LMFDB - Embedded newform 3680.2.i.b.1471.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1471,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3680.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.i (of order $2$, degree $1$, 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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.5
Character	$\chi$	$=$	3680.1471
Dual form			3680.2.i.b.1471.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77635i q^{3} -1.00000i q^{5} -2.12267 q^{7} -4.70810 q^{9} +O(q^{10})$	$q-2.77635i q^{3} -1.00000i q^{5} -2.12267 q^{7} -4.70810 q^{9} +3.88538 q^{11} +5.83215 q^{13} -2.77635 q^{15} +3.55858i q^{17} -0.118081 q^{19} +5.89326i q^{21} +(4.68963 - 1.00369i) q^{23} -1.00000 q^{25} +4.74226i q^{27} +6.55238 q^{29} -4.89662i q^{31} -10.7872i q^{33} +2.12267i q^{35} -5.10404i q^{37} -16.1921i q^{39} +9.44324 q^{41} +5.28813 q^{43} +4.70810i q^{45} -1.10156i q^{47} -2.49428 q^{49} +9.87984 q^{51} +6.02382i q^{53} -3.88538i q^{55} +0.327832i q^{57} -3.73175i q^{59} +3.23099i q^{61} +9.99372 q^{63} -5.83215i q^{65} -1.04280 q^{67} +(-2.78658 - 13.0200i) q^{69} -9.98084i q^{71} +3.55886 q^{73} +2.77635i q^{75} -8.24737 q^{77} -11.9599 q^{79} -0.958124 q^{81} +12.9805 q^{83} +3.55858 q^{85} -18.1917i q^{87} +11.0591i q^{89} -12.3797 q^{91} -13.5947 q^{93} +0.118081i q^{95} +7.80650i q^{97} -18.2927 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q + 8 * q^7 - 48 * q^9	$ 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) $ 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	2.77635i		−	1.60292i	−0.598046	−	0.801462i	$-0.704056\pi$
$3$		−	2.77635i		−	1.60292i	0.598046	−	0.801462i	$-0.295944\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−2.12267			−0.802293			−0.401147	−	0.916014i	$-0.631388\pi$
$7$	−2.12267			−0.802293			−0.401147	+	0.916014i	$0.631388\pi$
$8$		0			0
$9$	−4.70810			−1.56937
$10$		0			0
$11$	3.88538			1.17149			0.585743	−	0.810497i	$-0.300803\pi$
$11$	3.88538			1.17149			0.585743	+	0.810497i	$0.300803\pi$
$12$		0			0
$13$	5.83215			1.61755			0.808773	−	0.588121i	$-0.200132\pi$
$13$	5.83215			1.61755			0.808773	+	0.588121i	$0.200132\pi$
$14$		0			0
$15$	−2.77635			−0.716849
$16$		0			0
$17$			3.55858i			0.863082i	0.902093	+	0.431541i	$0.142030\pi$
$17$			3.55858i			0.863082i	−0.902093	+	0.431541i	$0.857970\pi$
$18$		0			0
$19$	−0.118081			−0.0270895			−0.0135448	−	0.999908i	$-0.504312\pi$
$19$	−0.118081			−0.0270895			−0.0135448	+	0.999908i	$0.504312\pi$
$20$		0			0
$21$			5.89326i			1.28601i
$22$		0			0
$23$	4.68963	−	1.00369i	0.977855	−	0.209283i
$23$	4.68963	−	1.00369i	0.977855	−	0.209283i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			4.74226i			0.912649i
$28$		0			0
$29$	6.55238			1.21675			0.608373	−	0.793651i	$-0.291822\pi$
$29$	6.55238			1.21675			0.608373	+	0.793651i	$0.291822\pi$
$30$		0			0
$31$		−	4.89662i		−	0.879458i	−0.898130	−	0.439729i	$-0.855074\pi$
$31$		−	4.89662i		−	0.879458i	0.898130	−	0.439729i	$-0.144926\pi$
$32$		0			0
$33$		−	10.7872i		−	1.87780i
$34$		0			0
$35$			2.12267i			0.358796i
$36$		0			0
$37$		−	5.10404i		−	0.839098i	−0.907733	−	0.419549i	$-0.862188\pi$
$37$		−	5.10404i		−	0.839098i	0.907733	−	0.419549i	$-0.137812\pi$
$38$		0			0
$39$		−	16.1921i		−	2.59280i
$40$		0			0
$41$	9.44324			1.47479			0.737393	−	0.675464i	$-0.236057\pi$
$41$	9.44324			1.47479			0.737393	+	0.675464i	$0.236057\pi$
$42$		0			0
$43$	5.28813			0.806432			0.403216	−	0.915105i	$-0.367892\pi$
$43$	5.28813			0.806432			0.403216	+	0.915105i	$0.367892\pi$
$44$		0			0
$45$			4.70810i			0.701841i
$46$		0			0
$47$		−	1.10156i		−	0.160679i	−0.996768	−	0.0803393i	$-0.974400\pi$
$47$		−	1.10156i		−	0.160679i	0.996768	−	0.0803393i	$-0.0256004\pi$
$48$		0			0
$49$	−2.49428			−0.356326
$50$		0			0
$51$	9.87984			1.38345
$52$		0			0
$53$			6.02382i			0.827436i	0.910405	+	0.413718i	$0.135770\pi$
$53$			6.02382i			0.827436i	−0.910405	+	0.413718i	$0.864230\pi$
$54$		0			0
$55$		−	3.88538i		−	0.523905i
$56$		0			0
$57$			0.327832i			0.0434225i
$58$		0			0
$59$		−	3.73175i		−	0.485832i	−0.970047	−	0.242916i	$-0.921896\pi$
$59$		−	3.73175i		−	0.485832i	0.970047	−	0.242916i	$-0.0781040\pi$
$60$		0			0
$61$			3.23099i			0.413686i	0.978374	+	0.206843i	$0.0663190\pi$
$61$			3.23099i			0.413686i	−0.978374	+	0.206843i	$0.933681\pi$
$62$		0			0
$63$	9.99372			1.25909
$64$		0			0
$65$		−	5.83215i		−	0.723389i
$66$		0			0
$67$	−1.04280			−0.127398			−0.0636991	−	0.997969i	$-0.520290\pi$
$67$	−1.04280			−0.127398			−0.0636991	+	0.997969i	$0.520290\pi$
$68$		0			0
$69$	−2.78658	−	13.0200i	−0.335465	−	1.56743i
$70$		0			0
$71$		−	9.98084i		−	1.18451i	−0.805752	−	0.592254i	$-0.798238\pi$
$71$		−	9.98084i		−	1.18451i	0.805752	−	0.592254i	$-0.201762\pi$
$72$		0			0
$73$	3.55886			0.416533			0.208267	−	0.978072i	$-0.433218\pi$
$73$	3.55886			0.416533			0.208267	+	0.978072i	$0.433218\pi$
$74$		0			0
$75$			2.77635i			0.320585i
$76$		0			0
$77$	−8.24737			−0.939876
$78$		0			0
$79$	−11.9599			−1.34560			−0.672800	−	0.739825i	$-0.734908\pi$
$79$	−11.9599			−1.34560			−0.672800	+	0.739825i	$0.734908\pi$
$80$		0			0
