LMFDB - Embedded newform 4140.2.f.a.829.2

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:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.2
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.a.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.21432 + 0.311108i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(-2.21432 + 0.311108i) q^{5} +1.00000i q^{7} -2.21432 q^{11} -5.11753i q^{13} +4.11753i q^{17} +5.11753 q^{19} -1.00000i q^{23} +(4.80642 - 1.37778i) q^{25} -2.93332 q^{29} -2.09679 q^{31} +(-0.311108 - 2.21432i) q^{35} -1.28100i q^{37} -0.458751 q^{41} +2.14764i q^{43} -2.40790i q^{47} +6.00000 q^{49} -3.83654i q^{53} +(4.90321 - 0.688892i) q^{55} +1.68889 q^{59} -15.5462 q^{61} +(1.59210 + 11.3319i) q^{65} +1.76986i q^{67} -4.39853 q^{71} +11.7397i q^{73} -2.21432i q^{77} +14.6637 q^{79} +11.4953i q^{83} +(-1.28100 - 9.11753i) q^{85} +0.755569 q^{89} +5.11753 q^{91} +(-11.3319 + 1.59210i) q^{95} +11.8938i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 4 q^{19} + 2 q^{25} - 18 q^{29} - 26 q^{31} - 2 q^{35} + 10 q^{41} + 36 q^{49} + 16 q^{55} + 10 q^{59} - 40 q^{61} - 4 q^{65} + 14 q^{71} + 8 q^{79} + 6 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) $ 6 * q + 4 * q^19 + 2 * q^25 - 18 * q^29 - 26 * q^31 - 2 * q^35 + 10 * q^41 + 36 * q^49 + 16 * q^55 + 10 * q^59 - 40 * q^61 - 4 * q^65 + 14 * q^71 + 8 * q^79 + 6 * q^85 + 4 * q^89 + 4 * q^91 - 28 * 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
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.21432			−0.667643			−0.333821	−	0.942636i	$-0.608338\pi$
$11$	−2.21432			−0.667643			−0.333821	+	0.942636i	$0.608338\pi$
$12$		0			0
$13$		−	5.11753i		−	1.41935i	−0.704530	−	0.709674i	$-0.748842\pi$
$13$		−	5.11753i		−	1.41935i	0.704530	−	0.709674i	$-0.251158\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.11753i			0.998648i	0.866415	+	0.499324i	$0.166418\pi$
$17$			4.11753i			0.998648i	−0.866415	+	0.499324i	$0.833582\pi$
$18$		0			0
$19$	5.11753			1.17404			0.587021	−	0.809572i	$-0.300301\pi$
$19$	5.11753			1.17404			0.587021	+	0.809572i	$0.300301\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.93332			−0.544704			−0.272352	−	0.962198i	$-0.587802\pi$
$29$	−2.93332			−0.544704			−0.272352	+	0.962198i	$0.587802\pi$
$30$		0			0
$31$	−2.09679			−0.376594			−0.188297	−	0.982112i	$-0.560297\pi$
$31$	−2.09679			−0.376594			−0.188297	+	0.982112i	$0.560297\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.311108	−	2.21432i	−0.0525868	−	0.374288i
$36$		0			0
$37$		−	1.28100i		−	0.210594i	−0.994441	−	0.105297i	$-0.966421\pi$
$37$		−	1.28100i		−	0.210594i	0.994441	−	0.105297i	$-0.0335794\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.458751			−0.0716449			−0.0358224	−	0.999358i	$-0.511405\pi$
$41$	−0.458751			−0.0716449			−0.0358224	+	0.999358i	$0.511405\pi$
$42$		0			0
$43$			2.14764i			0.327513i	0.986501	+	0.163756i	$0.0523611\pi$
$43$			2.14764i			0.327513i	−0.986501	+	0.163756i	$0.947639\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.40790i		−	0.351228i	−0.984459	−	0.175614i	$-0.943809\pi$
$47$		−	2.40790i		−	0.351228i	0.984459	−	0.175614i	$-0.0561910\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.83654i		−	0.526989i	−0.964661	−	0.263494i	$-0.915125\pi$
$53$		−	3.83654i		−	0.526989i	0.964661	−	0.263494i	$-0.0848750\pi$
$54$		0			0
$55$	4.90321	−	0.688892i	0.661149	−	0.0928902i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.68889			0.219875			0.109938	−	0.993939i	$-0.464935\pi$
$59$	1.68889			0.219875			0.109938	+	0.993939i	$0.464935\pi$
$60$		0			0
$61$	−15.5462			−1.99048			−0.995242	−	0.0974377i	$-0.968935\pi$
$61$	−15.5462			−1.99048			−0.995242	+	0.0974377i	$0.968935\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.59210	+	11.3319i	0.197476	+	1.40554i
$66$		0			0
$67$			1.76986i			0.216223i	0.994139	+	0.108111i	$0.0344803\pi$
$67$			1.76986i			0.216223i	−0.994139	+	0.108111i	$0.965520\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.39853			−0.522009			−0.261005	−	0.965338i	$-0.584054\pi$
$71$	−4.39853			−0.522009			−0.261005	+	0.965338i	$0.584054\pi$
$72$		0			0
$73$			11.7397i			1.37403i	0.726642	+	0.687017i	$0.241080\pi$
$73$			11.7397i			1.37403i	−0.726642	+	0.687017i	$0.758920\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.21432i		−	0.252345i
$78$		0			0
$79$	14.6637			1.64980			0.824898	−	0.565282i	$-0.191233\pi$
$79$	14.6637			1.64980			0.824898	+	0.565282i	$0.191233\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			11.4953i			1.26177i	0.775874	+	0.630887i	$0.217309\pi$
$83$			11.4953i			1.26177i	−0.775874	+	0.630887i	$0.782691\pi$
$84$		0			0
$85$	−1.28100	−	9.11753i	−0.138944	−	0.988935i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.755569			0.0800901			0.0400451	−	0.999198i	$-0.487250\pi$
$89$	0.755569			0.0800901			0.0400451	+	0.999198i	$0.487250\pi$
$90$		0			0
$91$	5.11753			0.536463
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.3319	+	1.59210i	−1.16262	+	0.163346i
