LMFDB - Embedded newform 4140.2.f.b.829.12

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 $ x^12 + 24x^10 + 188x^8 + 530x^6 + 508x^4 + 80*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.12
Root			$-1.65047i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.17393 + 0.523461i) q^{5} -4.50896i q^{7} +O(q^{10})$	$q+(2.17393 + 0.523461i) q^{5} -4.50896i q^{7} +4.10479 q^{11} -4.10245i q^{13} -2.26588i q^{17} -6.77484 q^{19} -1.00000i q^{23} +(4.45198 + 2.27594i) q^{25} +4.13863 q^{29} +1.84124 q^{31} +(2.36026 - 9.80218i) q^{35} -11.1155i q^{37} -8.36833 q^{41} +5.43473i q^{43} -0.593285i q^{47} -13.3307 q^{49} -1.70512i q^{53} +(8.92353 + 2.14870i) q^{55} +6.19772 q^{59} -11.3814 q^{61} +(2.14747 - 8.91846i) q^{65} +5.78978i q^{67} +11.9915 q^{71} -0.363592i q^{73} -18.5083i q^{77} -1.75692 q^{79} +9.72171i q^{83} +(1.18610 - 4.92587i) q^{85} -17.2208 q^{89} -18.4978 q^{91} +(-14.7281 - 3.54636i) q^{95} -4.38314i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) $ 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * 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
$5$	2.17393	+	0.523461i	0.972213	+	0.234099i
$5$	2.17393	+	0.523461i	0.972213	+	0.234099i
$6$		0			0
$7$		−	4.50896i		−	1.70423i	−0.523357	−	0.852113i	$-0.675321\pi$
$7$		−	4.50896i		−	1.70423i	0.523357	−	0.852113i	$-0.324679\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.10479			1.23764			0.618820	−	0.785533i	$-0.287611\pi$
$11$	4.10479			1.23764			0.618820	+	0.785533i	$0.287611\pi$
$12$		0			0
$13$		−	4.10245i		−	1.13781i	−0.822402	−	0.568907i	$-0.807366\pi$
$13$		−	4.10245i		−	1.13781i	0.822402	−	0.568907i	$-0.192634\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.26588i		−	0.549556i	−0.961508	−	0.274778i	$-0.911396\pi$
$17$		−	2.26588i		−	0.549556i	0.961508	−	0.274778i	$-0.0886044\pi$
$18$		0			0
$19$	−6.77484			−1.55425			−0.777127	−	0.629343i	$-0.783324\pi$
$19$	−6.77484			−1.55425			−0.777127	+	0.629343i	$0.783324\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	4.45198	+	2.27594i	0.890395	+	0.455188i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.13863			0.768525			0.384263	−	0.923224i	$-0.374456\pi$
$29$	4.13863			0.768525			0.384263	+	0.923224i	$0.374456\pi$
$30$		0			0
$31$	1.84124			0.330697			0.165349	−	0.986235i	$-0.447125\pi$
$31$	1.84124			0.330697			0.165349	+	0.986235i	$0.447125\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.36026	−	9.80218i	0.398958	−	1.65687i
$36$		0			0
$37$		−	11.1155i		−	1.82738i	−0.406411	−	0.913690i	$-0.633220\pi$
$37$		−	11.1155i		−	1.82738i	0.406411	−	0.913690i	$-0.366780\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.36833			−1.30691			−0.653457	−	0.756964i	$-0.726682\pi$
$41$	−8.36833			−1.30691			−0.653457	+	0.756964i	$0.726682\pi$
$42$		0			0
$43$			5.43473i			0.828789i	0.910097	+	0.414395i	$0.136007\pi$
$43$			5.43473i			0.828789i	−0.910097	+	0.414395i	$0.863993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.593285i		−	0.0865396i	−0.999063	−	0.0432698i	$-0.986222\pi$
$47$		−	0.593285i		−	0.0865396i	0.999063	−	0.0432698i	$-0.0137775\pi$
$48$		0			0
$49$	−13.3307			−1.90439
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.70512i		−	0.234216i	−0.993119	−	0.117108i	$-0.962638\pi$
$53$		−	1.70512i		−	0.234216i	0.993119	−	0.117108i	$-0.0373624\pi$
$54$		0			0
$55$	8.92353	+	2.14870i	1.20325	+	0.289730i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.19772			0.806874			0.403437	−	0.915007i	$-0.367815\pi$
$59$	6.19772			0.806874			0.403437	+	0.915007i	$0.367815\pi$
$60$		0			0
$61$	−11.3814			−1.45724			−0.728620	−	0.684919i	$-0.759838\pi$
$61$	−11.3814			−1.45724			−0.728620	+	0.684919i	$0.759838\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.14747	−	8.91846i	0.266361	−	1.10620i
$66$		0			0
$67$			5.78978i			0.707335i	0.935371	+	0.353667i	$0.115066\pi$
$67$			5.78978i			0.707335i	−0.935371	+	0.353667i	$0.884934\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.9915			1.42312			0.711562	−	0.702623i	$-0.247988\pi$
$71$	11.9915			1.42312			0.711562	+	0.702623i	$0.247988\pi$
$72$		0			0
$73$		−	0.363592i		−	0.0425552i	−0.999774	−	0.0212776i	$-0.993227\pi$
$73$		−	0.363592i		−	0.0425552i	0.999774	−	0.0212776i	$-0.00677338\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	18.5083i		−	2.10922i
$78$		0			0
$79$	−1.75692			−0.197669			−0.0988344	−	0.995104i	$-0.531511\pi$
$79$	−1.75692			−0.197669			−0.0988344	+	0.995104i	$0.531511\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.72171i			1.06710i	0.845770	+	0.533548i	$0.179142\pi$
$83$			9.72171i			1.06710i	−0.845770	+	0.533548i	$0.820858\pi$
$84$		0			0
$85$	1.18610	−	4.92587i	0.128650	−	0.534286i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.2208			−1.82540			−0.912698	−	0.408634i	$-0.866005\pi$
$89$	−17.2208			−1.82540			−0.912698	+	0.408634i	$0.866005\pi$
$90$		0			0
$91$	−18.4978			−1.93909
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.7281	−	3.54636i	−1.51107	−	0.363849i
