LMFDB - Embedded newform 2208.2.f.b.1105.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2208,2,Mod(1105,2208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2208.1105");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1105.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2208.1105
Dual form			2208.2.f.b.1105.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.41421i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.41421i q^{5} +2.00000 q^{7} -1.00000 q^{9} +4.24264i q^{11} +6.82843i q^{13} +3.41421 q^{15} -1.17157 q^{17} +2.24264i q^{19} +2.00000i q^{21} +1.00000 q^{23} -6.65685 q^{25} -1.00000i q^{27} -2.82843i q^{29} -10.8284 q^{31} -4.24264 q^{33} -6.82843i q^{35} +3.07107i q^{37} -6.82843 q^{39} +8.48528 q^{41} -1.75736i q^{43} +3.41421i q^{45} -6.00000 q^{47} -3.00000 q^{49} -1.17157i q^{51} +11.4142i q^{53} +14.4853 q^{55} -2.24264 q^{57} +5.65685i q^{59} +3.75736i q^{61} -2.00000 q^{63} +23.3137 q^{65} +9.07107i q^{67} +1.00000i q^{69} +11.6569 q^{71} -6.65685i q^{75} +8.48528i q^{77} -9.31371 q^{79} +1.00000 q^{81} -4.24264i q^{83} +4.00000i q^{85} +2.82843 q^{87} +5.17157 q^{89} +13.6569i q^{91} -10.8284i q^{93} +7.65685 q^{95} +3.65685 q^{97} -4.24264i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 4 * q^9	$ 4 q + 8 q^{7} - 4 q^{9} + 8 q^{15} - 16 q^{17} + 4 q^{23} - 4 q^{25} - 32 q^{31} - 16 q^{39} - 24 q^{47} - 12 q^{49} + 24 q^{55} + 8 q^{57} - 8 q^{63} + 48 q^{65} + 24 q^{71} + 8 q^{79} + 4 q^{81} + 32 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 - 4 * q^9 + 8 * q^15 - 16 * q^17 + 4 * q^23 - 4 * q^25 - 32 * q^31 - 16 * q^39 - 24 * q^47 - 12 * q^49 + 24 * q^55 + 8 * q^57 - 8 * q^63 + 48 * q^65 + 24 * q^71 + 8 * q^79 + 4 * q^81 + 32 * q^89 + 8 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 3.41421i	− 1.52688i	−0.645877	−	0.763441i	$-0.723508\pi$
$5$	− 3.41421i	− 1.52688i	0.645877	−	0.763441i	$-0.276492\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	6.82843i	1.89386i	0.321433	+	0.946932i	$0.395836\pi$
$13$	6.82843i	1.89386i	−0.321433	+	0.946932i	$0.604164\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	2.24264i	0.514497i	0.966345	+	0.257249i	$0.0828159\pi$
$19$	2.24264i	0.514497i	−0.966345	+	0.257249i	$0.917184\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−6.65685	−1.33137
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 2.82843i	− 0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	− 2.82843i	− 0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$	0	0
$31$	−10.8284	−1.94484	−0.972421	−	0.233231i	$-0.925070\pi$
$31$	−10.8284	−1.94484	−0.972421	+	0.233231i	$0.925070\pi$
$32$	0	0
$33$	−4.24264	−0.738549
$34$	0	0
$35$	− 6.82843i	− 1.15421i
$36$	0	0
$37$	3.07107i	0.504880i	0.967612	+	0.252440i	$0.0812331\pi$
$37$	3.07107i	0.504880i	−0.967612	+	0.252440i	$0.918767\pi$
$38$	0	0
$39$	−6.82843	−1.09342
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	− 1.75736i	− 0.267995i	−0.990982	−	0.133997i	$-0.957219\pi$
$43$	− 1.75736i	− 0.267995i	0.990982	−	0.133997i	$-0.0427814\pi$
$44$	0	0
$45$	3.41421i	0.508961i
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 1.17157i	− 0.164053i
$52$	0	0
$53$	11.4142i	1.56786i	0.620847	+	0.783931i	$0.286789\pi$
$53$	11.4142i	1.56786i	−0.620847	+	0.783931i	$0.713211\pi$
$54$	0	0
$55$	14.4853	1.95319
$56$	0	0
$57$	−2.24264	−0.297045
$58$	0	0
$59$	5.65685i	0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$	5.65685i	0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$	0	0
$61$	3.75736i	0.481081i	0.970639	+	0.240540i	$0.0773246\pi$
$61$	3.75736i	0.481081i	−0.970639	+	0.240540i	$0.922675\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	23.3137	2.89171
$66$	0	0
$67$	9.07107i	1.10821i	0.832448	+	0.554104i	$0.186939\pi$
$67$	9.07107i	1.10821i	−0.832448	+	0.554104i	$0.813061\pi$
$68$	0	0
$69$	1.00000i	0.120386i
$70$	0	0
$71$	11.6569	1.38341	0.691707	−	0.722178i	$-0.256859\pi$
$71$	11.6569	1.38341	0.691707	+	0.722178i	$0.256859\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	− 6.65685i	− 0.768667i
$76$	0	0
$77$	8.48528i	0.966988i
$78$	0	0
$79$	−9.31371	−1.04787	−0.523937	−	0.851757i	$-0.675537\pi$
$79$	−9.31371	−1.04787	−0.523937	+	0.851757i	$0.675537\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.24264i	− 0.465690i	−0.972514	−	0.232845i	$-0.925196\pi$
$83$	− 4.24264i	− 0.465690i	0.972514	−	0.232845i	$-0.0748035\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	5.17157	0.548186	0.274093	−	0.961703i	$-0.411622\pi$
