2.77259			0.00628705
$442$		0			0
$443$	−507.136			−1.14478			−0.572388	−	0.819983i	$-0.693983\pi$
$443$	−507.136			−1.14478			−0.572388	+	0.819983i	$0.693983\pi$
$444$		0			0
$445$	54.6971	+	13.2592i	0.122915	+	0.0297959i
$446$		0			0
$447$	−308.910			−0.691074
$448$		0			0
$449$	328.869			0.732447			0.366224	−	0.930527i	$-0.380651\pi$
$449$	328.869			0.732447			0.366224	+	0.930527i	$0.380651\pi$
$450$		0			0
$451$			111.249i			0.246672i
$452$		0			0
$453$		−	295.852i		−	0.653094i
$454$		0			0
$455$	−161.335	+	665.546i	−0.354583	+	1.46274i
$456$		0			0
$457$		−	640.422i		−	1.40136i	−0.713475	−	0.700681i	$-0.752880\pi$
$457$		−	640.422i		−	1.40136i	0.713475	−	0.700681i	$-0.247120\pi$
$458$		0			0
$459$		−	67.5859i		−	0.147246i
$460$		0			0
$461$	−447.177			−0.970014			−0.485007	−	0.874510i	$-0.661183\pi$
$461$	−447.177			−0.970014			−0.485007	+	0.874510i	$0.661183\pi$
$462$		0			0
$463$	−892.367			−1.92736			−0.963679	−	0.267064i	$-0.913947\pi$
$463$	−892.367			−1.92736			−0.963679	+	0.267064i	$0.913947\pi$
$464$		0			0
$465$	−123.539	+	509.626i	−0.265675	+	1.09597i
$466$		0			0
$467$	371.767			0.796075			0.398038	−	0.917369i	$-0.369691\pi$
$467$	371.767			0.796075			0.398038	+	0.917369i	$0.369691\pi$
$468$		0			0
$469$	−19.5502			−0.0416849
$470$		0			0
$471$			144.104i			0.305954i
$472$		0			0
$473$		−	34.4532i		−	0.0728398i
$474$		0			0
$475$	−151.471	+	294.066i	−0.318886	+	0.619087i
$476$		0			0
$477$		−	33.6004i		−	0.0704411i
$478$		0			0
$479$		−	105.771i		−	0.220817i	−0.993886	−	0.110408i	$-0.964784\pi$
$479$		−	105.771i		−	0.220817i	0.993886	−	0.110408i	$-0.0352159\pi$
$480$		0			0
$481$	−910.124			−1.89215
$482$		0			0
$483$	64.4207			0.133376
$484$		0			0
$485$	58.3601	−	240.749i	0.120330	−	0.496389i
$486$		0			0
$487$	519.395			1.06652			0.533259	−	0.845952i	$-0.320967\pi$
$487$	519.395			1.06652			0.533259	+	0.845952i	$0.320967\pi$
$488$		0			0
$489$	−140.793			−0.287921
$490$		0			0
$491$		−	805.321i		−	1.64016i	−0.572246	−	0.820082i	$-0.693928\pi$
$491$		−	805.321i		−	1.64016i	0.572246	−	0.820082i	$-0.306072\pi$
$492$		0			0
$493$		−	429.320i		−	0.870832i
$494$		0			0
$495$	−6.01827	+	24.8267i	−0.0121581	+	0.0501550i
$496$		0			0
$497$		−	477.891i		−	0.961551i
$498$		0			0
$499$		−	825.125i		−	1.65356i	−0.562528	−	0.826778i	$-0.690171\pi$
$499$		−	825.125i		−	1.65356i	0.562528	−	0.826778i	$-0.309829\pi$
$500$		0			0
$501$	545.019			1.08786
$502$		0			0
$503$	577.032			1.14718			0.573590	−	0.819142i	$-0.305550\pi$
$503$	577.032			1.14718			0.573590	+	0.819142i	$0.305550\pi$
$504$		0			0
$505$	−277.623	−	67.2988i	−0.549749	−	0.133265i
$506$		0			0
$507$	358.108			0.706327
$508$		0			0
$509$	240.707			0.472902			0.236451	−	0.971643i	$-0.424016\pi$
$509$	240.707			0.472902			0.236451	+	0.971643i	$0.424016\pi$
$510$		0			0
$511$			578.436i			1.13197i
$512$		0			0
$513$		−	68.7523i		−	0.134020i
$514$		0			0
$515$	−306.942	−	74.4061i	−0.596004	−	0.144478i
$516$		0			0
$517$		−	68.2304i		−	0.131974i
$518$		0			0
$519$			247.932i			0.477711i
