LMFDB - Embedded newform 624.4.bc.d.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(31,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))

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

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

chi := DirichletCharacter("624.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.bc (of order $4$, degree $2$, 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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.1
Character	$\chi$	$=$	624.31
Dual form			624.4.bc.d.463.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.00000i q^{3} +(-13.9138 + 13.9138i) q^{5} +(3.70358 - 3.70358i) q^{7} -9.00000 q^{9} +(47.9147 - 47.9147i) q^{11} +(-44.1297 + 15.7977i) q^{13} +(-41.7413 - 41.7413i) q^{15} -16.9316i q^{17} +(4.73512 + 4.73512i) q^{19} +(11.1107 + 11.1107i) q^{21} -185.659 q^{23} -262.186i q^{25} -27.0000i q^{27} -63.8055 q^{29} +(40.5024 + 40.5024i) q^{31} +(143.744 + 143.744i) q^{33} +103.062i q^{35} +(-153.900 - 153.900i) q^{37} +(-47.3932 - 132.389i) q^{39} +(244.160 - 244.160i) q^{41} +204.154 q^{43} +(125.224 - 125.224i) q^{45} +(163.090 - 163.090i) q^{47} +315.567i q^{49} +50.7948 q^{51} +745.175 q^{53} +1333.35i q^{55} +(-14.2054 + 14.2054i) q^{57} +(356.309 - 356.309i) q^{59} +633.349 q^{61} +(-33.3322 + 33.3322i) q^{63} +(394.205 - 833.817i) q^{65} +(254.501 + 254.501i) q^{67} -556.977i q^{69} +(-638.833 - 638.833i) q^{71} +(-239.652 - 239.652i) q^{73} +786.558 q^{75} -354.912i q^{77} +638.294i q^{79} +81.0000 q^{81} +(718.687 + 718.687i) q^{83} +(235.582 + 235.582i) q^{85} -191.416i q^{87} +(34.5610 + 34.5610i) q^{89} +(-104.930 + 221.946i) q^{91} +(-121.507 + 121.507i) q^{93} -131.767 q^{95} +(-346.192 + 346.192i) q^{97} +(-431.232 + 431.232i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) $ 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−13.9138	+	13.9138i	−1.24449	+	1.24449i	−0.286365	+	0.958121i	$0.592447\pi$
$5$	−13.9138	+	13.9138i	−1.24449	+	1.24449i	−0.958121	+	0.286365i	$0.907553\pi$
$6$		0			0
$7$	3.70358	−	3.70358i	0.199975	−	0.199975i	−0.600015	−	0.799989i	$-0.704839\pi$
$7$	3.70358	−	3.70358i	0.199975	−	0.199975i	0.799989	+	0.600015i	$0.204839\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	47.9147	−	47.9147i	1.31335	−	1.31335i	0.394416	−	0.918932i	$-0.370947\pi$
$11$	47.9147	−	47.9147i	1.31335	−	1.31335i	0.918932	−	0.394416i	$-0.129053\pi$
$12$		0			0
$13$	−44.1297	+	15.7977i	−0.941491	+	0.337038i
$13$	−44.1297	+	15.7977i	−0.941491	+	0.337038i
$14$		0			0
$15$	−41.7413	−	41.7413i	−0.718504	−	0.718504i
$16$		0			0
$17$		−	16.9316i		−	0.241560i	−0.992679	−	0.120780i	$-0.961460\pi$
$17$		−	16.9316i		−	0.241560i	0.992679	−	0.120780i	$-0.0385395\pi$
$18$		0			0
$19$	4.73512	+	4.73512i	0.0571743	+	0.0571743i	0.735116	−	0.677942i	$-0.237128\pi$
$19$	4.73512	+	4.73512i	0.0571743	+	0.0571743i	−0.677942	+	0.735116i	$0.737128\pi$
$20$		0			0
$21$	11.1107	+	11.1107i	0.115455	+	0.115455i
$22$		0			0
$23$	−185.659			−1.68316			−0.841578	−	0.540135i	$-0.818373\pi$
$23$	−185.659			−1.68316			−0.841578	+	0.540135i	$0.818373\pi$
$24$		0			0
$25$		−	262.186i		−	2.09749i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−63.8055			−0.408565			−0.204282	−	0.978912i	$-0.565486\pi$
$29$	−63.8055			−0.408565			−0.204282	+	0.978912i	$0.565486\pi$
$30$		0			0
$31$	40.5024	+	40.5024i	0.234660	+	0.234660i	0.814634	−	0.579975i	$-0.196938\pi$
$31$	40.5024	+	40.5024i	0.234660	+	0.234660i	−0.579975	+	0.814634i	$0.696938\pi$
$32$		0			0
$33$	143.744	+	143.744i	0.758262	+	0.758262i
$34$		0			0
$35$			103.062i			0.497731i
$36$		0			0
$37$	−153.900	−	153.900i	−0.683811	−	0.683811i	0.277046	−	0.960857i	$-0.410645\pi$
$37$	−153.900	−	153.900i	−0.683811	−	0.683811i	−0.960857	+	0.277046i	$0.910645\pi$
$38$		0			0
$39$	−47.3932	−	132.389i	−0.194589	−	0.543570i
$40$		0			0
$41$	244.160	−	244.160i	0.930034	−	0.930034i	−0.0676732	−	0.997708i	$-0.521558\pi$
$41$	244.160	−	244.160i	0.930034	−	0.930034i	0.997708	+	0.0676732i	$0.0215575\pi$
$42$		0			0
$43$	204.154			0.724030			0.362015	−	0.932172i	$-0.382089\pi$
$43$	204.154			0.724030			0.362015	+	0.932172i	$0.382089\pi$
$44$		0			0
$45$	125.224	−	125.224i	0.414828	−	0.414828i
$46$		0			0
$47$	163.090	−	163.090i	0.506151	−	0.506151i	−0.407192	−	0.913343i	$-0.633492\pi$
$47$	163.090	−	163.090i	0.506151	−	0.506151i	0.913343	+	0.407192i	$0.133492\pi$
$48$		0			0
$49$			315.567i			0.920020i
$50$		0			0
$51$	50.7948			0.139465
$52$		0			0
$53$	745.175			1.93128			0.965638	−	0.259890i	$-0.0836864\pi$
$53$	745.175			1.93128			0.965638	+	0.259890i	$0.0836864\pi$
$54$		0			0
$55$			1333.35i			3.26889i
$56$		0			0
$57$	−14.2054	+	14.2054i	−0.0330096	+	0.0330096i
$58$		0			0
$59$	356.309	−	356.309i	0.786228	−	0.786228i	−0.194646	−	0.980874i	$-0.562356\pi$
$59$	356.309	−	356.309i	0.786228	−	0.786228i	0.980874	+	0.194646i	$0.0623557\pi$
$60$		0			0
$61$	633.349			1.32938			0.664689	−	0.747120i	$-0.268564\pi$
$61$	633.349			1.32938			0.664689	+	0.747120i	$0.268564\pi$
$62$		0			0
$63$	−33.3322	+	33.3322i	−0.0666582	+	0.0666582i
$64$		0			0
$65$	394.205	−	833.817i	0.752232	−	1.59111i
$66$		0			0
$67$	254.501	+	254.501i	0.464063	+	0.464063i	0.899985	−	0.435922i	$-0.143577\pi$
$67$	254.501	+	254.501i	0.464063	+	0.464063i	−0.435922	+	0.899985i	$0.643577\pi$
$68$		0			0
$69$		−	556.977i		−	0.971771i
$70$		0			0
$71$	−638.833	−	638.833i	−1.06782	−	1.06782i	−0.997526	−	0.0702975i	$-0.977605\pi$
$71$	−638.833	−	638.833i	−1.06782	−	1.06782i	−0.0702975	−	0.997526i	$-0.522395\pi$
$72$		0			0
$73$	−239.652	−	239.652i	−0.384235	−	0.384235i	0.488390	−	0.872625i	$-0.337584\pi$
$73$	−239.652	−	239.652i	−0.384235	−	0.384235i	−0.872625	+	0.488390i	$0.837584\pi$
$74$		0			0
$75$	786.558			1.21099
$76$		0			0
$77$		−	354.912i		−	0.525273i
$78$		0			0
$79$			638.294i			0.909034i	0.890738	+	0.454517i	$0.150188\pi$
$79$			638.294i			0.909034i	−0.890738	+	0.454517i	$0.849812\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	718.687	+	718.687i	0.950435	+	0.950435i	0.998828	−	0.0483931i	$-0.0154100\pi$
