LMFDB - Embedded newform 96.4.f.b.47.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,Mod(47,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("96.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	96.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:	$5.66418336055$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 $ x^8 - 10x^6 + 120x^4 - 640*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.7
Root			$1.95291 - 2.04601i$ of defining polynomial
Character	$\chi$	$=$	96.47
Dual form			96.4.f.b.47.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.37228 + 2.80770i) q^{3} -13.1715 q^{5} +26.9490i q^{7} +(11.2337 + 24.5521i) q^{9} +O(q^{10})$	\(q+(4.37228 + 2.80770i) q^{3} -13.1715 q^{5} +26.9490i q^{7} +(11.2337 + 24.5521i) q^{9} +21.9834i q^{11} -10.0326i q^{13} +(-57.5896 - 36.9816i) q^{15} +6.09352i q^{17} +40.1902 q^{19} +(-75.6647 + 117.829i) q^{21} +9.80703 q^{23} +48.4891 q^{25} +(-19.8179 + 138.889i) q^{27} +164.501 q^{29} +47.0143i q^{31} +(-61.7228 + 96.1178i) q^{33} -354.960i q^{35} -205.560i q^{37} +(28.1685 - 43.8654i) q^{39} -419.120i q^{41} -205.038 q^{43} +(-147.965 - 323.388i) q^{45} +566.089 q^{47} -383.250 q^{49} +(-17.1087 + 26.6426i) q^{51} +342.173 q^{53} -289.555i q^{55} +(175.723 + 112.842i) q^{57} +3.70288i q^{59} +717.005i q^{61} +(-661.654 + 302.737i) q^{63} +132.145i q^{65} +238.701 q^{67} +(42.8791 + 27.5351i) q^{69} -517.054 q^{71} +984.195 q^{73} +(212.008 + 136.143i) q^{75} -592.432 q^{77} -329.100i q^{79} +(-476.609 + 551.621i) q^{81} +625.333i q^{83} -80.2609i q^{85} +(719.244 + 461.868i) q^{87} +238.583i q^{89} +270.369 q^{91} +(-132.002 + 205.560i) q^{93} -529.366 q^{95} -1001.01 q^{97} +(-539.739 + 246.955i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} - 48 q^{9}+O(q^{10}) $ 8 * q + 12 * q^3 - 48 * q^9	$ 8 q + 12 q^{3} - 48 q^{9} - 184 q^{19} + 296 q^{25} + 324 q^{27} - 264 q^{33} + 152 q^{43} - 952 q^{49} - 1056 q^{51} + 1176 q^{57} + 1496 q^{67} + 1072 q^{73} + 708 q^{75} - 504 q^{81} - 3168 q^{91} - 3872 q^{97} - 2112 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 - 48 * q^9 - 184 * q^19 + 296 * q^25 + 324 * q^27 - 264 * q^33 + 152 * q^43 - 952 * q^49 - 1056 * q^51 + 1176 * q^57 + 1496 * q^67 + 1072 * q^73 + 708 * q^75 - 504 * q^81 - 3168 * q^91 - 3872 * q^97 - 2112 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	4.37228	+	2.80770i	0.841446	+	0.540341i
$3$	4.37228	+	2.80770i	0.841446	+	0.540341i
$4$		0			0
$5$	−13.1715			−1.17810			−0.589049	−	0.808098i	$-0.700497\pi$
$5$	−13.1715			−1.17810			−0.589049	+	0.808098i	$0.700497\pi$
$6$		0			0
$7$			26.9490i			1.45511i	0.686049	+	0.727555i	$0.259344\pi$
$7$			26.9490i			1.45511i	−0.686049	+	0.727555i	$0.740656\pi$
$8$		0			0
$9$	11.2337	+	24.5521i	0.416063	+	0.909336i
$10$		0			0
$11$			21.9834i			0.602569i	0.953534	+	0.301284i	$0.0974153\pi$
$11$			21.9834i			0.602569i	−0.953534	+	0.301284i	$0.902585\pi$
$12$		0			0
$13$		−	10.0326i		−	0.214042i	−0.994257	−	0.107021i	$-0.965869\pi$
$13$		−	10.0326i		−	0.214042i	0.994257	−	0.107021i	$-0.0341312\pi$
$14$		0			0
$15$	−57.5896	−	36.9816i	−0.991305	−	0.636575i
$16$		0			0
$17$			6.09352i			0.0869350i	0.999055	+	0.0434675i	$0.0138405\pi$
$17$			6.09352i			0.0869350i	−0.999055	+	0.0434675i	$0.986160\pi$
$18$		0			0
$19$	40.1902			0.485277			0.242638	−	0.970117i	$-0.421987\pi$
$19$	40.1902			0.485277			0.242638	+	0.970117i	$0.421987\pi$
$20$		0			0
$21$	−75.6647	+	117.829i	−0.786256	+	1.22440i
$22$		0			0
$23$	9.80703			0.0889090			0.0444545	−	0.999011i	$-0.485845\pi$
$23$	9.80703			0.0889090			0.0444545	+	0.999011i	$0.485845\pi$
$24$		0			0
$25$	48.4891			0.387913
$26$		0			0
$27$	−19.8179	+	138.889i	−0.141258	+	0.989973i
$28$		0			0
$29$	164.501			1.05335			0.526673	−	0.850068i	$-0.323439\pi$
$29$	164.501			1.05335			0.526673	+	0.850068i	$0.323439\pi$
$30$		0			0
$31$			47.0143i			0.272387i	0.990682	+	0.136194i	$0.0434870\pi$
$31$			47.0143i			0.272387i	−0.990682	+	0.136194i	$0.956513\pi$
$32$		0			0
$33$	−61.7228	+	96.1178i	−0.325593	+	0.507029i
$34$		0			0
$35$		−	354.960i		−	1.71426i
$36$		0			0
$37$		−	205.560i		−	0.913346i	−0.889635	−	0.456673i	$-0.849041\pi$
$37$		−	205.560i		−	0.913346i	0.889635	−	0.456673i	$-0.150959\pi$
$38$		0			0
$39$	28.1685	−	43.8654i	0.115656	−	0.180105i
$40$		0			0
$41$		−	419.120i		−	1.59648i	−0.602342	−	0.798238i	$-0.705766\pi$
$41$		−	419.120i		−	1.59648i	0.602342	−	0.798238i	$-0.294234\pi$
$42$		0			0
$43$	−205.038			−0.727163			−0.363581	−	0.931562i	$-0.618446\pi$
$43$	−205.038			−0.727163			−0.363581	+	0.931562i	$0.618446\pi$
$44$		0			0
$45$	−147.965	−	323.388i	−0.490162	−	1.07129i
$46$		0			0
$47$	566.089			1.75686			0.878432	−	0.477868i	$-0.158590\pi$
$47$	566.089			1.75686			0.878432	+	0.477868i	$0.158590\pi$
$48$		0			0
$49$	−383.250			−1.11735
$50$		0			0
$51$	−17.1087	+	26.6426i	−0.0469746	+	0.0731511i
$52$		0			0
$53$	342.173			0.886813			0.443407	−	0.896321i	$-0.353770\pi$
$53$	342.173			0.886813			0.443407	+	0.896321i	$0.353770\pi$
$54$		0			0
$55$		−	289.555i		−	0.709885i
$56$		0			0
$57$	175.723	+	112.842i	0.408334	+	0.262215i
$58$		0			0
$59$			3.70288i			0.00817075i	0.999992	+	0.00408538i	$0.00130042\pi$
$59$			3.70288i			0.00817075i	−0.999992	+	0.00408538i	$0.998700\pi$
$60$		0			0
$61$			717.005i			1.50497i	0.658610	+	0.752484i	$0.271145\pi$
$61$			717.005i			1.50497i	−0.658610	+	0.752484i	$0.728855\pi$
$62$		0			0
$63$	−661.654	+	302.737i	−1.32318	+	0.605417i
$64$		0			0
$65$			132.145i			0.252162i
$66$		0			0
$67$	238.701			0.435253			0.217627	−	0.976032i	$-0.430168\pi$
$67$	238.701			0.435253			0.217627	+	0.976032i	$0.430168\pi$
$68$		0			0
$69$	42.8791	+	27.5351i	0.0748121	+	0.0480412i
$70$		0			0
$71$	−517.054			−0.864268			−0.432134	−	0.901809i	$-0.642239\pi$