$520$		0			0
$521$	−928.796			−1.78272			−0.891359	−	0.453299i	$-0.850247\pi$
$521$	−928.796			−1.78272			−0.891359	+	0.453299i	$0.850247\pi$
$522$		0			0
$523$	−1007.76			−1.92689			−0.963443	−	0.267914i	$-0.913666\pi$
$523$	−1007.76			−1.92689			−0.963443	+	0.267914i	$0.913666\pi$
$524$		0			0
$525$	−271.992	−	140.100i	−0.518080	−	0.266858i
$526$		0			0
$527$	787.580			1.49446
$528$		0			0
$529$	−501.291			−0.947620
$530$		0			0
$531$			119.659i			0.225346i
$532$		0			0
$533$			1266.25i			2.37571i
$534$		0			0
$535$	−287.444	−	69.6794i	−0.537278	−	0.130242i
$536$		0			0
$537$			396.742i			0.738811i
$538$		0			0
$539$			1.57395i			0.00292014i
$540$		0			0
$541$	733.141			1.35516			0.677580	−	0.735449i	$-0.263029\pi$
$541$	733.141			1.35516			0.677580	+	0.735449i	$0.263029\pi$
$542$		0			0
$543$	268.796			0.495021
$544$		0			0
$545$	775.815	+	188.066i	1.42351	+	0.345075i
$546$		0			0
$547$	−197.440			−0.360950			−0.180475	−	0.983580i	$-0.557764\pi$
$547$	−197.440			−0.360950			−0.180475	+	0.983580i	$0.557764\pi$
$548$		0			0
$549$	−96.6091			−0.175973
$550$		0			0
$551$		−	436.729i		−	0.792612i
$552$		0			0
$553$			1014.03i			1.83369i
$554$		0			0
$555$	95.7927	−	395.167i	0.172599	−	0.712012i
$556$		0			0
$557$			44.5650i			0.0800089i	0.999200	+	0.0400045i	$0.0127372\pi$
$557$			44.5650i			0.0800089i	−0.999200	+	0.0400045i	$0.987263\pi$
$558$		0			0
$559$		−	392.151i		−	0.701523i
$560$		0			0
$561$	38.3675			0.0683912
$562$		0			0
$563$	166.410			0.295577			0.147788	−	0.989019i	$-0.452785\pi$
$563$	166.410			0.295577			0.147788	+	0.989019i	$0.452785\pi$
$564$		0			0
$565$	−171.602	+	707.897i	−0.303720	+	1.25291i
$566$		0			0
$567$	63.5914			0.112154
$568$		0			0
$569$	−503.775			−0.885369			−0.442685	−	0.896677i	$-0.645974\pi$
$569$	−503.775			−0.885369			−0.442685	+	0.896677i	$0.645974\pi$
$570$		0			0
$571$		−	898.626i		−	1.57378i	−0.617096	−	0.786888i	$-0.711691\pi$
$571$		−	898.626i		−	1.57378i	0.617096	−	0.786888i	$-0.288309\pi$
$572$		0			0
$573$			517.947i			0.903922i
$574$		0			0
$575$	−116.990	−	60.2605i	−0.203461	−	0.104801i
$576$		0			0
$577$		−	371.658i		−	0.644121i	−0.946719	−	0.322061i	$-0.895624\pi$
$577$		−	371.658i		−	0.644121i	0.946719	−	0.322061i	$-0.104376\pi$
$578$		0			0
$579$		−	194.250i		−	0.335493i
$580$		0			0
$581$	894.412			1.53943
$582$		0			0
$583$	19.0744			0.0327177
$584$		0			0
$585$	−68.5008	+	282.581i	−0.117095	+	0.483045i
$586$		0			0
$587$	−1049.99			−1.78873			−0.894366	−	0.447335i	$-0.852373\pi$
$587$	−1049.99			−1.78873			−0.894366	+	0.447335i	$0.852373\pi$
$588$		0			0
$589$	801.172			1.36022
$590$		0			0
$591$			604.280i			1.02247i
$592$		0			0
$593$			553.135i			0.932774i	0.884581	+	0.466387i	$0.154445\pi$
$593$			553.135i			0.932774i	−0.884581	+	0.466387i	$0.845555\pi$
$594$		0			0
$595$	−108.256	+	446.581i	−0.181943	+	0.750557i
$596$		0			0
$597$		−	377.228i		−	0.631873i
$598$		0			0
$599$		−	240.892i		−	0.402156i	−0.979575	−	0.201078i	$-0.935555\pi$
$599$		−	240.892i		−	0.402156i	0.979575	−	0.201078i	$-0.0644445\pi$
