LMFDB - Embedded newform 4140.2.f.c.829.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.6
Root			$2.88373i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.792997 + 2.09073i) q^{5} -4.31589i q^{7} +O(q^{10})$	$q+(-0.792997 + 2.09073i) q^{5} -4.31589i q^{7} +2.26617 q^{11} +2.25590i q^{13} +0.535482i q^{17} +6.91942 q^{19} +1.00000i q^{23} +(-3.74231 - 3.31589i) q^{25} +5.64598 q^{29} -10.4240 q^{31} +(9.02336 + 3.42249i) q^{35} -8.10481i q^{37} -0.633143 q^{41} -1.30687i q^{43} +12.4159i q^{47} -11.6269 q^{49} -5.98659i q^{53} +(-1.79707 + 4.73796i) q^{55} +7.01063 q^{59} +5.98705 q^{61} +(-4.71649 - 1.78893i) q^{65} -6.32616i q^{67} -0.151090 q^{71} +8.88768i q^{73} -9.78055i q^{77} +10.3312 q^{79} +0.716944i q^{83} +(-1.11955 - 0.424636i) q^{85} +15.5781 q^{89} +9.73623 q^{91} +(-5.48708 + 14.4666i) q^{95} -16.2497i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.792997	+	2.09073i	−0.354639	+	0.935003i
$5$	−0.792997	+	2.09073i	−0.354639	+	0.935003i
$6$		0			0
$7$		−	4.31589i		−	1.63125i	−0.578579	−	0.815626i	$-0.696393\pi$
$7$		−	4.31589i		−	1.63125i	0.578579	−	0.815626i	$-0.303607\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.26617			0.683277			0.341638	−	0.939831i	$-0.389018\pi$
$11$	2.26617			0.683277			0.341638	+	0.939831i	$0.389018\pi$
$12$		0			0
$13$			2.25590i			0.625675i	0.949807	+	0.312838i	$0.101280\pi$
$13$			2.25590i			0.625675i	−0.949807	+	0.312838i	$0.898720\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.535482i			0.129874i	0.997889	+	0.0649368i	$0.0206846\pi$
$17$			0.535482i			0.129874i	−0.997889	+	0.0649368i	$0.979315\pi$
$18$		0			0
$19$	6.91942			1.58742			0.793712	−	0.608294i	$-0.208146\pi$
$19$	6.91942			1.58742			0.793712	+	0.608294i	$0.208146\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−3.74231	−	3.31589i	−0.748462	−	0.663178i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.64598			1.04843			0.524216	−	0.851585i	$-0.324358\pi$
$29$	5.64598			1.04843			0.524216	+	0.851585i	$0.324358\pi$
$30$		0			0
$31$	−10.4240			−1.87220			−0.936099	−	0.351736i	$-0.885591\pi$
$31$	−10.4240			−1.87220			−0.936099	+	0.351736i	$0.885591\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.02336	+	3.42249i	1.52523	+	0.578506i
$36$		0			0
$37$		−	8.10481i		−	1.33242i	−0.745763	−	0.666212i	$-0.767915\pi$
$37$		−	8.10481i		−	1.33242i	0.745763	−	0.666212i	$-0.232085\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.633143			−0.0988803			−0.0494402	−	0.998777i	$-0.515744\pi$
$41$	−0.633143			−0.0988803			−0.0494402	+	0.998777i	$0.515744\pi$
$42$		0			0
$43$		−	1.30687i		−	0.199296i	−0.995023	−	0.0996481i	$-0.968228\pi$
$43$		−	1.30687i		−	0.199296i	0.995023	−	0.0996481i	$-0.0317717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.4159i			1.81104i	0.424299	+	0.905522i	$0.360521\pi$
$47$			12.4159i			1.81104i	−0.424299	+	0.905522i	$0.639479\pi$
$48$		0			0
$49$	−11.6269			−1.66098
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.98659i		−	0.822322i	−0.911563	−	0.411161i	$-0.865123\pi$
$53$		−	5.98659i		−	0.822322i	0.911563	−	0.411161i	$-0.134877\pi$
$54$		0			0
$55$	−1.79707	+	4.73796i	−0.242317	+	0.638866i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.01063			0.912706			0.456353	−	0.889799i	$-0.349155\pi$
$59$	7.01063			0.912706			0.456353	+	0.889799i	$0.349155\pi$
$60$		0			0
$61$	5.98705			0.766563			0.383282	−	0.923632i	$-0.374794\pi$
$61$	5.98705			0.766563			0.383282	+	0.923632i	$0.374794\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.71649	−	1.78893i	−0.585008	−	0.221889i
$66$		0			0
$67$		−	6.32616i		−	0.772863i	−0.922318	−	0.386432i	$-0.873708\pi$
$67$		−	6.32616i		−	0.772863i	0.922318	−	0.386432i	$-0.126292\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.151090			−0.0179311			−0.00896553	−	0.999960i	$-0.502854\pi$
$71$	−0.151090			−0.0179311			−0.00896553	+	0.999960i	$0.502854\pi$
$72$		0			0
$73$			8.88768i			1.04022i	0.854098	+	0.520112i	$0.174110\pi$
$73$			8.88768i			1.04022i	−0.854098	+	0.520112i	$0.825890\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	9.78055i		−	1.11460i
$78$		0			0
$79$	10.3312			1.16235			0.581175	−	0.813779i	$-0.302593\pi$
$79$	10.3312			1.16235			0.581175	+	0.813779i	$0.302593\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.716944i			0.0786948i	0.999226	+	0.0393474i	$0.0125279\pi$
$83$			0.716944i			0.0786948i	−0.999226	+	0.0393474i	$0.987472\pi$
$84$		0			0
$85$	−1.11955	−	0.424636i	−0.121432	−	0.0460583i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.5781			1.65128			0.825639	−	0.564199i	$-0.190815\pi$
$89$	15.5781			1.65128			0.825639	+	0.564199i	$0.190815\pi$
$90$		0			0
$91$	9.73623			1.02063
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.48708	+	14.4666i	−0.562963	+	1.48425i
$96$		0			0
$97$		−	16.2497i		−	1.64990i	−0.565204	−	0.824951i	$-0.691203\pi$