$89$	5.17157	0.548186	0.274093	+	0.961703i	$0.411622\pi$
$90$	0	0
$91$	13.6569i	1.43163i
$92$	0	0
$93$	− 10.8284i	− 1.12286i
$94$	0	0
$95$	7.65685	0.785577
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0	0
$99$	− 4.24264i	− 0.426401i
$100$	0	0
$101$	10.1421i	1.00918i	0.863359	+	0.504590i	$0.168356\pi$
$101$	10.1421i	1.00918i	−0.863359	+	0.504590i	$0.831644\pi$
$102$	0	0
$103$	19.6569	1.93685	0.968424	−	0.249310i	$-0.0802038\pi$
$103$	19.6569	1.93685	0.968424	+	0.249310i	$0.0802038\pi$
$104$	0	0
$105$	6.82843	0.666386
$106$	0	0
$107$	− 2.58579i	− 0.249977i	−0.992158	−	0.124989i	$-0.960111\pi$
$107$	− 2.58579i	− 0.249977i	0.992158	−	0.124989i	$-0.0398895\pi$
$108$	0	0
$109$	12.7279i	1.21911i	0.792742	+	0.609557i	$0.208653\pi$
$109$	12.7279i	1.21911i	−0.792742	+	0.609557i	$0.791347\pi$
$110$	0	0
$111$	−3.07107	−0.291493
$112$	0	0
$113$	−18.8284	−1.77123	−0.885615	−	0.464421i	$-0.846263\pi$
$113$	−18.8284	−1.77123	−0.885615	+	0.464421i	$0.846263\pi$
$114$	0	0
$115$	− 3.41421i	− 0.318377i
$116$	0	0
$117$	− 6.82843i	− 0.631288i
$118$	0	0
$119$	−2.34315	−0.214796
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.48528i	0.765092i
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	13.3137	1.18140	0.590700	−	0.806891i	$-0.298852\pi$
$127$	13.3137	1.18140	0.590700	+	0.806891i	$0.298852\pi$
$128$	0	0
$129$	1.75736	0.154727
$130$	0	0
$131$	− 14.8284i	− 1.29557i	−0.761825	−	0.647783i	$-0.775696\pi$
$131$	− 14.8284i	− 1.29557i	0.761825	−	0.647783i	$-0.224304\pi$
$132$	0	0
$133$	4.48528i	0.388923i
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	14.8284	1.26688	0.633439	−	0.773793i	$-0.281643\pi$
$137$	14.8284	1.26688	0.633439	+	0.773793i	$0.281643\pi$
$138$	0	0
$139$	− 3.31371i	− 0.281065i	−0.990076	−	0.140533i	$-0.955119\pi$
$139$	− 3.31371i	− 0.281065i	0.990076	−	0.140533i	$-0.0448815\pi$
$140$	0	0
$141$	− 6.00000i	− 0.505291i
$142$	0	0
$143$	−28.9706	−2.42264
$144$	0	0
$145$	−9.65685	−0.801958
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	− 13.0711i	− 1.07082i	−0.844591	−	0.535412i	$-0.820156\pi$
$149$	− 13.0711i	− 1.07082i	0.844591	−	0.535412i	$-0.179844\pi$
$150$	0	0
$151$	3.51472	0.286024	0.143012	−	0.989721i	$-0.454321\pi$
$151$	3.51472	0.286024	0.143012	+	0.989721i	$0.454321\pi$
$152$	0	0
$153$	1.17157	0.0947161
$154$	0	0
$155$	36.9706i	2.96955i
$156$	0	0
$157$	16.7279i	1.33503i	0.744595	+	0.667517i	$0.232643\pi$
$157$	16.7279i	1.33503i	−0.744595	+	0.667517i	$0.767357\pi$
$158$	0	0
$159$	−11.4142	−0.905206
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	4.48528i	0.351314i	0.984451	+	0.175657i	$0.0562050\pi$
$163$	4.48528i	0.351314i	−0.984451	+	0.175657i	$0.943795\pi$
$164$	0	0
$165$	14.4853i	1.12768i
$166$	0	0
$167$	7.65685	0.592505	0.296253	−	0.955110i	$-0.404263\pi$
$167$	7.65685	0.592505	0.296253	+	0.955110i	$0.404263\pi$
$168$	0	0
$169$	−33.6274	−2.58672
$170$	0	0
$171$	− 2.24264i	− 0.171499i
$172$	0	0
$173$	18.8284i	1.43150i	0.698357	+	0.715749i	$0.253915\pi$
$173$	18.8284i	1.43150i	−0.698357	+	0.715749i	$0.746085\pi$
$174$	0	0
$175$	−13.3137	−1.00642
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	8.48528i	0.634220i	0.948389	+	0.317110i	$0.102712\pi$
$179$	8.48528i	0.634220i	−0.948389	+	0.317110i	$0.897288\pi$
$180$	0	0
$181$	− 12.2426i	− 0.909988i	−0.890494	−	0.454994i	$-0.849641\pi$
$181$	− 12.2426i	− 0.909988i	0.890494	−	0.454994i	$-0.150359\pi$
$182$	0	0
$183$	−3.75736	−0.277752
$184$	0	0
$185$	10.4853	0.770893
$186$	0	0
$187$	− 4.97056i	− 0.363484i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	9.65685	0.698745	0.349373	−	0.936984i	$-0.386395\pi$
$191$	9.65685	0.698745	0.349373	+	0.936984i	$0.386395\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	23.3137i	1.66953i
$196$	0	0
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	−10.9706	−0.777683	−0.388841	−	0.921305i	$-0.627125\pi$
$199$	−10.9706	−0.777683	−0.388841	+	0.921305i	$0.627125\pi$
$200$	0	0
$201$	−9.07107	−0.639824
$202$	0	0
$203$	− 5.65685i	− 0.397033i
$204$	0	0
$205$	− 28.9706i	− 2.02339i
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−9.51472	−0.658147
$210$	0	0
$211$	0.485281i	0.0334081i	0.999860	+	0.0167041i	$0.00531732\pi$
$211$	0.485281i	0.0334081i	−0.999860	+	0.0167041i	$0.994683\pi$
$212$	0	0
$213$	11.6569i	0.798714i
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−21.6569	−1.47016
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	−6.82843	−0.457265	−0.228633	−	0.973513i	$-0.573425\pi$