$600$		0			0
$601$	939.752			1.56365			0.781823	−	0.623500i	$-0.214290\pi$
$601$	939.752			1.56365			0.781823	+	0.623500i	$0.214290\pi$
$602$		0			0
$603$	−8.30076			−0.0137658
$604$		0			0
$605$	573.877	+	139.114i	0.948558	+	0.229941i
$606$		0			0
$607$	−159.561			−0.262868			−0.131434	−	0.991325i	$-0.541958\pi$
$607$	−159.561			−0.262868			−0.131434	+	0.991325i	$0.541958\pi$
$608$		0			0
$609$	403.946			0.663294
$610$		0			0
$611$		−	776.608i		−	1.27104i
$612$		0			0
$613$			405.464i			0.661442i	0.943729	+	0.330721i	$0.107292\pi$
$613$			405.464i			0.661442i	−0.943729	+	0.330721i	$0.892708\pi$
$614$		0			0
$615$	−549.794	−	133.276i	−0.893973	−	0.216709i
$616$		0			0
$617$			210.340i			0.340907i	0.985366	+	0.170453i	$0.0545232\pi$
$617$			210.340i			0.340907i	−0.985366	+	0.170453i	$0.945477\pi$
$618$		0			0
$619$		−	374.381i		−	0.604815i	−0.953179	−	0.302408i	$-0.902210\pi$
$619$		−	374.381i		−	0.604815i	0.953179	−	0.302408i	$-0.0977904\pi$
$620$		0			0
$621$	27.3521			0.0440453
$622$		0			0
$623$	79.5334			0.127662
$624$		0			0
$625$	362.894	+	508.854i	0.580631	+	0.814167i
$626$		0			0
$627$	39.0296			0.0622481
$628$		0			0
$629$	−610.694			−0.970897
$630$		0			0
$631$			543.402i			0.861176i	0.902549	+	0.430588i	$0.141694\pi$
$631$			543.402i			0.861176i	−0.902549	+	0.430588i	$0.858306\pi$
$632$		0			0
$633$		−	551.175i		−	0.870734i
$634$		0			0
$635$	1150.78	+	278.962i	1.81225	+	0.439310i
$636$		0			0
$637$			17.9150i			0.0281240i
$638$		0			0
$639$		−	202.906i		−	0.317536i
$640$		0			0
$641$	−27.0948			−0.0422696			−0.0211348	−	0.999777i	$-0.506728\pi$
$641$	−27.0948			−0.0422696			−0.0211348	+	0.999777i	$0.506728\pi$
$642$		0			0
$643$	933.931			1.45246			0.726229	−	0.687453i	$-0.241271\pi$
$643$	933.931			1.45246			0.726229	+	0.687453i	$0.241271\pi$
$644$		0			0
$645$	170.268	+	41.2748i	0.263982	+	0.0639920i
$646$		0			0
$647$	−824.289			−1.27402			−0.637009	−	0.770857i	$-0.719829\pi$
$647$	−824.289			−1.27402			−0.637009	+	0.770857i	$0.719829\pi$
$648$		0			0
$649$	−67.9283			−0.104666
$650$		0			0
$651$			741.031i			1.13830i
$652$		0			0
$653$			794.099i			1.21608i	0.793907	+	0.608039i	$0.208044\pi$
$653$			794.099i			1.21608i	−0.793907	+	0.608039i	$0.791956\pi$
$654$		0			0
$655$	125.269	−	516.764i	0.191251	−	0.788953i
$656$		0			0
$657$			245.596i			0.373814i
$658$		0			0
$659$		−	251.919i		−	0.382275i	−0.981563	−	0.191137i	$-0.938782\pi$
$659$		−	251.919i		−	0.382275i	0.981563	−	0.191137i	$-0.0612176\pi$
$660$		0			0
$661$	−190.361			−0.287989			−0.143995	−	0.989578i	$-0.545995\pi$
$661$	−190.361			−0.287989			−0.143995	+	0.989578i	$0.545995\pi$
$662$		0			0
$663$	436.703			0.658678
$664$		0			0
$665$	−110.124	+	454.288i	−0.165601	+	0.683140i
$666$		0			0
$667$	173.747			0.260490
$668$		0			0
$669$	−321.673			−0.480826
$670$		0			0
$671$		−	54.8434i		−	0.0817339i
$672$		0			0
$673$			1116.66i			1.65922i	0.558341	+	0.829612i	$0.311438\pi$
$673$			1116.66i			1.65922i	−0.558341	+	0.829612i	$0.688562\pi$
$674$		0			0