$96$		0			0
$97$		−	4.38314i		−	0.445040i	−0.974928	−	0.222520i	$-0.928572\pi$
$97$		−	4.38314i		−	0.445040i	0.974928	−	0.222520i	$-0.0714283\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.21428			0.319833			0.159917	−	0.987131i	$-0.448877\pi$
$101$	3.21428			0.319833			0.159917	+	0.987131i	$0.448877\pi$
$102$		0			0
$103$		−	2.09140i		−	0.206072i	−0.994678	−	0.103036i	$-0.967144\pi$
$103$		−	2.09140i		−	0.206072i	0.994678	−	0.103036i	$-0.0328556\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.81411i		−	0.175377i	−0.996148	−	0.0876885i	$-0.972052\pi$
$107$		−	1.81411i		−	0.175377i	0.996148	−	0.0876885i	$-0.0279480\pi$
$108$		0			0
$109$	−3.16716			−0.303359			−0.151679	−	0.988430i	$-0.548468\pi$
$109$	−3.16716			−0.303359			−0.151679	+	0.988430i	$0.548468\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.0407i			1.22677i	0.789785	+	0.613384i	$0.210192\pi$
$113$			13.0407i			1.22677i	−0.789785	+	0.613384i	$0.789808\pi$
$114$		0			0
$115$	0.523461	−	2.17393i	0.0488130	−	0.202720i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.2168			−0.936568
$120$		0			0
$121$	5.84927			0.531752
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.48694	+	7.27818i	0.759095	+	0.650980i
$126$		0			0
$127$			12.0290i			1.06740i	0.845674	+	0.533700i	$0.179199\pi$
$127$			12.0290i			1.06740i	−0.845674	+	0.533700i	$0.820801\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.33814			−0.116914			−0.0584571	−	0.998290i	$-0.518618\pi$
$131$	−1.33814			−0.116914			−0.0584571	+	0.998290i	$0.518618\pi$
$132$		0			0
$133$			30.5475i			2.64880i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.53176i			0.728917i	0.931220	+	0.364459i	$0.118746\pi$
$137$			8.53176i			0.728917i	−0.931220	+	0.364459i	$0.881254\pi$
$138$		0			0
$139$	19.2021			1.62870			0.814352	−	0.580371i	$-0.197092\pi$
$139$	19.2021			1.62870			0.814352	+	0.580371i	$0.197092\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	16.8397i		−	1.40820i
$144$		0			0
$145$	8.99712	+	2.16641i	0.747170	+	0.179911i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.9657			−1.14412			−0.572058	−	0.820213i	$-0.693855\pi$
$149$	−13.9657			−1.14412			−0.572058	+	0.820213i	$0.693855\pi$
$150$		0			0
$151$	−9.38572			−0.763799			−0.381900	−	0.924204i	$-0.624730\pi$
$151$	−9.38572			−0.763799			−0.381900	+	0.924204i	$0.624730\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00274	+	0.963819i	0.321508	+	0.0774158i
$156$		0			0
$157$		−	22.3649i		−	1.78491i	−0.451133	−	0.892457i	$-0.648980\pi$
$157$		−	22.3649i		−	1.78491i	0.451133	−	0.892457i	$-0.351020\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.50896			−0.355356
$162$		0			0
$163$		−	5.91327i		−	0.463163i	−0.972816	−	0.231581i	$-0.925610\pi$
$163$		−	5.91327i		−	0.463163i	0.972816	−	0.231581i	$-0.0743900\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.73957i		−	0.676288i	−0.941094	−	0.338144i	$-0.890201\pi$
$167$		−	8.73957i		−	0.676288i	0.941094	−	0.338144i	$-0.109799\pi$
$168$		0			0
$169$	−3.83010			−0.294623
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	10.5284i		−	0.800462i	−0.916414	−	0.400231i	$-0.868930\pi$
$173$		−	10.5284i		−	0.800462i	0.916414	−	0.400231i	$-0.131070\pi$
$174$		0			0
$175$	10.2621	−	20.0738i	0.775743	−	1.51744i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.29655			−0.246396			−0.123198	−	0.992382i	$-0.539315\pi$
$179$	−3.29655			−0.246396			−0.123198	+	0.992382i	$0.539315\pi$
$180$		0			0
$181$	15.7994			1.17436			0.587180	−	0.809456i	$-0.300238\pi$
$181$	15.7994			1.17436			0.587180	+	0.809456i	$0.300238\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.81854	−	24.1644i	0.427788	−	1.77660i
$186$		0			0
$187$		−	9.30095i		−	0.680153i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.9965			1.44689			0.723447	−	0.690380i	$-0.242557\pi$
$191$	19.9965			1.44689			0.723447	+	0.690380i	$0.242557\pi$
$192$		0			0
$193$		−	0.613407i		−	0.0441540i	−0.999756	−	0.0220770i	$-0.992972\pi$
$193$		−	0.613407i		−	0.0441540i	0.999756	−	0.0220770i	$-0.00702790\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.63996i			0.330584i	0.986245	+	0.165292i	$0.0528566\pi$
$197$			4.63996i			0.330584i	−0.986245	+	0.165292i	$0.947143\pi$
$198$		0			0
$199$	13.1446			0.931795			0.465898	−	0.884839i	$-0.345731\pi$
$199$	13.1446			0.931795			0.465898	+	0.884839i	$0.345731\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	18.6609i		−	1.30974i
$204$		0			0
$205$	−18.1922	−	4.38049i	−1.27060	−	0.305947i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−27.8093			−1.92361
$210$		0			0
$211$	−6.76863			−0.465972			−0.232986	−	0.972480i	$-0.574850\pi$
$211$	−6.76863			−0.465972			−0.232986	+	0.972480i	$0.574850\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.84487	+	11.8148i	−0.194019	+	0.805759i
$216$		0			0
$217$		−	8.30209i		−	0.563583i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.29565			−0.625293
