LMFDB - Embedded newform 480.4.h.b.191.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(191,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("480.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.h (of order $2$, degree $1$, 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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.16
Character	$\chi$	$=$	480.191
Dual form			480.4.h.b.191.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.28458 + 4.66698i) q^{3} -5.00000i q^{5} +8.51421i q^{7} +(-16.5614 + 21.3242i) q^{9} -38.7469 q^{11} +81.0585 q^{13} +(23.3349 - 11.4229i) q^{15} +1.42775i q^{17} +70.0862i q^{19} +(-39.7356 + 19.4514i) q^{21} -190.718 q^{23} -25.0000 q^{25} +(-137.355 - 28.5752i) q^{27} +96.7846i q^{29} +235.759i q^{31} +(-88.5202 - 180.831i) q^{33} +42.5710 q^{35} -342.821 q^{37} +(185.184 + 378.299i) q^{39} -158.352i q^{41} -161.739i q^{43} +(106.621 + 82.8071i) q^{45} +241.292 q^{47} +270.508 q^{49} +(-6.66328 + 3.26180i) q^{51} +240.675i q^{53} +193.734i q^{55} +(-327.091 + 160.117i) q^{57} -579.434 q^{59} -687.344 q^{61} +(-181.558 - 141.007i) q^{63} -405.293i q^{65} +607.711i q^{67} +(-435.710 - 890.077i) q^{69} +175.195 q^{71} -776.670 q^{73} +(-57.1144 - 116.675i) q^{75} -329.899i q^{77} +372.245i q^{79} +(-180.439 - 706.316i) q^{81} -191.879 q^{83} +7.13875 q^{85} +(-451.692 + 221.112i) q^{87} -1571.97i q^{89} +690.149i q^{91} +(-1100.28 + 538.609i) q^{93} +350.431 q^{95} +759.448 q^{97} +(641.703 - 826.245i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{9} + 72 q^{11} - 72 q^{13} + 20 q^{15} - 68 q^{21} + 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} - 504 q^{37} - 456 q^{39} - 220 q^{45} + 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57}+ \cdots + 3160 q^{99}+O(q^{100}) $ 24 * q + 20 * q^9 + 72 * q^11 - 72 * q^13 + 20 * q^15 - 68 * q^21 + 96 * q^23 - 600 * q^25 + 168 * q^27 - 80 * q^33 - 504 * q^37 - 456 * q^39 - 220 * q^45 + 432 * q^47 - 816 * q^49 + 1240 * q^51 + 40 * q^57 - 2184 * q^59 - 1080 * q^61 - 1720 * q^63 - 1548 * q^69 + 816 * q^71 - 1440 * q^73 - 2304 * q^81 - 2016 * q^83 - 2144 * q^87 - 112 * q^93 + 1800 * q^95 + 720 * q^97 + 3160 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	2.28458	+	4.66698i	0.439667	+	0.898161i
$3$	2.28458	+	4.66698i	0.439667	+	0.898161i
$4$		0			0
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$			8.51421i			0.459724i	0.973223	+	0.229862i	$0.0738275\pi$
$7$			8.51421i			0.459724i	−0.973223	+	0.229862i	$0.926173\pi$
$8$		0			0
$9$	−16.5614	+	21.3242i	−0.613386	+	0.789783i
$10$		0			0
$11$	−38.7469			−1.06206			−0.531029	−	0.847354i	$-0.678194\pi$
$11$	−38.7469			−1.06206			−0.531029	+	0.847354i	$0.678194\pi$
$12$		0			0
$13$	81.0585			1.72935			0.864677	−	0.502329i	$-0.167523\pi$
$13$	81.0585			1.72935			0.864677	+	0.502329i	$0.167523\pi$
$14$		0			0
$15$	23.3349	−	11.4229i	0.401670	−	0.196625i
$16$		0			0
$17$			1.42775i			0.0203694i	0.999948	+	0.0101847i	$0.00324195\pi$
$17$			1.42775i			0.0203694i	−0.999948	+	0.0101847i	$0.996758\pi$
$18$		0			0
$19$			70.0862i			0.846257i	0.906070	+	0.423128i	$0.139068\pi$
$19$			70.0862i			0.846257i	−0.906070	+	0.423128i	$0.860932\pi$
$20$		0			0
$21$	−39.7356	+	19.4514i	−0.412906	+	0.202125i
$22$		0			0
$23$	−190.718			−1.72902			−0.864509	−	0.502617i	$-0.832371\pi$
$23$	−190.718			−1.72902			−0.864509	+	0.502617i	$0.832371\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−137.355	−	28.5752i	−0.979038	−	0.203678i
$28$		0			0
$29$			96.7846i			0.619740i	0.950779	+	0.309870i	$0.100285\pi$
$29$			96.7846i			0.619740i	−0.950779	+	0.309870i	$0.899715\pi$
$30$		0			0
$31$			235.759i			1.36592i	0.730456	+	0.682960i	$0.239308\pi$
$31$			235.759i			1.36592i	−0.730456	+	0.682960i	$0.760692\pi$
$32$		0			0
$33$	−88.5202	−	180.831i	−0.466951	−	0.953898i
$34$		0			0
$35$	42.5710			0.205595
$36$		0			0
$37$	−342.821			−1.52323			−0.761613	−	0.648032i	$-0.775592\pi$
$37$	−342.821			−1.52323			−0.761613	+	0.648032i	$0.775592\pi$
$38$		0			0
$39$	185.184	+	378.299i	0.760340	+	1.55324i
$40$		0			0
$41$		−	158.352i		−	0.603181i	−0.953438	−	0.301591i	$-0.902482\pi$
$41$		−	158.352i		−	0.603181i	0.953438	−	0.301591i	$-0.0975175\pi$
$42$		0			0
$43$		−	161.739i		−	0.573604i	−0.957990	−	0.286802i	$-0.907408\pi$
$43$		−	161.739i		−	0.573604i	0.957990	−	0.286802i	$-0.0925922\pi$
$44$		0			0
$45$	106.621	+	82.8071i	0.353202	+	0.274315i
$46$		0			0
$47$	241.292			0.748851			0.374426	−	0.927257i	$-0.377840\pi$
$47$	241.292			0.748851			0.374426	+	0.927257i	$0.377840\pi$
$48$		0			0
$49$	270.508			0.788654
$50$		0			0
$51$	−6.66328	+	3.26180i	−0.0182950	+	0.00895576i
$52$		0			0
$53$			240.675i			0.623760i	0.950122	+	0.311880i	$0.100959\pi$
$53$			240.675i			0.623760i	−0.950122	+	0.311880i	$0.899041\pi$
$54$		0			0
$55$			193.734i			0.474966i
$56$		0			0
$57$	−327.091	+	160.117i	−0.760075	+	0.372071i
$58$		0			0
$59$	−579.434			−1.27857			−0.639287	−	0.768968i	$-0.720770\pi$
$59$	−579.434			−1.27857			−0.639287	+	0.768968i	$0.720770\pi$
$60$		0			0
$61$	−687.344			−1.44271			−0.721356	−	0.692565i	$-0.756481\pi$
$61$	−687.344			−1.44271			−0.721356	+	0.692565i	$0.756481\pi$
$62$		0			0
$63$	−181.558	−	141.007i	−0.363082	−	0.281988i
$64$		0			0
$65$		−	405.293i		−	0.773390i
$66$		0			0
$67$			607.711i			1.10811i	0.832479	+	0.554057i	$0.186921\pi$
$67$			607.711i			1.10811i	−0.832479	+	0.554057i	$0.813079\pi$
$68$		0			0
$69$	−435.710	−	890.077i	−0.760192	−	1.55294i
$70$		0			0
$71$	175.195			0.292843			0.146421	−	0.989222i	$-0.453224\pi$
$71$	175.195			0.292843			0.146421	+	0.989222i	$0.453224\pi$
$72$		0			0
$73$	−776.670			−1.24524			−0.622619	−	0.782525i	$-0.713931\pi$