$675$	−115.484	−	59.4847i	−0.171088	−	0.0881255i
$676$		0			0
$677$		−	170.279i		−	0.251520i	−0.992061	−	0.125760i	$-0.959863\pi$
$677$		−	170.279i		−	0.251520i	0.992061	−	0.125760i	$-0.0401370\pi$
$678$		0			0
$679$		−	350.065i		−	0.515560i
$680$		0			0
$681$	−496.365			−0.728876
$682$		0			0
$683$	−602.204			−0.881704			−0.440852	−	0.897580i	$-0.645324\pi$
$683$	−602.204			−0.881704			−0.440852	+	0.897580i	$0.645324\pi$
$684$		0			0
$685$	−163.498	+	674.467i	−0.238683	+	0.984623i
$686$		0			0
$687$	189.835			0.276324
$688$		0			0
$689$	217.107			0.315105
$690$		0			0
$691$		−	268.267i		−	0.388230i	−0.980979	−	0.194115i	$-0.937816\pi$
$691$		−	268.267i		−	0.388230i	0.980979	−	0.194115i	$-0.0621836\pi$
$692$		0			0
$693$			36.0998i			0.0520921i
$694$		0			0
$695$	−28.6281	+	118.097i	−0.0411915	+	0.169924i
$696$		0			0
$697$			849.655i			1.21902i
$698$		0			0
$699$			700.920i			1.00275i
$700$		0			0
$701$	1009.23			1.43970			0.719848	−	0.694132i	$-0.244212\pi$
$701$	1009.23			1.43970			0.719848	+	0.694132i	$0.244212\pi$
$702$		0			0
$703$	−621.233			−0.883688
$704$		0			0
$705$	337.195	+	81.7398i	0.478291	+	0.115943i
$706$		0			0
$707$	−403.683			−0.570980
$708$		0			0
$709$	−737.419			−1.04008			−0.520042	−	0.854141i	$-0.674084\pi$
$709$	−737.419			−1.04008			−0.520042	+	0.854141i	$0.674084\pi$
$710$		0			0
$711$			430.543i			0.605546i
$712$		0			0
$713$			318.735i			0.447034i
$714$		0			0
$715$	−160.417	−	38.8868i	−0.224359	−	0.0543871i
$716$		0			0
$717$			176.302i			0.245888i
$718$		0			0
$719$			747.093i			1.03907i	0.854448	+	0.519536i	$0.173895\pi$
$719$			747.093i			1.03907i	−0.854448	+	0.519536i	$0.826105\pi$
$720$		0			0
$721$	−446.315			−0.619022
$722$		0			0
$723$	−193.487			−0.267617
$724$		0			0
$725$	−733.579	−	377.859i	−1.01183	−	0.521185i
$726$		0			0
$727$	209.131			0.287663			0.143832	−	0.989602i	$-0.454058\pi$
$727$	209.131			0.287663			0.143832	+	0.989602i	$0.454058\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	263.134i		−	0.359964i
$732$		0			0
$733$		−	557.973i		−	0.761218i	−0.924736	−	0.380609i	$-0.875714\pi$
$733$		−	557.973i		−	0.761218i	0.924736	−	0.380609i	$-0.124286\pi$
$734$		0			0
$735$	−7.77849	−	1.88559i	−0.0105830	−	0.00256543i
$736$		0			0
$737$		−	4.71221i		−	0.00639377i
$738$		0			0
$739$		−	79.3912i		−	0.107431i	−0.998556	−	0.0537153i	$-0.982894\pi$
$739$		−	79.3912i		−	0.107431i	0.998556	−	0.0537153i	$-0.0171063\pi$
$740$		0			0
$741$	444.240			0.599514
$742$		0			0
$743$	606.208			0.815892			0.407946	−	0.913006i	$-0.366245\pi$
$743$	606.208			0.815892			0.407946	+	0.913006i	$0.366245\pi$
$744$		0			0
$745$	866.647	+	210.085i	1.16328	+	0.281993i
$746$		0			0
$747$	379.755			0.508373
$748$		0			0
$749$	−417.963			−0.558028
$750$		0			0
$751$			548.207i			0.729969i	0.931014	+	0.364984i	$0.118926\pi$
$751$			548.207i			0.729969i	−0.931014	+	0.364984i	$0.881074\pi$
$752$		0			0
$753$		−	292.306i		−	0.388189i
$754$		0			0
$755$	−201.204	+	830.011i	−0.266495	+	1.09935i
$756$		0			0
$757$			1299.81i			1.71705i	0.512770	+	0.858526i	$0.328619\pi$