$96$		0			0
$97$			11.8938i			1.20764i	0.797122	+	0.603818i	$0.206355\pi$
$97$			11.8938i			1.20764i	−0.797122	+	0.603818i	$0.793645\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.59210			0.854946			0.427473	−	0.904028i	$-0.359404\pi$
$101$	8.59210			0.854946			0.427473	+	0.904028i	$0.359404\pi$
$102$		0			0
$103$			4.90321i			0.483128i	0.970385	+	0.241564i	$0.0776603\pi$
$103$			4.90321i			0.483128i	−0.970385	+	0.241564i	$0.922340\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.82717i			0.853354i	0.904404	+	0.426677i	$0.140316\pi$
$107$			8.82717i			0.853354i	−0.904404	+	0.426677i	$0.859684\pi$
$108$		0			0
$109$	0.822245			0.0787568			0.0393784	−	0.999224i	$-0.487462\pi$
$109$	0.822245			0.0787568			0.0393784	+	0.999224i	$0.487462\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.26517i			0.589378i	0.955593	+	0.294689i	$0.0952161\pi$
$113$			6.26517i			0.589378i	−0.955593	+	0.294689i	$0.904784\pi$
$114$		0			0
$115$	0.311108	+	2.21432i	0.0290110	+	0.206486i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.11753			−0.377454
$120$		0			0
$121$	−6.09679			−0.554253
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.2143	+	4.54617i	−0.913597	+	0.406622i
$126$		0			0
$127$			19.1590i			1.70009i	0.526712	+	0.850044i	$0.323425\pi$
$127$			19.1590i			1.70009i	−0.526712	+	0.850044i	$0.676575\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.1526			−1.14914			−0.574572	−	0.818454i	$-0.694832\pi$
$131$	−13.1526			−1.14914			−0.574572	+	0.818454i	$0.694832\pi$
$132$		0			0
$133$			5.11753i			0.443746i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.76049i			0.492152i	0.969251	+	0.246076i	$0.0791413\pi$
$137$			5.76049i			0.492152i	−0.969251	+	0.246076i	$0.920859\pi$
$138$		0			0
$139$	1.42864			0.121176			0.0605878	−	0.998163i	$-0.480702\pi$
$139$	1.42864			0.121176			0.0605878	+	0.998163i	$0.480702\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			11.3319i			0.947617i
$144$		0			0
$145$	6.49532	−	0.912580i	0.539407	−	0.0757856i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.78568			−0.473981			−0.236991	−	0.971512i	$-0.576161\pi$
$149$	−5.78568			−0.473981			−0.236991	+	0.971512i	$0.576161\pi$
$150$		0			0
$151$	−3.52543			−0.286895			−0.143448	−	0.989658i	$-0.545819\pi$
$151$	−3.52543			−0.286895			−0.143448	+	0.989658i	$0.545819\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.64296	−	0.652327i	0.372931	−	0.0523962i
$156$		0			0
$157$			3.81579i			0.304533i	0.988339	+	0.152267i	$0.0486573\pi$
$157$			3.81579i			0.304533i	−0.988339	+	0.152267i	$0.951343\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$			7.05086i			0.552266i	0.961119	+	0.276133i	$0.0890530\pi$
$163$			7.05086i			0.552266i	−0.961119	+	0.276133i	$0.910947\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.3892i		−	1.42300i	−0.702688	−	0.711498i	$-0.748017\pi$
$167$		−	18.3892i		−	1.42300i	0.702688	−	0.711498i	$-0.251983\pi$
$168$		0			0
$169$	−13.1891			−1.01455
$170$		0			0
$171$		0			0
$172$		0			0
$173$			0.0602231i			0.00457867i	0.999997	+	0.00228934i	$0.000728719\pi$
$173$			0.0602231i			0.00457867i	−0.999997	+	0.00228934i	$0.999271\pi$
$174$		0			0
$175$	1.37778	+	4.80642i	0.104151	+	0.363331i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.33185			−0.548008			−0.274004	−	0.961729i	$-0.588348\pi$
$179$	−7.33185			−0.548008			−0.274004	+	0.961729i	$0.588348\pi$
$180$		0			0
$181$	8.04149			0.597719			0.298860	−	0.954297i	$-0.403394\pi$
$181$	8.04149			0.597719			0.298860	+	0.954297i	$0.403394\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.398528	+	2.83654i	0.0293004	+	0.208546i
$186$		0			0
$187$		−	9.11753i		−	0.666740i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.59210			0.549345			0.274673	−	0.961538i	$-0.411431\pi$
$191$	7.59210			0.549345			0.274673	+	0.961538i	$0.411431\pi$
$192$		0			0
$193$		−	12.7239i		−	0.915888i	−0.888981	−	0.457944i	$-0.848586\pi$
$193$		−	12.7239i		−	0.915888i	0.888981	−	0.457944i	$-0.151414\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.1526i			1.22207i	0.791604	+	0.611035i	$0.209246\pi$
$197$			17.1526i			1.22207i	−0.791604	+	0.611035i	$0.790754\pi$
$198$		0			0
$199$	−2.69535			−0.191068			−0.0955340	−	0.995426i	$-0.530456\pi$
$199$	−2.69535			−0.191068			−0.0955340	+	0.995426i	$0.530456\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.93332i		−	0.205879i
$204$		0			0
$205$	1.01582	−	0.142721i	0.0709480	−	0.00996807i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.3319			−0.783841
$210$		0			0
$211$	−21.3684			−1.47106			−0.735531	−	0.677491i	$-0.763067\pi$
$211$	−21.3684			−1.47106			−0.735531	+	0.677491i	$0.763067\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.668149	−	4.75557i	−0.0455674	−	0.324327i
$216$		0			0
$217$		−	2.09679i		−	0.142339i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	21.0716			1.41743
$222$		0			0
$223$			10.7556i			0.720246i	0.932905	+	0.360123i	$0.117265\pi$