$73$	−776.670			−1.24524			−0.622619	+	0.782525i	$0.713931\pi$
$74$		0			0
$75$	−57.1144	−	116.675i	−0.0879334	−	0.179632i
$76$		0			0
$77$		−	329.899i		−	0.488253i
$78$		0			0
$79$			372.245i			0.530137i	0.964230	+	0.265069i	$0.0853946\pi$
$79$			372.245i			0.530137i	−0.964230	+	0.265069i	$0.914605\pi$
$80$		0			0
$81$	−180.439	−	706.316i	−0.247516	−	0.968884i
$82$		0			0
$83$	−191.879			−0.253752			−0.126876	−	0.991919i	$-0.540495\pi$
$83$	−191.879			−0.253752			−0.126876	+	0.991919i	$0.540495\pi$
$84$		0			0
$85$	7.13875			0.00910948
$86$		0			0
$87$	−451.692	+	221.112i	−0.556626	+	0.272479i
$88$		0			0
$89$		−	1571.97i		−	1.87223i	−0.351697	−	0.936114i	$-0.614395\pi$
$89$		−	1571.97i		−	1.87223i	0.351697	−	0.936114i	$-0.385605\pi$
$90$		0			0
$91$			690.149i			0.795025i
$92$		0			0
$93$	−1100.28	+	538.609i	−1.22682	+	0.600550i
$94$		0			0
$95$	350.431			0.378458
$96$		0			0
$97$	759.448			0.794952			0.397476	−	0.917613i	$-0.369886\pi$
$97$	759.448			0.794952			0.397476	+	0.917613i	$0.369886\pi$
$98$		0			0
$99$	641.703	−	826.245i	0.651451	−	0.838795i
$100$		0			0
$101$			1179.56i			1.16209i	0.813873	+	0.581043i	$0.197355\pi$
$101$			1179.56i			1.16209i	−0.813873	+	0.581043i	$0.802645\pi$
$102$		0			0
$103$			980.381i			0.937862i	0.883235	+	0.468931i	$0.155361\pi$
$103$			980.381i			0.937862i	−0.883235	+	0.468931i	$0.844639\pi$
$104$		0			0
$105$	97.2568	+	198.678i	0.0903933	+	0.184657i
$106$		0			0
$107$	194.242			0.175496			0.0877482	−	0.996143i	$-0.472033\pi$
$107$	194.242			0.175496			0.0877482	+	0.996143i	$0.472033\pi$
$108$		0			0
$109$	791.809			0.695794			0.347897	−	0.937533i	$-0.386896\pi$
$109$	791.809			0.695794			0.347897	+	0.937533i	$0.386896\pi$
$110$		0			0
$111$	−783.200	−	1599.94i	−0.669713	−	1.36810i
$112$		0			0
$113$		−	474.323i		−	0.394872i	−0.980316	−	0.197436i	$-0.936739\pi$
$113$		−	474.323i		−	0.394872i	0.980316	−	0.197436i	$-0.0632615\pi$
$114$		0			0
$115$			953.589i			0.773241i
$116$		0			0
$117$	−1342.44	+	1728.50i	−1.06076	+	1.36581i
$118$		0			0
$119$	−12.1562			−0.00936431
$120$		0			0
$121$	170.322			0.127965
$122$		0			0
$123$	739.026	−	361.767i	0.541754	−	0.265199i
$124$		0			0
$125$			125.000i			0.0894427i
$126$		0			0
$127$		−	302.167i		−	0.211126i	−0.994413	−	0.105563i	$-0.966336\pi$
$127$		−	302.167i		−	0.211126i	0.994413	−	0.105563i	$-0.0336644\pi$
$128$		0			0
$129$	754.833	−	369.505i	0.515189	−	0.252195i
$130$		0			0
$131$	−1361.01			−0.907724			−0.453862	−	0.891072i	$-0.649954\pi$
$131$	−1361.01			−0.907724			−0.453862	+	0.891072i	$0.649954\pi$
$132$		0			0
$133$	−596.729			−0.389045
$134$		0			0
$135$	−142.876	+	686.776i	−0.0910873	+	0.437839i
$136$		0			0
$137$			786.617i			0.490549i	0.969454	+	0.245275i	$0.0788781\pi$
$137$			786.617i			0.490549i	−0.969454	+	0.245275i	$0.921122\pi$
$138$		0			0
$139$			3076.96i			1.87758i	0.344483	+	0.938792i	$0.388054\pi$
$139$			3076.96i			1.87758i	−0.344483	+	0.938792i	$0.611946\pi$
$140$		0			0
$141$	551.249	+	1126.10i	0.329245	+	0.672589i
$142$		0			0
$143$	−3140.77			−1.83667
$144$		0			0
$145$	483.923			0.277156
$146$		0			0
$147$	617.997	+	1262.46i	0.346745	+	0.708338i
$148$		0			0
$149$			969.009i			0.532780i	0.963865	+	0.266390i	$0.0858310\pi$
$149$			969.009i			0.532780i	−0.963865	+	0.266390i	$0.914169\pi$
$150$		0			0
$151$		−	639.744i		−	0.344779i	−0.985029	−	0.172390i	$-0.944851\pi$
$151$		−	639.744i		−	0.344779i	0.985029	−	0.172390i	$-0.0551488\pi$
$152$		0			0
$153$	−30.4455	−	23.6456i	−0.0160874	−	0.0124943i
$154$		0			0
$155$	1178.79			0.610858
$156$		0			0
$157$	1615.91			0.821426			0.410713	−	0.911765i	$-0.365280\pi$
$157$	1615.91			0.821426			0.410713	+	0.911765i	$0.365280\pi$
$158$		0			0
$159$	−1123.23	+	549.841i	−0.560237	+	0.274247i
$160$		0			0
$161$		−	1623.81i		−	0.794871i
$162$		0			0
$163$		−	2778.73i		−	1.33526i	−0.744495	−	0.667628i	$-0.767310\pi$
$163$		−	2778.73i		−	1.33526i	0.744495	−	0.667628i	$-0.232690\pi$
$164$		0			0
$165$	−904.155	+	442.601i	−0.426596	+	0.208827i
$166$		0			0
$167$	3626.17			1.68025			0.840124	−	0.542394i	$-0.182482\pi$
$167$	3626.17			1.68025			0.840124	+	0.542394i	$0.182482\pi$
$168$		0			0
$169$	4373.49			1.99066
$170$		0			0
$171$	−1494.53	−	1160.73i	−0.668359	−	0.519082i
$172$		0			0
$173$			592.810i			0.260523i	0.991480	+	0.130262i	$0.0415817\pi$
$173$			592.810i			0.260523i	−0.991480	+	0.130262i	$0.958418\pi$
$174$		0			0
$175$		−	212.855i		−	0.0919448i
$176$		0			0
$177$	−1323.76	−	2704.21i	−0.562147	−	1.14836i
$178$		0			0
$179$	1622.69			0.677575			0.338787	−	0.940863i	$-0.389983\pi$
$179$	1622.69			0.677575			0.338787	+	0.940863i	$0.389983\pi$
$180$		0			0
$181$	1693.90			0.695615			0.347808	−	0.937566i	$-0.386926\pi$
$181$	1693.90			0.695615			0.347808	+	0.937566i	$0.386926\pi$
$182$		0			0
$183$	−1570.29	−	3207.82i	−0.634313	−	1.29579i
$184$		0			0
$185$			1714.10i			0.681208i
$186$		0			0
$187$		−	55.3208i		−	0.0216335i
$188$		0			0
$189$	243.295	−	1169.47i	0.0936354	−	0.450087i
$190$		0			0
$191$	2963.96			1.12285			0.561427	−	0.827527i	$-0.310253\pi$
$191$	2963.96			1.12285			0.561427	+	0.827527i	$0.310253\pi$
$192$		0			0
$193$	4166.05			1.55378			0.776888	−	0.629638i	$-0.216797\pi$
$193$	4166.05			1.55378			0.776888	+	0.629638i	$0.216797\pi$
$194$		0			0
$195$	1891.49	−	925.922i	0.694629	−	0.340034i
$196$		0			0
$197$			5220.61i			1.88809i	0.329820	+	0.944044i	$0.393012\pi$
$197$			5220.61i			1.88809i	−0.329820	+	0.944044i	$0.606988\pi$
$198$		0			0
$199$		−	3043.91i		−	1.08431i	−0.840280	−	0.542153i	$-0.817609\pi$
$199$		−	3043.91i		−	1.08431i	0.840280	−	0.542153i	$-0.182391\pi$
$200$		0			0
$201$	−2836.17	+	1388.36i	−0.995265	+	0.487201i
$202$		0			0
$203$	−824.044			−0.284909
$204$		0			0
