LMFDB - Embedded newform 2368.2.g.l.961.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(961,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2368 = 2^{6} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2368.g (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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 8x^{4} + 16x^{2} + 1$ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 1184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.5
Root			$2.11491i$ of defining polynomial
Character	$\chi$	$=$	2368.961
Dual form			2368.2.g.l.961.6

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.47283 q^{3} -2.75698i q^{5} +1.58774 q^{7} +3.11491 q^{9} +6.11491 q^{11} +0.885092i q^{13} -6.81756i q^{15} +4.94567i q^{17} -2.94567i q^{19} +3.92622 q^{21} +2.16924i q^{23} -2.60095 q^{25} +0.284147 q^{27} -0.885092i q^{29} -7.47283i q^{31} +15.1212 q^{33} -4.37737i q^{35} +(4.87189 + 3.64207i) q^{37} +2.18869i q^{39} +1.18869 q^{41} +6.45963i q^{43} -8.58774i q^{45} -3.92622 q^{47} -4.47908 q^{49} +12.2298i q^{51} -7.81756 q^{53} -16.8587i q^{55} -7.28415i q^{57} +5.66152i q^{59} +0.418502i q^{61} +4.94567 q^{63} +2.44018 q^{65} +8.27094 q^{67} +5.36417i q^{69} -5.12811 q^{71} -10.0062 q^{73} -6.43171 q^{75} +9.70889 q^{77} +10.7764i q^{79} -8.64207 q^{81} -2.15604 q^{83} +13.6351 q^{85} -2.18869i q^{87} -17.1755i q^{89} +1.40530i q^{91} -18.4791i q^{93} -8.12115 q^{95} -18.5808i q^{97} +19.0474 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 4 q^{3} - 14 q^{7} + 6 q^{9} + 24 q^{11} + 18 q^{21} - 32 q^{25} - 2 q^{27} + 22 q^{33} + 2 q^{37} - 18 q^{47} + 40 q^{49} + 2 q^{53} + 8 q^{63} - 20 q^{65} + 6 q^{67} - 58 q^{71} - 4 q^{73} - 46 q^{75}+ \cdots + 40 q^{99}+O(q^{100})$ 6 * q + 4 * q^3 - 14 * q^7 + 6 * q^9 + 24 * q^11 + 18 * q^21 - 32 * q^25 - 2 * q^27 + 22 * q^33 + 2 * q^37 - 18 * q^47 + 40 * q^49 + 2 * q^53 + 8 * q^63 - 20 * q^65 + 6 * q^67 - 58 * q^71 - 4 * q^73 - 46 * q^75 - 34 * q^77 - 50 * q^81 + 18 * q^83 - 16 * q^85 + 20 * q^95 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$ .

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.47283			1.42769			0.713846	−	0.700303i	$-0.246952\pi$
$3$	2.47283			1.42769			0.713846	+	0.700303i	$0.246952\pi$
$4$		0			0
$5$		−	2.75698i		−	1.23296i	−0.787371	−	0.616480i	$-0.788558\pi$
$5$		−	2.75698i		−	1.23296i	0.787371	−	0.616480i	$-0.211442\pi$
$6$		0			0
$7$	1.58774			0.600110			0.300055	−	0.953922i	$-0.402995\pi$
$7$	1.58774			0.600110			0.300055	+	0.953922i	$0.402995\pi$
$8$		0			0
$9$	3.11491			1.03830
$10$		0			0
$11$	6.11491			1.84371			0.921857	−	0.387530i	$-0.126672\pi$
$11$	6.11491			1.84371			0.921857	+	0.387530i	$0.126672\pi$
$12$		0			0
$13$			0.885092i			0.245480i	0.992439	+	0.122740i	$0.0391682\pi$
$13$			0.885092i			0.245480i	−0.992439	+	0.122740i	$0.960832\pi$
$14$		0			0
$15$		−	6.81756i		−	1.76029i
$16$		0			0
$17$			4.94567i			1.19950i	0.800187	+	0.599750i	$0.204733\pi$
$17$			4.94567i			1.19950i	−0.800187	+	0.599750i	$0.795267\pi$
$18$		0			0
$19$		−	2.94567i		−	0.675783i	−0.941185	−	0.337891i	$-0.890286\pi$
$19$		−	2.94567i		−	0.675783i	0.941185	−	0.337891i	$-0.109714\pi$
$20$		0			0
$21$	3.92622			0.856772
$22$		0			0
$23$			2.16924i			0.452318i	0.974090	+	0.226159i	$0.0726169\pi$
$23$			2.16924i			0.452318i	−0.974090	+	0.226159i	$0.927383\pi$
$24$		0			0
$25$	−2.60095			−0.520189
$26$		0			0
$27$	0.284147			0.0546842
$28$		0			0
$29$		−	0.885092i		−	0.164358i	−0.996618	−	0.0821788i	$-0.973812\pi$
$29$		−	0.885092i		−	0.164358i	0.996618	−	0.0821788i	$-0.0261878\pi$
$30$		0			0
$31$		−	7.47283i		−	1.34216i	−0.741385	−	0.671080i	$-0.765831\pi$
$31$		−	7.47283i		−	1.34216i	0.741385	−	0.671080i	$-0.234169\pi$
$32$		0			0
$33$	15.1212			2.63225
$34$		0			0
$35$		−	4.37737i		−	0.739911i
$36$		0			0
$37$	4.87189	+	3.64207i	0.800934	+	0.598753i
$37$	4.87189	+	3.64207i	0.800934	+	0.598753i
$38$		0			0
$39$			2.18869i			0.350470i
$40$		0			0
$41$	1.18869			0.185642			0.0928208	−	0.995683i	$-0.470412\pi$
$41$	1.18869			0.185642			0.0928208	+	0.995683i	$0.470412\pi$
$42$		0			0
$43$			6.45963i			0.985084i	0.870289	+	0.492542i	$0.163932\pi$
$43$			6.45963i			0.985084i	−0.870289	+	0.492542i	$0.836068\pi$
$44$		0			0
$45$		−	8.58774i		−	1.28018i
$46$		0			0
$47$	−3.92622			−0.572698			−0.286349	−	0.958125i	$-0.592442\pi$
$47$	−3.92622			−0.572698			−0.286349	+	0.958125i	$0.592442\pi$
$48$		0			0
$49$	−4.47908			−0.639868
$50$		0			0
$51$			12.2298i			1.71252i
$52$		0			0
$53$	−7.81756			−1.07382			−0.536912	−	0.843638i	$-0.680409\pi$
$53$	−7.81756			−1.07382			−0.536912	+	0.843638i	$0.680409\pi$
$54$		0			0
$55$		−	16.8587i		−	2.27322i
$56$		0			0
$57$		−	7.28415i		−	0.964809i
$58$		0			0
$59$			5.66152i			0.737067i	0.929614	+	0.368534i	$0.120140\pi$
$59$			5.66152i			0.737067i	−0.929614	+	0.368534i	$0.879860\pi$
$60$		0			0
$61$			0.418502i			0.0535837i	0.999641	+	0.0267918i	$0.00852912\pi$
$61$			0.418502i			0.0535837i	−0.999641	+	0.0267918i	$0.991471\pi$
$62$		0			0
$63$	4.94567			0.623096
$64$		0			0
$65$	2.44018			0.302667
$66$		0			0
$67$	8.27094			1.01046			0.505228	−	0.862986i	$-0.331408\pi$