$757$			1299.81i			1.71705i	−0.512770	+	0.858526i	$0.671381\pi$
$758$		0			0
$759$			15.5274i			0.0204577i
$760$		0			0
$761$	−390.151			−0.512682			−0.256341	−	0.966586i	$-0.582517\pi$
$761$	−390.151			−0.512682			−0.256341	+	0.966586i	$0.582517\pi$
$762$		0			0
$763$	1128.09			1.47849
$764$		0			0
$765$	−45.9641	+	189.612i	−0.0600837	+	0.247859i
$766$		0			0
$767$	−773.169			−1.00804
$768$		0			0
$769$	−450.628			−0.585992			−0.292996	−	0.956114i	$-0.594652\pi$
$769$	−450.628			−0.585992			−0.292996	+	0.956114i	$0.594652\pi$
$770$		0			0
$771$			555.543i			0.720549i
$772$		0			0
$773$			226.657i			0.293217i	0.989195	+	0.146608i	$0.0468357\pi$
$773$			226.657i			0.293217i	−0.989195	+	0.146608i	$0.953164\pi$
$774$		0			0
$775$	693.176	−	1345.74i	0.894421	−	1.73644i
$776$		0			0
$777$		−	574.600i		−	0.739510i
$778$		0			0
$779$			864.318i			1.10952i
$780$		0			0
$781$	115.186			0.147486
$782$		0			0
$783$	171.510			0.219042
$784$		0			0
$785$	98.0028	−	404.284i	0.124844	−	0.515011i
$786$		0			0
$787$	204.623			0.260004			0.130002	−	0.991514i	$-0.458502\pi$
$787$	204.623			0.260004			0.130002	+	0.991514i	$0.458502\pi$
$788$		0			0
$789$	100.841			0.127809
$790$		0			0
$791$			1029.33i			1.30130i
$792$		0			0
$793$		−	624.235i		−	0.787182i
$794$		0			0
$795$	−22.8511	+	94.2658i	−0.0287435	+	0.118573i
$796$		0			0
$797$		−	431.987i		−	0.542016i	−0.962577	−	0.271008i	$-0.912643\pi$
$797$		−	431.987i		−	0.542016i	0.962577	−	0.271008i	$-0.0873570\pi$
$798$		0			0
$799$		−	521.104i		−	0.652195i
$800$		0			0
$801$	33.7688			0.0421583
$802$		0			0
$803$	−139.421			−0.173625
$804$		0			0
$805$	−180.732	−	43.8114i	−0.224512	−	0.0544242i
$806$		0			0
$807$	449.888			0.557482
$808$		0			0
$809$	1348.23			1.66654			0.833270	−	0.552867i	$-0.186466\pi$
$809$	1348.23			1.66654			0.833270	+	0.552867i	$0.186466\pi$
$810$		0			0
$811$			635.864i			0.784050i	0.919955	+	0.392025i	$0.128225\pi$
$811$			635.864i			0.784050i	−0.919955	+	0.392025i	$0.871775\pi$
$812$		0			0
$813$		−	108.738i		−	0.133750i
$814$		0			0
$815$	394.995	+	95.7510i	0.484656	+	0.117486i
$816$		0			0
$817$		−	267.675i		−	0.327631i
$818$		0			0
$819$			410.893i			0.501700i
$820$		0			0
$821$	−3.88250			−0.00472899			−0.00236449	−	0.999997i	$-0.500753\pi$
$821$	−3.88250			−0.00472899			−0.00236449	+	0.999997i	$0.500753\pi$
$822$		0			0
$823$	−799.122			−0.970987			−0.485494	−	0.874240i	$-0.661360\pi$
$823$	−799.122			−0.970987			−0.485494	+	0.874240i	$0.661360\pi$
$824$		0			0
$825$	33.7685	−	65.5585i	0.0409315	−	0.0794648i
$826$		0			0
$827$	1337.16			1.61688			0.808438	−	0.588582i	$-0.200313\pi$
$827$	1337.16			1.61688			0.808438	+	0.588582i	$0.200313\pi$
$828$		0			0
$829$	711.385			0.858124			0.429062	−	0.903275i	$-0.358844\pi$
$829$	711.385			0.858124			0.429062	+	0.903275i	$0.358844\pi$
$830$		0			0
$831$			726.101i			0.873767i
$832$		0			0
$833$			12.0209i			0.0144309i
$834$		0			0
$835$	−1529.05	−	370.658i	−1.83120	−	0.443902i
$836$		0			0
$837$			314.632i			0.375904i