$205$	−791.760			−0.269751
$206$		0			0
$207$	3158.56	−	4066.90i	1.06056	−	1.36555i
$208$		0			0
$209$		−	2715.62i		−	0.898773i
$210$		0			0
$211$			274.774i			0.0896503i	0.998995	+	0.0448251i	$0.0142731\pi$
$211$			274.774i			0.0896503i	−0.998995	+	0.0448251i	$0.985727\pi$
$212$		0			0
$213$	400.247	+	817.633i	0.128753	+	0.263020i
$214$		0			0
$215$	−808.695			−0.256524
$216$		0			0
$217$	−2007.30			−0.627946
$218$		0			0
$219$	−1774.36	−	3624.70i	−0.547490	−	1.11842i
$220$		0			0
$221$			115.731i			0.0352259i
$222$		0			0
$223$			2914.20i			0.875109i	0.899192	+	0.437554i	$0.144155\pi$
$223$			2914.20i			0.875109i	−0.899192	+	0.437554i	$0.855845\pi$
$224$		0			0
$225$	414.035	−	533.104i	0.122677	−	0.157957i
$226$		0			0
$227$	237.573			0.0694638			0.0347319	−	0.999397i	$-0.488942\pi$
$227$	237.573			0.0694638			0.0347319	+	0.999397i	$0.488942\pi$
$228$		0			0
$229$	614.568			0.177344			0.0886720	−	0.996061i	$-0.471738\pi$
$229$	614.568			0.177344			0.0886720	+	0.996061i	$0.471738\pi$
$230$		0			0
$231$	1539.63	−	753.680i	0.438530	−	0.214669i
$232$		0			0
$233$		−	4953.20i		−	1.39268i	−0.717711	−	0.696341i	$-0.754810\pi$
$233$		−	4953.20i		−	1.39268i	0.717711	−	0.696341i	$-0.245190\pi$
$234$		0			0
$235$		−	1206.46i		−	0.334896i
$236$		0			0
$237$	−1737.26	+	850.423i	−0.476149	+	0.233084i
$238$		0			0
$239$	−862.873			−0.233534			−0.116767	−	0.993159i	$-0.537253\pi$
$239$	−862.873			−0.233534			−0.116767	+	0.993159i	$0.537253\pi$
$240$		0			0
$241$	4330.52			1.15748			0.578741	−	0.815511i	$-0.303544\pi$
$241$	4330.52			1.15748			0.578741	+	0.815511i	$0.303544\pi$
$242$		0			0
$243$	2884.14	−	2455.74i	0.761389	−	0.648295i
$244$		0			0
$245$		−	1352.54i		−	0.352697i
$246$		0			0
$247$			5681.08i			1.46348i
$248$		0			0
$249$	−438.361	−	895.494i	−0.111566	−	0.227910i
$250$		0			0
$251$	−1921.40			−0.483178			−0.241589	−	0.970379i	$-0.577669\pi$
$251$	−1921.40			−0.483178			−0.241589	+	0.970379i	$0.577669\pi$
$252$		0			0
$253$	7389.72			1.83632
$254$		0			0
$255$	16.3090	+	33.3164i	0.00400514	+	0.00818178i
$256$		0			0
$257$			7373.22i			1.78961i	0.446460	+	0.894803i	$0.352684\pi$
$257$			7373.22i			1.78961i	−0.446460	+	0.894803i	$0.647316\pi$
$258$		0			0
$259$		−	2918.85i		−	0.700264i
$260$		0			0
$261$	−2063.85	−	1602.89i	−0.489460	−	0.380140i
$262$		0			0
$263$	5671.84			1.32981			0.664906	−	0.746927i	$-0.268472\pi$
$263$	5671.84			1.32981			0.664906	+	0.746927i	$0.268472\pi$
$264$		0			0
$265$	1203.38			0.278954
$266$		0			0
$267$	7336.34	−	3591.28i	1.68156	−	0.823157i
$268$		0			0
$269$		−	6899.13i		−	1.56374i	−0.623439	−	0.781872i	$-0.714265\pi$
$269$		−	6899.13i		−	1.56374i	0.623439	−	0.781872i	$-0.285735\pi$
$270$		0			0
$271$		−	186.576i		−	0.0418217i	−0.999781	−	0.0209108i	$-0.993343\pi$
$271$		−	186.576i		−	0.0418217i	0.999781	−	0.0209108i	$-0.00665661\pi$
$272$		0			0
$273$	−3220.91	+	1576.70i	−0.714061	+	0.349546i
$274$		0			0
$275$	968.672			0.212411
$276$		0			0
$277$	−1906.70			−0.413583			−0.206791	−	0.978385i	$-0.566302\pi$
$277$	−1906.70			−0.413583			−0.206791	+	0.978385i	$0.566302\pi$
$278$		0			0
$279$	−5027.36	−	3904.50i	−1.07878	−	0.837836i
$280$		0			0
$281$			1893.13i			0.401902i	0.979601	+	0.200951i	$0.0644032\pi$
$281$			1893.13i			0.401902i	−0.979601	+	0.200951i	$0.935597\pi$
$282$		0			0
$283$		−	4397.28i		−	0.923643i	−0.886973	−	0.461821i	$-0.847196\pi$
$283$		−	4397.28i		−	0.923643i	0.886973	−	0.461821i	$-0.152804\pi$
$284$		0			0
$285$	800.587	+	1635.45i	0.166395	+	0.339916i
$286$		0			0
$287$	1348.24			0.277297
$288$		0			0
$289$	4910.96			0.999585
$290$		0			0
$291$	1735.02	+	3544.33i	0.349514	+	0.713994i
$292$		0			0
$293$			1766.06i			0.352131i	0.984378	+	0.176066i	$0.0563371\pi$
$293$			1766.06i			0.352131i	−0.984378	+	0.176066i	$0.943663\pi$
$294$		0			0
$295$			2897.17i			0.571795i
$296$		0			0
$297$	5322.09	+	1107.20i	1.03979	+	0.216317i
$298$		0			0
$299$	−15459.3			−2.99008
$300$		0			0
$301$	1377.08			0.263700
$302$		0			0
$303$	−5504.99	+	2694.80i	−1.04374	+	0.510931i
$304$		0			0
$305$			3436.72i			0.645200i
$306$		0			0
$307$			8818.17i			1.63935i	0.572831	+	0.819674i	$0.305845\pi$
$307$			8818.17i			1.63935i	−0.572831	+	0.819674i	$0.694155\pi$
$308$		0			0
$309$	−4575.42	+	2239.76i	−0.842351	+	0.412347i
$310$		0			0
$311$	7900.58			1.44052			0.720258	−	0.693706i	$-0.244023\pi$
$311$	7900.58			1.44052			0.720258	+	0.693706i	$0.244023\pi$
$312$		0			0
$313$	−5815.07			−1.05012			−0.525059	−	0.851066i	$-0.675957\pi$
$313$	−5815.07			−1.05012			−0.525059	+	0.851066i	$0.675957\pi$
$314$		0			0
$315$	−705.037	+	907.791i	−0.126109	+	0.162375i
$316$		0			0
$317$			7734.93i			1.37046i	0.728325	+	0.685231i	$0.240299\pi$
$317$			7734.93i			1.37046i	−0.728325	+	0.685231i	$0.759701\pi$
$318$		0			0
$319$		−	3750.10i		−	0.658199i
$320$		0			0
$321$	443.762	+	906.526i	0.0771600	+	0.157624i
$322$		0			0
$323$	−100.066			−0.0172378
$324$		0			0
$325$	−2026.46			−0.345871
$326$		0			0
$327$	1808.95	+	3695.36i	0.305917	+	0.624935i
$328$		0			0
$329$			2054.41i			0.344265i
$330$		0			0
$331$			1555.42i			0.258290i	0.991626	+	0.129145i	$0.0412232\pi$
$331$			1555.42i			0.258290i	−0.991626	+	0.129145i	$0.958777\pi$
$332$		0			0
$333$	5677.60	−	7310.36i	0.934326	−	1.20302i
$334$		0			0
$335$	3038.55			0.495564
$336$		0			0
$337$	−5346.56			−0.864231			−0.432115	−	0.901818i	$-0.642233\pi$
$337$	−5346.56			−0.864231			−0.432115	+	0.901818i	$0.642233\pi$
$338$		0			0
$339$	2213.66	−	1083.63i	0.354659	−	0.173612i
$340$		0			0
$341$		−	9134.92i		−	1.45069i
$342$		0			0
$343$			5223.54i			0.822287i
$344$		0			0
$345$	−4450.38	+	2178.55i	−0.694494	+	0.339968i
$346$		0			0
$347$	−4367.18			−0.675627			−0.337813	−	0.941213i	$-0.609687\pi$