$83$	718.687	+	718.687i	0.950435	+	0.950435i	−0.0483931	+	0.998828i	$0.515410\pi$
$84$		0			0
$85$	235.582	+	235.582i	0.300618	+	0.300618i
$86$		0			0
$87$		−	191.416i		−	0.235885i
$88$		0			0
$89$	34.5610	+	34.5610i	0.0411625	+	0.0411625i	0.727388	−	0.686226i	$-0.240734\pi$
$89$	34.5610	+	34.5610i	0.0411625	+	0.0411625i	−0.686226	+	0.727388i	$0.740734\pi$
$90$		0			0
$91$	−104.930	+	221.946i	−0.120875	+	0.255673i
$92$		0			0
$93$	−121.507	+	121.507i	−0.135481	+	0.135481i
$94$		0			0
$95$	−131.767			−0.142305
$96$		0			0
$97$	−346.192	+	346.192i	−0.362376	+	0.362376i	−0.864687	−	0.502311i	$-0.832483\pi$
$97$	−346.192	+	346.192i	−0.362376	+	0.362376i	0.502311	+	0.864687i	$0.332483\pi$
$98$		0			0
$99$	−431.232	+	431.232i	−0.437783	+	0.437783i
$100$		0			0
$101$		−	1452.70i		−	1.43117i	−0.698523	−	0.715587i	$-0.746159\pi$
$101$		−	1452.70i		−	1.43117i	0.698523	−	0.715587i	$-0.253841\pi$
$102$		0			0
$103$	119.648			0.114458			0.0572292	−	0.998361i	$-0.481773\pi$
$103$	119.648			0.114458			0.0572292	+	0.998361i	$0.481773\pi$
$104$		0			0
$105$	−309.185			−0.287365
$106$		0			0
$107$			763.111i			0.689465i	0.938701	+	0.344732i	$0.112030\pi$
$107$			763.111i			0.689465i	−0.938701	+	0.344732i	$0.887970\pi$
$108$		0			0
$109$	−654.709	+	654.709i	−0.575319	+	0.575319i	−0.933610	−	0.358291i	$-0.883360\pi$
$109$	−654.709	+	654.709i	−0.575319	+	0.575319i	0.358291	+	0.933610i	$0.383360\pi$
$110$		0			0
$111$	461.700	−	461.700i	0.394798	−	0.394798i
$112$		0			0
$113$	−1855.80			−1.54495			−0.772474	−	0.635046i	$-0.780981\pi$
$113$	−1855.80			−1.54495			−0.772474	+	0.635046i	$0.780981\pi$
$114$		0			0
$115$	2583.22	−	2583.22i	2.09466	−	2.09466i
$116$		0			0
$117$	397.167	−	142.179i	0.313830	−	0.112346i
$118$		0			0
$119$	−62.7075	−	62.7075i	−0.0483058	−	0.0483058i
$120$		0			0
$121$		−	3260.64i		−	2.44977i
$122$		0			0
$123$	732.480	+	732.480i	0.536956	+	0.536956i
$124$		0			0
$125$	1908.77	+	1908.77i	1.36581	+	1.36581i
$126$		0			0
$127$	322.009			0.224990			0.112495	−	0.993652i	$-0.464116\pi$
$127$	322.009			0.224990			0.112495	+	0.993652i	$0.464116\pi$
$128$		0			0
$129$			612.463i			0.418019i
$130$		0			0
$131$		−	2282.35i		−	1.52221i	−0.648629	−	0.761105i	$-0.724657\pi$
$131$		−	2282.35i		−	1.52221i	0.648629	−	0.761105i	$-0.275343\pi$
$132$		0			0
$133$	35.0738			0.0228668
$134$		0			0
$135$	375.672	+	375.672i	0.239501	+	0.239501i
$136$		0			0
$137$	8.27752	+	8.27752i	0.00516201	+	0.00516201i	0.709683	−	0.704521i	$-0.248838\pi$
$137$	8.27752	+	8.27752i	0.00516201	+	0.00516201i	−0.704521	+	0.709683i	$0.748838\pi$
$138$		0			0
$139$			430.200i			0.262511i	0.991349	+	0.131256i	$0.0419009\pi$
$139$			430.200i			0.262511i	−0.991349	+	0.131256i	$0.958099\pi$
$140$		0			0
$141$	489.270	+	489.270i	0.292227	+	0.292227i
$142$		0			0
$143$	−1357.52	+	2871.41i	−0.793857	+	1.67915i
$144$		0			0
$145$	887.775	−	887.775i	0.508453	−	0.508453i
$146$		0			0
$147$	−946.701			−0.531174
$148$		0			0
$149$	1427.12	−	1427.12i	0.784661	−	0.784661i	−0.195953	−	0.980613i	$-0.562780\pi$
$149$	1427.12	−	1427.12i	0.784661	−	0.784661i	0.980613	+	0.195953i	$0.0627799\pi$
$150$		0			0
$151$	−855.610	+	855.610i	−0.461116	+	0.461116i	−0.899021	−	0.437905i	$-0.855721\pi$
$151$	−855.610	+	855.610i	−0.461116	+	0.461116i	0.437905	+	0.899021i	$0.355721\pi$
$152$		0			0
$153$			152.384i			0.0805199i
$154$		0			0
$155$	−1127.08			−0.584061
$156$		0			0
$157$	2438.43			1.23954			0.619772	−	0.784782i	$-0.287225\pi$
$157$	2438.43			1.23954			0.619772	+	0.784782i	$0.287225\pi$
$158$		0			0
$159$			2235.52i			1.11502i
$160$		0			0
$161$	−687.604	+	687.604i	−0.336589	+	0.336589i
$162$		0			0
$163$	236.605	−	236.605i	0.113695	−	0.113695i	−0.647970	−	0.761666i	$-0.724382\pi$
$163$	236.605	−	236.605i	0.113695	−	0.113695i	0.761666	+	0.647970i	$0.224382\pi$
$164$		0			0
$165$	−4000.05			−1.88729
$166$		0			0
$167$	130.558	−	130.558i	0.0604962	−	0.0604962i	−0.676211	−	0.736708i	$-0.736379\pi$
$167$	130.558	−	130.558i	0.0604962	−	0.0604962i	0.736708	+	0.676211i	$0.236379\pi$
$168$		0			0
$169$	1697.86	−	1394.30i	0.772810	−	0.634637i
$170$		0			0
$171$	−42.6161	−	42.6161i	−0.0190581	−	0.0190581i
$172$		0			0
$173$		−	4274.55i		−	1.87854i	−0.343173	−	0.939272i	$-0.611502\pi$
$173$		−	4274.55i		−	1.87854i	0.343173	−	0.939272i	$-0.388498\pi$
$174$		0			0
$175$	−971.027	−	971.027i	−0.419444	−	0.419444i
$176$		0			0
$177$	1068.93	+	1068.93i	0.453929	+	0.453929i
$178$		0			0
$179$	−2595.26			−1.08368			−0.541840	−	0.840482i	$-0.682272\pi$
$179$	−2595.26			−1.08368			−0.541840	+	0.840482i	$0.682272\pi$
$180$		0			0
$181$			68.1614i			0.0279911i	0.999902	+	0.0139956i	$0.00445507\pi$
$181$			68.1614i			0.0279911i	−0.999902	+	0.0139956i	$0.995545\pi$
$182$		0			0
$183$			1900.05i			0.767517i
$184$		0			0
$185$	4282.66			1.70198
$186$		0			0
$187$	−811.273	−	811.273i	−0.317252	−	0.317252i
$188$		0			0
$189$	−99.9967	−	99.9967i	−0.0384851	−	0.0384851i
$190$		0			0
$191$		−	4946.82i		−	1.87403i	−0.349293	−	0.937013i	$-0.613578\pi$
$191$		−	4946.82i		−	1.87403i	0.349293	−	0.937013i	$-0.386422\pi$
$192$		0			0
$193$	22.2809	+	22.2809i	0.00830990	+	0.00830990i	0.711250	−	0.702940i	$-0.248130\pi$
$193$	22.2809	+	22.2809i	0.00830990	+	0.00830990i	−0.702940	+	0.711250i	$0.748130\pi$
$194$		0			0
$195$	2501.45	+	1182.61i	0.918628	+	0.434302i
$196$		0			0
$197$	−2098.27	+	2098.27i	−0.758861	+	0.758861i	−0.976115	−	0.217254i	$-0.930290\pi$
$197$	−2098.27	+	2098.27i	−0.758861	+	0.758861i	0.217254	+	0.976115i	$0.430290\pi$
$198$		0			0
$199$	2678.59			0.954174			0.477087	−	0.878856i	$-0.341693\pi$
$199$	2678.59			0.954174			0.477087	+	0.878856i	$0.341693\pi$
$200$		0			0
$201$	−763.503	+	763.503i	−0.267927	+	0.267927i
$202$		0			0
$203$	−236.309	+	236.309i	−0.0817026	+	0.0817026i
$204$		0			0
$205$			6794.38i			2.31483i
$206$		0			0
$207$	1670.93			0.561052
$208$		0			0
$209$	453.764			0.150180
$210$		0			0