$97$		−	16.2497i		−	1.64990i	0.565204	−	0.824951i	$-0.308797\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.8428			−1.27790			−0.638952	−	0.769247i	$-0.720632\pi$
$101$	−12.8428			−1.27790			−0.638952	+	0.769247i	$0.720632\pi$
$102$		0			0
$103$		−	16.1002i		−	1.58640i	−0.608961	−	0.793200i	$-0.708413\pi$
$103$		−	16.1002i		−	1.58640i	0.608961	−	0.793200i	$-0.291587\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.3335i			1.28900i	0.764604	+	0.644501i	$0.222935\pi$
$107$			13.3335i			1.28900i	−0.764604	+	0.644501i	$0.777065\pi$
$108$		0			0
$109$	6.61883			0.633969			0.316984	−	0.948431i	$-0.397330\pi$
$109$	6.61883			0.633969			0.316984	+	0.948431i	$0.397330\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	17.5215i		−	1.64828i	−0.566383	−	0.824142i	$-0.691658\pi$
$113$		−	17.5215i		−	1.64828i	0.566383	−	0.824142i	$-0.308342\pi$
$114$		0			0
$115$	−2.09073	−	0.792997i	−0.194962	−	0.0739474i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.31108			0.211857
$120$		0			0
$121$	−5.86446			−0.533133
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.90027	−	5.19467i	0.885507	−	0.464626i
$126$		0			0
$127$			13.3001i			1.18019i	0.807334	+	0.590095i	$0.200910\pi$
$127$			13.3001i			1.18019i	−0.807334	+	0.590095i	$0.799090\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.99386			0.261575			0.130787	−	0.991410i	$-0.458249\pi$
$131$	2.99386			0.261575			0.130787	+	0.991410i	$0.458249\pi$
$132$		0			0
$133$		−	29.8634i		−	2.58949i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	17.5528i		−	1.49964i	−0.661643	−	0.749819i	$-0.730141\pi$
$137$		−	17.5528i		−	1.49964i	0.661643	−	0.749819i	$-0.269859\pi$
$138$		0			0
$139$	17.8735			1.51601			0.758004	−	0.652250i	$-0.226175\pi$
$139$	17.8735			1.51601			0.758004	+	0.652250i	$0.226175\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			5.11227i			0.427509i
$144$		0			0
$145$	−4.47725	+	11.8042i	−0.371815	+	0.980287i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.71607			0.304432			0.152216	−	0.988347i	$-0.451359\pi$
$149$	3.71607			0.304432			0.152216	+	0.988347i	$0.451359\pi$
$150$		0			0
$151$	14.3527			1.16800			0.584001	−	0.811753i	$-0.301486\pi$
$151$	14.3527			1.16800			0.584001	+	0.811753i	$0.301486\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.26617	−	21.7937i	0.663955	−	1.75051i
$156$		0			0
$157$			9.94566i			0.793750i	0.917873	+	0.396875i	$0.129905\pi$
$157$			9.94566i			0.793750i	−0.917873	+	0.396875i	$0.870095\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.31589			0.340140
$162$		0			0
$163$		−	9.87216i		−	0.773247i	−0.922238	−	0.386624i	$-0.873641\pi$
$163$		−	9.87216i		−	0.773247i	0.922238	−	0.386624i	$-0.126359\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.92476i			0.226325i	0.993577	+	0.113162i	$0.0360980\pi$
$167$			2.92476i			0.226325i	−0.993577	+	0.113162i	$0.963902\pi$
$168$		0			0
$169$	7.91090			0.608531
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	0.421083i		−	0.0320144i	−0.999872	−	0.0160072i	$-0.994905\pi$
$173$		−	0.421083i		−	0.0320144i	0.999872	−	0.0160072i	$-0.00509546\pi$
$174$		0			0
$175$	−14.3110	+	16.1514i	−1.08181	+	1.22093i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.3955			1.07597			0.537986	−	0.842954i	$-0.319185\pi$
$179$	14.3955			1.07597			0.537986	+	0.842954i	$0.319185\pi$
$180$		0			0
$181$	7.54278			0.560650			0.280325	−	0.959905i	$-0.409558\pi$
$181$	7.54278			0.560650			0.280325	+	0.959905i	$0.409558\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.9450	+	6.42710i	1.24582	+	0.472530i
$186$		0			0
$187$			1.21350i			0.0887396i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.9141			−1.36857			−0.684287	−	0.729213i	$-0.739887\pi$
$191$	−18.9141			−1.36857			−0.684287	+	0.729213i	$0.739887\pi$
$192$		0			0
$193$			3.56998i			0.256973i	0.991711	+	0.128487i	$0.0410119\pi$
$193$			3.56998i			0.256973i	−0.991711	+	0.128487i	$0.958988\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.90677i		−	0.135852i	−0.997690	−	0.0679258i	$-0.978362\pi$
$197$		−	1.90677i		−	0.135852i	0.997690	−	0.0679258i	$-0.0216381\pi$
$198$		0			0
$199$	−1.84829			−0.131022			−0.0655110	−	0.997852i	$-0.520868\pi$
$199$	−1.84829			−0.131022			−0.0655110	+	0.997852i	$0.520868\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	24.3674i		−	1.71026i
$204$		0			0
$205$	0.502081	−	1.32373i	0.0350668	−	0.0924534i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.6806			1.08465
$210$		0			0
$211$	14.0429			0.966753			0.483377	−	0.875413i	$-0.339410\pi$
$211$	14.0429			0.966753			0.483377	+	0.875413i	$0.339410\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.73232	+	1.03635i	0.186343	+	0.0706783i
$216$		0			0
$217$			44.9886i			3.05403i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.20800			−0.0812586
$222$		0			0
$223$		−	11.3081i		−	0.757245i	−0.925551	−	0.378623i	$-0.876398\pi$