$223$			10.7556i			0.720246i	−0.932905	+	0.360123i	$0.882735\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.36842i			0.422687i	0.977412	+	0.211343i	$0.0677838\pi$
$227$			6.36842i			0.422687i	−0.977412	+	0.211343i	$0.932216\pi$
$228$		0			0
$229$	−18.1432			−1.19894			−0.599468	−	0.800399i	$-0.704621\pi$
$229$	−18.1432			−1.19894			−0.599468	+	0.800399i	$0.704621\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			25.3274i			1.65925i	0.558318	+	0.829627i	$0.311447\pi$
$233$			25.3274i			1.65925i	−0.558318	+	0.829627i	$0.688553\pi$
$234$		0			0
$235$	0.749115	+	5.33185i	0.0488669	+	0.347812i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.8178			1.02317			0.511584	−	0.859233i	$-0.329059\pi$
$239$	15.8178			1.02317			0.511584	+	0.859233i	$0.329059\pi$
$240$		0			0
$241$	−25.8415			−1.66459			−0.832297	−	0.554330i	$-0.812975\pi$
$241$	−25.8415			−1.66459			−0.832297	+	0.554330i	$0.812975\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.2859	+	1.86665i	−0.848806	+	0.119256i
$246$		0			0
$247$		−	26.1891i		−	1.66637i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.4795			0.724579			0.362290	−	0.932066i	$-0.381995\pi$
$251$	11.4795			0.724579			0.362290	+	0.932066i	$0.381995\pi$
$252$		0			0
$253$			2.21432i			0.139213i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			24.7447i			1.54353i	0.635908	+	0.771765i	$0.280626\pi$
$257$			24.7447i			1.54353i	−0.635908	+	0.771765i	$0.719374\pi$
$258$		0			0
$259$	1.28100			0.0795972
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.3985i			1.25783i	0.777475	+	0.628914i	$0.216500\pi$
$263$			20.3985i			1.25783i	−0.777475	+	0.628914i	$0.783500\pi$
$264$		0			0
$265$	1.19358	+	8.49532i	0.0733208	+	0.521863i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.6702			0.955426			0.477713	−	0.878516i	$-0.341466\pi$
$269$	15.6702			0.955426			0.477713	+	0.878516i	$0.341466\pi$
$270$		0			0
$271$	−6.37778			−0.387423			−0.193711	−	0.981059i	$-0.562053\pi$
$271$	−6.37778			−0.387423			−0.193711	+	0.981059i	$0.562053\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.6430	+	3.05086i	−0.641795	+	0.183973i
$276$		0			0
$277$			9.80642i			0.589211i	0.955619	+	0.294605i	$0.0951882\pi$
$277$			9.80642i			0.589211i	−0.955619	+	0.294605i	$0.904812\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.4306			1.93465			0.967325	−	0.253539i	$-0.0815947\pi$
$281$	32.4306			1.93465			0.967325	+	0.253539i	$0.0815947\pi$
$282$		0			0
$283$			14.2859i			0.849210i	0.905379	+	0.424605i	$0.139587\pi$
$283$			14.2859i			0.849210i	−0.905379	+	0.424605i	$0.860413\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.458751i		−	0.0270792i
$288$		0			0
$289$	0.0459330			0.00270194
$290$		0			0
$291$		0			0
$292$		0			0
$293$			15.1684i			0.886147i	0.896485	+	0.443073i	$0.146112\pi$
$293$			15.1684i			0.886147i	−0.896485	+	0.443073i	$0.853888\pi$
$294$		0			0
$295$	−3.73975	+	0.525428i	−0.217737	+	0.0305916i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.11753			−0.295954
$300$		0			0
$301$	−2.14764			−0.123788
$302$		0			0
$303$		0			0
$304$		0			0
$305$	34.4242	−	4.83654i	1.97112	−	0.276939i
$306$		0			0
$307$		−	24.4035i		−	1.39278i	−0.717664	−	0.696389i	$-0.754789\pi$
$307$		−	24.4035i		−	1.39278i	0.717664	−	0.696389i	$-0.245211\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	26.2351			1.48765			0.743827	−	0.668372i	$-0.233009\pi$
$311$	26.2351			1.48765			0.743827	+	0.668372i	$0.233009\pi$
$312$		0			0
$313$			30.3733i			1.71680i	0.512979	+	0.858401i	$0.328542\pi$
$313$			30.3733i			1.71680i	−0.512979	+	0.858401i	$0.671458\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.85881i		−	0.553726i	−0.960909	−	0.276863i	$-0.910705\pi$
$317$		−	9.85881i		−	0.553726i	0.960909	−	0.276863i	$-0.0892948\pi$
$318$		0			0
$319$	6.49532			0.363668
$320$		0			0
$321$		0			0
$322$		0			0
$323$			21.0716i			1.17246i
$324$		0			0
$325$	−7.05086	−	24.5970i	−0.391111	−	1.36440i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.40790			0.132752
$330$		0			0
$331$	5.90321			0.324470			0.162235	−	0.986752i	$-0.448130\pi$
$331$	5.90321			0.324470			0.162235	+	0.986752i	$0.448130\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.550617	−	3.91903i	−0.0300834	−	0.214120i
$336$		0			0
$337$		−	16.1017i		−	0.877116i	−0.898703	−	0.438558i	$-0.855489\pi$
$337$		−	16.1017i		−	0.877116i	0.898703	−	0.438558i	$-0.144511\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.64296			0.251430
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	14.5303i		−	0.780030i	−0.920808	−	0.390015i	$-0.872470\pi$
$347$		−	14.5303i		−	0.780030i	0.920808	−	0.390015i	$-0.127530\pi$
$348$		0			0
$349$	7.32693			0.392202			0.196101	−	0.980584i	$-0.437172\pi$
$349$	7.32693			0.392202			0.196101	+	0.980584i	$0.437172\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.5827i		−	0.776161i	−0.921626	−	0.388080i	$-0.873138\pi$
$353$		−	14.5827i		−	0.776161i	0.921626	−	0.388080i	$-0.126862\pi$
