LMFDB - Embedded newform 3960.2.d.f.3169.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.d (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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.7
Root			$0.655762i$ of defining polynomial
Character	$\chi$	$=$	3960.3169
Dual form			3960.2.d.f.3169.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23285 - 0.119978i) q^{5} +0.415806i q^{7} +O(q^{10})$	$q+(2.23285 - 0.119978i) q^{5} +0.415806i q^{7} -1.00000 q^{11} +4.00000i q^{13} -6.51558i q^{17} -5.20406 q^{19} -8.54830i q^{23} +(4.97121 - 0.535784i) q^{25} +0.895717 q^{29} -6.73836 q^{31} +(0.0498875 + 0.928432i) q^{35} -8.96410i q^{37} -10.0998 q^{41} -4.78825i q^{43} +5.61986i q^{47} +6.82711 q^{49} -10.0357i q^{53} +(-2.23285 + 0.119978i) q^{55} -1.63408 q^{59} +7.10428 q^{61} +(0.479911 + 8.93139i) q^{65} -10.6914i q^{67} +6.19302 q^{71} +3.16839i q^{73} -0.415806i q^{77} +11.2682 q^{79} -16.2429i q^{83} +(-0.781725 - 14.5483i) q^{85} +9.56998 q^{89} -1.66323 q^{91} +(-11.6199 + 0.624371i) q^{95} -0.591657i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) $ 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.23285	−	0.119978i	0.998559	−	0.0536557i
$5$	2.23285	−	0.119978i	0.998559	−	0.0536557i
$6$		0			0
$7$			0.415806i			0.157160i	0.996908	+	0.0785800i	$0.0250386\pi$
$7$			0.415806i			0.157160i	−0.996908	+	0.0785800i	$0.974961\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.51558i		−	1.58026i	−0.612939	−	0.790130i	$-0.710013\pi$
$17$		−	6.51558i		−	1.58026i	0.612939	−	0.790130i	$-0.289987\pi$
$18$		0			0
$19$	−5.20406			−1.19389			−0.596946	−	0.802281i	$-0.703619\pi$
$19$	−5.20406			−1.19389			−0.596946	+	0.802281i	$0.703619\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.54830i		−	1.78244i	−0.453568	−	0.891221i	$-0.649849\pi$
$23$		−	8.54830i		−	1.78244i	0.453568	−	0.891221i	$-0.350151\pi$
$24$		0			0
$25$	4.97121	−	0.535784i	0.994242	−	0.107157i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.895717			0.166331			0.0831653	−	0.996536i	$-0.473497\pi$
$29$	0.895717			0.166331			0.0831653	+	0.996536i	$0.473497\pi$
$30$		0			0
$31$	−6.73836			−1.21025			−0.605123	−	0.796132i	$-0.706876\pi$
$31$	−6.73836			−1.21025			−0.605123	+	0.796132i	$0.706876\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.0498875	+	0.928432i	0.00843253	+	0.156934i
$36$		0			0
$37$		−	8.96410i		−	1.47369i	−0.676062	−	0.736845i	$-0.736315\pi$
$37$		−	8.96410i		−	1.47369i	0.676062	−	0.736845i	$-0.263685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.0998			−1.57732			−0.788660	−	0.614830i	$-0.789225\pi$
$41$	−10.0998			−1.57732			−0.788660	+	0.614830i	$0.789225\pi$
$42$		0			0
$43$		−	4.78825i		−	0.730201i	−0.930968	−	0.365101i	$-0.881035\pi$
$43$		−	4.78825i		−	0.730201i	0.930968	−	0.365101i	$-0.118965\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.61986i			0.819741i	0.912144	+	0.409871i	$0.134426\pi$
$47$			5.61986i			0.819741i	−0.912144	+	0.409871i	$0.865574\pi$
$48$		0			0
$49$	6.82711			0.975301
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.0357i		−	1.37851i	−0.724521	−	0.689253i	$-0.757939\pi$
$53$		−	10.0357i		−	1.37851i	0.724521	−	0.689253i	$-0.242061\pi$
$54$		0			0
$55$	−2.23285	+	0.119978i	−0.301077	+	0.0161778i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.63408			−0.212739			−0.106370	−	0.994327i	$-0.533923\pi$
$59$	−1.63408			−0.212739			−0.106370	+	0.994327i	$0.533923\pi$
$60$		0			0
$61$	7.10428			0.909610			0.454805	−	0.890591i	$-0.349709\pi$
$61$	7.10428			0.909610			0.454805	+	0.890591i	$0.349709\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.479911	+	8.93139i	0.0595257	+	1.10780i
$66$		0			0
$67$		−	10.6914i		−	1.30617i	−0.757286	−	0.653083i	$-0.773475\pi$
$67$		−	10.6914i		−	1.30617i	0.757286	−	0.653083i	$-0.226525\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.19302			0.734977			0.367488	−	0.930028i	$-0.380218\pi$
$71$	6.19302			0.734977			0.367488	+	0.930028i	$0.380218\pi$
