LMFDB - Embedded newform 867.2.d.c.577.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(577,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("867.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (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:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.c.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +1.00000i q^{3} +4.56155 q^{4} -3.56155i q^{5} +2.56155i q^{6} +6.56155 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.56155 q^{2} +1.00000i q^{3} +4.56155 q^{4} -3.56155i q^{5} +2.56155i q^{6} +6.56155 q^{8} -1.00000 q^{9} -9.12311i q^{10} +1.56155i q^{11} +4.56155i q^{12} +0.438447 q^{13} +3.56155 q^{15} +7.68466 q^{16} -2.56155 q^{18} +4.68466 q^{19} -16.2462i q^{20} +4.00000i q^{22} -2.43845i q^{23} +6.56155i q^{24} -7.68466 q^{25} +1.12311 q^{26} -1.00000i q^{27} +8.24621i q^{29} +9.12311 q^{30} -3.12311i q^{31} +6.56155 q^{32} -1.56155 q^{33} -4.56155 q^{36} +5.12311i q^{37} +12.0000 q^{38} +0.438447i q^{39} -23.3693i q^{40} -3.56155i q^{41} -4.68466 q^{43} +7.12311i q^{44} +3.56155i q^{45} -6.24621i q^{46} -11.1231 q^{47} +7.68466i q^{48} +7.00000 q^{49} -19.6847 q^{50} +2.00000 q^{52} -12.2462 q^{53} -2.56155i q^{54} +5.56155 q^{55} +4.68466i q^{57} +21.1231i q^{58} -7.12311 q^{59} +16.2462 q^{60} +9.12311i q^{61} -8.00000i q^{62} +1.43845 q^{64} -1.56155i q^{65} -4.00000 q^{66} +4.00000 q^{67} +2.43845 q^{69} +6.24621i q^{71} -6.56155 q^{72} +12.2462i q^{73} +13.1231i q^{74} -7.68466i q^{75} +21.3693 q^{76} +1.12311i q^{78} -9.36932i q^{79} -27.3693i q^{80} +1.00000 q^{81} -9.12311i q^{82} +0.876894 q^{83} -12.0000 q^{86} -8.24621 q^{87} +10.2462i q^{88} -1.12311 q^{89} +9.12311i q^{90} -11.1231i q^{92} +3.12311 q^{93} -28.4924 q^{94} -16.6847i q^{95} +6.56155i q^{96} +2.87689i q^{97} +17.9309 q^{98} -1.56155i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9	$ 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9} + 10 q^{13} + 6 q^{15} + 6 q^{16} - 2 q^{18} - 6 q^{19} - 6 q^{25} - 12 q^{26} + 20 q^{30} + 18 q^{32} + 2 q^{33} - 10 q^{36} + 48 q^{38} + 6 q^{43} - 28 q^{47} + 28 q^{49} - 54 q^{50} + 8 q^{52} - 16 q^{53} + 14 q^{55} - 12 q^{59} + 32 q^{60} + 14 q^{64} - 16 q^{66} + 16 q^{67} + 18 q^{69} - 18 q^{72} + 36 q^{76} + 4 q^{81} + 20 q^{83} - 48 q^{86} + 12 q^{89} - 4 q^{93} - 48 q^{94} + 14 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9 + 10 * q^13 + 6 * q^15 + 6 * q^16 - 2 * q^18 - 6 * q^19 - 6 * q^25 - 12 * q^26 + 20 * q^30 + 18 * q^32 + 2 * q^33 - 10 * q^36 + 48 * q^38 + 6 * q^43 - 28 * q^47 + 28 * q^49 - 54 * q^50 + 8 * q^52 - 16 * q^53 + 14 * q^55 - 12 * q^59 + 32 * q^60 + 14 * q^64 - 16 * q^66 + 16 * q^67 + 18 * q^69 - 18 * q^72 + 36 * q^76 + 4 * q^81 + 20 * q^83 - 48 * q^86 + 12 * q^89 - 4 * q^93 - 48 * q^94 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	4.56155	2.28078
$5$	− 3.56155i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 3.56155i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	2.56155i	1.04575i
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	6.56155	2.31986
$9$	−1.00000	−0.333333
$10$	− 9.12311i	− 2.88498i
$11$	1.56155i	0.470826i	0.971895	+	0.235413i	$0.0756443\pi$
$11$	1.56155i	0.470826i	−0.971895	+	0.235413i	$0.924356\pi$
$12$	4.56155i	1.31681i
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	3.56155	0.919589
$16$	7.68466	1.92116
$17$	0	0
$17$	0	0
$18$	−2.56155	−0.603764
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	− 16.2462i	− 3.63276i
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 2.43845i	− 0.508451i	−0.967145	−	0.254226i	$-0.918179\pi$
$23$	− 2.43845i	− 0.508451i	0.967145	−	0.254226i	$-0.0818206\pi$
$24$	6.56155i	1.33937i
$25$	−7.68466	−1.53693
$26$	1.12311	0.220259
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	8.24621i	1.53128i	0.643268	+	0.765641i	$0.277578\pi$
$29$	8.24621i	1.53128i	−0.643268	+	0.765641i	$0.722422\pi$
$30$	9.12311	1.66564
$31$	− 3.12311i	− 0.560926i	−0.959865	−	0.280463i	$-0.909512\pi$
$31$	− 3.12311i	− 0.560926i	0.959865	−	0.280463i	$-0.0904881\pi$
$32$	6.56155	1.15993
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	−4.56155	−0.760259
$37$	5.12311i	0.842233i	0.907006	+	0.421117i	$0.138362\pi$
$37$	5.12311i	0.842233i	−0.907006	+	0.421117i	$0.861638\pi$
$38$	12.0000	1.94666
$39$	0.438447i	0.0702077i
$40$	− 23.3693i	− 3.69501i
$41$	− 3.56155i	− 0.556221i	−0.960549	−	0.278111i	$-0.910292\pi$
$41$	− 3.56155i	− 0.556221i	0.960549	−	0.278111i	$-0.0897082\pi$
$42$	0	0
$43$	−4.68466	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$43$	−4.68466	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$44$	7.12311i	1.07385i
$45$	3.56155i	0.530925i
$46$	− 6.24621i	− 0.920954i
$47$	−11.1231	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$47$	−11.1231	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$48$	7.68466i	1.10918i
$49$	7.00000	1.00000
$50$	−19.6847	−2.78383
$51$	0	0
$52$	2.00000	0.277350
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	− 2.56155i	− 0.348583i
$55$	5.56155	0.749920
$56$	0	0
$57$	4.68466i	0.620498i
$58$	21.1231i	2.77360i
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	16.2462	2.09738
$61$	9.12311i	1.16809i	0.811720	+	0.584047i	$0.198532\pi$
$61$	9.12311i	1.16809i	−0.811720	+	0.584047i	$0.801468\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	1.43845	0.179806
$65$	− 1.56155i	− 0.193687i
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	2.43845	0.293555
$70$	0	0
$71$	6.24621i	0.741289i	0.928775	+	0.370644i	$0.120863\pi$