$347$	−4367.18			−0.675627			−0.337813	+	0.941213i	$0.609687\pi$
$348$		0			0
$349$	−5911.78			−0.906735			−0.453368	−	0.891324i	$-0.649777\pi$
$349$	−5911.78			−0.906735			−0.453368	+	0.891324i	$0.649777\pi$
$350$		0			0
$351$	−11133.8	−	2316.26i	−1.69310	−	0.352230i
$352$		0			0
$353$		−	5678.01i		−	0.856119i	−0.903751	−	0.428059i	$-0.859197\pi$
$353$		−	5678.01i		−	0.856119i	0.903751	−	0.428059i	$-0.140803\pi$
$354$		0			0
$355$		−	875.976i		−	0.130963i
$356$		0			0
$357$	−27.7717	−	56.7325i	−0.00411718	−	0.00841066i
$358$		0			0
$359$	−938.826			−0.138020			−0.0690102	−	0.997616i	$-0.521984\pi$
$359$	−938.826			−0.138020			−0.0690102	+	0.997616i	$0.521984\pi$
$360$		0			0
$361$	1946.92			0.283850
$362$		0			0
$363$	389.113	+	794.888i	0.0562620	+	0.114933i
$364$		0			0
$365$			3883.35i			0.556887i
$366$		0			0
$367$			5229.58i			0.743820i	0.928269	+	0.371910i	$0.121297\pi$
$367$			5229.58i			0.743820i	−0.928269	+	0.371910i	$0.878703\pi$
$368$		0			0
$369$	3376.72	+	2622.53i	0.476382	+	0.369983i
$370$		0			0
$371$	−2049.16			−0.286757
$372$		0			0
$373$	7484.25			1.03893			0.519464	−	0.854492i	$-0.326132\pi$
$373$	7484.25			1.03893			0.519464	+	0.854492i	$0.326132\pi$
$374$		0			0
$375$	−583.373	+	285.572i	−0.0803339	+	0.0393250i
$376$		0			0
$377$			7845.22i			1.07175i
$378$		0			0
$379$		−	2886.60i		−	0.391226i	−0.980681	−	0.195613i	$-0.937330\pi$
$379$		−	2886.60i		−	0.391226i	0.980681	−	0.195613i	$-0.0626697\pi$
$380$		0			0
$381$	1410.21	−	690.323i	0.189625	−	0.0928250i
$382$		0			0
$383$	−12453.0			−1.66141			−0.830705	−	0.556712i	$-0.812063\pi$
$383$	−12453.0			−1.66141			−0.830705	+	0.556712i	$0.812063\pi$
$384$		0			0
$385$	−1649.50			−0.218353
$386$		0			0
$387$	3448.95	+	2678.63i	0.453023	+	0.351841i
$388$		0			0
$389$		−	7099.46i		−	0.925339i	−0.886531	−	0.462669i	$-0.846892\pi$
$389$		−	7099.46i		−	0.925339i	0.886531	−	0.462669i	$-0.153108\pi$
$390$		0			0
$391$		−	272.297i		−	0.0352191i
$392$		0			0
$393$	−3109.33	−	6351.80i	−0.399096	−	0.815282i
$394$		0			0
$395$	1861.23			0.237085
$396$		0			0
$397$	1161.29			0.146809			0.0734047	−	0.997302i	$-0.476614\pi$
$397$	1161.29			0.146809			0.0734047	+	0.997302i	$0.476614\pi$
$398$		0			0
$399$	−1363.27	−	2784.92i	−0.171050	−	0.349425i
$400$		0			0
$401$		−	10688.1i		−	1.33101i	−0.746392	−	0.665507i	$-0.768216\pi$
$401$		−	10688.1i		−	1.33101i	0.746392	−	0.665507i	$-0.231784\pi$
$402$		0			0
$403$			19110.3i			2.36216i
$404$		0			0
$405$	−3531.58	+	902.194i	−0.433298	+	0.110692i
$406$		0			0
$407$	13283.2			1.61775
$408$		0			0
$409$	−9323.79			−1.12722			−0.563609	−	0.826042i	$-0.690587\pi$
$409$	−9323.79			−1.12722			−0.563609	+	0.826042i	$0.690587\pi$
$410$		0			0
$411$	−3671.13	+	1797.09i	−0.440592	+	0.215678i
$412$		0			0
$413$		−	4933.42i		−	0.587791i
$414$		0			0
$415$			959.393i			0.113481i
$416$		0			0
$417$	−14360.1	+	7029.55i	−1.68637	+	0.825512i
$418$		0			0
$419$	−8539.90			−0.995707			−0.497854	−	0.867261i	$-0.665878\pi$
$419$	−8539.90			−0.995707			−0.497854	+	0.867261i	$0.665878\pi$
$420$		0			0
$421$	12344.1			1.42901			0.714505	−	0.699630i	$-0.246652\pi$
$421$	12344.1			1.42901			0.714505	+	0.699630i	$0.246652\pi$
$422$		0			0
$423$	−3996.13	+	5145.34i	−0.459335	+	0.591430i
$424$		0			0
$425$		−	35.6937i		−	0.00407388i
$426$		0			0
$427$		−	5852.19i		−	0.663249i
$428$		0			0
$429$	−7175.32	−	14657.9i	−0.807524	−	1.64963i
$430$		0			0
$431$	1591.26			0.177839			0.0889194	−	0.996039i	$-0.471659\pi$
$431$	1591.26			0.177839			0.0889194	+	0.996039i	$0.471659\pi$
$432$		0			0
$433$	−7757.86			−0.861014			−0.430507	−	0.902587i	$-0.641665\pi$
$433$	−7757.86			−0.861014			−0.430507	+	0.902587i	$0.641665\pi$
$434$		0			0
$435$	1105.56	+	2258.46i	0.121856	+	0.248931i
$436$		0			0
$437$		−	13366.7i		−	1.46319i
$438$		0			0
$439$			6967.38i			0.757483i	0.925503	+	0.378741i	$0.123643\pi$
$439$			6967.38i			0.757483i	−0.925503	+	0.378741i	$0.876357\pi$
$440$		0			0
$441$	−4480.00	+	5768.36i	−0.483749	+	0.622866i
$442$		0			0
$443$	3077.88			0.330100			0.165050	−	0.986285i	$-0.447221\pi$
$443$	3077.88			0.330100			0.165050	+	0.986285i	$0.447221\pi$
$444$		0			0
$445$	−7859.84			−0.837286
$446$		0			0
$447$	−4522.35	+	2213.77i	−0.478523	+	0.234246i
$448$		0			0
$449$			14495.6i			1.52359i	0.647819	+	0.761795i	$0.275681\pi$
$449$			14495.6i			1.52359i	−0.647819	+	0.761795i	$0.724319\pi$
$450$		0			0
$451$			6135.65i			0.640613i
$452$		0			0
$453$	2985.67	−	1461.54i	0.309667	−	0.151588i
$454$		0			0
$455$	3450.75			0.355546
$456$		0			0
$457$	−4262.66			−0.436321			−0.218161	−	0.975913i	$-0.570006\pi$
$457$	−4262.66			−0.436321			−0.218161	+	0.975913i	$0.570006\pi$
$458$		0			0
$459$	40.7982	−	196.109i	0.00414879	−	0.0199424i
$460$		0			0
$461$			140.153i			0.0141596i	0.999975	+	0.00707982i	$0.00225359\pi$
$461$			140.153i			0.0141596i	−0.999975	+	0.00707982i	$0.997746\pi$
$462$		0			0
$463$		−	16063.2i		−	1.61235i	−0.591675	−	0.806177i	$-0.701533\pi$
$463$		−	16063.2i		−	1.61235i	0.591675	−	0.806177i	$-0.298467\pi$
$464$		0			0
$465$	2693.05	+	5501.41i	0.268574	+	0.548649i
$466$		0			0
$467$	16355.3			1.62063			0.810314	−	0.585996i	$-0.199297\pi$
$467$	16355.3			1.62063			0.810314	+	0.585996i	$0.199297\pi$
$468$		0			0
$469$	−5174.18			−0.509427
$470$		0			0
$471$	3691.68	+	7541.44i	0.361154	+	0.737773i
$472$		0			0
$473$			6266.88i			0.609200i
$474$		0			0
$475$		−	1752.16i		−	0.169251i
$476$		0			0
$477$	−5132.19	−	3985.92i	−0.492635	−	0.382605i
$478$		0			0
$479$	−15067.8			−1.43730			−0.718649	−	0.695373i	$-0.755239\pi$
$479$	−15067.8			−1.43730			−0.718649	+	0.695373i	$0.755239\pi$
$480$		0			0
$481$	−27788.6			−2.63420
$482$		0			0