$223$	−6.82843	−0.457265	−0.228633	+	0.973513i	$0.573425\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	16.2426i	1.07806i	0.842286	+	0.539031i	$0.181209\pi$
$227$	16.2426i	1.07806i	−0.842286	+	0.539031i	$0.818791\pi$
$228$	0	0
$229$	− 8.72792i	− 0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$229$	− 8.72792i	− 0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$230$	0	0
$231$	−8.48528	−0.558291
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	20.4853i	1.33631i
$236$	0	0
$237$	− 9.31371i	− 0.604990i
$238$	0	0
$239$	5.65685	0.365911	0.182956	−	0.983121i	$-0.441433\pi$
$239$	5.65685	0.365911	0.182956	+	0.983121i	$0.441433\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	10.2426i	0.654378i
$246$	0	0
$247$	−15.3137	−0.974388
$248$	0	0
$249$	4.24264	0.268866
$250$	0	0
$251$	− 13.8995i	− 0.877328i	−0.898651	−	0.438664i	$-0.855452\pi$
$251$	− 13.8995i	− 0.877328i	0.898651	−	0.438664i	$-0.144548\pi$
$252$	0	0
$253$	4.24264i	0.266733i
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	6.14214	0.383136	0.191568	−	0.981479i	$-0.438643\pi$
$257$	6.14214	0.383136	0.191568	+	0.981479i	$0.438643\pi$
$258$	0	0
$259$	6.14214i	0.381654i
$260$	0	0
$261$	2.82843i	0.175075i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	38.9706	2.39394
$266$	0	0
$267$	5.17157i	0.316495i
$268$	0	0
$269$	2.34315i	0.142864i	0.997445	+	0.0714321i	$0.0227569\pi$
$269$	2.34315i	0.142864i	−0.997445	+	0.0714321i	$0.977243\pi$
$270$	0	0
$271$	13.3137	0.808750	0.404375	−	0.914593i	$-0.367489\pi$
$271$	13.3137	0.808750	0.404375	+	0.914593i	$0.367489\pi$
$272$	0	0
$273$	−13.6569	−0.826550
$274$	0	0
$275$	− 28.2426i	− 1.70310i
$276$	0	0
$277$	− 22.1421i	− 1.33039i	−0.746669	−	0.665196i	$-0.768348\pi$
$277$	− 22.1421i	− 1.33039i	0.746669	−	0.665196i	$-0.231652\pi$
$278$	0	0
$279$	10.8284	0.648281
$280$	0	0
$281$	−11.7990	−0.703869	−0.351934	−	0.936025i	$-0.614476\pi$
$281$	−11.7990	−0.703869	−0.351934	+	0.936025i	$0.614476\pi$
$282$	0	0
$283$	− 19.8995i	− 1.18290i	−0.806341	−	0.591451i	$-0.798555\pi$
$283$	− 19.8995i	− 1.18290i	0.806341	−	0.591451i	$-0.201445\pi$
$284$	0	0
$285$	7.65685i	0.453553i
$286$	0	0
$287$	16.9706	1.00174
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	3.65685i	0.214369i
$292$	0	0
$293$	− 1.75736i	− 0.102666i	−0.998682	−	0.0513330i	$-0.983653\pi$
$293$	− 1.75736i	− 0.102666i	0.998682	−	0.0513330i	$-0.0163470\pi$
$294$	0	0
$295$	19.3137	1.12449
$296$	0	0
$297$	4.24264	0.246183
$298$	0	0
$299$	6.82843i	0.394898i
$300$	0	0
$301$	− 3.51472i	− 0.202585i
$302$	0	0
$303$	−10.1421	−0.582650
$304$	0	0
$305$	12.8284	0.734554
$306$	0	0
$307$	− 12.4853i	− 0.712573i	−0.934377	−	0.356286i	$-0.884043\pi$
$307$	− 12.4853i	− 0.712573i	0.934377	−	0.356286i	$-0.115957\pi$
$308$	0	0
$309$	19.6569i	1.11824i
$310$	0	0
$311$	21.6569	1.22805	0.614024	−	0.789288i	$-0.289550\pi$
$311$	21.6569	1.22805	0.614024	+	0.789288i	$0.289550\pi$
$312$	0	0
$313$	6.97056	0.394000	0.197000	−	0.980404i	$-0.436880\pi$
$313$	6.97056	0.394000	0.197000	+	0.980404i	$0.436880\pi$
$314$	0	0
$315$	6.82843i	0.384738i
$316$	0	0
$317$	− 6.14214i	− 0.344977i	−0.985012	−	0.172488i	$-0.944819\pi$
$317$	− 6.14214i	− 0.344977i	0.985012	−	0.172488i	$-0.0551807\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	2.58579	0.144325
$322$	0	0
$323$	− 2.62742i	− 0.146193i
$324$	0	0
$325$	− 45.4558i	− 2.52144i
$326$	0	0
$327$	−12.7279	−0.703856
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	24.4853i	1.34583i	0.739719	+	0.672916i	$0.234959\pi$
$331$	24.4853i	1.34583i	−0.739719	+	0.672916i	$0.765041\pi$
$332$	0	0
$333$	− 3.07107i	− 0.168293i
$334$	0	0
$335$	30.9706	1.69210
$336$	0	0
$337$	−15.6569	−0.852883	−0.426442	−	0.904515i	$-0.640233\pi$
$337$	−15.6569	−0.852883	−0.426442	+	0.904515i	$0.640233\pi$
$338$	0	0
$339$	− 18.8284i	− 1.02262i
$340$	0	0
$341$	− 45.9411i	− 2.48785i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	3.41421	0.183815
$346$	0	0
$347$	15.5147i	0.832874i	0.909165	+	0.416437i	$0.136721\pi$
$347$	15.5147i	0.832874i	−0.909165	+	0.416437i	$0.863279\pi$
$348$	0	0
$349$	− 28.9706i	− 1.55076i	−0.631496	−	0.775379i	$-0.717559\pi$
$349$	− 28.9706i	− 1.55076i	0.631496	−	0.775379i	$-0.282441\pi$
$350$	0	0
$351$	6.82843	0.364474
$352$	0	0
$353$	−3.65685	−0.194635	−0.0973174	−	0.995253i	$-0.531026\pi$
$353$	−3.65685	−0.194635	−0.0973174	+	0.995253i	$0.531026\pi$
$354$	0	0
$355$	− 39.7990i	− 2.11231i
$356$	0	0
$357$	− 2.34315i	− 0.124012i