$223$		−	11.3081i		−	0.757245i	0.925551	−	0.378623i	$-0.123602\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			26.3506i			1.74895i	0.485067	+	0.874477i	$0.338795\pi$
$227$			26.3506i			1.74895i	−0.485067	+	0.874477i	$0.661205\pi$
$228$		0			0
$229$	13.7704			0.909976			0.454988	−	0.890498i	$-0.349643\pi$
$229$	13.7704			0.909976			0.454988	+	0.890498i	$0.349643\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.05029i		−	0.396368i	−0.980165	−	0.198184i	$-0.936496\pi$
$233$		−	6.05029i		−	0.396368i	0.980165	−	0.198184i	$-0.0635043\pi$
$234$		0			0
$235$	−25.9583	−	9.84577i	−1.69333	−	0.642267i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.2468			−0.986234			−0.493117	−	0.869963i	$-0.664143\pi$
$239$	−15.2468			−0.986234			−0.493117	+	0.869963i	$0.664143\pi$
$240$		0			0
$241$	15.3001			0.985563			0.492782	−	0.870153i	$-0.335980\pi$
$241$	15.3001			0.985563			0.492782	+	0.870153i	$0.335980\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.22009	−	24.3087i	0.589050	−	1.55303i
$246$		0			0
$247$			15.6095i			0.993211i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.4096			1.28825			0.644123	−	0.764922i	$-0.277223\pi$
$251$	20.4096			1.28825			0.644123	+	0.764922i	$0.277223\pi$
$252$		0			0
$253$			2.26617i			0.142473i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	14.8072i		−	0.923649i	−0.886971	−	0.461825i	$-0.847195\pi$
$257$		−	14.8072i		−	0.923649i	0.886971	−	0.461825i	$-0.152805\pi$
$258$		0			0
$259$	−34.9795			−2.17352
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.5266i			0.895747i	0.894097	+	0.447873i	$0.147818\pi$
$263$			14.5266i			0.895747i	−0.894097	+	0.447873i	$0.852182\pi$
$264$		0			0
$265$	12.5164	+	4.74735i	0.768874	+	0.291628i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.4759			−1.00455			−0.502276	−	0.864708i	$-0.667504\pi$
$269$	−16.4759			−1.00455			−0.502276	+	0.864708i	$0.667504\pi$
$270$		0			0
$271$	22.0764			1.34105			0.670524	−	0.741888i	$-0.266069\pi$
$271$	22.0764			1.34105			0.670524	+	0.741888i	$0.266069\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.48072	−	7.51437i	−0.511407	−	0.453134i
$276$		0			0
$277$			6.70017i			0.402574i	0.979532	+	0.201287i	$0.0645125\pi$
$277$			6.70017i			0.402574i	−0.979532	+	0.201287i	$0.935488\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.6857			1.35332			0.676658	−	0.736298i	$-0.263428\pi$
$281$	22.6857			1.35332			0.676658	+	0.736298i	$0.263428\pi$
$282$		0			0
$283$			19.9470i			1.18572i	0.805304	+	0.592862i	$0.202002\pi$
$283$			19.9470i			1.18572i	−0.805304	+	0.592862i	$0.797998\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.73257i			0.161299i
$288$		0			0
$289$	16.7133			0.983133
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.9230i			0.871808i	0.899993	+	0.435904i	$0.143571\pi$
$293$			14.9230i			0.871808i	−0.899993	+	0.435904i	$0.856429\pi$
$294$		0			0
$295$	−5.55941	+	14.6573i	−0.323681	+	0.853383i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.25590			−0.130462
$300$		0			0
$301$	−5.64032			−0.325102
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.74772	+	12.5173i	−0.271853	+	0.716739i
$306$		0			0
$307$		−	12.9612i		−	0.739735i	−0.929084	−	0.369868i	$-0.879403\pi$
$307$		−	12.9612i		−	0.739735i	0.929084	−	0.369868i	$-0.120597\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.0709			0.741182			0.370591	−	0.928796i	$-0.379155\pi$
$311$	13.0709			0.741182			0.370591	+	0.928796i	$0.379155\pi$
$312$		0			0
$313$			0.224783i			0.0127055i	0.999980	+	0.00635273i	$0.00202215\pi$
$313$			0.224783i			0.0127055i	−0.999980	+	0.00635273i	$0.997978\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.03602i		−	0.395182i	−0.980285	−	0.197591i	$-0.936688\pi$
$317$		−	7.03602i		−	0.395182i	0.980285	−	0.197591i	$-0.0633119\pi$
$318$		0			0
$319$	12.7948			0.716369
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.70523i			0.206164i
$324$		0			0
$325$	7.48032	−	8.44229i	0.414934	−	0.468294i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	53.5856			2.95427
$330$		0			0
$331$	−13.7320			−0.754781			−0.377391	−	0.926054i	$-0.623179\pi$
$331$	−13.7320			−0.754781			−0.377391	+	0.926054i	$0.623179\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.2263	+	5.01663i	0.722629	+	0.274088i
$336$		0			0
$337$		−	10.4552i		−	0.569533i	−0.958597	−	0.284767i	$-0.908084\pi$
$337$		−	10.4552i		−	0.569533i	0.958597	−	0.284767i	$-0.0919161\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−23.6225			−1.27923
$342$		0			0
$343$			19.9691i			1.07823i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	35.4166i		−	1.90126i	−0.310321	−	0.950632i	$-0.600437\pi$
$347$		−	35.4166i		−	1.90126i	0.310321	−	0.950632i	$-0.399563\pi$
$348$		0			0
$349$	10.9753			0.587493			0.293747	−	0.955883i	$-0.405098\pi$
$349$	10.9753			0.587493			0.293747	+	0.955883i	$0.405098\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.0802i			0.642963i	0.946916	+	0.321481i	$0.104181\pi$
