LMFDB - Embedded newform 448.3.l.b.209.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(209,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("448.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.4
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.31524 + 3.31524i) q^{3} +(4.66000 + 4.66000i) q^{5} +(-2.13987 - 6.66491i) q^{7} -12.9816i q^{9} +O(q^{10})$	\(q+(-3.31524 + 3.31524i) q^{3} +(4.66000 + 4.66000i) q^{5} +(-2.13987 - 6.66491i) q^{7} -12.9816i q^{9} +(-8.29514 + 8.29514i) q^{11} +(-13.1395 + 13.1395i) q^{13} -30.8981 q^{15} -11.5210i q^{17} +(5.21978 - 5.21978i) q^{19} +(29.1899 + 15.0016i) q^{21} +20.5622i q^{23} +18.4313i q^{25} +(13.2001 + 13.2001i) q^{27} +(-39.7028 - 39.7028i) q^{29} +12.7004i q^{31} -55.0008i q^{33} +(21.0867 - 41.0303i) q^{35} +(5.59936 - 5.59936i) q^{37} -87.1213i q^{39} -22.7338 q^{41} +(22.2151 - 22.2151i) q^{43} +(60.4945 - 60.4945i) q^{45} -75.1143i q^{47} +(-39.8419 + 28.5240i) q^{49} +(38.1949 + 38.1949i) q^{51} +(33.8972 - 33.8972i) q^{53} -77.3108 q^{55} +34.6097i q^{57} +(-2.18100 - 2.18100i) q^{59} +(-42.9938 + 42.9938i) q^{61} +(-86.5215 + 27.7790i) q^{63} -122.460 q^{65} +(2.01453 + 2.01453i) q^{67} +(-68.1685 - 68.1685i) q^{69} -40.2019i q^{71} +53.0564 q^{73} +(-61.1042 - 61.1042i) q^{75} +(73.0368 + 37.5358i) q^{77} -36.5862 q^{79} +29.3116 q^{81} +(-90.3415 + 90.3415i) q^{83} +(53.6879 - 53.6879i) q^{85} +263.249 q^{87} -99.6505 q^{89} +(115.690 + 59.4568i) q^{91} +(-42.1050 - 42.1050i) q^{93} +48.6484 q^{95} +161.684i q^{97} +(107.685 + 107.685i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−3.31524	+	3.31524i	−1.10508	+	1.10508i	−0.111293	+	0.993788i	$0.535499\pi$
$3$	−3.31524	+	3.31524i	−1.10508	+	1.10508i	−0.993788	+	0.111293i	$0.964501\pi$
$4$		0			0
$5$	4.66000	+	4.66000i	0.932001	+	0.932001i	0.997831	−	0.0658299i	$-0.0209695\pi$
$5$	4.66000	+	4.66000i	0.932001	+	0.932001i	−0.0658299	+	0.997831i	$0.520969\pi$
$6$		0			0
$7$	−2.13987	−	6.66491i	−0.305695	−	0.952129i
$7$	−2.13987	−	6.66491i	−0.305695	−	0.952129i
$8$		0			0
$9$		−	12.9816i		−	1.44241i
$10$		0			0
$11$	−8.29514	+	8.29514i	−0.754103	+	0.754103i	−0.975242	−	0.221139i	$-0.929023\pi$
$11$	−8.29514	+	8.29514i	−0.754103	+	0.754103i	0.221139	+	0.975242i	$0.429023\pi$
$12$		0			0
$13$	−13.1395	+	13.1395i	−1.01073	+	1.01073i	−0.0107894	+	0.999942i	$0.503434\pi$
$13$	−13.1395	+	13.1395i	−1.01073	+	1.01073i	−0.999942	+	0.0107894i	$0.996566\pi$
$14$		0			0
$15$	−30.8981			−2.05987
$16$		0			0
$17$		−	11.5210i		−	0.677705i	−0.940839	−	0.338853i	$-0.889961\pi$
$17$		−	11.5210i		−	0.677705i	0.940839	−	0.338853i	$-0.110039\pi$
$18$		0			0
$19$	5.21978	−	5.21978i	0.274725	−	0.274725i	−0.556274	−	0.830999i	$-0.687769\pi$
$19$	5.21978	−	5.21978i	0.274725	−	0.274725i	0.830999	+	0.556274i	$0.187769\pi$
$20$		0			0
$21$	29.1899	+	15.0016i	1.39000	+	0.714362i
$22$		0			0
$23$			20.5622i			0.894007i	0.894532	+	0.447003i	$0.147509\pi$
$23$			20.5622i			0.894007i	−0.894532	+	0.447003i	$0.852491\pi$
$24$		0			0
$25$			18.4313i			0.737251i
$26$		0			0
$27$	13.2001	+	13.2001i	0.488893	+	0.488893i
$28$		0			0
$29$	−39.7028	−	39.7028i	−1.36906	−	1.36906i	−0.861780	−	0.507283i	$-0.830650\pi$
$29$	−39.7028	−	39.7028i	−1.36906	−	1.36906i	−0.507283	−	0.861780i	$-0.669350\pi$
$30$		0			0
$31$			12.7004i			0.409691i	0.978794	+	0.204846i	$0.0656693\pi$
$31$			12.7004i			0.409691i	−0.978794	+	0.204846i	$0.934331\pi$
$32$		0			0
$33$		−	55.0008i		−	1.66669i
$34$		0			0
$35$	21.0867	−	41.0303i	0.602477	−	1.17229i
$36$		0			0
$37$	5.59936	−	5.59936i	0.151334	−	0.151334i	−0.627380	−	0.778714i	$-0.715873\pi$
$37$	5.59936	−	5.59936i	0.151334	−	0.151334i	0.778714	+	0.627380i	$0.215873\pi$
$38$		0			0
$39$		−	87.1213i		−	2.23388i
$40$		0			0
$41$	−22.7338			−0.554482			−0.277241	−	0.960800i	$-0.589420\pi$
$41$	−22.7338			−0.554482			−0.277241	+	0.960800i	$0.589420\pi$
$42$		0			0
$43$	22.2151	−	22.2151i	0.516631	−	0.516631i	−0.399920	−	0.916550i	$-0.630962\pi$
$43$	22.2151	−	22.2151i	0.516631	−	0.516631i	0.916550	+	0.399920i	$0.130962\pi$
$44$		0			0
$45$	60.4945	−	60.4945i	1.34432	−	1.34432i
$46$		0			0
$47$		−	75.1143i		−	1.59818i	−0.601213	−	0.799089i	$-0.705316\pi$
$47$		−	75.1143i		−	1.59818i	0.601213	−	0.799089i	$-0.294684\pi$
$48$		0			0
$49$	−39.8419	+	28.5240i	−0.813101	+	0.582123i
$50$		0			0
$51$	38.1949	+	38.1949i	0.748919	+	0.748919i
$52$		0			0
$53$	33.8972	−	33.8972i	0.639570	−	0.639570i	−0.310880	−	0.950449i	$-0.600624\pi$
$53$	33.8972	−	33.8972i	0.639570	−	0.639570i	0.950449	+	0.310880i	$0.100624\pi$
$54$		0			0
$55$	−77.3108			−1.40565
$56$		0			0
$57$			34.6097i			0.607187i
$58$		0			0
$59$	−2.18100	−	2.18100i	−0.0369662	−	0.0369662i	0.688382	−	0.725348i	$-0.258321\pi$
$59$	−2.18100	−	2.18100i	−0.0369662	−	0.0369662i	−0.725348	+	0.688382i	$0.758321\pi$
$60$		0			0
$61$	−42.9938	+	42.9938i	−0.704816	+	0.704816i	−0.965440	−	0.260624i	$-0.916072\pi$
$61$	−42.9938	+	42.9938i	−0.704816	+	0.704816i	0.260624	+	0.965440i	$0.416072\pi$
$62$		0			0
$63$	−86.5215	+	27.7790i	−1.37336	+	0.440936i
$64$		0			0
$65$	−122.460			−1.88400
$66$		0			0
$67$	2.01453	+	2.01453i	0.0300676	+	0.0300676i	0.721981	−	0.691913i	$-0.243232\pi$
$67$	2.01453	+	2.01453i	0.0300676	+	0.0300676i	−0.691913	+	0.721981i	$0.743232\pi$
$68$		0			0
$69$	−68.1685	−	68.1685i	−0.987950	−	0.987950i
$70$		0			0
$71$		−	40.2019i		−	0.566224i	−0.959087	−	0.283112i	$-0.908633\pi$
$71$		−	40.2019i		−	0.566224i	0.959087	−	0.283112i	$-0.0913668\pi$
$72$		0			0
$73$	53.0564			0.726799			0.363400	−	0.931633i	$-0.381616\pi$
$73$	53.0564			0.726799			0.363400	+	0.931633i	$0.381616\pi$
$74$		0			0
$75$	−61.1042	−	61.1042i	−0.814722	−	0.814722i
$76$		0			0
$77$	73.0368	+	37.5358i	0.948530	+	0.487478i
$78$		0			0
$79$	−36.5862			−0.463117			−0.231558	−	0.972821i	$-0.574382\pi$
$79$	−36.5862			−0.463117			−0.231558	+	0.972821i	$0.574382\pi$
$80$		0			0
$81$	29.3116			0.361872
$82$		0			0
$83$	−90.3415	+	90.3415i	−1.08845	+	1.08845i	−0.0927635	+	0.995688i	$0.529570\pi$
$83$	−90.3415	+	90.3415i	−1.08845	+	1.08845i	−0.995688	+	0.0927635i	$0.970430\pi$