$81$	−0.958124			−0.106458
$82$		0			0
$83$	12.9805			1.42480			0.712398	−	0.701775i	$-0.247609\pi$
$83$	12.9805			1.42480			0.712398	+	0.701775i	$0.247609\pi$
$84$		0			0
$85$	3.55858			0.385982
$86$		0			0
$87$		−	18.1917i		−	1.95035i
$88$		0			0
$89$			11.0591i			1.17226i	0.810217	+	0.586130i	$0.199349\pi$
$89$			11.0591i			1.17226i	−0.810217	+	0.586130i	$0.800651\pi$
$90$		0			0
$91$	−12.3797			−1.29775
$92$		0			0
$93$	−13.5947			−1.40970
$94$		0			0
$95$			0.118081i			0.0121148i
$96$		0			0
$97$			7.80650i			0.792630i	0.918115	+	0.396315i	$0.129711\pi$
$97$			7.80650i			0.792630i	−0.918115	+	0.396315i	$0.870289\pi$
$98$		0			0
$99$	−18.2927			−1.83849
$100$		0			0
$101$	−7.84603			−0.780710			−0.390355	−	0.920665i	$-0.627648\pi$
$101$	−7.84603			−0.780710			−0.390355	+	0.920665i	$0.627648\pi$
$102$		0			0
$103$	9.45414			0.931544			0.465772	−	0.884905i	$-0.345777\pi$
$103$	9.45414			0.931544			0.465772	+	0.884905i	$0.345777\pi$
$104$		0			0
$105$	5.89326			0.575123
$106$		0			0
$107$	−5.03270			−0.486530			−0.243265	−	0.969960i	$-0.578218\pi$
$107$	−5.03270			−0.486530			−0.243265	+	0.969960i	$0.578218\pi$
$108$		0			0
$109$			5.36876i			0.514234i	0.966380	+	0.257117i	$0.0827726\pi$
$109$			5.36876i			0.514234i	−0.966380	+	0.257117i	$0.917227\pi$
$110$		0			0
$111$	−14.1706			−1.34501
$112$		0			0
$113$			4.06575i			0.382474i	0.981544	+	0.191237i	$0.0612499\pi$
$113$			4.06575i			0.382474i	−0.981544	+	0.191237i	$0.938750\pi$
$114$		0			0
$115$	−1.00369	−	4.68963i	−0.0935942	−	0.437310i
$116$		0			0
$117$	−27.4583			−2.53852
$118$		0			0
$119$		−	7.55368i		−	0.692444i
$120$		0			0
$121$	4.09618			0.372380
$122$		0			0
$123$		−	26.2177i		−	2.36397i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			7.46925i			0.662789i	0.943492	+	0.331394i	$0.107519\pi$
$127$			7.46925i			0.662789i	−0.943492	+	0.331394i	$0.892481\pi$
$128$		0			0
$129$		−	14.6817i		−	1.29265i
$130$		0			0
$131$		−	12.1579i		−	1.06224i	−0.847295	−	0.531122i	$-0.821771\pi$
$131$		−	12.1579i		−	1.06224i	0.847295	−	0.531122i	$-0.178229\pi$
$132$		0			0
$133$	0.250646			0.0217337
$134$		0			0
$135$	4.74226			0.408149
$136$		0			0
$137$			19.5016i			1.66613i	0.553173	+	0.833066i	$0.313417\pi$
$137$			19.5016i			1.66613i	−0.553173	+	0.833066i	$0.686583\pi$
$138$		0			0
$139$		−	4.86192i		−	0.412383i	−0.978512	−	0.206192i	$-0.933893\pi$
$139$		−	4.86192i		−	0.412383i	0.978512	−	0.206192i	$-0.0661070\pi$
$140$		0			0
$141$	−3.05830			−0.257556
$142$		0			0
$143$	22.6601			1.89493
$144$		0			0
$145$		−	6.55238i		−	0.544146i
$146$		0			0
$147$			6.92498i			0.571163i
$148$		0			0
$149$		−	13.7604i		−	1.12729i	−0.826016	−	0.563647i	$-0.809398\pi$
$149$		−	13.7604i		−	1.12729i	0.826016	−	0.563647i	$-0.190602\pi$
$150$		0			0
$151$		−	14.6647i		−	1.19340i	−0.802466	−	0.596698i	$-0.796479\pi$
$151$		−	14.6647i		−	1.19340i	0.802466	−	0.596698i	$-0.203521\pi$
$152$		0			0
$153$		−	16.7541i		−	1.35449i
$154$		0			0
$155$	−4.89662			−0.393306
$156$		0			0
$157$		−	16.5379i		−	1.31987i	−0.751324	−	0.659933i	$-0.770585\pi$
$157$		−	16.5379i		−	1.31987i	0.751324	−	0.659933i	$-0.229415\pi$
$158$		0			0
$159$	16.7242			1.32632
$160$		0			0
$161$	−9.95453	+	2.13049i	−0.784526	+	0.167906i
$162$		0			0
$163$			18.6985i			1.46458i	0.680995	+	0.732288i	$0.261547\pi$
$163$			18.6985i			1.46458i	−0.680995	+	0.732288i	$0.738453\pi$
$164$		0			0
$165$	−10.7872			−0.839779
$166$		0			0
$167$		−	16.4117i		−	1.26997i	−0.772523	−	0.634986i	$-0.781006\pi$
$167$		−	16.4117i		−	1.26997i	0.772523	−	0.634986i	$-0.218994\pi$
$168$		0			0
$169$	21.0139			1.61646
$170$		0			0
$171$	0.555934			0.0425134
$172$		0			0
$173$	−22.9622			−1.74578			−0.872892	−	0.487914i	$-0.837758\pi$
$173$	−22.9622			−1.74578			−0.872892	+	0.487914i	$0.837758\pi$
$174$		0			0
$175$	2.12267			0.160459
$176$		0			0
$177$	−10.3606			−0.778752
$178$		0			0
$179$			19.2382i			1.43793i	0.695045	+	0.718966i	$0.255385\pi$
$179$			19.2382i			1.43793i	−0.695045	+	0.718966i	$0.744615\pi$
$180$		0			0
$181$		−	1.03724i		−	0.0770977i	−0.999257	−	0.0385489i	$-0.987726\pi$
$181$		−	1.03724i		−	0.0770977i	0.999257	−	0.0385489i	$-0.0122735\pi$
$182$		0			0
$183$	8.97036			0.663108
$184$		0			0
$185$	−5.10404			−0.375256
$186$		0			0
$187$			13.8264i			1.01109i
$188$		0			0
$189$		−	10.0663i		−	0.732212i
$190$		0			0
$191$	15.9355			1.15305			0.576526	−	0.817079i	$-0.304408\pi$
$191$	15.9355			1.15305			0.576526	+	0.817079i	$0.304408\pi$
$192$		0			0
$193$	8.90975			0.641338			0.320669	−	0.947191i	$-0.396092\pi$
$193$	8.90975			0.641338			0.320669	+	0.947191i	$0.396092\pi$
$194$		0			0
$195$	−16.1921			−1.15954
$196$		0			0
$197$	−5.77839			−0.411693			−0.205847	−	0.978584i	$-0.565995\pi$
$197$	−5.77839			−0.411693			−0.205847	+	0.978584i	$0.565995\pi$
$198$		0			0
$199$	−21.1623			−1.50016			−0.750078	−	0.661349i	$-0.769984\pi$
$199$	−21.1623			−1.50016			−0.750078	+	0.661349i	$0.769984\pi$
$200$		0			0
$201$			2.89517i			0.204209i
$202$		0			0
$203$	−13.9085			−0.976187
$204$		0			0
$205$		−	9.44324i		−	0.659544i
$206$		0			0
$207$	−22.0792	+	4.72545i	−1.53461	+	0.328441i
$208$		0			0