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	13.9706	0.735293
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−30.9706	−1.61665	−0.808325	−	0.588736i	$-0.799626\pi$
$367$	−30.9706	−1.61665	−0.808325	+	0.588736i	$0.799626\pi$
$368$	0	0
$369$	−8.48528	−0.441726
$370$	0	0
$371$	22.8284i	1.18519i
$372$	0	0
$373$	36.7279i	1.90170i	0.309652	+	0.950850i	$0.399787\pi$
$373$	36.7279i	1.90170i	−0.309652	+	0.950850i	$0.600213\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	19.3137	0.994707
$378$	0	0
$379$	− 14.7279i	− 0.756523i	−0.925699	−	0.378261i	$-0.876522\pi$
$379$	− 14.7279i	− 0.756523i	0.925699	−	0.378261i	$-0.123478\pi$
$380$	0	0
$381$	13.3137i	0.682082i
$382$	0	0
$383$	19.3137	0.986884	0.493442	−	0.869779i	$-0.335738\pi$
$383$	19.3137	0.986884	0.493442	+	0.869779i	$0.335738\pi$
$384$	0	0
$385$	28.9706	1.47648
$386$	0	0
$387$	1.75736i	0.0893316i
$388$	0	0
$389$	− 14.0416i	− 0.711939i	−0.934498	−	0.355969i	$-0.884151\pi$
$389$	− 14.0416i	− 0.711939i	0.934498	−	0.355969i	$-0.115849\pi$
$390$	0	0
$391$	−1.17157	−0.0592490
$392$	0	0
$393$	14.8284	0.747995
$394$	0	0
$395$	31.7990i	1.59998i
$396$	0	0
$397$	− 0.970563i	− 0.0487111i	−0.999703	−	0.0243556i	$-0.992247\pi$
$397$	− 0.970563i	− 0.0487111i	0.999703	−	0.0243556i	$-0.00775339\pi$
$398$	0	0
$399$	−4.48528	−0.224545
$400$	0	0
$401$	−37.4558	−1.87046	−0.935228	−	0.354047i	$-0.884805\pi$
$401$	−37.4558	−1.87046	−0.935228	+	0.354047i	$0.884805\pi$
$402$	0	0
$403$	− 73.9411i	− 3.68327i
$404$	0	0
$405$	− 3.41421i	− 0.169654i
$406$	0	0
$407$	−13.0294	−0.645845
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	14.8284i	0.731432i
$412$	0	0
$413$	11.3137i	0.556711i
$414$	0	0
$415$	−14.4853	−0.711054
$416$	0	0
$417$	3.31371	0.162273
$418$	0	0
$419$	− 0.928932i	− 0.0453813i	−0.999743	−	0.0226907i	$-0.992777\pi$
$419$	− 0.928932i	− 0.0453813i	0.999743	−	0.0226907i	$-0.00722328\pi$
$420$	0	0
$421$	19.5563i	0.953118i	0.879142	+	0.476559i	$0.158116\pi$
$421$	19.5563i	0.953118i	−0.879142	+	0.476559i	$0.841884\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	7.79899	0.378307
$426$	0	0
$427$	7.51472i	0.363663i
$428$	0	0
$429$	− 28.9706i	− 1.39871i
$430$	0	0
$431$	20.2843	0.977059	0.488529	−	0.872547i	$-0.337533\pi$
$431$	20.2843	0.977059	0.488529	+	0.872547i	$0.337533\pi$
$432$	0	0
$433$	−10.9706	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$433$	−10.9706	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$434$	0	0
$435$	− 9.65685i	− 0.463011i
$436$	0	0
$437$	2.24264i	0.107280i
$438$	0	0
$439$	−36.6274	−1.74813	−0.874066	−	0.485808i	$-0.838525\pi$
$439$	−36.6274	−1.74813	−0.874066	+	0.485808i	$0.838525\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	16.2843i	0.773689i	0.922145	+	0.386845i	$0.126435\pi$
$443$	16.2843i	0.773689i	−0.922145	+	0.386845i	$0.873565\pi$
$444$	0	0
$445$	− 17.6569i	− 0.837015i
$446$	0	0
$447$	13.0711	0.618240
$448$	0	0
$449$	27.7990	1.31192	0.655958	−	0.754798i	$-0.272265\pi$
$449$	27.7990	1.31192	0.655958	+	0.754798i	$0.272265\pi$
$450$	0	0
$451$	36.0000i	1.69517i
$452$	0	0
$453$	3.51472i	0.165136i
$454$	0	0
$455$	46.6274	2.18593
$456$	0	0
$457$	27.6569	1.29373	0.646867	−	0.762603i	$-0.276079\pi$
$457$	27.6569	1.29373	0.646867	+	0.762603i	$0.276079\pi$
$458$	0	0
$459$	1.17157i	0.0546843i
$460$	0	0
$461$	28.2843i	1.31733i	0.752436	+	0.658665i	$0.228879\pi$
$461$	28.2843i	1.31733i	−0.752436	+	0.658665i	$0.771121\pi$
$462$	0	0
$463$	25.3137	1.17643	0.588214	−	0.808705i	$-0.299831\pi$
$463$	25.3137	1.17643	0.588214	+	0.808705i	$0.299831\pi$
$464$	0	0
$465$	−36.9706	−1.71447
$466$	0	0
$467$	− 35.5563i	− 1.64535i	−0.568511	−	0.822676i	$-0.692480\pi$
$467$	− 35.5563i	− 1.64535i	0.568511	−	0.822676i	$-0.307520\pi$
$468$	0	0
$469$	18.1421i	0.837726i
$470$	0	0
$471$	−16.7279	−0.770782
$472$	0	0
$473$	7.45584	0.342820
$474$	0	0
$475$	− 14.9289i	− 0.684986i
$476$	0	0
$477$	− 11.4142i	− 0.522621i
$478$	0	0
$479$	−17.6569	−0.806762	−0.403381	−	0.915032i	$-0.632165\pi$
$479$	−17.6569	−0.806762	−0.403381	+	0.915032i	$0.632165\pi$
$480$	0	0
$481$	−20.9706	−0.956175
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	− 12.4853i	− 0.566927i
$486$	0	0
$487$	35.9411	1.62865	0.814324	−	0.580411i	$-0.197108\pi$
$487$	35.9411	1.62865	0.814324	+	0.580411i	$0.197108\pi$
$488$	0	0
$489$	−4.48528	−0.202831
$490$	0	0
$491$	− 14.1421i	− 0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	− 14.1421i	− 0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$	0	0
$493$	3.31371i	0.149242i
$494$	0	0
