LMFDB - Embedded newform 4140.3.d.b.2161.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.16
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +13.2033i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +13.2033i q^{7} -11.2113i q^{11} -17.4654 q^{13} +22.0956i q^{17} -6.38843i q^{19} +(-22.3859 + 5.27944i) q^{23} -5.00000 q^{25} +26.6662 q^{29} -17.2415 q^{31} +29.5234 q^{35} -33.5497i q^{37} +9.39969 q^{41} +34.5042i q^{43} +0.525339 q^{47} -125.326 q^{49} +1.66679i q^{53} -25.0692 q^{55} +12.8063 q^{59} -74.9031i q^{61} +39.0538i q^{65} +42.5332i q^{67} +30.9853 q^{71} -21.1091 q^{73} +148.026 q^{77} -25.5678i q^{79} -31.7749i q^{83} +49.4073 q^{85} -90.7583i q^{89} -230.600i q^{91} -14.2850 q^{95} -51.5745i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) $ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

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
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			13.2033i			1.88618i	0.332535	+	0.943091i	$0.392096\pi$
$7$			13.2033i			1.88618i	−0.332535	+	0.943091i	$0.607904\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	11.2113i		−	1.01921i	−0.860409	−	0.509604i	$-0.829792\pi$
$11$		−	11.2113i		−	1.01921i	0.860409	−	0.509604i	$-0.170208\pi$
$12$		0			0
$13$	−17.4654			−1.34349			−0.671746	−	0.740782i	$-0.734455\pi$
$13$	−17.4654			−1.34349			−0.671746	+	0.740782i	$0.734455\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			22.0956i			1.29974i	0.760044	+	0.649871i	$0.225177\pi$
$17$			22.0956i			1.29974i	−0.760044	+	0.649871i	$0.774823\pi$
$18$		0			0
$19$		−	6.38843i		−	0.336233i	−0.985767	−	0.168117i	$-0.946232\pi$
$19$		−	6.38843i		−	0.336233i	0.985767	−	0.168117i	$-0.0537685\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−22.3859	+	5.27944i	−0.973299	+	0.229541i
$23$	−22.3859	+	5.27944i	−0.973299	+	0.229541i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	26.6662			0.919524			0.459762	−	0.888042i	$-0.347935\pi$
$29$	26.6662			0.919524			0.459762	+	0.888042i	$0.347935\pi$
$30$		0			0
$31$	−17.2415			−0.556178			−0.278089	−	0.960555i	$-0.589701\pi$
$31$	−17.2415			−0.556178			−0.278089	+	0.960555i	$0.589701\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	29.5234			0.843526
$36$		0			0
$37$		−	33.5497i		−	0.906747i	−0.891321	−	0.453374i	$-0.850220\pi$
$37$		−	33.5497i		−	0.906747i	0.891321	−	0.453374i	$-0.149780\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9.39969			0.229261			0.114630	−	0.993408i	$-0.463432\pi$
$41$	9.39969			0.229261			0.114630	+	0.993408i	$0.463432\pi$
$42$		0			0
$43$			34.5042i			0.802424i	0.915985	+	0.401212i	$0.131411\pi$
$43$			34.5042i			0.802424i	−0.915985	+	0.401212i	$0.868589\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.525339			0.0111774			0.00558872	−	0.999984i	$-0.498221\pi$
$47$	0.525339			0.0111774			0.00558872	+	0.999984i	$0.498221\pi$
$48$		0			0
$49$	−125.326			−2.55768
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.66679i			0.0314488i	0.999876	+	0.0157244i	$0.00500543\pi$
$53$			1.66679i			0.0314488i	−0.999876	+	0.0157244i	$0.994995\pi$
$54$		0			0
$55$	−25.0692			−0.455803
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.8063			0.217057			0.108528	−	0.994093i	$-0.465386\pi$
$59$	12.8063			0.217057			0.108528	+	0.994093i	$0.465386\pi$
$60$		0			0
$61$		−	74.9031i		−	1.22792i	−0.789337	−	0.613960i	$-0.789576\pi$
$61$		−	74.9031i		−	1.22792i	0.789337	−	0.613960i	$-0.210424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			39.0538i			0.600828i