$71$	−517.054			−0.864268			−0.432134	+	0.901809i	$0.642239\pi$
$72$		0			0
$73$	984.195			1.57796			0.788982	−	0.614417i	$-0.210609\pi$
$73$	984.195			1.57796			0.788982	+	0.614417i	$0.210609\pi$
$74$		0			0
$75$	212.008	+	136.143i	0.326408	+	0.209605i
$76$		0			0
$77$	−592.432			−0.876804
$78$		0			0
$79$		−	329.100i		−	0.468691i	−0.972153	−	0.234346i	$-0.924705\pi$
$79$		−	329.100i		−	0.468691i	0.972153	−	0.234346i	$-0.0752948\pi$
$80$		0			0
$81$	−476.609	+	551.621i	−0.653784	+	0.756681i
$82$		0			0
$83$			625.333i			0.826978i	0.910509	+	0.413489i	$0.135690\pi$
$83$			625.333i			0.826978i	−0.910509	+	0.413489i	$0.864310\pi$
$84$		0			0
$85$		−	80.2609i		−	0.102418i
$86$		0			0
$87$	719.244	+	461.868i	0.886334	+	0.569167i
$88$		0			0
$89$			238.583i			0.284154i	0.989856	+	0.142077i	$0.0453781\pi$
$89$			238.583i			0.284154i	−0.989856	+	0.142077i	$0.954622\pi$
$90$		0			0
$91$	270.369			0.311455
$92$		0			0
$93$	−132.002	+	205.560i	−0.147182	+	0.229199i
$94$		0			0
$95$	−529.366			−0.571703
$96$		0			0
$97$	−1001.01			−1.04781			−0.523903	−	0.851778i	$-0.675525\pi$
$97$	−1001.01			−1.04781			−0.523903	+	0.851778i	$0.675525\pi$
$98$		0			0
$99$	−539.739	+	246.955i	−0.547937	+	0.250706i
$100$		0			0
$101$	−598.875			−0.590003			−0.295001	−	0.955497i	$-0.595320\pi$
$101$	−598.875			−0.590003			−0.295001	+	0.955497i	$0.595320\pi$
$102$		0			0
$103$			1225.30i			1.17216i	0.810253	+	0.586080i	$0.199330\pi$
$103$			1225.30i			1.17216i	−0.810253	+	0.586080i	$0.800670\pi$
$104$		0			0
$105$	996.619	−	1551.98i	0.926286	−	1.44246i
$106$		0			0
$107$		−	324.370i		−	0.293066i	−0.989206	−	0.146533i	$-0.953189\pi$
$107$		−	324.370i		−	0.293066i	0.989206	−	0.146533i	$-0.0468114\pi$
$108$		0			0
$109$		−	1208.38i		−	1.06186i	−0.847417	−	0.530928i	$-0.821843\pi$
$109$		−	1208.38i		−	1.06186i	0.847417	−	0.530928i	$-0.178157\pi$
$110$		0			0
$111$	577.149	−	898.764i	0.493518	−	0.768531i
$112$		0			0
$113$			1823.44i			1.51800i	0.651089	+	0.759002i	$0.274313\pi$
$113$			1823.44i			1.51800i	−0.651089	+	0.759002i	$0.725687\pi$
$114$		0			0
$115$	−129.174			−0.104743
$116$		0			0
$117$	246.322	−	112.703i	0.194636	−	0.0890549i
$118$		0			0
$119$	−164.214			−0.126500
$120$		0			0
$121$	847.728			0.636911
$122$		0			0
$123$	1176.76	−	1832.51i	0.862642	−	1.34335i
$124$		0			0
$125$	1007.77			0.721098
$126$		0			0
$127$			543.520i			0.379760i	0.981807	+	0.189880i	$0.0608100\pi$
$127$			543.520i			0.379760i	−0.981807	+	0.189880i	$0.939190\pi$
$128$		0			0
$129$	−896.484	−	575.684i	−0.611868	−	0.392916i
$130$		0			0
$131$			1246.47i			0.831335i	0.909517	+	0.415668i	$0.136452\pi$
$131$			1246.47i			0.831335i	−0.909517	+	0.415668i	$0.863548\pi$
$132$		0			0
$133$			1083.09i			0.706132i
$134$		0			0
$135$	261.032	−	1829.38i	0.166415	−	1.16628i
$136$		0			0
$137$		−	2246.38i		−	1.40088i	−0.713709	−	0.700442i	$-0.752986\pi$
$137$		−	2246.38i		−	1.40088i	0.713709	−	0.700442i	$-0.247014\pi$
$138$		0			0
$139$	−1733.77			−1.05796			−0.528979	−	0.848635i	$-0.677425\pi$
$139$	−1733.77			−1.05796			−0.528979	+	0.848635i	$0.677425\pi$
$140$		0			0
$141$	2475.10	+	1589.41i	1.47831	+	0.949306i
$142$		0			0
$143$	220.551			0.128975
$144$		0			0
$145$	−2166.73			−1.24094
$146$		0			0
$147$	−1675.68	−	1076.05i	−0.940187	−	0.603749i
$148$		0			0
$149$	−1196.89			−0.658074			−0.329037	−	0.944317i	$-0.606724\pi$
$149$	−1196.89			−0.658074			−0.329037	+	0.944317i	$0.606724\pi$
$150$		0			0
$151$		−	2505.09i		−	1.35007i	−0.737784	−	0.675037i	$-0.764128\pi$
$151$		−	2505.09i		−	1.35007i	0.737784	−	0.675037i	$-0.235872\pi$
$152$		0			0
$153$	−149.609	+	68.4527i	−0.0790531	+	0.0361704i
$154$		0			0
$155$		−	619.250i		−	0.320899i
$156$		0			0
$157$		−	2717.53i		−	1.38142i	−0.723133	−	0.690709i	$-0.757299\pi$
$157$		−	2717.53i		−	1.38142i	0.723133	−	0.690709i	$-0.242701\pi$
$158$		0			0
$159$	1496.08	+	960.718i	0.746205	+	0.479182i
$160$		0			0
$161$			264.290i			0.129372i
$162$		0			0
$163$	2009.71			0.965723			0.482861	−	0.875697i	$-0.339598\pi$
$163$	2009.71			0.965723			0.482861	+	0.875697i	$0.339598\pi$
$164$		0			0
$165$	812.984	−	1266.02i	0.383580	−	0.597329i
$166$		0			0
$167$	4.79686			0.00222271			0.00111135	−	0.999999i	$-0.499646\pi$
$167$	4.79686			0.00222271			0.00111135	+	0.999999i	$0.499646\pi$
$168$		0			0
$169$	2096.35			0.954186
$170$		0			0
$171$	451.484	+	986.752i	0.201906	+	0.441280i
$172$		0			0
$173$	−1501.84			−0.660016			−0.330008	−	0.943978i	$-0.607052\pi$
$173$	−1501.84			−0.660016			−0.330008	+	0.943978i	$0.607052\pi$
$174$		0			0
$175$			1306.73i			0.564456i
$176$		0			0
$177$	−10.3966	+	16.1901i	−0.00441500	+	0.00687525i
$178$		0			0
$179$		−	360.443i		−	0.150507i	−0.997164	−	0.0752535i	$-0.976023\pi$
$179$		−	360.443i		−	0.150507i	0.997164	−	0.0752535i	$-0.0239766\pi$
$180$		0			0
$181$			1143.06i			0.469410i	0.972067	+	0.234705i	$0.0754125\pi$
$181$			1143.06i			0.469410i	−0.972067	+	0.234705i	$0.924588\pi$
$182$		0			0
$183$	−2013.13	+	3134.95i	−0.813197	+	1.26635i
$184$		0			0
$185$			2707.53i			1.07601i
$186$		0			0
$187$	−133.957			−0.0523843
$188$		0			0
$189$	−3742.93	−	534.073i	−1.44052	−	0.205546i
$190$		0			0
$191$	−1904.00			−0.721302			−0.360651	−	0.932701i	$-0.617446\pi$
$191$	−1904.00			−0.721302			−0.360651	+	0.932701i	$0.617446\pi$
$192$		0			0
$193$	934.152			0.348403			0.174201	−	0.984710i	$-0.444266\pi$
$193$	934.152			0.348403			0.174201	+	0.984710i	$0.444266\pi$
$194$		0			0
$195$	−371.023	+	577.775i	−0.136254	+	0.212181i
$196$		0			0
$197$	1268.62			0.458808			0.229404	−	0.973331i	$-0.426322\pi$
$197$	1268.62			0.458808			0.229404	+	0.973331i	$0.426322\pi$
$198$		0			0
$199$			2293.01i			0.816820i	0.912798	+	0.408410i	$0.133917\pi$
$199$			2293.01i			0.816820i	−0.912798	+	0.408410i	$0.866083\pi$
$200$		0			0
$201$	1043.67	+	670.200i	0.366242	+	0.235185i