$209$	−0.458788			−0.0317350
$210$		0			0
$211$			7.81849i			0.538247i	0.963106	+	0.269123i	$0.0867340\pi$
$211$			7.81849i			0.538247i	−0.963106	+	0.269123i	$0.913266\pi$
$212$		0			0
$213$	−27.7103			−1.89868
$214$		0			0
$215$		−	5.28813i		−	0.360647i
$216$		0			0
$217$			10.3939i			0.705583i
$218$		0			0
$219$		−	9.88064i		−	0.667671i
$220$		0			0
$221$			20.7541i			1.39607i
$222$		0			0
$223$			14.8662i			0.995515i	0.867316	+	0.497757i	$0.165843\pi$
$223$			14.8662i			0.995515i	−0.867316	+	0.497757i	$0.834157\pi$
$224$		0			0
$225$	4.70810			0.313873
$226$		0			0
$227$	−16.5284			−1.09703			−0.548513	−	0.836142i	$-0.684806\pi$
$227$	−16.5284			−1.09703			−0.548513	+	0.836142i	$0.684806\pi$
$228$		0			0
$229$		−	28.4773i		−	1.88183i	−0.338640	−	0.940916i	$-0.609967\pi$
$229$		−	28.4773i		−	1.88183i	0.338640	−	0.940916i	$-0.390033\pi$
$230$		0			0
$231$			22.8976i			1.50655i
$232$		0			0
$233$	11.8269			0.774806			0.387403	−	0.921910i	$-0.373372\pi$
$233$	11.8269			0.774806			0.387403	+	0.921910i	$0.373372\pi$
$234$		0			0
$235$	−1.10156			−0.0718577
$236$		0			0
$237$			33.2049i			2.15689i
$238$		0			0
$239$			4.47853i			0.289692i	0.989454	+	0.144846i	$0.0462687\pi$
$239$			4.47853i			0.289692i	−0.989454	+	0.144846i	$0.953731\pi$
$240$		0			0
$241$			17.6301i			1.13566i	0.823147	+	0.567828i	$0.192216\pi$
$241$			17.6301i			1.13566i	−0.823147	+	0.567828i	$0.807784\pi$
$242$		0			0
$243$			16.8869i			1.08329i
$244$		0			0
$245$			2.49428i			0.159354i
$246$		0			0
$247$	−0.688663			−0.0438186
$248$		0			0
$249$		−	36.0384i		−	2.28384i
$250$		0			0
$251$	16.6788			1.05276			0.526379	−	0.850250i	$-0.323549\pi$
$251$	16.6788			1.05276			0.526379	+	0.850250i	$0.323549\pi$
$252$		0			0
$253$	18.2210	−	3.89970i	1.14554	−	0.245172i
$254$		0			0
$255$		−	9.87984i		−	0.618699i
$256$		0			0
$257$	−14.7737			−0.921559			−0.460779	−	0.887515i	$-0.652430\pi$
$257$	−14.7737			−0.921559			−0.460779	+	0.887515i	$0.652430\pi$
$258$		0			0
$259$			10.8342i			0.673203i
$260$		0			0
$261$	−30.8492			−1.90952
$262$		0			0
$263$	−16.8314			−1.03787			−0.518935	−	0.854814i	$-0.673671\pi$
$263$	−16.8314			−1.03787			−0.518935	+	0.854814i	$0.673671\pi$
$264$		0			0
$265$	6.02382			0.370041
$266$		0			0
$267$	30.7038			1.87904
$268$		0			0
$269$	−23.0569			−1.40581			−0.702903	−	0.711286i	$-0.748113\pi$
$269$	−23.0569			−1.40581			−0.702903	+	0.711286i	$0.748113\pi$
$270$		0			0
$271$		−	13.8980i		−	0.844241i	−0.906540	−	0.422120i	$-0.861286\pi$
$271$		−	13.8980i		−	0.844241i	0.906540	−	0.422120i	$-0.138714\pi$
$272$		0			0
$273$			34.3704i			2.08019i
$274$		0			0
$275$	−3.88538			−0.234297
$276$		0			0
$277$	17.2926			1.03901			0.519507	−	0.854466i	$-0.326116\pi$
$277$	17.2926			1.03901			0.519507	+	0.854466i	$0.326116\pi$
$278$		0			0
$279$			23.0537i			1.38019i
$280$		0			0
$281$		−	29.7634i		−	1.77554i	−0.460290	−	0.887769i	$-0.652255\pi$
$281$		−	29.7634i		−	1.77554i	0.460290	−	0.887769i	$-0.347745\pi$
$282$		0			0
$283$	−13.7925			−0.819880			−0.409940	−	0.912113i	$-0.634450\pi$
$283$	−13.7925			−0.819880			−0.409940	+	0.912113i	$0.634450\pi$
$284$		0			0
$285$	0.327832			0.0194191
$286$		0			0
$287$	−20.0449			−1.18321
$288$		0			0
$289$	4.33654			0.255090
$290$		0			0
$291$	21.6735			1.27053
$292$		0			0
$293$			27.0369i			1.57951i	0.613421	+	0.789756i	$0.289793\pi$
$293$			27.0369i			1.57951i	−0.613421	+	0.789756i	$0.710207\pi$
$294$		0			0
$295$	−3.73175			−0.217271
$296$		0			0
$297$			18.4255i			1.06916i
$298$		0			0
$299$	27.3506	−	5.85364i	1.58173	−	0.338525i
$300$		0			0
$301$	−11.2249			−0.646995
$302$		0			0
$303$			21.7833i			1.25142i
$304$		0			0
$305$	3.23099			0.185006
$306$		0			0
$307$		−	17.3641i		−	0.991020i	−0.868602	−	0.495510i	$-0.834981\pi$
$307$		−	17.3641i		−	0.991020i	0.868602	−	0.495510i	$-0.165019\pi$
$308$		0			0
$309$		−	26.2480i		−	1.49319i
$310$		0			0
$311$		−	9.29954i		−	0.527328i	−0.964615	−	0.263664i	$-0.915069\pi$
$311$		−	9.29954i		−	0.527328i	0.964615	−	0.263664i	$-0.0849311\pi$
$312$		0			0
$313$			13.2617i			0.749593i	0.927107	+	0.374797i	$0.122288\pi$
$313$			13.2617i			0.749593i	−0.927107	+	0.374797i	$0.877712\pi$
$314$		0			0
$315$		−	9.99372i		−	0.563083i
$316$		0			0
$317$	16.5351			0.928705			0.464352	−	0.885651i	$-0.346287\pi$
$317$	16.5351			0.928705			0.464352	+	0.885651i	$0.346287\pi$
$318$		0			0
$319$	25.4585			1.42540
$320$		0			0
$321$			13.9725i			0.779870i
$322$		0			0
$323$		−	0.420199i		−	0.0233805i
$324$		0			0
$325$	−5.83215			−0.323509
$326$		0			0
$327$	14.9055			0.824278
$328$		0			0
$329$			2.33824i			0.128911i
$330$		0			0
$331$			13.6012i			0.747589i	0.927512	+	0.373794i	$0.121943\pi$
$331$			13.6012i			0.747589i	−0.927512	+	0.373794i	$0.878057\pi$
$332$		0			0
$333$			24.0303i			1.31685i
$334$		0			0
$335$			1.04280i			0.0569742i
$336$		0			0
$337$		−	26.9957i		−	1.47055i	−0.677770	−	0.735274i	$-0.737054\pi$
$337$		−	26.9957i		−	1.47055i	0.677770	−	0.735274i	$-0.262946\pi$
$338$		0			0
$339$	11.2879			0.613076
$340$		0			0
$341$		−	19.0252i		−	1.03027i
$342$		0			0
$343$	20.1532			1.08817
$344$		0			0
$345$	−13.0200	+	2.78658i	−0.700975	+	0.150024i