$211$		−	4048.59i		−	1.32093i	−0.750856	−	0.660466i	$-0.770359\pi$
$211$		−	4048.59i		−	1.32093i	0.750856	−	0.660466i	$-0.229641\pi$
$212$		0			0
$213$	1916.50	−	1916.50i	0.616508	−	0.616508i
$214$		0			0
$215$	−2840.56	+	2840.56i	−0.901044	+	0.901044i
$216$		0			0
$217$	300.008			0.0938520
$218$		0			0
$219$	718.957	−	718.957i	0.221838	−	0.221838i
$220$		0			0
$221$	267.481	+	747.186i	0.0814149	+	0.227426i
$222$		0			0
$223$	157.101	+	157.101i	0.0471760	+	0.0471760i	0.730301	−	0.683125i	$-0.239380\pi$
$223$	157.101	+	157.101i	0.0471760	+	0.0471760i	−0.683125	+	0.730301i	$0.739380\pi$
$224$		0			0
$225$			2359.67i			0.699163i
$226$		0			0
$227$	−2012.14	−	2012.14i	−0.588327	−	0.588327i	0.348851	−	0.937178i	$-0.386572\pi$
$227$	−2012.14	−	2012.14i	−0.588327	−	0.588327i	−0.937178	+	0.348851i	$0.886572\pi$
$228$		0			0
$229$	1174.51	+	1174.51i	0.338926	+	0.338926i	0.855963	−	0.517037i	$-0.172965\pi$
$229$	1174.51	+	1174.51i	0.338926	+	0.338926i	−0.517037	+	0.855963i	$0.672965\pi$
$230$		0			0
$231$	1064.74			0.303266
$232$		0			0
$233$		−	2022.45i		−	0.568648i	−0.958728	−	0.284324i	$-0.908231\pi$
$233$		−	2022.45i		−	0.568648i	0.958728	−	0.284324i	$-0.0917692\pi$
$234$		0			0
$235$			4538.39i			1.25980i
$236$		0			0
$237$	−1914.88			−0.524831
$238$		0			0
$239$	−3061.00	−	3061.00i	−0.828452	−	0.828452i	0.158851	−	0.987303i	$-0.449221\pi$
$239$	−3061.00	−	3061.00i	−0.828452	−	0.828452i	−0.987303	+	0.158851i	$0.949221\pi$
$240$		0			0
$241$	−2138.76	−	2138.76i	−0.571659	−	0.571659i	0.360933	−	0.932592i	$-0.382458\pi$
$241$	−2138.76	−	2138.76i	−0.571659	−	0.571659i	−0.932592	+	0.360933i	$0.882458\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−4390.73	−	4390.73i	−1.14495	−	1.14495i
$246$		0			0
$247$	−283.764	−	134.155i	−0.0730990	−	0.0345591i
$248$		0			0
$249$	−2156.06	+	2156.06i	−0.548734	+	0.548734i
$250$		0			0
$251$	1878.74			0.472449			0.236225	−	0.971698i	$-0.424090\pi$
$251$	1878.74			0.472449			0.236225	+	0.971698i	$0.424090\pi$
$252$		0			0
$253$	−8895.80	+	8895.80i	−2.21057	+	2.21057i
$254$		0			0
$255$	−706.747	+	706.747i	−0.173562	+	0.173562i
$256$		0			0
$257$		−	1025.01i		−	0.248787i	−0.992233	−	0.124393i	$-0.960301\pi$
$257$		−	1025.01i		−	0.248787i	0.992233	−	0.124393i	$-0.0396985\pi$
$258$		0			0
$259$	−1139.96			−0.273490
$260$		0			0
$261$	574.249			0.136188
$262$		0			0
$263$			873.537i			0.204808i	0.994743	+	0.102404i	$0.0326535\pi$
$263$			873.537i			0.204808i	−0.994743	+	0.102404i	$0.967347\pi$
$264$		0			0
$265$	−10368.2	+	10368.2i	−2.40345	+	2.40345i
$266$		0			0
$267$	−103.683	+	103.683i	−0.0237652	+	0.0237652i
$268$		0			0
$269$	−5959.94			−1.35087			−0.675435	−	0.737420i	$-0.736044\pi$
$269$	−5959.94			−1.35087			−0.675435	+	0.737420i	$0.736044\pi$
$270$		0			0
$271$	3050.09	−	3050.09i	0.683690	−	0.683690i	−0.277139	−	0.960830i	$-0.589386\pi$
$271$	3050.09	−	3050.09i	0.683690	−	0.683690i	0.960830	+	0.277139i	$0.0893864\pi$
$272$		0			0
$273$	−665.838	−	314.790i	−0.147613	−	0.0697873i
$274$		0			0
$275$	−12562.6	−	12562.6i	−2.75473	−	2.75473i
$276$		0			0
$277$		−	6987.78i		−	1.51572i	−0.652416	−	0.757861i	$-0.726245\pi$
$277$		−	6987.78i		−	1.51572i	0.652416	−	0.757861i	$-0.273755\pi$
$278$		0			0
$279$	−364.522	−	364.522i	−0.0782199	−	0.0782199i
$280$		0			0
$281$	2229.86	+	2229.86i	0.473390	+	0.473390i	0.903010	−	0.429620i	$-0.141352\pi$
$281$	2229.86	+	2229.86i	0.473390	+	0.473390i	−0.429620	+	0.903010i	$0.641352\pi$
$282$		0			0
$283$	−6950.34			−1.45991			−0.729955	−	0.683495i	$-0.760459\pi$
$283$	−6950.34			−1.45991			−0.729955	+	0.683495i	$0.760459\pi$
$284$		0			0
$285$		−	395.300i		−	0.0821599i
$286$		0			0
$287$		−	1808.53i		−	0.371966i
$288$		0			0
$289$	4626.32			0.941649
$290$		0			0
$291$	−1038.58	−	1038.58i	−0.209218	−	0.209218i
$292$		0			0
$293$	5684.18	+	5684.18i	1.13336	+	1.13336i	0.989616	+	0.143740i	$0.0459129\pi$
$293$	5684.18	+	5684.18i	1.13336	+	1.13336i	0.143740	+	0.989616i	$0.454087\pi$
$294$		0			0
$295$			9915.20i			1.95690i
$296$		0			0
$297$	−1293.70	−	1293.70i	−0.252754	−	0.252754i
$298$		0			0
$299$	8193.08	−	2932.99i	1.58468	−	0.567288i
$300$		0			0
$301$	756.103	−	756.103i	0.144787	−	0.144787i
$302$		0			0
$303$	4358.09			0.826289
$304$		0			0
$305$	−8812.28	+	8812.28i	−1.65439	+	1.65439i
$306$		0			0
$307$	−248.466	+	248.466i	−0.0461912	+	0.0461912i	−0.729825	−	0.683634i	$-0.760399\pi$
$307$	−248.466	+	248.466i	−0.0461912	+	0.0461912i	0.683634	+	0.729825i	$0.260399\pi$
$308$		0			0
$309$			358.943i			0.0660826i
$310$		0			0
$311$	510.392			0.0930600			0.0465300	−	0.998917i	$-0.485184\pi$
$311$	510.392			0.0930600			0.0465300	+	0.998917i	$0.485184\pi$
$312$		0			0
$313$	−7639.87			−1.37965			−0.689826	−	0.723975i	$-0.742313\pi$
$313$	−7639.87			−1.37965			−0.689826	+	0.723975i	$0.742313\pi$
$314$		0			0
$315$		−	927.554i		−	0.165910i
$316$		0			0
$317$	4695.69	−	4695.69i	0.831975	−	0.831975i	−0.155812	−	0.987787i	$-0.549799\pi$
$317$	4695.69	−	4695.69i	0.831975	−	0.831975i	0.987787	+	0.155812i	$0.0497993\pi$
$318$		0			0
$319$	−3057.22	+	3057.22i	−0.536588	+	0.536588i
$320$		0			0
$321$	−2289.33			−0.398063
$322$		0			0
$323$	80.1732	−	80.1732i	0.0138110	−	0.0138110i
$324$		0			0
$325$	4141.94	+	11570.2i	0.706934	+	1.97477i
$326$		0			0
$327$	−1964.13	−	1964.13i	−0.332161	−	0.332161i
$328$		0			0
$329$		−	1208.03i		−	0.202435i
$330$		0			0
$331$	7147.38	+	7147.38i	1.18688	+	1.18688i	0.977927	+	0.208949i	$0.0670042\pi$
$331$	7147.38	+	7147.38i	1.18688	+	1.18688i	0.208949	+	0.977927i	$0.432996\pi$
$332$		0			0
$333$	1385.10	+	1385.10i	0.227937	+	0.227937i
$334$		0			0
$335$	−7082.13			−1.15504
$336$		0			0
$337$		−	606.819i		−	0.0980877i	−0.998797	−	0.0490438i	$-0.984383\pi$
$337$		−	606.819i		−	0.0980877i	0.998797	−	0.0490438i	$-0.0156174\pi$
$338$		0			0
$339$		−	5567.41i		−	0.891976i
$340$		0			0
$341$	3881.33			0.616380
$342$		0			0
$343$	2439.06	+	2439.06i	0.383955	+	0.383955i
$344$		0			0
$345$	7749.65	+	7749.65i	1.20935	+	1.20935i
$346$		0			0