$72$		0			0
$73$			3.16839i			0.370832i	0.982660	+	0.185416i	$0.0593632\pi$
$73$			3.16839i			0.370832i	−0.982660	+	0.185416i	$0.940637\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.415806i		−	0.0473855i
$78$		0			0
$79$	11.2682			1.26777			0.633884	−	0.773428i	$-0.281460\pi$
$79$	11.2682			1.26777			0.633884	+	0.773428i	$0.281460\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.2429i		−	1.78289i	−0.453128	−	0.891446i	$-0.649692\pi$
$83$		−	16.2429i		−	1.78289i	0.453128	−	0.891446i	$-0.350308\pi$
$84$		0			0
$85$	−0.781725	−	14.5483i	−0.0847900	−	1.57798i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.56998			1.01442			0.507208	−	0.861824i	$-0.330678\pi$
$89$	9.56998			1.01442			0.507208	+	0.861824i	$0.330678\pi$
$90$		0			0
$91$	−1.66323			−0.174353
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.6199	+	0.624371i	−1.19217	+	0.0640591i
$96$		0			0
$97$		−	0.591657i		−	0.0600737i	−0.999549	−	0.0300368i	$-0.990438\pi$
$97$		−	0.591657i		−	0.0600737i	0.999549	−	0.0300368i	$-0.00956246\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.62305			0.460010			0.230005	−	0.973189i	$-0.426126\pi$
$101$	4.62305			0.460010			0.230005	+	0.973189i	$0.426126\pi$
$102$		0			0
$103$		−	8.24291i		−	0.812198i	−0.913829	−	0.406099i	$-0.866889\pi$
$103$		−	8.24291i		−	0.812198i	0.913829	−	0.406099i	$-0.133111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			17.6199i			1.70338i	0.524049	+	0.851688i	$0.324421\pi$
$107$			17.6199i			1.70338i	−0.524049	+	0.851688i	$0.675579\pi$
$108$		0			0
$109$	−3.66323			−0.350873			−0.175437	−	0.984491i	$-0.556134\pi$
$109$	−3.66323			−0.350873			−0.175437	+	0.984491i	$0.556134\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.4797i			1.07992i	0.841691	+	0.539959i	$0.181560\pi$
$113$			11.4797i			1.07992i	−0.841691	+	0.539959i	$0.818440\pi$
$114$		0			0
$115$	−1.02561	−	19.0870i	−0.0956382	−	1.77988i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.70922			0.248354
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0357	−	1.79276i	0.987060	−	0.160349i
$126$		0			0
$127$			9.13995i			0.811040i	0.914086	+	0.405520i	$0.132909\pi$
$127$			9.13995i			0.811040i	−0.914086	+	0.405520i	$0.867091\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.48123			0.216787			0.108393	−	0.994108i	$-0.465429\pi$
$131$	2.48123			0.216787			0.108393	+	0.994108i	$0.465429\pi$
$132$		0			0
$133$		−	2.16388i		−	0.187632i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.40834i			0.632937i	0.948603	+	0.316469i	$0.102497\pi$
$137$			7.40834i			0.632937i	−0.948603	+	0.316469i	$0.897503\pi$
$138$		0			0
$139$	1.69166			0.143485			0.0717424	−	0.997423i	$-0.477144\pi$
$139$	1.69166			0.143485			0.0717424	+	0.997423i	$0.477144\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	2.00000	−	0.107466i	0.166091	−	0.00892458i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.74940			0.307163			0.153581	−	0.988136i	$-0.450919\pi$
$149$	3.74940			0.307163			0.153581	+	0.988136i	$0.450919\pi$
$150$		0			0
$151$	−15.1400			−1.23207			−0.616036	−	0.787718i	$-0.711262\pi$
$151$	−15.1400			−1.23207			−0.616036	+	0.787718i	$0.711262\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.0457	+	0.808454i	−1.20850	+	0.0649366i
$156$		0			0
$157$		−	11.5871i		−	0.924755i	−0.886683	−	0.462378i	$-0.846996\pi$
$157$		−	11.5871i		−	0.924755i	0.886683	−	0.462378i	$-0.153004\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.55444			0.280129
$162$		0			0
$163$			6.82392i			0.534491i	0.963629	+	0.267245i	$0.0861134\pi$
$163$			6.82392i			0.534491i	−0.963629	+	0.267245i	$0.913887\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.10428i			0.549746i	0.961481	+	0.274873i	$0.0886358\pi$
$167$			7.10428i			0.549746i	−0.961481	+	0.274873i	$0.911364\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	9.31152i		−	0.707942i	−0.935256	−	0.353971i	$-0.884831\pi$
$173$		−	9.31152i		−	0.707942i	0.935256	−	0.353971i	$-0.115169\pi$
$174$		0			0