$354$		0			0
$355$	9.73975	−	1.36842i	0.516932	−	0.0726280i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.3289			−1.60070			−0.800350	−	0.599533i	$-0.795353\pi$
$359$	−30.3289			−1.60070			−0.800350	+	0.599533i	$0.795353\pi$
$360$		0			0
$361$	7.18913			0.378375
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.65233	−	25.9956i	−0.191172	−	1.36067i
$366$		0			0
$367$			7.94470i			0.414710i	0.978266	+	0.207355i	$0.0664855\pi$
$367$			7.94470i			0.414710i	−0.978266	+	0.207355i	$0.933514\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.83654			0.199183
$372$		0			0
$373$			29.2716i			1.51563i	0.652471	+	0.757814i	$0.273732\pi$
$373$			29.2716i			1.51563i	−0.652471	+	0.757814i	$0.726268\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.0114i			0.773125i
$378$		0			0
$379$	−6.48886			−0.333310			−0.166655	−	0.986015i	$-0.553297\pi$
$379$	−6.48886			−0.333310			−0.166655	+	0.986015i	$0.553297\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5.13981i		−	0.262632i	−0.991341	−	0.131316i	$-0.958080\pi$
$383$		−	5.13981i		−	0.262632i	0.991341	−	0.131316i	$-0.0419202\pi$
$384$		0			0
$385$	0.688892	+	4.90321i	0.0351092	+	0.249891i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.9813			−1.31730			−0.658651	−	0.752448i	$-0.728873\pi$
$389$	−25.9813			−1.31730			−0.658651	+	0.752448i	$0.728873\pi$
$390$		0			0
$391$	4.11753			0.208233
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−32.4701	+	4.56199i	−1.63375	+	0.229539i
$396$		0			0
$397$			0.663703i			0.0333103i	0.999861	+	0.0166551i	$0.00530174\pi$
$397$			0.663703i			0.0333103i	−0.999861	+	0.0166551i	$0.994698\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.2351			1.21024			0.605121	−	0.796134i	$-0.293125\pi$
$401$	24.2351			1.21024			0.605121	+	0.796134i	$0.293125\pi$
$402$		0			0
$403$			10.7304i			0.534518i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.83654i			0.140602i
$408$		0			0
$409$	24.0192			1.18767			0.593837	−	0.804586i	$-0.297612\pi$
$409$	24.0192			1.18767			0.593837	+	0.804586i	$0.297612\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.68889i			0.0831050i
$414$		0			0
$415$	−3.57628	−	25.4543i	−0.175553	−	1.24950i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	29.9605			1.46367			0.731834	−	0.681483i	$-0.238665\pi$
$419$	29.9605			1.46367			0.731834	+	0.681483i	$0.238665\pi$
$420$		0			0
$421$	−5.04440			−0.245849			−0.122925	−	0.992416i	$-0.539227\pi$
$421$	−5.04440			−0.245849			−0.122925	+	0.992416i	$0.539227\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.67307	+	19.7906i	0.275184	+	0.959985i
$426$		0			0
$427$		−	15.5462i		−	0.752332i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.1017			0.775592			0.387796	−	0.921745i	$-0.373236\pi$
$431$	16.1017			0.775592			0.387796	+	0.921745i	$0.373236\pi$
$432$		0			0
$433$		−	18.0321i		−	0.866568i	−0.901257	−	0.433284i	$-0.857355\pi$
$433$		−	18.0321i		−	0.866568i	0.901257	−	0.433284i	$-0.142645\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5.11753i		−	0.244805i
$438$		0			0
$439$	−32.1575			−1.53479			−0.767397	−	0.641173i	$-0.778448\pi$
$439$	−32.1575			−1.53479			−0.767397	+	0.641173i	$0.778448\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	35.3481i		−	1.67944i	−0.543019	−	0.839721i	$-0.682719\pi$
$443$		−	35.3481i		−	1.67944i	0.543019	−	0.839721i	$-0.317281\pi$
$444$		0			0
$445$	−1.67307	+	0.235063i	−0.0793112	+	0.0111431i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.2603			−0.814562			−0.407281	−	0.913303i	$-0.633523\pi$
$449$	−17.2603			−0.814562			−0.407281	+	0.913303i	$0.633523\pi$
$450$		0			0
$451$	1.01582			0.0478332
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−11.3319	+	1.59210i	−0.531245	+	0.0746390i
$456$		0			0
$457$			3.38271i			0.158236i	0.996865	+	0.0791182i	$0.0252104\pi$
$457$			3.38271i			0.158236i	−0.996865	+	0.0791182i	$0.974790\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.94914			0.323654			0.161827	−	0.986819i	$-0.448261\pi$
$461$	6.94914			0.323654			0.161827	+	0.986819i	$0.448261\pi$
$462$		0			0
$463$		−	28.6800i		−	1.33287i	−0.745562	−	0.666436i	$-0.767819\pi$
$463$		−	28.6800i		−	1.33287i	0.745562	−	0.666436i	$-0.232181\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	33.7862i		−	1.56344i	−0.623632	−	0.781718i	$-0.714343\pi$
$467$		−	33.7862i		−	1.56344i	0.623632	−	0.781718i	$-0.285657\pi$
$468$		0			0
$469$	−1.76986			−0.0817245
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.75557i		−	0.218661i
$474$		0			0
$475$	24.5970	−	7.05086i	1.12859	−	0.323515i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.78568			0.355737			0.177868	−	0.984054i	$-0.443080\pi$
$479$	7.78568			0.355737			0.177868	+	0.984054i	$0.443080\pi$
$480$		0			0
$481$	−6.55554			−0.298907
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.70027	−	26.3368i	−0.168021	−	1.19589i
$486$		0			0
$487$			11.7397i			0.531979i	0.963976	+	0.265989i	$0.0856986\pi$