$202$		0			0
$203$			4433.14i			1.53274i
$204$		0			0
$205$			5520.45i			1.88080i
$206$		0			0
$207$	110.169	+	240.783i	0.0369917	+	0.0808481i
$208$		0			0
$209$			883.519i			0.292413i
$210$		0			0
$211$	2663.70			0.869084			0.434542	−	0.900652i	$-0.356910\pi$
$211$	2663.70			0.869084			0.434542	+	0.900652i	$0.356910\pi$
$212$		0			0
$213$	−2260.71	−	1451.73i	−0.727235	−	0.467000i
$214$		0			0
$215$	2700.66			0.856668
$216$		0			0
$217$	−1266.99			−0.396354
$218$		0			0
$219$	4303.18	+	2763.32i	1.32777	+	0.852639i
$220$		0			0
$221$	61.1339			0.0186078
$222$		0			0
$223$		−	5227.96i		−	1.56991i	−0.619552	−	0.784956i	$-0.712686\pi$
$223$		−	5227.96i		−	1.56991i	0.619552	−	0.784956i	$-0.287314\pi$
$224$		0			0
$225$	544.712	+	1190.51i	0.161396	+	0.352743i
$226$		0			0
$227$		−	5531.45i		−	1.61734i	−0.588265	−	0.808669i	$-0.700189\pi$
$227$		−	5531.45i		−	1.61734i	0.588265	−	0.808669i	$-0.299811\pi$
$228$		0			0
$229$		−	782.326i		−	0.225754i	−0.993609	−	0.112877i	$-0.963993\pi$
$229$		−	782.326i		−	0.225754i	0.993609	−	0.112877i	$-0.0360065\pi$
$230$		0			0
$231$	−2590.28	−	1663.37i	−0.737783	−	0.473774i
$232$		0			0
$233$		−	3918.64i		−	1.10180i	−0.834573	−	0.550898i	$-0.814285\pi$
$233$		−	3918.64i		−	1.10180i	0.834573	−	0.550898i	$-0.185715\pi$
$234$		0			0
$235$	−7456.26			−2.06976
$236$		0			0
$237$	924.012	−	1438.92i	0.253253	−	0.394378i
$238$		0			0
$239$	−4134.14			−1.11889			−0.559446	−	0.828867i	$-0.688986\pi$
$239$	−4134.14			−1.11889			−0.559446	+	0.828867i	$0.688986\pi$
$240$		0			0
$241$	1221.11			0.326384			0.163192	−	0.986594i	$-0.447821\pi$
$241$	1221.11			0.326384			0.163192	+	0.986594i	$0.447821\pi$
$242$		0			0
$243$	−3632.65	+	1073.67i	−0.958990	+	0.283440i
$244$		0			0
$245$	5047.99			1.31634
$246$		0			0
$247$		−	403.213i		−	0.103870i
$248$		0			0
$249$	−1755.74	+	2734.13i	−0.446850	+	0.695857i
$250$		0			0
$251$		−	4399.34i		−	1.10631i	−0.833078	−	0.553156i	$-0.813423\pi$
$251$		−	4399.34i		−	1.10631i	0.833078	−	0.553156i	$-0.186577\pi$
$252$		0			0
$253$			215.592i			0.0535738i
$254$		0			0
$255$	225.348	−	350.923i	0.0553406	−	0.0861791i
$256$		0			0
$257$		−	3796.23i		−	0.921409i	−0.887553	−	0.460705i	$-0.847597\pi$
$257$		−	3796.23i		−	0.921409i	0.887553	−	0.460705i	$-0.152403\pi$
$258$		0			0
$259$	5539.63			1.32902
$260$		0			0
$261$	1847.95	+	4038.84i	0.438258	+	0.957846i
$262$		0			0
$263$	6525.99			1.53007			0.765037	−	0.643986i	$-0.222720\pi$
$263$	6525.99			1.53007			0.765037	+	0.643986i	$0.222720\pi$
$264$		0			0
$265$	−4506.94			−1.04475
$266$		0			0
$267$	−669.868	+	1043.15i	−0.153540	+	0.239100i
$268$		0			0
$269$	772.537			0.175102			0.0875509	−	0.996160i	$-0.472096\pi$
$269$	772.537			0.175102			0.0875509	+	0.996160i	$0.472096\pi$
$270$		0			0
$271$			3732.58i			0.836673i	0.908292	+	0.418336i	$0.137387\pi$
$271$			3732.58i			0.836673i	−0.908292	+	0.418336i	$0.862613\pi$
$272$		0			0
$273$	1182.13	+	759.115i	0.262072	+	0.168292i
$274$		0			0
$275$			1065.96i			0.233744i
$276$		0			0
$277$		−	6502.55i		−	1.41047i	−0.708973	−	0.705236i	$-0.750841\pi$
$277$		−	6502.55i		−	1.41047i	0.708973	−	0.705236i	$-0.249159\pi$
$278$		0			0
$279$	−1154.30	+	528.144i	−0.247692	+	0.113330i
$280$		0			0
$281$		−	548.243i		−	0.116389i	−0.998305	−	0.0581947i	$-0.981466\pi$
$281$		−	548.243i		−	0.116389i	0.998305	−	0.0581947i	$-0.0185344\pi$
$282$		0			0
$283$	664.623			0.139603			0.0698017	−	0.997561i	$-0.477763\pi$
$283$	664.623			0.139603			0.0698017	+	0.997561i	$0.477763\pi$
$284$		0			0
$285$	−2314.54	−	1486.30i	−0.481057	−	0.308915i
$286$		0			0
$287$	11294.9			2.32305
$288$		0			0
$289$	4875.87			0.992442
$290$		0			0
$291$	−4376.70	−	2810.53i	−0.881673	−	0.566173i
$292$		0			0
$293$	−5112.98			−1.01947			−0.509733	−	0.860332i	$-0.670256\pi$
$293$	−5112.98			−1.01947			−0.509733	+	0.860332i	$0.670256\pi$
$294$		0			0
$295$		−	48.7726i		−	0.00962594i
$296$		0			0
$297$	−3053.27	−	435.666i	−0.596527	−	0.0851175i
$298$		0			0
$299$		−	98.3902i		−	0.0190303i
$300$		0			0
$301$		−	5525.57i		−	1.05810i
$302$		0			0
$303$	−2618.45	−	1681.46i	−0.496455	−	0.318803i
$304$		0			0
$305$		−	9444.05i		−	1.77300i
$306$		0			0
$307$	−594.602			−0.110540			−0.0552699	−	0.998471i	$-0.517602\pi$
$307$	−594.602			−0.110540			−0.0552699	+	0.998471i	$0.517602\pi$
$308$		0			0
$309$	−3440.27	+	5357.36i	−0.633367	+	0.986310i
$310$		0			0
$311$	−7168.24			−1.30699			−0.653495	−	0.756931i	$-0.726698\pi$
$311$	−7168.24			−1.30699			−0.653495	+	0.756931i	$0.726698\pi$
$312$		0			0
$313$	−3774.51			−0.681623			−0.340811	−	0.940132i	$-0.610702\pi$
$313$	−3774.51			−0.681623			−0.340811	+	0.940132i	$0.610702\pi$
$314$		0			0
$315$	8715.00	−	3987.51i	1.55884	−	0.713240i
$316$		0			0
$317$	9400.75			1.66561			0.832805	−	0.553566i	$-0.186733\pi$
$317$	9400.75			1.66561			0.832805	+	0.553566i	$0.186733\pi$
$318$		0			0
$319$			3616.29i			0.634714i
$320$		0			0
$321$	910.733	−	1418.24i	0.158356	−	0.246599i
$322$		0			0
$323$			244.900i			0.0421876i
$324$		0			0
$325$		−	486.473i		−	0.0830297i
$326$		0			0
$327$	3392.78	−	5283.40i	0.573765	−	0.893494i
$328$		0			0
$329$			15255.6i			2.55643i
$330$		0			0
$331$	−9218.16			−1.53074			−0.765371	−	0.643589i	$-0.777445\pi$
$331$	−9218.16			−1.53074			−0.765371	+	0.643589i	$0.777445\pi$
$332$		0			0
$333$	5046.91	−	2309.19i	0.830538	−	0.380009i
$334$		0			0
$335$	−3144.06			−0.512771
$336$		0			0
$337$	2977.58			0.481302			0.240651	−	0.970612i	$-0.422639\pi$
$337$	2977.58			0.481302			0.240651	+	0.970612i	$0.422639\pi$
$338$		0			0
$339$	−5119.65	+	7972.57i	−0.820240	+	1.27732i
$340$		0			0
$341$	−1033.54			−0.164132
$342$		0			0
$343$		−	1084.70i		−	0.170752i
$344$		0			0
$345$	−564.783	−	362.680i	−0.0881359	−	0.0565972i
$346$		0			0
$347$		−	2497.80i		−	0.386424i	−0.981157	−	0.193212i	$-0.938110\pi$