$175$	0.222782	+	2.06706i	0.0168408	+	0.156255i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−26.6368			−1.99093			−0.995464	−	0.0951359i	$-0.969671\pi$
$179$	−26.6368			−1.99093			−0.995464	+	0.0951359i	$0.969671\pi$
$180$		0			0
$181$	1.01103			0.0751495			0.0375748	−	0.999294i	$-0.488037\pi$
$181$	1.01103			0.0751495			0.0375748	+	0.999294i	$0.488037\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.07549	−	20.0155i	−0.0790718	−	1.47157i
$186$		0			0
$187$			6.51558i			0.476466i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.5655			−0.764490			−0.382245	−	0.924061i	$-0.624849\pi$
$191$	−10.5655			−0.764490			−0.382245	+	0.924061i	$0.624849\pi$
$192$		0			0
$193$			11.8271i			0.851334i	0.904880	+	0.425667i	$0.139960\pi$
$193$			11.8271i			0.851334i	−0.904880	+	0.425667i	$0.860040\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.31152i		−	0.0934422i	−0.998908	−	0.0467211i	$-0.985123\pi$
$197$		−	1.31152i		−	0.0934422i	0.998908	−	0.0467211i	$-0.0148772\pi$
$198$		0			0
$199$	11.8271			0.838401			0.419201	−	0.907894i	$-0.362310\pi$
$199$	11.8271			0.838401			0.419201	+	0.907894i	$0.362310\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.372445i			0.0261405i
$204$		0			0
$205$	−22.5513	+	1.21175i	−1.57505	+	0.0846322i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.20406			0.359972
$210$		0			0
$211$	−8.37244			−0.576383			−0.288191	−	0.957573i	$-0.593054\pi$
$211$	−8.37244			−0.576383			−0.288191	+	0.957573i	$0.593054\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.574484	−	10.6914i	−0.0391795	−	0.729150i
$216$		0			0
$217$		−	2.80185i		−	0.190202i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	26.0623			1.75314
$222$		0			0
$223$		−	22.1461i		−	1.48301i	−0.670946	−	0.741506i	$-0.734112\pi$
$223$		−	22.1461i		−	1.48301i	0.670946	−	0.741506i	$-0.265888\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	21.8194i		−	1.44821i	−0.689692	−	0.724103i	$-0.742254\pi$
$227$		−	21.8194i		−	1.44821i	0.689692	−	0.724103i	$-0.257746\pi$
$228$		0			0
$229$	−2.80247			−0.185192			−0.0925962	−	0.995704i	$-0.529517\pi$
$229$	−2.80247			−0.185192			−0.0925962	+	0.995704i	$0.529517\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.24424i		−	0.540098i	−0.962847	−	0.270049i	$-0.912960\pi$
$233$		−	8.24424i		−	0.540098i	0.962847	−	0.270049i	$-0.0870399\pi$
$234$		0			0
$235$	0.674259	+	12.5483i	0.0439838	+	0.818561i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.2682			−0.728877			−0.364438	−	0.931227i	$-0.618739\pi$
$239$	−11.2682			−0.728877			−0.364438	+	0.931227i	$0.618739\pi$
$240$		0			0
$241$	22.3860			1.44201			0.721006	−	0.692929i	$-0.243680\pi$
$241$	22.3860			1.44201			0.721006	+	0.692929i	$0.243680\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.2439	−	0.819101i	0.973896	−	0.0523304i
$246$		0			0
$247$		−	20.8162i		−	1.32451i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.84265			−0.242546			−0.121273	−	0.992619i	$-0.538698\pi$
$251$	−3.84265			−0.242546			−0.121273	+	0.992619i	$0.538698\pi$
$252$		0			0
$253$			8.54830i			0.537427i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			10.6230i			0.662648i	0.943517	+	0.331324i	$0.107495\pi$
$257$			10.6230i			0.662648i	−0.943517	+	0.331324i	$0.892505\pi$
$258$		0			0
$259$	3.72733			0.231605
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.79276i		−	0.357197i	−0.983922	−	0.178598i	$-0.942844\pi$
$263$		−	5.79276i		−	0.357197i	0.983922	−	0.178598i	$-0.0571563\pi$
$264$		0			0
$265$	−1.20406	−	22.4081i	−0.0739647	−	1.37652i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.20857			−0.500485			−0.250243	−	0.968183i	$-0.580510\pi$
$269$	−8.20857			−0.500485			−0.250243	+	0.968183i	$0.580510\pi$
$270$		0			0
$271$	27.3111			1.65903			0.829515	−	0.558485i	$-0.188617\pi$
$271$	27.3111			1.65903			0.829515	+	0.558485i	$0.188617\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.97121	+	0.535784i	−0.299775	+	0.0323090i
$276$		0			0