$487$			11.7397i			0.531979i	−0.963976	+	0.265989i	$0.914301\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−43.9610			−1.98393			−0.991966	−	0.126505i	$-0.959624\pi$
$491$	−43.9610			−1.98393			−0.991966	+	0.126505i	$0.959624\pi$
$492$		0			0
$493$		−	12.0781i		−	0.543968i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	4.39853i		−	0.197301i
$498$		0			0
$499$	11.7382			0.525475			0.262737	−	0.964867i	$-0.415375\pi$
$499$	11.7382			0.525475			0.262737	+	0.964867i	$0.415375\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			19.7304i			0.879734i	0.898063	+	0.439867i	$0.144974\pi$
$503$			19.7304i			0.879734i	−0.898063	+	0.439867i	$0.855026\pi$
$504$		0			0
$505$	−19.0257	+	2.67307i	−0.846631	+	0.118950i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.1289			0.537604			0.268802	−	0.963195i	$-0.413372\pi$
$509$	12.1289			0.537604			0.268802	+	0.963195i	$0.413372\pi$
$510$		0			0
$511$	−11.7397			−0.519336
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.52543	−	10.8573i	−0.0672184	−	0.478429i
$516$		0			0
$517$			5.33185i			0.234495i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.5605			−0.944581			−0.472290	−	0.881443i	$-0.656573\pi$
$521$	−21.5605			−0.944581			−0.472290	+	0.881443i	$0.656573\pi$
$522$		0			0
$523$			29.9684i			1.31042i	0.755445	+	0.655212i	$0.227421\pi$
$523$			29.9684i			1.31042i	−0.755445	+	0.655212i	$0.772579\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.63359i		−	0.376085i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.34767i			0.101689i
$534$		0			0
$535$	−2.74620	−	19.5462i	−0.118729	−	0.845055i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13.2859			−0.572265
$540$		0			0
$541$	−27.1985			−1.16935			−0.584677	−	0.811266i	$-0.698779\pi$
$541$	−27.1985			−1.16935			−0.584677	+	0.811266i	$0.698779\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.82071	+	0.255807i	−0.0779908	+	0.0109576i
$546$		0			0
$547$		−	28.5433i		−	1.22042i	−0.792239	−	0.610211i	$-0.791085\pi$
$547$		−	28.5433i		−	1.22042i	0.792239	−	0.610211i	$-0.208915\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.0114			−0.639506
$552$		0			0
$553$			14.6637i			0.623564i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			4.87755i			0.206668i	0.994647	+	0.103334i	$0.0329511\pi$
$557$			4.87755i			0.206668i	−0.994647	+	0.103334i	$0.967049\pi$
$558$		0			0
$559$	10.9906			0.464854
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	33.5195i		−	1.41268i	−0.707875	−	0.706338i	$-0.750346\pi$
$563$		−	33.5195i		−	1.41268i	0.707875	−	0.706338i	$-0.249654\pi$
$564$		0			0
$565$	−1.94914	−	13.8731i	−0.0820011	−	0.583646i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.44738			−0.186444			−0.0932218	−	0.995645i	$-0.529717\pi$
$569$	−4.44738			−0.186444			−0.0932218	+	0.995645i	$0.529717\pi$
$570$		0			0
$571$	40.3718			1.68951			0.844754	−	0.535155i	$-0.179747\pi$
$571$	40.3718			1.68951			0.844754	+	0.535155i	$0.179747\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.37778	−	4.80642i	−0.0574576	−	0.200442i
$576$		0			0
$577$			22.9403i			0.955015i	0.878628	+	0.477508i	$0.158460\pi$
$577$			22.9403i			0.955015i	−0.878628	+	0.477508i	$0.841540\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11.4953			−0.476906
$582$		0			0
$583$			8.49532i			0.351840i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	13.0509i		−	0.538666i	−0.963047	−	0.269333i	$-0.913197\pi$
$587$		−	13.0509i		−	0.538666i	0.963047	−	0.269333i	$-0.0868033\pi$
$588$		0			0
$589$	−10.7304			−0.442138
$590$		0			0
$591$		0			0
$592$		0			0
$593$			43.3165i			1.77880i	0.457134	+	0.889398i	$0.348876\pi$
$593$			43.3165i			1.77880i	−0.457134	+	0.889398i	$0.651124\pi$
$594$		0			0
$595$	9.11753	−	1.28100i	0.373782	−	0.0525157i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	33.7560			1.37923			0.689617	−	0.724174i	$-0.257779\pi$
$599$	33.7560			1.37923			0.689617	+	0.724174i	$0.257779\pi$
$600$		0			0
$601$	−26.2114			−1.06919			−0.534593	−	0.845110i	$-0.679535\pi$
$601$	−26.2114			−1.06919			−0.534593	+	0.845110i	$0.679535\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.5002	−	1.89676i	0.548863	−	0.0771142i
$606$		0			0
$607$		−	22.1367i		−	0.898503i	−0.893405	−	0.449251i	$-0.851691\pi$
$607$		−	22.1367i		−	0.898503i	0.893405	−	0.449251i	$-0.148309\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.3225			−0.498514
$612$		0			0
$613$		−	1.65433i		−	0.0668180i	−0.999442	−	0.0334090i	$-0.989364\pi$
$613$		−	1.65433i		−	0.0668180i	0.999442	−	0.0334090i	$-0.0106364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			36.6608i			1.47591i	0.674851	+	0.737954i	$0.264208\pi$
$617$			36.6608i			1.47591i	−0.674851	+	0.737954i	$0.735792\pi$
$618$		0			0
$619$	9.13383			0.367120			0.183560	−	0.983009i	$-0.441238\pi$
$619$	9.13383			0.367120			0.183560	+	0.983009i	$0.441238\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			0.755569i			0.0302712i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.27454			0.210310