$222$		0			0
$223$		−	5.46851i		−	0.366199i	−0.983094	−	0.183099i	$-0.941387\pi$
$223$		−	5.46851i		−	0.366199i	0.983094	−	0.183099i	$-0.0586130\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	5.55927i		−	0.368981i	−0.982834	−	0.184491i	$-0.940936\pi$
$227$		−	5.55927i		−	0.368981i	0.982834	−	0.184491i	$-0.0590636\pi$
$228$		0			0
$229$	6.33693			0.418756			0.209378	−	0.977835i	$-0.432856\pi$
$229$	6.33693			0.418756			0.209378	+	0.977835i	$0.432856\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.84557i		−	0.645005i	−0.946569	−	0.322502i	$-0.895476\pi$
$233$		−	9.84557i		−	0.645005i	0.946569	−	0.322502i	$-0.104524\pi$
$234$		0			0
$235$	0.310562	−	1.28976i	0.0202588	−	0.0841349i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.9526			1.67873			0.839366	−	0.543567i	$-0.182926\pi$
$239$	25.9526			1.67873			0.839366	+	0.543567i	$0.182926\pi$
$240$		0			0
$241$	−12.3220			−0.793729			−0.396865	−	0.917877i	$-0.629902\pi$
$241$	−12.3220			−0.793729			−0.396865	+	0.917877i	$0.629902\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−28.9801	−	6.97811i	−1.85147	−	0.445815i
$246$		0			0
$247$			27.7934i			1.76845i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.68797			0.485261			0.242630	−	0.970119i	$-0.421990\pi$
$251$	7.68797			0.485261			0.242630	+	0.970119i	$0.421990\pi$
$252$		0			0
$253$		−	4.10479i		−	0.258066i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.7126i		−	1.22964i	−0.788669	−	0.614818i	$-0.789229\pi$
$257$		−	19.7126i		−	1.22964i	0.788669	−	0.614818i	$-0.210771\pi$
$258$		0			0
$259$	−50.1195			−3.11427
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.96836i			0.183037i	0.995803	+	0.0915183i	$0.0291720\pi$
$263$			2.96836i			0.183037i	−0.995803	+	0.0915183i	$0.970828\pi$
$264$		0			0
$265$	0.892564	−	3.70682i	0.0548297	−	0.227708i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.417916			0.0254808			0.0127404	−	0.999919i	$-0.495944\pi$
$269$	0.417916			0.0254808			0.0127404	+	0.999919i	$0.495944\pi$
$270$		0			0
$271$	10.1776			0.618245			0.309122	−	0.951022i	$-0.399965\pi$
$271$	10.1776			0.618245			0.309122	+	0.951022i	$0.399965\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.2744	+	9.34224i	1.10199	+	0.563358i
$276$		0			0
$277$		−	18.8176i		−	1.13064i	−0.824871	−	0.565321i	$-0.808752\pi$
$277$		−	18.8176i		−	1.13064i	0.824871	−	0.565321i	$-0.191248\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	25.9609			1.54870			0.774350	−	0.632757i	$-0.218077\pi$
$281$	25.9609			1.54870			0.774350	+	0.632757i	$0.218077\pi$
$282$		0			0
$283$			22.2612i			1.32329i	0.749817	+	0.661646i	$0.230142\pi$
$283$			22.2612i			1.32329i	−0.749817	+	0.661646i	$0.769858\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			37.7325i			2.22728i
$288$		0			0
$289$	11.8658			0.697988
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.21032i		−	0.129128i	−0.997914	−	0.0645641i	$-0.979434\pi$
$293$		−	2.21032i		−	0.129128i	0.997914	−	0.0645641i	$-0.0205657\pi$
$294$		0			0
$295$	13.4734	+	3.24426i	0.784453	+	0.188888i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.10245			−0.237251
$300$		0			0
$301$	24.5050			1.41244
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.7424	−	5.95772i	−1.41675	−	0.341138i
$306$		0			0
$307$		−	19.7730i		−	1.12851i	−0.825602	−	0.564253i	$-0.809164\pi$
$307$		−	19.7730i		−	1.12851i	0.825602	−	0.564253i	$-0.190836\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.1593			−0.689490			−0.344745	−	0.938696i	$-0.612035\pi$
$311$	−12.1593			−0.689490			−0.344745	+	0.938696i	$0.612035\pi$
$312$		0			0
$313$		−	34.0793i		−	1.92628i	−0.269003	−	0.963139i	$-0.586694\pi$
$313$		−	34.0793i		−	1.92628i	0.269003	−	0.963139i	$-0.413306\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			3.66161i			0.205657i	0.994699	+	0.102828i	$0.0327892\pi$
$317$			3.66161i			0.205657i	−0.994699	+	0.102828i	$0.967211\pi$
$318$		0			0
$319$	16.9882			0.951157
$320$		0			0
$321$		0			0
$322$		0			0
$323$			15.3510i			0.854150i
$324$		0			0
$325$	9.33693	−	18.2640i	0.517920	−	1.01311i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.67510			−0.147483
$330$		0			0
$331$	3.50062			0.192412			0.0962058	−	0.995361i	$-0.469329\pi$
$331$	3.50062			0.192412			0.0962058	+	0.995361i	$0.469329\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.03073	+	12.5866i	−0.165586	+	0.687680i
$336$		0			0
$337$			33.3643i			1.81747i	0.417373	+	0.908735i	$0.362951\pi$
$337$			33.3643i			1.81747i	−0.417373	+	0.908735i	$0.637049\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.55791			0.409284
$342$		0			0
$343$			28.5450i			1.54128i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5.80416i			0.311584i	0.987790	+	0.155792i	$0.0497929\pi$
$347$			5.80416i			0.311584i	−0.987790	+	0.155792i	$0.950207\pi$
$348$		0			0
$349$	0.496058			0.0265534			0.0132767	−	0.999912i	$-0.495774\pi$