$84$		0			0
$85$	53.6879	−	53.6879i	0.631622	−	0.631622i
$86$		0			0
$87$	263.249			3.02585
$88$		0			0
$89$	−99.6505			−1.11967			−0.559834	−	0.828605i	$-0.689135\pi$
$89$	−99.6505			−1.11967			−0.559834	+	0.828605i	$0.689135\pi$
$90$		0			0
$91$	115.690	+	59.4568i	1.27132	+	0.653371i
$92$		0			0
$93$	−42.1050	−	42.1050i	−0.452742	−	0.452742i
$94$		0			0
$95$	48.6484			0.512089
$96$		0			0
$97$			161.684i			1.66685i	0.552633	+	0.833425i	$0.313623\pi$
$97$			161.684i			1.66685i	−0.552633	+	0.833425i	$0.686377\pi$
$98$		0			0
$99$	107.685	+	107.685i	1.08772	+	1.08772i
$100$		0			0
$101$	−19.7670	−	19.7670i	−0.195713	−	0.195713i	0.602446	−	0.798159i	$-0.294193\pi$
$101$	−19.7670	−	19.7670i	−0.195713	−	0.195713i	−0.798159	+	0.602446i	$0.794193\pi$
$102$		0			0
$103$	51.6767			0.501716			0.250858	−	0.968024i	$-0.419287\pi$
$103$	51.6767			0.501716			0.250858	+	0.968024i	$0.419287\pi$
$104$		0			0
$105$	66.1177	+	205.933i	0.629693	+	1.96126i
$106$		0			0
$107$	−20.4315	+	20.4315i	−0.190949	+	0.190949i	−0.796106	−	0.605157i	$-0.793110\pi$
$107$	−20.4315	+	20.4315i	−0.190949	+	0.190949i	0.605157	+	0.796106i	$0.293110\pi$
$108$		0			0
$109$	−85.8458	−	85.8458i	−0.787576	−	0.787576i	0.193520	−	0.981096i	$-0.438009\pi$
$109$	−85.8458	−	85.8458i	−0.787576	−	0.787576i	−0.981096	+	0.193520i	$0.938009\pi$
$110$		0			0
$111$			37.1265i			0.334473i
$112$		0			0
$113$	43.1097			0.381502			0.190751	−	0.981638i	$-0.438908\pi$
$113$	43.1097			0.381502			0.190751	+	0.981638i	$0.438908\pi$
$114$		0			0
$115$	−95.8198	+	95.8198i	−0.833215	+	0.833215i
$116$		0			0
$117$	170.572	+	170.572i	1.45788	+	1.45788i
$118$		0			0
$119$	−76.7863	+	24.6534i	−0.645263	+	0.207171i
$120$		0			0
$121$		−	16.6186i		−	0.137344i
$122$		0			0
$123$	75.3679	−	75.3679i	0.612747	−	0.612747i
$124$		0			0
$125$	30.6102	−	30.6102i	0.244882	−	0.244882i
$126$		0			0
$127$	−32.1368			−0.253046			−0.126523	−	0.991964i	$-0.540382\pi$
$127$	−32.1368			−0.253046			−0.126523	+	0.991964i	$0.540382\pi$
$128$		0			0
$129$			147.297i			1.14184i
$130$		0			0
$131$	−125.429	+	125.429i	−0.957471	+	0.957471i	−0.999132	−	0.0416609i	$-0.986735\pi$
$131$	−125.429	+	125.429i	−0.957471	+	0.957471i	0.0416609	+	0.999132i	$0.486735\pi$
$132$		0			0
$133$	−45.9590	−	23.6197i	−0.345556	−	0.177592i
$134$		0			0
$135$			123.025i			0.911298i
$136$		0			0
$137$			53.1819i			0.388189i	0.980983	+	0.194095i	$0.0621769\pi$
$137$			53.1819i			0.388189i	−0.980983	+	0.194095i	$0.937823\pi$
$138$		0			0
$139$	−113.897	−	113.897i	−0.819401	−	0.819401i	0.166620	−	0.986021i	$-0.446715\pi$
$139$	−113.897	−	113.897i	−0.819401	−	0.819401i	−0.986021	+	0.166620i	$0.946715\pi$
$140$		0			0
$141$	249.022	+	249.022i	1.76611	+	1.76611i
$142$		0			0
$143$		−	217.988i		−	1.52439i
$144$		0			0
$145$		−	370.031i		−	2.55193i
$146$		0			0
$147$	37.5217	−	226.650i	0.255250	−	1.54183i
$148$		0			0
$149$	−6.75799	+	6.75799i	−0.0453556	+	0.0453556i	−0.729421	−	0.684065i	$-0.760210\pi$
$149$	−6.75799	+	6.75799i	−0.0453556	+	0.0453556i	0.684065	+	0.729421i	$0.260210\pi$
$150$		0			0
$151$			165.702i			1.09736i	0.836032	+	0.548681i	$0.184870\pi$
$151$			165.702i			1.09736i	−0.836032	+	0.548681i	$0.815130\pi$
$152$		0			0
$153$	−149.561			−0.977526
$154$		0			0
$155$	−59.1841	+	59.1841i	−0.381833	+	0.381833i
$156$		0			0
$157$	−82.7381	+	82.7381i	−0.526994	+	0.526994i	−0.919675	−	0.392681i	$-0.871548\pi$
$157$	−82.7381	+	82.7381i	−0.526994	+	0.526994i	0.392681	+	0.919675i	$0.371548\pi$
$158$		0			0
$159$			224.755i			1.41355i
$160$		0			0
$161$	137.045	−	44.0003i	0.851210	−	0.273294i
$162$		0			0
$163$	198.331	+	198.331i	1.21675	+	1.21675i	0.968763	+	0.247989i	$0.0797696\pi$
$163$	198.331	+	198.331i	1.21675	+	1.21675i	0.247989	+	0.968763i	$0.420230\pi$
$164$		0			0
$165$	256.304	−	256.304i	1.55336	−	1.55336i
$166$		0			0
$167$	−234.300			−1.40299			−0.701497	−	0.712673i	$-0.747484\pi$
$167$	−234.300			−1.40299			−0.701497	+	0.712673i	$0.747484\pi$
$168$		0			0
$169$		−	176.293i		−	1.04316i
$170$		0			0
$171$	−67.7614	−	67.7614i	−0.396265	−	0.396265i
$172$		0			0
$173$	−113.853	+	113.853i	−0.658112	+	0.658112i	−0.954933	−	0.296821i	$-0.904074\pi$
$173$	−113.853	+	113.853i	−0.658112	+	0.658112i	0.296821	+	0.954933i	$0.404074\pi$
$174$		0			0
$175$	122.843	−	39.4405i	0.701959	−	0.225374i
$176$		0			0
$177$	14.4611			0.0817012
$178$		0			0
$179$	54.7105	+	54.7105i	0.305645	+	0.305645i	0.843218	−	0.537572i	$-0.180659\pi$
$179$	54.7105	+	54.7105i	0.305645	+	0.305645i	−0.537572	+	0.843218i	$0.680659\pi$
$180$		0			0
$181$	−137.091	−	137.091i	−0.757406	−	0.757406i	0.218443	−	0.975850i	$-0.429902\pi$
$181$	−137.091	−	137.091i	−0.757406	−	0.757406i	−0.975850	+	0.218443i	$0.929902\pi$
$182$		0			0
$183$		−	285.070i		−	1.55776i
$184$		0			0
$185$	52.1861			0.282087
$186$		0			0
$187$	95.5682	+	95.5682i	0.511060	+	0.511060i
$188$		0			0
$189$	59.7311	−	116.224i	0.316038	−	0.614942i
$190$		0			0
$191$	−40.7416			−0.213307			−0.106653	−	0.994296i	$-0.534014\pi$
$191$	−40.7416			−0.213307			−0.106653	+	0.994296i	$0.534014\pi$
$192$		0			0
$193$	81.8084			0.423878			0.211939	−	0.977283i	$-0.432022\pi$
$193$	81.8084			0.423878			0.211939	+	0.977283i	$0.432022\pi$
$194$		0			0
$195$	405.985	−	405.985i	2.08198	−	2.08198i
$196$		0			0
$197$	−254.441	+	254.441i	−1.29158	+	1.29158i	−0.357768	+	0.933810i	$0.616462\pi$
$197$	−254.441	+	254.441i	−1.29158	+	1.29158i	−0.933810	+	0.357768i	$0.883538\pi$
$198$		0			0
$199$	−201.967			−1.01491			−0.507455	−	0.861678i	$-0.669414\pi$
$199$	−201.967			−1.01491			−0.507455	+	0.861678i	$0.669414\pi$
$200$		0			0
$201$	−13.3573			−0.0664542
$202$		0			0
$203$	−179.657	+	349.574i	−0.885009	+	1.72204i
$204$		0			0
$205$	−105.939	−	105.939i	−0.516778	−	0.516778i
$206$		0			0
$207$	266.931			1.28952
$208$		0			0
$209$			86.5976i			0.414343i
$210$		0			0
$211$	−162.003	−	162.003i	−0.767785	−	0.767785i	0.209932	−	0.977716i	$-0.432676\pi$
$211$	−162.003	−	162.003i	−0.767785	−	0.767785i	−0.977716	+	0.209932i	$0.932676\pi$