$347$		−	857.609i		−	0.132677i	−0.997797	−	0.0663384i	$-0.978868\pi$
$347$		−	857.609i		−	0.132677i	0.997797	−	0.0663384i	$-0.0211317\pi$
$348$		0			0
$349$	−5371.11	−	5371.11i	−0.823808	−	0.823808i	0.162844	−	0.986652i	$-0.447933\pi$
$349$	−5371.11	−	5371.11i	−0.823808	−	0.823808i	−0.986652	+	0.162844i	$0.947933\pi$
$350$		0			0
$351$	426.538	+	1191.50i	0.0648631	+	0.181190i
$352$		0			0
$353$	4317.37	−	4317.37i	0.650965	−	0.650965i	−0.302260	−	0.953225i	$-0.597741\pi$
$353$	4317.37	−	4317.37i	0.650965	−	0.650965i	0.953225	+	0.302260i	$0.0977412\pi$
$354$		0			0
$355$	17777.1			2.65778
$356$		0			0
$357$	188.123	−	188.123i	0.0278894	−	0.0278894i
$358$		0			0
$359$	−8644.26	+	8644.26i	−1.27083	+	1.27083i	−0.325171	+	0.945655i	$0.605422\pi$
$359$	−8644.26	+	8644.26i	−1.27083	+	1.27083i	−0.945655	+	0.325171i	$0.894578\pi$
$360$		0			0
$361$		−	6814.16i		−	0.993462i
$362$		0			0
$363$	9781.92			1.41437
$364$		0			0
$365$	6668.93			0.956350
$366$		0			0
$367$		−	2722.03i		−	0.387164i	−0.981084	−	0.193582i	$-0.937990\pi$
$367$		−	2722.03i		−	0.387164i	0.981084	−	0.193582i	$-0.0620105\pi$
$368$		0			0
$369$	−2197.44	+	2197.44i	−0.310011	+	0.310011i
$370$		0			0
$371$	2759.82	−	2759.82i	0.386206	−	0.386206i
$372$		0			0
$373$	−9.55862			−0.00132688			−0.000663440	−	1.00000i	$-0.500211\pi$
$373$	−9.55862			−0.00132688			−0.000663440		1.00000i	$0.500211\pi$
$374$		0			0
$375$	−5726.32	+	5726.32i	−0.788550	+	0.788550i
$376$		0			0
$377$	2815.72	−	1007.98i	0.384660	−	0.137702i
$378$		0			0
$379$	7403.27	+	7403.27i	1.00338	+	1.00338i	0.999994	+	0.00338379i	$0.00107710\pi$
$379$	7403.27	+	7403.27i	1.00338	+	1.00338i	0.00338379	+	0.999994i	$0.498923\pi$
$380$		0			0
$381$			966.028i			0.129898i
$382$		0			0
$383$	−2840.27	−	2840.27i	−0.378932	−	0.378932i	0.491785	−	0.870717i	$-0.336345\pi$
$383$	−2840.27	−	2840.27i	−0.378932	−	0.378932i	−0.870717	+	0.491785i	$0.836345\pi$
$384$		0			0
$385$	4938.17	+	4938.17i	0.653694	+	0.653694i
$386$		0			0
$387$	−1837.39			−0.241343
$388$		0			0
$389$			2814.71i			0.366868i	0.983032	+	0.183434i	$0.0587214\pi$
$389$			2814.71i			0.366868i	−0.983032	+	0.183434i	$0.941279\pi$
$390$		0			0
$391$			3143.50i			0.406583i
$392$		0			0
$393$	6847.04			0.878848
$394$		0			0
$395$	−8881.07	−	8881.07i	−1.13128	−	1.13128i
$396$		0			0
$397$	−6247.09	−	6247.09i	−0.789754	−	0.789754i	0.191699	−	0.981454i	$-0.438600\pi$
$397$	−6247.09	−	6247.09i	−0.789754	−	0.789754i	−0.981454	+	0.191699i	$0.938600\pi$
$398$		0			0
$399$			105.221i			0.0132022i
$400$		0			0
$401$	2415.38	+	2415.38i	0.300794	+	0.300794i	0.841324	−	0.540531i	$-0.181776\pi$
$401$	2415.38	+	2415.38i	0.300794	+	0.300794i	−0.540531	+	0.841324i	$0.681776\pi$
$402$		0			0
$403$	−2427.21	−	1147.52i	−0.300019	−	0.141841i
$404$		0			0
$405$	−1127.02	+	1127.02i	−0.138276	+	0.138276i
$406$		0			0
$407$	−14748.1			−1.79616
$408$		0			0
$409$	9169.65	−	9169.65i	1.10858	−	1.10858i	0.115245	−	0.993337i	$-0.463235\pi$
$409$	9169.65	−	9169.65i	1.10858	−	1.10858i	0.993337	−	0.115245i	$-0.0367652\pi$
$410$		0			0
$411$	−24.8325	+	24.8325i	−0.00298029	+	0.00298029i
$412$		0			0
$413$		−	2639.24i		−	0.314451i
$414$		0			0
$415$	−19999.3			−2.36561
$416$		0			0
$417$	−1290.60			−0.151561
$418$		0			0
$419$		−	5630.26i		−	0.656458i	−0.944598	−	0.328229i	$-0.893548\pi$
$419$		−	5630.26i		−	0.656458i	0.944598	−	0.328229i	$-0.106452\pi$
$420$		0			0
$421$	−5823.08	+	5823.08i	−0.674108	+	0.674108i	−0.958661	−	0.284552i	$-0.908155\pi$
$421$	−5823.08	+	5823.08i	−0.674108	+	0.674108i	0.284552	+	0.958661i	$0.408155\pi$
$422$		0			0
$423$	−1467.81	+	1467.81i	−0.168717	+	0.168717i
$424$		0			0
$425$	−4439.23			−0.506669
$426$		0			0
$427$	2345.66	−	2345.66i	0.265842	−	0.265842i
$428$		0			0
$429$	−8614.22	−	4072.56i	−0.969460	−	0.458333i
$430$		0			0
$431$	4828.81	+	4828.81i	0.539665	+	0.539665i	0.923431	−	0.383766i	$-0.125373\pi$
$431$	4828.81	+	4828.81i	0.539665	+	0.539665i	−0.383766	+	0.923431i	$0.625373\pi$
$432$		0			0
$433$		−	8140.76i		−	0.903510i	−0.892142	−	0.451755i	$-0.850798\pi$
$433$		−	8140.76i		−	0.903510i	0.892142	−	0.451755i	$-0.149202\pi$
$434$		0			0
$435$	2663.32	+	2663.32i	0.293555	+	0.293555i
$436$		0			0
$437$	−879.119	−	879.119i	−0.0962333	−	0.0962333i
$438$		0			0
$439$	−5202.19			−0.565574			−0.282787	−	0.959183i	$-0.591259\pi$
$439$	−5202.19			−0.565574			−0.282787	+	0.959183i	$0.591259\pi$
$440$		0			0
$441$		−	2840.10i		−	0.306673i
$442$		0			0
$443$		−	4780.70i		−	0.512727i	−0.966580	−	0.256364i	$-0.917476\pi$
$443$		−	4780.70i		−	0.512727i	0.966580	−	0.256364i	$-0.0825244\pi$
$444$		0			0
$445$	−961.748			−0.102452
$446$		0			0
$447$	4281.37	+	4281.37i	0.453024	+	0.453024i
$448$		0			0
$449$	−5127.82	−	5127.82i	−0.538969	−	0.538969i	0.384257	−	0.923226i	$-0.374458\pi$
$449$	−5127.82	−	5127.82i	−0.538969	−	0.538969i	−0.923226	+	0.384257i	$0.874458\pi$
$450$		0			0
$451$		−	23397.7i		−	2.44292i
$452$		0			0
$453$	−2566.83	−	2566.83i	−0.266225	−	0.266225i
$454$		0			0
$455$	−1628.14	−	4548.08i	−0.167754	−	0.468609i
$456$		0			0
$457$	−10547.1	+	10547.1i	−1.07959	+	1.07959i	−0.0830452	+	0.996546i	$0.526465\pi$
$457$	−10547.1	+	10547.1i	−1.07959	+	1.07959i	−0.996546	+	0.0830452i	$0.973535\pi$
$458$		0			0
$459$	−457.153			−0.0464882
$460$		0			0
$461$	7482.40	−	7482.40i	0.755944	−	0.755944i	−0.219638	−	0.975582i	$-0.570487\pi$
$461$	7482.40	−	7482.40i	0.755944	−	0.755944i	0.975582	+	0.219638i	$0.0704875\pi$
$462$		0			0
$463$	−2861.40	+	2861.40i	−0.287215	+	0.287215i	−0.835978	−	0.548763i	$-0.815099\pi$
$463$	−2861.40	+	2861.40i	−0.287215	+	0.287215i	0.548763	+	0.835978i	$0.315099\pi$
$464$		0			0
$465$		−	3381.25i		−	0.337208i
$466$		0			0
$467$	14319.2			1.41887			0.709437	−	0.704768i	$-0.248949\pi$
$467$	14319.2			1.41887			0.709437	+	0.704768i	$0.248949\pi$
$468$		0			0
$469$	1885.13			0.185602
$470$		0			0
$471$			7315.30i			0.715651i
$472$		0			0
$473$	9782.00	−	9782.00i	0.950903	−	0.950903i
$474$		0			0
$475$	1241.48	−	1241.48i	0.119922	−	0.119922i