$349$	0.496058			0.0265534			0.0132767	+	0.999912i	$0.495774\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			5.80547i			0.308994i	0.987993	+	0.154497i	$0.0493757\pi$
$353$			5.80547i			0.308994i	−0.987993	+	0.154497i	$0.950624\pi$
$354$		0			0
$355$	26.0686	+	6.27706i	1.38358	+	0.333152i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.3473			1.49611			0.748057	−	0.663635i	$-0.230987\pi$
$359$	28.3473			1.49611			0.748057	+	0.663635i	$0.230987\pi$
$360$		0			0
$361$	26.8984			1.41571
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.190326	−	0.790424i	0.00996212	−	0.0413727i
$366$		0			0
$367$		−	24.2615i		−	1.26644i	−0.773972	−	0.633219i	$-0.781733\pi$
$367$		−	24.2615i		−	1.26644i	0.773972	−	0.633219i	$-0.218267\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.68832			−0.399158
$372$		0			0
$373$			22.6261i			1.17153i	0.810479	+	0.585767i	$0.199207\pi$
$373$			22.6261i			1.17153i	−0.810479	+	0.585767i	$0.800793\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16.9785i		−	0.874439i
$378$		0			0
$379$	−6.43791			−0.330693			−0.165347	−	0.986236i	$-0.552874\pi$
$379$	−6.43791			−0.330693			−0.165347	+	0.986236i	$0.552874\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	14.5145i		−	0.741657i	−0.928701	−	0.370829i	$-0.879074\pi$
$383$		−	14.5145i		−	0.741657i	0.928701	−	0.370829i	$-0.120926\pi$
$384$		0			0
$385$	9.68838	−	40.2359i	0.493766	−	2.05061i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−26.1850			−1.32763			−0.663815	−	0.747897i	$-0.731064\pi$
$389$	−26.1850			−1.32763			−0.663815	+	0.747897i	$0.731064\pi$
$390$		0			0
$391$	−2.26588			−0.114590
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.81942	−	0.919678i	−0.192176	−	0.0462740i
$396$		0			0
$397$		−	20.6074i		−	1.03426i	−0.855908	−	0.517129i	$-0.827001\pi$
$397$		−	20.6074i		−	1.03426i	0.855908	−	0.517129i	$-0.172999\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.4900			−0.623719			−0.311859	−	0.950128i	$-0.600952\pi$
$401$	−12.4900			−0.623719			−0.311859	+	0.950128i	$0.600952\pi$
$402$		0			0
$403$		−	7.55361i		−	0.376272i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	45.6268i		−	2.26164i
$408$		0			0
$409$	−9.11740			−0.450826			−0.225413	−	0.974263i	$-0.572373\pi$
$409$	−9.11740			−0.450826			−0.225413	+	0.974263i	$0.572373\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	27.9453i		−	1.37510i
$414$		0			0
$415$	−5.08894	+	21.1344i	−0.249806	+	1.03744i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.12358			−0.348010			−0.174005	−	0.984745i	$-0.555671\pi$
$419$	−7.12358			−0.348010			−0.174005	+	0.984745i	$0.555671\pi$
$420$		0			0
$421$	7.13707			0.347840			0.173920	−	0.984760i	$-0.444357\pi$
$421$	7.13707			0.347840			0.173920	+	0.984760i	$0.444357\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.15700	−	10.0876i	0.250151	−	0.489322i
$426$		0			0
$427$			51.3183i			2.48347i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.2840			0.688036			0.344018	−	0.938963i	$-0.388212\pi$
$431$	14.2840			0.688036			0.344018	+	0.938963i	$0.388212\pi$
$432$		0			0
$433$		−	17.5102i		−	0.841485i	−0.907180	−	0.420743i	$-0.861770\pi$
$433$		−	17.5102i		−	0.841485i	0.907180	−	0.420743i	$-0.138230\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.77484i			0.324084i
$438$		0			0
$439$	−0.991828			−0.0473374			−0.0236687	−	0.999720i	$-0.507535\pi$
$439$	−0.991828			−0.0473374			−0.0236687	+	0.999720i	$0.507535\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.99095i		−	0.142105i	−0.997473	−	0.0710523i	$-0.977364\pi$
$443$		−	2.99095i		−	0.142105i	0.997473	−	0.0710523i	$-0.0226357\pi$
$444$		0			0
$445$	−37.4368	−	9.01440i	−1.77467	−	0.427323i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.8981			0.986244			0.493122	−	0.869960i	$-0.335856\pi$
$449$	20.8981			0.986244			0.493122	+	0.869960i	$0.335856\pi$
$450$		0			0
$451$	−34.3502			−1.61749
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−40.2130	−	9.68287i	−1.88521	−	0.453940i
$456$		0			0
$457$			33.4482i			1.56464i	0.622877	+	0.782320i	$0.285964\pi$
$457$			33.4482i			1.56464i	−0.622877	+	0.782320i	$0.714036\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.53331			−0.397436			−0.198718	−	0.980057i	$-0.563678\pi$
$461$	−8.53331			−0.397436			−0.198718	+	0.980057i	$0.563678\pi$
$462$		0			0
$463$			13.5937i			0.631751i	0.948801	+	0.315876i	$0.102298\pi$
$463$			13.5937i			0.631751i	−0.948801	+	0.315876i	$0.897702\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.3442i			1.45044i	0.688518	+	0.725219i	$0.258262\pi$
$467$			31.3442i			1.45044i	−0.688518	+	0.725219i	$0.741738\pi$
$468$		0			0
$469$	26.1059			1.20546
$470$		0			0
$471$		0			0
$472$		0			0
$473$			22.3084i			1.02574i
$474$		0			0
$475$	−30.1614	−	15.4191i	−1.38390	−	0.707478i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.0701			0.917027			0.458513	−	0.888687i	$-0.348382\pi$
$479$	20.0701			0.917027			0.458513	+	0.888687i	$0.348382\pi$
$480$		0			0