$71$	6.24621i	0.741289i	−0.928775	+	0.370644i	$0.879137\pi$
$72$	−6.56155	−0.773286
$73$	12.2462i	1.43331i	0.697428	+	0.716655i	$0.254328\pi$
$73$	12.2462i	1.43331i	−0.697428	+	0.716655i	$0.745672\pi$
$74$	13.1231i	1.52553i
$75$	− 7.68466i	− 0.887348i
$76$	21.3693	2.45123
$77$	0	0
$78$	1.12311i	0.127167i
$79$	− 9.36932i	− 1.05413i	−0.849825	−	0.527065i	$-0.823292\pi$
$79$	− 9.36932i	− 1.05413i	0.849825	−	0.527065i	$-0.176708\pi$
$80$	− 27.3693i	− 3.05998i
$81$	1.00000	0.111111
$82$	− 9.12311i	− 1.00748i
$83$	0.876894	0.0962517	0.0481258	−	0.998841i	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−8.24621	−0.884087
$88$	10.2462i	1.09225i
$89$	−1.12311	−0.119049	−0.0595245	−	0.998227i	$-0.518958\pi$
$89$	−1.12311	−0.119049	−0.0595245	+	0.998227i	$0.518958\pi$
$90$	9.12311i	0.961660i
$91$	0	0
$92$	− 11.1231i	− 1.15966i
$93$	3.12311	0.323851
$94$	−28.4924	−2.93877
$95$	− 16.6847i	− 1.71181i
$96$	6.56155i	0.669686i
$97$	2.87689i	0.292104i	0.989277	+	0.146052i	$0.0466567\pi$
$97$	2.87689i	0.292104i	−0.989277	+	0.146052i	$0.953343\pi$
$98$	17.9309	1.81129
$99$	− 1.56155i	− 0.156942i
$100$	−35.0540	−3.50540
$101$	10.8769	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$102$	0	0
$103$	16.6847	1.64399	0.821994	−	0.569496i	$-0.192862\pi$
$103$	16.6847	1.64399	0.821994	+	0.569496i	$0.192862\pi$
$104$	2.87689	0.282103
$105$	0	0
$106$	−31.3693	−3.04686
$107$	4.68466i	0.452883i	0.974025	+	0.226442i	$0.0727092\pi$
$107$	4.68466i	0.452883i	−0.974025	+	0.226442i	$0.927291\pi$
$108$	− 4.56155i	− 0.438936i
$109$	− 6.87689i	− 0.658687i	−0.944210	−	0.329344i	$-0.893173\pi$
$109$	− 6.87689i	− 0.658687i	0.944210	−	0.329344i	$-0.106827\pi$
$110$	14.2462	1.35832
$111$	−5.12311	−0.486264
$112$	0	0
$113$	− 0.438447i	− 0.0412456i	−0.999787	−	0.0206228i	$-0.993435\pi$
$113$	− 0.438447i	− 0.0412456i	0.999787	−	0.0206228i	$-0.00656491\pi$
$114$	12.0000i	1.12390i
$115$	−8.68466	−0.809849
$116$	37.6155i	3.49251i
$117$	−0.438447	−0.0405345
$118$	−18.2462	−1.67970
$119$	0	0
$120$	23.3693	2.13332
$121$	8.56155	0.778323
$122$	23.3693i	2.11576i
$123$	3.56155	0.321134
$124$	− 14.2462i	− 1.27935i
$125$	9.56155i	0.855211i
$126$	0	0
$127$	−19.8078	−1.75765	−0.878827	−	0.477140i	$-0.841674\pi$
$127$	−19.8078	−1.75765	−0.878827	+	0.477140i	$0.841674\pi$
$128$	−9.43845	−0.834249
$129$	− 4.68466i	− 0.412461i
$130$	− 4.00000i	− 0.350823i
$131$	− 14.4384i	− 1.26149i	−0.775989	−	0.630746i	$-0.782749\pi$
$131$	− 14.4384i	− 1.26149i	0.775989	−	0.630746i	$-0.217251\pi$
$132$	−7.12311	−0.619987
$133$	0	0
$134$	10.2462	0.885138
$135$	−3.56155	−0.306530
$136$	0	0
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	6.24621	0.531713
$139$	0.876894i	0.0743772i	0.999308	+	0.0371886i	$0.0118402\pi$
$139$	0.876894i	0.0743772i	−0.999308	+	0.0371886i	$0.988160\pi$
$140$	0	0
$141$	− 11.1231i	− 0.936734i
$142$	16.0000i	1.34269i
$143$	0.684658i	0.0572540i
$144$	−7.68466	−0.640388
$145$	29.3693	2.43899
$146$	31.3693i	2.59614i
$147$	7.00000i	0.577350i
$148$	23.3693i	1.92095i
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	− 19.6847i	− 1.60725i
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	30.7386	2.49323
$153$	0	0
$154$	0	0
$155$	−11.1231	−0.893429
$156$	2.00000i	0.160128i
$157$	6.68466	0.533494	0.266747	−	0.963767i	$-0.414051\pi$
$157$	6.68466	0.533494	0.266747	+	0.963767i	$0.414051\pi$
$158$	− 24.0000i	− 1.90934i
$159$	− 12.2462i	− 0.971188i
$160$	− 23.3693i	− 1.84751i
$161$	0	0
$162$	2.56155	0.201255
$163$	− 15.1231i	− 1.18453i	−0.805742	−	0.592267i	$-0.798233\pi$
$163$	− 15.1231i	− 1.18453i	0.805742	−	0.592267i	$-0.201767\pi$
$164$	− 16.2462i	− 1.26862i
$165$	5.56155i	0.432966i
$166$	2.24621	0.174340
$167$	− 19.8078i	− 1.53277i	−0.642381	−	0.766385i	$-0.722053\pi$
$167$	− 19.8078i	− 1.53277i	0.642381	−	0.766385i	$-0.277947\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	−4.68466	−0.358245
$172$	−21.3693	−1.62940
$173$	− 1.80776i	− 0.137442i	−0.997636	−	0.0687209i	$-0.978108\pi$
$173$	− 1.80776i	− 0.137442i	0.997636	−	0.0687209i	$-0.0218918\pi$
$174$	−21.1231	−1.60134
$175$	0	0
$176$	12.0000i	0.904534i
$177$	− 7.12311i	− 0.535405i
$178$	−2.87689	−0.215632
$179$	0.876894	0.0655422	0.0327711	−	0.999463i	$-0.489567\pi$
$179$	0.876894	0.0655422	0.0327711	+	0.999463i	$0.489567\pi$
$180$	16.2462i	1.21092i
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	−9.12311	−0.674399
$184$	− 16.0000i	− 1.17954i
$185$	18.2462	1.34149
$186$	8.00000	0.586588
$187$	0	0
$188$	−50.7386	−3.70050
$189$	0	0
$190$	− 42.7386i	− 3.10059i
$191$	−4.87689	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$191$	−4.87689	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$192$	1.43845i	0.103811i
$193$	− 7.75379i	− 0.558130i	−0.960272	−	0.279065i	$-0.909976\pi$
$193$	− 7.75379i	− 0.558130i	0.960272	−	0.279065i	$-0.0900245\pi$
$194$	7.36932i	0.529086i
$195$	1.56155	0.111825
$196$	31.9309	2.28078
$197$	− 8.93087i	− 0.636298i	−0.948041	−	0.318149i	$-0.896939\pi$
$197$	− 8.93087i	− 0.636298i	0.948041	−	0.318149i	$-0.103061\pi$
$198$	− 4.00000i	− 0.284268i
$199$	− 16.0000i	− 1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$	− 16.0000i	− 1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$	−50.4233	−3.56547
$201$	4.00000i	0.282138i
$202$	27.8617	1.96035
$203$	0	0
$204$	0	0
$205$	−12.6847	−0.885935
$206$	42.7386	2.97774
$207$	2.43845i	0.169484i
$208$	3.36932	0.233620
$209$	7.31534i	0.506013i
$210$	0	0
$211$	13.3693i	0.920382i	0.887820	+	0.460191i	$0.152219\pi$