$347$		−	2497.80i		−	0.386424i	0.981157	−	0.193212i	$-0.0618905\pi$
$348$		0			0
$349$			8874.07i			1.36108i	0.732710	+	0.680541i	$0.238255\pi$
$349$			8874.07i			1.36108i	−0.732710	+	0.680541i	$0.761745\pi$
$350$		0			0
$351$	1393.42	+	198.826i	0.211896	+	0.0302351i
$352$		0			0
$353$			2525.66i			0.380814i	0.981705	+	0.190407i	$0.0609808\pi$
$353$			2525.66i			0.380814i	−0.981705	+	0.190407i	$0.939019\pi$
$354$		0			0
$355$	6810.39			1.01819
$356$		0			0
$357$	−717.991	−	461.064i	−0.106443	−	0.0683532i
$358$		0			0
$359$	−9422.15			−1.38519			−0.692594	−	0.721328i	$-0.743532\pi$
$359$	−9422.15			−1.38519			−0.692594	+	0.721328i	$0.743532\pi$
$360$		0			0
$361$	−5243.75			−0.764506
$362$		0			0
$363$	3706.51	+	2380.16i	0.535926	+	0.344149i
$364$		0			0
$365$	−12963.4			−1.85899
$366$		0			0
$367$			613.530i			0.0872643i	0.999048	+	0.0436322i	$0.0138930\pi$
$367$			613.530i			0.0872643i	−0.999048	+	0.0436322i	$0.986107\pi$
$368$		0			0
$369$	10290.3	−	4708.26i	1.45173	−	0.664234i
$370$		0			0
$371$			9221.23i			1.29041i
$372$		0			0
$373$			10759.8i			1.49362i	0.665039	+	0.746809i	$0.268415\pi$
$373$			10759.8i			1.49362i	−0.665039	+	0.746809i	$0.731585\pi$
$374$		0			0
$375$	4406.23	+	2829.50i	0.606765	+	0.389639i
$376$		0			0
$377$		−	1650.37i		−	0.225460i
$378$		0			0
$379$	10132.1			1.37322			0.686610	−	0.727026i	$-0.259098\pi$
$379$	10132.1			1.37322			0.686610	+	0.727026i	$0.259098\pi$
$380$		0			0
$381$	−1526.04	+	2376.42i	−0.205200	+	0.319548i
$382$		0			0
$383$	−9452.43			−1.26109			−0.630544	−	0.776154i	$-0.717168\pi$
$383$	−9452.43			−1.26109			−0.630544	+	0.776154i	$0.717168\pi$
$384$		0			0
$385$	7803.24			1.03296
$386$		0			0
$387$	−2303.33	−	5034.11i	−0.302545	−	0.661235i
$388$		0			0
$389$	−3385.08			−0.441209			−0.220605	−	0.975363i	$-0.570803\pi$
$389$	−3385.08			−0.441209			−0.220605	+	0.975363i	$0.570803\pi$
$390$		0			0
$391$			59.7593i			0.00772930i
$392$		0			0
$393$	−3499.72	+	5449.94i	−0.449205	+	0.699524i
$394$		0			0
$395$			4334.75i			0.552164i
$396$		0			0
$397$		−	4125.91i		−	0.521596i	−0.965393	−	0.260798i	$-0.916014\pi$
$397$		−	4125.91i		−	0.521596i	0.965393	−	0.260798i	$-0.0839857\pi$
$398$		0			0
$399$	−3040.98	+	4735.56i	−0.381552	+	0.594172i
$400$		0			0
$401$			8829.27i			1.09953i	0.835318	+	0.549767i	$0.185283\pi$
$401$			8829.27i			1.09953i	−0.835318	+	0.549767i	$0.814717\pi$
$402$		0			0
$403$	471.676			0.0583024
$404$		0			0
$405$	6277.66	−	7265.69i	0.770221	−	0.891444i
$406$		0			0
$407$	4518.91			0.550354
$408$		0			0
$409$	1648.22			0.199264			0.0996320	−	0.995024i	$-0.468233\pi$
$409$	1648.22			0.199264			0.0996320	+	0.995024i	$0.468233\pi$
$410$		0			0
$411$	6307.15	−	9821.81i	0.756956	−	1.17877i
$412$		0			0
$413$	−99.7891			−0.0118893
$414$		0			0
$415$		−	8236.59i		−	0.974261i
$416$		0			0
$417$	−7580.51	−	4867.89i	−0.890214	−	0.571658i
$418$		0			0
$419$			11553.7i			1.34710i	0.739143	+	0.673549i	$0.235231\pi$
$419$			11553.7i			1.34710i	−0.739143	+	0.673549i	$0.764769\pi$
$420$		0			0
$421$		−	7651.40i		−	0.885763i	−0.896580	−	0.442882i	$-0.853956\pi$
$421$		−	7651.40i		−	0.885763i	0.896580	−	0.442882i	$-0.146044\pi$
$422$		0			0
$423$	6359.27	+	13898.7i	0.730965	+	1.59758i
$424$		0			0
$425$			295.469i			0.0337232i
$426$		0			0
$427$	−19322.6			−2.18990
$428$		0			0
$429$	964.313	+	619.241i	0.108526	+	0.0696906i
$430$		0			0
$431$	14803.7			1.65445			0.827224	−	0.561873i	$-0.189919\pi$
$431$	14803.7			1.65445			0.827224	+	0.561873i	$0.189919\pi$
$432$		0			0
$433$	1938.03			0.215094			0.107547	−	0.994200i	$-0.465700\pi$
$433$	1938.03			0.215094			0.107547	+	0.994200i	$0.465700\pi$
$434$		0			0
$435$	−9473.54	−	6083.51i	−1.04419	−	0.670533i
$436$		0			0
$437$	394.146			0.0431455
$438$		0			0
$439$			5276.73i			0.573678i	0.957979	+	0.286839i	$0.0926045\pi$
$439$			5276.73i			0.573678i	−0.957979	+	0.286839i	$0.907395\pi$
$440$		0			0
$441$	−4305.31	−	9409.58i	−0.464886	−	1.01604i
$442$		0			0
$443$		−	9203.98i		−	0.987121i	−0.869712	−	0.493560i	$-0.835695\pi$
$443$		−	9203.98i		−	0.987121i	0.869712	−	0.493560i	$-0.164305\pi$
$444$		0			0
$445$		−	3142.50i		−	0.334761i
$446$		0			0
$447$	−5233.14	−	3360.50i	−0.553734	−	0.355585i
$448$		0			0
$449$		−	3221.17i		−	0.338567i	−0.985567	−	0.169283i	$-0.945855\pi$
$449$		−	3221.17i		−	0.338567i	0.985567	−	0.169283i	$-0.0541453\pi$
$450$		0			0
$451$	9213.69			0.961986
$452$		0			0
$453$	7033.52	−	10952.9i	0.729500	−	1.13601i
$454$		0			0
$455$	−3561.18			−0.366924
$456$		0			0
$457$	−12015.1			−1.22985			−0.614925	−	0.788585i	$-0.710814\pi$
$457$	−12015.1			−1.22985			−0.614925	+	0.788585i	$0.710814\pi$
$458$		0			0
$459$	−846.325	−	120.761i	−0.0860633	−	0.0122802i
$460$		0			0
$461$	−4758.79			−0.480778			−0.240389	−	0.970677i	$-0.577275\pi$
$461$	−4758.79			−0.480778			−0.240389	+	0.970677i	$0.577275\pi$
$462$		0			0
$463$		−	12568.3i		−	1.26156i	−0.775963	−	0.630778i	$-0.782736\pi$
$463$		−	12568.3i		−	1.26156i	0.775963	−	0.630778i	$-0.217264\pi$
$464$		0			0
$465$	1738.66	−	2707.53i	0.173395	−	0.270019i
$466$		0			0
$467$		−	15360.6i		−	1.52207i	−0.648712	−	0.761034i	$-0.724692\pi$
$467$		−	15360.6i		−	1.52207i	0.648712	−	0.761034i	$-0.275308\pi$
$468$		0			0
$469$			6432.76i			0.633342i
$470$		0			0
$471$	7630.00	−	11881.8i	0.746437	−	1.16239i
$472$		0			0
$473$		−	4507.44i		−	0.438165i
$474$		0			0
$475$	1948.79			0.188245
$476$		0			0
$477$	3843.87	+	8401.06i	0.368970	+	0.806411i
$478$		0			0
$479$	−2268.94			−0.216431			−0.108216	−	0.994127i	$-0.534514\pi$
$479$	−2268.94			−0.216431			−0.108216	+	0.994127i	$0.534514\pi$
$480$		0			0
$481$	−2062.30			−0.195494
$482$		0			0
$483$	−742.045	+	1155.55i	−0.0699052	+	0.108860i
$484$		0			0
$485$	13184.8			1.23442
$486$		0			0