$495$	−14.4853	−0.651065
$496$	0	0
$497$	23.3137	1.04576
$498$	0	0
$499$	6.82843i	0.305682i	0.988251	+	0.152841i	$0.0488423\pi$
$499$	6.82843i	0.305682i	−0.988251	+	0.152841i	$0.951158\pi$
$500$	0	0
$501$	7.65685i	0.342083i
$502$	0	0
$503$	−1.65685	−0.0738755	−0.0369377	−	0.999318i	$-0.511760\pi$
$503$	−1.65685	−0.0738755	−0.0369377	+	0.999318i	$0.511760\pi$
$504$	0	0
$505$	34.6274	1.54090
$506$	0	0
$507$	− 33.6274i	− 1.49345i
$508$	0	0
$509$	− 2.82843i	− 0.125368i	−0.998033	−	0.0626839i	$-0.980034\pi$
$509$	− 2.82843i	− 0.125368i	0.998033	−	0.0626839i	$-0.0199660\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.24264	0.0990150
$514$	0	0
$515$	− 67.1127i	− 2.95734i
$516$	0	0
$517$	− 25.4558i	− 1.11955i
$518$	0	0
$519$	−18.8284	−0.826476
$520$	0	0
$521$	−22.8284	−1.00013	−0.500066	−	0.865987i	$-0.666691\pi$
$521$	−22.8284	−1.00013	−0.500066	+	0.865987i	$0.666691\pi$
$522$	0	0
$523$	− 1.75736i	− 0.0768440i	−0.999262	−	0.0384220i	$-0.987767\pi$
$523$	− 1.75736i	− 0.0768440i	0.999262	−	0.0384220i	$-0.0122331\pi$
$524$	0	0
$525$	− 13.3137i	− 0.581058i
$526$	0	0
$527$	12.6863	0.552624
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	− 5.65685i	− 0.245487i
$532$	0	0
$533$	57.9411i	2.50971i
$534$	0	0
$535$	−8.82843	−0.381686
$536$	0	0
$537$	−8.48528	−0.366167
$538$	0	0
$539$	− 12.7279i	− 0.548230i
$540$	0	0
$541$	36.0000i	1.54776i	0.633332	+	0.773880i	$0.281687\pi$
$541$	36.0000i	1.54776i	−0.633332	+	0.773880i	$0.718313\pi$
$542$	0	0
$543$	12.2426	0.525382
$544$	0	0
$545$	43.4558	1.86144
$546$	0	0
$547$	15.5147i	0.663361i	0.943392	+	0.331681i	$0.107616\pi$
$547$	15.5147i	0.663361i	−0.943392	+	0.331681i	$0.892384\pi$
$548$	0	0
$549$	− 3.75736i	− 0.160360i
$550$	0	0
$551$	6.34315	0.270227
$552$	0	0
$553$	−18.6274	−0.792118
$554$	0	0
$555$	10.4853i	0.445075i
$556$	0	0
$557$	− 23.8995i	− 1.01265i	−0.862342	−	0.506327i	$-0.831003\pi$
$557$	− 23.8995i	− 1.01265i	0.862342	−	0.506327i	$-0.168997\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	4.97056	0.209857
$562$	0	0
$563$	0.727922i	0.0306783i	0.999882	+	0.0153391i	$0.00488279\pi$
$563$	0.727922i	0.0306783i	−0.999882	+	0.0153391i	$0.995117\pi$
$564$	0	0
$565$	64.2843i	2.70446i
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	26.8284	1.12471	0.562353	−	0.826897i	$-0.309896\pi$
$569$	26.8284	1.12471	0.562353	+	0.826897i	$0.309896\pi$
$570$	0	0
$571$	− 12.5858i	− 0.526699i	−0.964701	−	0.263349i	$-0.915173\pi$
$571$	− 12.5858i	− 0.526699i	0.964701	−	0.263349i	$-0.0848272\pi$
$572$	0	0
$573$	9.65685i	0.403421i
$574$	0	0
$575$	−6.65685	−0.277610
$576$	0	0
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	0	0
$579$	− 10.0000i	− 0.415586i
$580$	0	0
$581$	− 8.48528i	− 0.352029i
$582$	0	0
$583$	−48.4264	−2.00562
$584$	0	0
$585$	−23.3137	−0.963903
$586$	0	0
$587$	− 15.7990i	− 0.652094i	−0.945354	−	0.326047i	$-0.894283\pi$
$587$	− 15.7990i	− 0.652094i	0.945354	−	0.326047i	$-0.105717\pi$
$588$	0	0
$589$	− 24.2843i	− 1.00062i
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	−22.1421	−0.909269	−0.454634	−	0.890678i	$-0.650230\pi$
$593$	−22.1421	−0.909269	−0.454634	+	0.890678i	$0.650230\pi$
$594$	0	0
$595$	8.00000i	0.327968i
$596$	0	0
$597$	− 10.9706i	− 0.448995i
$598$	0	0
$599$	3.31371	0.135394	0.0676972	−	0.997706i	$-0.478435\pi$
$599$	3.31371	0.135394	0.0676972	+	0.997706i	$0.478435\pi$
$600$	0	0
$601$	−42.6274	−1.73881	−0.869404	−	0.494101i	$-0.835497\pi$
$601$	−42.6274	−1.73881	−0.869404	+	0.494101i	$0.835497\pi$
$602$	0	0
$603$	− 9.07107i	− 0.369402i
$604$	0	0
$605$	23.8995i	0.971653i
$606$	0	0
$607$	−20.4853	−0.831472	−0.415736	−	0.909485i	$-0.636476\pi$
$607$	−20.4853	−0.831472	−0.415736	+	0.909485i	$0.636476\pi$
$608$	0	0
$609$	5.65685	0.229227
$610$	0	0
$611$	− 40.9706i	− 1.65749i
$612$	0	0
$613$	11.7574i	0.474875i	0.971403	+	0.237438i	$0.0763075\pi$
$613$	11.7574i	0.474875i	−0.971403	+	0.237438i	$0.923692\pi$
$614$	0	0
$615$	28.9706	1.16821
$616$	0	0
$617$	−18.1421	−0.730375	−0.365187	−	0.930934i	$-0.618995\pi$
$617$	−18.1421	−0.730375	−0.365187	+	0.930934i	$0.618995\pi$
$618$	0	0
$619$	− 8.38478i	− 0.337013i	−0.985701	−	0.168506i	$-0.946106\pi$
$619$	− 8.38478i	− 0.337013i	0.985701	−	0.168506i	$-0.0538944\pi$
$620$	0	0
$621$	− 1.00000i	− 0.0401286i
$622$	0	0
$623$	10.3431	0.414389
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	− 9.51472i	− 0.379981i
$628$	0	0
$629$	− 3.59798i	− 0.143461i
$630$	0	0
$631$	−26.2843	−1.04636	−0.523180	−	0.852222i	$-0.675255\pi$