$211$	13.3693i	0.920382i	−0.887820	+	0.460191i	$0.847781\pi$
$212$	−55.8617	−3.83660
$213$	−6.24621	−0.427983
$214$	12.0000i	0.820303i
$215$	16.6847i	1.13788i
$216$	− 6.56155i	− 0.446457i
$217$	0	0
$218$	− 17.6155i	− 1.19307i
$219$	−12.2462	−0.827522
$220$	25.3693	1.71040
$221$	0	0
$222$	−13.1231	−0.880765
$223$	14.9309	0.999845	0.499922	−	0.866070i	$-0.333362\pi$
$223$	14.9309	0.999845	0.499922	+	0.866070i	$0.333362\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	− 1.12311i	− 0.0747079i
$227$	− 14.0540i	− 0.932795i	−0.884575	−	0.466398i	$-0.845552\pi$
$227$	− 14.0540i	− 0.932795i	0.884575	−	0.466398i	$-0.154448\pi$
$228$	21.3693i	1.41522i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−22.2462	−1.46687
$231$	0	0
$232$	54.1080i	3.55236i
$233$	3.56155i	0.233325i	0.993172	+	0.116663i	$0.0372196\pi$
$233$	3.56155i	0.233325i	−0.993172	+	0.116663i	$0.962780\pi$
$234$	−1.12311	−0.0734197
$235$	39.6155i	2.58423i
$236$	−32.4924	−2.11508
$237$	9.36932	0.608603
$238$	0	0
$239$	−6.24621	−0.404034	−0.202017	−	0.979382i	$-0.564750\pi$
$239$	−6.24621	−0.404034	−0.202017	+	0.979382i	$0.564750\pi$
$240$	27.3693	1.76668
$241$	− 3.36932i	− 0.217037i	−0.994094	−	0.108518i	$-0.965389\pi$
$241$	− 3.36932i	− 0.217037i	0.994094	−	0.108518i	$-0.0346106\pi$
$242$	21.9309	1.40977
$243$	1.00000i	0.0641500i
$244$	41.6155i	2.66416i
$245$	− 24.9309i	− 1.59277i
$246$	9.12311	0.581668
$247$	2.05398	0.130691
$248$	− 20.4924i	− 1.30127i
$249$	0.876894i	0.0555709i
$250$	24.4924i	1.54904i
$251$	8.49242	0.536037	0.268018	−	0.963414i	$-0.413631\pi$
$251$	8.49242	0.536037	0.268018	+	0.963414i	$0.413631\pi$
$252$	0	0
$253$	3.80776	0.239392
$254$	−50.7386	−3.18363
$255$	0	0
$256$	−27.0540	−1.69087
$257$	15.3693	0.958712	0.479356	−	0.877621i	$-0.340870\pi$
$257$	15.3693	0.958712	0.479356	+	0.877621i	$0.340870\pi$
$258$	− 12.0000i	− 0.747087i
$259$	0	0
$260$	− 7.12311i	− 0.441756i
$261$	− 8.24621i	− 0.510428i
$262$	− 36.9848i	− 2.28493i
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	−10.2462	−0.630611
$265$	43.6155i	2.67928i
$266$	0	0
$267$	− 1.12311i	− 0.0687329i
$268$	18.2462	1.11456
$269$	− 16.4384i	− 1.00227i	−0.865369	−	0.501135i	$-0.832916\pi$
$269$	− 16.4384i	− 1.00227i	0.865369	−	0.501135i	$-0.167084\pi$
$270$	−9.12311	−0.555215
$271$	19.8078	1.20324	0.601618	−	0.798784i	$-0.294523\pi$
$271$	19.8078	1.20324	0.601618	+	0.798784i	$0.294523\pi$
$272$	0	0
$273$	0	0
$274$	0.630683	0.0381010
$275$	− 12.0000i	− 0.723627i
$276$	11.1231	0.669532
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	2.24621i	0.134719i
$279$	3.12311i	0.186975i
$280$	0	0
$281$	10.8769	0.648861	0.324431	−	0.945910i	$-0.394827\pi$
$281$	10.8769	0.648861	0.324431	+	0.945910i	$0.394827\pi$
$282$	− 28.4924i	− 1.69670i
$283$	21.3693i	1.27027i	0.772399	+	0.635137i	$0.219056\pi$
$283$	21.3693i	1.27027i	−0.772399	+	0.635137i	$0.780944\pi$
$284$	28.4924i	1.69071i
$285$	16.6847	0.988314
$286$	1.75379i	0.103704i
$287$	0	0
$288$	−6.56155	−0.386643
$289$	0	0
$290$	75.2311	4.41772
$291$	−2.87689	−0.168647
$292$	55.8617i	3.26906i
$293$	1.12311	0.0656125	0.0328063	−	0.999462i	$-0.489556\pi$
$293$	1.12311	0.0656125	0.0328063	+	0.999462i	$0.489556\pi$
$294$	17.9309i	1.04575i
$295$	25.3693i	1.47706i
$296$	33.6155i	1.95386i
$297$	1.56155	0.0906105
$298$	31.3693	1.81718
$299$	− 1.06913i	− 0.0618294i
$300$	− 35.0540i	− 2.02384i
$301$	0	0
$302$	−20.4924	−1.17921
$303$	10.8769i	0.624861i
$304$	36.0000	2.06474
$305$	32.4924	1.86051
$306$	0	0
$307$	32.4924	1.85444	0.927220	−	0.374516i	$-0.122191\pi$
$307$	32.4924	1.85444	0.927220	+	0.374516i	$0.122191\pi$
$308$	0	0
$309$	16.6847i	0.949157i
$310$	−28.4924	−1.61826
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	2.87689i	0.162872i
$313$	33.6155i	1.90006i	0.312156	+	0.950031i	$0.398949\pi$
$313$	33.6155i	1.90006i	−0.312156	+	0.950031i	$0.601051\pi$
$314$	17.1231	0.966313
$315$	0	0
$316$	− 42.7386i	− 2.40424i
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 31.3693i	− 1.75910i
$319$	−12.8769	−0.720968
$320$	− 5.12311i	− 0.286390i
$321$	−4.68466	−0.261472
$322$	0	0
$323$	0	0
$324$	4.56155	0.253420
$325$	−3.36932	−0.186896
$326$	− 38.7386i	− 2.14553i
$327$	6.87689	0.380293
$328$	− 23.3693i	− 1.29035i
$329$	0	0
$330$	14.2462i	0.784228i
$331$	34.9309	1.91997	0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309	1.91997	0.959987	+	0.280044i	$0.0903491\pi$
$332$	4.00000	0.219529
$333$	− 5.12311i	− 0.280744i
$334$	− 50.7386i	− 2.77629i
$335$	− 14.2462i	− 0.778354i
$336$	0	0
$337$	16.7386i	0.911811i	0.890028	+	0.455906i	$0.150685\pi$
$337$	16.7386i	0.911811i	−0.890028	+	0.455906i	$0.849315\pi$
$338$	−32.8078	−1.78451
$339$	0.438447	0.0238132
$340$	0	0
$341$	4.87689	0.264099
$342$	−12.0000	−0.648886
$343$	0	0
$344$	−30.7386	−1.65732
$345$	− 8.68466i	− 0.467566i
$346$	− 4.63068i	− 0.248947i
$347$	− 8.49242i	− 0.455897i	−0.973673	−	0.227949i	$-0.926798\pi$
$347$	− 8.49242i	− 0.455897i	0.973673	−	0.227949i	$-0.0732018\pi$
$348$	−37.6155	−2.01640
$349$	−11.5616	−0.618876	−0.309438	−	0.950920i	$-0.600141\pi$
$349$	−11.5616	−0.618876	−0.309438	+	0.950920i	$0.600141\pi$
$350$	0	0
$351$	− 0.438447i	− 0.0234026i
$352$	10.2462i	0.546125i
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	− 18.2462i	− 0.969775i
$355$	22.2462	1.18071
$356$	−5.12311	−0.271524
$357$	0	0
$358$	2.24621	0.118716
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	23.3693i	1.23167i
$361$	2.94602	0.155054
$362$	15.3693i	0.807793i
$363$	8.56155i	0.449365i
$364$	0	0
$365$	43.6155	2.28294
$366$	−23.3693	−1.22153