$277$			14.4735i			0.869631i	0.900520	+	0.434815i	$0.143186\pi$
$277$			14.4735i			0.869631i	−0.900520	+	0.434815i	$0.856814\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.2799			−0.971178			−0.485589	−	0.874187i	$-0.661395\pi$
$281$	−16.2799			−0.971178			−0.485589	+	0.874187i	$0.661395\pi$
$282$		0			0
$283$			10.5169i			0.625165i	0.949891	+	0.312583i	$0.101194\pi$
$283$			10.5169i			0.625165i	−0.949891	+	0.312583i	$0.898806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.19955i		−	0.247892i
$288$		0			0
$289$	−25.4528			−1.49722
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.37695i			0.547807i	0.961757	+	0.273904i	$0.0883150\pi$
$293$			9.37695i			0.547807i	−0.961757	+	0.273904i	$0.911685\pi$
$294$		0			0
$295$	−3.64865	+	0.196053i	−0.212433	+	0.0114147i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	34.1932			1.97744
$300$		0			0
$301$	1.99099			0.114758
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.8628	−	0.852356i	0.908300	−	0.0488058i
$306$		0			0
$307$		−	2.09978i		−	0.119840i	−0.998203	−	0.0599202i	$-0.980915\pi$
$307$		−	2.09978i		−	0.119840i	0.998203	−	0.0599202i	$-0.0190846\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	20.3724			1.15522			0.577608	−	0.816314i	$-0.303986\pi$
$311$	20.3724			1.15522			0.577608	+	0.816314i	$0.303986\pi$
$312$		0			0
$313$		−	23.8968i		−	1.35073i	−0.737485	−	0.675364i	$-0.763987\pi$
$313$		−	23.8968i		−	1.35073i	0.737485	−	0.675364i	$-0.236013\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	5.86114i		−	0.329195i	−0.986361	−	0.164597i	$-0.947368\pi$
$317$		−	5.86114i		−	0.329195i	0.986361	−	0.164597i	$-0.0526325\pi$
$318$		0			0
$319$	−0.895717			−0.0501506
$320$		0			0
$321$		0			0
$322$		0			0
$323$			33.9075i			1.88666i
$324$		0			0
$325$	2.14314	+	19.8848i	0.118880	+	1.10301i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.33677			−0.128831
$330$		0			0
$331$	−0.574484			−0.0315765			−0.0157882	−	0.999875i	$-0.505026\pi$
$331$	−0.574484			−0.0315765			−0.0157882	+	0.999875i	$0.505026\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.28273	−	23.8723i	−0.0700833	−	1.30428i
$336$		0			0
$337$			4.10747i			0.223748i	0.993722	+	0.111874i	$0.0356853\pi$
$337$			4.10747i			0.223748i	−0.993722	+	0.111874i	$0.964315\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.73836			0.364903
$342$		0			0
$343$			5.74940i			0.310438i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.30834i		−	0.446015i	−0.974817	−	0.223008i	$-0.928413\pi$
$347$		−	8.30834i		−	0.446015i	0.974817	−	0.223008i	$-0.0715875\pi$
$348$		0			0
$349$	18.4858			0.989523			0.494762	−	0.869029i	$-0.335255\pi$
$349$	18.4858			0.989523			0.494762	+	0.869029i	$0.335255\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	26.5199i		−	1.41151i	−0.708456	−	0.705755i	$-0.750608\pi$
$353$		−	26.5199i		−	1.41151i	0.708456	−	0.705755i	$-0.249392\pi$
$354$		0			0
$355$	13.8281	−	0.743025i	0.733918	−	0.0394357i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.8226			−0.993419			−0.496709	−	0.867917i	$-0.665459\pi$
$359$	−18.8226			−0.993419			−0.496709	+	0.867917i	$0.665459\pi$
$360$		0			0
$361$	8.08222			0.425380
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.380136	+	7.07452i	0.0198972	+	0.370298i
$366$		0			0
$367$			10.2115i			0.533037i	0.963830	+	0.266519i	$0.0858734\pi$
$367$			10.2115i			0.533037i	−0.963830	+	0.266519i	$0.914127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.17289			0.216646
$372$		0			0
$373$		−	32.3363i		−	1.67431i	−0.546965	−	0.837156i	$-0.684217\pi$
$373$		−	32.3363i		−	1.67431i	0.546965	−	0.837156i	$-0.315783\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.58287i			0.184527i
$378$		0			0
$379$	−15.3686			−0.789434			−0.394717	−	0.918803i	$-0.629157\pi$
$379$	−15.3686			−0.789434			−0.394717	+	0.918803i	$0.629157\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			13.6662i			0.698309i	0.937065	+	0.349155i	$0.113531\pi$
$383$			13.6662i			0.698309i	−0.937065	+	0.349155i	$0.886469\pi$