$353$			12.0802i			0.642963i	−0.946916	+	0.321481i	$0.895819\pi$
$354$		0			0
$355$	0.119814	−	0.315888i	0.00635906	−	0.0167656i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.1356			1.69605			0.848026	−	0.529954i	$-0.177791\pi$
$359$	32.1356			1.69605			0.848026	+	0.529954i	$0.177791\pi$
$360$		0			0
$361$	28.8784			1.51991
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.5817	−	7.04791i	−0.972613	−	0.368904i
$366$		0			0
$367$			10.7278i			0.559985i	0.960002	+	0.279992i	$0.0903320\pi$
$367$			10.7278i			0.559985i	−0.960002	+	0.279992i	$0.909668\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−25.8375			−1.34141
$372$		0			0
$373$		−	23.1722i		−	1.19981i	−0.800071	−	0.599905i	$-0.795205\pi$
$373$		−	23.1722i		−	1.19981i	0.800071	−	0.599905i	$-0.204795\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.7368i			0.655978i
$378$		0			0
$379$	−19.2361			−0.988090			−0.494045	−	0.869436i	$-0.664482\pi$
$379$	−19.2361			−0.988090			−0.494045	+	0.869436i	$0.664482\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	14.7618i		−	0.754292i	−0.926154	−	0.377146i	$-0.876905\pi$
$383$		−	14.7618i		−	0.754292i	0.926154	−	0.377146i	$-0.123095\pi$
$384$		0			0
$385$	20.4485	+	7.75595i	1.04215	+	0.395280i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0615			−0.712948			−0.356474	−	0.934305i	$-0.616021\pi$
$389$	−14.0615			−0.712948			−0.356474	+	0.934305i	$0.616021\pi$
$390$		0			0
$391$	−0.535482			−0.0270805
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.19260	+	21.5997i	−0.412215	+	1.08680i
$396$		0			0
$397$			13.9662i			0.700945i	0.936573	+	0.350473i	$0.113979\pi$
$397$			13.9662i			0.700945i	−0.936573	+	0.350473i	$0.886021\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−35.3183			−1.76371			−0.881855	−	0.471521i	$-0.843705\pi$
$401$	−35.3183			−1.76371			−0.881855	+	0.471521i	$0.843705\pi$
$402$		0			0
$403$		−	23.5154i		−	1.17139i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	18.3669i		−	0.910414i
$408$		0			0
$409$	−17.7067			−0.875541			−0.437770	−	0.899087i	$-0.644232\pi$
$409$	−17.7067			−0.875541			−0.437770	+	0.899087i	$0.644232\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	30.2571i		−	1.48885i
$414$		0			0
$415$	−1.49894	−	0.568535i	−0.0735799	−	0.0279083i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.6432			−1.35046			−0.675229	−	0.737608i	$-0.735955\pi$
$419$	−27.6432			−1.35046			−0.675229	+	0.737608i	$0.735955\pi$
$420$		0			0
$421$	−15.3867			−0.749902			−0.374951	−	0.927045i	$-0.622340\pi$
$421$	−15.3867			−0.749902			−0.374951	+	0.927045i	$0.622340\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.77560	−	2.00394i	0.0861292	−	0.0972054i
$426$		0			0
$427$		−	25.8394i		−	1.25046i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.0437			−1.35082			−0.675409	−	0.737443i	$-0.736033\pi$
$431$	−28.0437			−1.35082			−0.675409	+	0.737443i	$0.736033\pi$
$432$		0			0
$433$			10.5613i			0.507544i	0.967264	+	0.253772i	$0.0816713\pi$
$433$			10.5613i			0.507544i	−0.967264	+	0.253772i	$0.918329\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.91942i			0.331001i
$438$		0			0
$439$	−16.9599			−0.809452			−0.404726	−	0.914438i	$-0.632633\pi$
$439$	−16.9599			−0.809452			−0.404726	+	0.914438i	$0.632633\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			7.15493i			0.339941i	0.985449	+	0.169970i	$0.0543672\pi$
$443$			7.15493i			0.339941i	−0.985449	+	0.169970i	$0.945633\pi$
$444$		0			0
$445$	−12.3534	+	32.5697i	−0.585608	+	1.54395i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.2411			0.672079			0.336039	−	0.941848i	$-0.390912\pi$
$449$	14.2411			0.672079			0.336039	+	0.941848i	$0.390912\pi$
$450$		0			0
$451$	−1.43481			−0.0675626
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.72080	+	20.3558i	−0.361957	+	0.954296i
$456$		0			0
$457$		−	24.9658i		−	1.16785i	−0.811807	−	0.583926i	$-0.801516\pi$
$457$		−	24.9658i		−	1.16785i	0.811807	−	0.583926i	$-0.198484\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.88444			−0.274066			−0.137033	−	0.990566i	$-0.543757\pi$
$461$	−5.88444			−0.274066			−0.137033	+	0.990566i	$0.543757\pi$
$462$		0			0
$463$			10.4047i			0.483546i	0.970333	+	0.241773i	$0.0777289\pi$
$463$			10.4047i			0.483546i	−0.970333	+	0.241773i	$0.922271\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.8260i		−	1.19508i	−0.801838	−	0.597541i	$-0.796144\pi$
$467$		−	25.8260i		−	1.19508i	0.801838	−	0.597541i	$-0.203856\pi$
$468$		0			0
$469$	−27.3030			−1.26073
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.96160i		−	0.136174i
$474$		0			0
$475$	−25.8946	−	22.9440i	−1.18813	−	1.05274i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.2144			−0.877930			−0.438965	−	0.898504i	$-0.644655\pi$
$479$	−19.2144			−0.877930			−0.438965	+	0.898504i	$0.644655\pi$
$480$		0			0
$481$	18.2837			0.833664
$482$		0			0
$483$		0			0