$367$	− 1.75379i	− 0.0915470i	−0.998952	−	0.0457735i	$-0.985425\pi$
$367$	− 1.75379i	− 0.0915470i	0.998952	−	0.0457735i	$-0.0145753\pi$
$368$	− 18.7386i	− 0.976819i
$369$	3.56155i	0.185407i
$370$	46.7386	2.42983
$371$	0	0
$372$	14.2462	0.738632
$373$	−0.246211	−0.0127483	−0.00637417	−	0.999980i	$-0.502029\pi$
$373$	−0.246211	−0.0127483	−0.00637417	+	0.999980i	$0.502029\pi$
$374$	0	0
$375$	−9.56155	−0.493756
$376$	−72.9848	−3.76391
$377$	3.61553i	0.186209i
$378$	0	0
$379$	12.0000i	0.616399i	0.951322	+	0.308199i	$0.0997264\pi$
$379$	12.0000i	0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$	− 76.1080i	− 3.90426i
$381$	− 19.8078i	− 1.01478i
$382$	−12.4924	−0.639168
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	− 9.43845i	− 0.481654i
$385$	0	0
$386$	− 19.8617i	− 1.01094i
$387$	4.68466	0.238135
$388$	13.1231i	0.666225i
$389$	−35.8617	−1.81826	−0.909131	−	0.416510i	$-0.863253\pi$
$389$	−35.8617	−1.81826	−0.909131	+	0.416510i	$0.863253\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	45.9309	2.31986
$393$	14.4384	0.728323
$394$	− 22.8769i	− 1.15252i
$395$	−33.3693	−1.67899
$396$	− 7.12311i	− 0.357950i
$397$	− 19.3693i	− 0.972118i	−0.873926	−	0.486059i	$-0.838434\pi$
$397$	− 19.3693i	− 0.972118i	0.873926	−	0.486059i	$-0.161566\pi$
$398$	− 40.9848i	− 2.05438i
$399$	0	0
$400$	−59.0540	−2.95270
$401$	39.1771i	1.95641i	0.207641	+	0.978205i	$0.433421\pi$
$401$	39.1771i	1.95641i	−0.207641	+	0.978205i	$0.566579\pi$
$402$	10.2462i	0.511035i
$403$	− 1.36932i	− 0.0682105i
$404$	49.6155	2.46846
$405$	− 3.56155i	− 0.176975i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	−32.4924	−1.60469
$411$	0.246211i	0.0121447i
$412$	76.1080	3.74957
$413$	0	0
$414$	6.24621i	0.306985i
$415$	− 3.12311i	− 0.153307i
$416$	2.87689	0.141051
$417$	−0.876894	−0.0429417
$418$	18.7386i	0.916537i
$419$	− 0.492423i	− 0.0240564i	−0.999928	−	0.0120282i	$-0.996171\pi$
$419$	− 0.492423i	− 0.0240564i	0.999928	−	0.0120282i	$-0.00382879\pi$
$420$	0	0
$421$	24.4384	1.19106	0.595529	−	0.803334i	$-0.296943\pi$
$421$	24.4384	1.19106	0.595529	+	0.803334i	$0.296943\pi$
$422$	34.2462i	1.66708i
$423$	11.1231	0.540824
$424$	−80.3542	−3.90234
$425$	0	0
$426$	−16.0000	−0.775203
$427$	0	0
$428$	21.3693i	1.03292i
$429$	−0.684658	−0.0330556
$430$	42.7386i	2.06104i
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	− 7.68466i	− 0.369728i
$433$	−26.6847	−1.28238	−0.641191	−	0.767381i	$-0.721560\pi$
$433$	−26.6847	−1.28238	−0.641191	+	0.767381i	$0.721560\pi$
$434$	0	0
$435$	29.3693i	1.40815i
$436$	− 31.3693i	− 1.50232i
$437$	− 11.4233i	− 0.546450i
$438$	−31.3693	−1.49888
$439$	22.2462i	1.06175i	0.847449	+	0.530877i	$0.178137\pi$
$439$	22.2462i	1.06175i	−0.847449	+	0.530877i	$0.821863\pi$
$440$	36.4924	1.73971
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−31.1231	−1.47870	−0.739352	−	0.673319i	$-0.764868\pi$
$443$	−31.1231	−1.47870	−0.739352	+	0.673319i	$0.764868\pi$
$444$	−23.3693	−1.10906
$445$	4.00000i	0.189618i
$446$	38.2462	1.81101
$447$	12.2462i	0.579226i
$448$	0	0
$449$	36.7386i	1.73380i	0.498479	+	0.866902i	$0.333892\pi$
$449$	36.7386i	1.73380i	−0.498479	+	0.866902i	$0.666108\pi$
$450$	19.6847	0.927944
$451$	5.56155	0.261883
$452$	− 2.00000i	− 0.0940721i
$453$	− 8.00000i	− 0.375873i
$454$	− 36.0000i	− 1.68956i
$455$	0	0
$456$	30.7386i	1.43947i
$457$	−13.8078	−0.645900	−0.322950	−	0.946416i	$-0.604675\pi$
$457$	−13.8078	−0.645900	−0.322950	+	0.946416i	$0.604675\pi$
$458$	−15.3693	−0.718161
$459$	0	0
$460$	−39.6155	−1.84708
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	−40.9848	−1.90473	−0.952364	−	0.304965i	$-0.901355\pi$
$463$	−40.9848	−1.90473	−0.952364	+	0.304965i	$0.901355\pi$
$464$	63.3693i	2.94185i
$465$	− 11.1231i	− 0.515822i
$466$	9.12311i	0.422620i
$467$	−21.3693	−0.988854	−0.494427	−	0.869219i	$-0.664622\pi$
$467$	−21.3693	−0.988854	−0.494427	+	0.869219i	$0.664622\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	101.477i	4.68080i
$471$	6.68466i	0.308013i
$472$	−46.7386	−2.15132
$473$	− 7.31534i	− 0.336360i
$474$	24.0000	1.10236
$475$	−36.0000	−1.65179
$476$	0	0
$477$	12.2462	0.560715
$478$	−16.0000	−0.731823
$479$	− 24.3002i	− 1.11030i	−0.831749	−	0.555152i	$-0.812660\pi$
$479$	− 24.3002i	− 1.11030i	0.831749	−	0.555152i	$-0.187340\pi$
$480$	23.3693	1.06666
$481$	2.24621i	0.102418i
$482$	− 8.63068i	− 0.393117i
$483$	0	0
$484$	39.0540	1.77518
$485$	10.2462	0.465256
$486$	2.56155i	0.116194i
$487$	17.3693i	0.787079i	0.919308	+	0.393539i	$0.128750\pi$
$487$	17.3693i	0.787079i	−0.919308	+	0.393539i	$0.871250\pi$
$488$	59.8617i	2.70981i
$489$	15.1231	0.683890
$490$	− 63.8617i	− 2.88498i
$491$	21.3693	0.964384	0.482192	−	0.876066i	$-0.339841\pi$
$491$	21.3693	0.964384	0.482192	+	0.876066i	$0.339841\pi$
$492$	16.2462	0.732436
$493$	0	0
$494$	5.26137	0.236720
$495$	−5.56155	−0.249973
$496$	− 24.0000i	− 1.07763i
$497$	0	0
$498$	2.24621i	0.100655i
$499$	13.3693i	0.598493i	0.954176	+	0.299246i	$0.0967353\pi$
$499$	13.3693i	0.598493i	−0.954176	+	0.299246i	$0.903265\pi$
$500$	43.6155i	1.95055i
$501$	19.8078	0.884946
$502$	21.7538	0.970919
$503$	− 29.5616i	− 1.31808i	−0.752106	−	0.659042i	$-0.770962\pi$
$503$	− 29.5616i	− 1.31808i	0.752106	−	0.659042i	$-0.229038\pi$
$504$	0	0
$505$	− 38.7386i	− 1.72385i
$506$	9.75379	0.433609
$507$	− 12.8078i	− 0.568813i
$508$	−90.3542	−4.00882
$509$	25.1231	1.11356	0.556781	−	0.830659i	$-0.312036\pi$
$509$	25.1231	1.11356	0.556781	+	0.830659i	$0.312036\pi$
$510$	0	0
$511$	0	0
$512$	−50.4233	−2.22842
$513$	− 4.68466i	− 0.206833i