$476$		0			0
$477$	−6706.57			−0.643759
$478$		0			0
$479$	5724.20	−	5724.20i	0.546023	−	0.546023i	−0.379265	−	0.925288i	$-0.623823\pi$
$479$	5724.20	−	5724.20i	0.546023	−	0.546023i	0.925288	+	0.379265i	$0.123823\pi$
$480$		0			0
$481$	9222.83	+	4360.29i	0.874272	+	0.413331i
$482$		0			0
$483$	−2062.81	−	2062.81i	−0.194329	−	0.194329i
$484$		0			0
$485$		−	9633.67i		−	0.901943i
$486$		0			0
$487$	9997.97	+	9997.97i	0.930290	+	0.930290i	0.997724	−	0.0674341i	$-0.0214812\pi$
$487$	9997.97	+	9997.97i	0.930290	+	0.930290i	−0.0674341	+	0.997724i	$0.521481\pi$
$488$		0			0
$489$	709.816	+	709.816i	0.0656421	+	0.0656421i
$490$		0			0
$491$	9597.29			0.882117			0.441059	−	0.897478i	$-0.354603\pi$
$491$	9597.29			0.882117			0.441059	+	0.897478i	$0.354603\pi$
$492$		0			0
$493$			1080.33i			0.0986928i
$494$		0			0
$495$		−	12000.1i		−	1.08963i
$496$		0			0
$497$	−4731.94			−0.427075
$498$		0			0
$499$	6746.94	+	6746.94i	0.605279	+	0.605279i	0.941709	−	0.336429i	$-0.109219\pi$
$499$	6746.94	+	6746.94i	0.605279	+	0.605279i	−0.336429	+	0.941709i	$0.609219\pi$
$500$		0			0
$501$	391.673	+	391.673i	0.0349275	+	0.0349275i
$502$		0			0
$503$			18174.2i			1.61102i	0.592579	+	0.805512i	$0.298110\pi$
$503$			18174.2i			1.61102i	−0.592579	+	0.805512i	$0.701890\pi$
$504$		0			0
$505$	20212.5	+	20212.5i	1.78108	+	1.78108i
$506$		0			0
$507$	4182.89	+	5093.59i	0.366408	+	0.446182i
$508$		0			0
$509$	−4943.56	+	4943.56i	−0.430490	+	0.430490i	−0.888795	−	0.458305i	$-0.848457\pi$
$509$	−4943.56	+	4943.56i	−0.430490	+	0.430490i	0.458305	+	0.888795i	$0.348457\pi$
$510$		0			0
$511$	−1775.14			−0.153675
$512$		0			0
$513$	127.848	−	127.848i	0.0110032	−	0.0110032i
$514$		0			0
$515$	−1664.75	+	1664.75i	−0.142442	+	0.142442i
$516$		0			0
$517$		−	15628.8i		−	1.32951i
$518$		0			0
$519$	12823.7			1.08458
$520$		0			0
$521$	−13138.6			−1.10482			−0.552410	−	0.833573i	$-0.686292\pi$
$521$	−13138.6			−1.10482			−0.552410	+	0.833573i	$0.686292\pi$
$522$		0			0
$523$		−	14057.7i		−	1.17533i	−0.809104	−	0.587666i	$-0.800047\pi$
$523$		−	14057.7i		−	1.17533i	0.809104	−	0.587666i	$-0.199953\pi$
$524$		0			0
$525$	2913.08	−	2913.08i	0.242166	−	0.242166i
$526$		0			0
$527$	685.771	−	685.771i	0.0566843	−	0.0566843i
$528$		0			0
$529$	22302.3			1.83302
$530$		0			0
$531$	−3206.78	+	3206.78i	−0.262076	+	0.262076i
$532$		0			0
$533$	−6917.55	+	14631.9i	−0.562162	+	1.18908i
$534$		0			0
$535$	−10617.8	−	10617.8i	−0.858029	−	0.858029i
$536$		0			0
$537$		−	7785.78i		−	0.625663i
$538$		0			0
$539$	15120.3	+	15120.3i	1.20831	+	1.20831i
$540$		0			0
$541$	−11871.4	−	11871.4i	−0.943420	−	0.943420i	0.0550625	−	0.998483i	$-0.482464\pi$
$541$	−11871.4	−	11871.4i	−0.943420	−	0.943420i	−0.998483	+	0.0550625i	$0.982464\pi$
$542$		0			0
$543$	−204.484			−0.0161607
$544$		0			0
$545$		−	18219.0i		−	1.43195i
$546$		0			0
$547$		−	5827.38i		−	0.455504i	−0.973719	−	0.227752i	$-0.926862\pi$
$547$		−	5827.38i		−	0.455504i	0.973719	−	0.227752i	$-0.0731376\pi$
$548$		0			0
$549$	−5700.14			−0.443126
$550$		0			0
$551$	−302.127	−	302.127i	−0.0233594	−	0.0233594i
$552$		0			0
$553$	2363.97	+	2363.97i	0.181784	+	0.181784i
$554$		0			0
$555$			12848.0i			0.982641i
$556$		0			0
$557$	−4672.59	−	4672.59i	−0.355447	−	0.355447i	0.506685	−	0.862131i	$-0.330871\pi$
$557$	−4672.59	−	4672.59i	−0.355447	−	0.355447i	−0.862131	+	0.506685i	$0.830871\pi$
$558$		0			0
$559$	−9009.28	+	3225.17i	−0.681667	+	0.244026i
$560$		0			0
$561$	2433.82	−	2433.82i	0.183166	−	0.183166i
$562$		0			0
$563$	−1509.10			−0.112968			−0.0564839	−	0.998404i	$-0.517989\pi$
$563$	−1509.10			−0.112968			−0.0564839	+	0.998404i	$0.517989\pi$
$564$		0			0
$565$	25821.2	−	25821.2i	1.92267	−	1.92267i
$566$		0			0
$567$	299.990	−	299.990i	0.0222194	−	0.0222194i
$568$		0			0
$569$		−	14420.9i		−	1.06249i	−0.847220	−	0.531243i	$-0.821725\pi$
$569$		−	14420.9i		−	1.06249i	0.847220	−	0.531243i	$-0.178275\pi$
$570$		0			0
$571$	−9487.02			−0.695305			−0.347653	−	0.937623i	$-0.613021\pi$
$571$	−9487.02			−0.695305			−0.347653	+	0.937623i	$0.613021\pi$
$572$		0			0
$573$	14840.5			1.08197
$574$		0			0
$575$			48677.2i			3.53040i
$576$		0			0
$577$	−9411.62	+	9411.62i	−0.679048	+	0.679048i	−0.959785	−	0.280737i	$-0.909421\pi$
$577$	−9411.62	+	9411.62i	−0.679048	+	0.679048i	0.280737	+	0.959785i	$0.409421\pi$
$578$		0			0
$579$	−66.8426	+	66.8426i	−0.00479772	+	0.00479772i
$580$		0			0
$581$	5323.43			0.380126
$582$		0			0
$583$	35704.8	−	35704.8i	2.53644	−	2.53644i
$584$		0			0
$585$	−3547.84	+	7504.35i	−0.250744	+	0.530370i
$586$		0			0
$587$	10597.4	+	10597.4i	0.745151	+	0.745151i	0.973564	−	0.228414i	$-0.0733538\pi$
$587$	10597.4	+	10597.4i	0.745151	+	0.745151i	−0.228414	+	0.973564i	$0.573354\pi$
$588$		0			0
$589$			383.568i			0.0268330i
$590$		0			0
$591$	−6294.82	−	6294.82i	−0.438129	−	0.438129i
$592$		0			0
$593$	−10426.2	−	10426.2i	−0.722013	−	0.722013i	0.247002	−	0.969015i	$-0.420555\pi$
$593$	−10426.2	−	10426.2i	−0.722013	−	0.722013i	−0.969015	+	0.247002i	$0.920555\pi$
$594$		0			0
$595$	1745.00			0.120232
$596$		0			0
$597$			8035.78i			0.550892i
$598$		0			0
$599$		−	22692.0i		−	1.54786i	−0.633270	−	0.773931i	$-0.718288\pi$
$599$		−	22692.0i		−	1.54786i	0.633270	−	0.773931i	$-0.281712\pi$
$600$		0			0
$601$	7977.98			0.541479			0.270739	−	0.962653i	$-0.412732\pi$
$601$	7977.98			0.541479			0.270739	+	0.962653i	$0.412732\pi$
$602$		0			0
$603$	−2290.51	−	2290.51i	−0.154688	−	0.154688i
$604$		0			0
$605$	45367.8	+	45367.8i	3.04870	+	3.04870i
$606$		0			0
$607$			22309.1i			1.49176i	0.666082	+	0.745879i	$0.267970\pi$
$607$			22309.1i			1.49176i	−0.666082	+	0.745879i	$0.732030\pi$
$608$		0			0
$609$	−708.926	−	708.926i	−0.0471710	−	0.0471710i
$610$		0			0
$611$	−4620.66	+	9773.56i	−0.305944	+	0.647129i
$612$		0			0
$613$	14934.7	−	14934.7i	0.984025	−	0.984025i	−0.0158491	−	0.999874i	$-0.505045\pi$
$613$	14934.7	−	14934.7i	0.984025	−	0.984025i	0.999874	+	0.0158491i	$0.00504512\pi$
$614$		0			0