$66$		0			0
$67$			42.5332i			0.634824i	0.948288	+	0.317412i	$0.102814\pi$
$67$			42.5332i			0.634824i	−0.948288	+	0.317412i	$0.897186\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	30.9853			0.436412			0.218206	−	0.975903i	$-0.429979\pi$
$71$	30.9853			0.436412			0.218206	+	0.975903i	$0.429979\pi$
$72$		0			0
$73$	−21.1091			−0.289166			−0.144583	−	0.989493i	$-0.546184\pi$
$73$	−21.1091			−0.289166			−0.144583	+	0.989493i	$0.546184\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	148.026			1.92241
$78$		0			0
$79$		−	25.5678i		−	0.323643i	−0.986820	−	0.161821i	$-0.948263\pi$
$79$		−	25.5678i		−	0.323643i	0.986820	−	0.161821i	$-0.0517369\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	31.7749i		−	0.382830i	−0.981509	−	0.191415i	$-0.938692\pi$
$83$		−	31.7749i		−	0.382830i	0.981509	−	0.191415i	$-0.0613076\pi$
$84$		0			0
$85$	49.4073			0.581263
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	90.7583i		−	1.01976i	−0.860247	−	0.509878i	$-0.829690\pi$
$89$		−	90.7583i		−	1.01976i	0.860247	−	0.509878i	$-0.170310\pi$
$90$		0			0
$91$		−	230.600i		−	2.53407i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.2850			−0.150368
$96$		0			0
$97$		−	51.5745i		−	0.531696i	−0.964015	−	0.265848i	$-0.914348\pi$
$97$		−	51.5745i		−	0.531696i	0.964015	−	0.265848i	$-0.0856520\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.4508			−0.152979			−0.0764893	−	0.997070i	$-0.524371\pi$
$101$	−15.4508			−0.152979			−0.0764893	+	0.997070i	$0.524371\pi$
$102$		0			0
$103$		−	9.31402i		−	0.0904274i	−0.998977	−	0.0452137i	$-0.985603\pi$
$103$		−	9.31402i		−	0.0904274i	0.998977	−	0.0452137i	$-0.0143969\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	157.134i		−	1.46855i	−0.678854	−	0.734273i	$-0.737523\pi$
$107$		−	157.134i		−	1.46855i	0.678854	−	0.734273i	$-0.262477\pi$
$108$		0			0
$109$		−	117.792i		−	1.08066i	−0.841453	−	0.540330i	$-0.818299\pi$
$109$		−	117.792i		−	1.08066i	0.841453	−	0.540330i	$-0.181701\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			221.153i			1.95710i	0.206004	+	0.978551i	$0.433954\pi$
$113$			221.153i			1.95710i	−0.206004	+	0.978551i	$0.566046\pi$
$114$		0			0
$115$	11.8052	+	50.0563i	0.102654	+	0.435273i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−291.735			−2.45155
$120$		0			0
$121$	−4.69276			−0.0387832
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	35.6835			0.280973			0.140486	−	0.990083i	$-0.455133\pi$
$127$	35.6835			0.280973			0.140486	+	0.990083i	$0.455133\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.5344			0.118583			0.0592917	−	0.998241i	$-0.481116\pi$
$131$	15.5344			0.118583			0.0592917	+	0.998241i	$0.481116\pi$
$132$		0			0
$133$	84.3482			0.634197
$134$		0			0
$135$		0			0
$136$		0			0
$137$			104.295i			0.761278i	0.924724	+	0.380639i	$0.124296\pi$
$137$			104.295i			0.761278i	−0.924724	+	0.380639i	$0.875704\pi$
$138$		0			0
$139$	109.260			0.786047			0.393023	−	0.919528i	$-0.371429\pi$
$139$	109.260			0.786047			0.393023	+	0.919528i	$0.371429\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			195.809i			1.36930i
$144$		0			0
$145$		−	59.6275i		−	0.411224i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	28.2723i		−	0.189747i	−0.995489	−	0.0948734i	$-0.969755\pi$
$149$		−	28.2723i		−	0.189747i	0.995489	−	0.0948734i	$-0.0302446\pi$
$150$		0			0
$151$	193.433			1.28102			0.640508	−	0.767951i	$-0.278724\pi$