$514$	39.3693	1.73651
$515$	− 59.4233i	− 2.61850i
$516$	− 21.3693i	− 0.940732i
$517$	− 17.3693i	− 0.763902i
$518$	0	0
$519$	1.80776	0.0793520
$520$	− 10.2462i	− 0.449326i
$521$	− 35.5616i	− 1.55798i	−0.627036	−	0.778990i	$-0.715732\pi$
$521$	− 35.5616i	− 1.55798i	0.627036	−	0.778990i	$-0.284268\pi$
$522$	− 21.1231i	− 0.924533i
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	− 65.8617i	− 2.87718i
$525$	0	0
$526$	52.4924	2.28878
$527$	0	0
$528$	−12.0000	−0.522233
$529$	17.0540	0.741477
$530$	111.723i	4.85296i
$531$	7.12311	0.309116
$532$	0	0
$533$	− 1.56155i	− 0.0676384i
$534$	− 2.87689i	− 0.124495i
$535$	16.6847	0.721341
$536$	26.2462	1.13366
$537$	0.876894i	0.0378408i
$538$	− 42.1080i	− 1.81540i
$539$	10.9309i	0.470826i
$540$	−16.2462	−0.699126
$541$	− 34.1080i	− 1.46642i	−0.680005	−	0.733208i	$-0.738022\pi$
$541$	− 34.1080i	− 1.46642i	0.680005	−	0.733208i	$-0.261978\pi$
$542$	50.7386	2.17941
$543$	−6.00000	−0.257485
$544$	0	0
$545$	−24.4924	−1.04914
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	1.12311	0.0479767
$549$	− 9.12311i	− 0.389365i
$550$	− 30.7386i	− 1.31070i
$551$	38.6307i	1.64572i
$552$	16.0000	0.681005
$553$	0	0
$554$	− 15.3693i	− 0.652980i
$555$	18.2462i	0.774509i
$556$	4.00000i	0.169638i
$557$	26.4924	1.12252	0.561260	−	0.827640i	$-0.310317\pi$
$557$	26.4924	1.12252	0.561260	+	0.827640i	$0.310317\pi$
$558$	8.00000i	0.338667i
$559$	−2.05398	−0.0868739
$560$	0	0
$561$	0	0
$562$	27.8617	1.17528
$563$	−31.1231	−1.31168	−0.655841	−	0.754899i	$-0.727686\pi$
$563$	−31.1231	−1.31168	−0.655841	+	0.754899i	$0.727686\pi$
$564$	− 50.7386i	− 2.13648i
$565$	−1.56155	−0.0656950
$566$	54.7386i	2.30084i
$567$	0	0
$568$	40.9848i	1.71969i
$569$	−21.1231	−0.885527	−0.442763	−	0.896639i	$-0.646002\pi$
$569$	−21.1231	−0.885527	−0.442763	+	0.896639i	$0.646002\pi$
$570$	42.7386	1.79012
$571$	− 30.7386i	− 1.28637i	−0.765710	−	0.643186i	$-0.777612\pi$
$571$	− 30.7386i	− 1.28637i	0.765710	−	0.643186i	$-0.222388\pi$
$572$	3.12311i	0.130584i
$573$	− 4.87689i	− 0.203735i
$574$	0	0
$575$	18.7386i	0.781455i
$576$	−1.43845	−0.0599353
$577$	−3.94602	−0.164275	−0.0821376	−	0.996621i	$-0.526175\pi$
$577$	−3.94602	−0.164275	−0.0821376	+	0.996621i	$0.526175\pi$
$578$	0	0
$579$	7.75379	0.322236
$580$	133.970	5.56279
$581$	0	0
$582$	−7.36932	−0.305468
$583$	− 19.1231i	− 0.791998i
$584$	80.3542i	3.32508i
$585$	1.56155i	0.0645623i
$586$	2.87689	0.118843
$587$	28.9848	1.19633	0.598166	−	0.801372i	$-0.295896\pi$
$587$	28.9848	1.19633	0.598166	+	0.801372i	$0.295896\pi$
$588$	31.9309i	1.31681i
$589$	− 14.6307i	− 0.602847i
$590$	64.9848i	2.67538i
$591$	8.93087	0.367367
$592$	39.3693i	1.61807i
$593$	−27.7538	−1.13971	−0.569856	−	0.821745i	$-0.693001\pi$
$593$	−27.7538	−1.13971	−0.569856	+	0.821745i	$0.693001\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	55.8617	2.28819
$597$	16.0000	0.654836
$598$	− 2.73863i	− 0.111991i
$599$	−0.384472	−0.0157091	−0.00785455	−	0.999969i	$-0.502500\pi$
$599$	−0.384472	−0.0157091	−0.00785455	+	0.999969i	$0.502500\pi$
$600$	− 50.4233i	− 2.05852i
$601$	− 30.9848i	− 1.26390i	−0.775010	−	0.631949i	$-0.782255\pi$
$601$	− 30.9848i	− 1.26390i	0.775010	−	0.631949i	$-0.217745\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−36.4924	−1.48486
$605$	− 30.4924i	− 1.23969i
$606$	27.8617i	1.13181i
$607$	9.36932i	0.380289i	0.981756	+	0.190144i	$0.0608956\pi$
$607$	9.36932i	0.380289i	−0.981756	+	0.190144i	$0.939104\pi$
$608$	30.7386	1.24662
$609$	0	0
$610$	83.2311	3.36993
$611$	−4.87689	−0.197298
$612$	0	0
$613$	14.6847	0.593108	0.296554	−	0.955016i	$-0.404163\pi$
$613$	14.6847	0.593108	0.296554	+	0.955016i	$0.404163\pi$
$614$	83.2311	3.35893
$615$	− 12.6847i	− 0.511495i
$616$	0	0
$617$	44.2462i	1.78129i	0.454704	+	0.890643i	$0.349745\pi$
$617$	44.2462i	1.78129i	−0.454704	+	0.890643i	$0.650255\pi$
$618$	42.7386i	1.71920i
$619$	5.36932i	0.215811i	0.994161	+	0.107906i	$0.0344144\pi$
$619$	5.36932i	0.215811i	−0.994161	+	0.107906i	$0.965586\pi$
$620$	−50.7386	−2.03771
$621$	−2.43845	−0.0978515
$622$	0	0
$623$	0	0
$624$	3.36932i	0.134881i
$625$	−4.36932	−0.174773
$626$	86.1080i	3.44157i
$627$	−7.31534	−0.292147
$628$	30.4924	1.21678
$629$	0	0
$630$	0	0
$631$	0.684658	0.0272558	0.0136279	−	0.999907i	$-0.495662\pi$
$631$	0.684658	0.0272558	0.0136279	+	0.999907i	$0.495662\pi$
$632$	− 61.4773i	− 2.44543i
$633$	−13.3693	−0.531383
$634$	− 46.1080i	− 1.83118i
$635$	70.5464i	2.79955i
$636$	− 55.8617i	− 2.21506i
$637$	3.06913	0.121603
$638$	−32.9848	−1.30588
$639$	− 6.24621i	− 0.247096i
$640$	33.6155i	1.32877i
$641$	28.9309i	1.14270i	0.820706	+	0.571350i	$0.193580\pi$
$641$	28.9309i	1.14270i	−0.820706	+	0.571350i	$0.806420\pi$
$642$	−12.0000	−0.473602
$643$	13.7538i	0.542396i	0.962524	+	0.271198i	$0.0874199\pi$
$643$	13.7538i	0.542396i	−0.962524	+	0.271198i	$0.912580\pi$
$644$	0	0
$645$	−16.6847	−0.656958
$646$	0	0
$647$	−9.36932	−0.368346	−0.184173	−	0.982894i	$-0.558961\pi$
$647$	−9.36932	−0.368346	−0.184173	+	0.982894i	$0.558961\pi$
$648$	6.56155	0.257762
$649$	− 11.1231i	− 0.436620i
$650$	−8.63068	−0.338523
$651$	0	0
$652$	− 68.9848i	− 2.70166i
$653$	− 32.9309i	− 1.28868i	−0.764737	−	0.644342i	$-0.777131\pi$
$653$	− 32.9309i	− 1.28868i	0.764737	−	0.644342i	$-0.222869\pi$
$654$	17.6155	0.688822
$655$	−51.4233	−2.00927
$656$	− 27.3693i	− 1.06859i
$657$	− 12.2462i	− 0.477770i
$658$	0	0
$659$	−9.86174	−0.384159	−0.192079	−	0.981379i	$-0.561523\pi$
$659$	−9.86174	−0.384159	−0.192079	+	0.981379i	$0.561523\pi$
$660$	25.3693i	0.987499i