$615$	−20383.1			−1.33647
$616$		0			0
$617$	15097.1	−	15097.1i	0.985064	−	0.985064i	−0.0148260	−	0.999890i	$-0.504719\pi$
$617$	15097.1	−	15097.1i	0.985064	−	0.985064i	0.999890	+	0.0148260i	$0.00471942\pi$
$618$		0			0
$619$	−4256.83	+	4256.83i	−0.276408	+	0.276408i	−0.831673	−	0.555265i	$-0.812617\pi$
$619$	−4256.83	+	4256.83i	−0.276408	+	0.276408i	0.555265	+	0.831673i	$0.312617\pi$
$620$		0			0
$621$			5012.80i			0.323924i
$622$		0			0
$623$	255.999			0.0164629
$624$		0			0
$625$	−20343.3			−1.30197
$626$		0			0
$627$			1361.29i			0.0867062i
$628$		0			0
$629$	−2605.77	+	2605.77i	−0.165181	+	0.165181i
$630$		0			0
$631$	−14308.8	+	14308.8i	−0.902731	+	0.902731i	−0.995672	−	0.0929410i	$-0.970373\pi$
$631$	−14308.8	+	14308.8i	−0.902731	+	0.902731i	0.0929410	+	0.995672i	$0.470373\pi$
$632$		0			0
$633$	12145.8			0.762640
$634$		0			0
$635$	−4480.37	+	4480.37i	−0.279997	+	0.279997i
$636$		0			0
$637$	−4985.24	−	13925.9i	−0.310082	−	0.866191i
$638$		0			0
$639$	5749.49	+	5749.49i	0.355941	+	0.355941i
$640$		0			0
$641$			7305.38i			0.450149i	0.974342	+	0.225074i	$0.0722625\pi$
$641$			7305.38i			0.450149i	−0.974342	+	0.225074i	$0.927737\pi$
$642$		0			0
$643$	−5342.98	−	5342.98i	−0.327693	−	0.327693i	0.524016	−	0.851709i	$-0.324433\pi$
$643$	−5342.98	−	5342.98i	−0.327693	−	0.327693i	−0.851709	+	0.524016i	$0.824433\pi$
$644$		0			0
$645$	−8521.68	−	8521.68i	−0.520218	−	0.520218i
$646$		0			0
$647$	29769.7			1.80891			0.904457	−	0.426565i	$-0.140277\pi$
$647$	29769.7			1.80891			0.904457	+	0.426565i	$0.140277\pi$
$648$		0			0
$649$		−	34144.9i		−	2.06518i
$650$		0			0
$651$			900.024i			0.0541855i
$652$		0			0
$653$	17597.7			1.05460			0.527299	−	0.849680i	$-0.323205\pi$
$653$	17597.7			1.05460			0.527299	+	0.849680i	$0.323205\pi$
$654$		0			0
$655$	31756.0	+	31756.0i	1.89437	+	1.89437i
$656$		0			0
$657$	2156.87	+	2156.87i	0.128078	+	0.128078i
$658$		0			0
$659$			11707.4i			0.692041i	0.938227	+	0.346021i	$0.112467\pi$
$659$			11707.4i			0.692041i	−0.938227	+	0.346021i	$0.887533\pi$
$660$		0			0
$661$	7126.50	+	7126.50i	0.419347	+	0.419347i	0.884979	−	0.465631i	$-0.154173\pi$
$661$	7126.50	+	7126.50i	0.419347	+	0.419347i	−0.465631	+	0.884979i	$0.654173\pi$
$662$		0			0
$663$	−2241.56	+	802.442i	−0.131305	+	0.0470049i
$664$		0			0
$665$	−488.009	+	488.009i	−0.0284574	+	0.0284574i
$666$		0			0
$667$	11846.1			0.687679
$668$		0			0
$669$	−471.302	+	471.302i	−0.0272371	+	0.0272371i
$670$		0			0
$671$	30346.7	−	30346.7i	1.74594	−	1.74594i
$672$		0			0
$673$			10774.8i			0.617147i	0.951201	+	0.308573i	$0.0998516\pi$
$673$			10774.8i			0.617147i	−0.951201	+	0.308573i	$0.900148\pi$
$674$		0			0
$675$	−7079.02			−0.403662
$676$		0			0
$677$	−25958.2			−1.47364			−0.736821	−	0.676088i	$-0.763674\pi$
$677$	−25958.2			−1.47364			−0.736821	+	0.676088i	$0.763674\pi$
$678$		0			0
$679$			2564.30i			0.144932i
$680$		0			0
$681$	6036.41	−	6036.41i	0.339671	−	0.339671i
$682$		0			0
$683$	−9964.30	+	9964.30i	−0.558233	+	0.558233i	−0.928804	−	0.370571i	$-0.879162\pi$
$683$	−9964.30	+	9964.30i	−0.558233	+	0.558233i	0.370571	+	0.928804i	$0.379162\pi$
$684$		0			0
$685$	−230.343			−0.0128481
$686$		0			0
$687$	−3523.54	+	3523.54i	−0.195679	+	0.195679i
$688$		0			0
$689$	−32884.4	+	11772.1i	−1.81828	+	0.650914i
$690$		0			0
$691$	6936.58	+	6936.58i	0.381881	+	0.381881i	0.871780	−	0.489898i	$-0.162966\pi$
$691$	6936.58	+	6936.58i	0.381881	+	0.381881i	−0.489898	+	0.871780i	$0.662966\pi$
$692$		0			0
$693$			3194.21i			0.175091i
$694$		0			0
$695$	−5985.70	−	5985.70i	−0.326692	−	0.326692i
$696$		0			0
$697$	−4134.02	−	4134.02i	−0.224659	−	0.224659i
$698$		0			0
$699$	6067.35			0.328309
$700$		0			0
$701$			10708.3i			0.576957i	0.957486	+	0.288478i	$0.0931493\pi$
$701$			10708.3i			0.576957i	−0.957486	+	0.288478i	$0.906851\pi$
$702$		0			0
$703$		−	1457.47i		−	0.0781928i
$704$		0			0
$705$	−13615.2			−0.727343
$706$		0			0
$707$	−5380.18	−	5380.18i	−0.286199	−	0.286199i
$708$		0			0
$709$	3306.12	+	3306.12i	0.175125	+	0.175125i	0.789227	−	0.614102i	$-0.210482\pi$
$709$	3306.12	+	3306.12i	0.175125	+	0.175125i	−0.614102	+	0.789227i	$0.710482\pi$
$710$		0			0
$711$		−	5744.64i		−	0.303011i
$712$		0			0
$713$	−7519.65	−	7519.65i	−0.394969	−	0.394969i
$714$		0			0
$715$	−21063.9	−	58840.3i	−1.10174	−	3.07763i
$716$		0			0
$717$	9183.01	−	9183.01i	0.478307	−	0.478307i
$718$		0			0
$719$	9968.69			0.517065			0.258532	−	0.966003i	$-0.416761\pi$
$719$	9968.69			0.517065			0.258532	+	0.966003i	$0.416761\pi$
$720$		0			0
$721$	443.124	−	443.124i	0.0228888	−	0.0228888i
$722$		0			0
$723$	6416.29	−	6416.29i	0.330048	−	0.330048i
$724$		0			0
$725$			16728.9i			0.856960i
$726$		0			0
$727$	−27575.5			−1.40676			−0.703382	−	0.710812i	$-0.748328\pi$
$727$	−27575.5			−1.40676			−0.703382	+	0.710812i	$0.748328\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$		−	3456.66i		−	0.174896i
$732$		0			0
$733$	−1500.14	+	1500.14i	−0.0755922	+	0.0755922i	−0.743892	−	0.668300i	$-0.767022\pi$
$733$	−1500.14	+	1500.14i	−0.0755922	+	0.0755922i	0.668300	+	0.743892i	$0.267022\pi$
$734$		0			0
$735$	13172.2	−	13172.2i	0.661038	−	0.661038i
$736$		0			0
$737$	24388.7			1.21895
$738$		0			0
$739$	−24477.1	+	24477.1i	−1.21841	+	1.21841i	−0.250219	+	0.968189i	$0.580503\pi$
$739$	−24477.1	+	24477.1i	−1.21841	+	1.21841i	−0.968189	+	0.250219i	$0.919497\pi$
$740$		0			0
$741$	402.466	−	851.291i	0.0199527	−	0.0422037i
$742$		0			0
$743$	−12943.9	−	12943.9i	−0.639117	−	0.639117i	0.311221	−	0.950338i	$-0.399262\pi$
$743$	−12943.9	−	12943.9i	−0.639117	−	0.639117i	−0.950338	+	0.311221i	$0.899262\pi$
$744$		0			0
$745$			39713.3i			1.95300i
$746$		0			0
$747$	−6468.18	−	6468.18i	−0.316812	−	0.316812i
$748$		0			0
$749$	2826.24	+	2826.24i	0.137875	+	0.137875i
$750$		0			0
$751$	27554.5			1.33885			0.669427	−	0.742878i	$-0.266540\pi$
$751$	27554.5			1.33885			0.669427	+	0.742878i	$0.266540\pi$
$752$		0			0
$753$			5636.21i			0.272769i
$754$		0			0
$755$		−	23809.5i		−	1.14770i
$756$		0			0