$483$	7578.30	−	3709.72i	0.713922	−	0.349479i
$484$		0			0
$485$		−	3797.24i		−	0.355513i
$486$		0			0
$487$		−	7052.41i		−	0.656212i	−0.944641	−	0.328106i	$-0.893590\pi$
$487$		−	7052.41i		−	0.656212i	0.944641	−	0.328106i	$-0.106410\pi$
$488$		0			0
$489$	12968.3	−	6348.22i	1.19927	−	0.587068i
$490$		0			0
$491$	−4162.07			−0.382549			−0.191274	−	0.981537i	$-0.561262\pi$
$491$	−4162.07			−0.382549			−0.191274	+	0.981537i	$0.561262\pi$
$492$		0			0
$493$	−138.184			−0.0126237
$494$		0			0
$495$	−4131.22	−	3208.52i	−0.375121	−	0.291338i
$496$		0			0
$497$			1491.65i			0.134627i
$498$		0			0
$499$		−	2429.78i		−	0.217980i	−0.994043	−	0.108990i	$-0.965238\pi$
$499$		−	2429.78i		−	0.217980i	0.994043	−	0.108990i	$-0.0347616\pi$
$500$		0			0
$501$	8284.26	+	16923.3i	0.738750	+	1.50913i
$502$		0			0
$503$	11530.6			1.02212			0.511058	−	0.859546i	$-0.329254\pi$
$503$	11530.6			1.02212			0.511058	+	0.859546i	$0.329254\pi$
$504$		0			0
$505$	5897.81			0.519701
$506$		0			0
$507$	9991.56	+	20411.0i	0.875229	+	1.78794i
$508$		0			0
$509$		−	10714.8i		−	0.933054i	−0.884507	−	0.466527i	$-0.845505\pi$
$509$		−	10714.8i		−	0.933054i	0.884507	−	0.466527i	$-0.154495\pi$
$510$		0			0
$511$		−	6612.73i		−	0.572466i
$512$		0			0
$513$	2002.72	−	9626.71i	0.172363	−	0.828517i
$514$		0			0
$515$	4901.90			0.419425
$516$		0			0
$517$	−9349.30			−0.795323
$518$		0			0
$519$	−2766.63	+	1354.32i	−0.233992	+	0.114543i
$520$		0			0
$521$		−	12048.5i		−	1.01316i	−0.862193	−	0.506580i	$-0.830909\pi$
$521$		−	12048.5i		−	1.01316i	0.862193	−	0.506580i	$-0.169091\pi$
$522$		0			0
$523$		−	1324.33i		−	0.110724i	−0.998466	−	0.0553621i	$-0.982369\pi$
$523$		−	1324.33i		−	0.110724i	0.998466	−	0.0553621i	$-0.0176313\pi$
$524$		0			0
$525$	993.391	−	486.284i	0.0825812	−	0.0404251i
$526$		0			0
$527$	−336.604			−0.0278230
$528$		0			0
$529$	24206.3			1.98950
$530$		0			0
$531$	9596.24	−	12355.9i	0.784259	−	1.00980i
$532$		0			0
$533$		−	12835.8i		−	1.04311i
$534$		0			0
$535$		−	971.212i		−	0.0784844i
$536$		0			0
$537$	3707.17	+	7573.08i	0.297907	+	0.608571i
$538$		0			0
$539$	−10481.4			−0.837595
$540$		0			0
$541$	−17585.0			−1.39748			−0.698742	−	0.715374i	$-0.746256\pi$
$541$	−17585.0			−1.39748			−0.698742	+	0.715374i	$0.746256\pi$
$542$		0			0
$543$	3869.84	+	7905.38i	0.305839	+	0.624774i
$544$		0			0
$545$		−	3959.04i		−	0.311168i
$546$		0			0
$547$			2713.12i			0.212074i	0.994362	+	0.106037i	$0.0338162\pi$
$547$			2713.12i			0.212074i	−0.994362	+	0.106037i	$0.966184\pi$
$548$		0			0
$549$	11383.4	−	14657.0i	0.884939	−	1.13943i
$550$		0			0
$551$	−6783.27			−0.524459
$552$		0			0
$553$	−3169.37			−0.243717
$554$		0			0
$555$	−7999.69	+	3916.00i	−0.611834	+	0.299505i
$556$		0			0
$557$			24100.1i			1.83331i	0.399681	+	0.916654i	$0.369121\pi$
$557$			24100.1i			1.83331i	−0.399681	+	0.916654i	$0.630879\pi$
$558$		0			0
$559$		−	13110.3i		−	0.991964i
$560$		0			0
$561$	258.181	−	126.385i	0.0194303	−	0.00951153i
$562$		0			0
$563$	−24652.3			−1.84542			−0.922710	−	0.385495i	$-0.874031\pi$
$563$	−24652.3			−1.84542			−0.922710	+	0.385495i	$0.874031\pi$
$564$		0			0
$565$	−2371.61			−0.176592
$566$		0			0
$567$	6013.72	−	1536.29i	0.445419	−	0.113789i
$568$		0			0
$569$		−	3448.99i		−	0.254111i	−0.991896	−	0.127056i	$-0.959447\pi$
$569$		−	3448.99i		−	0.254111i	0.991896	−	0.127056i	$-0.0405527\pi$
$570$		0			0
$571$			754.536i			0.0553001i	0.999618	+	0.0276501i	$0.00880241\pi$
$571$			754.536i			0.0553001i	−0.999618	+	0.0276501i	$0.991198\pi$
$572$		0			0
$573$	6771.41	+	13832.8i	0.493682	+	1.00850i
$574$		0			0
$575$	4767.95			0.345804
$576$		0			0
$577$	−7469.10			−0.538896			−0.269448	−	0.963015i	$-0.586841\pi$
$577$	−7469.10			−0.538896			−0.269448	+	0.963015i	$0.586841\pi$
$578$		0			0
$579$	9517.66	+	19442.9i	0.683144	+	1.39554i
$580$		0			0
$581$		−	1633.69i		−	0.116656i
$582$		0			0
$583$		−	9325.41i		−	0.662468i
$584$		0			0
$585$	8642.52	+	6712.22i	0.610811	+	0.474387i
$586$		0			0
$587$	−24744.1			−1.73986			−0.869929	−	0.493176i	$-0.835836\pi$
$587$	−24744.1			−1.73986			−0.869929	+	0.493176i	$0.835836\pi$
$588$		0			0
$589$	−16523.4			−1.15592
$590$		0			0
$591$	−24364.5	+	11926.9i	−1.69581	+	0.830130i
$592$		0			0
$593$			21917.9i			1.51781i	0.651204	+	0.758903i	$0.274264\pi$
$593$			21917.9i			1.51781i	−0.651204	+	0.758903i	$0.725736\pi$
$594$		0			0
$595$			60.7808i			0.00418785i
$596$		0			0
$597$	14205.9	−	6954.04i	0.973882	−	0.476734i
$598$		0			0
$599$	6639.97			0.452925			0.226462	−	0.974020i	$-0.427284\pi$
$599$	6639.97			0.452925			0.226462	+	0.974020i	$0.427284\pi$
$600$		0			0
$601$	6968.78			0.472982			0.236491	−	0.971634i	$-0.424003\pi$
$601$	6968.78			0.472982			0.236491	+	0.971634i	$0.424003\pi$
$602$		0			0
$603$	−12958.9	−	10064.6i	−0.875171	−	0.679702i
$604$		0			0
$605$		−	851.608i		−	0.0572277i
$606$		0			0
$607$			10741.4i			0.718252i	0.933289	+	0.359126i	$0.116925\pi$
$607$			10741.4i			0.718252i	−0.933289	+	0.359126i	$0.883075\pi$
$608$		0			0
$609$	−1882.59	−	3845.80i	−0.125265	−	0.255894i
$610$		0			0
$611$	19558.7			1.29503
$612$		0			0
$613$	12625.9			0.831898			0.415949	−	0.909388i	$-0.363450\pi$
$613$	12625.9			0.831898			0.415949	+	0.909388i	$0.363450\pi$
$614$		0			0
$615$	−1808.84	−	3695.13i	−0.118601	−	0.242280i
$616$		0			0
$617$			12503.3i			0.815822i	0.913022	+	0.407911i	$0.133743\pi$
$617$			12503.3i			0.815822i	−0.913022	+	0.407911i	$0.866257\pi$
$618$		0			0
$619$		−	20479.8i		−	1.32981i	−0.746928	−	0.664905i	$-0.768472\pi$
$619$		−	20479.8i		−	1.32981i	0.746928	−	0.664905i	$-0.231528\pi$
$620$		0			0
$621$	26196.1	+	5449.79i	1.69277	+	0.352162i
$622$		0			0
$623$	13384.1			0.860708