$838$		0			0
$839$		−	903.214i		−	1.07654i	−0.842774	−	0.538268i	$-0.819079\pi$
$839$		−	903.214i		−	1.07654i	0.842774	−	0.538268i	$-0.180921\pi$
$840$		0			0
$841$	248.466			0.295441
$842$		0			0
$843$	−744.333			−0.882957
$844$		0			0
$845$	−1004.67	−	243.543i	−1.18896	−	0.288217i
$846$		0			0
$847$	834.457			0.985191
$848$		0			0
$849$	−358.799			−0.422613
$850$		0			0
$851$		−	247.149i		−	0.290422i
$852$		0			0
$853$		−	473.826i		−	0.555482i	−0.960656	−	0.277741i	$-0.910414\pi$
$853$		−	473.826i		−	0.555482i	0.960656	−	0.277741i	$-0.0895857\pi$
$854$		0			0
$855$	−46.7573	+	192.884i	−0.0546869	+	0.225596i
$856$		0			0
$857$			357.541i			0.417201i	0.978001	+	0.208601i	$0.0668909\pi$
$857$			357.541i			0.417201i	−0.978001	+	0.208601i	$0.933109\pi$
$858$		0			0
$859$		−	312.604i		−	0.363917i	−0.983306	−	0.181958i	$-0.941756\pi$
$859$		−	312.604i		−	0.363917i	0.983306	−	0.181958i	$-0.0582436\pi$
$860$		0			0
$861$	−799.438			−0.928499
$862$		0			0
$863$	−1106.26			−1.28187			−0.640937	−	0.767594i	$-0.721454\pi$
$863$	−1106.26			−1.28187			−0.640937	+	0.767594i	$0.721454\pi$
$864$		0			0
$865$	168.614	−	695.572i	0.194930	−	0.804130i
$866$		0			0
$867$	−207.534			−0.239371
$868$		0			0
$869$	−244.412			−0.281257
$870$		0			0
$871$		−	53.6350i		−	0.0615786i
$872$		0			0
$873$		−	148.633i		−	0.170255i
$874$		0			0
$875$	667.794	+	578.029i	0.763193	+	0.660604i
$876$		0			0
$877$		−	150.342i		−	0.171428i	−0.996320	−	0.0857140i	$-0.972683\pi$
$877$		−	150.342i		−	0.171428i	0.996320	−	0.0857140i	$-0.0273171\pi$
$878$		0			0
$879$		−	273.639i		−	0.311307i
$880$		0			0
$881$	717.680			0.814619			0.407310	−	0.913290i	$-0.366467\pi$
$881$	717.680			0.814619			0.407310	+	0.913290i	$0.366467\pi$
$882$		0			0
$883$	−900.807			−1.02017			−0.510083	−	0.860125i	$-0.670385\pi$
$883$	−900.807			−1.02017			−0.510083	+	0.860125i	$0.670385\pi$
$884$		0			0
$885$	81.3778	−	335.702i	0.0919524	−	0.379324i
$886$		0			0
$887$	−794.320			−0.895513			−0.447757	−	0.894155i	$-0.647777\pi$
$887$	−794.320			−0.895513			−0.447757	+	0.894155i	$0.647777\pi$
$888$		0			0
$889$	1673.31			1.88224
$890$		0			0
$891$			15.3275i			0.0172025i
$892$		0			0
$893$		−	530.097i		−	0.593614i
$894$		0			0
$895$	269.817	−	1113.06i	0.301472	−	1.24364i
$896$		0			0
$897$			176.735i			0.197029i
$898$		0			0
$899$			1998.61i			2.22314i
$900$		0			0
$901$	145.679			0.161686
$902$		0			0
$903$	247.582			0.274177
$904$		0			0
$905$	−754.107	−	182.804i	−0.833268	−	0.201993i
$906$		0			0
$907$	−340.876			−0.375829			−0.187914	−	0.982185i	$-0.560173\pi$
$907$	−340.876			−0.375829			−0.187914	+	0.982185i	$0.560173\pi$
$908$		0			0
$909$	−171.398			−0.188557
$910$		0			0
$911$		−	1364.47i		−	1.49777i	−0.662697	−	0.748887i	$-0.730588\pi$
$911$		−	1364.47i		−	1.49777i	0.662697	−	0.748887i	$-0.269412\pi$
$912$		0			0
$913$			215.581i			0.236123i
$914$		0			0
$915$	271.037	+	65.7022i	0.296215	+	0.0718057i
$916$		0			0
$917$		−	751.410i		−	0.819422i
$918$		0			0
$919$		−	903.899i		−	0.983568i	−0.870717	−	0.491784i	$-0.836345\pi$