$661$	−13.3153	−0.517907	−0.258953	−	0.965890i	$-0.583378\pi$
$661$	−13.3153	−0.517907	−0.258953	+	0.965890i	$0.583378\pi$
$662$	89.4773	3.47763
$663$	0	0
$664$	5.75379	0.223290
$665$	0	0
$666$	− 13.1231i	− 0.508510i
$667$	20.1080	0.778583
$668$	− 90.3542i	− 3.49591i
$669$	14.9309i	0.577261i
$670$	− 36.4924i	− 1.40983i
$671$	−14.2462	−0.549969
$672$	0	0
$673$	− 0.738634i	− 0.0284722i	−0.999899	−	0.0142361i	$-0.995468\pi$
$673$	− 0.738634i	− 0.0284722i	0.999899	−	0.0142361i	$-0.00453165\pi$
$674$	42.8769i	1.65156i
$675$	7.68466i	0.295783i
$676$	−58.4233	−2.24705
$677$	1.31534i	0.0505527i	0.999681	+	0.0252763i	$0.00804657\pi$
$677$	1.31534i	0.0505527i	−0.999681	+	0.0252763i	$0.991953\pi$
$678$	1.12311	0.0431326
$679$	0	0
$680$	0	0
$681$	14.0540	0.538550
$682$	12.4924	0.478360
$683$	9.56155i	0.365863i	0.983126	+	0.182931i	$0.0585586\pi$
$683$	9.56155i	0.365863i	−0.983126	+	0.182931i	$0.941441\pi$
$684$	−21.3693	−0.817076
$685$	− 0.876894i	− 0.0335044i
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	−36.0000	−1.37249
$689$	−5.36932	−0.204555
$690$	− 22.2462i	− 0.846899i
$691$	− 28.9848i	− 1.10264i	−0.834295	−	0.551318i	$-0.814125\pi$
$691$	− 28.9848i	− 1.10264i	0.834295	−	0.551318i	$-0.185875\pi$
$692$	− 8.24621i	− 0.313474i
$693$	0	0
$694$	− 21.7538i	− 0.825763i
$695$	3.12311	0.118466
$696$	−54.1080	−2.05096
$697$	0	0
$698$	−29.6155	−1.12096
$699$	−3.56155	−0.134710
$700$	0	0
$701$	15.3693	0.580491	0.290246	−	0.956952i	$-0.406263\pi$
$701$	15.3693	0.580491	0.290246	+	0.956952i	$0.406263\pi$
$702$	− 1.12311i	− 0.0423889i
$703$	24.0000i	0.905177i
$704$	2.24621i	0.0846573i
$705$	−39.6155	−1.49201
$706$	−26.8769	−1.01153
$707$	0	0
$708$	− 32.4924i	− 1.22114i
$709$	44.7386i	1.68019i	0.542436	+	0.840097i	$0.317502\pi$
$709$	44.7386i	1.68019i	−0.542436	+	0.840097i	$0.682498\pi$
$710$	56.9848	2.13860
$711$	9.36932i	0.351377i
$712$	−7.36932	−0.276177
$713$	−7.61553	−0.285204
$714$	0	0
$715$	2.43845	0.0911928
$716$	4.00000	0.149487
$717$	− 6.24621i	− 0.233269i
$718$	−36.4924	−1.36189
$719$	11.8078i	0.440355i	0.975460	+	0.220178i	$0.0706637\pi$
$719$	11.8078i	0.440355i	−0.975460	+	0.220178i	$0.929336\pi$
$720$	27.3693i	1.01999i
$721$	0	0
$722$	7.54640	0.280848
$723$	3.36932	0.125306
$724$	27.3693i	1.01717i
$725$	− 63.3693i	− 2.35348i
$726$	21.9309i	0.813931i
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	111.723	4.13507
$731$	0	0
$732$	−41.6155	−1.53815
$733$	11.7538	0.434136	0.217068	−	0.976156i	$-0.430351\pi$
$733$	11.7538	0.434136	0.217068	+	0.976156i	$0.430351\pi$
$734$	− 4.49242i	− 0.165818i
$735$	24.9309	0.919589
$736$	− 16.0000i	− 0.589768i
$737$	6.24621i	0.230082i
$738$	9.12311i	0.335826i
$739$	20.6847	0.760897	0.380449	−	0.924802i	$-0.375770\pi$
$739$	20.6847	0.760897	0.380449	+	0.924802i	$0.375770\pi$
$740$	83.2311	3.05963
$741$	2.05398i	0.0754547i
$742$	0	0
$743$	− 28.4924i	− 1.04529i	−0.852552	−	0.522643i	$-0.824946\pi$
$743$	− 28.4924i	− 1.04529i	0.852552	−	0.522643i	$-0.175054\pi$
$744$	20.4924	0.751289
$745$	− 43.6155i	− 1.59795i
$746$	−0.630683	−0.0230909
$747$	−0.876894	−0.0320839
$748$	0	0
$749$	0	0
$750$	−24.4924	−0.894337
$751$	25.3693i	0.925740i	0.886426	+	0.462870i	$0.153180\pi$
$751$	25.3693i	0.925740i	−0.886426	+	0.462870i	$0.846820\pi$
$752$	−85.4773	−3.11704
$753$	8.49242i	0.309481i
$754$	9.26137i	0.337279i
$755$	28.4924i	1.03695i
$756$	0	0
$757$	−16.0540	−0.583492	−0.291746	−	0.956496i	$-0.594236\pi$
$757$	−16.0540	−0.583492	−0.291746	+	0.956496i	$0.594236\pi$
$758$	30.7386i	1.11648i
$759$	3.80776i	0.138213i
$760$	− 109.477i	− 3.97116i
$761$	−15.7538	−0.571074	−0.285537	−	0.958368i	$-0.592172\pi$
$761$	−15.7538	−0.571074	−0.285537	+	0.958368i	$0.592172\pi$
$762$	− 50.7386i	− 1.83807i
$763$	0	0
$764$	−22.2462	−0.804840
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−3.12311	−0.112769
$768$	− 27.0540i	− 0.976226i
$769$	40.5464	1.46214	0.731070	−	0.682302i	$-0.239021\pi$
$769$	40.5464	1.46214	0.731070	+	0.682302i	$0.239021\pi$
$770$	0	0
$771$	15.3693i	0.553512i
$772$	− 35.3693i	− 1.27297i
$773$	8.63068	0.310424	0.155212	−	0.987881i	$-0.450394\pi$
$773$	8.63068	0.310424	0.155212	+	0.987881i	$0.450394\pi$
$774$	12.0000	0.431331
$775$	24.0000i	0.862105i
$776$	18.8769i	0.677641i
$777$	0	0
$778$	−91.8617	−3.29340
$779$	− 16.6847i	− 0.597790i
$780$	7.12311	0.255048
$781$	−9.75379	−0.349018
$782$	0	0
$783$	8.24621	0.294696
$784$	53.7926	1.92116
$785$	− 23.8078i	− 0.849736i
$786$	36.9848	1.31921
$787$	10.2462i	0.365238i	0.983184	+	0.182619i	$0.0584575\pi$
$787$	10.2462i	0.365238i	−0.983184	+	0.182619i	$0.941543\pi$
$788$	− 40.7386i	− 1.45125i
$789$	20.4924i	0.729550i
$790$	−85.4773	−3.04114
$791$	0	0
$792$	− 10.2462i	− 0.364083i
$793$	4.00000i	0.142044i
$794$	− 49.6155i	− 1.76079i
$795$	−43.6155	−1.54688
$796$	− 72.9848i	− 2.58688i
$797$	9.61553	0.340599	0.170300	−	0.985392i	$-0.445526\pi$
$797$	9.61553	0.340599	0.170300	+	0.985392i	$0.445526\pi$
$798$	0	0
$799$	0	0
$800$	−50.4233	−1.78273
$801$	1.12311	0.0396830
$802$	100.354i	3.54363i
$803$	−19.1231	−0.674840
$804$	18.2462i	0.643494i
$805$	0	0
$806$	− 3.50758i	− 0.123549i
$807$	16.4384	0.578661
$808$	71.3693	2.51076
$809$	15.9460i	0.560632i	0.959908	+	0.280316i	$0.0904393\pi$
$809$	15.9460i	0.560632i	−0.959908	+	0.280316i	$0.909561\pi$
$810$	− 9.12311i	− 0.320553i
$811$	45.3693i	1.59313i	0.604551	+	0.796566i	$0.293352\pi$
$811$	45.3693i	1.59313i	−0.604551	+	0.796566i	$0.706648\pi$
$812$	0	0
$813$	19.8078i	0.694689i
$814$	−20.4924	−0.718259
$815$	−53.8617	−1.88669
$816$	0	0