$346$		0			0
$347$		−	0.820623i		−	0.0440533i	−0.999757	−	0.0220267i	$-0.992988\pi$
$347$		−	0.820623i		−	0.0440533i	0.999757	−	0.0220267i	$-0.00701188\pi$
$348$		0			0
$349$	−23.3542			−1.25012			−0.625061	−	0.780576i	$-0.714926\pi$
$349$	−23.3542			−1.25012			−0.625061	+	0.780576i	$0.714926\pi$
$350$		0			0
$351$			27.6576i			1.47625i
$352$		0			0
$353$	−24.4525			−1.30148			−0.650739	−	0.759302i	$-0.725541\pi$
$353$	−24.4525			−1.30148			−0.650739	+	0.759302i	$0.725541\pi$
$354$		0			0
$355$	−9.98084			−0.529728
$356$		0			0
$357$	−20.9716			−1.10994
$358$		0			0
$359$	−21.7221			−1.14645			−0.573225	−	0.819398i	$-0.694308\pi$
$359$	−21.7221			−1.14645			−0.573225	+	0.819398i	$0.694308\pi$
$360$		0			0
$361$	−18.9861			−0.999266
$362$		0			0
$363$		−	11.3724i		−	0.596897i
$364$		0			0
$365$		−	3.55886i		−	0.186279i
$366$		0			0
$367$	9.42878			0.492178			0.246089	−	0.969247i	$-0.420854\pi$
$367$	9.42878			0.492178			0.246089	+	0.969247i	$0.420854\pi$
$368$		0			0
$369$	−44.4597			−2.31448
$370$		0			0
$371$		−	12.7866i		−	0.663846i
$372$		0			0
$373$			7.05335i			0.365208i	0.983186	+	0.182604i	$0.0584527\pi$
$373$			7.05335i			0.365208i	−0.983186	+	0.182604i	$0.941547\pi$
$374$		0			0
$375$	2.77635			0.143370
$376$		0			0
$377$	38.2144			1.96814
$378$		0			0
$379$	9.23178			0.474205			0.237102	−	0.971485i	$-0.423802\pi$
$379$	9.23178			0.474205			0.237102	+	0.971485i	$0.423802\pi$
$380$		0			0
$381$	20.7372			1.06240
$382$		0			0
$383$	15.4246			0.788161			0.394080	−	0.919076i	$-0.371063\pi$
$383$	15.4246			0.788161			0.394080	+	0.919076i	$0.371063\pi$
$384$		0			0
$385$			8.24737i			0.420325i
$386$		0			0
$387$	−24.8970			−1.26559
$388$		0			0
$389$		−	22.4903i		−	1.14030i	−0.821540	−	0.570152i	$-0.806884\pi$
$389$		−	22.4903i		−	1.14030i	0.821540	−	0.570152i	$-0.193116\pi$
$390$		0			0
$391$	3.57169	+	16.6884i	0.180628	+	0.843969i
$392$		0			0
$393$	−33.7546			−1.70270
$394$		0			0
$395$			11.9599i			0.601770i
$396$		0			0
$397$	11.3276			0.568517			0.284259	−	0.958748i	$-0.408253\pi$
$397$	11.3276			0.568517			0.284259	+	0.958748i	$0.408253\pi$
$398$		0			0
$399$		−	0.695879i		−	0.0348375i
$400$		0			0
$401$		−	11.6144i		−	0.579994i	−0.957028	−	0.289997i	$-0.906346\pi$
$401$		−	11.6144i		−	0.579994i	0.957028	−	0.289997i	$-0.0936542\pi$
$402$		0			0
$403$		−	28.5578i		−	1.42256i
$404$		0			0
$405$			0.958124i			0.0476096i
$406$		0			0
$407$		−	19.8311i		−	0.982992i
$408$		0			0
$409$	−8.26980			−0.408915			−0.204458	−	0.978875i	$-0.565543\pi$
$409$	−8.26980			−0.408915			−0.204458	+	0.978875i	$0.565543\pi$
$410$		0			0
$411$	54.1431			2.67068
$412$		0			0
$413$			7.92127i			0.389780i
$414$		0			0
$415$		−	12.9805i		−	0.637188i
$416$		0			0
$417$	−13.4984			−0.661019
$418$		0			0
$419$	−15.1987			−0.742506			−0.371253	−	0.928532i	$-0.621072\pi$
$419$	−15.1987			−0.742506			−0.371253	+	0.928532i	$0.621072\pi$
$420$		0			0
$421$			14.7721i			0.719946i	0.932963	+	0.359973i	$0.117214\pi$
$421$			14.7721i			0.719946i	−0.932963	+	0.359973i	$0.882786\pi$
$422$		0			0
$423$			5.18624i			0.252163i
$424$		0			0
$425$		−	3.55858i		−	0.172616i
$426$		0			0
$427$		−	6.85833i		−	0.331898i
$428$		0			0
$429$		−	62.9123i		−	3.03743i
$430$		0			0
$431$	−19.3888			−0.933926			−0.466963	−	0.884277i	$-0.654652\pi$
$431$	−19.3888			−0.933926			−0.466963	+	0.884277i	$0.654652\pi$
$432$		0			0
$433$		−	8.46235i		−	0.406674i	−0.979109	−	0.203337i	$-0.934821\pi$
$433$		−	8.46235i		−	0.406674i	0.979109	−	0.203337i	$-0.0651788\pi$
$434$		0			0
$435$	−18.1917			−0.872224
$436$		0			0
$437$	−0.553754	+	0.118516i	−0.0264896	+	0.00566938i
$438$		0			0
$439$		−	28.4816i		−	1.35935i	−0.733511	−	0.679677i	$-0.762120\pi$
$439$		−	28.4816i		−	1.35935i	0.733511	−	0.679677i	$-0.237880\pi$
$440$		0			0
$441$	11.7433			0.559205
$442$		0			0
$443$		−	27.0185i		−	1.28369i	−0.766835	−	0.641845i	$-0.778169\pi$
$443$		−	27.0185i		−	1.28369i	0.766835	−	0.641845i	$-0.221831\pi$
$444$		0			0
$445$	11.0591			0.524251
$446$		0			0
$447$	−38.2035			−1.80697
$448$		0			0
$449$	30.3093			1.43038			0.715192	−	0.698928i	$-0.246339\pi$
$449$	30.3093			1.43038			0.715192	+	0.698928i	$0.246339\pi$
$450$		0			0
$451$	36.6906			1.72769
$452$		0			0
$453$	−40.7143			−1.91292
$454$		0			0
$455$			12.3797i			0.580370i
$456$		0			0
$457$			10.0871i			0.471856i	0.971771	+	0.235928i	$0.0758130\pi$
$457$			10.0871i			0.471856i	−0.971771	+	0.235928i	$0.924187\pi$
$458$		0			0
$459$	−16.8757			−0.787690
$460$		0			0
$461$	27.6122			1.28603			0.643014	−	0.765854i	$-0.277684\pi$
$461$	27.6122			1.28603			0.643014	+	0.765854i	$0.277684\pi$
$462$		0			0
$463$		−	24.8679i		−	1.15571i	−0.816140	−	0.577855i	$-0.803890\pi$
$463$		−	24.8679i		−	1.15571i	0.816140	−	0.577855i	$-0.196110\pi$
$464$		0			0
$465$			13.5947i			0.630439i
$466$		0			0
$467$	−7.52128			−0.348043			−0.174022	−	0.984742i	$-0.555676\pi$
$467$	−7.52128			−0.348043			−0.174022	+	0.984742i	$0.555676\pi$
$468$		0			0
$469$	2.21352			0.102211
$470$		0			0
$471$	−45.9149			−2.11565
$472$		0			0
$473$	20.5464			0.944724
$474$		0			0
$475$	0.118081			0.00541791