$487$		−	11478.8i		−	1.06808i	−0.845459	−	0.534040i	$-0.820673\pi$
$487$		−	11478.8i		−	1.06808i	0.845459	−	0.534040i	$-0.179327\pi$
$488$		0			0
$489$	8787.02	+	5642.66i	0.812603	+	0.521820i
$490$		0			0
$491$			20306.4i			1.86642i	0.359325	+	0.933212i	$0.383007\pi$
$491$			20306.4i			1.86642i	−0.359325	+	0.933212i	$0.616993\pi$
$492$		0			0
$493$			1002.39i			0.0915727i
$494$		0			0
$495$	7109.19	−	3252.78i	0.645524	−	0.295356i
$496$		0			0
$497$		−	13934.1i		−	1.25761i
$498$		0			0
$499$	−1024.04			−0.0918681			−0.0459340	−	0.998944i	$-0.514626\pi$
$499$	−1024.04			−0.0918681			−0.0459340	+	0.998944i	$0.514626\pi$
$500$		0			0
$501$	20.9732	+	13.4681i	0.00187029	+	0.00120102i
$502$		0			0
$503$	−7799.62			−0.691388			−0.345694	−	0.938347i	$-0.612356\pi$
$503$	−7799.62			−0.691388			−0.345694	+	0.938347i	$0.612356\pi$
$504$		0			0
$505$	7888.10			0.695080
$506$		0			0
$507$	9165.82	+	5885.90i	0.802896	+	0.515586i
$508$		0			0
$509$	−298.935			−0.0260315			−0.0130158	−	0.999915i	$-0.504143\pi$
$509$	−298.935			−0.0260315			−0.0130158	+	0.999915i	$0.504143\pi$
$510$		0			0
$511$			26523.1i			2.29611i
$512$		0			0
$513$	−796.485	+	5581.99i	−0.0685491	+	0.480411i
$514$		0			0
$515$		−	16139.1i		−	1.38092i
$516$		0			0
$517$			12444.6i			1.05863i
$518$		0			0
$519$	−6566.47	−	4216.71i	−0.555368	−	0.356634i
$520$		0			0
$521$			9553.38i			0.803342i	0.915784	+	0.401671i	$0.131571\pi$
$521$			9553.38i			0.803342i	−0.915784	+	0.401671i	$0.868429\pi$
$522$		0			0
$523$	−13485.0			−1.12745			−0.563726	−	0.825962i	$-0.690633\pi$
$523$	−13485.0			−1.12745			−0.563726	+	0.825962i	$0.690633\pi$
$524$		0			0
$525$	−3668.91	+	5713.41i	−0.304999	+	0.474959i
$526$		0			0
$527$	−286.482			−0.0236800
$528$		0			0
$529$	−12070.8			−0.992095
$530$		0			0
$531$	−90.9135	+	41.5970i	−0.00742996	+	0.00339954i
$532$		0			0
$533$	−4204.87			−0.341713
$534$		0			0
$535$			4272.45i			0.345260i
$536$		0			0
$537$	1012.01	−	1575.96i	0.0813251	−	0.126643i
$538$		0			0
$539$		−	8425.15i		−	0.673278i
$540$		0			0
$541$		−	14897.0i		−	1.18387i	−0.805986	−	0.591934i	$-0.798365\pi$
$541$		−	14897.0i		−	1.18387i	0.805986	−	0.591934i	$-0.201635\pi$
$542$		0			0
$543$	−3209.38	+	4997.80i	−0.253642	+	0.394983i
$544$		0			0
$545$			15916.3i			1.25097i
$546$		0			0
$547$	8674.79			0.678076			0.339038	−	0.940773i	$-0.389899\pi$
$547$	8674.79			0.678076			0.339038	+	0.940773i	$0.389899\pi$
$548$		0			0
$549$	−17604.0	+	8054.61i	−1.36852	+	0.626161i
$550$		0			0
$551$	6611.32			0.511165
$552$		0			0
$553$	8868.92			0.681998
$554$		0			0
$555$	−7601.93	+	11838.1i	−0.581413	+	0.905404i
$556$		0			0
$557$	22222.8			1.69050			0.845251	−	0.534369i	$-0.179451\pi$
$557$	22222.8			1.69050			0.845251	+	0.534369i	$0.179451\pi$
$558$		0			0
$559$			2057.07i			0.155643i
$560$		0			0
$561$	−585.696	−	376.109i	−0.0440786	−	0.0283054i
$562$		0			0
$563$			8321.78i			0.622950i	0.950254	+	0.311475i	$0.100823\pi$
$563$			8321.78i			0.622950i	−0.950254	+	0.311475i	$0.899177\pi$
$564$		0			0
$565$		−	24017.4i		−	1.78836i
$566$		0			0
$567$	−14865.6	−	12844.1i	−1.10105	−	0.951328i
$568$		0			0
$569$		−	12051.5i		−	0.887915i	−0.896048	−	0.443957i	$-0.853574\pi$
$569$		−	12051.5i		−	0.887915i	0.896048	−	0.443957i	$-0.146426\pi$
$570$		0			0
$571$	11319.5			0.829608			0.414804	−	0.909911i	$-0.363850\pi$
$571$	11319.5			0.829608			0.414804	+	0.909911i	$0.363850\pi$
$572$		0			0
$573$	−8324.83	−	5345.86i	−0.606937	−	0.389749i
$574$		0			0
$575$	475.534			0.0344889
$576$		0			0
$577$	3145.14			0.226922			0.113461	−	0.993542i	$-0.463806\pi$
$577$	3145.14			0.226922			0.113461	+	0.993542i	$0.463806\pi$
$578$		0			0
$579$	4084.37	+	2622.81i	0.293162	+	0.188256i
$580$		0			0
$581$	−16852.1			−1.20334
$582$		0			0
$583$			7522.14i			0.534366i
$584$		0			0
$585$	−3244.43	+	1484.47i	−0.229300	+	0.104915i
$586$		0			0
$587$			8484.84i			0.596604i	0.954472	+	0.298302i	$0.0964203\pi$
$587$			8484.84i			0.596604i	−0.954472	+	0.298302i	$0.903580\pi$
$588$		0			0
$589$			1889.51i			0.132183i
$590$		0			0
$591$	5546.75	+	3561.89i	0.386062	+	0.247913i
$592$		0			0
$593$			6571.68i			0.455087i	0.973768	+	0.227543i	$0.0730693\pi$
$593$			6571.68i			0.455087i	−0.973768	+	0.227543i	$0.926931\pi$
$594$		0			0
$595$	2162.95			0.149029
$596$		0			0
$597$	−6438.08	+	10025.7i	−0.441362	+	0.687310i
$598$		0			0
$599$	17732.4			1.20956			0.604779	−	0.796393i	$-0.293261\pi$
$599$	17732.4			1.20956			0.604779	+	0.796393i	$0.293261\pi$
$600$		0			0
$601$	22182.6			1.50557			0.752785	−	0.658266i	$-0.228710\pi$
$601$	22182.6			1.50557			0.752785	+	0.658266i	$0.228710\pi$
$602$		0			0
$603$	2681.49	+	5860.61i	0.181093	+	0.395792i
$604$		0			0
$605$	−11165.9			−0.750343
$606$		0			0
$607$		−	4360.03i		−	0.291546i	−0.989318	−	0.145773i	$-0.953433\pi$
$607$		−	4360.03i		−	0.291546i	0.989318	−	0.145773i	$-0.0465668\pi$
$608$		0			0
$609$	−12446.9	+	19382.9i	−0.828200	+	1.28971i
$610$		0			0
$611$		−	5679.36i		−	0.376043i
$612$		0			0
$613$			14917.5i			0.982892i	0.870908	+	0.491446i	$0.163532\pi$
$613$			14917.5i			0.982892i	−0.870908	+	0.491446i	$0.836468\pi$
$614$		0			0
$615$	−15499.7	+	24136.9i	−1.01628	+	1.58259i
$616$		0			0
$617$		−	16400.2i		−	1.07010i	−0.844822	−	0.535048i	$-0.820294\pi$
$617$		−	16400.2i		−	1.07010i	0.844822	−	0.535048i	$-0.179706\pi$
$618$		0			0
$619$	−3457.20			−0.224486			−0.112243	−	0.993681i	$-0.535803\pi$
$619$	−3457.20			−0.224486			−0.112243	+	0.993681i	$0.535803\pi$
$620$		0			0
$621$	−194.355	+	1362.09i	−0.0125591	+	0.0880175i
$622$		0			0
$623$	−6429.57			−0.413476
$624$		0			0
$625$	−19334.9			−1.23744
$626$		0			0
$627$	−2480.65	+	3862.99i	−0.158003	+	0.246050i
$628$		0			0
$629$	1252.58			0.0794017
$630$		0			0
$631$			10766.4i			0.679247i	0.940562	+	0.339623i	$0.110300\pi$
$631$			10766.4i			0.679247i	−0.940562	+	0.339623i	$0.889700\pi$