$919$		−	903.899i		−	0.983568i	0.870717	−	0.491784i	$-0.163655\pi$
$920$		0			0
$921$	−29.4354			−0.0319602
$922$		0			0
$923$	1311.07			1.42044
$924$		0			0
$925$	−537.493	+	1043.49i	−0.581073	+	1.12810i
$926$		0			0
$927$	−189.499			−0.204422
$928$		0			0
$929$	267.291			0.287719			0.143860	−	0.989598i	$-0.454049\pi$
$929$	267.291			0.287719			0.143860	+	0.989598i	$0.454049\pi$
$930$		0			0
$931$			12.2284i			0.0131347i
$932$		0			0
$933$			745.237i			0.798753i
$934$		0			0
$935$	−107.640	−	26.0931i	−0.115123	−	0.0279070i
$936$		0			0
$937$		−	1612.36i		−	1.72076i	−0.509649	−	0.860382i	$-0.670225\pi$
$937$		−	1612.36i		−	1.72076i	0.509649	−	0.860382i	$-0.329775\pi$
$938$		0			0
$939$		−	589.533i		−	0.627831i
$940$		0			0
$941$	−878.212			−0.933276			−0.466638	−	0.884448i	$-0.654535\pi$
$941$	−878.212			−0.933276			−0.466638	+	0.884448i	$0.654535\pi$
$942$		0			0
$943$	−343.857			−0.364641
$944$		0			0
$945$	−178.405	−	43.2474i	−0.188789	−	0.0457645i
$946$		0			0
$947$	726.974			0.767660			0.383830	−	0.923404i	$-0.374605\pi$
$947$	726.974			0.767660			0.383830	+	0.923404i	$0.374605\pi$
$948$		0			0
$949$	−1586.91			−1.67219
$950$		0			0
$951$			602.055i			0.633076i
$952$		0			0
$953$			603.417i			0.633177i	0.948563	+	0.316588i	$0.102537\pi$
$953$			603.417i			0.633177i	−0.948563	+	0.316588i	$0.897463\pi$
$954$		0			0
$955$	352.247	−	1453.10i	0.368845	−	1.52157i
$956$		0			0
$957$			97.3633i			0.101738i
$958$		0			0
$959$			980.721i			1.02265i
$960$		0			0
$961$	−2705.41			−2.81520
$962$		0			0
$963$	−177.461			−0.184279
$964$		0			0
$965$	−132.106	+	544.969i	−0.136898	+	0.564734i
$966$		0			0
$967$	−408.083			−0.422009			−0.211004	−	0.977485i	$-0.567673\pi$
$967$	−408.083			−0.422009			−0.211004	+	0.977485i	$0.567673\pi$
$968$		0			0
$969$	298.085			0.307621
$970$		0			0
$971$		−	963.723i		−	0.992506i	−0.868178	−	0.496253i	$-0.834709\pi$
$971$		−	963.723i		−	0.992506i	0.868178	−	0.496253i	$-0.165291\pi$
$972$		0			0
$973$			171.722i			0.176487i
$974$		0			0
$975$	384.358	−	746.195i	0.394213	−	0.765328i
$976$		0			0
$977$			498.460i			0.510195i	0.966915	+	0.255097i	$0.0821075\pi$
$977$			498.460i			0.510195i	−0.966915	+	0.255097i	$0.917892\pi$
$978$		0			0
$979$			19.1700i			0.0195812i
$980$		0			0
$981$	478.970			0.488247
$982$		0			0
$983$	1328.10			1.35107			0.675535	−	0.737328i	$-0.263913\pi$
$983$	1328.10			1.35107			0.675535	+	0.737328i	$0.263913\pi$
$984$		0			0
$985$	410.960	−	1695.31i	0.417219	−	1.72112i
$986$		0			0
$987$	490.305			0.496763
$988$		0			0
$989$	106.491			0.107675
$990$		0			0
$991$		−	1160.02i		−	1.17055i	−0.810834	−	0.585277i	$-0.800986\pi$
$991$		−	1160.02i		−	1.17055i	0.810834	−	0.585277i	$-0.199014\pi$
$992$		0			0
$993$		−	902.863i		−	0.909228i
$994$		0			0
$995$	−256.546	+	1058.31i	−0.257836	+	1.06363i
$996$		0			0
$997$		−	806.436i		−	0.808862i	−0.914568	−	0.404431i	$-0.867470\pi$
$997$		−	806.436i		−	0.808862i	0.914568	−	0.404431i	$-0.132530\pi$
$998$		0			0