$817$	−21.9460	−0.767794
$818$	−37.6155	−1.31520
$819$	0	0
$820$	−57.8617	−2.02062
$821$	12.4384i	0.434105i	0.976160	+	0.217052i	$0.0696443\pi$
$821$	12.4384i	0.434105i	−0.976160	+	0.217052i	$0.930356\pi$
$822$	0.630683i	0.0219976i
$823$	3.50758i	0.122266i	0.998130	+	0.0611332i	$0.0194715\pi$
$823$	3.50758i	0.122266i	−0.998130	+	0.0611332i	$0.980529\pi$
$824$	109.477	3.81382
$825$	12.0000	0.417786
$826$	0	0
$827$	47.4233i	1.64907i	0.565811	+	0.824535i	$0.308563\pi$
$827$	47.4233i	1.64907i	−0.565811	+	0.824535i	$0.691437\pi$
$828$	11.1231i	0.386555i
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	− 8.00000i	− 0.277684i
$831$	6.00000	0.208138
$832$	0.630683	0.0218650
$833$	0	0
$834$	−2.24621	−0.0777799
$835$	−70.5464	−2.44136
$836$	33.3693i	1.15410i
$837$	−3.12311	−0.107950
$838$	− 1.26137i	− 0.0435732i
$839$	− 26.0540i	− 0.899483i	−0.893159	−	0.449742i	$-0.851516\pi$
$839$	− 26.0540i	− 0.899483i	0.893159	−	0.449742i	$-0.148484\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	62.6004	2.15735
$843$	10.8769i	0.374620i
$844$	60.9848i	2.09918i
$845$	45.6155i	1.56922i
$846$	28.4924	0.979590
$847$	0	0
$848$	−94.1080	−3.23168
$849$	−21.3693	−0.733393
$850$	0	0
$851$	12.4924	0.428235
$852$	−28.4924	−0.976134
$853$	28.7386i	0.983992i	0.870597	+	0.491996i	$0.163733\pi$
$853$	28.7386i	0.983992i	−0.870597	+	0.491996i	$0.836267\pi$
$854$	0	0
$855$	16.6847i	0.570603i
$856$	30.7386i	1.05062i
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	−1.75379	−0.0598734
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	76.1080i	2.59526i
$861$	0	0
$862$	− 61.4773i	− 2.09392i
$863$	−9.75379	−0.332023	−0.166011	−	0.986124i	$-0.553089\pi$
$863$	−9.75379	−0.332023	−0.166011	+	0.986124i	$0.553089\pi$
$864$	− 6.56155i	− 0.223229i
$865$	−6.43845	−0.218914
$866$	−68.3542	−2.32277
$867$	0	0
$868$	0	0
$869$	14.6307	0.496312
$870$	75.2311i	2.55057i
$871$	1.75379	0.0594249
$872$	− 45.1231i	− 1.52806i
$873$	− 2.87689i	− 0.0973681i
$874$	− 29.2614i	− 0.989780i
$875$	0	0
$876$	−55.8617	−1.88739
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	56.9848i	1.92315i
$879$	1.12311i	0.0378814i
$880$	42.7386	1.44072
$881$	− 40.2462i	− 1.35593i	−0.735095	−	0.677965i	$-0.762862\pi$
$881$	− 40.2462i	− 1.35593i	0.735095	−	0.677965i	$-0.237138\pi$
$882$	−17.9309	−0.603764
$883$	−23.4233	−0.788257	−0.394128	−	0.919055i	$-0.628953\pi$
$883$	−23.4233	−0.788257	−0.394128	+	0.919055i	$0.628953\pi$
$884$	0	0
$885$	−25.3693	−0.852780
$886$	−79.7235	−2.67836
$887$	− 18.4384i	− 0.619102i	−0.950883	−	0.309551i	$-0.899821\pi$
$887$	− 18.4384i	− 0.619102i	0.950883	−	0.309551i	$-0.100179\pi$
$888$	−33.6155	−1.12806
$889$	0	0
$890$	10.2462i	0.343454i
$891$	1.56155i	0.0523140i
$892$	68.1080	2.28042
$893$	−52.1080	−1.74373
$894$	31.3693i	1.04915i
$895$	− 3.12311i	− 0.104394i
$896$	0	0
$897$	1.06913	0.0356972
$898$	94.1080i	3.14042i
$899$	25.7538	0.858937
$900$	35.0540	1.16847
$901$	0	0
$902$	14.2462	0.474347
$903$	0	0
$904$	− 2.87689i	− 0.0956841i
$905$	21.3693	0.710340
$906$	− 20.4924i	− 0.680815i
$907$	− 9.86174i	− 0.327454i	−0.986506	−	0.163727i	$-0.947648\pi$
$907$	− 9.86174i	− 0.327454i	0.986506	−	0.163727i	$-0.0523516\pi$
$908$	− 64.1080i	− 2.12750i
$909$	−10.8769	−0.360764
$910$	0	0
$911$	24.3002i	0.805101i	0.915398	+	0.402551i	$0.131876\pi$
$911$	24.3002i	0.805101i	−0.915398	+	0.402551i	$0.868124\pi$
$912$	36.0000i	1.19208i
$913$	1.36932i	0.0453178i
$914$	−35.3693	−1.16991
$915$	32.4924i	1.07417i
$916$	−27.3693	−0.904308
$917$	0	0
$918$	0	0
$919$	−16.6847	−0.550376	−0.275188	−	0.961390i	$-0.588740\pi$
$919$	−16.6847	−0.550376	−0.275188	+	0.961390i	$0.588740\pi$
$920$	−56.9848	−1.87873
$921$	32.4924i	1.07066i
$922$	21.1231	0.695652
$923$	2.73863i	0.0901432i
$924$	0	0
$925$	− 39.3693i	− 1.29446i
$926$	−104.985	−3.45002
$927$	−16.6847	−0.547996
$928$	54.1080i	1.77618i
$929$	3.06913i	0.100695i	0.998732	+	0.0503474i	$0.0160329\pi$
$929$	3.06913i	0.100695i	−0.998732	+	0.0503474i	$0.983967\pi$
$930$	− 28.4924i	− 0.934303i
$931$	32.7926	1.07473
$932$	16.2462i	0.532162i
$933$	0	0
$934$	−54.7386	−1.79110
$935$	0	0
$936$	−2.87689	−0.0940342
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−33.6155	−1.09700
$940$	180.708i	5.89406i
$941$	30.0000i	0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$	30.0000i	0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	17.1231i	0.557901i
$943$	−8.68466	−0.282811
$944$	−54.7386	−1.78159
$945$	0	0
$946$	− 18.7386i	− 0.609246i
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	42.7386	1.38809
$949$	5.36932i	0.174295i
$950$	−92.2159	−2.99188
$951$	18.0000	0.583690
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	31.3693	1.01562
$955$	17.3693i	0.562058i
$956$	−28.4924	−0.921511
$957$	− 12.8769i	− 0.416251i
$958$	− 62.2462i	− 2.01108i
$959$	0	0
$960$	5.12311	0.165348
$961$	21.2462	0.685362
$962$	5.75379i	0.185510i
$963$	− 4.68466i	− 0.150961i
$964$	− 15.3693i	− 0.495012i
$965$	−27.6155	−0.888975
$966$	0	0
$967$	−42.4384	−1.36473	−0.682364	−	0.731012i	$-0.739048\pi$
$967$	−42.4384	−1.36473	−0.682364	+	0.731012i	$0.739048\pi$
$968$	56.1771	1.80560
$969$	0	0
$970$	26.2462	0.842715
$971$	43.6155	1.39969	0.699844	−	0.714295i	$-0.253253\pi$
$971$	43.6155	1.39969	0.699844	+	0.714295i	$0.253253\pi$
$972$	4.56155i	0.146312i
$973$	0	0
$974$	44.4924i	1.42563i
$975$	− 3.36932i	− 0.107904i
$976$	70.1080i	2.24410i
$977$	−8.24621	−0.263820	−0.131910	−	0.991262i	$-0.542111\pi$
$977$	−8.24621	−0.263820	−0.131910	+	0.991262i	$0.542111\pi$
$978$	38.7386	1.23872