$384$		0			0
$385$	−0.0498875	−	0.928432i	−0.00254250	−	0.0473173i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.01103			−0.456878			−0.228439	−	0.973558i	$-0.573362\pi$
$389$	−9.01103			−0.456878			−0.228439	+	0.973558i	$0.573362\pi$
$390$		0			0
$391$	−55.6971			−2.81672
$392$		0			0
$393$		0			0
$394$		0			0
$395$	25.1601	−	1.35193i	1.26594	−	0.0680229i
$396$		0			0
$397$			3.45466i			0.173384i	0.996235	+	0.0866922i	$0.0276297\pi$
$397$			3.45466i			0.173384i	−0.996235	+	0.0866922i	$0.972370\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.62756			0.0812762			0.0406381	−	0.999174i	$-0.487061\pi$
$401$	1.62756			0.0812762			0.0406381	+	0.999174i	$0.487061\pi$
$402$		0			0
$403$		−	26.9535i		−	1.34265i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.96410i			0.444334i
$408$		0			0
$409$	−6.54534			−0.323646			−0.161823	−	0.986820i	$-0.551737\pi$
$409$	−6.54534			−0.323646			−0.161823	+	0.986820i	$0.551737\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	0.679461i		−	0.0334341i
$414$		0			0
$415$	−1.94879	−	36.2679i	−0.0956623	−	1.78032i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−34.4795			−1.68443			−0.842216	−	0.539141i	$-0.818749\pi$
$419$	−34.4795			−1.68443			−0.842216	+	0.539141i	$0.818749\pi$
$420$		0			0
$421$	6.54534			0.319000			0.159500	−	0.987198i	$-0.449012\pi$
$421$	6.54534			0.319000			0.159500	+	0.987198i	$0.449012\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.49094	−	32.3903i	−0.169336	−	1.57116i
$426$		0			0
$427$			2.95401i			0.142954i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−36.1711			−1.74230			−0.871151	−	0.491016i	$-0.836626\pi$
$431$	−36.1711			−1.74230			−0.871151	+	0.491016i	$0.836626\pi$
$432$		0			0
$433$		−	38.1027i		−	1.83110i	−0.402204	−	0.915550i	$-0.631756\pi$
$433$		−	38.1027i		−	1.83110i	0.402204	−	0.915550i	$-0.368244\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			44.4858i			2.12805i
$438$		0			0
$439$	−12.1282			−0.578848			−0.289424	−	0.957201i	$-0.593464\pi$
$439$	−12.1282			−0.578848			−0.289424	+	0.957201i	$0.593464\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			16.5483i			0.786233i	0.919489	+	0.393117i	$0.128603\pi$
$443$			16.5483i			0.786233i	−0.919489	+	0.393117i	$0.871397\pi$
$444$		0			0
$445$	21.3683	−	1.14818i	1.01295	−	0.0544292i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.6056			0.736476			0.368238	−	0.929732i	$-0.379961\pi$
$449$	15.6056			0.736476			0.368238	+	0.929732i	$0.379961\pi$
$450$		0			0
$451$	10.0998			0.475580
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.71373	+	0.199550i	−0.174102	+	0.00935505i
$456$		0			0
$457$			5.87761i			0.274943i	0.990506	+	0.137471i	$0.0438975\pi$
$457$			5.87761i			0.274943i	−0.990506	+	0.137471i	$0.956102\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.7495			−1.71159			−0.855797	−	0.517312i	$-0.826933\pi$
$461$	−36.7495			−1.71159			−0.855797	+	0.517312i	$0.826933\pi$
$462$		0			0
$463$		−	19.3081i		−	0.897324i	−0.893702	−	0.448662i	$-0.851901\pi$
$463$		−	19.3081i		−	0.897324i	0.893702	−	0.448662i	$-0.148099\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	11.4129i		−	0.528127i	−0.964505	−	0.264064i	$-0.914937\pi$
$467$		−	11.4129i		−	0.528127i	0.964505	−	0.264064i	$-0.0850629\pi$
$468$		0			0
$469$	4.44556			0.205277
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.78825i			0.220164i
$474$		0			0
$475$	−25.8705	+	2.78825i	−1.18702	+	0.127934i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	29.2307			1.33559			0.667793	−	0.744347i	$-0.267239\pi$
$479$	29.2307			1.33559			0.667793	+	0.744347i	$0.267239\pi$
$480$		0			0
$481$	35.8564			1.63491
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.0709857	−	1.32108i	−0.00322329	−	0.0599871i
$486$		0			0
$487$			33.2932i			1.50866i	0.656496	+	0.754329i	$0.272038\pi$
$487$			33.2932i			1.50866i	−0.656496	+	0.754329i	$0.727962\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.5151			1.10635			0.553176	−	0.833064i	$-0.313416\pi$