$631$	−26.2843	−1.04636	−0.523180	+	0.852222i	$0.675255\pi$
$632$	0	0
$633$	−0.485281	−0.0192882
$634$	0	0
$635$	− 45.4558i	− 1.80386i
$636$	0	0
$637$	− 20.4853i	− 0.811656i
$638$	0	0
$639$	−11.6569	−0.461138
$640$	0	0
$641$	−35.7990	−1.41398	−0.706988	−	0.707226i	$-0.749946\pi$
$641$	−35.7990	−1.41398	−0.706988	+	0.707226i	$0.749946\pi$
$642$	0	0
$643$	− 18.9289i	− 0.746484i	−0.927734	−	0.373242i	$-0.878246\pi$
$643$	− 18.9289i	− 0.746484i	0.927734	−	0.373242i	$-0.121754\pi$
$644$	0	0
$645$	− 6.00000i	− 0.236250i
$646$	0	0
$647$	−19.3137	−0.759300	−0.379650	−	0.925130i	$-0.623956\pi$
$647$	−19.3137	−0.759300	−0.379650	+	0.925130i	$0.623956\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	− 21.6569i	− 0.848799i
$652$	0	0
$653$	− 10.1421i	− 0.396892i	−0.980112	−	0.198446i	$-0.936410\pi$
$653$	− 10.1421i	− 0.396892i	0.980112	−	0.198446i	$-0.0635895\pi$
$654$	0	0
$655$	−50.6274	−1.97818
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.8701i	1.04671i	0.852115	+	0.523354i	$0.175320\pi$
$659$	26.8701i	1.04671i	−0.852115	+	0.523354i	$0.824680\pi$
$660$	0	0
$661$	21.6985i	0.843973i	0.906602	+	0.421987i	$0.138667\pi$
$661$	21.6985i	0.843973i	−0.906602	+	0.421987i	$0.861333\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	15.3137	0.593840
$666$	0	0
$667$	− 2.82843i	− 0.109517i
$668$	0	0
$669$	− 6.82843i	− 0.264002i
$670$	0	0
$671$	−15.9411	−0.615400
$672$	0	0
$673$	21.6569	0.834810	0.417405	−	0.908720i	$-0.362940\pi$
$673$	21.6569	0.834810	0.417405	+	0.908720i	$0.362940\pi$
$674$	0	0
$675$	6.65685i	0.256222i
$676$	0	0
$677$	3.89949i	0.149870i	0.997188	+	0.0749349i	$0.0238749\pi$
$677$	3.89949i	0.149870i	−0.997188	+	0.0749349i	$0.976125\pi$
$678$	0	0
$679$	7.31371	0.280674
$680$	0	0
$681$	−16.2426	−0.622419
$682$	0	0
$683$	− 43.1127i	− 1.64966i	−0.565380	−	0.824831i	$-0.691270\pi$
$683$	− 43.1127i	− 1.64966i	0.565380	−	0.824831i	$-0.308730\pi$
$684$	0	0
$685$	− 50.6274i	− 1.93437i
$686$	0	0
$687$	8.72792	0.332991
$688$	0	0
$689$	−77.9411	−2.96932
$690$	0	0
$691$	15.5147i	0.590208i	0.955465	+	0.295104i	$0.0953543\pi$
$691$	15.5147i	0.590208i	−0.955465	+	0.295104i	$0.904646\pi$
$692$	0	0
$693$	− 8.48528i	− 0.322329i
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	−9.94113	−0.376547
$698$	0	0
$699$	18.0000i	0.680823i
$700$	0	0
$701$	− 33.5563i	− 1.26741i	−0.773577	−	0.633703i	$-0.781534\pi$
$701$	− 33.5563i	− 1.26741i	0.773577	−	0.633703i	$-0.218466\pi$
$702$	0	0
$703$	−6.88730	−0.259760
$704$	0	0
$705$	−20.4853	−0.771520
$706$	0	0
$707$	20.2843i	0.762869i
$708$	0	0
$709$	42.5858i	1.59934i	0.600438	+	0.799671i	$0.294993\pi$
$709$	42.5858i	1.59934i	−0.600438	+	0.799671i	$0.705007\pi$
$710$	0	0
$711$	9.31371	0.349291
$712$	0	0
$713$	−10.8284	−0.405528
$714$	0	0
$715$	98.9117i	3.69909i
$716$	0	0
$717$	5.65685i	0.211259i
$718$	0	0
$719$	−1.37258	−0.0511887	−0.0255944	−	0.999672i	$-0.508148\pi$
$719$	−1.37258	−0.0511887	−0.0255944	+	0.999672i	$0.508148\pi$
$720$	0	0
$721$	39.3137	1.46412
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	18.8284i	0.699270i
$726$	0	0
$727$	42.2843	1.56824	0.784118	−	0.620611i	$-0.213115\pi$
$727$	42.2843	1.56824	0.784118	+	0.620611i	$0.213115\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.05887i	0.0761502i
$732$	0	0
$733$	− 15.7574i	− 0.582011i	−0.956721	−	0.291006i	$-0.906010\pi$
$733$	− 15.7574i	− 0.582011i	0.956721	−	0.291006i	$-0.0939899\pi$
$734$	0	0
$735$	−10.2426	−0.377805
$736$	0	0
$737$	−38.4853	−1.41762
$738$	0	0
$739$	− 38.3431i	− 1.41048i	−0.708971	−	0.705238i	$-0.750840\pi$
$739$	− 38.3431i	− 1.41048i	0.708971	−	0.705238i	$-0.249160\pi$
$740$	0	0
$741$	− 15.3137i	− 0.562563i
$742$	0	0
$743$	−27.5980	−1.01247	−0.506236	−	0.862395i	$-0.668963\pi$
$743$	−27.5980	−1.01247	−0.506236	+	0.862395i	$0.668963\pi$
$744$	0	0
$745$	−44.6274	−1.63502
$746$	0	0
$747$	4.24264i	0.155230i
$748$	0	0
$749$	− 5.17157i	− 0.188965i
$750$	0	0
$751$	−1.02944	−0.0375647	−0.0187823	−	0.999824i	$-0.505979\pi$
$751$	−1.02944	−0.0375647	−0.0187823	+	0.999824i	$0.505979\pi$
$752$	0	0
$753$	13.8995	0.506526
$754$	0	0
$755$	− 12.0000i	− 0.436725i
$756$	0	0
$757$	− 9.21320i	− 0.334860i	−0.985884	−	0.167430i	$-0.946453\pi$
$757$	− 9.21320i	− 0.334860i	0.985884	−	0.167430i	$-0.0535467\pi$
$758$	0	0
$759$	−4.24264	−0.153998
$760$	0	0
$761$	3.65685	0.132561	0.0662804	−	0.997801i	$-0.478887\pi$
$761$	3.65685	0.132561	0.0662804	+	0.997801i	$0.478887\pi$
$762$	0	0
$763$	25.4558i	0.921563i
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	−38.6274	−1.39476