$481$	−45.6009			−2.07922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.29440	−	9.52866i	0.104183	−	0.432674i
$486$		0			0
$487$		−	26.8679i		−	1.21750i	−0.793362	−	0.608750i	$-0.791671\pi$
$487$		−	26.8679i		−	1.21750i	0.793362	−	0.608750i	$-0.208329\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−33.2592			−1.50097			−0.750483	−	0.660890i	$-0.770179\pi$
$491$	−33.2592			−1.50097			−0.750483	+	0.660890i	$0.770179\pi$
$492$		0			0
$493$		−	9.37764i		−	0.422348i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	54.0690i		−	2.42533i
$498$		0			0
$499$	24.4726			1.09554			0.547772	−	0.836627i	$-0.315476\pi$
$499$	24.4726			1.09554			0.547772	+	0.836627i	$0.315476\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			39.8607i			1.77730i	0.458585	+	0.888650i	$0.348356\pi$
$503$			39.8607i			1.77730i	−0.458585	+	0.888650i	$0.651644\pi$
$504$		0			0
$505$	6.98764	+	1.68255i	0.310946	+	0.0748726i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.4456			0.906237			0.453119	−	0.891450i	$-0.350311\pi$
$509$	20.4456			0.906237			0.453119	+	0.891450i	$0.350311\pi$
$510$		0			0
$511$	−1.63942			−0.0725237
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.09477	−	4.54656i	0.0482411	−	0.200345i
$516$		0			0
$517$		−	2.43531i		−	0.107105i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.3985			1.33178			0.665892	−	0.746048i	$-0.268051\pi$
$521$	30.3985			1.33178			0.665892	+	0.746048i	$0.268051\pi$
$522$		0			0
$523$		−	4.27912i		−	0.187113i	−0.995614	−	0.0935564i	$-0.970176\pi$
$523$		−	4.27912i		−	0.187113i	0.995614	−	0.0935564i	$-0.0298235\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4.17203i		−	0.181737i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			34.3306i			1.48703i
$534$		0			0
$535$	0.949618	−	3.94376i	0.0410556	−	0.170504i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−54.7198			−2.35695
$540$		0			0
$541$	−3.60876			−0.155153			−0.0775764	−	0.996986i	$-0.524718\pi$
$541$	−3.60876			−0.155153			−0.0775764	+	0.996986i	$0.524718\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.88519	−	1.65788i	−0.294929	−	0.0710159i
$546$		0			0
$547$		−	30.6519i		−	1.31058i	−0.755377	−	0.655290i	$-0.772546\pi$
$547$		−	30.6519i		−	1.31058i	0.755377	−	0.655290i	$-0.227454\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.0386			−1.19448
$552$		0			0
$553$			7.92187i			0.336872i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	45.9504i		−	1.94698i	−0.228725	−	0.973491i	$-0.573456\pi$
$557$		−	45.9504i		−	1.94698i	0.228725	−	0.973491i	$-0.426544\pi$
$558$		0			0
$559$	22.2957			0.943009
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.30591i		−	0.181472i	−0.995875	−	0.0907362i	$-0.971078\pi$
$563$		−	4.30591i		−	0.181472i	0.995875	−	0.0907362i	$-0.0289220\pi$
$564$		0			0
$565$	−6.82631	+	28.3497i	−0.287185	+	1.19268i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.4627			1.23514			0.617571	−	0.786515i	$-0.288117\pi$
$569$	29.4627			1.23514			0.617571	+	0.786515i	$0.288117\pi$
$570$		0			0
$571$	−31.7317			−1.32793			−0.663966	−	0.747763i	$-0.731128\pi$
$571$	−31.7317			−1.32793			−0.663966	+	0.747763i	$0.731128\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.27594	−	4.45198i	0.0949132	−	0.185660i
$576$		0			0
$577$		−	24.1386i		−	1.00490i	−0.864606	−	0.502451i	$-0.832432\pi$
$577$		−	24.1386i		−	1.00490i	0.864606	−	0.502451i	$-0.167568\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	43.8348			1.81857
$582$		0			0
$583$		−	6.99915i		−	0.289875i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			33.9551i			1.40148i	0.713419	+	0.700738i	$0.247146\pi$
$587$			33.9551i			1.40148i	−0.713419	+	0.700738i	$0.752854\pi$
$588$		0			0
$589$	−12.4741			−0.513988
$590$		0			0
$591$		0			0
$592$		0			0
$593$			6.84201i			0.280968i	0.990083	+	0.140484i	$0.0448658\pi$
$593$			6.84201i			0.280968i	−0.990083	+	0.140484i	$0.955134\pi$
$594$		0			0
$595$	−22.2105	−	5.34807i	−0.910544	−	0.219250i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.6172			1.08755			0.543775	−	0.839231i	$-0.316995\pi$
$599$	26.6172			1.08755			0.543775	+	0.839231i	$0.316995\pi$
$600$		0			0
$601$	15.0702			0.614728			0.307364	−	0.951592i	$-0.400553\pi$
$601$	15.0702			0.614728			0.307364	+	0.951592i	$0.400553\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.7159	+	3.06186i	0.516976	+	0.124482i
$606$		0			0
$607$		−	43.4205i		−	1.76238i	−0.472761	−	0.881191i	$-0.656742\pi$
$607$		−	43.4205i		−	1.76238i	0.472761	−	0.881191i	$-0.343258\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.43392			−0.0984660
$612$		0			0
$613$			4.59633i			0.185644i	0.995683	+	0.0928220i	$0.0295888\pi$
$613$			4.59633i			0.185644i	−0.995683	+	0.0928220i	$0.970411\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			30.4809i			1.22712i	0.789650	+	0.613558i	$0.210262\pi$
$617$			30.4809i			1.22712i	−0.789650	+	0.613558i	$0.789738\pi$
$618$		0			0
$619$	37.0511			1.48921			0.744605	−	0.667506i	$-0.232638\pi$