$67$	8.27094			1.01046			0.505228	+	0.862986i	$0.331408\pi$
$68$		0			0
$69$			5.36417i			0.645770i
$70$		0			0
$71$	−5.12811			−0.608595			−0.304297	−	0.952577i	$-0.598422\pi$
$71$	−5.12811			−0.608595			−0.304297	+	0.952577i	$0.598422\pi$
$72$		0			0
$73$	−10.0062			−1.17114			−0.585571	−	0.810621i	$-0.699130\pi$
$73$	−10.0062			−1.17114			−0.585571	+	0.810621i	$0.699130\pi$
$74$		0			0
$75$	−6.43171			−0.742669
$76$		0			0
$77$	9.70889			1.10643
$78$		0			0
$79$			10.7764i			1.21244i	0.795296	+	0.606221i	$0.207315\pi$
$79$			10.7764i			1.21244i	−0.795296	+	0.606221i	$0.792685\pi$
$80$		0			0
$81$	−8.64207			−0.960230
$82$		0			0
$83$	−2.15604			−0.236656			−0.118328	−	0.992975i	$-0.537753\pi$
$83$	−2.15604			−0.236656			−0.118328	+	0.992975i	$0.537753\pi$
$84$		0			0
$85$	13.6351			1.47894
$86$		0			0
$87$		−	2.18869i		−	0.234652i
$88$		0			0
$89$		−	17.1755i		−	1.82060i	−0.413952	−	0.910299i	$-0.635852\pi$
$89$		−	17.1755i		−	1.82060i	0.413952	−	0.910299i	$-0.364148\pi$
$90$		0			0
$91$			1.40530i			0.147315i
$92$		0			0
$93$		−	18.4791i		−	1.91619i
$94$		0			0
$95$	−8.12115			−0.833212
$96$		0			0
$97$		−	18.5808i		−	1.88659i	−0.331952	−	0.943296i	$-0.607707\pi$
$97$		−	18.5808i		−	1.88659i	0.331952	−	0.943296i	$-0.392293\pi$
$98$		0			0
$99$	19.0474			1.91433
$100$		0			0
$101$	5.81756			0.578869			0.289434	−	0.957198i	$-0.406533\pi$
$101$	5.81756			0.578869			0.289434	+	0.957198i	$0.406533\pi$
$102$		0			0
$103$		−	7.32528i		−	0.721781i	−0.932608	−	0.360890i	$-0.882473\pi$
$103$		−	7.32528i		−	0.721781i	0.932608	−	0.360890i	$-0.117527\pi$
$104$		0			0
$105$		−	10.8245i		−	1.05636i
$106$		0			0
$107$	0.316798			0.0306260			0.0153130	−	0.999883i	$-0.495126\pi$
$107$	0.316798			0.0306260			0.0153130	+	0.999883i	$0.495126\pi$
$108$		0			0
$109$			17.1755i			1.64511i	0.568683	+	0.822556i	$0.307453\pi$
$109$			17.1755i			1.64511i	−0.568683	+	0.822556i	$0.692547\pi$
$110$		0			0
$111$	12.0474	+	9.00624i	1.14349	+	0.854835i
$112$		0			0
$113$		−	8.68945i		−	0.817434i	−0.912661	−	0.408717i	$-0.865976\pi$
$113$		−	8.68945i		−	0.817434i	0.912661	−	0.408717i	$-0.134024\pi$
$114$		0			0
$115$	5.98055			0.557689
$116$		0			0
$117$			2.75698i			0.254883i
$118$		0			0
$119$			7.85244i			0.719832i
$120$		0			0
$121$	26.3921			2.39928
$122$		0			0
$123$	2.93942			0.265039
$124$		0			0
$125$		−	6.61415i		−	0.591587i
$126$		0			0
$127$	−18.7632			−1.66497			−0.832483	−	0.554050i	$-0.813082\pi$
$127$	−18.7632			−1.66497			−0.832483	+	0.554050i	$0.813082\pi$
$128$		0			0
$129$			15.9736i			1.40640i
$130$		0			0
$131$		−	19.2577i		−	1.68256i	−0.540602	−	0.841278i	$-0.681804\pi$
$131$		−	19.2577i		−	1.68256i	0.540602	−	0.841278i	$-0.318196\pi$
$132$		0			0
$133$		−	4.67696i		−	0.405544i
$134$		0			0
$135$		−	0.783389i		−	0.0674234i
$136$		0			0
$137$	−0.297351			−0.0254044			−0.0127022	−	0.999919i	$-0.504043\pi$
$137$	−0.297351			−0.0254044			−0.0127022	+	0.999919i	$0.504043\pi$
$138$		0			0
$139$	−19.4659			−1.65107			−0.825537	−	0.564348i	$-0.809128\pi$
$139$	−19.4659			−1.65107			−0.825537	+	0.564348i	$0.809128\pi$
$140$		0			0
$141$	−9.70889			−0.817636
$142$		0			0
$143$			5.41226i			0.452596i
$144$		0			0
$145$	−2.44018			−0.202646
$146$		0			0
$147$	−11.0760			−0.913534
$148$		0			0
$149$	19.5613			1.60253			0.801263	−	0.598312i	$-0.204162\pi$
$149$	19.5613			1.60253			0.801263	+	0.598312i	$0.204162\pi$
$150$		0			0
$151$	12.7981			1.04150			0.520748	−	0.853711i	$-0.325653\pi$
$151$	12.7981			1.04150			0.520748	+	0.853711i	$0.325653\pi$
$152$		0			0
$153$			15.4053i			1.24544i
$154$		0			0
$155$	−20.6025			−1.65483
$156$		0			0
$157$	−11.3315			−0.904354			−0.452177	−	0.891928i	$-0.649352\pi$
$157$	−11.3315			−0.904354			−0.452177	+	0.891928i	$0.649352\pi$
$158$		0			0
$159$	−19.3315			−1.53309
$160$		0			0
$161$			3.44419i			0.271440i
$162$		0			0
$163$		−	8.12115i		−	0.636098i	−0.948074	−	0.318049i	$-0.896972\pi$
$163$		−	8.12115i		−	0.636098i	0.948074	−	0.318049i	$-0.103028\pi$
$164$		0			0
$165$		−	41.6887i		−	3.24546i
$166$		0			0
$167$		−	14.8781i		−	1.15130i	−0.817695	−	0.575652i	$-0.804748\pi$
$167$		−	14.8781i		−	1.15130i	0.817695	−	0.575652i	$-0.195252\pi$
$168$		0			0
$169$	12.2166			0.939739
$170$		0			0
$171$		−	9.17548i		−	0.701667i
$172$		0			0
$173$	−14.9108			−1.13365			−0.566823	−	0.823840i	$-0.691828\pi$
$173$	−14.9108			−1.13365			−0.566823	+	0.823840i	$0.691828\pi$
$174$		0			0
$175$	−4.12963			−0.312171
$176$		0			0
$177$			14.0000i			1.05230i
$178$		0			0
$179$			8.68945i			0.649480i	0.945803	+	0.324740i	$0.105277\pi$
$179$			8.68945i			0.649480i	−0.945803	+	0.324740i	$0.894723\pi$
$180$		0			0
$181$	16.6421			1.23700			0.618498	−	0.785787i	$-0.287742\pi$
$181$	16.6421			1.23700			0.618498	+	0.785787i	$0.287742\pi$
$182$		0			0
$183$			1.03489i			0.0765009i
$184$		0			0
$185$	10.0411	−	13.4317i	0.738238	−	0.987519i
$186$		0			0
$187$			30.2423i			2.21154i
$188$		0			0
$189$	0.451152			0.0328165
$190$		0			0
$191$			20.9993i			1.51945i	0.650242	+	0.759727i	$0.274668\pi$
$191$			20.9993i			1.51945i	−0.650242	+	0.759727i	$0.725332\pi$
$192$		0			0