$757$	−2871.45			−0.137866			−0.0689330	−	0.997621i	$-0.521959\pi$
$757$	−2871.45			−0.137866			−0.0689330	+	0.997621i	$0.521959\pi$
$758$		0			0
$759$	−26687.4	−	26687.4i	−1.27627	−	1.27627i
$760$		0			0
$761$	−20998.4	−	20998.4i	−1.00025	−	1.00025i	−1.00000		0.000250668i	$-0.999920\pi$
$761$	−20998.4	−	20998.4i	−1.00025	−	1.00025i	−0.000250668	−	1.00000i	$-0.500080\pi$
$762$		0			0
$763$			4849.54i			0.230098i
$764$		0			0
$765$	−2120.24	−	2120.24i	−0.100206	−	0.100206i
$766$		0			0
$767$	−10094.9	+	21352.7i	−0.475237	+	1.00522i
$768$		0			0
$769$	−2899.25	+	2899.25i	−0.135955	+	0.135955i	−0.771809	−	0.635854i	$-0.780648\pi$
$769$	−2899.25	+	2899.25i	−0.135955	+	0.135955i	0.635854	+	0.771809i	$0.280648\pi$
$770$		0			0
$771$	3075.02			0.143637
$772$		0			0
$773$	9595.72	−	9595.72i	0.446487	−	0.446487i	−0.447698	−	0.894185i	$-0.647756\pi$
$773$	9595.72	−	9595.72i	0.446487	−	0.446487i	0.894185	+	0.447698i	$0.147756\pi$
$774$		0			0
$775$	10619.2	−	10619.2i	0.492196	−	0.492196i
$776$		0			0
$777$		−	3419.89i		−	0.157899i
$778$		0			0
$779$	2312.26			0.106348
$780$		0			0
$781$	−61219.0			−2.80485
$782$		0			0
$783$			1722.75i			0.0786283i
$784$		0			0
$785$	−33927.8	+	33927.8i	−1.54259	+	1.54259i
$786$		0			0
$787$	−30533.5	+	30533.5i	−1.38298	+	1.38298i	−0.543692	+	0.839285i	$0.682974\pi$
$787$	−30533.5	+	30533.5i	−1.38298	+	1.38298i	−0.839285	+	0.543692i	$0.817026\pi$
$788$		0			0
$789$	−2620.61			−0.118246
$790$		0			0
$791$	−6873.11	+	6873.11i	−0.308950	+	0.308950i
$792$		0			0
$793$	−27949.5	+	10005.5i	−1.25160	+	0.448051i
$794$		0			0
$795$	−31104.6	−	31104.6i	−1.38763	−	1.38763i
$796$		0			0
$797$		−	9629.74i		−	0.427984i	−0.976835	−	0.213992i	$-0.931353\pi$
$797$		−	9629.74i		−	0.427984i	0.976835	−	0.213992i	$-0.0686466\pi$
$798$		0			0
$799$	−2761.37	−	2761.37i	−0.122266	−	0.122266i
$800$		0			0
$801$	−311.049	−	311.049i	−0.0137208	−	0.0137208i
$802$		0			0
$803$	−22965.7			−1.00927
$804$		0			0
$805$		−	19134.3i		−	0.837759i
$806$		0			0
$807$		−	17879.8i		−	0.779925i
$808$		0			0
$809$	−34151.9			−1.48420			−0.742099	−	0.670290i	$-0.766170\pi$
$809$	−34151.9			−1.48420			−0.742099	+	0.670290i	$0.766170\pi$
$810$		0			0
$811$	−6272.73	−	6272.73i	−0.271597	−	0.271597i	0.558146	−	0.829743i	$-0.311513\pi$
$811$	−6272.73	−	6272.73i	−0.271597	−	0.271597i	−0.829743	+	0.558146i	$0.811513\pi$
$812$		0			0
$813$	9150.28	+	9150.28i	0.394729	+	0.394729i
$814$		0			0
$815$			6584.15i			0.282985i
$816$		0			0
$817$	966.696	+	966.696i	0.0413959	+	0.0413959i
$818$		0			0
$819$	944.369	−	1997.52i	0.0402917	−	0.0852245i
$820$		0			0
$821$	234.902	−	234.902i	0.00998553	−	0.00998553i	−0.702096	−	0.712082i	$-0.747752\pi$
$821$	234.902	−	234.902i	0.00998553	−	0.00998553i	0.712082	+	0.702096i	$0.247752\pi$
$822$		0			0
$823$	−33988.5			−1.43957			−0.719785	−	0.694197i	$-0.755760\pi$
$823$	−33988.5			−1.43957			−0.719785	+	0.694197i	$0.755760\pi$
$824$		0			0
$825$	37687.7	−	37687.7i	1.59045	−	1.59045i
$826$		0			0
$827$	15068.0	−	15068.0i	0.633575	−	0.633575i	−0.315388	−	0.948963i	$-0.602135\pi$
$827$	15068.0	−	15068.0i	0.633575	−	0.633575i	0.948963	+	0.315388i	$0.102135\pi$
$828$		0			0
$829$			2485.12i			0.104115i	0.998644	+	0.0520577i	$0.0165780\pi$
$829$			2485.12i			0.104115i	−0.998644	+	0.0520577i	$0.983422\pi$
$830$		0			0
$831$	20963.3			0.875102
$832$		0			0
$833$	5343.05			0.222240
$834$		0			0
$835$			3633.10i			0.150573i
$836$		0			0
$837$	1093.57	−	1093.57i	0.0451603	−	0.0451603i
$838$		0			0
$839$	19119.8	−	19119.8i	0.786758	−	0.786758i	−0.194204	−	0.980961i	$-0.562212\pi$
$839$	19119.8	−	19119.8i	0.786758	−	0.786758i	0.980961	+	0.194204i	$0.0622122\pi$
$840$		0			0
$841$	−20317.9			−0.833075
$842$		0			0
$843$	−6689.59	+	6689.59i	−0.273312	+	0.273312i
$844$		0			0
$845$	−4223.76	+	43023.6i	−0.171955	+	1.75155i
$846$		0			0
$847$	−12076.0	−	12076.0i	−0.489891	−	0.489891i
$848$		0			0
$849$		−	20851.0i		−	0.842880i
$850$		0			0
$851$	28572.9	+	28572.9i	1.15096	+	1.15096i
$852$		0			0
$853$	−6270.09	−	6270.09i	−0.251681	−	0.251681i	0.569979	−	0.821659i	$-0.306951\pi$
$853$	−6270.09	−	6270.09i	−0.251681	−	0.251681i	−0.821659	+	0.569979i	$0.806951\pi$
$854$		0			0
$855$	1185.90			0.0474351
$856$		0			0
$857$			33972.2i			1.35410i	0.735935	+	0.677052i	$0.236743\pi$
$857$			33972.2i			1.35410i	−0.735935	+	0.677052i	$0.763257\pi$
$858$		0			0
$859$			27055.8i			1.07466i	0.843372	+	0.537330i	$0.180567\pi$
$859$			27055.8i			1.07466i	−0.843372	+	0.537330i	$0.819433\pi$
$860$		0			0
$861$	5425.60			0.214755
$862$		0			0
$863$	18238.2	+	18238.2i	0.719391	+	0.719391i	0.968480	−	0.249090i	$-0.0801314\pi$
$863$	18238.2	+	18238.2i	0.719391	+	0.719391i	−0.249090	+	0.968480i	$0.580131\pi$
$864$		0			0
$865$	59475.1	+	59475.1i	2.33782	+	2.33782i
$866$		0			0
$867$			13879.0i			0.543661i
$868$		0			0
$869$	30583.7	+	30583.7i	1.19388	+	1.19388i
$870$		0			0
$871$	−15251.6	−	7210.52i	−0.593318	−	0.280504i
$872$		0			0
$873$	3115.73	−	3115.73i	0.120792	−	0.120792i
$874$		0			0
$875$	14138.6			0.546254
$876$		0			0
$877$	−49.2535	+	49.2535i	−0.00189643	+	0.00189643i	−0.708054	−	0.706158i	$-0.750427\pi$
$877$	−49.2535	+	49.2535i	−0.00189643	+	0.00189643i	0.706158	+	0.708054i	$0.250427\pi$
$878$		0			0
$879$	−17052.5	+	17052.5i	−0.654343	+	0.654343i
$880$		0			0
$881$		−	19071.9i		−	0.729339i	−0.931137	−	0.364669i	$-0.881182\pi$
$881$		−	19071.9i		−	0.729339i	0.931137	−	0.364669i	$-0.118818\pi$
$882$		0			0
$883$	−21611.7			−0.823659			−0.411830	−	0.911261i	$-0.635110\pi$
$883$	−21611.7			−0.823659			−0.411830	+	0.911261i	$0.635110\pi$
$884$		0			0
$885$	−29745.6			−1.12982
$886$		0			0
$887$			8838.65i			0.334581i	0.985908	+	0.167290i	$0.0535017\pi$
$887$			8838.65i			0.334581i	−0.985908	+	0.167290i	$0.946498\pi$
$888$		0			0
$889$	1192.59	−	1192.59i	0.0449923	−	0.0449923i
$890$		0			0
$891$	3881.09	−	3881.09i	0.145928	−	0.145928i
$892$		0			0
$893$	1544.50			0.0578777
$894$		0			0
$895$	36109.8	−	36109.8i	1.34862	−	1.34862i