$476$		0			0
$477$		−	28.3607i		−	1.29855i
$478$		0			0
$479$	−22.0591			−1.00791			−0.503953	−	0.863731i	$-0.668122\pi$
$479$	−22.0591			−1.00791			−0.503953	+	0.863731i	$0.668122\pi$
$480$		0			0
$481$		−	29.7675i		−	1.35728i
$482$		0			0
$483$	5.91498	+	27.6372i	0.269141	+	1.25754i
$484$		0			0
$485$	7.80650			0.354475
$486$		0			0
$487$			15.7131i			0.712030i	0.934480	+	0.356015i	$0.115865\pi$
$487$			15.7131i			0.712030i	−0.934480	+	0.356015i	$0.884135\pi$
$488$		0			0
$489$	51.9134			2.34760
$490$		0			0
$491$		−	12.8581i		−	0.580277i	−0.956985	−	0.290139i	$-0.906299\pi$
$491$		−	12.8581i		−	0.580277i	0.956985	−	0.290139i	$-0.0937014\pi$
$492$		0			0
$493$			23.3171i			1.05015i
$494$		0			0
$495$			18.2927i			0.822198i
$496$		0			0
$497$			21.1860i			0.950322i
$498$		0			0
$499$		−	22.8652i		−	1.02359i	−0.859108	−	0.511794i	$-0.828981\pi$
$499$		−	22.8652i		−	1.02359i	0.859108	−	0.511794i	$-0.171019\pi$
$500$		0			0
$501$	−45.5644			−2.03567
$502$		0			0
$503$	44.3523			1.97757			0.988785	−	0.149348i	$-0.0477175\pi$
$503$	44.3523			1.97757			0.988785	+	0.149348i	$0.0477175\pi$
$504$		0			0
$505$			7.84603i			0.349144i
$506$		0			0
$507$		−	58.3419i		−	2.59106i
$508$		0			0
$509$	8.04463			0.356572			0.178286	−	0.983979i	$-0.442945\pi$
$509$	8.04463			0.356572			0.178286	+	0.983979i	$0.442945\pi$
$510$		0			0
$511$	−7.55429			−0.334182
$512$		0			0
$513$		−	0.559969i		−	0.0247232i
$514$		0			0
$515$		−	9.45414i		−	0.416599i
$516$		0			0
$517$		−	4.27997i		−	0.188233i
$518$		0			0
$519$			63.7510i			2.79836i
$520$		0			0
$521$		−	33.5780i		−	1.47108i	−0.677481	−	0.735540i	$-0.736928\pi$
$521$		−	33.5780i		−	1.47108i	0.677481	−	0.735540i	$-0.263072\pi$
$522$		0			0
$523$	−26.0721			−1.14005			−0.570026	−	0.821627i	$-0.693067\pi$
$523$	−26.0721			−1.14005			−0.570026	+	0.821627i	$0.693067\pi$
$524$		0			0
$525$		−	5.89326i		−	0.257203i
$526$		0			0
$527$	17.4250			0.759044
$528$		0			0
$529$	20.9852	−	9.41383i	0.912401	−	0.409297i
$530$		0			0
$531$			17.5694i			0.762448i
$532$		0			0
$533$	55.0743			2.38553
$534$		0			0
$535$			5.03270i			0.217583i
$536$		0			0
$537$	53.4120			2.30490
$538$		0			0
$539$	−9.69123			−0.417431
$540$		0			0
$541$	24.9247			1.07160			0.535798	−	0.844346i	$-0.320011\pi$
$541$	24.9247			1.07160			0.535798	+	0.844346i	$0.320011\pi$
$542$		0			0
$543$	−2.87975			−0.123582
$544$		0			0
$545$	5.36876			0.229973
$546$		0			0
$547$			36.8872i			1.57718i	0.614918	+	0.788591i	$0.289189\pi$
$547$			36.8872i			1.57718i	−0.614918	+	0.788591i	$0.710811\pi$
$548$		0			0
$549$		−	15.2118i		−	0.649225i
$550$		0			0
$551$	−0.773709			−0.0329611
$552$		0			0
$553$	25.3870			1.07957
$554$		0			0
$555$			14.1706i			0.601507i
$556$		0			0
$557$			44.0639i			1.86705i	0.358514	+	0.933524i	$0.383284\pi$
$557$			44.0639i			1.86705i	−0.358514	+	0.933524i	$0.616716\pi$
$558$		0			0
$559$	30.8411			1.30444
$560$		0			0
$561$	38.3869			1.62070
$562$		0			0
$563$	−37.5573			−1.58285			−0.791427	−	0.611264i	$-0.790661\pi$
$563$	−37.5573			−1.58285			−0.791427	+	0.611264i	$0.790661\pi$
$564$		0			0
$565$	4.06575			0.171047
$566$		0			0
$567$	2.03378			0.0854107
$568$		0			0
$569$		−	0.817680i		−	0.0342789i	−0.999853	−	0.0171395i	$-0.994544\pi$
$569$		−	0.817680i		−	0.0342789i	0.999853	−	0.0171395i	$-0.00545593\pi$
$570$		0			0
$571$	−32.7278			−1.36961			−0.684807	−	0.728724i	$-0.740114\pi$
$571$	−32.7278			−1.36961			−0.684807	+	0.728724i	$0.740114\pi$
$572$		0			0
$573$		−	44.2424i		−	1.84825i
$574$		0			0
$575$	−4.68963	+	1.00369i	−0.195571	+	0.0418566i
$576$		0			0
$577$	5.24581			0.218386			0.109193	−	0.994021i	$-0.465173\pi$
$577$	5.24581			0.218386			0.109193	+	0.994021i	$0.465173\pi$
$578$		0			0
$579$		−	24.7365i		−	1.02802i
$580$		0			0
$581$	−27.5533			−1.14310
$582$		0			0
$583$			23.4049i			0.969330i
$584$		0			0
$585$			27.4583i			1.13526i
$586$		0			0
$587$		−	3.51658i		−	0.145145i	−0.997363	−	0.0725724i	$-0.976879\pi$
$587$		−	3.51658i		−	0.145145i	0.997363	−	0.0725724i	$-0.0231208\pi$
$588$		0			0
$589$			0.578195i			0.0238241i
$590$		0			0
$591$			16.0428i			0.659913i
$592$		0			0
$593$	−6.57776			−0.270116			−0.135058	−	0.990838i	$-0.543122\pi$
$593$	−6.57776			−0.270116			−0.135058	+	0.990838i	$0.543122\pi$
$594$		0			0
$595$	−7.55368			−0.309671
$596$		0			0
$597$			58.7539i			2.40464i
$598$		0			0
$599$		−	10.4320i		−	0.426240i	−0.977026	−	0.213120i	$-0.931637\pi$
$599$		−	10.4320i		−	0.426240i	0.977026	−	0.213120i	$-0.0683625\pi$
$600$		0			0
$601$	42.8739			1.74886			0.874432	−	0.485147i	$-0.161234\pi$
$601$	42.8739			1.74886			0.874432	+	0.485147i	$0.161234\pi$
$602$		0			0
$603$	4.90959			0.199934
$604$		0			0
$605$		−	4.09618i		−	0.166534i
$606$		0			0
$607$			12.6628i			0.513969i	0.966416	+	0.256984i	$0.0827289\pi$
$607$			12.6628i			0.513969i	−0.966416	+	0.256984i	$0.917271\pi$
$608$		0			0
$609$			38.6149i			1.56475i
$610$		0			0
$611$		−	6.42444i		−	0.259905i
$612$		0			0
$613$		−	16.2560i		−	0.656572i	−0.944578	−	0.328286i	$-0.893529\pi$
$613$		−	16.2560i		−	0.656572i	0.944578	−	0.328286i	$-0.106471\pi$
$614$		0			0
$615$	−26.2177			−1.05720