$193$			19.7827i			1.42399i	0.702186	+	0.711994i	$0.252208\pi$
$193$			19.7827i			1.42399i	−0.702186	+	0.711994i	$0.747792\pi$
$194$		0			0
$195$	6.03417			0.432116
$196$		0			0
$197$	14.3859			1.02495			0.512475	−	0.858702i	$-0.328729\pi$
$197$	14.3859			1.02495			0.512475	+	0.858702i	$0.328729\pi$
$198$		0			0
$199$			19.6351i			1.39190i	0.718092	+	0.695948i	$0.245016\pi$
$199$			19.6351i			1.39190i	−0.718092	+	0.695948i	$0.754984\pi$
$200$		0			0
$201$	20.4527			1.44262
$202$		0			0
$203$		−	1.40530i		−	0.0986326i
$204$		0			0
$205$		−	3.27719i		−	0.228889i
$206$		0			0
$207$			6.75698i			0.469643i
$208$		0			0
$209$		−	18.0125i		−	1.24595i
$210$		0			0
$211$	−0.600945			−0.0413708			−0.0206854	−	0.999786i	$-0.506585\pi$
$211$	−0.600945			−0.0413708			−0.0206854	+	0.999786i	$0.506585\pi$
$212$		0			0
$213$	−12.6810			−0.868886
$214$		0			0
$215$	17.8091			1.21457
$216$		0			0
$217$		−	11.8649i		−	0.805444i
$218$		0			0
$219$	−24.7438			−1.67203
$220$		0			0
$221$	−4.37737			−0.294454
$222$		0			0
$223$	−19.3315			−1.29453			−0.647267	−	0.762263i	$-0.724088\pi$
$223$	−19.3315			−1.29453			−0.647267	+	0.762263i	$0.724088\pi$
$224$		0			0
$225$	−8.10170			−0.540114
$226$		0			0
$227$			3.20189i			0.212517i	0.994339	+	0.106258i	$0.0338871\pi$
$227$			3.20189i			0.212517i	−0.994339	+	0.106258i	$0.966113\pi$
$228$		0			0
$229$	−13.3579			−0.882717			−0.441358	−	0.897331i	$-0.645503\pi$
$229$	−13.3579			−0.882717			−0.441358	+	0.897331i	$0.645503\pi$
$230$		0			0
$231$	24.0085			1.57964
$232$		0			0
$233$	−11.4923			−0.752884			−0.376442	−	0.926440i	$-0.622853\pi$
$233$	−11.4923			−0.752884			−0.376442	+	0.926440i	$0.622853\pi$
$234$		0			0
$235$			10.8245i			0.706114i
$236$		0			0
$237$			26.6483i			1.73099i
$238$		0			0
$239$			23.5676i			1.52446i	0.647306	+	0.762230i	$0.275895\pi$
$239$			23.5676i			1.52446i	−0.647306	+	0.762230i	$0.724105\pi$
$240$		0			0
$241$			25.8649i			1.66611i	0.553193	+	0.833053i	$0.313409\pi$
$241$			25.8649i			1.66611i	−0.553193	+	0.833053i	$0.686591\pi$
$242$		0			0
$243$	−22.2229			−1.42560
$244$		0			0
$245$			12.3487i			0.788931i
$246$		0			0
$247$	2.60719			0.165891
$248$		0			0
$249$	−5.33152			−0.337871
$250$		0			0
$251$			3.66152i			0.231113i	0.993301	+	0.115557i	$0.0368652\pi$
$251$			3.66152i			0.231113i	−0.993301	+	0.115557i	$0.963135\pi$
$252$		0			0
$253$			13.2647i			0.833945i
$254$		0			0
$255$	33.7174			2.11146
$256$		0			0
$257$			2.33848i			0.145870i	0.997337	+	0.0729352i	$0.0232366\pi$
$257$			2.33848i			0.145870i	−0.997337	+	0.0729352i	$0.976763\pi$
$258$		0			0
$259$	7.73530	+	5.78267i	0.480648	+	0.359318i
$260$		0			0
$261$		−	2.75698i		−	0.170653i
$262$		0			0
$263$	−15.2229			−0.938681			−0.469341	−	0.883017i	$-0.655508\pi$
$263$	−15.2229			−0.938681			−0.469341	+	0.883017i	$0.655508\pi$
$264$		0			0
$265$			21.5529i			1.32398i
$266$		0			0
$267$		−	42.4721i		−	2.59925i
$268$		0			0
$269$	17.6615			1.07684			0.538421	−	0.842676i	$-0.319021\pi$
$269$	17.6615			1.07684			0.538421	+	0.842676i	$0.319021\pi$
$270$		0			0
$271$	−11.4791			−0.697304			−0.348652	−	0.937252i	$-0.613361\pi$
$271$	−11.4791			−0.697304			−0.348652	+	0.937252i	$0.613361\pi$
$272$		0			0
$273$			3.47507i			0.210321i
$274$		0			0
$275$	−15.9045			−0.959080
$276$		0			0
$277$			12.6483i			0.759964i	0.924994	+	0.379982i	$0.124070\pi$
$277$			12.6483i			0.759964i	−0.924994	+	0.379982i	$0.875930\pi$
$278$		0			0
$279$		−	23.2772i		−	1.39357i
$280$		0			0
$281$		−	11.0279i		−	0.657871i	−0.944352	−	0.328935i	$-0.893310\pi$
$281$		−	11.0279i		−	0.657871i	0.944352	−	0.328935i	$-0.106690\pi$
$282$		0			0
$283$			11.6351i			0.691636i	0.938302	+	0.345818i	$0.112398\pi$
$283$			11.6351i			0.691636i	−0.938302	+	0.345818i	$0.887602\pi$
$284$		0			0
$285$	−20.0823			−1.18957
$286$		0			0
$287$	1.88733			0.111405
$288$		0			0
$289$	−7.45963			−0.438802
$290$		0			0
$291$		−	45.9472i		−	2.69347i
$292$		0			0
$293$	4.86341			0.284124			0.142062	−	0.989858i	$-0.454627\pi$
$293$	4.86341			0.284124			0.142062	+	0.989858i	$0.454627\pi$
$294$		0			0
$295$	15.6087			0.908774
$296$		0			0
$297$	1.73753			0.100822
$298$		0			0
$299$	−1.91998			−0.111035
$300$		0			0
$301$			10.2562i			0.591159i
$302$		0			0
$303$	14.3859			0.826446
$304$		0			0
$305$	1.15380			0.0660665
$306$		0			0
$307$	8.55509			0.488265			0.244132	−	0.969742i	$-0.421497\pi$
$307$	8.55509			0.488265			0.244132	+	0.969742i	$0.421497\pi$
$308$		0			0
$309$		−	18.1142i		−	1.03048i
$310$		0			0
$311$			7.93942i			0.450203i	0.974335	+	0.225102i	$0.0722714\pi$
$311$			7.93942i			0.450203i	−0.974335	+	0.225102i	$0.927729\pi$
$312$		0			0
$313$			19.2144i			1.08606i	0.839713	+	0.543030i	$0.182723\pi$
$313$			19.2144i			1.08606i	−0.839713	+	0.543030i	$0.817277\pi$
$314$		0			0
$315$		−	13.6351i		−	0.768252i
$316$		0			0
$317$	−4.90677			−0.275592			−0.137796	−	0.990461i	$-0.544002\pi$
$317$	−4.90677			−0.275592			−0.137796	+	0.990461i	$0.544002\pi$
$318$		0			0
$319$		−	5.41226i		−	0.303028i
$320$		0			0
$321$	0.783389			0.0437245
$322$		0			0
$323$	14.5683			0.810602
$324$		0			0
$325$		−	2.30208i		−	0.127696i