$768$	0	0
$769$	26.9706	0.972583	0.486292	−	0.873797i	$-0.338349\pi$
$769$	26.9706	0.972583	0.486292	+	0.873797i	$0.338349\pi$
$770$	0	0
$771$	6.14214i	0.221204i
$772$	0	0
$773$	− 45.0711i	− 1.62109i	−0.585674	−	0.810547i	$-0.699170\pi$
$773$	− 45.0711i	− 1.62109i	0.585674	−	0.810547i	$-0.300830\pi$
$774$	0	0
$775$	72.0833	2.58931
$776$	0	0
$777$	−6.14214	−0.220348
$778$	0	0
$779$	19.0294i	0.681800i
$780$	0	0
$781$	49.4558i	1.76967i
$782$	0	0
$783$	−2.82843	−0.101080
$784$	0	0
$785$	57.1127	2.03844
$786$	0	0
$787$	− 4.10051i	− 0.146167i	−0.997326	−	0.0730836i	$-0.976716\pi$
$787$	− 4.10051i	− 0.146167i	0.997326	−	0.0730836i	$-0.0232840\pi$
$788$	0	0
$789$	− 12.0000i	− 0.427211i
$790$	0	0
$791$	−37.6569	−1.33892
$792$	0	0
$793$	−25.6569	−0.911102
$794$	0	0
$795$	38.9706i	1.38214i
$796$	0	0
$797$	− 17.3553i	− 0.614758i	−0.951587	−	0.307379i	$-0.900548\pi$
$797$	− 17.3553i	− 0.614758i	0.951587	−	0.307379i	$-0.0994519\pi$
$798$	0	0
$799$	7.02944	0.248684
$800$	0	0
$801$	−5.17157	−0.182729
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 6.82843i	− 0.240670i
$806$	0	0
$807$	−2.34315	−0.0824826
$808$	0	0
$809$	12.3431	0.433962	0.216981	−	0.976176i	$-0.430379\pi$
$809$	12.3431	0.433962	0.216981	+	0.976176i	$0.430379\pi$
$810$	0	0
$811$	30.1421i	1.05843i	0.848487	+	0.529217i	$0.177514\pi$
$811$	30.1421i	1.05843i	−0.848487	+	0.529217i	$0.822486\pi$
$812$	0	0
$813$	13.3137i	0.466932i
$814$	0	0
$815$	15.3137	0.536416
$816$	0	0
$817$	3.94113	0.137883
$818$	0	0
$819$	− 13.6569i	− 0.477209i
$820$	0	0
$821$	− 50.1421i	− 1.74997i	−0.484148	−	0.874986i	$-0.660870\pi$
$821$	− 50.1421i	− 1.74997i	0.484148	−	0.874986i	$-0.339130\pi$
$822$	0	0
$823$	−25.4558	−0.887335	−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558	−0.887335	−0.443667	+	0.896191i	$0.646323\pi$
$824$	0	0
$825$	28.2426	0.983283
$826$	0	0
$827$	32.5269i	1.13107i	0.824724	+	0.565536i	$0.191331\pi$
$827$	32.5269i	1.13107i	−0.824724	+	0.565536i	$0.808669\pi$
$828$	0	0
$829$	− 36.0000i	− 1.25033i	−0.780492	−	0.625166i	$-0.785031\pi$
$829$	− 36.0000i	− 1.25033i	0.780492	−	0.625166i	$-0.214969\pi$
$830$	0	0
$831$	22.1421	0.768102
$832$	0	0
$833$	3.51472	0.121778
$834$	0	0
$835$	− 26.1421i	− 0.904686i
$836$	0	0
$837$	10.8284i	0.374285i
$838$	0	0
$839$	−15.3137	−0.528688	−0.264344	−	0.964428i	$-0.585155\pi$
$839$	−15.3137	−0.528688	−0.264344	+	0.964428i	$0.585155\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	− 11.7990i	− 0.406379i
$844$	0	0
$845$	114.811i	3.94962i
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	19.8995	0.682949
$850$	0	0
$851$	3.07107i	0.105275i
$852$	0	0
$853$	12.2843i	0.420605i	0.977636	+	0.210303i	$0.0674450\pi$
$853$	12.2843i	0.420605i	−0.977636	+	0.210303i	$0.932555\pi$
$854$	0	0
$855$	−7.65685	−0.261859
$856$	0	0
$857$	21.3137	0.728062	0.364031	−	0.931387i	$-0.381400\pi$
$857$	21.3137	0.728062	0.364031	+	0.931387i	$0.381400\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	16.9706i	0.578355i
$862$	0	0
$863$	16.9706	0.577685	0.288842	−	0.957377i	$-0.406730\pi$
$863$	16.9706	0.577685	0.288842	+	0.957377i	$0.406730\pi$
$864$	0	0
$865$	64.2843	2.18573
$866$	0	0
$867$	− 15.6274i	− 0.530735i
$868$	0	0
$869$	− 39.5147i	− 1.34045i
$870$	0	0
$871$	−61.9411	−2.09879
$872$	0	0
$873$	−3.65685	−0.123766
$874$	0	0
$875$	11.3137i	0.382473i
$876$	0	0
$877$	8.48528i	0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$	8.48528i	0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$	0	0
$879$	1.75736	0.0592743
$880$	0	0
$881$	37.4558	1.26192	0.630960	−	0.775816i	$-0.282661\pi$
$881$	37.4558	1.26192	0.630960	+	0.775816i	$0.282661\pi$
$882$	0	0
$883$	8.48528i	0.285552i	0.989755	+	0.142776i	$0.0456029\pi$
$883$	8.48528i	0.285552i	−0.989755	+	0.142776i	$0.954397\pi$
$884$	0	0
$885$	19.3137i	0.649223i
$886$	0	0
$887$	−15.9411	−0.535251	−0.267625	−	0.963523i	$-0.586239\pi$
$887$	−15.9411	−0.535251	−0.267625	+	0.963523i	$0.586239\pi$
$888$	0	0
$889$	26.6274	0.893055
$890$	0	0
$891$	4.24264i	0.142134i
$892$	0	0
$893$	− 13.4558i	− 0.450283i
$894$	0	0
$895$	28.9706	0.968379
$896$	0	0
$897$	−6.82843	−0.227995
$898$	0	0
$899$	30.6274i	1.02148i
$900$	0	0
$901$	− 13.3726i	− 0.445505i
$902$	0	0
$903$	3.51472	0.116963
$904$	0	0
$905$	−41.7990	−1.38945
$906$	0	0
$907$	37.5563i	1.24704i	0.781808	+	0.623519i	$0.214298\pi$
$907$	37.5563i	1.24704i	−0.781808	+	0.623519i	$0.785702\pi$
$908$	0	0
$909$	− 10.1421i	− 0.336393i
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	0	0