$484$		0			0
$485$	33.9736	+	12.8859i	1.54266	+	0.585120i
$486$		0			0
$487$		−	33.4334i		−	1.51501i	−0.652828	−	0.757506i	$-0.726417\pi$
$487$		−	33.4334i		−	1.51501i	0.652828	−	0.757506i	$-0.273583\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.17956			0.233750			0.116875	−	0.993147i	$-0.462712\pi$
$491$	5.17956			0.233750			0.116875	+	0.993147i	$0.462712\pi$
$492$		0			0
$493$			3.02332i			0.136164i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			0.652087i			0.0292501i
$498$		0			0
$499$	−10.4240			−0.466641			−0.233320	−	0.972400i	$-0.574959\pi$
$499$	−10.4240			−0.466641			−0.233320	+	0.972400i	$0.574959\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	33.7019i		−	1.50269i	−0.659908	−	0.751346i	$-0.729405\pi$
$503$		−	33.7019i		−	1.50269i	0.659908	−	0.751346i	$-0.270595\pi$
$504$		0			0
$505$	10.1843	−	26.8508i	0.453195	−	1.19484i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−23.7289			−1.05177			−0.525883	−	0.850557i	$-0.676265\pi$
$509$	−23.7289			−1.05177			−0.525883	+	0.850557i	$0.676265\pi$
$510$		0			0
$511$	38.3582			1.69687
$512$		0			0
$513$		0			0
$514$		0			0
$515$	33.6612	+	12.7674i	1.48329	+	0.562600i
$516$		0			0
$517$			28.1366i			1.23744i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.0975			−0.486189			−0.243095	−	0.970003i	$-0.578163\pi$
$521$	−11.0975			−0.486189			−0.243095	+	0.970003i	$0.578163\pi$
$522$		0			0
$523$		−	10.9113i		−	0.477117i	−0.971128	−	0.238558i	$-0.923325\pi$
$523$		−	10.9113i		−	0.477117i	0.971128	−	0.238558i	$-0.0766749\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	5.58185i		−	0.243149i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.42831i		−	0.0618669i
$534$		0			0
$535$	−27.8768	−	10.5735i	−1.20522	−	0.457130i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−26.3485			−1.13491
$540$		0			0
$541$	16.1780			0.695548			0.347774	−	0.937578i	$-0.386938\pi$
$541$	16.1780			0.695548			0.347774	+	0.937578i	$0.386938\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.24871	+	13.8382i	−0.224830	+	0.592763i
$546$		0			0
$547$			24.2259i			1.03583i	0.855434	+	0.517913i	$0.173291\pi$
$547$			24.2259i			1.03583i	−0.855434	+	0.517913i	$0.826709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	39.0669			1.66431
$552$		0			0
$553$		−	44.5882i		−	1.89608i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			20.7973i			0.881211i	0.897701	+	0.440606i	$0.145236\pi$
$557$			20.7973i			0.881211i	−0.897701	+	0.440606i	$0.854764\pi$
$558$		0			0
$559$	2.94818			0.124695
$560$		0			0
$561$		0			0
$562$		0			0
$563$			10.4382i			0.439918i	0.975509	+	0.219959i	$0.0705924\pi$
$563$			10.4382i			0.439918i	−0.975509	+	0.219959i	$0.929408\pi$
$564$		0			0
$565$	36.6328	+	13.8945i	1.54115	+	0.584546i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.8974			−0.540687			−0.270344	−	0.962764i	$-0.587137\pi$
$569$	−12.8974			−0.540687			−0.270344	+	0.962764i	$0.587137\pi$
$570$		0			0
$571$	−4.59496			−0.192293			−0.0961465	−	0.995367i	$-0.530652\pi$
$571$	−4.59496			−0.192293			−0.0961465	+	0.995367i	$0.530652\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.31589	−	3.74231i	0.138282	−	0.156065i
$576$		0			0
$577$			20.9660i			0.872825i	0.899747	+	0.436413i	$0.143751\pi$
$577$			20.9660i			0.872825i	−0.899747	+	0.436413i	$0.856249\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.09425			0.128371
$582$		0			0
$583$		−	13.5667i		−	0.561873i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			6.60723i			0.272710i	0.990660	+	0.136355i	$0.0435387\pi$
$587$			6.60723i			0.272710i	−0.990660	+	0.136355i	$0.956461\pi$
$588$		0			0
$589$	−72.1277			−2.97197
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.94467i			0.203053i	0.994833	+	0.101527i	$0.0323727\pi$
$593$			4.94467i			0.203053i	−0.994833	+	0.101527i	$0.967627\pi$
$594$		0			0
$595$	−1.83268	+	4.83185i	−0.0751326	+	0.198087i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.1634			−0.864712			−0.432356	−	0.901703i	$-0.642318\pi$
$599$	−21.1634			−0.864712			−0.432356	+	0.901703i	$0.642318\pi$
$600$		0			0
$601$	−6.65198			−0.271340			−0.135670	−	0.990754i	$-0.543319\pi$
$601$	−6.65198			−0.271340			−0.135670	+	0.990754i	$0.543319\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.65050	−	12.2610i	0.189070	−	0.498481i
$606$		0			0
$607$		−	1.87232i		−	0.0759953i	−0.999278	−	0.0379976i	$-0.987902\pi$
$607$		−	1.87232i		−	0.0759953i	0.999278	−	0.0379976i	$-0.0120979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−28.0091			−1.13313
$612$		0			0
$613$		−	29.2195i		−	1.18017i	−0.807342	−	0.590083i	$-0.799095\pi$
$613$		−	29.2195i		−	1.18017i	0.807342	−	0.590083i	$-0.200905\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.3342i		−	0.778365i	−0.921161	−	0.389182i	$-0.872758\pi$
$617$		−	19.3342i		−	0.778365i	0.921161	−	0.389182i	$-0.127242\pi$
$618$		0			0