$616$		0			0
$617$		−	6.99134i		−	0.281461i	−0.990048	−	0.140730i	$-0.955055\pi$
$617$		−	6.99134i		−	0.281461i	0.990048	−	0.140730i	$-0.0449451\pi$
$618$		0			0
$619$	26.7735			1.07612			0.538060	−	0.842907i	$-0.319157\pi$
$619$	26.7735			1.07612			0.538060	+	0.842907i	$0.319157\pi$
$620$		0			0
$621$	4.75974	+	22.2395i	0.191002	+	0.892438i
$622$		0			0
$623$		−	23.4748i		−	0.940497i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			1.27375i			0.0508688i
$628$		0			0
$629$	18.1631			0.724210
$630$		0			0
$631$	−39.3060			−1.56475			−0.782373	−	0.622810i	$-0.785991\pi$
$631$	−39.3060			−1.56475			−0.782373	+	0.622810i	$0.785991\pi$
$632$		0			0
$633$	21.7068			0.862769
$634$		0			0
$635$	7.46925			0.296408
$636$		0			0
$637$	−14.5470			−0.576373
$638$		0			0
$639$			46.9907i			1.85892i
$640$		0			0
$641$		−	2.41297i		−	0.0953065i	−0.998864	−	0.0476533i	$-0.984826\pi$
$641$		−	2.41297i		−	0.0953065i	0.998864	−	0.0476533i	$-0.0151743\pi$
$642$		0			0
$643$	−17.3612			−0.684658			−0.342329	−	0.939580i	$-0.611216\pi$
$643$	−17.3612			−0.684658			−0.342329	+	0.939580i	$0.611216\pi$
$644$		0			0
$645$	−14.6817			−0.578090
$646$		0			0
$647$		−	17.3058i		−	0.680363i	−0.940360	−	0.340181i	$-0.889512\pi$
$647$		−	17.3058i		−	0.680363i	0.940360	−	0.340181i	$-0.110488\pi$
$648$		0			0
$649$		−	14.4993i		−	0.569146i
$650$		0			0
$651$	28.8570			1.13100
$652$		0			0
$653$	7.02936			0.275080			0.137540	−	0.990496i	$-0.456080\pi$
$653$	7.02936			0.275080			0.137540	+	0.990496i	$0.456080\pi$
$654$		0			0
$655$	−12.1579			−0.475050
$656$		0			0
$657$	−16.7555			−0.653693
$658$		0			0
$659$	21.5565			0.839723			0.419861	−	0.907588i	$-0.362079\pi$
$659$	21.5565			0.839723			0.419861	+	0.907588i	$0.362079\pi$
$660$		0			0
$661$			6.28224i			0.244351i	0.992509	+	0.122175i	$0.0389870\pi$
$661$			6.28224i			0.244351i	−0.992509	+	0.122175i	$0.961013\pi$
$662$		0			0
$663$	57.6207			2.23780
$664$		0			0
$665$		−	0.250646i		−	0.00971963i
$666$		0			0
$667$	30.7282	−	6.57653i	1.18980	−	0.254644i
$668$		0			0
$669$	41.2737			1.59573
$670$		0			0
$671$			12.5536i			0.484628i
$672$		0			0
$673$	37.2293			1.43508			0.717542	−	0.696515i	$-0.245267\pi$
$673$	37.2293			1.43508			0.717542	+	0.696515i	$0.245267\pi$
$674$		0			0
$675$		−	4.74226i		−	0.182530i
$676$		0			0
$677$			45.6005i			1.75257i	0.481794	+	0.876284i	$0.339985\pi$
$677$			45.6005i			1.75257i	−0.481794	+	0.876284i	$0.660015\pi$
$678$		0			0
$679$		−	16.5706i		−	0.635921i
$680$		0			0
$681$			45.8884i			1.75845i
$682$		0			0
$683$			39.0466i			1.49408i	0.664782	+	0.747038i	$0.268525\pi$
$683$			39.0466i			1.49408i	−0.664782	+	0.747038i	$0.731475\pi$
$684$		0			0
$685$	19.5016			0.745117
$686$		0			0
$687$	−79.0628			−3.01643
$688$		0			0
$689$			35.1318i			1.33842i
$690$		0			0
$691$			20.1445i			0.766333i	0.923679	+	0.383167i	$0.125167\pi$
$691$			20.1445i			0.766333i	−0.923679	+	0.383167i	$0.874833\pi$
$692$		0			0
$693$	38.8294			1.47501
$694$		0			0
$695$	−4.86192			−0.184423
$696$		0			0
$697$			33.6045i			1.27286i
$698$		0			0
$699$		−	32.8356i		−	1.24196i
$700$		0			0
$701$		−	1.12335i		−	0.0424284i	−0.999775	−	0.0212142i	$-0.993247\pi$
$701$		−	1.12335i		−	0.0424284i	0.999775	−	0.0212142i	$-0.00675320\pi$
$702$		0			0
$703$			0.602687i			0.0227308i
$704$		0			0
$705$			3.05830i			0.115182i
$706$		0			0
$707$	16.6545			0.626358
$708$		0			0
$709$			27.7585i			1.04249i	0.853406	+	0.521247i	$0.174533\pi$
$709$			27.7585i			1.04249i	−0.853406	+	0.521247i	$0.825467\pi$
$710$		0			0
$711$	56.3086			2.11174
$712$		0			0
$713$	−4.91467	−	22.9633i	−0.184056	−	0.859983i
$714$		0			0
$715$		−	22.6601i		−	0.847440i
$716$		0			0
$717$	12.4339			0.464354
$718$		0			0
$719$			36.5318i			1.36240i	0.732095	+	0.681202i	$0.238543\pi$
$719$			36.5318i			1.36240i	−0.732095	+	0.681202i	$0.761457\pi$
$720$		0			0
$721$	−20.0680			−0.747371
$722$		0			0
$723$	48.9473			1.82037
$724$		0			0
$725$	−6.55238			−0.243349
$726$		0			0
$727$	−16.1333			−0.598351			−0.299176	−	0.954198i	$-0.596712\pi$
$727$	−16.1333			−0.598351			−0.299176	+	0.954198i	$0.596712\pi$
$728$		0			0
$729$	44.0094			1.62998
$730$		0			0
$731$			18.8182i			0.696017i
$732$		0			0
$733$			30.5646i			1.12893i	0.825457	+	0.564465i	$0.190918\pi$
$733$			30.5646i			1.12893i	−0.825457	+	0.564465i	$0.809082\pi$
$734$		0			0
$735$	6.92498			0.255432
$736$		0			0
$737$	−4.05167			−0.149245
$738$		0			0
$739$			2.68089i			0.0986183i	0.998784	+	0.0493091i	$0.0157020\pi$
$739$			2.68089i			0.0986183i	−0.998784	+	0.0493091i	$0.984298\pi$
$740$		0			0
$741$			1.91197i			0.0702378i
$742$		0			0
$743$	6.25282			0.229394			0.114697	−	0.993401i	$-0.463410\pi$
$743$	6.25282			0.229394			0.114697	+	0.993401i	$0.463410\pi$
$744$		0			0
$745$	−13.7604			−0.504141
$746$		0			0
$747$	−61.1135			−2.23603
$748$		0			0
$749$	10.6828			0.390340
$750$		0			0
$751$	35.8459			1.30804			0.654018	−	0.756479i	$-0.273082\pi$
$751$	35.8459			1.30804			0.654018	+	0.756479i	$0.273082\pi$
$752$		0			0
$753$		−	46.3062i		−	1.68749i
$754$		0			0
$755$	−14.6647			−0.533703
$756$		0			0
$757$			20.1331i			0.731750i	0.930664	+	0.365875i	$0.119230\pi$