$212$		0			0
$213$	133.279	+	133.279i	0.625723	+	0.625723i
$214$		0			0
$215$	207.045			0.963000
$216$		0			0
$217$	84.6472	−	27.1772i	0.390079	−	0.125241i
$218$		0			0
$219$	−175.895	+	175.895i	−0.803172	+	0.803172i
$220$		0			0
$221$	151.380	+	151.380i	0.684978	+	0.684978i
$222$		0			0
$223$		−	16.8442i		−	0.0755344i	−0.999287	−	0.0377672i	$-0.987975\pi$
$223$		−	16.8442i		−	0.0755344i	0.999287	−	0.0377672i	$-0.0120245\pi$
$224$		0			0
$225$	239.268			1.06342
$226$		0			0
$227$	84.3452	−	84.3452i	0.371565	−	0.371565i	−0.496482	−	0.868047i	$-0.665375\pi$
$227$	84.3452	−	84.3452i	0.371565	−	0.371565i	0.868047	+	0.496482i	$0.165375\pi$
$228$		0			0
$229$	153.690	+	153.690i	0.671136	+	0.671136i	0.957978	−	0.286842i	$-0.0926053\pi$
$229$	153.690	+	153.690i	0.671136	+	0.671136i	−0.286842	+	0.957978i	$0.592605\pi$
$230$		0			0
$231$	−366.575	+	117.694i	−1.58690	+	0.509499i
$232$		0			0
$233$			268.552i			1.15258i	0.817245	+	0.576291i	$0.195501\pi$
$233$			268.552i			1.15258i	−0.817245	+	0.576291i	$0.804499\pi$
$234$		0			0
$235$	350.033	−	350.033i	1.48950	−	1.48950i
$236$		0			0
$237$	121.292	−	121.292i	0.511781	−	0.511781i
$238$		0			0
$239$	−223.803			−0.936414			−0.468207	−	0.883619i	$-0.655100\pi$
$239$	−223.803			−0.936414			−0.468207	+	0.883619i	$0.655100\pi$
$240$		0			0
$241$		−	103.352i		−	0.428846i	−0.976741	−	0.214423i	$-0.931213\pi$
$241$		−	103.352i		−	0.428846i	0.976741	−	0.214423i	$-0.0687872\pi$
$242$		0			0
$243$	−215.976	+	215.976i	−0.888791	+	0.888791i
$244$		0			0
$245$	−318.586	−	52.7417i	−1.30035	−	0.215272i
$246$		0			0
$247$			137.171i			0.555347i
$248$		0			0
$249$		−	599.008i		−	2.40565i
$250$		0			0
$251$	332.455	+	332.455i	1.32452	+	1.32452i	0.910072	+	0.414450i	$0.136026\pi$
$251$	332.455	+	332.455i	1.32452	+	1.32452i	0.414450	+	0.910072i	$0.363974\pi$
$252$		0			0
$253$	−170.566	−	170.566i	−0.674174	−	0.674174i
$254$		0			0
$255$			355.977i			1.39599i
$256$		0			0
$257$			90.4052i			0.351771i	0.984411	+	0.175886i	$0.0562789\pi$
$257$			90.4052i			0.351771i	−0.984411	+	0.175886i	$0.943721\pi$
$258$		0			0
$259$	−49.3011	−	25.3373i	−0.190352	−	0.0978276i
$260$		0			0
$261$	−515.408	+	515.408i	−1.97474	+	1.97474i
$262$		0			0
$263$		−	197.271i		−	0.750079i	−0.927009	−	0.375040i	$-0.877629\pi$
$263$		−	197.271i		−	0.750079i	0.927009	−	0.375040i	$-0.122371\pi$
$264$		0			0
$265$	315.922			1.19216
$266$		0			0
$267$	330.365	−	330.365i	1.23732	−	1.23732i
$268$		0			0
$269$	−75.0832	+	75.0832i	−0.279120	+	0.279120i	−0.832758	−	0.553638i	$-0.813239\pi$
$269$	−75.0832	+	75.0832i	−0.279120	+	0.279120i	0.553638	+	0.832758i	$0.313239\pi$
$270$		0			0
$271$		−	157.382i		−	0.580747i	−0.956913	−	0.290373i	$-0.906220\pi$
$271$		−	157.382i		−	0.580747i	0.956913	−	0.290373i	$-0.0937795\pi$
$272$		0			0
$273$	−580.655	+	186.428i	−2.12694	+	0.682886i
$274$		0			0
$275$	−152.890	−	152.890i	−0.555964	−	0.555964i
$276$		0			0
$277$	107.714	−	107.714i	0.388859	−	0.388859i	−0.485421	−	0.874280i	$-0.661334\pi$
$277$	107.714	−	107.714i	0.388859	−	0.388859i	0.874280	+	0.485421i	$0.161334\pi$
$278$		0			0
$279$	164.873			0.590941
$280$		0			0
$281$		−	141.437i		−	0.503335i	−0.967814	−	0.251667i	$-0.919021\pi$
$281$		−	141.437i		−	0.503335i	0.967814	−	0.251667i	$-0.0809789\pi$
$282$		0			0
$283$	26.9231	+	26.9231i	0.0951347	+	0.0951347i	0.753072	−	0.657938i	$-0.228571\pi$
$283$	26.9231	+	26.9231i	0.0951347	+	0.0951347i	−0.657938	+	0.753072i	$0.728571\pi$
$284$		0			0
$285$	−161.281	+	161.281i	−0.565899	+	0.565899i
$286$		0			0
$287$	48.6472	+	151.518i	0.169502	+	0.527939i
$288$		0			0
$289$	156.267			0.540715
$290$		0			0
$291$	−536.023	−	536.023i	−1.84200	−	1.84200i
$292$		0			0
$293$	32.0487	+	32.0487i	0.109381	+	0.109381i	0.759679	−	0.650298i	$-0.225356\pi$
$293$	32.0487	+	32.0487i	0.109381	+	0.109381i	−0.650298	+	0.759679i	$0.725356\pi$
$294$		0			0
$295$		−	20.3270i		−	0.0689050i
$296$		0			0
$297$	−218.994			−0.737352
$298$		0			0
$299$	−270.177	−	270.177i	−0.903601	−	0.903601i
$300$		0			0
$301$	−195.599	−	100.524i	−0.649831	−	0.333968i
$302$		0			0
$303$	131.065			0.432558
$304$		0			0
$305$	−400.703			−1.31378
$306$		0			0
$307$	−90.4013	+	90.4013i	−0.294467	+	0.294467i	−0.838842	−	0.544375i	$-0.816767\pi$
$307$	−90.4013	+	90.4013i	−0.294467	+	0.294467i	0.544375	+	0.838842i	$0.316767\pi$
$308$		0			0
$309$	−171.321	+	171.321i	−0.554436	+	0.554436i
$310$		0			0
$311$	84.3194			0.271123			0.135562	−	0.990769i	$-0.456716\pi$
$311$	84.3194			0.271123			0.135562	+	0.990769i	$0.456716\pi$
$312$		0			0
$313$	172.576			0.551362			0.275681	−	0.961249i	$-0.411097\pi$
$313$	172.576			0.551362			0.275681	+	0.961249i	$0.411097\pi$
$314$		0			0
$315$	−532.641	−	273.740i	−1.69092	−	0.869017i
$316$		0			0
$317$	−59.8683	−	59.8683i	−0.188859	−	0.188859i	0.606344	−	0.795203i	$-0.292636\pi$
$317$	−59.8683	−	59.8683i	−0.188859	−	0.188859i	−0.795203	+	0.606344i	$0.792636\pi$
$318$		0			0
$319$	658.681			2.06483
$320$		0			0
$321$		−	135.471i		−	0.422027i
$322$		0			0
$323$	−60.1371	−	60.1371i	−0.186183	−	0.186183i
$324$		0			0
$325$	−242.178	−	242.178i	−0.745163	−	0.745163i
$326$		0			0
$327$	569.199			1.74067
$328$		0			0
$329$	−500.630	+	160.735i	−1.52167	+	0.488555i
$330$		0			0
$331$	−186.575	+	186.575i	−0.563672	+	0.563672i	−0.930348	−	0.366677i	$-0.880496\pi$
$331$	−186.575	+	186.575i	−0.563672	+	0.563672i	0.366677	+	0.930348i	$0.380496\pi$
$332$		0			0
$333$	−72.6889	−	72.6889i	−0.218285	−	0.218285i
$334$		0			0
$335$			18.7754i			0.0560461i
$336$		0			0
$337$	505.780			1.50083			0.750416	−	0.660966i	$-0.229853\pi$
$337$	505.780			1.50083			0.750416	+	0.660966i	$0.229853\pi$
$338$		0			0
$339$	−142.919	+	142.919i	−0.421590	+	0.421590i
$340$		0			0
$341$	−105.352	−	105.352i	−0.308950	−	0.308950i
$342$		0			0
$343$	275.366	+	204.505i	0.802817	+	0.596225i
$344$		0			0
$345$		−	635.331i		−	1.84154i
$346$		0			0
$347$	−347.916	+	347.916i	−1.00264	+	1.00264i	−0.00264331	+	0.999997i	$0.500841\pi$
$347$	−347.916	+	347.916i	−1.00264	+	1.00264i	−0.999997	+	0.00264331i	$0.999159\pi$