$915$	12.8284i	0.424095i
$916$	0	0
$917$	− 29.6569i	− 0.979356i
$918$	0	0
$919$	27.9411	0.921693	0.460846	−	0.887480i	$-0.347546\pi$
$919$	27.9411	0.921693	0.460846	+	0.887480i	$0.347546\pi$
$920$	0	0
$921$	12.4853	0.411404
$922$	0	0
$923$	79.5980i	2.62000i
$924$	0	0
$925$	− 20.4437i	− 0.672183i
$926$	0	0
$927$	−19.6569	−0.645616
$928$	0	0
$929$	45.1716	1.48203	0.741016	−	0.671488i	$-0.234344\pi$
$929$	45.1716	1.48203	0.741016	+	0.671488i	$0.234344\pi$
$930$	0	0
$931$	− 6.72792i	− 0.220499i
$932$	0	0
$933$	21.6569i	0.709014i
$934$	0	0
$935$	−16.9706	−0.554997
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	6.97056i	0.227476i
$940$	0	0
$941$	− 11.8995i	− 0.387912i	−0.981010	−	0.193956i	$-0.937868\pi$
$941$	− 11.8995i	− 0.387912i	0.981010	−	0.193956i	$-0.0621320\pi$
$942$	0	0
$943$	8.48528	0.276319
$944$	0	0
$945$	−6.82843	−0.222129
$946$	0	0
$947$	− 4.20101i	− 0.136514i	−0.997668	−	0.0682572i	$-0.978256\pi$
$947$	− 4.20101i	− 0.136514i	0.997668	−	0.0682572i	$-0.0217439\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	6.14214	0.199172
$952$	0	0
$953$	50.1421	1.62426	0.812132	−	0.583474i	$-0.198307\pi$
$953$	50.1421	1.62426	0.812132	+	0.583474i	$0.198307\pi$
$954$	0	0
$955$	− 32.9706i	− 1.06690i
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	29.6569	0.957670
$960$	0	0
$961$	86.2548	2.78241
$962$	0	0
$963$	2.58579i	0.0833258i
$964$	0	0
$965$	34.1421i	1.09907i
$966$	0	0
$967$	8.48528	0.272868	0.136434	−	0.990649i	$-0.456436\pi$
$967$	8.48528	0.272868	0.136434	+	0.990649i	$0.456436\pi$
$968$	0	0
$969$	2.62742	0.0844048
$970$	0	0
$971$	30.3848i	0.975094i	0.873097	+	0.487547i	$0.162108\pi$
$971$	30.3848i	0.975094i	−0.873097	+	0.487547i	$0.837892\pi$
$972$	0	0
$973$	− 6.62742i	− 0.212465i
$974$	0	0
$975$	45.4558	1.45575
$976$	0	0
$977$	−20.4853	−0.655382	−0.327691	−	0.944785i	$-0.606271\pi$
$977$	−20.4853	−0.655382	−0.327691	+	0.944785i	$0.606271\pi$
$978$	0	0
$979$	21.9411i	0.701241i
$980$	0	0
$981$	− 12.7279i	− 0.406371i
$982$	0	0
$983$	28.9706	0.924017	0.462009	−	0.886875i	$-0.347129\pi$
$983$	28.9706	0.924017	0.462009	+	0.886875i	$0.347129\pi$
$984$	0	0
$985$	−81.9411	−2.61086
$986$	0	0
$987$	− 12.0000i	− 0.381964i
$988$	0	0
$989$	− 1.75736i	− 0.0558808i
$990$	0	0
$991$	42.4264	1.34772	0.673860	−	0.738859i	$-0.264635\pi$
$991$	42.4264	1.34772	0.673860	+	0.738859i	$0.264635\pi$
$992$	0	0
$993$	−24.4853	−0.777017
$994$	0	0
$995$	37.4558i	1.18743i
$996$	0	0
$997$	− 18.6274i	− 0.589936i	−0.955507	−	0.294968i	$-0.904691\pi$
$997$	− 18.6274i	− 0.589936i	0.955507	−	0.294968i	$-0.0953090\pi$
$998$	0	0
$999$	3.07107	0.0971643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.f.b.1105.3		4
4.3	odd	2		552.2.f.b.277.3	✓	4
8.3	odd	2		552.2.f.b.277.4	yes	4
8.5	even	2	inner	2208.2.f.b.1105.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.f.b.277.3	✓	4	4.3	odd	2
552.2.f.b.277.4	yes	4	8.3	odd	2
2208.2.f.b.1105.2		4	8.5	even	2	inner
2208.2.f.b.1105.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.41421i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.41421i q^{5} +2.00000 q^{7} -1.00000 q^{9} +4.24264i q^{11} +6.82843i q^{13} +3.41421 q^{15} -1.17157 q^{17} +2.24264i q^{19} +2.00000i q^{21} +1.00000 q^{23} -6.65685 q^{25} -1.00000i q^{27} -2.82843i q^{29} -10.8284 q^{31} -4.24264 q^{33} -6.82843i q^{35} +3.07107i q^{37} -6.82843 q^{39} +8.48528 q^{41} -1.75736i q^{43} +3.41421i q^{45} -6.00000 q^{47} -3.00000 q^{49} -1.17157i q^{51} +11.4142i q^{53} +14.4853 q^{55} -2.24264 q^{57} +5.65685i q^{59} +3.75736i q^{61} -2.00000 q^{63} +23.3137 q^{65} +9.07107i q^{67} +1.00000i q^{69} +11.6569 q^{71} -6.65685i q^{75} +8.48528i q^{77} -9.31371 q^{79} +1.00000 q^{81} -4.24264i q^{83} +4.00000i q^{85} +2.82843 q^{87} +5.17157 q^{89} +13.6569i q^{91} -10.8284i q^{93} +7.65685 q^{95} +3.65685 q^{97} -4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 4 * q^9	\( 4 q + 8 q^{7} - 4 q^{9} + 8 q^{15} - 16 q^{17} + 4 q^{23} - 4 q^{25} - 32 q^{31} - 16 q^{39} - 24 q^{47} - 12 q^{49} + 24 q^{55} + 8 q^{57} - 8 q^{63} + 48 q^{65} + 24 q^{71} + 8 q^{79} + 4 q^{81} + 32 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 - 4 * q^9 + 8 * q^15 - 16 * q^17 + 4 * q^23 - 4 * q^25 - 32 * q^31 - 16 * q^39 - 24 * q^47 - 12 * q^49 + 24 * q^55 + 8 * q^57 - 8 * q^63 + 48 * q^65 + 24 * q^71 + 8 * q^79 + 4 * q^81 + 32 * q^89 + 8 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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