$491$	24.5151			1.10635			0.553176	+	0.833064i	$0.313416\pi$
$492$		0			0
$493$		−	5.83612i		−	0.262846i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.57510i			0.115509i
$498$		0			0
$499$	18.9093			0.846497			0.423249	−	0.906014i	$-0.360890\pi$
$499$	18.9093			0.846497			0.423249	+	0.906014i	$0.360890\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.42031i		−	0.420031i	−0.977698	−	0.210016i	$-0.932649\pi$
$503$		−	9.42031i		−	0.420031i	0.977698	−	0.210016i	$-0.0673515\pi$
$504$		0			0
$505$	10.3226	−	0.554663i	0.459348	−	0.0246822i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.7804			−0.743778			−0.371889	−	0.928277i	$-0.621290\pi$
$509$	−16.7804			−0.743778			−0.371889	+	0.928277i	$0.621290\pi$
$510$		0			0
$511$	−1.31744			−0.0582799
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.988966	−	18.4052i	−0.0435791	−	0.811028i
$516$		0			0
$517$		−	5.61986i		−	0.247161i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	37.8273			1.65724			0.828621	−	0.559810i	$-0.189126\pi$
$521$	37.8273			1.65724			0.828621	+	0.559810i	$0.189126\pi$
$522$		0			0
$523$		−	10.0998i		−	0.441632i	−0.975315	−	0.220816i	$-0.929128\pi$
$523$		−	10.0998i		−	0.441632i	0.975315	−	0.220816i	$-0.0708721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			43.9044i			1.91250i
$528$		0			0
$529$	−50.0734			−2.17710
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	40.3991i		−	1.74988i
$534$		0			0
$535$	2.11399	+	39.3425i	0.0913959	+	1.70092i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.82711			−0.294064
$540$		0			0
$541$	−29.9269			−1.28666			−0.643329	−	0.765590i	$-0.722447\pi$
$541$	−29.9269			−1.28666			−0.643329	+	0.765590i	$0.722447\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.17942	+	0.439506i	−0.350368	+	0.0188264i
$546$		0			0
$547$			23.7630i			1.01603i	0.861347	+	0.508016i	$0.169621\pi$
$547$			23.7630i			1.01603i	−0.861347	+	0.508016i	$0.830379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.66137			−0.198581
$552$		0			0
$553$			4.68537i			0.199242i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			24.3363i			1.03116i	0.856841	+	0.515581i	$0.172424\pi$
$557$			24.3363i			1.03116i	−0.856841	+	0.515581i	$0.827576\pi$
$558$		0			0
$559$	19.1530			0.810086
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.85368i		−	0.373138i	−0.982442	−	0.186569i	$-0.940263\pi$
$563$		−	8.85368i		−	0.373138i	0.982442	−	0.186569i	$-0.0597368\pi$
$564$		0			0
$565$	1.37731	+	25.6324i	0.0579437	+	1.07836i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.3860			1.10616			0.553080	−	0.833128i	$-0.313452\pi$
$569$	26.3860			1.10616			0.553080	+	0.833128i	$0.313452\pi$
$570$		0			0
$571$	6.45280			0.270041			0.135021	−	0.990843i	$-0.456890\pi$
$571$	6.45280			0.270041			0.135021	+	0.990843i	$0.456890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.58004	−	42.4954i	−0.191001	−	1.77218i
$576$		0			0
$577$			12.1745i			0.506832i	0.967357	+	0.253416i	$0.0815542\pi$
$577$			12.1745i			0.506832i	−0.967357	+	0.253416i	$0.918446\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.75390			0.280199
$582$		0			0
$583$			10.0357i			0.415635i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			22.9521i			0.947336i	0.880704	+	0.473668i	$0.157070\pi$
$587$			22.9521i			0.947336i	−0.880704	+	0.473668i	$0.842930\pi$
$588$		0			0
$589$	35.0668			1.44490
$590$		0			0
$591$		0			0
$592$		0			0
$593$			10.4081i			0.427410i	0.976898	+	0.213705i	$0.0685531\pi$
$593$			10.4081i			0.427410i	−0.976898	+	0.213705i	$0.931447\pi$
$594$		0			0
$595$	6.04927	−	0.325046i	0.247996	−	0.0133256i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.98913			−0.367286			−0.183643	−	0.982993i	$-0.558789\pi$
$599$	−8.98913			−0.367286			−0.183643	+	0.982993i	$0.558789\pi$
$600$		0			0
$601$	−3.57650			−0.145889			−0.0729443	−	0.997336i	$-0.523240\pi$
$601$	−3.57650			−0.145889			−0.0729443	+	0.997336i	$0.523240\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.23285	−	0.119978i	0.0907781	−	0.00487779i