$348$		0			0
$349$	279.398	−	279.398i	0.800567	−	0.800567i	−0.182617	−	0.983184i	$-0.558457\pi$
$349$	279.398	−	279.398i	0.800567	−	0.800567i	0.983184	+	0.182617i	$0.0584568\pi$
$350$		0			0
$351$	−346.886			−0.988280
$352$		0			0
$353$		−	428.422i		−	1.21366i	−0.794832	−	0.606830i	$-0.792441\pi$
$353$		−	428.422i		−	1.21366i	0.794832	−	0.606830i	$-0.207559\pi$
$354$		0			0
$355$	187.341	−	187.341i	0.527721	−	0.527721i
$356$		0			0
$357$	172.833	−	336.297i	0.484127	−	0.942009i
$358$		0			0
$359$			5.80972i			0.0161831i	0.999967	+	0.00809154i	$0.00257564\pi$
$359$			5.80972i			0.0161831i	−0.999967	+	0.00809154i	$0.997424\pi$
$360$		0			0
$361$			306.508i			0.849052i
$362$		0			0
$363$	55.0947	+	55.0947i	0.151776	+	0.151776i
$364$		0			0
$365$	247.243	+	247.243i	0.677378	+	0.677378i
$366$		0			0
$367$		−	99.5445i		−	0.271238i	−0.990761	−	0.135619i	$-0.956698\pi$
$367$		−	99.5445i		−	0.271238i	0.990761	−	0.135619i	$-0.0433024\pi$
$368$		0			0
$369$			295.122i			0.799788i
$370$		0			0
$371$	−298.457	−	153.386i	−0.804467	−	0.413440i
$372$		0			0
$373$	239.494	−	239.494i	0.642076	−	0.642076i	−0.308990	−	0.951065i	$-0.599991\pi$
$373$	239.494	−	239.494i	0.642076	−	0.642076i	0.951065	+	0.308990i	$0.0999908\pi$
$374$		0			0
$375$			202.961i			0.541228i
$376$		0			0
$377$	1043.35			2.76751
$378$		0			0
$379$	120.835	−	120.835i	0.318825	−	0.318825i	−0.529491	−	0.848316i	$-0.677617\pi$
$379$	120.835	−	120.835i	0.318825	−	0.318825i	0.848316	+	0.529491i	$0.177617\pi$
$380$		0			0
$381$	106.541	−	106.541i	0.279636	−	0.279636i
$382$		0			0
$383$		−	715.292i		−	1.86760i	−0.357792	−	0.933801i	$-0.616470\pi$
$383$		−	715.292i		−	1.86760i	0.357792	−	0.933801i	$-0.383530\pi$
$384$		0			0
$385$	165.435	+	515.269i	0.429700	+	1.33836i
$386$		0			0
$387$	−288.389	−	288.389i	−0.745191	−	0.745191i
$388$		0			0
$389$	234.994	−	234.994i	0.604096	−	0.604096i	−0.337301	−	0.941397i	$-0.609514\pi$
$389$	234.994	−	234.994i	0.604096	−	0.604096i	0.941397	+	0.337301i	$0.109514\pi$
$390$		0			0
$391$	236.896			0.605873
$392$		0			0
$393$		−	831.653i		−	2.11616i
$394$		0			0
$395$	−170.492	−	170.492i	−0.431625	−	0.431625i
$396$		0			0
$397$	463.022	−	463.022i	1.16630	−	1.16630i	0.183234	−	0.983069i	$-0.441343\pi$
$397$	463.022	−	463.022i	1.16630	−	1.16630i	0.983069	−	0.183234i	$-0.0586565\pi$
$398$		0			0
$399$	230.670	−	74.0601i	0.578121	−	0.185614i
$400$		0			0
$401$	−379.791			−0.947111			−0.473555	−	0.880764i	$-0.657030\pi$
$401$	−379.791			−0.947111			−0.473555	+	0.880764i	$0.657030\pi$
$402$		0			0
$403$	−166.877	−	166.877i	−0.414088	−	0.414088i
$404$		0			0
$405$	136.592	+	136.592i	0.337265	+	0.337265i
$406$		0			0
$407$			92.8950i			0.228243i
$408$		0			0
$409$	−489.166			−1.19601			−0.598003	−	0.801494i	$-0.704039\pi$
$409$	−489.166			−1.19601			−0.598003	+	0.801494i	$0.704039\pi$
$410$		0			0
$411$	−176.311	−	176.311i	−0.428980	−	0.428980i
$412$		0			0
$413$	−9.86913	+	19.2032i	−0.0238962	+	0.0464969i
$414$		0			0
$415$	−841.983			−2.02888
$416$		0			0
$417$	755.191			1.81101
$418$		0			0
$419$	26.0850	−	26.0850i	0.0622553	−	0.0622553i	−0.675294	−	0.737549i	$-0.735983\pi$
$419$	26.0850	−	26.0850i	0.0622553	−	0.0622553i	0.737549	+	0.675294i	$0.235983\pi$
$420$		0			0
$421$	−355.644	+	355.644i	−0.844760	+	0.844760i	−0.989474	−	0.144714i	$-0.953774\pi$
$421$	−355.644	+	355.644i	−0.844760	+	0.844760i	0.144714	+	0.989474i	$0.453774\pi$
$422$		0			0
$423$	−975.108			−2.30522
$424$		0			0
$425$	212.347			0.499639
$426$		0			0
$427$	378.551	+	194.549i	0.886535	+	0.455617i
$428$		0			0
$429$	722.683	+	722.683i	1.68458	+	1.68458i
$430$		0			0
$431$	−90.4146			−0.209779			−0.104889	−	0.994484i	$-0.533449\pi$
$431$	−90.4146			−0.209779			−0.104889	+	0.994484i	$0.533449\pi$
$432$		0			0
$433$			359.503i			0.830262i	0.909762	+	0.415131i	$0.136264\pi$
$433$			359.503i			0.830262i	−0.909762	+	0.415131i	$0.863736\pi$
$434$		0			0
$435$	1226.74	+	1226.74i	2.82009	+	2.82009i
$436$		0			0
$437$	107.330	+	107.330i	0.245606	+	0.245606i
$438$		0			0
$439$	−427.544			−0.973904			−0.486952	−	0.873429i	$-0.661891\pi$
$439$	−427.544			−0.973904			−0.486952	+	0.873429i	$0.661891\pi$
$440$		0			0
$441$	370.289	+	517.214i	0.839657	+	1.17282i
$442$		0			0
$443$	216.567	−	216.567i	0.488865	−	0.488865i	−0.419083	−	0.907948i	$-0.637648\pi$
$443$	216.567	−	216.567i	0.488865	−	0.488865i	0.907948	+	0.419083i	$0.137648\pi$
$444$		0			0
$445$	−464.372	−	464.372i	−1.04353	−	1.04353i
$446$		0			0
$447$		−	44.8087i		−	0.100243i
$448$		0			0
$449$	−167.920			−0.373987			−0.186994	−	0.982361i	$-0.559874\pi$
$449$	−167.920			−0.373987			−0.186994	+	0.982361i	$0.559874\pi$
$450$		0			0
$451$	188.580	−	188.580i	0.418137	−	0.418137i
$452$		0			0
$453$	−549.341	−	549.341i	−1.21267	−	1.21267i
$454$		0			0
$455$	262.049	+	816.187i	0.575931	+	1.79382i
$456$		0			0
$457$			509.760i			1.11545i	0.830026	+	0.557725i	$0.188325\pi$
$457$			509.760i			1.11545i	−0.830026	+	0.557725i	$0.811675\pi$
$458$		0			0
$459$	152.079	−	152.079i	0.331326	−	0.331326i
$460$		0			0
$461$	−604.068	+	604.068i	−1.31034	+	1.31034i	−0.389181	+	0.921161i	$0.627242\pi$
$461$	−604.068	+	604.068i	−1.31034	+	1.31034i	−0.921161	+	0.389181i	$0.872758\pi$
$462$		0			0
$463$	248.442			0.536592			0.268296	−	0.963336i	$-0.413539\pi$
$463$	248.442			0.536592			0.268296	+	0.963336i	$0.413539\pi$
$464$		0			0
$465$		−	392.419i		−	0.843912i
$466$		0			0
$467$	241.814	−	241.814i	0.517802	−	0.517802i	−0.399103	−	0.916906i	$-0.630679\pi$
$467$	241.814	−	241.814i	0.517802	−	0.517802i	0.916906	+	0.399103i	$0.130679\pi$
$468$		0			0
$469$	9.11582	−	17.7375i	0.0194367	−	0.0378198i
$470$		0			0
$471$		−	548.594i		−	1.16474i
$472$		0			0
$473$			368.555i			0.779186i
$474$		0			0
$475$	96.2073	+	96.2073i	0.202542	+	0.202542i
$476$		0			0
$477$	−440.041	−	440.041i	−0.922519	−	0.922519i
$478$		0			0
$479$			747.728i			1.56102i	0.625144	+	0.780509i	$0.285040\pi$
$479$			747.728i			1.56102i	−0.625144	+	0.780509i	$0.714960\pi$
$480$		0			0
$481$			147.146i			0.305916i
$482$		0			0