$619$	37.0511			1.48921			0.744605	+	0.667506i	$0.232638\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			77.6477i			3.11089i
$624$		0			0
$625$	14.6402	+	20.2649i	0.585608	+	0.810594i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25.1864			−1.00425
$630$		0			0
$631$	−29.3035			−1.16655			−0.583276	−	0.812274i	$-0.698230\pi$
$631$	−29.3035			−1.16655			−0.583276	+	0.812274i	$0.698230\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.29671	+	26.1502i	−0.249877	+	1.03774i
$636$		0			0
$637$			54.6886i			2.16684i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.3307			0.447536			0.223768	−	0.974642i	$-0.428164\pi$
$641$	11.3307			0.447536			0.223768	+	0.974642i	$0.428164\pi$
$642$		0			0
$643$			28.2340i			1.11344i	0.830700	+	0.556721i	$0.187941\pi$
$643$			28.2340i			1.11344i	−0.830700	+	0.556721i	$0.812059\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.0192i			1.18018i	0.807338	+	0.590089i	$0.200907\pi$
$647$			30.0192i			1.18018i	−0.807338	+	0.590089i	$0.799093\pi$
$648$		0			0
$649$	25.4403			0.998619
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	23.7192i		−	0.928204i	−0.885782	−	0.464102i	$-0.846377\pi$
$653$		−	23.7192i		−	0.928204i	0.885782	−	0.464102i	$-0.153623\pi$
$654$		0			0
$655$	−2.90904	−	0.700467i	−0.113666	−	0.0273695i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.8488			−0.656336			−0.328168	−	0.944619i	$-0.606431\pi$
$659$	−16.8488			−0.656336			−0.328168	+	0.944619i	$0.606431\pi$
$660$		0			0
$661$	−23.4941			−0.913816			−0.456908	−	0.889514i	$-0.651043\pi$
$661$	−23.4941			−0.913816			−0.456908	+	0.889514i	$0.651043\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.9904	+	66.4082i	−0.620082	+	2.57520i
$666$		0			0
$667$		−	4.13863i		−	0.160249i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−46.7182			−1.80354
$672$		0			0
$673$		−	19.9329i		−	0.768358i	−0.923259	−	0.384179i	$-0.874485\pi$
$673$		−	19.9329i		−	0.768358i	0.923259	−	0.384179i	$-0.125515\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	33.9965i		−	1.30659i	−0.757102	−	0.653296i	$-0.773386\pi$
$677$		−	33.9965i		−	1.30659i	0.757102	−	0.653296i	$-0.226614\pi$
$678$		0			0
$679$	−19.7634			−0.758450
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	11.8223i		−	0.452367i	−0.974085	−	0.226183i	$-0.927375\pi$
$683$		−	11.8223i		−	0.452367i	0.974085	−	0.226183i	$-0.0726248\pi$
$684$		0			0
$685$	−4.46604	+	18.5475i	−0.170639	+	0.708663i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.99517			−0.266495
$690$		0			0
$691$	−39.0765			−1.48654			−0.743271	−	0.668991i	$-0.766727\pi$
$691$	−39.0765			−1.48654			−0.743271	+	0.668991i	$0.766727\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	41.7442	+	10.0516i	1.58345	+	0.381278i
$696$		0			0
$697$			18.9616i			0.718222i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36.3454			1.37275			0.686373	−	0.727250i	$-0.259202\pi$
$701$	36.3454			1.37275			0.686373	+	0.727250i	$0.259202\pi$
$702$		0			0
$703$			75.3059i			2.84021i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14.4931i		−	0.545068i
$708$		0			0
$709$	16.6781			0.626359			0.313179	−	0.949694i	$-0.398606\pi$
$709$	16.6781			0.626359			0.313179	+	0.949694i	$0.398606\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.84124i		−	0.0689551i
$714$		0			0
$715$	8.81492	−	36.6084i	0.329659	−	1.36907i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.1134			−0.899278			−0.449639	−	0.893210i	$-0.648447\pi$
$719$	−24.1134			−0.899278			−0.449639	+	0.893210i	$0.648447\pi$
$720$		0			0
$721$	−9.43003			−0.351193
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.4251	+	9.41928i	0.684291	+	0.349823i
$726$		0			0
$727$		−	3.05594i		−	0.113339i	−0.998393	−	0.0566693i	$-0.981952\pi$
$727$		−	3.05594i		−	0.113339i	0.998393	−	0.0566693i	$-0.0180481\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.3144			0.455466
$732$		0			0
$733$			49.2230i			1.81809i	0.416696	+	0.909046i	$0.363188\pi$
$733$			49.2230i			1.81809i	−0.416696	+	0.909046i	$0.636812\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			23.7658i			0.875425i
$738$		0			0
$739$	−13.1179			−0.482548			−0.241274	−	0.970457i	$-0.577565\pi$
$739$	−13.1179			−0.482548			−0.241274	+	0.970457i	$0.577565\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			14.9149i			0.547173i	0.961847	+	0.273587i	$0.0882100\pi$
$743$			14.9149i			0.547173i	−0.961847	+	0.273587i	$0.911790\pi$
$744$		0			0
$745$	−30.3605	−	7.31050i	−1.11232	−	0.267836i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.17977			−0.298882
$750$		0			0
$751$	7.40642			0.270264			0.135132	−	0.990828i	$-0.456854\pi$
$751$	7.40642			0.270264			0.135132	+	0.990828i	$0.456854\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.4039	−	4.91306i	−0.742575	−	0.178804i
$756$		0			0
$757$			1.37246i			0.0498831i	0.999689	+	0.0249415i	$0.00793996\pi$