$606$		0			0
$607$			27.0235i			1.09685i	0.836200	+	0.548424i	$0.184772\pi$
$607$			27.0235i			1.09685i	−0.836200	+	0.548424i	$0.815228\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.4795			−0.909421
$612$		0			0
$613$			44.5572i			1.79965i	0.436254	+	0.899823i	$0.356305\pi$
$613$			44.5572i			1.79965i	−0.436254	+	0.899823i	$0.643695\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.2966i			0.615818i	0.951416	+	0.307909i	$0.0996292\pi$
$617$			15.2966i			0.615818i	−0.951416	+	0.307909i	$0.900371\pi$
$618$		0			0
$619$	−14.5655			−0.585436			−0.292718	−	0.956199i	$-0.594560\pi$
$619$	−14.5655			−0.585436			−0.292718	+	0.956199i	$0.594560\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			3.97926i			0.159426i
$624$		0			0
$625$	24.4259	−	5.32699i	0.977035	−	0.213080i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−58.4063			−2.32881
$630$		0			0
$631$	−22.0779			−0.878906			−0.439453	−	0.898266i	$-0.644828\pi$
$631$	−22.0779			−0.878906			−0.439453	+	0.898266i	$0.644828\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.09659	+	20.4081i	0.0435169	+	0.809871i
$636$		0			0
$637$			27.3084i			1.08200i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.2152			1.23293			0.616463	−	0.787384i	$-0.288565\pi$
$641$	31.2152			1.23293			0.616463	+	0.787384i	$0.288565\pi$
$642$		0			0
$643$			10.4498i			0.412102i	0.978541	+	0.206051i	$0.0660612\pi$
$643$			10.4498i			0.412102i	−0.978541	+	0.206051i	$0.933939\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.1307i		−	1.02730i	−0.857999	−	0.513652i	$-0.828292\pi$
$647$		−	26.1307i		−	1.02730i	0.857999	−	0.513652i	$-0.171708\pi$
$648$		0			0
$649$	1.63408			0.0641433
$650$		0			0
$651$		0			0
$652$		0			0
$653$			44.2097i			1.73006i	0.501719	+	0.865030i	$0.332701\pi$
$653$			44.2097i			1.73006i	−0.501719	+	0.865030i	$0.667299\pi$
$654$		0			0
$655$	5.54022	−	0.297693i	0.216474	−	0.0116318i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.96706			0.271398			0.135699	−	0.990750i	$-0.456672\pi$
$659$	6.96706			0.271398			0.135699	+	0.990750i	$0.456672\pi$
$660$		0			0
$661$	−9.19753			−0.357743			−0.178871	−	0.983872i	$-0.557245\pi$
$661$	−9.19753			−0.357743			−0.178871	+	0.983872i	$0.557245\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.259618	−	4.83161i	−0.0100675	−	0.187362i
$666$		0			0
$667$		−	7.65686i		−	0.296475i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.10428			−0.274258
$672$		0			0
$673$			8.72415i			0.336291i	0.985762	+	0.168146i	$0.0537779\pi$
$673$			8.72415i			0.336291i	−0.985762	+	0.168146i	$0.946222\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	8.76618i		−	0.336912i	−0.985709	−	0.168456i	$-0.946122\pi$
$677$		−	8.76618i		−	0.336912i	0.985709	−	0.168456i	$-0.0538781\pi$
$678$		0			0
$679$	0.246015			0.00944118
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	49.2320i		−	1.88381i	−0.335877	−	0.941906i	$-0.609033\pi$
$683$		−	49.2320i		−	1.88381i	0.335877	−	0.941906i	$-0.390967\pi$
$684$		0			0
$685$	0.888837	+	16.5417i	0.0339607	+	0.632026i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	40.1427			1.52931
$690$		0			0
$691$	−18.1483			−0.690395			−0.345198	−	0.938530i	$-0.612188\pi$
$691$	−18.1483			−0.690395			−0.345198	+	0.938530i	$0.612188\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.77722	−	0.202962i	0.143278	−	0.00769877i
$696$		0			0
$697$			65.8059i			2.49258i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.6587			−0.855808			−0.427904	−	0.903824i	$-0.640748\pi$
$701$	−22.6587			−0.855808			−0.427904	+	0.903824i	$0.640748\pi$
$702$		0			0
$703$			46.6497i			1.75943i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.92229i			0.0722952i
$708$		0			0
$709$	36.7366			1.37967			0.689836	−	0.723966i	$-0.257683\pi$
$709$	36.7366			1.37967			0.689836	+	0.723966i	$0.257683\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			57.6015i			2.15719i
$714$		0			0