$896$		0			0
$897$	8798.97	+	24579.3i	0.327524	+	0.914913i
$898$		0			0
$899$	−2584.28	−	2584.28i	−0.0958737	−	0.0958737i
$900$		0			0
$901$		−	12617.0i		−	0.466519i
$902$		0			0
$903$	2268.31	+	2268.31i	0.0835931	+	0.0835931i
$904$		0			0
$905$	−948.382	−	948.382i	−0.0348346	−	0.0348346i
$906$		0			0
$907$	−16267.8			−0.595549			−0.297774	−	0.954636i	$-0.596244\pi$
$907$	−16267.8			−0.595549			−0.297774	+	0.954636i	$0.596244\pi$
$908$		0			0
$909$			13074.3i			0.477058i
$910$		0			0
$911$		−	23946.0i		−	0.870874i	−0.900219	−	0.435437i	$-0.856594\pi$
$911$		−	23946.0i		−	0.870874i	0.900219	−	0.435437i	$-0.143406\pi$
$912$		0			0
$913$	68871.3			2.49650
$914$		0			0
$915$	−26436.8	−	26436.8i	−0.955163	−	0.955163i
$916$		0			0
$917$	−8452.85	−	8452.85i	−0.304403	−	0.304403i
$918$		0			0
$919$		−	48773.1i		−	1.75068i	−0.483508	−	0.875340i	$-0.660637\pi$
$919$		−	48773.1i		−	1.75068i	0.483508	−	0.875340i	$-0.339363\pi$
$920$		0			0
$921$	−745.397	−	745.397i	−0.0266685	−	0.0266685i
$922$		0			0
$923$	38283.6	+	18099.4i	1.36524	+	0.645449i
$924$		0			0
$925$	−40350.4	+	40350.4i	−1.43428	+	1.43428i
$926$		0			0
$927$	−1076.83			−0.0381528
$928$		0			0
$929$	21546.4	−	21546.4i	0.760943	−	0.760943i	−0.215550	−	0.976493i	$-0.569154\pi$
$929$	21546.4	−	21546.4i	0.760943	−	0.760943i	0.976493	+	0.215550i	$0.0691544\pi$
$930$		0			0
$931$	−1494.25	+	1494.25i	−0.0526015	+	0.0526015i
$932$		0			0
$933$			1531.18i			0.0537282i
$934$		0			0
$935$	22575.7			0.789631
$936$		0			0
$937$	−27077.8			−0.944070			−0.472035	−	0.881580i	$-0.656480\pi$
$937$	−27077.8			−0.944070			−0.472035	+	0.881580i	$0.656480\pi$
$938$		0			0
$939$		−	22919.6i		−	0.796543i
$940$		0			0
$941$	−4511.90	+	4511.90i	−0.156306	+	0.156306i	−0.780928	−	0.624622i	$-0.785253\pi$
$941$	−4511.90	+	4511.90i	−0.156306	+	0.156306i	0.624622	+	0.780928i	$0.285253\pi$
$942$		0			0
$943$	−45330.6	+	45330.6i	−1.56539	+	1.56539i
$944$		0			0
$945$	2782.66			0.0957884
$946$		0			0
$947$	−31678.3	+	31678.3i	−1.08702	+	1.08702i	−0.0911855	+	0.995834i	$0.529066\pi$
$947$	−31678.3	+	31678.3i	−1.08702	+	1.08702i	−0.995834	+	0.0911855i	$0.970934\pi$
$948$		0			0
$949$	14361.7	+	6789.83i	0.491256	+	0.232252i
$950$		0			0
$951$	14087.1	+	14087.1i	0.480341	+	0.480341i
$952$		0			0
$953$			27802.9i			0.945042i	0.881319	+	0.472521i	$0.156656\pi$
$953$			27802.9i			0.945042i	−0.881319	+	0.472521i	$0.843344\pi$
$954$		0			0
$955$	68828.9	+	68828.9i	2.33220	+	2.33220i
$956$		0			0
$957$	−9171.66	−	9171.66i	−0.309799	−	0.309799i
$958$		0			0
$959$	61.3129			0.00206454
$960$		0			0
$961$		−	26510.1i		−	0.889870i
$962$		0			0
$963$		−	6868.00i		−	0.229822i
$964$		0			0
$965$	−620.021			−0.0206831
$966$		0			0
$967$	−32797.4	−	32797.4i	−1.09068	−	1.09068i	−0.995455	−	0.0952289i	$-0.969642\pi$
$967$	−32797.4	−	32797.4i	−1.09068	−	1.09068i	−0.0952289	−	0.995455i	$-0.530358\pi$
$968$		0			0
$969$	240.520	+	240.520i	0.00797379	+	0.00797379i
$970$		0			0
$971$		−	3417.73i		−	0.112956i	−0.998404	−	0.0564779i	$-0.982013\pi$
$971$		−	3417.73i		−	0.112956i	0.998404	−	0.0564779i	$-0.0179871\pi$
$972$		0			0
$973$	1593.28	+	1593.28i	0.0524956	+	0.0524956i
$974$		0			0
$975$	−34710.6	+	12425.8i	−1.14013	+	0.408148i
$976$		0			0
$977$	−40024.3	+	40024.3i	−1.31064	+	1.31064i	−0.389691	+	0.920946i	$0.627418\pi$
$977$	−40024.3	+	40024.3i	−1.31064	+	1.31064i	−0.920946	+	0.389691i	$0.872582\pi$
$978$		0			0
$979$	3311.96			0.108121
$980$		0			0
$981$	5892.38	−	5892.38i	0.191773	−	0.191773i
$982$		0			0
$983$	−7505.62	+	7505.62i	−0.243532	+	0.243532i	−0.818310	−	0.574778i	$-0.805089\pi$
$983$	−7505.62	+	7505.62i	−0.243532	+	0.243532i	0.574778	+	0.818310i	$0.305089\pi$
$984$		0			0
$985$		−	58389.7i		−	1.88878i
$986$		0			0
$987$	3624.10			0.116876
$988$		0			0
$989$	−37903.1			−1.21866
$990$		0			0
$991$		−	48879.6i		−	1.56681i	−0.621510	−	0.783406i	$-0.713481\pi$
$991$		−	48879.6i		−	1.56681i	0.621510	−	0.783406i	$-0.286519\pi$
$992$		0			0
$993$	−21442.2	+	21442.2i	−0.685243	+	0.685243i
$994$		0			0
$995$	−37269.4	+	37269.4i	−1.18746	+	1.18746i
$996$		0			0
$997$	16635.7			0.528443			0.264221	−	0.964462i	$-0.414885\pi$
$997$	16635.7			0.528443			0.264221	+	0.964462i	$0.414885\pi$
$998$		0			0
$999$	−4155.30	+	4155.30i	−0.131599	+	0.131599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.bc.d.31.1	yes	28
4.3	odd	2		624.4.bc.c.31.1	✓	28
13.8	odd	4		624.4.bc.c.463.1	yes	28
52.47	even	4	inner	624.4.bc.d.463.1	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.bc.c.31.1	✓	28	4.3	odd	2
624.4.bc.c.463.1	yes	28	13.8	odd	4
624.4.bc.d.31.1	yes	28	1.1	even	1	trivial
624.4.bc.d.463.1	yes	28	52.47	even	4	inner

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(-13.9138 + 13.9138i) q^{5} +(3.70358 - 3.70358i) q^{7} -9.00000 q^{9} +(47.9147 - 47.9147i) q^{11} +(-44.1297 + 15.7977i) q^{13} +(-41.7413 - 41.7413i) q^{15} -16.9316i q^{17} +(4.73512 + 4.73512i) q^{19} +(11.1107 + 11.1107i) q^{21} -185.659 q^{23} -262.186i q^{25} -27.0000i q^{27} -63.8055 q^{29} +(40.5024 + 40.5024i) q^{31} +(143.744 + 143.744i) q^{33} +103.062i q^{35} +(-153.900 - 153.900i) q^{37} +(-47.3932 - 132.389i) q^{39} +(244.160 - 244.160i) q^{41} +204.154 q^{43} +(125.224 - 125.224i) q^{45} +(163.090 - 163.090i) q^{47} +315.567i q^{49} +50.7948 q^{51} +745.175 q^{53} +1333.35i q^{55} +(-14.2054 + 14.2054i) q^{57} +(356.309 - 356.309i) q^{59} +633.349 q^{61} +(-33.3322 + 33.3322i) q^{63} +(394.205 - 833.817i) q^{65} +(254.501 + 254.501i) q^{67} -556.977i q^{69} +(-638.833 - 638.833i) q^{71} +(-239.652 - 239.652i) q^{73} +786.558 q^{75} -354.912i q^{77} +638.294i q^{79} +81.0000 q^{81} +(718.687 + 718.687i) q^{83} +(235.582 + 235.582i) q^{85} -191.416i q^{87} +(34.5610 + 34.5610i) q^{89} +(-104.930 + 221.946i) q^{91} +(-121.507 + 121.507i) q^{93} -131.767 q^{95} +(-346.192 + 346.192i) q^{97} +(-431.232 + 431.232i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Introduction

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