$483$	−308.465	+	600.208i	−0.638645	+	1.24267i
$484$		0			0
$485$	−753.450	+	753.450i	−1.55351	+	1.55351i
$486$		0			0
$487$		−	479.744i		−	0.985100i	−0.870284	−	0.492550i	$-0.836065\pi$
$487$		−	479.744i		−	0.985100i	0.870284	−	0.492550i	$-0.163935\pi$
$488$		0			0
$489$	−1315.03			−2.68922
$490$		0			0
$491$	−515.730	+	515.730i	−1.05037	+	1.05037i	−0.0517031	+	0.998663i	$0.516465\pi$
$491$	−515.730	+	515.730i	−1.05037	+	1.05037i	−0.998663	+	0.0517031i	$0.983535\pi$
$492$		0			0
$493$	−457.416	+	457.416i	−0.927821	+	0.927821i
$494$		0			0
$495$			1003.62i			2.02752i
$496$		0			0
$497$	−267.942	+	86.0266i	−0.539118	+	0.173092i
$498$		0			0
$499$	−113.064	−	113.064i	−0.226581	−	0.226581i	0.584682	−	0.811263i	$-0.301219\pi$
$499$	−113.064	−	113.064i	−0.226581	−	0.226581i	−0.811263	+	0.584682i	$0.801219\pi$
$500$		0			0
$501$	776.760	−	776.760i	1.55042	−	1.55042i
$502$		0			0
$503$	−91.7304			−0.182367			−0.0911833	−	0.995834i	$-0.529065\pi$
$503$	−91.7304			−0.182367			−0.0911833	+	0.995834i	$0.529065\pi$
$504$		0			0
$505$		−	184.229i		−	0.364810i
$506$		0			0
$507$	584.455	+	584.455i	1.15277	+	1.15277i
$508$		0			0
$509$	282.406	−	282.406i	0.554826	−	0.554826i	−0.373004	−	0.927830i	$-0.621672\pi$
$509$	282.406	−	282.406i	0.554826	−	0.554826i	0.927830	+	0.373004i	$0.121672\pi$
$510$		0			0
$511$	−113.533	−	353.616i	−0.222179	−	0.692007i
$512$		0			0
$513$	137.804			0.268623
$514$		0			0
$515$	240.814	+	240.814i	0.467599	+	0.467599i
$516$		0			0
$517$	623.084	+	623.084i	1.20519	+	1.20519i
$518$		0			0
$519$		−	754.903i		−	1.45453i
$520$		0			0
$521$	258.579			0.496313			0.248156	−	0.968720i	$-0.420175\pi$
$521$	258.579			0.496313			0.248156	+	0.968720i	$0.420175\pi$
$522$		0			0
$523$	533.576	+	533.576i	1.02022	+	1.02022i	0.999791	+	0.0204296i	$0.00650339\pi$
$523$	533.576	+	533.576i	1.02022	+	1.02022i	0.0204296	+	0.999791i	$0.493497\pi$
$524$		0			0
$525$	−276.499	+	538.008i	−0.526664	+	1.02478i
$526$		0			0
$527$	146.322			0.277650
$528$		0			0
$529$	106.198			0.200752
$530$		0			0
$531$	−28.3130	+	28.3130i	−0.0533202	+	0.0533202i
$532$		0			0
$533$	298.711	−	298.711i	0.560432	−	0.560432i
$534$		0			0
$535$	−190.422			−0.355928
$536$		0			0
$537$	−362.757			−0.675525
$538$		0			0
$539$	93.8839	−	567.105i	0.174182	−	1.05214i
$540$		0			0
$541$	318.101	+	318.101i	0.587987	+	0.587987i	0.937086	−	0.349099i	$-0.113512\pi$
$541$	318.101	+	318.101i	0.587987	+	0.587987i	−0.349099	+	0.937086i	$0.613512\pi$
$542$		0			0
$543$	908.976			1.67399
$544$		0			0
$545$		−	800.084i		−	1.46804i
$546$		0			0
$547$	591.111	+	591.111i	1.08064	+	1.08064i	0.996450	+	0.0841918i	$0.0268308\pi$
$547$	591.111	+	591.111i	1.08064	+	1.08064i	0.0841918	+	0.996450i	$0.473169\pi$
$548$		0			0
$549$	558.130	+	558.130i	1.01663	+	1.01663i
$550$		0			0
$551$	−414.480			−0.752233
$552$		0			0
$553$	78.2896	+	243.844i	0.141572	+	0.440947i
$554$		0			0
$555$	−173.010	+	173.010i	−0.311729	+	0.311729i
$556$		0			0
$557$	337.829	+	337.829i	0.606516	+	0.606516i	0.942034	−	0.335518i	$-0.108911\pi$
$557$	337.829	+	337.829i	0.606516	+	0.606516i	−0.335518	+	0.942034i	$0.608911\pi$
$558$		0			0
$559$			583.791i			1.04435i
$560$		0			0
$561$	−633.663			−1.12952
$562$		0			0
$563$	531.679	−	531.679i	0.944368	−	0.944368i	−0.0541639	−	0.998532i	$-0.517249\pi$
$563$	531.679	−	531.679i	0.944368	−	0.944368i	0.998532	+	0.0541639i	$0.0172493\pi$
$564$		0			0
$565$	200.891	+	200.891i	0.355560	+	0.355560i
$566$		0			0
$567$	−62.7230	−	195.359i	−0.110623	−	0.344549i
$568$		0			0
$569$			464.888i			0.817026i	0.912753	+	0.408513i	$0.133952\pi$
$569$			464.888i			0.817026i	−0.912753	+	0.408513i	$0.866048\pi$
$570$		0			0
$571$	354.278	−	354.278i	0.620453	−	0.620453i	−0.325195	−	0.945647i	$-0.605430\pi$
$571$	354.278	−	354.278i	0.620453	−	0.620453i	0.945647	+	0.325195i	$0.105430\pi$
$572$		0			0
$573$	135.068	−	135.068i	0.235721	−	0.235721i
$574$		0			0
$575$	−378.987			−0.659108
$576$		0			0
$577$			477.529i			0.827606i	0.910366	+	0.413803i	$0.135800\pi$
$577$			477.529i			0.827606i	−0.910366	+	0.413803i	$0.864200\pi$
$578$		0			0
$579$	−271.215	+	271.215i	−0.468419	+	0.468419i
$580$		0			0
$581$	795.436	+	408.799i	1.36908	+	0.703613i
$582$		0			0
$583$			562.364i			0.964603i
$584$		0			0
$585$			1589.74i			2.71750i
$586$		0			0
$587$	−407.567	−	407.567i	−0.694322	−	0.694322i	0.268858	−	0.963180i	$-0.413354\pi$
$587$	−407.567	−	407.567i	−0.694322	−	0.694322i	−0.963180	+	0.268858i	$0.913354\pi$
$588$		0			0
$589$	66.2935	+	66.2935i	0.112553	+	0.112553i
$590$		0			0
$591$		−	1687.07i		−	2.85460i
$592$		0			0
$593$		−	293.102i		−	0.494269i	−0.968981	−	0.247135i	$-0.920511\pi$
$593$		−	293.102i		−	0.494269i	0.968981	−	0.247135i	$-0.0794890\pi$
$594$		0			0
$595$	−472.710	−	242.940i	−0.794470	−	0.408302i
$596$		0			0
$597$	669.570	−	669.570i	1.12156	−	1.12156i
$598$		0			0
$599$			29.9752i			0.0500420i	0.999687	+	0.0250210i	$0.00796526\pi$
$599$			29.9752i			0.0500420i	−0.999687	+	0.0250210i	$0.992035\pi$
$600$		0			0
$601$	−770.010			−1.28121			−0.640607	−	0.767869i	$-0.721317\pi$
$601$	−770.010			−1.28121			−0.640607	+	0.767869i	$0.721317\pi$
$602$		0			0
$603$	26.1519	−	26.1519i	0.0433697	−	0.0433697i
$604$		0			0
$605$	77.4428	−	77.4428i	0.128005	−	0.128005i
$606$		0			0
$607$		−	374.183i		−	0.616447i	−0.951314	−	0.308223i	$-0.900266\pi$
$607$		−	374.183i		−	0.616447i	0.951314	−	0.308223i	$-0.0997344\pi$
$608$		0			0
$609$	−563.317	−	1754.53i	−0.924987	−	2.88100i
$610$		0			0
$611$	986.965	+	986.965i	1.61533	+	1.61533i
$612$		0			0
$613$	−523.959	+	523.959i	−0.854746	+	0.854746i	−0.990713	−	0.135968i	$-0.956586\pi$
$613$	−523.959	+	523.959i	−0.854746	+	0.854746i	0.135968	+	0.990713i	$0.456586\pi$
$614$		0			0
$615$	702.430			1.14216
$616$		0			0
$617$		−	599.458i		−	0.971569i	−0.874079	−	0.485785i	$-0.838534\pi$
$617$		−	599.458i		−	0.971569i	0.874079	−	0.485785i	$-0.161466\pi$
$618$		0			0
$619$	−135.489	−	135.489i	−0.218883	−	0.218883i	0.589145	−	0.808028i	$-0.299465\pi$
$619$	−135.489	−	135.489i	−0.218883	−	0.218883i	−0.808028	+	0.589145i	$0.799465\pi$