$757$			1.37246i			0.0498831i	−0.999689	+	0.0249415i	$0.992060\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.9083			1.22918			0.614588	−	0.788849i	$-0.289322\pi$
$761$	33.9083			1.22918			0.614588	+	0.788849i	$0.289322\pi$
$762$		0			0
$763$			14.2806i			0.516992i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	25.4258i		−	0.918073i
$768$		0			0
$769$	−7.17267			−0.258653			−0.129327	−	0.991602i	$-0.541282\pi$
$769$	−7.17267			−0.258653			−0.129327	+	0.991602i	$0.541282\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			3.13634i			0.112806i	0.998408	+	0.0564031i	$0.0179632\pi$
$773$			3.13634i			0.112806i	−0.998408	+	0.0564031i	$0.982037\pi$
$774$		0			0
$775$	8.19718	+	4.19056i	0.294451	+	0.150529i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	56.6941			2.03128
$780$		0			0
$781$	49.2224			1.76131
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.7072	−	48.6198i	0.417846	−	1.73532i
$786$		0			0
$787$		−	0.546252i		−	0.0194718i	−0.999953	−	0.00973589i	$-0.996901\pi$
$787$		−	0.546252i		−	0.0194718i	0.999953	−	0.00973589i	$-0.00309908\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	58.8001			2.09069
$792$		0			0
$793$			46.6916i			1.65807i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			45.0453i			1.59559i	0.602931	+	0.797794i	$0.294000\pi$
$797$			45.0453i			1.59559i	−0.602931	+	0.797794i	$0.706000\pi$
$798$		0			0
$799$	−1.34431			−0.0475584
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.49247i		−	0.0526680i
$804$		0			0
$805$	−9.80218	−	2.36026i	−0.345481	−	0.0831884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.0227			1.47744			0.738719	−	0.674013i	$-0.235431\pi$
$809$	42.0227			1.47744			0.738719	+	0.674013i	$0.235431\pi$
$810$		0			0
$811$	14.9218			0.523974			0.261987	−	0.965071i	$-0.415622\pi$
$811$	14.9218			0.523974			0.261987	+	0.965071i	$0.415622\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.09537	−	12.8551i	0.108426	−	0.450293i
$816$		0			0
$817$		−	36.8194i		−	1.28815i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32.2317			−1.12489			−0.562447	−	0.826833i	$-0.690140\pi$
$821$	−32.2317			−1.12489			−0.562447	+	0.826833i	$0.690140\pi$
$822$		0			0
$823$			31.3609i			1.09317i	0.837403	+	0.546586i	$0.184073\pi$
$823$			31.3609i			1.09317i	−0.837403	+	0.546586i	$0.815927\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			29.6787i			1.03203i	0.856580	+	0.516015i	$0.172585\pi$
$827$			29.6787i			1.03203i	−0.856580	+	0.516015i	$0.827415\pi$
$828$		0			0
$829$	−17.4868			−0.607342			−0.303671	−	0.952777i	$-0.598212\pi$
$829$	−17.4868			−0.607342			−0.303671	+	0.952777i	$0.598212\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			30.2058i			1.04657i
$834$		0			0
$835$	4.57482	−	18.9992i	0.158318	−	0.657496i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.2177			0.767039			0.383520	−	0.923533i	$-0.374712\pi$
$839$	22.2177			0.767039			0.383520	+	0.923533i	$0.374712\pi$
$840$		0			0
$841$	−11.8717			−0.409369
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.32638	−	2.00491i	−0.286436	−	0.0689709i
$846$		0			0
$847$		−	26.3741i		−	0.906225i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.1155			−0.381035
$852$		0			0
$853$			26.5885i			0.910374i	0.890396	+	0.455187i	$0.150428\pi$
$853$			26.5885i			0.910374i	−0.890396	+	0.455187i	$0.849572\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			21.0394i			0.718692i	0.933204	+	0.359346i	$0.117000\pi$
$857$			21.0394i			0.718692i	−0.933204	+	0.359346i	$0.883000\pi$
$858$		0			0
$859$	42.7924			1.46006			0.730028	−	0.683417i	$-0.239507\pi$
$859$	42.7924			1.46006			0.730028	+	0.683417i	$0.239507\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44.5471i			1.51640i	0.652022	+	0.758200i	$0.273921\pi$
$863$			44.5471i			1.51640i	−0.652022	+	0.758200i	$0.726079\pi$
$864$		0			0
$865$	5.51123	−	22.8881i	0.187387	−	0.778220i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.21177			−0.244643
$870$		0			0
$871$	23.7523			0.804816
$872$		0			0
$873$		0			0
$874$		0			0
$875$	32.8170	−	38.2673i	1.10942	−	1.29367i
$876$		0			0
$877$			12.1814i			0.411336i	0.978622	+	0.205668i	$0.0659367\pi$
$877$			12.1814i			0.411336i	−0.978622	+	0.205668i	$0.934063\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.3075			0.515723			0.257862	−	0.966182i	$-0.416982\pi$
$881$	15.3075			0.515723			0.257862	+	0.966182i	$0.416982\pi$
$882$		0			0
$883$			24.1308i			0.812065i	0.913859	+	0.406033i	$0.133088\pi$
$883$			24.1308i			0.812065i	−0.913859	+	0.406033i	$0.866912\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.9353i		−	0.602210i	−0.953591	−	0.301105i	$-0.902645\pi$
$887$		−	17.9353i		−	0.602210i	0.953591	−	0.301105i	$-0.0973555\pi$
$888$		0			0
$889$	54.2383			1.81909
$890$		0			0
$891$		0			0
$892$		0			0
$893$			4.01941i			0.134505i
$894$		0			0
$895$	−7.16648	−	1.72561i	−0.239549	−	0.0576809i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.62024			0.254149
$900$		0			0
$901$	−3.86359			−0.128715