$632$		0			0
$633$	11646.4	+	7478.86i	0.731287	+	0.469602i
$634$		0			0
$635$		−	7158.99i		−	0.447395i
$636$		0			0
$637$			3845.00i			0.239159i
$638$		0			0
$639$	−5808.42	−	12694.7i	−0.359590	−	0.785910i
$640$		0			0
$641$		−	4116.88i		−	0.253677i	−0.991923	−	0.126839i	$-0.959517\pi$
$641$		−	4116.88i		−	0.253677i	0.991923	−	0.126839i	$-0.0404830\pi$
$642$		0			0
$643$	7733.58			0.474312			0.237156	−	0.971472i	$-0.423785\pi$
$643$	7733.58			0.474312			0.237156	+	0.971472i	$0.423785\pi$
$644$		0			0
$645$	11808.1	+	7582.64i	0.720840	+	0.462893i
$646$		0			0
$647$	2769.95			0.168312			0.0841561	−	0.996453i	$-0.473181\pi$
$647$	2769.95			0.168312			0.0841561	+	0.996453i	$0.473181\pi$
$648$		0			0
$649$	−81.4021			−0.00492344
$650$		0			0
$651$	−5539.63	−	3557.32i	−0.333510	−	0.214166i
$652$		0			0
$653$	13463.7			0.806855			0.403428	−	0.915012i	$-0.367819\pi$
$653$	13463.7			0.806855			0.403428	+	0.915012i	$0.367819\pi$
$654$		0			0
$655$		−	16418.0i		−	0.979394i
$656$		0			0
$657$	11056.1	+	24164.0i	0.656532	+	1.43490i
$658$		0			0
$659$			5523.87i			0.326524i	0.986583	+	0.163262i	$0.0522017\pi$
$659$			5523.87i			0.326524i	−0.986583	+	0.163262i	$0.947798\pi$
$660$		0			0
$661$			5189.81i			0.305386i	0.988274	+	0.152693i	$0.0487946\pi$
$661$			5189.81i			0.305386i	−0.988274	+	0.152693i	$0.951205\pi$
$662$		0			0
$663$	267.295	+	171.646i	0.0156574	+	0.0100545i
$664$		0			0
$665$		−	14265.9i		−	0.831892i
$666$		0			0
$667$	1613.26			0.0936519
$668$		0			0
$669$	14678.5	−	22858.1i	0.848288	−	1.32100i
$670$		0			0
$671$	−15762.2			−0.906847
$672$		0			0
$673$	−3878.74			−0.222161			−0.111081	−	0.993811i	$-0.535431\pi$
$673$	−3878.74			−0.222161			−0.111081	+	0.993811i	$0.535431\pi$
$674$		0			0
$675$	−960.953	+	6734.62i	−0.0547957	+	0.384023i
$676$		0			0
$677$	24431.3			1.38696			0.693480	−	0.720476i	$-0.256077\pi$
$677$	24431.3			1.38696			0.693480	+	0.720476i	$0.256077\pi$
$678$		0			0
$679$		−	26976.3i		−	1.52467i
$680$		0			0
$681$	15530.6	−	24185.1i	0.873914	−	1.36090i
$682$		0			0
$683$		−	18958.5i		−	1.06212i	−0.847335	−	0.531058i	$-0.821795\pi$
$683$		−	18958.5i		−	1.06212i	0.847335	−	0.531058i	$-0.178205\pi$
$684$		0			0
$685$			29588.3i			1.65038i
$686$		0			0
$687$	2196.53	−	3420.55i	0.121984	−	0.189959i
$688$		0			0
$689$		−	3432.89i		−	0.189815i
$690$		0			0
$691$	−25328.5			−1.39442			−0.697209	−	0.716868i	$-0.745575\pi$
$691$	−25328.5			−1.39442			−0.697209	+	0.716868i	$0.745575\pi$
$692$		0			0
$693$	−6655.20	−	14545.4i	−0.364805	−	0.797310i
$694$		0			0
$695$	22836.3			1.24638
$696$		0			0
$697$	2553.91			0.138790
$698$		0			0
$699$	11002.3	−	17133.4i	0.595346	−	0.927102i
$700$		0			0
$701$	−11295.0			−0.608569			−0.304284	−	0.952581i	$-0.598417\pi$
$701$	−11295.0			−0.608569			−0.304284	+	0.952581i	$0.598417\pi$
$702$		0			0
$703$		−	8261.48i		−	0.443226i
$704$		0			0
$705$	−32600.9	−	20934.9i	−1.74159	−	1.11837i
$706$		0			0
$707$		−	16139.1i		−	0.858519i
$708$		0			0
$709$		−	29445.8i		−	1.55975i	−0.625936	−	0.779874i	$-0.715283\pi$
$709$		−	29445.8i		−	1.55975i	0.625936	−	0.779874i	$-0.284717\pi$
$710$		0			0
$711$	8080.08	−	3697.00i	0.426198	−	0.195005i
$712$		0			0
$713$			461.070i			0.0242177i
$714$		0			0
$715$	−2905.00			−0.151945
$716$		0			0
$717$	−18075.6	−	11607.4i	−0.941488	−	0.604584i
$718$		0			0
$719$	12404.3			0.643396			0.321698	−	0.946842i	$-0.395746\pi$
$719$	12404.3			0.643396			0.321698	+	0.946842i	$0.395746\pi$
$720$		0			0
$721$	−33020.7			−1.70562
$722$		0			0
$723$	5339.03	+	3428.50i	0.274634	+	0.176359i
$724$		0			0
$725$	7976.50			0.408607
$726$		0			0
$727$			7076.17i			0.360991i	0.983576	+	0.180496i	$0.0577702\pi$
$727$			7076.17i			0.360991i	−0.983576	+	0.180496i	$0.942230\pi$
$728$		0			0
$729$	−18897.5	−	5504.99i	−0.960093	−	0.279683i
$730$		0			0
$731$		−	1249.40i		−	0.0632159i
$732$		0			0
$733$			10430.6i			0.525600i	0.964850	+	0.262800i	$0.0846460\pi$
$733$			10430.6i			0.525600i	−0.964850	+	0.262800i	$0.915354\pi$
$734$		0			0
$735$	22071.2	+	14173.2i	1.10763	+	0.711274i
$736$		0			0
$737$			5247.47i			0.262270i
$738$		0			0
$739$	−30256.6			−1.50610			−0.753050	−	0.657963i	$-0.771418\pi$
$739$	−30256.6			−1.50610			−0.753050	+	0.657963i	$0.771418\pi$
$740$		0			0
$741$	1132.10	−	1762.96i	0.0561251	−	0.0874007i
$742$		0			0
$743$	5577.21			0.275381			0.137690	−	0.990475i	$-0.456032\pi$
$743$	5577.21			0.275381			0.137690	+	0.990475i	$0.456032\pi$
$744$		0			0
$745$	15764.9			0.775275
$746$		0			0
$747$	−15353.2	+	7024.79i	−0.752001	+	0.344075i
$748$		0			0
$749$	8741.46			0.426443
$750$		0			0
$751$			10687.5i			0.519296i	0.965703	+	0.259648i	$0.0836066\pi$
$751$			10687.5i			0.519296i	−0.965703	+	0.259648i	$0.916393\pi$
$752$		0			0
$753$	12352.0	−	19235.2i	0.597786	−	0.930901i
$754$		0			0
$755$			32995.8i			1.59052i
$756$		0			0
$757$		−	9377.01i		−	0.450215i	−0.974334	−	0.225108i	$-0.927727\pi$
$757$		−	9377.01i		−	0.450215i	0.974334	−	0.225108i	$-0.0722734\pi$
$758$		0			0
$759$	−605.317	+	942.630i	−0.0289481	+	0.0450794i
$760$		0			0
$761$			15663.5i			0.746124i	0.927806	+	0.373062i	$0.121692\pi$
$761$			15663.5i			0.746124i	−0.927806	+	0.373062i	$0.878308\pi$
$762$		0			0
$763$	32564.8			1.54512
$764$		0			0
$765$	1970.57	−	901.626i	0.0931323	−	0.0426123i
$766$		0			0
$767$	37.1496			0.00174889
$768$		0			0
$769$	18293.6			0.857848			0.428924	−	0.903341i	$-0.358893\pi$
$769$	18293.6			0.857848			0.428924	+	0.903341i	$0.358893\pi$
$770$		0			0
$771$	10658.7	−	16598.2i	0.497876	−	0.775316i
$772$		0			0
$773$	−7495.03			−0.348742			−0.174371	−	0.984680i	$-0.555789\pi$
$773$	−7495.03			−0.348742			−0.174371	+	0.984680i	$0.555789\pi$
$774$		0			0
$775$			2279.68i			0.105663i
$776$		0			0
$777$	24220.8	+	15553.6i	1.11830	+	0.718124i