$620$		0			0
$621$	−271.423	+	271.423i	−0.437074	+	0.437074i
$622$		0			0
$623$	213.239	+	664.161i	0.342277	+	1.06607i
$624$		0			0
$625$	746.070			1.19371
$626$		0			0
$627$	−287.092	−	287.092i	−0.457882	−	0.457882i
$628$		0			0
$629$	−64.5102	−	64.5102i	−0.102560	−	0.102560i
$630$		0			0
$631$			317.510i			0.503186i	0.967833	+	0.251593i	$0.0809544\pi$
$631$			317.510i			0.503186i	−0.967833	+	0.251593i	$0.919046\pi$
$632$		0			0
$633$	1074.15			1.69693
$634$		0			0
$635$	−149.758	−	149.758i	−0.235839	−	0.235839i
$636$		0			0
$637$	148.712	−	898.295i	0.233457	−	1.41020i
$638$		0			0
$639$	−521.887			−0.816724
$640$		0			0
$641$	−294.765			−0.459852			−0.229926	−	0.973208i	$-0.573848\pi$
$641$	−294.765			−0.459852			−0.229926	+	0.973208i	$0.573848\pi$
$642$		0			0
$643$	−596.748	+	596.748i	−0.928068	+	0.928068i	−0.997581	−	0.0695126i	$-0.977856\pi$
$643$	−596.748	+	596.748i	−0.928068	+	0.928068i	0.0695126	+	0.997581i	$0.477856\pi$
$644$		0			0
$645$	−686.404	+	686.404i	−1.06419	+	1.06419i
$646$		0			0
$647$	958.028			1.48072			0.740362	−	0.672208i	$-0.234654\pi$
$647$	958.028			1.48072			0.740362	+	0.672208i	$0.234654\pi$
$648$		0			0
$649$	36.1835			0.0557526
$650$		0			0
$651$	−190.527	+	370.725i	−0.292668	+	0.569470i
$652$		0			0
$653$	−418.776	−	418.776i	−0.641310	−	0.641310i	0.309567	−	0.950878i	$-0.399816\pi$
$653$	−418.776	−	418.776i	−0.641310	−	0.641310i	−0.950878	+	0.309567i	$0.899816\pi$
$654$		0			0
$655$	−1169.00			−1.78473
$656$		0			0
$657$		−	688.759i		−	1.04834i
$658$		0			0
$659$	359.378	+	359.378i	0.545338	+	0.545338i	0.925089	−	0.379750i	$-0.123990\pi$
$659$	359.378	+	359.378i	0.545338	+	0.545338i	−0.379750	+	0.925089i	$0.623990\pi$
$660$		0			0
$661$	−286.993	−	286.993i	−0.434181	−	0.434181i	0.455867	−	0.890048i	$-0.349329\pi$
$661$	−286.993	−	286.993i	−0.434181	−	0.434181i	−0.890048	+	0.455867i	$0.849329\pi$
$662$		0			0
$663$	−1003.72			−1.51391
$664$		0			0
$665$	−104.101	−	324.237i	−0.156543	−	0.487575i
$666$		0			0
$667$	816.375	−	816.375i	1.22395	−	1.22395i
$668$		0			0
$669$	55.8425	+	55.8425i	0.0834716	+	0.0834716i
$670$		0			0
$671$		−	713.279i		−	1.06301i
$672$		0			0
$673$	1166.76			1.73367			0.866837	−	0.498591i	$-0.166149\pi$
$673$	1166.76			1.73367			0.866837	+	0.498591i	$0.166149\pi$
$674$		0			0
$675$	−243.295	+	243.295i	−0.360437	+	0.360437i
$676$		0			0
$677$	−950.932	−	950.932i	−1.40463	−	1.40463i	−0.784509	−	0.620117i	$-0.787085\pi$
$677$	−950.932	−	950.932i	−1.40463	−	1.40463i	−0.620117	−	0.784509i	$-0.712915\pi$
$678$		0			0
$679$	1077.61	−	345.983i	1.58706	−	0.509548i
$680$		0			0
$681$			559.250i			0.821218i
$682$		0			0
$683$	−598.548	+	598.548i	−0.876351	+	0.876351i	−0.993155	−	0.116804i	$-0.962735\pi$
$683$	−598.548	+	598.548i	−0.876351	+	0.876351i	0.116804	+	0.993155i	$0.462735\pi$
$684$		0			0
$685$	−247.828	+	247.828i	−0.361793	+	0.361793i
$686$		0			0
$687$	−1019.04			−1.48332
$688$		0			0
$689$			890.785i			1.29287i
$690$		0			0
$691$	758.106	−	758.106i	1.09711	−	1.09711i	0.102368	−	0.994747i	$-0.467358\pi$
$691$	758.106	−	758.106i	1.09711	−	1.09711i	0.994747	−	0.102368i	$-0.0326419\pi$
$692$		0			0
$693$	487.277	−	948.138i	0.703141	−	1.36816i
$694$		0			0
$695$		−	1061.52i		−	1.52737i
$696$		0			0
$697$			261.916i			0.375776i
$698$		0			0
$699$	−890.313	−	890.313i	−1.27370	−	1.27370i
$700$		0			0
$701$	−149.764	−	149.764i	−0.213643	−	0.213643i	0.592170	−	0.805813i	$-0.298271\pi$
$701$	−149.764	−	149.764i	−0.213643	−	0.213643i	−0.805813	+	0.592170i	$0.798271\pi$
$702$		0			0
$703$		−	58.4549i		−	0.0831507i
$704$		0			0
$705$			2320.89i			3.29204i
$706$		0			0
$707$	−89.4467	+	174.044i	−0.126516	+	0.246173i
$708$		0			0
$709$	938.869	−	938.869i	1.32422	−	1.32422i	0.413888	−	0.910328i	$-0.364170\pi$
$709$	938.869	−	938.869i	1.32422	−	1.32422i	0.910328	−	0.413888i	$-0.135830\pi$
$710$		0			0
$711$			474.949i			0.668002i
$712$		0			0
$713$	−261.148			−0.366267
$714$		0			0
$715$	1015.83	−	1015.83i	1.42073	−	1.42073i
$716$		0			0
$717$	741.961	−	741.961i	1.03481	−	1.03481i
$718$		0			0
$719$			524.039i			0.728844i	0.931234	+	0.364422i	$0.118733\pi$
$719$			524.039i			0.728844i	−0.931234	+	0.364422i	$0.881267\pi$
$720$		0			0
$721$	−110.581	−	344.420i	−0.153372	−	0.477698i
$722$		0			0
$723$	342.637	+	342.637i	0.473910	+	0.473910i
$724$		0			0
$725$	731.774	−	731.774i	1.00934	−	1.00934i
$726$		0			0
$727$	−550.877			−0.757739			−0.378870	−	0.925450i	$-0.623687\pi$
$727$	−550.877			−0.757739			−0.378870	+	0.925450i	$0.623687\pi$
$728$		0			0
$729$		−	1168.22i		−	1.60250i
$730$		0			0
$731$	−255.940	−	255.940i	−0.350123	−	0.350123i
$732$		0			0
$733$	106.728	−	106.728i	0.145604	−	0.145604i	−0.630547	−	0.776151i	$-0.717169\pi$
$733$	106.728	−	106.728i	0.145604	−	0.145604i	0.776151	+	0.630547i	$0.217169\pi$
$734$		0			0
$735$	1231.04	−	881.337i	1.67488	−	1.19910i
$736$		0			0
$737$	−33.4216			−0.0453482
$738$		0			0
$739$	−788.242	−	788.242i	−1.06663	−	1.06663i	−0.997615	−	0.0690175i	$-0.978014\pi$
$739$	−788.242	−	788.242i	−1.06663	−	1.06663i	−0.0690175	−	0.997615i	$-0.521986\pi$
$740$		0			0
$741$	−454.754	−	454.754i	−0.613703	−	0.613703i
$742$		0			0
$743$			6.36550i			0.00856729i	0.999991	+	0.00428365i	$0.00136353\pi$
$743$			6.36550i			0.00856729i	−0.999991	+	0.00428365i	$0.998636\pi$
$744$		0			0
$745$	−62.9845			−0.0845430
$746$		0			0
$747$	1172.78	+	1172.78i	1.56999	+	1.56999i
$748$		0			0
$749$	179.895	+	92.4533i	0.240180	+	0.123436i
$750$		0			0
$751$	−738.817			−0.983778			−0.491889	−	0.870658i	$-0.663693\pi$
$751$	−738.817			−0.983778			−0.491889	+	0.870658i	$0.663693\pi$
$752$		0			0
$753$	−2204.34			−2.92741
$754$		0			0
$755$	−772.171	+	772.171i	−1.02274	+	1.02274i
$756$		0			0
$757$	−538.649	+	538.649i	−0.711557	+	0.711557i	−0.966861	−	0.255304i	$-0.917824\pi$
$757$	−538.649	+	538.649i	−0.711557	+	0.711557i	0.255304	+	0.966861i	$0.417824\pi$
$758$		0			0
$759$	1130.93			1.49003
$760$		0			0
$761$	−304.961			−0.400737			−0.200369	−	0.979721i	$-0.564214\pi$
$761$	−304.961			−0.400737			−0.200369	+	0.979721i	$0.564214\pi$