$619$	44.9507			1.80672			0.903360	−	0.428883i	$-0.141093\pi$
$619$	44.9507			1.80672			0.903360	+	0.428883i	$0.141093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	67.2335i		−	2.69365i
$624$		0			0
$625$	3.00977	+	24.8182i	0.120391	+	0.992727i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.33998			0.173047
$630$		0			0
$631$	−7.86088			−0.312937			−0.156468	−	0.987683i	$-0.550011\pi$
$631$	−7.86088			−0.312937			−0.156468	+	0.987683i	$0.550011\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.8068	−	10.5469i	−1.10348	−	0.418541i
$636$		0			0
$637$		−	26.2291i		−	1.03924i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.20734			0.166180			0.0830900	−	0.996542i	$-0.473521\pi$
$641$	4.20734			0.166180			0.0830900	+	0.996542i	$0.473521\pi$
$642$		0			0
$643$		−	28.2418i		−	1.11375i	−0.830597	−	0.556874i	$-0.812001\pi$
$643$		−	28.2418i		−	1.11375i	0.830597	−	0.556874i	$-0.187999\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			22.7760i			0.895417i	0.894180	+	0.447709i	$0.147760\pi$
$647$			22.7760i			0.895417i	−0.894180	+	0.447709i	$0.852240\pi$
$648$		0			0
$649$	15.8873			0.623631
$650$		0			0
$651$		0			0
$652$		0			0
$653$			13.9316i			0.545187i	0.962129	+	0.272593i	$0.0878813\pi$
$653$			13.9316i			0.545187i	−0.962129	+	0.272593i	$0.912119\pi$
$654$		0			0
$655$	−2.37412	+	6.25936i	−0.0927647	+	0.244573i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−47.9771			−1.86892			−0.934461	−	0.356065i	$-0.884118\pi$
$659$	−47.9771			−1.86892			−0.934461	+	0.356065i	$0.884118\pi$
$660$		0			0
$661$	−4.19509			−0.163170			−0.0815851	−	0.996666i	$-0.525998\pi$
$661$	−4.19509			−0.163170			−0.0815851	+	0.996666i	$0.525998\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	62.4364	+	23.6816i	2.42118	+	0.918334i
$666$		0			0
$667$			5.64598i			0.218613i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.5677			0.523775
$672$		0			0
$673$			25.0494i			0.965585i	0.875735	+	0.482793i	$0.160378\pi$
$673$			25.0494i			0.965585i	−0.875735	+	0.482793i	$0.839622\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			6.13989i			0.235975i	0.993015	+	0.117988i	$0.0376443\pi$
$677$			6.13989i			0.235975i	−0.993015	+	0.117988i	$0.962356\pi$
$678$		0			0
$679$	−70.1317			−2.69141
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28.9930i			1.10939i	0.832055	+	0.554693i	$0.187164\pi$
$683$			28.9930i			1.10939i	−0.832055	+	0.554693i	$0.812836\pi$
$684$		0			0
$685$	36.6982	+	13.9193i	1.40217	+	0.531830i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.5052			0.514506
$690$		0			0
$691$	12.4418			0.473309			0.236654	−	0.971594i	$-0.423949\pi$
$691$	12.4418			0.473309			0.236654	+	0.971594i	$0.423949\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.1736	+	37.3686i	−0.537636	+	1.41747i
$696$		0			0
$697$		−	0.339037i		−	0.0128419i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	49.4593			1.86805			0.934025	−	0.357207i	$-0.116271\pi$
$701$	49.4593			1.86805			0.934025	+	0.357207i	$0.116271\pi$
$702$		0			0
$703$		−	56.0806i		−	2.11512i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			55.4280i			2.08458i
$708$		0			0
$709$	−20.5265			−0.770889			−0.385444	−	0.922731i	$-0.625952\pi$
$709$	−20.5265			−0.770889			−0.385444	+	0.922731i	$0.625952\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	10.4240i		−	0.390380i
$714$		0			0
$715$	−10.6884	−	4.05401i	−0.399723	−	0.151612i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.25057			0.121226			0.0606129	−	0.998161i	$-0.480694\pi$
$719$	3.25057			0.121226			0.0606129	+	0.998161i	$0.480694\pi$
$720$		0			0
$721$	−69.4867			−2.58782
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.1290	−	18.7214i	−0.784712	−	0.695297i
$726$		0			0
$727$			28.9446i			1.07350i	0.843742	+	0.536748i	$0.180348\pi$
$727$			28.9446i			1.07350i	−0.843742	+	0.536748i	$0.819652\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.699807			0.0258833
$732$		0			0
$733$			13.8681i			0.512232i	0.966646	+	0.256116i	$0.0824429\pi$
$733$			13.8681i			0.512232i	−0.966646	+	0.256116i	$0.917557\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	14.3362i		−	0.528079i
$738$		0			0
$739$	−14.4772			−0.532552			−0.266276	−	0.963897i	$-0.585793\pi$
$739$	−14.4772			−0.532552			−0.266276	+	0.963897i	$0.585793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27.4335i			1.00644i	0.864159	+	0.503218i	$0.167851\pi$
$743$			27.4335i			1.00644i	−0.864159	+	0.503218i	$0.832149\pi$
$744$		0			0
$745$	−2.94684	+	7.76931i	−0.107964	+	0.284645i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	57.5460			2.10269
$750$		0			0
$751$	26.1414			0.953913			0.476956	−	0.878927i	$-0.341740\pi$
$751$	26.1414			0.953913			0.476956	+	0.878927i	$0.341740\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.3816	+	30.0075i	−0.414219	+	1.09209i
$756$		0			0
$757$		−	14.8127i		−	0.538379i	−0.963087	−	0.269189i	$-0.913244\pi$