$630$		0			0
$631$	−10.7620			−0.428429			−0.214215	−	0.976787i	$-0.568719\pi$
$631$	−10.7620			−0.428429			−0.214215	+	0.976787i	$0.568719\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.96052	−	42.4242i	−0.236536	−	1.68355i
$636$		0			0
$637$		−	30.7052i		−	1.21658i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.4177			−0.727457			−0.363729	−	0.931505i	$-0.618496\pi$
$641$	−18.4177			−0.727457			−0.363729	+	0.931505i	$0.618496\pi$
$642$		0			0
$643$		−	12.4750i		−	0.491968i	−0.969274	−	0.245984i	$-0.920889\pi$
$643$		−	12.4750i		−	0.491968i	0.969274	−	0.245984i	$-0.0791111\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.78568i		−	0.0702023i	−0.999384	−	0.0351012i	$-0.988825\pi$
$647$		−	1.78568i		−	0.0702023i	0.999384	−	0.0351012i	$-0.0111753\pi$
$648$		0			0
$649$	−3.73975			−0.146798
$650$		0			0
$651$		0			0
$652$		0			0
$653$			10.3793i			0.406174i	0.979161	+	0.203087i	$0.0650974\pi$
$653$			10.3793i			0.406174i	−0.979161	+	0.203087i	$0.934903\pi$
$654$		0			0
$655$	29.1240	−	4.09187i	1.13797	−	0.159882i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.7353			−0.612960			−0.306480	−	0.951877i	$-0.599151\pi$
$659$	−15.7353			−0.612960			−0.306480	+	0.951877i	$0.599151\pi$
$660$		0			0
$661$	45.6829			1.77686			0.888430	−	0.459013i	$-0.151797\pi$
$661$	45.6829			1.77686			0.888430	+	0.459013i	$0.151797\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.59210	−	11.3319i	−0.0617391	−	0.439430i
$666$		0			0
$667$			2.93332i			0.113579i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	34.4242			1.32893
$672$		0			0
$673$			36.7906i			1.41817i	0.705121	+	0.709087i	$0.250893\pi$
$673$			36.7906i			1.41817i	−0.705121	+	0.709087i	$0.749107\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	15.7032i		−	0.603522i	−0.953384	−	0.301761i	$-0.902425\pi$
$677$		−	15.7032i		−	0.603522i	0.953384	−	0.301761i	$-0.0975745\pi$
$678$		0			0
$679$	−11.8938			−0.456444
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.7067i			0.907112i	0.891228	+	0.453556i	$0.149845\pi$
$683$			23.7067i			0.907112i	−0.891228	+	0.453556i	$0.850155\pi$
$684$		0			0
$685$	−1.79213	−	12.7556i	−0.0684739	−	0.487365i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.6336			−0.747980
$690$		0			0
$691$	22.2810			0.847609			0.423805	−	0.905754i	$-0.360694\pi$
$691$	22.2810			0.847609			0.423805	+	0.905754i	$0.360694\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.16346	+	0.444461i	−0.119997	+	0.0168594i
$696$		0			0
$697$		−	1.88892i		−	0.0715480i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	33.7955			1.27644			0.638220	−	0.769854i	$-0.279671\pi$
$701$	33.7955			1.27644			0.638220	+	0.769854i	$0.279671\pi$
$702$		0			0
$703$		−	6.55554i		−	0.247247i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			8.59210i			0.323139i
$708$		0			0
$709$	50.7817			1.90715			0.953574	−	0.301160i	$-0.0973739\pi$
$709$	50.7817			1.90715			0.953574	+	0.301160i	$0.0973739\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.09679i			0.0785253i
$714$		0			0
$715$	−3.52543	−	25.0923i	−0.131844	−	0.938400i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.3812			1.20761			0.603807	−	0.797131i	$-0.293650\pi$
$719$	32.3812			1.20761			0.603807	+	0.797131i	$0.293650\pi$
$720$		0			0
$721$	−4.90321			−0.182605
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.0988	+	4.04149i	−0.523616	+	0.150097i
$726$		0			0
$727$			19.3783i			0.718700i	0.933203	+	0.359350i	$0.117002\pi$
$727$			19.3783i			0.718700i	−0.933203	+	0.359350i	$0.882998\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.84299			−0.327070
$732$		0			0
$733$			16.9857i			0.627382i	0.949525	+	0.313691i	$0.101566\pi$
$733$			16.9857i			0.627382i	−0.949525	+	0.313691i	$0.898434\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.91903i		−	0.144359i
$738$		0			0
$739$	12.6271			0.464497			0.232248	−	0.972657i	$-0.425392\pi$
$739$	12.6271			0.464497			0.232248	+	0.972657i	$0.425392\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	28.8484i		−	1.05834i	−0.848514	−	0.529172i	$-0.822502\pi$
$743$		−	28.8484i		−	1.05834i	0.848514	−	0.529172i	$-0.177498\pi$
$744$		0			0
$745$	12.8113	−	1.79997i	0.469371	−	0.0659458i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.82717			−0.322538
$750$		0			0
$751$	−16.6099			−0.606105			−0.303053	−	0.952974i	$-0.598006\pi$
$751$	−16.6099			−0.606105			−0.303053	+	0.952974i	$0.598006\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.80642	−	1.09679i	0.284105	−	0.0399162i
$756$		0			0
$757$			25.0098i			0.908998i	0.890747	+	0.454499i	$0.150182\pi$
$757$			25.0098i			0.908998i	−0.890747	+	0.454499i	$0.849818\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−27.2830			−0.989008			−0.494504	−	0.869175i	$-0.664650\pi$
$761$	−27.2830			−0.989008			−0.494504	+	0.869175i	$0.664650\pi$
$762$		0			0
$763$			0.822245i			0.0297673i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	8.64296i		−	0.312079i
$768$		0			0