$326$		0			0
$327$			42.4721i			2.34871i
$328$		0			0
$329$	−6.23382			−0.343682
$330$		0			0
$331$			6.26871i			0.344559i	0.985048	+	0.172280i	$0.0551133\pi$
$331$			6.26871i			0.344559i	−0.985048	+	0.172280i	$0.944887\pi$
$332$		0			0
$333$	15.1755	+	11.3447i	0.831611	+	0.621687i
$334$		0			0
$335$		−	22.8028i		−	1.24585i
$336$		0			0
$337$	−30.4659			−1.65958			−0.829791	−	0.558074i	$-0.811540\pi$
$337$	−30.4659			−1.65958			−0.829791	+	0.558074i	$0.811540\pi$
$338$		0			0
$339$		−	21.4876i		−	1.16704i
$340$		0			0
$341$		−	45.6957i		−	2.47456i
$342$		0			0
$343$	−18.2258			−0.984101
$344$		0			0
$345$	14.7889			0.796208
$346$		0			0
$347$		−	29.8649i		−	1.60323i	−0.597838	−	0.801617i	$-0.703974\pi$
$347$		−	29.8649i		−	1.60323i	0.597838	−	0.801617i	$-0.296026\pi$
$348$		0			0
$349$	27.3789			1.46556			0.732779	−	0.680466i	$-0.238223\pi$
$349$	27.3789			1.46556			0.732779	+	0.680466i	$0.238223\pi$
$350$		0			0
$351$			0.251497i			0.0134239i
$352$		0			0
$353$		−	23.2577i		−	1.23788i	−0.785436	−	0.618942i	$-0.787561\pi$
$353$		−	23.2577i		−	1.23788i	0.785436	−	0.618942i	$-0.212439\pi$
$354$		0			0
$355$			14.1381i			0.750373i
$356$		0			0
$357$			19.4178i			1.02770i
$358$		0			0
$359$	16.7243			0.882676			0.441338	−	0.897341i	$-0.354504\pi$
$359$	16.7243			0.882676			0.441338	+	0.897341i	$0.354504\pi$
$360$		0			0
$361$	10.3230			0.543318
$362$		0			0
$363$	65.2633			3.42543
$364$		0			0
$365$			27.5870i			1.44397i
$366$		0			0
$367$	−15.2019			−0.793532			−0.396766	−	0.917920i	$-0.629868\pi$
$367$	−15.2019			−0.793532			−0.396766	+	0.917920i	$0.629868\pi$
$368$		0			0
$369$	3.70265			0.192752
$370$		0			0
$371$	−12.4123			−0.644412
$372$		0			0
$373$	−4.42474			−0.229105			−0.114552	−	0.993417i	$-0.536543\pi$
$373$	−4.42474			−0.229105			−0.114552	+	0.993417i	$0.536543\pi$
$374$		0			0
$375$		−	16.3557i		−	0.844604i
$376$		0			0
$377$	0.783389			0.0403466
$378$		0			0
$379$	−5.01472			−0.257589			−0.128794	−	0.991671i	$-0.541111\pi$
$379$	−5.01472			−0.257589			−0.128794	+	0.991671i	$0.541111\pi$
$380$		0			0
$381$	−46.3983			−2.37706
$382$		0			0
$383$		−	23.4876i		−	1.20016i	−0.799941	−	0.600079i	$-0.795136\pi$
$383$		−	23.4876i		−	1.20016i	0.799941	−	0.600079i	$-0.204864\pi$
$384$		0			0
$385$		−	26.7672i		−	1.36418i
$386$		0			0
$387$			20.1212i			1.02282i
$388$		0			0
$389$		−	1.15380i		−	0.0585001i	−0.999572	−	0.0292500i	$-0.990688\pi$
$389$		−	1.15380i		−	0.0585001i	0.999572	−	0.0292500i	$-0.00931190\pi$
$390$		0			0
$391$	−10.7283			−0.542555
$392$		0			0
$393$		−	47.6212i		−	2.40217i
$394$		0			0
$395$	29.7104			1.49489
$396$		0			0
$397$	7.00696			0.351669			0.175835	−	0.984420i	$-0.443738\pi$
$397$	7.00696			0.351669			0.175835	+	0.984420i	$0.443738\pi$
$398$		0			0
$399$		−	11.5653i		−	0.578991i
$400$		0			0
$401$			29.6087i			1.47859i	0.673383	+	0.739294i	$0.264841\pi$
$401$			29.6087i			1.47859i	−0.673383	+	0.739294i	$0.735159\pi$
$402$		0			0
$403$	6.61415			0.329474
$404$		0			0
$405$			23.8260i			1.18393i
$406$		0			0
$407$	29.7911	+	22.2709i	1.47669	+	1.10393i
$408$		0			0
$409$		−	9.52645i		−	0.471053i	−0.971868	−	0.235526i	$-0.924319\pi$
$409$		−	9.52645i		−	0.471053i	0.971868	−	0.235526i	$-0.0756814\pi$
$410$		0			0
$411$	−0.735300			−0.0362697
$412$		0			0
$413$			8.98903i			0.442321i
$414$		0			0
$415$			5.94415i			0.291787i
$416$		0			0
$417$	−48.1359			−2.35722
$418$		0			0
$419$	−25.4310			−1.24239			−0.621193	−	0.783658i	$-0.713352\pi$
$419$	−25.4310			−1.24239			−0.621193	+	0.783658i	$0.713352\pi$
$420$		0			0
$421$		−	4.69417i		−	0.228780i	−0.993436	−	0.114390i	$-0.963509\pi$
$421$		−	4.69417i		−	0.228780i	0.993436	−	0.114390i	$-0.0364913\pi$
$422$		0			0
$423$	−12.2298			−0.594634
$424$		0			0
$425$		−	12.8634i		−	0.623967i
$426$		0			0
$427$			0.664473i			0.0321561i
$428$		0			0
$429$			13.3836i			0.646167i
$430$		0			0
$431$		−	7.85244i		−	0.378239i	−0.981954	−	0.189119i	$-0.939437\pi$
$431$		−	7.85244i		−	0.378239i	0.981954	−	0.189119i	$-0.0605633\pi$
$432$		0			0
$433$	−19.2709			−0.926102			−0.463051	−	0.886332i	$-0.653245\pi$
$433$	−19.2709			−0.926102			−0.463051	+	0.886332i	$0.653245\pi$
$434$		0			0
$435$	−6.03417			−0.289316
$436$		0			0
$437$	6.38986			0.305668
$438$		0			0
$439$		−	4.58998i		−	0.219068i	−0.993983	−	0.109534i	$-0.965064\pi$
$439$		−	4.58998i		−	0.219068i	0.993983	−	0.109534i	$-0.0349358\pi$
$440$		0			0
$441$	−13.9519			−0.664377
$442$		0			0
$443$	−17.1428			−0.814481			−0.407240	−	0.913321i	$-0.633509\pi$
$443$	−17.1428			−0.814481			−0.407240	+	0.913321i	$0.633509\pi$
$444$		0			0
$445$	−47.3525			−2.24472
$446$		0			0
$447$	48.3719			2.28791
$448$		0			0
$449$		−	2.60719i		−	0.123041i	−0.998106	−	0.0615204i	$-0.980405\pi$
$449$		−	2.60719i		−	0.123041i	0.998106	−	0.0615204i	$-0.0195949\pi$
$450$		0			0
$451$	7.26871			0.342270
$452$		0			0
$453$	31.6476			1.48693
$454$		0			0
$455$	3.87438			0.181634
$456$		0			0
$457$			28.5374i			1.33492i	0.744644	+	0.667462i	$0.232619\pi$
$457$			28.5374i			1.33492i	−0.744644	+	0.667462i	$0.767381\pi$
$458$		0			0
$459$			1.40530i			0.0655937i
$460$		0			0