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.3469	+	8.27037i	1.14173	+	0.274916i
$906$		0			0
$907$			52.7750i			1.75236i	0.481980	+	0.876182i	$0.339918\pi$
$907$			52.7750i			1.75236i	−0.481980	+	0.876182i	$0.660082\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.2848			−0.771461			−0.385731	−	0.922611i	$-0.626051\pi$
$911$	−23.2848			−0.771461			−0.385731	+	0.922611i	$0.626051\pi$
$912$		0			0
$913$			39.9055i			1.32068i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.03364i			0.199248i
$918$		0			0
$919$	9.79940			0.323253			0.161626	−	0.986852i	$-0.448326\pi$
$919$	9.79940			0.323253			0.161626	+	0.986852i	$0.448326\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	49.1944i		−	1.61925i
$924$		0			0
$925$	25.2983	−	49.4861i	0.831802	−	1.62709i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.0707			−1.08501			−0.542507	−	0.840051i	$-0.682525\pi$
$929$	−33.0707			−1.08501			−0.542507	+	0.840051i	$0.682525\pi$
$930$		0			0
$931$	90.3135			2.95990
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.86868	−	20.2196i	0.159223	−	0.661253i
$936$		0			0
$937$			31.0078i			1.01298i	0.862245	+	0.506491i	$0.169058\pi$
$937$			31.0078i			1.01298i	−0.862245	+	0.506491i	$0.830942\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16.6560			−0.542971			−0.271486	−	0.962442i	$-0.587515\pi$
$941$	−16.6560			−0.542971			−0.271486	+	0.962442i	$0.587515\pi$
$942$		0			0
$943$			8.36833i			0.272510i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			16.3503i			0.531312i	0.964068	+	0.265656i	$0.0855885\pi$
$947$			16.3503i			0.531312i	−0.964068	+	0.265656i	$0.914411\pi$
$948$		0			0
$949$	−1.49162			−0.0484199
$950$		0			0
$951$		0			0
$952$		0			0
$953$			14.2223i			0.460704i	0.973107	+	0.230352i	$0.0739877\pi$
$953$			14.2223i			0.460704i	−0.973107	+	0.230352i	$0.926012\pi$
$954$		0			0
$955$	43.4710	+	10.4674i	1.40669	+	0.338716i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.4693			1.24224
$960$		0			0
$961$	−27.6098			−0.890639
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.321095	−	1.33351i	0.0103364	−	0.0429271i
$966$		0			0
$967$			31.2933i			1.00632i	0.864192	+	0.503162i	$0.167830\pi$
$967$			31.2933i			1.00632i	−0.864192	+	0.503162i	$0.832170\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	31.4671			1.00983			0.504914	−	0.863170i	$-0.331524\pi$
$971$	31.4671			1.00983			0.504914	+	0.863170i	$0.331524\pi$
$972$		0			0
$973$		−	86.5817i		−	2.77568i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			27.8139i			0.889846i	0.895569	+	0.444923i	$0.146769\pi$
$977$			27.8139i			0.889846i	−0.895569	+	0.444923i	$0.853231\pi$
$978$		0			0
$979$	−70.6875			−2.25918
$980$		0			0
$981$		0			0
$982$		0			0
$983$			57.5702i			1.83621i	0.396343	+	0.918103i	$0.370279\pi$
$983$			57.5702i			1.83621i	−0.396343	+	0.918103i	$0.629721\pi$
$984$		0			0
$985$	−2.42884	+	10.0870i	−0.0773893	+	0.321398i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.43473			0.172814
$990$		0			0
$991$	−48.4772			−1.53993			−0.769964	−	0.638088i	$-0.779726\pi$
$991$	−48.4772			−1.53993			−0.769964	+	0.638088i	$0.779726\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	28.5755	+	6.88068i	0.905903	+	0.218132i
$996$		0			0
$997$		−	20.6703i		−	0.654634i	−0.944915	−	0.327317i	$-0.893856\pi$
$997$		−	20.6703i		−	0.654634i	0.944915	−	0.327317i	$-0.106144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.b.829.12		12
3.2	odd	2		460.2.c.a.369.11	yes	12
5.4	even	2	inner	4140.2.f.b.829.11		12
12.11	even	2		1840.2.e.f.369.2		12
15.2	even	4		2300.2.a.o.1.5		6
15.8	even	4		2300.2.a.n.1.2		6
15.14	odd	2		460.2.c.a.369.2	✓	12
60.23	odd	4		9200.2.a.cy.1.5		6
60.47	odd	4		9200.2.a.cx.1.2		6
60.59	even	2		1840.2.e.f.369.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.2	✓	12	15.14	odd	2
460.2.c.a.369.11	yes	12	3.2	odd	2
1840.2.e.f.369.2		12	12.11	even	2
1840.2.e.f.369.11		12	60.59	even	2
2300.2.a.n.1.2		6	15.8	even	4
2300.2.a.o.1.5		6	15.2	even	4
4140.2.f.b.829.11		12	5.4	even	2	inner
4140.2.f.b.829.12		12	1.1	even	1	trivial
9200.2.a.cx.1.2		6	60.47	odd	4
9200.2.a.cy.1.5		6	60.23	odd	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)

\(f(q)\)	\(=\)	\(q+(2.17393 + 0.523461i) q^{5} -4.50896i q^{7} +O(q^{10})\)	\(q+(2.17393 + 0.523461i) q^{5} -4.50896i q^{7} +4.10479 q^{11} -4.10245i q^{13} -2.26588i q^{17} -6.77484 q^{19} -1.00000i q^{23} +(4.45198 + 2.27594i) q^{25} +4.13863 q^{29} +1.84124 q^{31} +(2.36026 - 9.80218i) q^{35} -11.1155i q^{37} -8.36833 q^{41} +5.43473i q^{43} -0.593285i q^{47} -13.3307 q^{49} -1.70512i q^{53} +(8.92353 + 2.14870i) q^{55} +6.19772 q^{59} -11.3814 q^{61} +(2.14747 - 8.91846i) q^{65} +5.78978i q^{67} +11.9915 q^{71} -0.363592i q^{73} -18.5083i q^{77} -1.75692 q^{79} +9.72171i q^{83} +(1.18610 - 4.92587i) q^{85} -17.2208 q^{89} -18.4978 q^{91} +(-14.7281 - 3.54636i) q^{95} -4.38314i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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