$757$			20.1331i			0.731750i	−0.930664	+	0.365875i	$0.880770\pi$
$758$		0			0
$759$	−10.8269	−	50.5878i	−0.392992	−	1.83622i
$760$		0			0
$761$	28.6194			1.03745			0.518726	−	0.854940i	$-0.326406\pi$
$761$	28.6194			1.03745			0.518726	+	0.854940i	$0.326406\pi$
$762$		0			0
$763$		−	11.3961i		−	0.412567i
$764$		0			0
$765$	−16.7541			−0.605746
$766$		0			0
$767$		−	21.7641i		−	0.785856i
$768$		0			0
$769$		−	22.0989i		−	0.796905i	−0.917189	−	0.398453i	$-0.869547\pi$
$769$		−	22.0989i		−	0.796905i	0.917189	−	0.398453i	$-0.130453\pi$
$770$		0			0
$771$			41.0169i			1.47719i
$772$		0			0
$773$		−	54.1160i		−	1.94642i	−0.229924	−	0.973209i	$-0.573848\pi$
$773$		−	54.1160i		−	1.94642i	0.229924	−	0.973209i	$-0.426152\pi$
$774$		0			0
$775$			4.89662i			0.175892i
$776$		0			0
$777$	30.0794			1.07909
$778$		0			0
$779$	−1.11506			−0.0399513
$780$		0			0
$781$		−	38.7793i		−	1.38763i
$782$		0			0
$783$			31.0731i			1.11046i
$784$		0			0
$785$	−16.5379			−0.590262
$786$		0			0
$787$	53.9663			1.92369			0.961845	−	0.273595i	$-0.0882127\pi$
$787$	53.9663			1.92369			0.961845	+	0.273595i	$0.0882127\pi$
$788$		0			0
$789$			46.7298i			1.66363i
$790$		0			0
$791$		−	8.63024i		−	0.306856i
$792$		0			0
$793$			18.8436i			0.669157i
$794$		0			0
$795$		−	16.7242i		−	0.593147i
$796$		0			0
$797$			2.33197i			0.0826027i	0.999147	+	0.0413013i	$0.0131504\pi$
$797$			2.33197i			0.0826027i	−0.999147	+	0.0413013i	$0.986850\pi$
$798$		0			0
$799$	3.91998			0.138679
$800$		0			0
$801$		−	52.0672i		−	1.83971i
$802$		0			0
$803$	13.8275			0.487963
$804$		0			0
$805$	2.13049	+	9.95453i	0.0750900	+	0.350851i
$806$		0			0
$807$			64.0140i			2.25340i
$808$		0			0
$809$	−43.3752			−1.52499			−0.762496	−	0.646993i	$-0.776026\pi$
$809$	−43.3752			−1.52499			−0.762496	+	0.646993i	$0.776026\pi$
$810$		0			0
$811$			13.7687i			0.483485i	0.970340	+	0.241742i	$0.0777189\pi$
$811$			13.7687i			0.483485i	−0.970340	+	0.241742i	$0.922281\pi$
$812$		0			0
$813$	−38.5855			−1.35325
$814$		0			0
$815$	18.6985			0.654978
$816$		0			0
$817$	−0.624425			−0.0218459
$818$		0			0
$819$	58.2849			2.03664
$820$		0			0
$821$	17.6132			0.614707			0.307353	−	0.951595i	$-0.400557\pi$
$821$	17.6132			0.614707			0.307353	+	0.951595i	$0.400557\pi$
$822$		0			0
$823$			3.87004i			0.134901i	0.997723	+	0.0674505i	$0.0214865\pi$
$823$			3.87004i			0.134901i	−0.997723	+	0.0674505i	$0.978514\pi$
$824$		0			0
$825$			10.7872i			0.375561i
$826$		0			0
$827$	−42.1264			−1.46488			−0.732439	−	0.680833i	$-0.761618\pi$
$827$	−42.1264			−1.46488			−0.732439	+	0.680833i	$0.761618\pi$
$828$		0			0
$829$	16.8804			0.586280			0.293140	−	0.956070i	$-0.405300\pi$
$829$	16.8804			0.586280			0.293140	+	0.956070i	$0.405300\pi$
$830$		0			0
$831$		−	48.0103i		−	1.66546i
$832$		0			0
$833$		−	8.87608i		−	0.307538i
$834$		0			0
$835$	−16.4117			−0.567949
$836$		0			0
$837$	23.2210			0.802637
$838$		0			0
$839$	−3.62775			−0.125244			−0.0626219	−	0.998037i	$-0.519946\pi$
$839$	−3.62775			−0.125244			−0.0626219	+	0.998037i	$0.519946\pi$
$840$		0			0
$841$	13.9337			0.480472
$842$		0			0
$843$	−82.6336			−2.84605
$844$		0			0
$845$		−	21.0139i		−	0.722901i
$846$		0			0
$847$	−8.69484			−0.298758
$848$		0			0
$849$			38.2928i			1.31420i
$850$		0			0
$851$	−5.12285	−	23.9360i	−0.175609	−	0.820517i
$852$		0			0
$853$	18.8498			0.645404			0.322702	−	0.946501i	$-0.395409\pi$
$853$	18.8498			0.645404			0.322702	+	0.946501i	$0.395409\pi$
$854$		0			0
$855$		−	0.555934i		−	0.0190126i
$856$		0			0
$857$	−35.0704			−1.19798			−0.598992	−	0.800755i	$-0.704432\pi$
$857$	−35.0704			−1.19798			−0.598992	+	0.800755i	$0.704432\pi$
$858$		0			0
$859$			32.1271i			1.09616i	0.836425	+	0.548081i	$0.184642\pi$
$859$			32.1271i			1.09616i	−0.836425	+	0.548081i	$0.815358\pi$
$860$		0			0
$861$			55.6515i			1.89660i
$862$		0			0
$863$			18.6407i			0.634538i	0.948336	+	0.317269i	$0.102766\pi$
$863$			18.6407i			0.634538i	−0.948336	+	0.317269i	$0.897234\pi$
$864$		0			0
$865$			22.9622i			0.780738i
$866$		0			0
$867$		−	12.0397i		−	0.408890i
$868$		0			0
$869$	−46.4689			−1.57635
$870$		0			0
$871$	−6.08175			−0.206072
$872$		0			0
$873$		−	36.7537i		−	1.24393i
$874$		0			0
$875$		−	2.12267i		−	0.0717593i
$876$		0			0
$877$	−30.0439			−1.01451			−0.507254	−	0.861796i	$-0.669340\pi$
$877$	−30.0439			−1.01451			−0.507254	+	0.861796i	$0.669340\pi$
$878$		0			0
$879$	75.0638			2.53184
$880$		0			0
$881$			52.8825i			1.78166i	0.454339	+	0.890829i	$0.349875\pi$
$881$			52.8825i			1.78166i	−0.454339	+	0.890829i	$0.650125\pi$
$882$		0			0
$883$			16.0949i			0.541636i	0.962631	+	0.270818i	$0.0872941\pi$
$883$			16.0949i			0.541636i	−0.962631	+	0.270818i	$0.912706\pi$
$884$		0			0
$885$			10.3606i			0.348269i
$886$		0			0
$887$		−	37.6520i		−	1.26423i	−0.774874	−	0.632116i	$-0.782187\pi$
$887$		−	37.6520i		−	1.26423i	0.774874	−	0.632116i	$-0.217813\pi$
$888$		0			0
$889$		−	15.8547i		−	0.531751i
$890$		0			0
$891$	−3.72268			−0.124714
$892$		0			0
$893$			0.130072i			0.00435271i
$894$		0			0
$895$	19.2382			0.643063
$896$		0			0
$897$	−16.2517	−	75.9347i	−0.542630	−	2.53539i