$715$	−0.479911	−	8.93139i	−0.0179477	−	0.334015i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.74738			0.326222			0.163111	−	0.986608i	$-0.447847\pi$
$719$	8.74738			0.326222			0.163111	+	0.986608i	$0.447847\pi$
$720$		0			0
$721$	3.42745			0.127645
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.45280	−	0.479911i	0.165373	−	0.0178235i
$726$		0			0
$727$		−	2.20887i		−	0.0819226i	−0.999161	−	0.0409613i	$-0.986958\pi$
$727$		−	2.20887i		−	0.0819226i	0.999161	−	0.0409613i	$-0.0130420\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−31.1982			−1.15391
$732$		0			0
$733$		−	3.58552i		−	0.132434i	−0.997805	−	0.0662171i	$-0.978907\pi$
$733$		−	3.58552i		−	0.132434i	0.997805	−	0.0662171i	$-0.0210930\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.6914i			0.393824i
$738$		0			0
$739$	24.9314			0.917116			0.458558	−	0.888665i	$-0.348366\pi$
$739$	24.9314			0.917116			0.458558	+	0.888665i	$0.348366\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			21.4257i			0.786032i	0.919532	+	0.393016i	$0.128568\pi$
$743$			21.4257i			0.786032i	−0.919532	+	0.393016i	$0.871432\pi$
$744$		0			0
$745$	8.37183	−	0.449844i	0.306720	−	0.0164810i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.32645			−0.267703
$750$		0			0
$751$	28.1582			1.02751			0.513754	−	0.857938i	$-0.328254\pi$
$751$	28.1582			1.02751			0.513754	+	0.857938i	$0.328254\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−33.8052	+	1.81646i	−1.23030	+	0.0661077i
$756$		0			0
$757$		−	10.7513i		−	0.390761i	−0.980728	−	0.195381i	$-0.937406\pi$
$757$		−	10.7513i		−	0.390761i	0.980728	−	0.195381i	$-0.0625942\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.0429			1.37905			0.689527	−	0.724260i	$-0.257818\pi$
$761$	38.0429			1.37905			0.689527	+	0.724260i	$0.257818\pi$
$762$		0			0
$763$		−	1.52319i		−	0.0551433i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	6.53632i		−	0.236013i
$768$		0			0
$769$	−26.2280			−0.945805			−0.472903	−	0.881115i	$-0.656794\pi$
$769$	−26.2280			−0.945805			−0.472903	+	0.881115i	$0.656794\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	8.94499i		−	0.321729i	−0.986977	−	0.160864i	$-0.948572\pi$
$773$		−	8.94499i		−	0.321729i	0.986977	−	0.160864i	$-0.0514282\pi$
$774$		0			0
$775$	−33.4978	+	3.61031i	−1.20328	+	0.129686i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	52.5598			1.88315
$780$		0			0
$781$	−6.19302			−0.221604
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.39020	−	25.8723i	−0.0496184	−	0.923423i
$786$		0			0
$787$		−	29.9716i		−	1.06837i	−0.845367	−	0.534185i	$-0.820618\pi$
$787$		−	29.9716i		−	1.06837i	0.845367	−	0.534185i	$-0.179382\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.77332			−0.169720
$792$		0			0
$793$			28.4171i			1.00912i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.80022i			0.240876i	0.992721	+	0.120438i	$0.0384299\pi$
$797$			6.80022i			0.240876i	−0.992721	+	0.120438i	$0.961570\pi$
$798$		0			0
$799$	36.6167			1.29541
$800$		0			0
$801$		0

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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.23285 - 0.119978i) q^{5} +0.415806i q^{7} +O(q^{10})\)	\(q+(2.23285 - 0.119978i) q^{5} +0.415806i q^{7} -1.00000 q^{11} +4.00000i q^{13} -6.51558i q^{17} -5.20406 q^{19} -8.54830i q^{23} +(4.97121 - 0.535784i) q^{25} +0.895717 q^{29} -6.73836 q^{31} +(0.0498875 + 0.928432i) q^{35} -8.96410i q^{37} -10.0998 q^{41} -4.78825i q^{43} +5.61986i q^{47} +6.82711 q^{49} -10.0357i q^{53} +(-2.23285 + 0.119978i) q^{55} -1.63408 q^{59} +7.10428 q^{61} +(0.479911 + 8.93139i) q^{65} -10.6914i q^{67} +6.19302 q^{71} +3.16839i q^{73} -0.415806i q^{77} +11.2682 q^{79} -16.2429i q^{83} +(-0.781725 - 14.5483i) q^{85} +9.56998 q^{89} -1.66323 q^{91} +(-11.6199 + 0.624371i) q^{95} -0.591657i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) \) 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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