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	12673.8	−	6204.05i	0.807243	−	0.395161i
$628$		0			0
$629$		−	489.462i		−	0.0310272i
$630$		0			0
$631$			12485.1i			0.787679i	0.919179	+	0.393840i	$0.128854\pi$
$631$			12485.1i			0.787679i	−0.919179	+	0.393840i	$0.871146\pi$
$632$		0			0
$633$	−1282.36	+	627.742i	−0.0805204	+	0.0394163i
$634$		0			0
$635$	−1510.83			−0.0944183
$636$		0			0
$637$	21927.0			1.36386
$638$		0			0
$639$	−2901.48	+	3735.89i	−0.179626	+	0.231282i
$640$		0			0
$641$			21790.7i			1.34272i	0.741134	+	0.671358i	$0.234288\pi$
$641$			21790.7i			1.34272i	−0.741134	+	0.671358i	$0.765712\pi$
$642$		0			0
$643$		−	26174.8i		−	1.60534i	−0.596422	−	0.802671i	$-0.703412\pi$
$643$		−	26174.8i		−	1.60534i	0.596422	−	0.802671i	$-0.296588\pi$
$644$		0			0
$645$	−1847.53	−	3774.16i	−0.112785	−	0.230399i
$646$		0			0
$647$	1314.54			0.0798762			0.0399381	−	0.999202i	$-0.487284\pi$
$647$	1314.54			0.0798762			0.0399381	+	0.999202i	$0.487284\pi$
$648$		0			0
$649$	22451.3			1.35792
$650$		0			0
$651$	−4585.83	−	9368.03i	−0.276087	−	0.563997i
$652$		0			0
$653$			22950.8i			1.37539i	0.725998	+	0.687697i	$0.241378\pi$
$653$			22950.8i			1.37539i	−0.725998	+	0.687697i	$0.758622\pi$
$654$		0			0
$655$			6805.04i			0.405946i
$656$		0			0
$657$	12862.8	−	16561.8i	0.763811	−	0.983468i
$658$		0			0
$659$	21378.6			1.26372			0.631859	−	0.775083i	$-0.282292\pi$
$659$	21378.6			1.26372			0.631859	+	0.775083i	$0.282292\pi$
$660$		0			0
$661$	13757.4			0.809531			0.404765	−	0.914421i	$-0.367353\pi$
$661$	13757.4			0.809531			0.404765	+	0.914421i	$0.367353\pi$
$662$		0			0
$663$	−540.116	+	264.397i	−0.0316385	+	0.0154877i
$664$		0			0
$665$			2983.64i			0.173986i
$666$		0			0
$667$		−	18458.6i		−	1.07154i
$668$		0			0
$669$	−13600.5	+	6657.71i	−0.785989	+	0.384756i
$670$		0			0
$671$	26632.4			1.53224
$672$		0			0
$673$	−19681.5			−1.12729			−0.563645	−	0.826017i	$-0.690602\pi$
$673$	−19681.5			−1.12729			−0.563645	+	0.826017i	$0.690602\pi$
$674$		0			0
$675$	3433.88	+	714.379i	0.195808	+	0.0407355i
$676$		0			0
$677$			19113.4i			1.08506i	0.840035	+	0.542532i	$0.182534\pi$
$677$			19113.4i			1.08506i	−0.840035	+	0.542532i	$0.817466\pi$
$678$		0			0
$679$			6466.10i			0.365458i
$680$		0			0
$681$	542.754	+	1108.75i	0.0305409	+	0.0623897i
$682$		0			0
$683$	−31199.4			−1.74789			−0.873947	−	0.486022i	$-0.838448\pi$
$683$	−31199.4			−1.74789			−0.873947	+	0.486022i	$0.838448\pi$
$684$		0			0
$685$	3933.09			0.219380
$686$		0			0
$687$	1404.03	+	2868.18i	0.0779723	+	0.159283i
$688$		0			0
$689$			19508.8i			1.07870i
$690$		0			0
$691$		−	23853.3i		−	1.31320i	−0.754238	−	0.656602i	$-0.771993\pi$
$691$		−	23853.3i		−	1.31320i	0.754238	−	0.656602i	$-0.228007\pi$
$692$		0			0
$693$	7034.82	+	5463.60i	0.385614	+	0.299488i
$694$		0			0
$695$	15384.8			0.839681
$696$		0			0
$697$	226.087			0.0122864
$698$		0			0
$699$	23116.5	−	11316.0i	1.25085	−	0.612316i
$700$		0			0
$701$		−	19255.9i		−	1.03750i	−0.854927	−	0.518748i	$-0.826398\pi$
$701$		−	19255.9i		−	1.03750i	0.854927	−	0.518748i	$-0.173602\pi$
$702$		0			0
$703$		−	24027.0i		−	1.28904i
$704$		0			0
$705$	5630.52	−	2756.25i	0.300791	−	0.147243i
$706$		0			0
$707$	−10043.0			−0.534239
$708$		0			0
$709$	−7359.10			−0.389812			−0.194906	−	0.980822i	$-0.562440\pi$
$709$	−7359.10			−0.389812			−0.194906	+	0.980822i	$0.562440\pi$
$710$		0			0
$711$	−7937.81	−	6164.91i	−0.418694	−	0.325179i
$712$		0			0
$713$		−	44963.4i		−	2.36170i
$714$		0			0
$715$			15703.8i			0.821385i
$716$		0			0
$717$	−1971.30	−	4027.01i	−0.102677	−	0.209751i
$718$		0			0
$719$	10261.2			0.532239			0.266120	−	0.963940i	$-0.414258\pi$
$719$	10261.2			0.532239			0.266120	+	0.963940i	$0.414258\pi$
$720$		0			0
$721$	−8347.17			−0.431158
$722$		0			0
$723$	9893.40	+	20210.4i	0.508907	+	1.03961i
$724$		0			0
$725$		−	2419.62i		−	0.123948i
$726$		0			0
$727$		−	10965.3i		−	0.559395i	−0.960088	−	0.279698i	$-0.909766\pi$
$727$		−	10965.3i		−	0.559395i	0.960088	−	0.279698i	$-0.0902342\pi$
$728$		0			0
$729$	18049.9	+	7849.90i	0.917031	+	0.398816i
$730$		0			0
$731$	230.923			0.0116840
$732$		0			0
$733$	7910.63			0.398616			0.199308	−	0.979937i	$-0.436131\pi$
$733$	7910.63			0.398616			0.199308	+	0.979937i	$0.436131\pi$
$734$		0			0
$735$	6312.28	−	3089.98i	0.316778	−	0.155069i
$736$		0			0
$737$		−	23546.9i		−	1.17688i
$738$		0			0
$739$			16479.4i			0.820304i	0.912017	+	0.410152i	$0.134524\pi$
$739$			16479.4i			0.820304i	−0.912017	+	0.410152i	$0.865476\pi$
$740$		0			0
$741$	−26513.5	+	12978.9i	−1.31444	+	0.643442i
$742$		0			0
$743$	15889.8			0.784575			0.392287	−	0.919843i	$-0.371684\pi$
$743$	15889.8			0.784575			0.392287	+	0.919843i	$0.371684\pi$
$744$		0			0
$745$	4845.04			0.238267
$746$		0			0
$747$	3177.78	−	4091.65i	0.155648	−	0.200409i
$748$		0			0
$749$			1653.82i			0.0806799i
$750$		0			0
$751$		−	34574.7i		−	1.67996i	−0.542618	−	0.839979i	$-0.682567\pi$
$751$		−	34574.7i		−	1.67996i	0.542618	−	0.839979i	$-0.317433\pi$
$752$		0			0
$753$	−4389.58	−	8967.13i	−0.212437	−	0.433971i
$754$		0			0
$755$	−3198.72			−0.154190
$756$		0			0
$757$	−25472.9			−1.22302			−0.611511	−	0.791236i	$-0.709438\pi$
$757$	−25472.9			−1.22302			−0.611511	+	0.791236i	$0.709438\pi$
$758$		0			0
$759$	16882.4	+	34487.7i	0.807368	+	1.64931i
$760$		0			0
$761$		−	38864.3i		−	1.85129i	−0.378398	−	0.925643i	$-0.623525\pi$
$761$		−	38864.3i		−	1.85129i	0.378398	−	0.925643i	$-0.376475\pi$
$762$		0			0
$763$			6741.62i			0.319873i
$764$		0			0
$765$	−118.228	+	152.228i	−0.00558763	+	0.00719452i
$766$		0			0
$767$	−46968.0			−2.21111
$768$		0			0
$769$	26642.5			1.24935			0.624677	−	0.780883i	$-0.285230\pi$
$769$	26642.5			1.24935			0.624677	+	0.780883i	$0.285230\pi$