$762$		0			0
$763$	−388.456	+	755.853i	−0.509116	+	0.990633i
$764$		0			0
$765$	−696.957	−	696.957i	−0.911055	−	0.911055i
$766$		0			0
$767$	57.3146			0.0747257
$768$		0			0
$769$			41.4537i			0.0539059i	0.999637	+	0.0269530i	$0.00858043\pi$
$769$			41.4537i			0.0539059i	−0.999637	+	0.0269530i	$0.991420\pi$
$770$		0			0
$771$	−299.715	−	299.715i	−0.388735	−	0.388735i
$772$		0			0
$773$	−500.899	−	500.899i	−0.647993	−	0.647993i	0.304514	−	0.952508i	$-0.401506\pi$
$773$	−500.899	−	500.899i	−0.647993	−	0.647993i	−0.952508	+	0.304514i	$0.901506\pi$
$774$		0			0
$775$	−234.085			−0.302046
$776$		0			0
$777$	247.444	−	79.4457i	0.318461	−	0.102247i
$778$		0			0
$779$	−118.665	+	118.665i	−0.152330	+	0.152330i
$780$		0			0
$781$	333.480	+	333.480i	0.426991	+	0.426991i
$782$		0			0
$783$		−	1048.16i		−	1.33865i
$784$		0			0
$785$	−771.120			−0.982318
$786$		0			0
$787$	−328.763	+	328.763i	−0.417742	+	0.417742i	−0.884425	−	0.466683i	$-0.845449\pi$
$787$	−328.763	+	328.763i	−0.417742	+	0.417742i	0.466683	+	0.884425i	$0.345449\pi$
$788$		0			0
$789$	654.001	+	654.001i	0.828898	+	0.828898i
$790$		0			0
$791$	−92.2489	−	287.322i	−0.116623	−	0.363239i
$792$		0			0
$793$		−	1129.83i		−	1.42476i
$794$		0			0
$795$	−1047.36	+	1047.36i	−1.31743	+	1.31743i
$796$		0			0
$797$	−350.489	+	350.489i	−0.439760	+	0.439760i	−0.891931	−	0.452171i	$-0.850650\pi$
$797$	−350.489	+	350.489i	−0.439760	+	0.439760i	0.452171	+	0.891931i	$0.350650\pi$
$798$		0			0
$799$	−865.392			−1.08309
$800$		0			0
$801$			1293.63i			1.61502i
$802$		0			0
$803$	−440.110	+	440.110i	−0.548082	+	0.548082i
$804$		0			0
$805$	843.671	+	433.588i	1.04804	+	0.538619i
$806$		0			0
$807$		−	497.838i		−	0.616900i
$808$		0			0
$809$		−	499.525i		−	0.617460i	−0.951150	−	0.308730i	$-0.900096\pi$
$809$		−	499.525i		−	0.617460i	0.951150	−	0.308730i	$-0.0999039\pi$
$810$		0			0
$811$	−458.310	−	458.310i	−0.565117	−	0.565117i	0.365639	−	0.930757i	$-0.380850\pi$
$811$	−458.310	−	458.310i	−0.565117	−	0.565117i	−0.930757	+	0.365639i	$0.880850\pi$
$812$		0			0
$813$	521.761	+	521.761i	0.641772	+	0.641772i
$814$		0			0
$815$			1848.44i			2.26803i
$816$		0			0
$817$		−	231.916i		−	0.283863i
$818$		0			0
$819$	771.847	−	1501.85i	0.942426	−	1.83376i
$820$		0			0
$821$	−799.288	+	799.288i	−0.973554	+	0.973554i	−0.999659	−	0.0261048i	$-0.991690\pi$
$821$	−799.288	+	799.288i	−0.973554	+	0.973554i	0.0261048	+	0.999659i	$0.491690\pi$
$822$		0			0
$823$			1017.99i			1.23693i	0.785812	+	0.618465i	$0.212245\pi$
$823$			1017.99i			1.23693i	−0.785812	+	0.618465i	$0.787755\pi$
$824$		0			0
$825$	1013.73			1.22877
$826$		0			0
$827$	−135.830	+	135.830i	−0.164245	+	0.164245i	−0.784444	−	0.620199i	$-0.787052\pi$
$827$	−135.830	+	135.830i	−0.164245	+	0.164245i	0.620199	+	0.784444i	$0.287052\pi$
$828$		0			0
$829$	240.155	−	240.155i	0.289693	−	0.289693i	−0.547266	−	0.836959i	$-0.684331\pi$
$829$	240.155	−	240.155i	0.289693	−	0.289693i	0.836959	+	0.547266i	$0.184331\pi$
$830$		0			0
$831$			714.195i			0.859441i
$832$		0			0
$833$	328.625	+	459.019i	0.394508	+	0.551043i
$834$		0			0
$835$	−1091.84	−	1091.84i	−1.30759	−	1.30759i
$836$		0			0
$837$	−167.647	+	167.647i	−0.200295	+	0.200295i
$838$		0			0
$839$	28.3581			0.0337999			0.0168999	−	0.999857i	$-0.494620\pi$
$839$	28.3581			0.0337999			0.0168999	+	0.999857i	$0.494620\pi$
$840$		0			0
$841$			2311.63i			2.74866i
$842$		0			0
$843$	468.898	+	468.898i	0.556225	+	0.556225i
$844$		0			0
$845$	821.527	−	821.527i	0.972222	−	0.972222i
$846$		0			0
$847$	−110.762	+	35.5616i	−0.130769	+	0.0419854i
$848$		0			0
$849$	−178.513			−0.210263
$850$		0			0
$851$	115.135	+	115.135i	0.135294	+	0.135294i
$852$		0			0
$853$	−351.304	−	351.304i	−0.411846	−	0.411846i	0.470535	−	0.882381i	$-0.344061\pi$
$853$	−351.304	−	351.304i	−0.411846	−	0.411846i	−0.882381	+	0.470535i	$0.844061\pi$
$854$		0			0
$855$		−	631.537i		−	0.738640i
$856$		0			0
$857$	−847.908			−0.989391			−0.494696	−	0.869066i	$-0.664720\pi$
$857$	−847.908			−0.989391			−0.494696	+	0.869066i	$0.664720\pi$
$858$		0			0
$859$	−578.524	−	578.524i	−0.673486	−	0.673486i	0.285032	−	0.958518i	$-0.407996\pi$
$859$	−578.524	−	578.524i	−0.673486	−	0.673486i	−0.958518	+	0.285032i	$0.907996\pi$
$860$		0			0
$861$	−663.597	−	341.043i	−0.770729	−	0.396101i
$862$		0			0
$863$	1082.66			1.25453			0.627266	−	0.778805i	$-0.284174\pi$
$863$	1082.66			1.25453			0.627266	+	0.778805i	$0.284174\pi$
$864$		0			0
$865$	−1061.11			−1.22672
$866$		0			0
$867$	−518.062	+	518.062i	−0.597534	+	0.597534i
$868$		0			0
$869$	303.488	−	303.488i	0.349238	−	0.349238i
$870$		0			0
$871$	−52.9398			−0.0607805
$872$		0			0
$873$	2098.93			2.40427
$874$		0			0
$875$	−269.516	−	138.513i	−0.308018	−	0.158300i
$876$		0			0
$877$	−378.296	−	378.296i	−0.431353	−	0.431353i	0.457736	−	0.889088i	$-0.348661\pi$
$877$	−378.296	−	378.296i	−0.431353	−	0.431353i	−0.889088	+	0.457736i	$0.848661\pi$
$878$		0			0
$879$	−212.498			−0.241750
$880$		0			0
$881$			348.803i			0.395917i	0.980210	+	0.197958i	$0.0634311\pi$
$881$			348.803i			0.395917i	−0.980210	+	0.197958i	$0.936569\pi$
$882$		0			0
$883$	93.9565	+	93.9565i	0.106406	+	0.106406i	0.758305	−	0.651899i	$-0.226028\pi$
$883$	93.9565	+	93.9565i	0.106406	+	0.106406i	−0.651899	+	0.758305i	$0.726028\pi$
$884$		0			0
$885$	67.3888	+	67.3888i	0.0761456	+	0.0761456i
$886$		0			0
$887$	−1418.82			−1.59957			−0.799784	−	0.600288i	$-0.795053\pi$
$887$	−1418.82			−1.59957			−0.799784	+	0.600288i	$0.795053\pi$
$888$		0			0
$889$	68.7685	+	214.189i	0.0773549	+	0.240932i
$890$		0			0
$891$	−243.144	+	243.144i	−0.272889	+	0.272889i
$892$		0			0
$893$	−392.081	−	392.081i	−0.439060	−	0.439060i
$894$		0			0
$895$			509.902i			0.569723i
$896$		0			0
$897$	1791.40			1.99710
$898$		0			0
$899$	504.243	−	504.243i	0.560893	−	0.560893i
$900$		0			0
$901$	−390.529	−	390.529i	−0.433440	−	0.433440i
$902$		0			0
$903$	981.720	−	315.196i	1.08718	−	0.349054i
$904$		0			0
$905$		−	1277.69i		−	1.41181i
$906$		0			0
$907$	1066.03	−	1066.03i	1.17533	−	1.17533i	0.194414	−	0.980920i	$-0.437720\pi$