$461$		−	13.2702i		−	0.618056i	−0.951053	−	0.309028i	$-0.899996\pi$
$461$		−	13.2702i		−	0.618056i	0.951053	−	0.309028i	$-0.100004\pi$
$462$		0			0
$463$		−	5.46587i		−	0.254021i	−0.991901	−	0.127010i	$-0.959462\pi$
$463$		−	5.46587i		−	0.254021i	0.991901	−	0.127010i	$-0.0405381\pi$
$464$		0			0
$465$	−50.9465			−2.36259
$466$		0			0
$467$			15.6615i			0.724729i	0.932037	+	0.362364i	$0.118030\pi$
$467$			15.6615i			0.724729i	−0.932037	+	0.362364i	$0.881970\pi$
$468$		0			0
$469$	13.1321			0.606385
$470$		0			0
$471$	−28.0210			−1.29114
$472$		0			0
$473$			39.5000i			1.81621i
$474$		0			0
$475$			7.66152i			0.351535i
$476$		0			0
$477$	−24.3510			−1.11495
$478$		0			0
$479$			35.3572i			1.61551i	0.589517	+	0.807756i	$0.299318\pi$
$479$			35.3572i			1.61551i	−0.589517	+	0.807756i	$0.700682\pi$
$480$		0			0
$481$	−3.22357	+	4.31207i	−0.146982	+	0.196614i
$482$		0			0
$483$			8.51691i			0.387533i
$484$		0			0
$485$	−51.2269			−2.32609
$486$		0			0
$487$		−	29.8216i		−	1.35134i	−0.737202	−	0.675672i	$-0.763854\pi$
$487$		−	29.8216i		−	1.35134i	0.737202	−	0.675672i	$-0.236146\pi$
$488$		0			0
$489$		−	20.0823i		−	0.908151i
$490$		0			0
$491$	25.8168			1.16510			0.582549	−	0.812796i	$-0.302055\pi$
$491$	25.8168			1.16510			0.582549	+	0.812796i	$0.302055\pi$
$492$		0			0
$493$	4.37737			0.197147
$494$		0			0
$495$		−	52.5132i		−	2.36029i
$496$		0			0
$497$	−8.14211			−0.365224
$498$		0			0
$499$			18.2298i			0.816079i	0.912964	+	0.408039i	$0.133787\pi$
$499$			18.2298i			0.816079i	−0.912964	+	0.408039i	$0.866213\pi$
$500$		0			0
$501$		−	36.7911i		−	1.64371i
$502$		0			0
$503$			4.30984i			0.192166i	0.995373	+	0.0960831i	$0.0306314\pi$
$503$			4.30984i			0.192166i	−0.995373	+	0.0960831i	$0.969369\pi$
$504$		0			0
$505$		−	16.0389i		−	0.713721i
$506$		0			0
$507$	30.2097			1.34166
$508$		0			0
$509$	34.4247			1.52585			0.762925	−	0.646487i	$-0.223763\pi$
$509$	34.4247			1.52585			0.762925	+	0.646487i	$0.223763\pi$
$510$		0			0
$511$	−15.8873			−0.702814
$512$		0			0
$513$		−	0.837003i		−	0.0369546i
$514$		0			0
$515$	−20.1956			−0.889927
$516$		0			0
$517$	−24.0085			−1.05589
$518$		0			0
$519$	−36.8719			−1.61850
$520$		0			0
$521$	20.8580			0.913804			0.456902	−	0.889517i	$-0.348959\pi$
$521$	20.8580			0.913804			0.456902	+	0.889517i	$0.348959\pi$
$522$		0			0
$523$		−	0.295116i		−	0.0129045i	−0.999979	−	0.00645227i	$-0.997946\pi$
$523$		−	0.295116i		−	0.0129045i	0.999979	−	0.00645227i	$-0.00205384\pi$
$524$		0			0
$525$	−10.2119			−0.445683
$526$		0			0
$527$	36.9582			1.60992
$528$		0			0
$529$	18.2944			0.795409
$530$		0			0
$531$			17.6351i			0.765299i
$532$		0			0
$533$			1.05210i			0.0455714i
$534$		0			0
$535$		−	0.873406i		−	0.0377606i
$536$		0			0
$537$			21.4876i			0.927256i
$538$		0			0
$539$	−27.3891			−1.17973
$540$		0			0
$541$			15.0521i			0.647140i	0.946204	+	0.323570i	$0.104883\pi$
$541$			15.0521i			0.647140i	−0.946204	+	0.323570i	$0.895117\pi$
$542$		0			0
$543$	41.1531			1.76605
$544$		0			0
$545$	47.3525			2.02836
$546$		0			0
$547$		−	33.5529i		−	1.43462i	−0.696756	−	0.717308i	$-0.745374\pi$
$547$		−	33.5529i		−	1.43462i	0.696756	−	0.717308i	$-0.254626\pi$
$548$		0			0
$549$			1.30359i			0.0556360i
$550$		0			0
$551$	−2.60719			−0.111070
$552$		0			0
$553$			17.1102i			0.727599i
$554$		0			0
$555$	24.8300	−	33.2144i	1.05398	−	1.40987i
$556$		0			0
$557$		−	33.5676i		−	1.42230i	−0.703038	−	0.711152i	$-0.748174\pi$
$557$		−	33.5676i		−	1.42230i	0.703038	−	0.711152i	$-0.251826\pi$
$558$		0			0
$559$	−5.71737			−0.241819
$560$		0			0
$561$			74.7842i			3.15739i
$562$		0			0
$563$			19.7563i			0.832627i	0.909221	+	0.416314i	$0.136678\pi$
$563$			19.7563i			0.832627i	−0.909221	+	0.416314i	$0.863322\pi$
$564$		0			0
$565$	−23.9566			−1.00786
$566$		0			0
$567$	−13.7214			−0.576244
$568$		0			0
$569$			33.7827i			1.41624i	0.706090	+	0.708122i	$0.250457\pi$
$569$			33.7827i			1.41624i	−0.706090	+	0.708122i	$0.749543\pi$
$570$		0			0
$571$	−35.3223			−1.47819			−0.739097	−	0.673599i	$-0.764747\pi$
$571$	−35.3223			−1.47819			−0.739097	+	0.673599i	$0.764747\pi$
$572$		0			0
$573$			51.9277i			2.16931i
$574$		0			0
$575$		−	5.64207i		−	0.235291i
$576$		0			0
$577$		−	36.3246i		−	1.51221i	−0.654450	−	0.756106i	$-0.727100\pi$
$577$		−	36.3246i		−	1.51221i	0.654450	−	0.756106i	$-0.272900\pi$
$578$		0			0
$579$			48.9193i			2.03302i
$580$		0			0
$581$	−3.42323			−0.142019
$582$		0			0
$583$	−47.8036			−1.97982
$584$		0			0
$585$	7.60095			0.314260
$586$		0			0
$587$			17.0015i			0.701728i	0.936426	+	0.350864i	$0.114112\pi$
$587$			17.0015i			0.701728i	−0.936426	+	0.350864i	$0.885888\pi$
$588$		0			0
$589$	−22.0125			−0.907009
$590$		0			0
$591$	35.5738			1.46331
$592$		0			0
$593$	−7.58998			−0.311683			−0.155841	−	0.987782i	$-0.549809\pi$
$593$	−7.58998			−0.311683			−0.155841	+	0.987782i	$0.549809\pi$
$594$		0			0
$595$	21.6490			0.887524
$596$		0			0
$597$			48.5544i			1.98720i
$598$		0			0
$599$	−6.59871			−0.269616			−0.134808	−	0.990872i	$-0.543042\pi$
$599$	−6.59871			−0.269616			−0.134808	+	0.990872i	$0.543042\pi$
$600$		0			0
$601$	−13.9714			−0.569904			−0.284952	−	0.958542i	$-0.591978\pi$