$770$		0			0
$771$	−34410.7	+	16844.7i	−1.60735	+	0.786831i
$772$		0			0
$773$			38743.6i			1.80273i	0.433062	+	0.901364i	$0.357433\pi$
$773$			38743.6i			1.80273i	−0.433062	+	0.901364i	$0.642567\pi$
$774$		0			0
$775$		−	5893.97i		−	0.273184i
$776$		0			0
$777$	13622.2	−	6668.33i	0.628950	−	0.307883i
$778$		0			0
$779$	11098.3			0.510446
$780$		0			0
$781$	−6788.27			−0.311016
$782$		0			0
$783$	2765.64	−	13293.9i	0.126227	−	0.606749i
$784$		0			0
$785$		−	8079.57i		−	0.367353i
$786$		0			0
$787$		−	41561.1i		−	1.88246i	−0.337772	−	0.941228i	$-0.609673\pi$
$787$		−	41561.1i		−	1.88246i	0.337772	−	0.941228i	$-0.390327\pi$
$788$		0			0
$789$	12957.8	+	26470.4i	0.584674	+	1.19439i
$790$		0			0
$791$	4038.48			0.181532
$792$		0			0
$793$	−55715.1			−2.49496
$794$		0			0
$795$	2749.20	+	5616.13i	0.122647	+	0.250545i
$796$		0			0
$797$			25762.2i			1.14497i	0.819914	+	0.572486i	$0.194021\pi$
$797$			25762.2i			1.14497i	−0.819914	+	0.572486i	$0.805979\pi$
$798$		0			0
$799$			344.504i			0.0152537i
$800$		0			0
$801$	33520.9	+	26034.0i	1.47865	+	1.14840i
$802$		0			0
$803$	30093.5			1.32251
$804$		0			0
$805$	−8119.06			−0.355477
$806$		0			0
$807$	32198.1	−	15761.6i	1.40449	−	0.687527i
$808$		0			0
$809$		−	21955.0i		−	0.954139i	−0.878866	−	0.477069i	$-0.841699\pi$
$809$		−	21955.0i		−	0.954139i	0.878866	−	0.477069i	$-0.158301\pi$
$810$		0			0
$811$		−	37665.1i		−	1.63083i	−0.578880	−	0.815413i	$-0.696510\pi$
$811$		−	37665.1i		−	1.63083i	0.578880	−	0.815413i	$-0.303490\pi$
$812$		0			0
$813$	870.746	−	426.247i	0.0375626	−	0.0183876i
$814$		0			0
$815$	−13893.6			−0.597145
$816$		0			0
$817$	11335.7			0.485416
$818$		0			0
$819$	−14716.8	−	11429.8i	−0.627898	−	0.487657i
$820$		0			0
$821$			4457.24i			0.189475i	0.995502	+	0.0947373i	$0.0302011\pi$
$821$			4457.24i			0.189475i	−0.995502	+	0.0947373i	$0.969799\pi$
$822$		0			0
$823$		−	6620.35i		−	0.280402i	−0.990123	−	0.140201i	$-0.955225\pi$
$823$		−	6620.35i		−	0.280402i	0.990123	−	0.140201i	$-0.0447749\pi$
$824$		0			0
$825$	2213.01	+	4520.77i	0.0933903	+	0.190780i
$826$		0			0
$827$	−4052.32			−0.170390			−0.0851952	−	0.996364i	$-0.527151\pi$
$827$	−4052.32			−0.170390			−0.0851952	+	0.996364i	$0.527151\pi$
$828$		0			0
$829$	−22750.2			−0.953132			−0.476566	−	0.879139i	$-0.658119\pi$
$829$	−22750.2			−0.953132			−0.476566	+	0.879139i	$0.658119\pi$
$830$		0			0
$831$	−4356.00	−	8898.52i	−0.181839	−	0.371464i
$832$		0			0
$833$			386.218i			0.0160644i
$834$		0			0
$835$		−	18130.9i		−	0.751430i
$836$		0			0
$837$	6736.85	−	32382.7i	0.278207	−	1.33729i
$838$		0			0
$839$	15461.9			0.636238			0.318119	−	0.948051i	$-0.396949\pi$
$839$	15461.9			0.636238			0.318119	+	0.948051i	$0.396949\pi$
$840$		0			0
$841$	15021.7			0.615923
$842$		0			0
$843$	−8835.18	+	4324.99i	−0.360973	+	0.176703i
$844$		0			0
$845$		−	21867.4i		−	0.890251i
$846$		0			0
$847$			1450.15i			0.0588286i
$848$		0			0
$849$	20522.0	−	10045.9i	0.829580	−	0.406095i
$850$		0			0
$851$	65382.1			2.63369
$852$		0			0
$853$	17947.8			0.720424			0.360212	−	0.932870i	$-0.382704\pi$
$853$	17947.8			0.720424			0.360212	+	0.932870i	$0.382704\pi$
$854$		0			0
$855$	−5803.63	+	7472.64i	−0.232140	+	0.298899i
$856$		0			0
$857$			12603.9i			0.502380i	0.967938	+	0.251190i	$0.0808220\pi$
$857$			12603.9i			0.502380i	−0.967938	+	0.251190i	$0.919178\pi$
$858$		0			0
$859$			35217.8i			1.39886i	0.714703	+	0.699428i	$0.246562\pi$
$859$			35217.8i			1.39886i	−0.714703	+	0.699428i	$0.753438\pi$
$860$		0			0
$861$	3080.16	+	6292.22i	0.121918	+	0.249057i
$862$		0			0
$863$	−17265.3			−0.681016			−0.340508	−	0.940242i	$-0.610599\pi$
$863$	−17265.3			−0.681016			−0.340508	+	0.940242i	$0.610599\pi$
$864$		0			0
$865$	2964.05			0.116510
$866$		0			0
$867$	11219.5	+	22919.4i	0.439485	+	0.897788i
$868$		0			0
$869$		−	14423.3i		−	0.563036i
$870$		0			0
$871$			49260.1i			1.91632i
$872$		0			0
$873$	−12577.5	+	16194.6i	−0.487612	+	0.627839i
$874$		0			0
$875$	−1064.28			−0.0411190
$876$		0			0
$877$	−2608.17			−0.100424			−0.0502120	−	0.998739i	$-0.515990\pi$
$877$	−2608.17			−0.100424			−0.0502120	+	0.998739i	$0.515990\pi$
$878$		0			0
$879$	−8242.18	+	4034.71i	−0.316271	+	0.154821i
$880$		0			0
$881$		−	17510.1i		−	0.669613i	−0.942287	−	0.334806i	$-0.891329\pi$
$881$		−	17510.1i		−	0.669613i	0.942287	−	0.334806i	$-0.108671\pi$
$882$		0			0
$883$			44103.9i			1.68088i	0.541907	+	0.840439i	$0.317703\pi$
$883$			44103.9i			1.68088i	−0.541907	+	0.840439i	$0.682297\pi$
$884$		0			0
$885$	−13521.0	+	6618.80i	−0.513564	+	0.251400i
$886$		0			0
$887$	−33475.0			−1.26717			−0.633586	−	0.773672i	$-0.718418\pi$
$887$	−33475.0			−1.26717			−0.633586	+	0.773672i	$0.718418\pi$
$888$		0			0
$889$	2572.71			0.0970595
$890$		0			0
$891$	6991.44	+	27367.6i	0.262876	+	1.02901i
$892$		0			0
$893$			16911.2i			0.633720i
$894$		0			0
$895$		−	8113.47i		−	0.303021i
$896$		0			0
$897$	−35318.0	−	72148.3i	−1.31464	−	2.68558i
$898$		0			0
$899$	−22817.8			−0.846515
$900$		0			0
$901$	−343.624			−0.0127056
$902$		0			0
$903$	3146.04	+	6426.80i	0.115940	+	0.236845i
$904$		0			0
$905$		−	8469.48i		−	0.311089i
$906$		0			0
$907$			1136.49i			0.0416057i	0.999784	+	0.0208029i	$0.00662223\pi$
$907$			1136.49i			0.0416057i	−0.999784	+	0.0208029i	$0.993378\pi$
$908$		0			0
$909$	−25153.2	−	19535.2i	−0.917797	−	0.712808i
$910$		0			0
$911$	−21643.4			−0.787133			−0.393567	−	0.919296i	$-0.628759\pi$
$911$	−21643.4			−0.787133			−0.393567	+	0.919296i	$0.628759\pi$
$912$		0			0
$913$	7434.70			0.269499
$914$		0			0
$915$	−16039.1	+	7851.45i	−0.579494	+	0.283673i
$916$		0			0
$917$		−	11587.9i		−	0.417302i
$918$		0			0
$919$		−	41646.5i		−	1.49488i	−0.664331	−	0.747439i	$-0.731283\pi$