$769$	30.8736			1.11333			0.556665	−	0.830737i	$-0.312081\pi$
$769$	30.8736			1.11333			0.556665	+	0.830737i	$0.312081\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	32.0513i		−	1.15281i	−0.817165	−	0.576403i	$-0.804456\pi$
$773$		−	32.0513i		−	1.15281i	0.817165	−	0.576403i	$-0.195544\pi$
$774$		0			0
$775$	−10.0781	+	2.88892i	−0.362014	+	0.103773i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.34767			−0.0841141
$780$		0			0
$781$	9.73975			0.348516
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.18712	−	8.44938i	−0.0423702	−	0.301571i
$786$		0			0
$787$		−	46.9452i		−	1.67341i	−0.547650	−	0.836707i	$-0.684478\pi$
$787$		−	46.9452i		−	1.67341i	0.547650	−	0.836707i	$-0.315522\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.26517			−0.222764
$792$		0			0
$793$			79.5580i			2.82519i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12.7955i			0.453241i	0.973983	+	0.226620i	$0.0727677\pi$
$797$			12.7955i			0.453241i	−0.973983	+	0.226620i	$0.927232\pi$
$798$		0			0
$799$	9.91459			0.350753
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	25.9956i		−	0.917363i
$804$		0			0
$805$	−2.21432	+	0.311108i	−0.0780445	+	0.0109651i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−33.6528			−1.18317			−0.591585	−	0.806243i	$-0.701498\pi$
$809$	−33.6528			−1.18317			−0.591585	+	0.806243i	$0.701498\pi$
$810$		0			0
$811$	8.49240			0.298209			0.149104	−	0.988821i	$-0.452361\pi$
$811$	8.49240			0.298209			0.149104	+	0.988821i	$0.452361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.19358	−	15.6128i	−0.0768376	−	0.546894i
$816$		0			0
$817$			10.9906i			0.384514i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.0558			−0.909353			−0.454676	−	0.890657i	$-0.650245\pi$
$821$	−26.0558			−0.909353			−0.454676	+	0.890657i	$0.650245\pi$
$822$		0			0
$823$		−	7.03212i		−	0.245124i	−0.992461	−	0.122562i	$-0.960889\pi$
$823$		−	7.03212i		−	0.245124i	0.992461	−	0.122562i	$-0.0391111\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	19.0622i		−	0.662859i	−0.943480	−	0.331429i	$-0.892469\pi$
$827$		−	19.0622i		−	0.662859i	0.943480	−	0.331429i	$-0.107531\pi$
$828$		0			0
$829$	−6.73191			−0.233809			−0.116904	−	0.993143i	$-0.537297\pi$
$829$	−6.73191			−0.233809			−0.116904	+	0.993143i	$0.537297\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			24.7052i			0.855984i
$834$		0			0
$835$	5.72101	+	40.7195i	0.197984	+	1.40916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.5812			0.606971			0.303485	−	0.952836i	$-0.401850\pi$
$839$	17.5812			0.606971			0.303485	+	0.952836i	$0.401850\pi$
$840$		0			0
$841$	−20.3956			−0.703297
$842$		0			0
$843$		0			0
$844$		0			0
$845$	29.2050	−	4.10324i	1.00468	−	0.141156i
$846$		0			0
$847$		−	6.09679i		−	0.209488i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.28100			−0.0439120
$852$		0			0
$853$			42.8069i			1.46568i	0.680401	+	0.732840i	$0.261806\pi$
$853$			42.8069i			1.46568i	−0.680401	+	0.732840i	$0.738194\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			26.1146i			0.892058i	0.895018	+	0.446029i	$0.147162\pi$
$857$			26.1146i			0.892058i	−0.895018	+	0.446029i	$0.852838\pi$
$858$		0			0
$859$	−33.3497			−1.13788			−0.568938	−	0.822380i	$-0.692646\pi$
$859$	−33.3497			−1.13788			−0.568938	+	0.822380i	$0.692646\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	32.2864i		−	1.09904i	−0.835480	−	0.549521i	$-0.814810\pi$
$863$		−	32.2864i		−	1.09904i	0.835480	−	0.549521i	$-0.185190\pi$
$864$		0			0
$865$	−0.0187359	−	0.133353i	−0.000637038	−	0.00453414i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−32.4701			−1.10147
$870$		0			0
$871$	9.05731			0.306895
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.54617	−	10.2143i	−0.153689	−	0.345307i
$876$		0			0
$877$		−	26.3466i		−	0.889662i	−0.895615	−	0.444831i	$-0.853264\pi$
$877$		−	26.3466i		−	0.889662i	0.895615	−	0.444831i	$-0.146736\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.7576			−1.23839			−0.619197	−	0.785236i	$-0.712542\pi$
$881$	−36.7576			−1.23839			−0.619197	+	0.785236i	$0.712542\pi$
$882$		0			0
$883$		−	35.5560i		−	1.19656i	−0.801289	−	0.598278i	$-0.795852\pi$
$883$		−	35.5560i		−	1.19656i	0.801289	−	0.598278i	$-0.204148\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	26.9906i		−	0.906257i	−0.891445	−	0.453128i	$-0.850308\pi$
$887$		−	26.9906i		−	0.906257i	0.891445	−	0.453128i	$-0.149692\pi$
$888$		0			0
$889$	−19.1590			−0.642573
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	12.3225i		−	0.412356i
$894$		0			0
$895$	16.2351	−	2.28100i	0.542678	−	0.0762453i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.15056			0.205133
$900$		0			0
$901$	15.7971			0.526276
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.8064	+	2.50177i	−0.591906	+	0.0831616i
$906$		0			0
$907$			17.6780i			0.586988i	0.955961	+	0.293494i	$0.0948181\pi$
$907$			17.6780i			0.586988i	−0.955961	+	0.293494i	$0.905182\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	39.8448			1.32012			0.660059	−	0.751213i	$-0.270531\pi$