$999$		−	243.967i		−	0.244211i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.j.f.319.6		12
4.3	odd	2		960.3.j.g.319.12		12
5.4	even	2		960.3.j.g.319.11		12
8.3	odd	2		480.3.j.b.319.1	yes	12
8.5	even	2		480.3.j.a.319.7	✓	12
20.19	odd	2	inner	960.3.j.f.319.5		12
24.5	odd	2		1440.3.j.d.1279.12		12
24.11	even	2		1440.3.j.c.1279.12		12
40.3	even	4		2400.3.e.h.1951.5		12
40.13	odd	4		2400.3.e.h.1951.8		12
40.19	odd	2		480.3.j.a.319.8	yes	12
40.27	even	4		2400.3.e.i.1951.8		12
40.29	even	2		480.3.j.b.319.2	yes	12
40.37	odd	4		2400.3.e.i.1951.5		12
120.29	odd	2		1440.3.j.c.1279.11		12
120.59	even	2		1440.3.j.d.1279.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.j.a.319.7	✓	12	8.5	even	2
480.3.j.a.319.8	yes	12	40.19	odd	2
480.3.j.b.319.1	yes	12	8.3	odd	2
480.3.j.b.319.2	yes	12	40.29	even	2
960.3.j.f.319.5		12	20.19	odd	2	inner
960.3.j.f.319.6		12	1.1	even	1	trivial
960.3.j.g.319.11		12	5.4	even	2
960.3.j.g.319.12		12	4.3	odd	2
1440.3.j.c.1279.11		12	120.29	odd	2
1440.3.j.c.1279.12		12	24.11	even	2
1440.3.j.d.1279.11		12	120.59	even	2
1440.3.j.d.1279.12		12	24.5	odd	2
2400.3.e.h.1951.5		12	40.3	even	4
2400.3.e.h.1951.8		12	40.13	odd	4
2400.3.e.i.1951.5		12	40.37	odd	4
2400.3.e.i.1951.8		12	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(4.85927 + 1.17794i) q^{5} +7.06571 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +(4.85927 + 1.17794i) q^{5} +7.06571 q^{7} +3.00000 q^{9} +1.70305i q^{11} +19.3844i q^{13} +(-8.41649 - 2.04025i) q^{15} +13.0069i q^{17} +13.2314i q^{19} -12.2382 q^{21} -5.26392 q^{23} +(22.2249 + 11.4478i) q^{25} -5.19615 q^{27} -33.0071 q^{29} -60.5509i q^{31} -2.94977i q^{33} +(34.3341 + 8.32297i) q^{35} +46.9515i q^{37} -33.5747i q^{39} +65.3233 q^{41} -20.2303 q^{43} +(14.5778 + 3.53382i) q^{45} -40.0636 q^{47} +0.924196 q^{49} -22.5286i q^{51} -11.2001i q^{53} +(-2.00609 + 8.27558i) q^{55} -22.9174i q^{57} +39.8862i q^{59} -32.2030 q^{61} +21.1971 q^{63} +(-22.8336 + 94.1938i) q^{65} -2.76692 q^{67} +9.11738 q^{69} -67.6353i q^{71} +81.8653i q^{73} +(-38.4947 - 19.8282i) q^{75} +12.0333i q^{77} +143.514i q^{79} +9.00000 q^{81} +126.585 q^{83} +(-15.3214 + 63.2041i) q^{85} +57.1699 q^{87} +11.2563 q^{89} +136.964i q^{91} +104.877i q^{93} +(-15.5858 + 64.2948i) q^{95} -49.5443i q^{97} +5.10916i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} + 36 q^{9}+O(q^{10}) \) 12 * q - 4 * q^5 + 36 * q^9	\( 12 q - 4 q^{5} + 36 q^{9} - 24 q^{15} - 80 q^{23} + 28 q^{25} + 40 q^{29} + 144 q^{35} + 136 q^{41} + 224 q^{43} - 12 q^{45} + 208 q^{47} + 212 q^{49} - 192 q^{55} - 40 q^{61} + 96 q^{65} - 352 q^{67} + 192 q^{75} + 108 q^{81} + 64 q^{85} + 48 q^{87} - 8 q^{89} - 176 q^{95}+O(q^{100}) \) 12 * q - 4 * q^5 + 36 * q^9 - 24 * q^15 - 80 * q^23 + 28 * q^25 + 40 * q^29 + 144 * q^35 + 136 * q^41 + 224 * q^43 - 12 * q^45 + 208 * q^47 + 212 * q^49 - 192 * q^55 - 40 * q^61 + 96 * q^65 - 352 * q^67 + 192 * q^75 + 108 * q^81 + 64 * q^85 + 48 * q^87 - 8 * q^89 - 176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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