$778$		0			0
$779$		−	16844.5i		−	0.774733i
$780$		0			0
$781$		−	11366.6i		−	0.520781i
$782$		0			0
$783$	−3260.06	+	22847.4i	−0.148793	+	1.04278i
$784$		0			0
$785$			35794.0i			1.62744i
$786$		0			0
$787$	16884.3			0.764753			0.382377	−	0.924007i	$-0.375106\pi$
$787$	16884.3			0.764753			0.382377	+	0.924007i	$0.375106\pi$
$788$		0			0
$789$	28533.5	+	18323.0i	1.28748	+	0.826763i
$790$		0			0
$791$	−49139.8			−2.20886
$792$		0			0
$793$	7193.44			0.322127
$794$		0			0
$795$	−19705.6	−	12654.1i	−0.879103	−	0.564523i
$796$		0			0
$797$	−29212.0			−1.29830			−0.649148	−	0.760662i	$-0.724875\pi$
$797$	−29212.0			−1.29830			−0.649148	+	0.760662i	$0.724875\pi$
$798$		0			0
$799$			3449.48i			0.152733i
$800$		0			0
$801$	−5857.70	+	2680.16i	−0.258392	+	0.118226i
$802$		0			0
$803$			21636.0i			0.950832i
$804$		0			0
$805$		−	3481.10i		−	0.152413i
$806$		0			0
$807$	3377.75	+	2169.05i	0.147339	+	0.0946147i
$808$		0			0
$809$		−	28157.8i		−	1.22370i	−0.790973	−	0.611851i	$-0.790425\pi$
$809$		−	28157.8i		−	1.22370i	0.790973	−	0.611851i	$-0.209575\pi$
$810$		0			0
$811$	24658.2			1.06765			0.533827	−	0.845594i	$-0.320753\pi$
$811$	24658.2			1.06765			0.533827	+	0.845594i	$0.320753\pi$
$812$		0			0
$813$	−10480.0	+	16319.9i	−0.452089	+	0.704015i
$814$		0			0
$815$	−26471.0			−1.13771
$816$		0			0
$817$	−8240.51			−0.352875
$818$		0			0
$819$	3037.24	+	6638.13i	0.129585	+	0.283217i
$820$		0			0
$821$	−23980.9			−1.01941			−0.509707	−	0.860348i	$-0.670246\pi$
$821$	−23980.9			−1.01941			−0.509707	+	0.860348i	$0.670246\pi$
$822$		0			0
$823$		−	7533.72i		−	0.319087i	−0.987191	−	0.159544i	$-0.948998\pi$
$823$		−	7533.72i		−	0.319087i	0.987191	−	0.159544i	$-0.0510023\pi$
$824$		0			0
$825$	−2992.89	+	4660.67i	−0.126302	+	0.196683i
$826$		0			0
$827$			25167.2i			1.05822i	0.848553	+	0.529110i	$0.177474\pi$
$827$			25167.2i			1.05822i	−0.848553	+	0.529110i	$0.822526\pi$
$828$		0			0
$829$			4440.96i			0.186057i	0.995663	+	0.0930283i	$0.0296547\pi$
$829$			4440.96i			0.186057i	−0.995663	+	0.0930283i	$0.970345\pi$
$830$		0			0
$831$	18257.2	−	28431.0i	0.762136	−	1.18684i
$832$		0			0
$833$		−	2335.34i		−	0.0971365i
$834$		0			0
$835$	−63.1820			−0.00261857
$836$		0			0
$837$	−6529.78	−	931.724i	−0.269656	−	0.0384768i
$838$		0			0
$839$	−8795.36			−0.361918			−0.180959	−	0.983491i	$-0.557920\pi$
$839$	−8795.36			−0.361918			−0.180959	+	0.983491i	$0.557920\pi$
$840$		0			0
$841$	2671.53			0.109538
$842$		0			0
$843$	1539.30	−	2397.07i	0.0628900	−	0.0979353i
$844$		0			0
$845$	−27612.1			−1.12412
$846$		0			0
$847$			22845.5i			0.926776i
$848$		0			0
$849$	2905.92	+	1866.06i	0.117469	+	0.0754335i
$850$		0			0
$851$		−	2015.93i		−	0.0812046i
$852$		0			0
$853$			35919.6i			1.44181i	0.693034	+	0.720905i	$0.256274\pi$
$853$			35919.6i			1.44181i	−0.693034	+	0.720905i	$0.743726\pi$
$854$		0			0
$855$	−5946.73	−	12997.0i	−0.237864	−	0.519870i
$856$		0			0
$857$			35984.3i			1.43431i	0.696916	+	0.717153i	$0.254555\pi$
$857$			35984.3i			1.43431i	−0.696916	+	0.717153i	$0.745445\pi$
$858$		0			0
$859$	−19362.0			−0.769060			−0.384530	−	0.923112i	$-0.625636\pi$
$859$	−19362.0			−0.769060			−0.384530	+	0.923112i	$0.625636\pi$
$860$		0			0
$861$	49384.3	+	31712.6i	1.95472	+	1.25524i
$862$		0			0
$863$	25590.4			1.00940			0.504698	−	0.863296i	$-0.331604\pi$
$863$	25590.4			1.00940			0.504698	+	0.863296i	$0.331604\pi$
$864$		0			0
$865$	19781.5			0.777563
$866$		0			0
$867$	21318.7	+	13690.0i	0.835087	+	0.536258i
$868$		0			0
$869$	7234.75			0.282419
$870$		0			0
$871$		−	2394.80i		−	0.0931626i
$872$		0			0
$873$	−11245.0	−	24576.9i	−0.435953	−	0.952808i
$874$		0			0
$875$			27158.3i			1.04928i
$876$		0			0
$877$		−	28481.1i		−	1.09662i	−0.836275	−	0.548311i	$-0.815271\pi$
$877$		−	28481.1i		−	1.09662i	0.836275	−	0.548311i	$-0.184729\pi$
$878$		0			0
$879$	−22355.4	−	14355.7i	−0.857826	−	0.550860i
$880$		0			0
$881$			24952.4i			0.954219i	0.878844	+	0.477109i	$0.158315\pi$
$881$			24952.4i			0.954219i	−0.878844	+	0.477109i	$0.841685\pi$
$882$		0			0
$883$	−28650.5			−1.09192			−0.545961	−	0.837811i	$-0.683835\pi$
$883$	−28650.5			−1.09192			−0.545961	+	0.837811i	$0.683835\pi$
$884$		0			0
$885$	136.939	−	213.248i	0.00520129	−	0.00809971i
$886$		0			0
$887$	542.531			0.0205371			0.0102686	−	0.999947i	$-0.496731\pi$
$887$	542.531			0.0205371			0.0102686	+	0.999947i	$0.496731\pi$
$888$		0			0
$889$	−14647.3			−0.552593
$890$		0			0
$891$	−12126.5	−	10477.5i	−0.455952	−	0.393950i
$892$		0			0
$893$	22751.2			0.852565
$894$		0			0
$895$			4747.58i			0.177312i
$896$		0			0
$897$	276.250	−	430.189i	0.0102828	−	0.0160129i
$898$		0			0
$899$			7733.89i			0.286918i
$900$		0			0
$901$			2085.04i			0.0770951i
$902$		0			0
$903$	15514.1	−	24159.4i	0.571736	−	0.890336i
$904$		0			0
$905$		−	15055.9i		−	0.553011i
$906$		0			0
$907$	17915.8			0.655880			0.327940	−	0.944699i	$-0.393646\pi$
$907$	17915.8			0.655880			0.327940	+	0.944699i	$0.393646\pi$
$908$		0			0
$909$	−6727.57	−	14703.6i	−0.245478	−	0.536511i
$910$		0			0
$911$	−30577.4			−1.11205			−0.556023	−	0.831167i	$-0.687673\pi$
$911$	−30577.4			−1.11205			−0.556023	+	0.831167i	$0.687673\pi$
$912$		0			0
$913$	−13747.0			−0.498311
$914$		0			0
$915$	26516.0	−	41292.0i	0.958025	−	1.49188i
$916$		0			0
$917$	−33591.3			−1.20968
$918$		0			0
$919$			21519.6i			0.772435i	0.922408	+	0.386217i	$0.126219\pi$
$919$			21519.6i			0.772435i	−0.922408	+	0.386217i	$0.873781\pi$
$920$		0			0
$921$	−2599.77	−	1669.46i	−0.0930133	−	0.0597292i
$922$		0			0
$923$			5187.41i			0.184990i
$924$		0			0
$925$		−	9967.40i		−	0.354299i
$926$		0			0
$927$	−30083.7	+	13764.7i	−1.06589	+	0.487692i
$928$		0			0
$929$		−	23763.7i		−	0.839247i	−0.907698	−	0.419624i	$-0.862162\pi$
$929$		−	23763.7i		−	0.839247i	0.907698	−	0.419624i	$-0.137838\pi$