$911$	39.8448			1.32012			0.660059	+	0.751213i	$0.270531\pi$
$912$		0			0
$913$		−	25.4543i		−	0.842415i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	13.1526i		−	0.434336i
$918$		0			0
$919$	−37.7877			−1.24650			−0.623250	−	0.782022i	$-0.714188\pi$
$919$	−37.7877			−1.24650			−0.623250	+	0.782022i	$0.714188\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			22.5096i			0.740913i
$924$		0			0
$925$	−1.76494	−	6.15701i	−0.0580308	−	0.202441i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29.1684			−0.956984			−0.478492	−	0.878092i	$-0.658816\pi$
$929$	−29.1684			−0.956984			−0.478492	+	0.878092i	$0.658816\pi$
$930$		0			0
$931$	30.7052			1.00632
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.83654	+	20.1891i	0.0927646	+	0.660255i
$936$		0			0
$937$		−	16.7511i		−	0.547235i	−0.961839	−	0.273618i	$-0.911780\pi$
$937$		−	16.7511i		−	0.547235i	0.961839	−	0.273618i	$-0.0882203\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.85881			−0.125794			−0.0628968	−	0.998020i	$-0.520034\pi$
$941$	−3.85881			−0.125794			−0.0628968	+	0.998020i	$0.520034\pi$
$942$		0			0
$943$			0.458751i			0.0149390i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	34.2953i		−	1.11445i	−0.830363	−	0.557224i	$-0.811867\pi$
$947$		−	34.2953i		−	1.11445i	0.830363	−	0.557224i	$-0.188133\pi$
$948$		0			0
$949$	60.0785			1.95023
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	22.0830i		−	0.715338i	−0.933848	−	0.357669i	$-0.883572\pi$
$953$		−	22.0830i		−	0.715338i	0.933848	−	0.357669i	$-0.116428\pi$
$954$		0			0
$955$	−16.8113	+	2.36196i	−0.544002	+	0.0764313i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.76049			−0.186016
$960$		0			0
$961$	−26.6035			−0.858177
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.95851	+	28.1748i	0.127429	+	0.906980i
$966$		0			0
$967$		−	41.1274i		−	1.32257i	−0.750136	−	0.661284i	$-0.770012\pi$
$967$		−	41.1274i		−	1.32257i	0.750136	−	0.661284i	$-0.229988\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17.2948			−0.555017			−0.277508	−	0.960723i	$-0.589509\pi$
$971$	−17.2948			−0.555017			−0.277508	+	0.960723i	$0.589509\pi$
$972$		0			0
$973$			1.42864i			0.0458001i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	49.3531i		−	1.57894i	−0.613786	−	0.789472i	$-0.710354\pi$
$977$		−	49.3531i		−	1.57894i	0.613786	−	0.789472i	$-0.289646\pi$
$978$		0			0
$979$	−1.67307			−0.0534716
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	15.7719i		−	0.503044i	−0.967852	−	0.251522i	$-0.919069\pi$
$983$		−	15.7719i		−	0.503044i	0.967852	−	0.251522i	$-0.0809311\pi$
$984$		0			0
$985$	−5.33630	−	37.9813i	−0.170029	−	1.21018i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.14764			0.0682911
$990$		0			0
$991$	−29.0928			−0.924164			−0.462082	−	0.886837i	$-0.652897\pi$
$991$	−29.0928			−0.924164			−0.462082	+	0.886837i	$0.652897\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.96836	−	0.838543i	0.189210	−	0.0265836i
$996$		0			0
$997$			22.0415i			0.698061i	0.937111	+	0.349030i	$0.113489\pi$
$997$			22.0415i			0.698061i	−0.937111	+	0.349030i	$0.886511\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.a.829.2		6
3.2	odd	2		1380.2.f.a.829.6	yes	6
5.4	even	2	inner	4140.2.f.a.829.1		6
15.2	even	4		6900.2.a.z.1.3		3
15.8	even	4		6900.2.a.y.1.3		3
15.14	odd	2		1380.2.f.a.829.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.3	✓	6	15.14	odd	2
1380.2.f.a.829.6	yes	6	3.2	odd	2
4140.2.f.a.829.1		6	5.4	even	2	inner
4140.2.f.a.829.2		6	1.1	even	1	trivial
6900.2.a.y.1.3		3	15.8	even	4
6900.2.a.z.1.3		3	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)

\(f(q)\)	\(=\)	\(q+(-2.21432 + 0.311108i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(-2.21432 + 0.311108i) q^{5} +1.00000i q^{7} -2.21432 q^{11} -5.11753i q^{13} +4.11753i q^{17} +5.11753 q^{19} -1.00000i q^{23} +(4.80642 - 1.37778i) q^{25} -2.93332 q^{29} -2.09679 q^{31} +(-0.311108 - 2.21432i) q^{35} -1.28100i q^{37} -0.458751 q^{41} +2.14764i q^{43} -2.40790i q^{47} +6.00000 q^{49} -3.83654i q^{53} +(4.90321 - 0.688892i) q^{55} +1.68889 q^{59} -15.5462 q^{61} +(1.59210 + 11.3319i) q^{65} +1.76986i q^{67} -4.39853 q^{71} +11.7397i q^{73} -2.21432i q^{77} +14.6637 q^{79} +11.4953i q^{83} +(-1.28100 - 9.11753i) q^{85} +0.755569 q^{89} +5.11753 q^{91} +(-11.3319 + 1.59210i) q^{95} +11.8938i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 4 q^{19} + 2 q^{25} - 18 q^{29} - 26 q^{31} - 2 q^{35} + 10 q^{41} + 36 q^{49} + 16 q^{55} + 10 q^{59} - 40 q^{61} - 4 q^{65} + 14 q^{71} + 8 q^{79} + 6 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) 6 * q + 4 * q^19 + 2 * q^25 - 18 * q^29 - 26 * q^31 - 2 * q^35 + 10 * q^41 + 36 * q^49 + 16 * q^55 + 10 * q^59 - 40 * q^61 - 4 * q^65 + 14 * q^71 + 8 * q^79 + 6 * q^85 + 4 * q^89 + 4 * q^91 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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