$907$	1066.03	−	1066.03i	1.17533	−	1.17533i	0.980920	−	0.194414i	$-0.0622804\pi$
$908$		0			0
$909$	−256.609	+	256.609i	−0.282298	+	0.282298i
$910$		0			0
$911$	−526.652			−0.578103			−0.289051	−	0.957314i	$-0.593340\pi$
$911$	−526.652			−0.578103			−0.289051	+	0.957314i	$0.593340\pi$
$912$		0			0
$913$		−	1498.79i		−	1.64161i
$914$		0			0
$915$	1328.43	−	1328.43i	1.45183	−	1.45183i
$916$		0			0
$917$	1104.37	+	567.570i	1.20433	+	0.618942i
$918$		0			0
$919$			306.233i			0.333224i	0.986023	+	0.166612i	$0.0532828\pi$
$919$			306.233i			0.333224i	−0.986023	+	0.166612i	$0.946717\pi$
$920$		0			0
$921$		−	599.404i		−	0.650819i
$922$		0			0
$923$	528.233	+	528.233i	0.572300	+	0.572300i
$924$		0			0
$925$	103.203	+	103.203i	0.111571	+	0.111571i
$926$		0			0
$927$		−	670.849i		−	0.723677i
$928$		0			0
$929$		−	644.517i		−	0.693776i	−0.937907	−	0.346888i	$-0.887238\pi$
$929$		−	644.517i		−	0.693776i	0.937907	−	0.346888i	$-0.112762\pi$
$930$		0			0
$931$	−59.0772	+	356.855i	−0.0634556	+	0.383303i
$932$		0			0
$933$	−279.539	+	279.539i	−0.299613	+	0.299613i
$934$		0			0
$935$			890.697i			0.952617i
$936$		0			0
$937$	−707.989			−0.755591			−0.377796	−	0.925889i	$-0.623318\pi$
$937$	−707.989			−0.755591			−0.377796	+	0.925889i	$0.623318\pi$
$938$		0			0
$939$	−572.132	+	572.132i	−0.609299	+	0.609299i
$940$		0			0
$941$	895.058	−	895.058i	0.951177	−	0.951177i	−0.0476851	−	0.998862i	$-0.515184\pi$
$941$	895.058	−	895.058i	0.951177	−	0.951177i	0.998862	+	0.0476851i	$0.0151844\pi$
$942$		0			0
$943$		−	467.455i		−	0.495711i
$944$		0			0
$945$	819.952	−	263.258i	0.867674	−	0.278579i
$946$		0			0
$947$	−622.940	−	622.940i	−0.657804	−	0.657804i	0.297056	−	0.954860i	$-0.403995\pi$
$947$	−622.940	−	622.940i	−0.657804	−	0.657804i	−0.954860	+	0.297056i	$0.903995\pi$
$948$		0			0
$949$	−697.134	+	697.134i	−0.734599	+	0.734599i
$950$		0			0
$951$	396.956			0.417409
$952$		0			0
$953$		−	1574.91i		−	1.65258i	−0.563243	−	0.826291i	$-0.690447\pi$
$953$		−	1574.91i		−	1.65258i	0.563243	−	0.826291i	$-0.309553\pi$
$954$		0			0
$955$	−189.856	−	189.856i	−0.198802	−	0.198802i
$956$		0			0
$957$	−2183.68	+	2183.68i	−2.28180	+	2.28180i
$958$		0			0
$959$	354.452	−	113.802i	0.369606	−	0.118668i
$960$		0			0
$961$	799.699			0.832153
$962$		0			0
$963$	265.234	+	265.234i	0.275425	+	0.275425i
$964$		0			0
$965$	381.227	+	381.227i	0.395054	+	0.395054i
$966$		0			0
$967$			114.443i			0.118348i	0.998248	+	0.0591741i	$0.0188467\pi$
$967$			114.443i			0.118348i	−0.998248	+	0.0591741i	$0.981153\pi$
$968$		0			0
$969$	398.738			0.411494
$970$		0			0
$971$	693.409	+	693.409i	0.714118	+	0.714118i	0.967394	−	0.253276i	$-0.0815081\pi$
$971$	693.409	+	693.409i	0.714118	+	0.714118i	−0.253276	+	0.967394i	$0.581508\pi$
$972$		0			0
$973$	−515.388	+	1002.84i	−0.529689	+	1.03066i
$974$		0			0
$975$	1605.76			1.64693
$976$		0			0
$977$	1198.67			1.22689			0.613446	−	0.789736i	$-0.289783\pi$
$977$	1198.67			1.22689			0.613446	+	0.789736i	$0.289783\pi$
$978$		0			0
$979$	826.615	−	826.615i	0.844346	−	0.844346i
$980$		0			0
$981$	−1114.42	+	1114.42i	−1.13600	+	1.13600i
$982$		0			0
$983$	381.555			0.388153			0.194077	−	0.980986i	$-0.437829\pi$
$983$	381.555			0.388153			0.194077	+	0.980986i	$0.437829\pi$
$984$		0			0
$985$	−2371.39			−2.40751
$986$		0			0
$987$	1126.84	−	2192.58i	1.14168	−	2.22146i
$988$		0			0
$989$	456.791	+	456.791i	0.461871	+	0.461871i
$990$		0			0
$991$	472.063			0.476350			0.238175	−	0.971222i	$-0.423451\pi$
$991$	472.063			0.476350			0.238175	+	0.971222i	$0.423451\pi$
$992$		0			0
$993$		−	1237.08i		−	1.24581i
$994$		0			0
$995$	−941.168	−	941.168i	−0.945898	−	0.945898i
$996$		0			0
$997$	−1090.50	−	1090.50i	−1.09378	−	1.09378i	−0.995121	−	0.0986584i	$-0.968545\pi$
$997$	−1090.50	−	1090.50i	−1.09378	−	1.09378i	−0.0986584	−	0.995121i	$-0.531455\pi$
$998$		0			0
$999$	147.825			0.147973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.4		56
4.3	odd	2		112.3.l.b.13.2	yes	56
7.6	odd	2	inner	448.3.l.b.209.25		56
16.5	even	4	inner	448.3.l.b.433.25		56
16.11	odd	4		112.3.l.b.69.1	yes	56
28.27	even	2		112.3.l.b.13.1	✓	56
112.27	even	4		112.3.l.b.69.2	yes	56
112.69	odd	4	inner	448.3.l.b.433.4		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.1	✓	56	28.27	even	2
112.3.l.b.13.2	yes	56	4.3	odd	2
112.3.l.b.69.1	yes	56	16.11	odd	4
112.3.l.b.69.2	yes	56	112.27	even	4
448.3.l.b.209.4		56	1.1	even	1	trivial
448.3.l.b.209.25		56	7.6	odd	2	inner
448.3.l.b.433.4		56	112.69	odd	4	inner
448.3.l.b.433.25		56	16.5	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.31524 + 3.31524i) q^{3} +(4.66000 + 4.66000i) q^{5} +(-2.13987 - 6.66491i) q^{7} -12.9816i q^{9} +O(q^{10})\)	\(q+(-3.31524 + 3.31524i) q^{3} +(4.66000 + 4.66000i) q^{5} +(-2.13987 - 6.66491i) q^{7} -12.9816i q^{9} +(-8.29514 + 8.29514i) q^{11} +(-13.1395 + 13.1395i) q^{13} -30.8981 q^{15} -11.5210i q^{17} +(5.21978 - 5.21978i) q^{19} +(29.1899 + 15.0016i) q^{21} +20.5622i q^{23} +18.4313i q^{25} +(13.2001 + 13.2001i) q^{27} +(-39.7028 - 39.7028i) q^{29} +12.7004i q^{31} -55.0008i q^{33} +(21.0867 - 41.0303i) q^{35} +(5.59936 - 5.59936i) q^{37} -87.1213i q^{39} -22.7338 q^{41} +(22.2151 - 22.2151i) q^{43} +(60.4945 - 60.4945i) q^{45} -75.1143i q^{47} +(-39.8419 + 28.5240i) q^{49} +(38.1949 + 38.1949i) q^{51} +(33.8972 - 33.8972i) q^{53} -77.3108 q^{55} +34.6097i q^{57} +(-2.18100 - 2.18100i) q^{59} +(-42.9938 + 42.9938i) q^{61} +(-86.5215 + 27.7790i) q^{63} -122.460 q^{65} +(2.01453 + 2.01453i) q^{67} +(-68.1685 - 68.1685i) q^{69} -40.2019i q^{71} +53.0564 q^{73} +(-61.1042 - 61.1042i) q^{75} +(73.0368 + 37.5358i) q^{77} -36.5862 q^{79} +29.3116 q^{81} +(-90.3415 + 90.3415i) q^{83} +(53.6879 - 53.6879i) q^{85} +263.249 q^{87} -99.6505 q^{89} +(115.690 + 59.4568i) q^{91} +(-42.1050 - 42.1050i) q^{93} +48.6484 q^{95} +161.684i q^{97} +(107.685 + 107.685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

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Modular forms

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