$898$		0			0
$899$		−	32.0845i		−	1.07008i
$900$		0			0
$901$	−21.4362			−0.714145
$902$		0			0
$903$			31.1643i			1.03708i
$904$		0			0
$905$	−1.03724			−0.0344791
$906$		0			0
$907$	8.91747			0.296100			0.148050	−	0.988980i	$-0.452700\pi$
$907$	8.91747			0.296100			0.148050	+	0.988980i	$0.452700\pi$
$908$		0			0
$909$	36.9399			1.22522
$910$		0			0
$911$	−49.5446			−1.64148			−0.820742	−	0.571299i	$-0.806440\pi$
$911$	−49.5446			−1.64148			−0.820742	+	0.571299i	$0.806440\pi$
$912$		0			0
$913$	50.4342			1.66913
$914$		0			0
$915$		−	8.97036i		−	0.296551i
$916$		0			0
$917$			25.8073i			0.852231i
$918$		0			0
$919$	−59.9002			−1.97593			−0.987963	−	0.154689i	$-0.950563\pi$
$919$	−59.9002			−1.97593			−0.987963	+	0.154689i	$0.950563\pi$
$920$		0			0
$921$	−48.2087			−1.58853
$922$		0			0
$923$		−	58.2097i		−	1.91600i
$924$		0			0
$925$			5.10404i			0.167820i
$926$		0			0
$927$	−44.5110			−1.46193
$928$		0			0
$929$	−30.8215			−1.01122			−0.505610	−	0.862762i	$-0.668732\pi$
$929$	−30.8215			−1.01122			−0.505610	+	0.862762i	$0.668732\pi$
$930$		0			0
$931$	0.294526			0.00965269
$932$		0			0
$933$	−25.8187			−0.845267
$934$		0			0
$935$	13.8264			0.452172
$936$		0			0
$937$			57.2315i			1.86967i	0.355081	+	0.934835i	$0.384453\pi$
$937$			57.2315i			1.86967i	−0.355081	+	0.934835i	$0.615547\pi$
$938$		0			0
$939$	36.8190			1.20154
$940$		0			0
$941$		−	40.4063i		−	1.31721i	−0.752490	−	0.658604i	$-0.771147\pi$
$941$		−	40.4063i		−	1.31721i	0.752490	−	0.658604i	$-0.228853\pi$
$942$		0			0
$943$	44.2853	−	9.47805i	1.44213	−	0.308648i
$944$		0			0
$945$	−10.0663			−0.327455
$946$		0			0
$947$			34.6705i			1.12664i	0.826239	+	0.563320i	$0.190476\pi$
$947$			34.6705i			1.12664i	−0.826239	+	0.563320i	$0.809524\pi$
$948$		0			0
$949$	20.7558			0.673762
$950$		0			0
$951$		−	45.9072i		−	1.48864i
$952$		0			0
$953$		−	13.3367i		−	0.432018i	−0.976391	−	0.216009i	$-0.930696\pi$
$953$		−	13.3367i		−	0.432018i	0.976391	−	0.216009i	$-0.0693041\pi$
$954$		0			0
$955$		−	15.9355i		−	0.515660i
$956$		0			0
$957$		−	70.6816i		−	2.28481i
$958$		0			0
$959$		−	41.3954i		−	1.33673i
$960$		0			0
$961$	7.02315			0.226553
$962$		0			0
$963$	23.6944			0.763543
$964$		0			0
$965$		−	8.90975i		−	0.286815i
$966$		0			0
$967$		−	14.9203i		−	0.479806i	−0.970797	−	0.239903i	$-0.922884\pi$
$967$		−	14.9203i		−	0.479806i	0.970797	−	0.239903i	$-0.0771156\pi$
$968$		0			0
$969$	−1.16662			−0.0374771
$970$		0			0
$971$	−12.7555			−0.409344			−0.204672	−	0.978831i	$-0.565613\pi$
$971$	−12.7555			−0.409344			−0.204672	+	0.978831i	$0.565613\pi$
$972$		0			0
$973$			10.3203i			0.330852i
$974$		0			0
$975$			16.1921i			0.518561i
$976$		0			0
$977$		−	40.8545i		−	1.30705i	−0.756904	−	0.653526i	$-0.773289\pi$
$977$		−	40.8545i		−	1.30705i	0.756904	−	0.653526i	$-0.226711\pi$
$978$		0			0
$979$			42.9688i			1.37329i
$980$		0			0
$981$		−	25.2766i		−	0.807021i
$982$		0			0
$983$	−10.7349			−0.342390			−0.171195	−	0.985237i	$-0.554763\pi$
$983$	−10.7349			−0.342390			−0.171195	+	0.985237i	$0.554763\pi$
$984$		0			0
$985$			5.77839i			0.184115i
$986$		0			0
$987$	6.49176			0.206635
$988$		0			0
$989$	24.7994	−	5.30762i	0.788574	−	0.168773i
$990$		0			0
$991$			26.4302i			0.839582i	0.907621	+	0.419791i	$0.137897\pi$
$991$			26.4302i			0.839582i	−0.907621	+	0.419791i	$0.862103\pi$
$992$		0			0
$993$	37.7616			1.19833
$994$		0			0
$995$			21.1623i			0.670890i
$996$		0			0
$997$	−33.0183			−1.04570			−0.522849	−	0.852425i	$-0.675131\pi$
$997$	−33.0183			−1.04570			−0.522849	+	0.852425i	$0.675131\pi$
$998$		0			0
$999$	24.2047			0.765802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.i.b.1471.5	yes	48
4.3	odd	2		3680.2.i.a.1471.44	yes	48
23.22	odd	2		3680.2.i.a.1471.5	✓	48
92.91	even	2	inner	3680.2.i.b.1471.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.i.a.1471.5	✓	48	23.22	odd	2
3680.2.i.a.1471.44	yes	48	4.3	odd	2
3680.2.i.b.1471.5	yes	48	1.1	even	1	trivial
3680.2.i.b.1471.44	yes	48	92.91	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.77635i q^{3} -1.00000i q^{5} -2.12267 q^{7} -4.70810 q^{9} +O(q^{10})\)	\(q-2.77635i q^{3} -1.00000i q^{5} -2.12267 q^{7} -4.70810 q^{9} +3.88538 q^{11} +5.83215 q^{13} -2.77635 q^{15} +3.55858i q^{17} -0.118081 q^{19} +5.89326i q^{21} +(4.68963 - 1.00369i) q^{23} -1.00000 q^{25} +4.74226i q^{27} +6.55238 q^{29} -4.89662i q^{31} -10.7872i q^{33} +2.12267i q^{35} -5.10404i q^{37} -16.1921i q^{39} +9.44324 q^{41} +5.28813 q^{43} +4.70810i q^{45} -1.10156i q^{47} -2.49428 q^{49} +9.87984 q^{51} +6.02382i q^{53} -3.88538i q^{55} +0.327832i q^{57} -3.73175i q^{59} +3.23099i q^{61} +9.99372 q^{63} -5.83215i q^{65} -1.04280 q^{67} +(-2.78658 - 13.0200i) q^{69} -9.98084i q^{71} +3.55886 q^{73} +2.77635i q^{75} -8.24737 q^{77} -11.9599 q^{79} -0.958124 q^{81} +12.9805 q^{83} +3.55858 q^{85} -18.1917i q^{87} +11.0591i q^{89} -12.3797 q^{91} -13.5947 q^{93} +0.118081i q^{95} +7.80650i q^{97} -18.2927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 8 * q^7 - 48 * q^9	\( 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) \) 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more