$601$	−13.9714			−0.569904			−0.284952	+	0.958542i	$0.591978\pi$
$602$		0			0
$603$	25.7632			1.04916
$604$		0			0
$605$		−	72.7625i		−	2.95822i
$606$		0			0
$607$		−	3.26247i		−	0.132419i	−0.997806	−	0.0662097i	$-0.978909\pi$
$607$		−	3.26247i		−	0.132419i	0.997806	−	0.0662097i	$-0.0210906\pi$
$608$		0			0
$609$		−	3.47507i		−	0.140817i
$610$		0			0
$611$		−	3.47507i		−	0.140586i
$612$		0			0
$613$	−39.4652			−1.59398			−0.796991	−	0.603991i	$-0.793576\pi$
$613$	−39.4652			−1.59398			−0.796991	+	0.603991i	$0.793576\pi$
$614$		0			0
$615$		−	8.10394i		−	0.326782i
$616$		0			0
$617$	−1.52717			−0.0614814			−0.0307407	−	0.999527i	$-0.509787\pi$
$617$	−1.52717			−0.0614814			−0.0307407	+	0.999527i	$0.509787\pi$
$618$		0			0
$619$	20.6832			0.831328			0.415664	−	0.909518i	$-0.363549\pi$
$619$	20.6832			0.831328			0.415664	+	0.909518i	$0.363549\pi$
$620$		0			0
$621$			0.616384i			0.0247346i
$622$		0			0
$623$		−	27.2702i		−	1.09256i
$624$		0			0
$625$	−31.2398			−1.24959
$626$		0			0
$627$		−	44.5419i		−	1.77883i
$628$		0			0
$629$	−18.0125	+	24.0947i	−0.718205	+	0.960720i
$630$		0			0
$631$		−	14.6289i		−	0.582366i	−0.956667	−	0.291183i	$-0.905951\pi$
$631$		−	14.6289i		−	0.582366i	0.956667	−	0.291183i	$-0.0940489\pi$
$632$		0			0
$633$	−1.48604			−0.0590647
$634$		0			0
$635$			51.7299i			2.05284i
$636$		0			0
$637$		−	3.96440i		−	0.157075i
$638$		0			0
$639$	−15.9736			−0.631906
$640$		0			0
$641$	32.6498			1.28959			0.644795	−	0.764355i	$-0.276943\pi$
$641$	32.6498			1.28959			0.644795	+	0.764355i	$0.276943\pi$
$642$		0			0
$643$			46.7019i			1.84174i	0.389865	+	0.920872i	$0.372522\pi$
$643$			46.7019i			1.84174i	−0.389865	+	0.920872i	$0.627478\pi$
$644$		0			0
$645$	44.0389			1.73403
$646$		0			0
$647$		−	41.4659i		−	1.63019i	−0.579326	−	0.815096i	$-0.696684\pi$
$647$		−	41.4659i		−	1.63019i	0.579326	−	0.815096i	$-0.303316\pi$
$648$		0			0
$649$			34.6197i			1.35894i
$650$		0			0
$651$		−	29.3400i		−	1.14993i
$652$		0			0
$653$			39.0451i			1.52795i	0.645243	+	0.763977i	$0.276756\pi$
$653$			39.0451i			1.52795i	−0.645243	+	0.763977i	$0.723244\pi$
$654$		0			0
$655$	−53.0932			−2.07452
$656$		0			0
$657$	−31.1685			−1.21600
$658$		0			0
$659$	4.24302			0.165285			0.0826423	−	0.996579i	$-0.473664\pi$
$659$	4.24302			0.165285			0.0826423	+	0.996579i	$0.473664\pi$
$660$		0			0
$661$		−	24.3462i		−	0.946959i	−0.880805	−	0.473479i	$-0.842998\pi$
$661$		−	24.3462i		−	0.946959i	0.880805	−	0.473479i	$-0.157002\pi$
$662$		0			0
$663$	−10.8245			−0.420389
$664$		0			0
$665$	−12.8943			−0.500019
$666$		0			0
$667$	1.91998			0.0743418
$668$		0			0
$669$	−47.8036			−1.84820
$670$		0			0
$671$			2.55910i			0.0987929i
$672$		0			0
$673$	−11.5397			−0.444821			−0.222410	−	0.974953i	$-0.571393\pi$
$673$	−11.5397			−0.444821			−0.222410	+	0.974953i	$0.571393\pi$
$674$		0			0
$675$	−0.739051			−0.0284461
$676$		0			0
$677$	19.2368			0.739329			0.369665	−	0.929165i	$-0.379473\pi$
$677$	19.2368			0.739329			0.369665	+	0.929165i	$0.379473\pi$
$678$		0			0
$679$		−	29.5015i		−	1.13216i
$680$		0			0
$681$			7.91774i			0.303409i
$682$		0			0
$683$		−	24.9457i		−	0.954519i	−0.878762	−	0.477260i	$-0.841630\pi$
$683$		−	24.9457i		−	0.954519i	0.878762	−	0.477260i	$-0.158370\pi$
$684$		0			0
$685$			0.819791i			0.0313226i
$686$		0			0
$687$	−33.0319			−1.26025
$688$		0			0
$689$		−	6.91926i		−	0.263603i
$690$		0			0
$691$	4.67696			0.177920			0.0889600	−	0.996035i	$-0.471646\pi$
$691$	4.67696			0.177920			0.0889600	+	0.996035i	$0.471646\pi$
$692$		0			0
$693$	30.2423			1.14881
$694$		0			0
$695$			53.6670i			2.03571i
$696$		0			0
$697$			5.87885i			0.222677i
$698$		0			0
$699$	−28.4185			−1.07489
$700$		0			0
$701$		−	3.12187i		−	0.117911i	−0.998261	−	0.0589557i	$-0.981223\pi$
$701$		−	3.12187i		−	0.117911i	0.998261	−	0.0589557i	$-0.0187771\pi$
$702$		0			0
$703$	10.7283	−	14.3510i	0.404627	−	0.541257i
$704$		0			0
$705$			26.7672i			1.00811i
$706$		0			0
$707$	9.23678			0.347385
$708$		0			0
$709$		−	41.1204i		−	1.54431i	−0.635434	−	0.772155i	$-0.719179\pi$
$709$		−	41.1204i		−	1.54431i	0.635434	−	0.772155i	$-0.280821\pi$
$710$		0			0
$711$			33.5676i			1.25888i
$712$		0			0
$713$	16.2104			0.607083
$714$		0			0
$715$	14.9215			0.558032
$716$		0			0
$717$			58.2787i			2.17646i
$718$		0			0
$719$	−40.2508			−1.50110			−0.750550	−	0.660813i	$-0.770212\pi$
$719$	−40.2508			−1.50110			−0.750550	+	0.660813i	$0.770212\pi$
$720$		0			0
$721$		−	11.6306i		−	0.433148i
$722$		0			0
$723$			63.9597i			2.37869i
$724$		0			0
$725$			2.30208i			0.0854970i
$726$		0			0
$727$			33.4589i			1.24092i	0.784237	+	0.620461i	$0.213055\pi$
$727$			33.4589i			1.24092i	−0.784237	+	0.620461i	$0.786945\pi$
$728$		0			0
$729$	−29.0272			−1.07508
$730$		0			0
$731$	−31.9472			−1.18161
$732$		0			0
$733$	38.6546			1.42774			0.713869	−	0.700279i	$-0.246941\pi$
$733$	38.6546			1.42774			0.713869	+	0.700279i	$0.246941\pi$
$734$		0			0
$735$			30.5364i			1.12635i
$736$		0			0
$737$	50.5761			1.86299
$738$		0			0
$739$	31.1120			1.14447			0.572236	−	0.820089i	$-0.306076\pi$
$739$	31.1120			1.14447			0.572236	+	0.820089i	$0.306076\pi$