$979$	− 1.75379i	− 0.0560513i
$980$	− 113.723i	− 3.63276i
$981$	6.87689i	0.219562i
$982$	54.7386	1.74678
$983$	− 30.9309i	− 0.986542i	−0.869876	−	0.493271i	$-0.835801\pi$
$983$	− 30.9309i	− 0.986542i	0.869876	−	0.493271i	$-0.164199\pi$
$984$	23.3693	0.744987
$985$	−31.8078	−1.01348
$986$	0	0
$987$	0	0
$988$	9.36932	0.298078
$989$	11.4233i	0.363240i
$990$	−14.2462	−0.452774
$991$	− 42.7386i	− 1.35764i	−0.734306	−	0.678819i	$-0.762492\pi$
$991$	− 42.7386i	− 1.35764i	0.734306	−	0.678819i	$-0.237508\pi$
$992$	− 20.4924i	− 0.650635i
$993$	34.9309i	1.10850i
$994$	0	0
$995$	−56.9848	−1.80654
$996$	4.00000i	0.126745i
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	34.2462i	1.08404i
$999$	5.12311	0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.c.577.4		4
17.2	even	8		867.2.e.f.829.4		8
17.3	odd	16		867.2.h.j.688.3		16
17.4	even	4		867.2.a.f.1.1		2
17.5	odd	16		867.2.h.j.757.1		16
17.6	odd	16		867.2.h.j.712.4		16
17.7	odd	16		867.2.h.j.733.2		16
17.8	even	8		867.2.e.f.616.2		8
17.9	even	8		867.2.e.f.616.1		8
17.10	odd	16		867.2.h.j.733.1		16
17.11	odd	16		867.2.h.j.712.3		16
17.12	odd	16		867.2.h.j.757.2		16
17.13	even	4		51.2.a.b.1.1	✓	2
17.14	odd	16		867.2.h.j.688.4		16
17.15	even	8		867.2.e.f.829.3		8
17.16	even	2	inner	867.2.d.c.577.3		4
51.38	odd	4		2601.2.a.t.1.2		2
51.47	odd	4		153.2.a.e.1.2		2
68.47	odd	4		816.2.a.m.1.2		2
85.13	odd	4		1275.2.b.d.1174.4		4
85.47	odd	4		1275.2.b.d.1174.1		4
85.64	even	4		1275.2.a.n.1.2		2
119.13	odd	4		2499.2.a.o.1.1		2
136.13	even	4		3264.2.a.bl.1.1		2
136.115	odd	4		3264.2.a.bg.1.1		2
187.98	odd	4		6171.2.a.p.1.2		2
204.47	even	4		2448.2.a.v.1.1		2
221.64	even	4		8619.2.a.q.1.2		2
255.149	odd	4		3825.2.a.s.1.1		2
357.251	even	4		7497.2.a.v.1.2		2
408.149	odd	4		9792.2.a.cy.1.2		2
408.251	even	4		9792.2.a.cz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.1	✓	2	17.13	even	4
153.2.a.e.1.2		2	51.47	odd	4
816.2.a.m.1.2		2	68.47	odd	4
867.2.a.f.1.1		2	17.4	even	4
867.2.d.c.577.3		4	17.16	even	2	inner
867.2.d.c.577.4		4	1.1	even	1	trivial
867.2.e.f.616.1		8	17.9	even	8
867.2.e.f.616.2		8	17.8	even	8
867.2.e.f.829.3		8	17.15	even	8
867.2.e.f.829.4		8	17.2	even	8
867.2.h.j.688.3		16	17.3	odd	16
867.2.h.j.688.4		16	17.14	odd	16
867.2.h.j.712.3		16	17.11	odd	16
867.2.h.j.712.4		16	17.6	odd	16
867.2.h.j.733.1		16	17.10	odd	16
867.2.h.j.733.2		16	17.7	odd	16
867.2.h.j.757.1		16	17.5	odd	16
867.2.h.j.757.2		16	17.12	odd	16
1275.2.a.n.1.2		2	85.64	even	4
1275.2.b.d.1174.1		4	85.47	odd	4
1275.2.b.d.1174.4		4	85.13	odd	4
2448.2.a.v.1.1		2	204.47	even	4
2499.2.a.o.1.1		2	119.13	odd	4
2601.2.a.t.1.2		2	51.38	odd	4
3264.2.a.bg.1.1		2	136.115	odd	4
3264.2.a.bl.1.1		2	136.13	even	4
3825.2.a.s.1.1		2	255.149	odd	4
6171.2.a.p.1.2		2	187.98	odd	4
7497.2.a.v.1.2		2	357.251	even	4
8619.2.a.q.1.2		2	221.64	even	4
9792.2.a.cy.1.2		2	408.149	odd	4
9792.2.a.cz.1.2		2	408.251	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +1.00000i q^{3} +4.56155 q^{4} -3.56155i q^{5} +2.56155i q^{6} +6.56155 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.56155 q^{2} +1.00000i q^{3} +4.56155 q^{4} -3.56155i q^{5} +2.56155i q^{6} +6.56155 q^{8} -1.00000 q^{9} -9.12311i q^{10} +1.56155i q^{11} +4.56155i q^{12} +0.438447 q^{13} +3.56155 q^{15} +7.68466 q^{16} -2.56155 q^{18} +4.68466 q^{19} -16.2462i q^{20} +4.00000i q^{22} -2.43845i q^{23} +6.56155i q^{24} -7.68466 q^{25} +1.12311 q^{26} -1.00000i q^{27} +8.24621i q^{29} +9.12311 q^{30} -3.12311i q^{31} +6.56155 q^{32} -1.56155 q^{33} -4.56155 q^{36} +5.12311i q^{37} +12.0000 q^{38} +0.438447i q^{39} -23.3693i q^{40} -3.56155i q^{41} -4.68466 q^{43} +7.12311i q^{44} +3.56155i q^{45} -6.24621i q^{46} -11.1231 q^{47} +7.68466i q^{48} +7.00000 q^{49} -19.6847 q^{50} +2.00000 q^{52} -12.2462 q^{53} -2.56155i q^{54} +5.56155 q^{55} +4.68466i q^{57} +21.1231i q^{58} -7.12311 q^{59} +16.2462 q^{60} +9.12311i q^{61} -8.00000i q^{62} +1.43845 q^{64} -1.56155i q^{65} -4.00000 q^{66} +4.00000 q^{67} +2.43845 q^{69} +6.24621i q^{71} -6.56155 q^{72} +12.2462i q^{73} +13.1231i q^{74} -7.68466i q^{75} +21.3693 q^{76} +1.12311i q^{78} -9.36932i q^{79} -27.3693i q^{80} +1.00000 q^{81} -9.12311i q^{82} +0.876894 q^{83} -12.0000 q^{86} -8.24621 q^{87} +10.2462i q^{88} -1.12311 q^{89} +9.12311i q^{90} -11.1231i q^{92} +3.12311 q^{93} -28.4924 q^{94} -16.6847i q^{95} +6.56155i q^{96} +2.87689i q^{97} +17.9309 q^{98} -1.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9	\( 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9} + 10 q^{13} + 6 q^{15} + 6 q^{16} - 2 q^{18} - 6 q^{19} - 6 q^{25} - 12 q^{26} + 20 q^{30} + 18 q^{32} + 2 q^{33} - 10 q^{36} + 48 q^{38} + 6 q^{43} - 28 q^{47} + 28 q^{49} - 54 q^{50} + 8 q^{52} - 16 q^{53} + 14 q^{55} - 12 q^{59} + 32 q^{60} + 14 q^{64} - 16 q^{66} + 16 q^{67} + 18 q^{69} - 18 q^{72} + 36 q^{76} + 4 q^{81} + 20 q^{83} - 48 q^{86} + 12 q^{89} - 4 q^{93} - 48 q^{94} + 14 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9 + 10 * q^13 + 6 * q^15 + 6 * q^16 - 2 * q^18 - 6 * q^19 - 6 * q^25 - 12 * q^26 + 20 * q^30 + 18 * q^32 + 2 * q^33 - 10 * q^36 + 48 * q^38 + 6 * q^43 - 28 * q^47 + 28 * q^49 - 54 * q^50 + 8 * q^52 - 16 * q^53 + 14 * q^55 - 12 * q^59 + 32 * q^60 + 14 * q^64 - 16 * q^66 + 16 * q^67 + 18 * q^69 - 18 * q^72 + 36 * q^76 + 4 * q^81 + 20 * q^83 - 48 * q^86 + 12 * q^89 - 4 * q^93 - 48 * q^94 + 14 * q^98

Introduction

L-functions

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Database

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