$757$		−	14.8127i		−	0.538379i	0.963087	−	0.269189i	$-0.0867557\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.0330			0.508695			0.254348	−	0.967113i	$-0.418139\pi$
$761$	14.0330			0.508695			0.254348	+	0.967113i	$0.418139\pi$
$762$		0			0
$763$		−	28.5661i		−	1.03416i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			15.8153i			0.571058i
$768$		0			0
$769$	12.8084			0.461881			0.230941	−	0.972968i	$-0.425820\pi$
$769$	12.8084			0.461881			0.230941	+	0.972968i	$0.425820\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	5.10123i		−	0.183479i	−0.995783	−	0.0917393i	$-0.970757\pi$
$773$		−	5.10123i		−	0.183479i	0.995783	−	0.0917393i	$-0.0292427\pi$
$774$		0			0
$775$	39.0097	+	34.5647i	1.40127	+	1.24160i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.38098			−0.156965
$780$		0			0
$781$	−0.342396			−0.0122519
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20.7937	−	7.88688i	−0.742159	−	0.281495i
$786$		0			0
$787$			38.2717i			1.36424i	0.731241	+	0.682119i	$0.238941\pi$
$787$			38.2717i			1.36424i	−0.731241	+	0.682119i	$0.761059\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−75.6209			−2.68877
$792$		0			0
$793$			13.5062i			0.479620i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			47.8787i			1.69595i	0.530037	+	0.847975i	$0.322178\pi$
$797$			47.8787i			1.69595i	−0.530037	+	0.847975i	$0.677822\pi$
$798$		0			0
$799$	−6.64849			−0.235207
$800$		0			0
$801$		0			0
$802$		0			0
$803$			20.1410i			0.710761i
$804$		0			0
$805$	−3.42249	+	9.02336i	−0.120627	+	0.318032i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	29.0384			1.02093			0.510467	−	0.859897i	$-0.329472\pi$
$809$	29.0384			1.02093			0.510467	+	0.859897i	$0.329472\pi$
$810$		0			0
$811$	−54.1645			−1.90197			−0.950986	−	0.309235i	$-0.899927\pi$
$811$	−54.1645			−1.90197			−0.950986	+	0.309235i	$0.899927\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	20.6400	+	7.82860i	0.722989	+	0.274224i
$816$		0			0
$817$		−	9.04280i		−	0.316367i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.7322			−0.828258			−0.414129	−	0.910218i	$-0.635914\pi$
$821$	−23.7322			−0.828258			−0.414129	+	0.910218i	$0.635914\pi$
$822$		0			0
$823$			49.4716i			1.72447i	0.506507	+	0.862236i	$0.330936\pi$
$823$			49.4716i			1.72447i	−0.506507	+	0.862236i	$0.669064\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			36.3893i			1.26538i	0.774405	+	0.632691i	$0.218049\pi$
$827$			36.3893i			1.26538i	−0.774405	+	0.632691i	$0.781951\pi$
$828$		0			0
$829$	−22.0316			−0.765188			−0.382594	−	0.923917i	$-0.624969\pi$
$829$	−22.0316			−0.765188			−0.382594	+	0.923917i	$0.624969\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	6.22599i		−	0.215718i
$834$		0			0
$835$	−6.11489	−	2.31933i	−0.211614	−	0.0802636i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.4980			−0.845765			−0.422882	−	0.906185i	$-0.638982\pi$
$839$	−24.4980			−0.845765			−0.422882	+	0.906185i	$0.638982\pi$
$840$		0			0
$841$	2.87708			0.0992097
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.27332	+	16.5396i	−0.215809	+	0.568978i
$846$		0			0
$847$			25.3104i			0.869674i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.10481			0.277829
$852$		0			0
$853$		−	35.3227i		−	1.20943i	−0.796443	−	0.604713i	$-0.793288\pi$
$853$		−	35.3227i		−	1.20943i	0.796443	−	0.604713i	$-0.206712\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			42.2075i			1.44178i	0.693049	+	0.720891i	$0.256267\pi$
$857$			42.2075i			1.44178i	−0.693049	+	0.720891i	$0.743733\pi$
$858$		0			0
$859$	31.3797			1.07066			0.535331	−	0.844643i	$-0.320187\pi$
$859$	31.3797			1.07066			0.535331	+	0.844643i	$0.320187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			27.0104i			0.919446i	0.888062	+	0.459723i	$0.152051\pi$
$863$			27.0104i			0.919446i	−0.888062	+	0.459723i	$0.847949\pi$
$864$		0			0
$865$	0.880372	+	0.333918i	0.0299335	+	0.0113536i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	23.4122			0.794206
$870$		0			0
$871$	14.2712			0.483561
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.4196	−	42.7285i	−0.757922	−	1.44449i
$876$		0			0
$877$			3.74969i			0.126618i	0.997994	+	0.0633090i	$0.0201654\pi$
$877$			3.74969i			0.126618i	−0.997994	+	0.0633090i	$0.979835\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−48.1065			−1.62075			−0.810375	−	0.585912i	$-0.800737\pi$
$881$	−48.1065			−1.62075			−0.810375	+	0.585912i	$0.800737\pi$
$882$		0			0
$883$		−	48.6331i		−	1.63664i	−0.574766	−	0.818318i	$-0.694907\pi$
$883$		−	48.6331i		−	1.63664i	0.574766	−	0.818318i	$-0.305093\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.9068i		−	1.23921i	−0.784914	−	0.619604i	$-0.787293\pi$
$887$		−	36.9068i		−	1.23921i	0.784914	−	0.619604i	$-0.212707\pi$
$888$		0			0
$889$	57.4016			1.92519
$890$		0			0
$891$		0			0
$892$		0			0
$893$			85.9108i			2.87489i
$894$		0			0