$740$		0			0
$741$	6.44714			0.236842
$742$		0			0
$743$	−40.0474			−1.46920			−0.734598	−	0.678503i	$-0.762629\pi$
$743$	−40.0474			−1.46920			−0.734598	+	0.678503i	$0.762629\pi$
$744$		0			0
$745$		−	53.9302i		−	1.97585i
$746$		0			0
$747$	−6.71585			−0.245720
$748$		0			0
$749$	0.502993			0.0183790
$750$		0			0
$751$	15.9526			0.582120			0.291060	−	0.956705i	$-0.405992\pi$
$751$	15.9526			0.582120			0.291060	+	0.956705i	$0.405992\pi$
$752$		0			0
$753$			9.05433i			0.329958i
$754$		0			0
$755$		−	35.2841i		−	1.28412i
$756$		0			0
$757$			0.418502i			0.0152107i	0.999971	+	0.00760535i	$0.00242088\pi$
$757$			0.418502i			0.0152107i	−0.999971	+	0.00760535i	$0.997579\pi$
$758$		0			0
$759$			32.8014i			1.19062i
$760$		0			0
$761$	23.7306			0.860233			0.430116	−	0.902773i	$-0.358473\pi$
$761$	23.7306			0.860233			0.430116	+	0.902773i	$0.358473\pi$
$762$		0			0
$763$			27.2702i			0.987248i
$764$		0			0
$765$	42.4721			1.53558
$766$		0			0
$767$	−5.01097			−0.180936
$768$		0			0
$769$			23.8913i			0.861544i	0.902461	+	0.430772i	$0.141759\pi$
$769$			23.8913i			0.861544i	−0.902461	+	0.430772i	$0.858241\pi$
$770$		0			0
$771$			5.78267i			0.208258i
$772$		0			0
$773$	35.0878			1.26202			0.631010	−	0.775775i	$-0.282641\pi$
$773$	35.0878			1.26202			0.631010	+	0.775775i	$0.282641\pi$
$774$		0			0
$775$			19.4364i			0.698177i
$776$		0			0
$777$	19.1281	+	14.2996i	0.686217	+	0.512995i
$778$		0			0
$779$		−	3.50148i		−	0.125453i
$780$		0			0
$781$	−31.3579			−1.12207
$782$		0			0
$783$		−	0.251497i		−	0.00898776i
$784$		0			0
$785$			31.2408i			1.11503i
$786$		0			0
$787$	18.3983			0.655830			0.327915	−	0.944707i	$-0.393654\pi$
$787$	18.3983			0.655830			0.327915	+	0.944707i	$0.393654\pi$
$788$		0			0
$789$	−37.6436			−1.34015
$790$		0			0
$791$		−	13.7966i		−	0.490550i
$792$		0			0
$793$	−0.370413			−0.0131537
$794$		0			0
$795$			53.2966i			1.89024i
$796$		0			0
$797$		−	4.92846i		−	0.174575i	−0.996183	−	0.0872874i	$-0.972180\pi$
$797$		−	4.92846i		−	0.174575i	0.996183	−	0.0872874i	$-0.0278199\pi$
$798$		0			0
$799$		−	19.4178i		−	0.686952i
$800$		0			0
$801$		−	53.5000i		−	1.89033i
$802$		0			0
$803$	−61.1873			−2.15925
$804$		0			0
$805$	9.49557			0.334675
$806$		0			0
$807$	43.6740			1.53740
$808$		0			0
$809$		−	47.1491i		−	1.65767i	−0.559491	−	0.828837i	$-0.689003\pi$
$809$		−	47.1491i		−	1.65767i	0.559491	−	0.828837i	$-0.310997\pi$
$810$		0			0
$811$	−12.4659			−0.437736			−0.218868	−	0.975754i	$-0.570236\pi$
$811$	−12.4659			−0.437736			−0.218868	+	0.975754i	$0.570236\pi$
$812$		0			0
$813$	−28.3859			−0.995535
$814$		0			0
$815$	−22.3899			−0.784283
$816$		0			0
$817$	19.0279			0.665703
$818$		0			0
$819$			4.37737i			0.152958i
$820$		0			0
$821$	36.7243			1.28169			0.640844	−	0.767671i	$-0.278585\pi$
$821$	36.7243			1.28169			0.640844	+	0.767671i	$0.278585\pi$
$822$		0			0
$823$	3.54037			0.123410			0.0617048	−	0.998094i	$-0.480346\pi$
$823$	3.54037			0.123410			0.0617048	+	0.998094i	$0.480346\pi$
$824$		0			0
$825$	−39.3293			−1.36927
$826$		0			0
$827$		−	16.4457i		−	0.571873i	−0.958249	−	0.285937i	$-0.907695\pi$
$827$		−	16.4457i		−	0.571873i	0.958249	−	0.285937i	$-0.0923047\pi$
$828$		0			0
$829$			44.3976i			1.54199i	0.636839	+	0.770997i	$0.280241\pi$
$829$			44.3976i			1.54199i	−0.636839	+	0.770997i	$0.719759\pi$
$830$		0			0
$831$			31.2772i			1.08499i
$832$		0			0
$833$		−	22.1520i		−	0.767522i
$834$		0			0
$835$	−41.0187			−1.41951
$836$		0			0
$837$		−	2.12339i		−	0.0733949i
$838$		0			0
$839$	−34.5544			−1.19295			−0.596475	−	0.802632i	$-0.703432\pi$
$839$	−34.5544			−1.19295			−0.596475	+	0.802632i	$0.703432\pi$
$840$		0			0
$841$	28.2166			0.972987
$842$		0			0
$843$		−	27.2702i		−	0.939236i
$844$		0			0
$845$		−	33.6810i		−	1.15866i
$846$		0			0
$847$	41.9038			1.43983
$848$		0			0
$849$			28.7717i			0.987442i
$850$		0			0
$851$	−7.90053	+	10.5683i	−0.270827	+	0.362276i
$852$		0			0
$853$			18.9993i			0.650523i	0.945624	+	0.325262i	$0.105452\pi$
$853$			18.9993i			0.650523i	−0.945624	+	0.325262i	$0.894548\pi$
$854$		0			0
$855$	−25.2966			−0.865127
$856$		0			0
$857$		−	23.8913i		−	0.816112i	−0.912957	−	0.408056i	$-0.866207\pi$
$857$		−	23.8913i		−	0.816112i	0.912957	−	0.408056i	$-0.133793\pi$
$858$		0			0
$859$			16.1336i			0.550473i	0.961377	+	0.275236i	$0.0887561\pi$
$859$			16.1336i			0.550473i	−0.961377	+	0.275236i	$0.911244\pi$
$860$		0			0
$861$	4.66705			0.159053
$862$		0			0
$863$	−28.8370			−0.981623			−0.490812	−	0.871266i	$-0.663300\pi$
$863$	−28.8370			−0.981623			−0.490812	+	0.871266i	$0.663300\pi$
$864$		0			0
$865$			41.1087i			1.39774i
$866$		0			0
$867$	−18.4464			−0.626473
$868$		0			0
$869$			65.8969i			2.23540i
$870$		0			0
$871$			7.32055i			0.248047i
$872$		0			0
$873$		−	57.8774i		−	1.95885i
$874$		0			0
$875$		−	10.5016i		−	0.355017i
$876$		0			0
$877$	7.31055			0.246860			0.123430	−	0.992353i	$-0.460611\pi$
$877$	7.31055			0.246860			0.123430	+	0.992353i	$0.460611\pi$
$878$		0			0
$879$	12.0264			0.405641
$880$		0			0
$881$	4.37961			0.147553			0.0737764	−	0.997275i	$-0.476495\pi$