$919$		−	41646.5i		−	1.49488i	0.664331	−	0.747439i	$-0.268717\pi$
$920$		0			0
$921$	−41154.2	+	20145.8i	−1.47240	+	0.720767i
$922$		0			0
$923$	14201.1			0.506429
$924$		0			0
$925$	8570.52			0.304645
$926$		0			0
$927$	−20905.8	−	16236.5i	−0.740708	−	0.575271i
$928$		0			0
$929$			46474.2i			1.64130i	0.571430	+	0.820651i	$0.306389\pi$
$929$			46474.2i			1.64130i	−0.571430	+	0.820651i	$0.693611\pi$
$930$		0			0
$931$			18958.9i			0.667404i
$932$		0			0
$933$	18049.5	+	36871.8i	0.633348	+	1.29382i
$934$		0			0
$935$	−276.604			−0.00967479
$936$		0			0
$937$	20056.0			0.699253			0.349626	−	0.936889i	$-0.386309\pi$
$937$	20056.0			0.699253			0.349626	+	0.936889i	$0.386309\pi$
$938$		0			0
$939$	−13285.0	−	27138.8i	−0.461702	−	0.943175i
$940$		0			0
$941$			23411.9i			0.811057i	0.914082	+	0.405529i	$0.132913\pi$
$941$			23411.9i			0.811057i	−0.914082	+	0.405529i	$0.867087\pi$
$942$		0			0
$943$			30200.5i			1.04291i
$944$		0			0
$945$	−5847.36	−	1216.47i	−0.201285	−	0.0418750i
$946$		0			0
$947$	−23063.9			−0.791423			−0.395711	−	0.918375i	$-0.629502\pi$
$947$	−23063.9			−0.791423			−0.395711	+	0.918375i	$0.629502\pi$
$948$		0			0
$949$	−62955.7			−2.15346
$950$		0			0
$951$	−36098.8	+	17671.0i	−1.23090	+	0.602547i
$952$		0			0
$953$		−	9943.82i		−	0.337998i	−0.985616	−	0.168999i	$-0.945947\pi$
$953$		−	9943.82i		−	0.337998i	0.985616	−	0.168999i	$-0.0540534\pi$
$954$		0			0
$955$		−	14819.8i		−	0.502155i
$956$		0			0
$957$	17501.7	−	8567.40i	0.591168	−	0.289388i
$958$		0			0
$959$	−6697.42			−0.225517
$960$		0			0
$961$	−25791.2			−0.865738
$962$		0			0
$963$	−3216.93	+	4142.05i	−0.107647	+	0.138604i
$964$		0			0
$965$		−	20830.3i		−	0.694870i
$966$		0			0
$967$			31385.5i			1.04373i	0.853028	+	0.521866i	$0.174764\pi$
$967$			31385.5i			1.04373i	−0.853028	+	0.521866i	$0.825236\pi$
$968$		0			0
$969$	−228.607	−	467.004i	−0.00757887	−	0.0154823i
$970$		0			0
$971$	−9898.13			−0.327133			−0.163566	−	0.986532i	$-0.552300\pi$
$971$	−9898.13			−0.327133			−0.163566	+	0.986532i	$0.552300\pi$
$972$		0			0
$973$	−26197.9			−0.863171
$974$		0			0
$975$	−4629.61	−	9457.47i	−0.152068	−	0.310647i
$976$		0			0
$977$		−	20061.2i		−	0.656924i	−0.944517	−	0.328462i	$-0.893470\pi$
$977$		−	20061.2i		−	0.656924i	0.944517	−	0.328462i	$-0.106530\pi$
$978$		0			0
$979$			60908.9i			1.98841i
$980$		0			0
$981$	−13113.5	+	16884.6i	−0.426790	+	0.549526i
$982$		0			0
$983$	−5560.50			−0.180419			−0.0902097	−	0.995923i	$-0.528754\pi$
$983$	−5560.50			−0.180419			−0.0902097	+	0.995923i	$0.528754\pi$
$984$		0			0
$985$	26103.1			0.844379
$986$		0			0
$987$	−9587.88	+	4693.45i	−0.309205	+	0.151362i
$988$		0			0
$989$			30846.5i			0.991772i
$990$		0			0
$991$			8795.87i			0.281948i	0.990013	+	0.140974i	$0.0450233\pi$
$991$			8795.87i			0.281948i	−0.990013	+	0.140974i	$0.954977\pi$
$992$		0			0
$993$	−7259.14	+	3553.49i	−0.231986	+	0.113561i
$994$		0			0
$995$	−15219.5			−0.484917
$996$		0			0
$997$	3763.22			0.119541			0.0597704	−	0.998212i	$-0.480963\pi$
$997$	3763.22			0.119541			0.0597704	+	0.998212i	$0.480963\pi$
$998$		0			0
$999$	47088.2	+	9796.16i	1.49130	+	0.310247i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.h.b.191.16	yes	24
3.2	odd	2		480.4.h.a.191.10	yes	24
4.3	odd	2		480.4.h.a.191.9	✓	24
8.3	odd	2		960.4.h.e.191.16		24
8.5	even	2		960.4.h.c.191.9		24
12.11	even	2	inner	480.4.h.b.191.15	yes	24
24.5	odd	2		960.4.h.e.191.15		24
24.11	even	2		960.4.h.c.191.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.h.a.191.9	✓	24	4.3	odd	2
480.4.h.a.191.10	yes	24	3.2	odd	2
480.4.h.b.191.15	yes	24	12.11	even	2	inner
480.4.h.b.191.16	yes	24	1.1	even	1	trivial
960.4.h.c.191.9		24	8.5	even	2
960.4.h.c.191.10		24	24.11	even	2
960.4.h.e.191.15		24	24.5	odd	2
960.4.h.e.191.16		24	8.3	odd	2

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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.28458 + 4.66698i) q^{3} -5.00000i q^{5} +8.51421i q^{7} +(-16.5614 + 21.3242i) q^{9} -38.7469 q^{11} +81.0585 q^{13} +(23.3349 - 11.4229i) q^{15} +1.42775i q^{17} +70.0862i q^{19} +(-39.7356 + 19.4514i) q^{21} -190.718 q^{23} -25.0000 q^{25} +(-137.355 - 28.5752i) q^{27} +96.7846i q^{29} +235.759i q^{31} +(-88.5202 - 180.831i) q^{33} +42.5710 q^{35} -342.821 q^{37} +(185.184 + 378.299i) q^{39} -158.352i q^{41} -161.739i q^{43} +(106.621 + 82.8071i) q^{45} +241.292 q^{47} +270.508 q^{49} +(-6.66328 + 3.26180i) q^{51} +240.675i q^{53} +193.734i q^{55} +(-327.091 + 160.117i) q^{57} -579.434 q^{59} -687.344 q^{61} +(-181.558 - 141.007i) q^{63} -405.293i q^{65} +607.711i q^{67} +(-435.710 - 890.077i) q^{69} +175.195 q^{71} -776.670 q^{73} +(-57.1144 - 116.675i) q^{75} -329.899i q^{77} +372.245i q^{79} +(-180.439 - 706.316i) q^{81} -191.879 q^{83} +7.13875 q^{85} +(-451.692 + 221.112i) q^{87} -1571.97i q^{89} +690.149i q^{91} +(-1100.28 + 538.609i) q^{93} +350.431 q^{95} +759.448 q^{97} +(641.703 - 826.245i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9} + 72 q^{11} - 72 q^{13} + 20 q^{15} - 68 q^{21} + 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} - 504 q^{37} - 456 q^{39} - 220 q^{45} + 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57}+ \cdots + 3160 q^{99}+O(q^{100}) \) 24 * q + 20 * q^9 + 72 * q^11 - 72 * q^13 + 20 * q^15 - 68 * q^21 + 96 * q^23 - 600 * q^25 + 168 * q^27 - 80 * q^33 - 504 * q^37 - 456 * q^39 - 220 * q^45 + 432 * q^47 - 816 * q^49 + 1240 * q^51 + 40 * q^57 - 2184 * q^59 - 1080 * q^61 - 1720 * q^63 - 1548 * q^69 + 816 * q^71 - 1440 * q^73 - 2304 * q^81 - 2016 * q^83 - 2144 * q^87 - 112 * q^93 + 1800 * q^95 + 720 * q^97 + 3160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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