$930$		0			0
$931$	−15402.9			−0.542223
$932$		0			0
$933$	−31341.6	−	20126.2i	−1.09976	−	0.706220i
$934$		0			0
$935$	1764.41			0.0617138
$936$		0			0
$937$	−11835.5			−0.412644			−0.206322	−	0.978484i	$-0.566149\pi$
$937$	−11835.5			−0.412644			−0.206322	+	0.978484i	$0.566149\pi$
$938$		0			0
$939$	−16503.2	−	10597.7i	−0.573549	−	0.368309i
$940$		0			0
$941$	41547.4			1.43933			0.719664	−	0.694323i	$-0.244296\pi$
$941$	41547.4			1.43933			0.719664	+	0.694323i	$0.244296\pi$
$942$		0			0
$943$		−	4110.32i		−	0.141941i
$944$		0			0
$945$	49300.1	+	7034.56i	1.69707	+	0.242153i
$946$		0			0
$947$		−	41519.6i		−	1.42472i	−0.701816	−	0.712358i	$-0.747627\pi$
$947$		−	41519.6i		−	1.42472i	0.701816	−	0.712358i	$-0.252373\pi$
$948$		0			0
$949$		−	9874.06i		−	0.337751i
$950$		0			0
$951$	41102.7	+	26394.4i	1.40152	+	0.899998i
$952$		0			0
$953$			31542.2i			1.07214i	0.844173	+	0.536071i	$0.180092\pi$
$953$			31542.2i			1.07214i	−0.844173	+	0.536071i	$0.819908\pi$
$954$		0			0
$955$	25078.6			0.849764
$956$		0			0
$957$	−10153.5	+	15811.5i	−0.342962	+	0.534077i
$958$		0			0
$959$	60537.8			2.03844
$960$		0			0
$961$	27580.7			0.925805
$962$		0			0
$963$	7963.96	−	3643.87i	0.266495	−	0.121934i
$964$		0			0
$965$	−12304.2			−0.410452
$966$		0			0
$967$			51845.5i			1.72414i	0.506793	+	0.862068i	$0.330831\pi$
$967$			51845.5i			1.72414i	−0.506793	+	0.862068i	$0.669169\pi$
$968$		0			0
$969$	−687.604	+	1070.77i	−0.0227957	+	0.0354986i
$970$		0			0
$971$			16303.5i			0.538830i	0.963024	+	0.269415i	$0.0868304\pi$
$971$			16303.5i			0.538830i	−0.963024	+	0.269415i	$0.913170\pi$
$972$		0			0
$973$		−	46723.3i		−	1.53945i
$974$		0			0
$975$	1365.87	−	2127.00i	0.0448644	−	0.0698650i
$976$		0			0
$977$		−	46997.0i		−	1.53896i	−0.638668	−	0.769482i	$-0.720514\pi$
$977$		−	46997.0i		−	1.53896i	0.638668	−	0.769482i	$-0.279486\pi$
$978$		0			0
$979$	−5244.87			−0.171222
$980$		0			0
$981$	29668.4	−	13574.6i	0.965584	−	0.441798i
$982$		0			0
$983$	−10730.9			−0.348180			−0.174090	−	0.984730i	$-0.555698\pi$
$983$	−10730.9			−0.348180			−0.174090	+	0.984730i	$0.555698\pi$
$984$		0			0
$985$	−16709.6			−0.540521
$986$		0			0
$987$	−42832.9	+	66701.6i	−1.38135	+	2.15110i
$988$		0			0
$989$	−2010.81			−0.0646513
$990$		0			0
$991$		−	53985.5i		−	1.73048i	−0.501358	−	0.865240i	$-0.667166\pi$
$991$		−	53985.5i		−	1.73048i	0.501358	−	0.865240i	$-0.332834\pi$
$992$		0			0
$993$	−40304.4	−	25881.8i	−1.28804	−	0.827123i
$994$		0			0
$995$		−	30202.5i		−	0.962294i
$996$		0			0
$997$			49281.6i			1.56546i	0.622361	+	0.782731i	$0.286174\pi$
$997$			49281.6i			1.56546i	−0.622361	+	0.782731i	$0.713826\pi$
$998$		0			0
$999$	28550.0	+	4073.76i	0.904187	+	0.129017i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.4.f.b.47.7		8
3.2	odd	2	inner	96.4.f.b.47.6		8
4.3	odd	2		24.4.f.b.11.3	✓	8
8.3	odd	2	inner	96.4.f.b.47.8		8
8.5	even	2		24.4.f.b.11.5	yes	8
12.11	even	2		24.4.f.b.11.6	yes	8
16.3	odd	4		768.4.c.v.767.10		16
16.5	even	4		768.4.c.v.767.9		16
16.11	odd	4		768.4.c.v.767.7		16
16.13	even	4		768.4.c.v.767.8		16
24.5	odd	2		24.4.f.b.11.4	yes	8
24.11	even	2	inner	96.4.f.b.47.5		8
48.5	odd	4		768.4.c.v.767.6		16
48.11	even	4		768.4.c.v.767.12		16
48.29	odd	4		768.4.c.v.767.11		16
48.35	even	4		768.4.c.v.767.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.f.b.11.3	✓	8	4.3	odd	2
24.4.f.b.11.4	yes	8	24.5	odd	2
24.4.f.b.11.5	yes	8	8.5	even	2
24.4.f.b.11.6	yes	8	12.11	even	2
96.4.f.b.47.5		8	24.11	even	2	inner
96.4.f.b.47.6		8	3.2	odd	2	inner
96.4.f.b.47.7		8	1.1	even	1	trivial
96.4.f.b.47.8		8	8.3	odd	2	inner
768.4.c.v.767.5		16	48.35	even	4
768.4.c.v.767.6		16	48.5	odd	4
768.4.c.v.767.7		16	16.11	odd	4
768.4.c.v.767.8		16	16.13	even	4
768.4.c.v.767.9		16	16.5	even	4
768.4.c.v.767.10		16	16.3	odd	4
768.4.c.v.767.11		16	48.29	odd	4
768.4.c.v.767.12		16	48.11	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:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	96.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.37228 + 2.80770i) q^{3} -13.1715 q^{5} +26.9490i q^{7} +(11.2337 + 24.5521i) q^{9} +O(q^{10})\)	\(q+(4.37228 + 2.80770i) q^{3} -13.1715 q^{5} +26.9490i q^{7} +(11.2337 + 24.5521i) q^{9} +21.9834i q^{11} -10.0326i q^{13} +(-57.5896 - 36.9816i) q^{15} +6.09352i q^{17} +40.1902 q^{19} +(-75.6647 + 117.829i) q^{21} +9.80703 q^{23} +48.4891 q^{25} +(-19.8179 + 138.889i) q^{27} +164.501 q^{29} +47.0143i q^{31} +(-61.7228 + 96.1178i) q^{33} -354.960i q^{35} -205.560i q^{37} +(28.1685 - 43.8654i) q^{39} -419.120i q^{41} -205.038 q^{43} +(-147.965 - 323.388i) q^{45} +566.089 q^{47} -383.250 q^{49} +(-17.1087 + 26.6426i) q^{51} +342.173 q^{53} -289.555i q^{55} +(175.723 + 112.842i) q^{57} +3.70288i q^{59} +717.005i q^{61} +(-661.654 + 302.737i) q^{63} +132.145i q^{65} +238.701 q^{67} +(42.8791 + 27.5351i) q^{69} -517.054 q^{71} +984.195 q^{73} +(212.008 + 136.143i) q^{75} -592.432 q^{77} -329.100i q^{79} +(-476.609 + 551.621i) q^{81} +625.333i q^{83} -80.2609i q^{85} +(719.244 + 461.868i) q^{87} +238.583i q^{89} +270.369 q^{91} +(-132.002 + 205.560i) q^{93} -529.366 q^{95} -1001.01 q^{97} +(-539.739 + 246.955i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} - 48 q^{9}+O(q^{10}) \) 8 * q + 12 * q^3 - 48 * q^9	\( 8 q + 12 q^{3} - 48 q^{9} - 184 q^{19} + 296 q^{25} + 324 q^{27} - 264 q^{33} + 152 q^{43} - 952 q^{49} - 1056 q^{51} + 1176 q^{57} + 1496 q^{67} + 1072 q^{73} + 708 q^{75} - 504 q^{81} - 3168 q^{91} - 3872 q^{97} - 2112 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 - 48 * q^9 - 184 * q^19 + 296 * q^25 + 324 * q^27 - 264 * q^33 + 152 * q^43 - 952 * q^49 - 1056 * q^51 + 1176 * q^57 + 1496 * q^67 + 1072 * q^73 + 708 * q^75 - 504 * q^81 - 3168 * q^91 - 3872 * q^97 - 2112 * q^99

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