$881$	4.37961			0.147553			0.0737764	+	0.997275i	$0.476495\pi$
$882$		0			0
$883$			5.14908i			0.173280i	0.996240	+	0.0866401i	$0.0276130\pi$
$883$			5.14908i			0.173280i	−0.996240	+	0.0866401i	$0.972387\pi$
$884$		0			0
$885$	38.5977			1.29745
$886$		0			0
$887$	6.80212			0.228393			0.114196	−	0.993458i	$-0.463571\pi$
$887$	6.80212			0.228393			0.114196	+	0.993458i	$0.463571\pi$
$888$		0			0
$889$	−29.7911			−0.999163
$890$		0			0
$891$	−52.8455			−1.77039
$892$		0			0
$893$			11.5653i			0.387019i
$894$		0			0
$895$	23.9566			0.800782
$896$		0			0
$897$	−4.74779			−0.158524
$898$		0			0
$899$	−6.61415			−0.220594
$900$		0			0
$901$		−	38.6630i		−	1.28805i
$902$		0			0
$903$			25.3619i			0.843992i
$904$		0			0
$905$		−	45.8819i		−	1.52517i
$906$		0			0
$907$			7.34000i			0.243721i	0.992547	+	0.121860i	$0.0388860\pi$
$907$			7.34000i			0.243721i	−0.992547	+	0.121860i	$0.961114\pi$
$908$		0			0
$909$	18.1212			0.601041
$910$		0			0
$911$			22.8804i			0.758060i	0.925384	+	0.379030i	$0.123742\pi$
$911$			22.8804i			0.758060i	−0.925384	+	0.379030i	$0.876258\pi$
$912$		0			0
$913$	−13.1840			−0.436325
$914$		0			0
$915$	2.85316			0.0943225
$916$		0			0
$917$		−	30.5763i		−	1.00972i
$918$		0			0
$919$		−	34.6461i		−	1.14287i	−0.820648	−	0.571434i	$-0.806387\pi$
$919$		−	34.6461i		−	1.14287i	0.820648	−	0.571434i	$-0.193613\pi$
$920$		0			0
$921$	21.1553			0.697091
$922$		0			0
$923$		−	4.53885i		−	0.149398i
$924$		0			0
$925$	−12.6715	−	9.47283i	−0.416637	−	0.311465i
$926$		0			0
$927$		−	22.8176i		−	0.749427i
$928$		0			0
$929$	−53.2166			−1.74598			−0.872990	−	0.487738i	$-0.837822\pi$
$929$	−53.2166			−1.74598			−0.872990	+	0.487738i	$0.837822\pi$
$930$		0			0
$931$			13.1939i			0.432412i
$932$		0			0
$933$			19.6329i			0.642752i
$934$		0			0
$935$	83.3775			2.72673
$936$		0			0
$937$	−41.2834			−1.34867			−0.674335	−	0.738425i	$-0.735570\pi$
$937$	−41.2834			−1.34867			−0.674335	+	0.738425i	$0.735570\pi$
$938$		0			0
$939$			47.5140i			1.55056i
$940$		0			0
$941$	30.2034			0.984603			0.492301	−	0.870425i	$-0.336156\pi$
$941$	30.2034			0.984603			0.492301	+	0.870425i	$0.336156\pi$
$942$		0			0
$943$			2.57855i			0.0839690i
$944$		0			0
$945$		−	1.24382i		−	0.0404614i
$946$		0			0
$947$		−	36.2857i		−	1.17913i	−0.807723	−	0.589563i	$-0.799300\pi$
$947$		−	36.2857i		−	1.17913i	0.807723	−	0.589563i	$-0.200700\pi$
$948$		0			0
$949$		−	8.85645i		−	0.287493i
$950$		0			0
$951$	−12.1336			−0.393460
$952$		0			0
$953$	5.70666			0.184857			0.0924284	−	0.995719i	$-0.470537\pi$
$953$	5.70666			0.184857			0.0924284	+	0.995719i	$0.470537\pi$
$954$		0			0
$955$	57.8946			1.87343
$956$		0			0
$957$		−	13.3836i		−	0.432631i
$958$		0			0
$959$	−0.472117			−0.0152454
$960$		0			0
$961$	−24.8432			−0.801395
$962$		0			0
$963$	0.986796			0.0317991
$964$		0			0
$965$	54.5405			1.75572
$966$		0			0
$967$		−	61.0287i		−	1.96255i	−0.192608	−	0.981276i	$-0.561695\pi$
$967$		−	61.0287i		−	1.96255i	0.192608	−	0.981276i	$-0.438305\pi$
$968$		0			0
$969$	36.0250			1.15729
$970$		0			0
$971$	42.6219			1.36780			0.683901	−	0.729575i	$-0.260282\pi$
$971$	42.6219			1.36780			0.683901	+	0.729575i	$0.260282\pi$
$972$		0			0
$973$	−30.9068			−0.990826
$974$		0			0
$975$		−	5.69265i		−	0.182311i
$976$		0			0
$977$			9.25774i			0.296181i	0.988974	+	0.148091i	$0.0473127\pi$
$977$			9.25774i			0.296181i	−0.988974	+	0.148091i	$0.952687\pi$
$978$		0			0
$979$		−	105.026i		−	3.35666i
$980$		0			0
$981$			53.5000i			1.70812i
$982$		0			0
$983$	−18.3983			−0.586816			−0.293408	−	0.955987i	$-0.594789\pi$
$983$	−18.3983			−0.586816			−0.293408	+	0.955987i	$0.594789\pi$
$984$		0			0
$985$		−	39.6615i		−	1.26372i
$986$		0			0
$987$	−15.4152			−0.490672
$988$		0			0
$989$	−14.0125			−0.445571
$990$		0			0
$991$			23.4270i			0.744183i	0.928196	+	0.372091i	$0.121359\pi$
$991$			23.4270i			0.744183i	−0.928196	+	0.372091i	$0.878641\pi$
$992$		0			0
$993$			15.5015i			0.491924i
$994$		0			0
$995$	54.1336			1.71615
$996$		0			0
$997$			8.05585i			0.255131i	0.991830	+	0.127566i	$0.0407164\pi$
$997$			8.05585i			0.255131i	−0.991830	+	0.127566i	$0.959284\pi$
$998$		0			0
$999$	1.38433	+	1.03489i	0.0437984	+	0.0327423i

Display

a_p

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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

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.g.l.961.5		6
4.3	odd	2		2368.2.g.k.961.1		6
8.3	odd	2		1184.2.g.f.961.6	yes	6
8.5	even	2		1184.2.g.e.961.2	yes	6
37.36	even	2	inner	2368.2.g.l.961.6		6
148.147	odd	2		2368.2.g.k.961.2		6
296.147	odd	2		1184.2.g.f.961.5	yes	6
296.221	even	2		1184.2.g.e.961.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.g.e.961.1	✓	6	296.221	even	2
1184.2.g.e.961.2	yes	6	8.5	even	2
1184.2.g.f.961.5	yes	6	296.147	odd	2
1184.2.g.f.961.6	yes	6	8.3	odd	2
2368.2.g.k.961.1		6	4.3	odd	2
2368.2.g.k.961.2		6	148.147	odd	2
2368.2.g.l.961.5		6	1.1	even	1	trivial
2368.2.g.l.961.6		6	37.36	even	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2368.2.g.l.961.5

Level	$2368$
Weight	$2$
Character	2368.961
Analytic conductor	$18.909$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$