$895$	−11.4156	+	30.0972i	−0.381582	+	1.00604i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−58.8535			−1.96287
$900$		0			0
$901$	3.20572			0.106798
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.98141	+	15.7699i	−0.198829	+	0.524210i
$906$		0			0
$907$		−	24.5797i		−	0.816155i	−0.912947	−	0.408078i	$-0.866199\pi$
$907$		−	24.5797i		−	0.816155i	0.912947	−	0.408078i	$-0.133801\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.64580			0.153922			0.0769611	−	0.997034i	$-0.475478\pi$
$911$	4.64580			0.153922			0.0769611	+	0.997034i	$0.475478\pi$
$912$		0			0
$913$			1.62472i			0.0537703i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	12.9212i		−	0.426695i
$918$		0			0
$919$	−23.6616			−0.780523			−0.390261	−	0.920704i	$-0.627615\pi$
$919$	−23.6616			−0.780523			−0.390261	+	0.920704i	$0.627615\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.340844i		−	0.0112190i
$924$		0			0
$925$	−26.8747	+	30.3307i	−0.883633	+	0.997268i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−45.9814			−1.50860			−0.754301	−	0.656529i	$-0.772024\pi$
$929$	−45.9814			−1.50860			−0.754301	+	0.656529i	$0.772024\pi$
$930$		0			0
$931$	−80.4513			−2.63669
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.53709	−	0.962299i	−0.0829718	−	0.0314705i
$936$		0			0
$937$		−	5.34076i		−	0.174475i	−0.996188	−	0.0872376i	$-0.972196\pi$
$937$		−	5.34076i		−	0.174475i	0.996188	−	0.0872376i	$-0.0278039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	57.4920			1.87419			0.937093	−	0.349079i	$-0.113505\pi$
$941$	57.4920			1.87419			0.937093	+	0.349079i	$0.113505\pi$
$942$		0			0
$943$		−	0.633143i		−	0.0206180i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			34.2375i			1.11257i	0.830992	+	0.556285i	$0.187774\pi$
$947$			34.2375i			1.11257i	−0.830992	+	0.556285i	$0.812226\pi$
$948$		0			0
$949$	−20.0498			−0.650842
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	0.00135437i		−	4.38723e-5i	−1.00000		2.19361e-5i	$-0.999993\pi$
$953$		−	0.00135437i		−	4.38723e-5i	1.00000		2.19361e-5i	$-6.98249e-6\pi$
$954$		0			0
$955$	14.9988	−	39.5442i	0.485350	−	1.27962i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−75.7560			−2.44629
$960$		0			0
$961$	77.6589			2.50513
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.46387	−	2.83099i	−0.240271	−	0.0911327i
$966$		0			0
$967$			36.9612i			1.18859i	0.804246	+	0.594296i	$0.202569\pi$
$967$			36.9612i			1.18859i	−0.804246	+	0.594296i	$0.797431\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−39.1406			−1.25608			−0.628041	−	0.778180i	$-0.716143\pi$
$971$	−39.1406			−1.25608			−0.628041	+	0.778180i	$0.716143\pi$
$972$		0			0
$973$		−	77.1399i		−	2.47299i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18.9260i			0.605498i	0.953070	+	0.302749i	$0.0979044\pi$
$977$			18.9260i			0.605498i	−0.953070	+	0.302749i	$0.902096\pi$
$978$		0			0
$979$	35.3027			1.12828
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	54.1666i		−	1.72765i	−0.503795	−	0.863823i	$-0.668063\pi$
$983$		−	54.1666i		−	1.72765i	0.503795	−	0.863823i	$-0.331937\pi$
$984$		0			0
$985$	3.98654	+	1.51206i	0.127022	+	0.0481783i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.30687			0.0415561
$990$		0			0
$991$	−2.28568			−0.0726069			−0.0363035	−	0.999341i	$-0.511558\pi$
$991$	−2.28568			−0.0726069			−0.0363035	+	0.999341i	$0.511558\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.46569	−	3.86428i	0.0464655	−	0.122506i
$996$		0			0
$997$			2.29171i			0.0725792i	0.999341	+	0.0362896i	$0.0115539\pi$
$997$			2.29171i			0.0725792i	−0.999341	+	0.0362896i	$0.988446\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.6		14
3.2	odd	2		1380.2.f.b.829.12	yes	14
5.4	even	2	inner	4140.2.f.c.829.5		14
15.2	even	4		6900.2.a.bd.1.7		7
15.8	even	4		6900.2.a.bc.1.1		7
15.14	odd	2		1380.2.f.b.829.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.5	✓	14	15.14	odd	2
1380.2.f.b.829.12	yes	14	3.2	odd	2
4140.2.f.c.829.5		14	5.4	even	2	inner
4140.2.f.c.829.6		14	1.1	even	1	trivial
6900.2.a.bc.1.1		7	15.8	even	4
6900.2.a.bd.1.7		7	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.792997 + 2.09073i) q^{5} -4.31589i q^{7} +O(q^{10})\)	\(q+(-0.792997 + 2.09073i) q^{5} -4.31589i q^{7} +2.26617 q^{11} +2.25590i q^{13} +0.535482i q^{17} +6.91942 q^{19} +1.00000i q^{23} +(-3.74231 - 3.31589i) q^{25} +5.64598 q^{29} -10.4240 q^{31} +(9.02336 + 3.42249i) q^{35} -8.10481i q^{37} -0.633143 q^{41} -1.30687i q^{43} +12.4159i q^{47} -11.6269 q^{49} -5.98659i q^{53} +(-1.79707 + 4.73796i) q^{55} +7.01063 q^{59} +5.98705 q^{61} +(-4.71649 - 1.78893i) q^{65} -6.32616i q^{67} -0.151090 q^{71} +8.88768i q^{73} -9.78055i q^{77} +10.3312 q^{79} +0.716944i q^{83} +(-1.11955 - 0.424636i) q^{85} +15.5781 q^{89} +9.73623 q^{91} +(-5.48708 + 14.4666i) q^{95} -16.2497i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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