$151$	193.433			1.28102			0.640508	+	0.767951i	$0.278724\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			38.5532i			0.248730i
$156$		0			0
$157$			69.1919i			0.440713i	0.975419	+	0.220356i	$0.0707221\pi$
$157$			69.1919i			0.440713i	−0.975419	+	0.220356i	$0.929278\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−69.7059	−	295.567i	−0.432956	−	1.83582i
$162$		0			0
$163$	−37.3290			−0.229013			−0.114506	−	0.993423i	$-0.536529\pi$
$163$	−37.3290			−0.229013			−0.114506	+	0.993423i	$0.536529\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	73.2056			0.438357			0.219178	−	0.975685i	$-0.429662\pi$
$167$	73.2056			0.438357			0.219178	+	0.975685i	$0.429662\pi$
$168$		0			0
$169$	136.040			0.804970
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−209.810			−1.21277			−0.606387	−	0.795170i	$-0.707382\pi$
$173$	−209.810			−1.21277			−0.606387	+	0.795170i	$0.707382\pi$
$174$		0			0
$175$		−	66.0164i		−	0.377236i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−125.483			−0.701022			−0.350511	−	0.936559i	$-0.613992\pi$
$179$	−125.483			−0.701022			−0.350511	+	0.936559i	$0.613992\pi$
$180$		0			0
$181$		−	236.495i		−	1.30660i	−0.757098	−	0.653301i	$-0.773384\pi$
$181$		−	236.495i		−	1.30660i	0.757098	−	0.653301i	$-0.226616\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−75.0193			−0.405510
$186$		0			0
$187$	247.720			1.32471
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	219.291i		−	1.14812i	−0.818813	−	0.574061i	$-0.805367\pi$
$191$		−	219.291i		−	1.14812i	0.818813	−	0.574061i	$-0.194633\pi$
$192$		0			0
$193$	−177.218			−0.918227			−0.459114	−	0.888378i	$-0.651833\pi$
$193$	−177.218			−0.918227			−0.459114	+	0.888378i	$0.651833\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	323.818			1.64375			0.821873	−	0.569671i	$-0.192929\pi$
$197$	323.818			1.64375			0.821873	+	0.569671i	$0.192929\pi$
$198$		0			0
$199$			135.346i			0.680132i	0.940402	+	0.340066i	$0.110449\pi$
$199$			135.346i			0.680132i	−0.940402	+	0.340066i	$0.889551\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			352.081i			1.73439i
$204$		0			0
$205$		−	21.0183i		−	0.102528i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−71.6225			−0.342691
$210$		0			0
$211$	345.381			1.63688			0.818438	−	0.574594i	$-0.194840\pi$
$211$	345.381			1.63688			0.818438	+	0.574594i	$0.194840\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	77.1538			0.358855
$216$		0			0
$217$		−	227.644i		−	1.04905i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	385.909i		−	1.74619i
$222$		0			0
$223$	300.887			1.34927			0.674634	−	0.738153i	$-0.264302\pi$
$223$	300.887			1.34927			0.674634	+	0.738153i	$0.264302\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	349.586i		−	1.54003i	−0.638028	−	0.770013i	$-0.720250\pi$
$227$		−	349.586i		−	1.54003i	0.638028	−	0.770013i	$-0.279750\pi$
$228$		0			0
$229$		−	6.84200i		−	0.0298777i	−0.999888	−	0.0149389i	$-0.995245\pi$
$229$		−	6.84200i		−	0.0298777i	0.999888	−	0.0149389i	$-0.00475537\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−115.673			−0.496449			−0.248224	−	0.968703i	$-0.579847\pi$
$233$	−115.673			−0.496449			−0.248224	+	0.968703i	$0.579847\pi$
$234$		0			0
$235$		−	1.17469i		−	0.00499870i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	147.933			0.618966			0.309483	−	0.950905i	$-0.399844\pi$
$239$	147.933			0.618966			0.309483	+	0.950905i	$0.399844\pi$
$240$		0			0
$241$		−	249.228i		−	1.03414i	−0.855942	−	0.517071i	$-0.827022\pi$
$241$		−	249.228i		−	1.03414i	0.855942	−	0.517071i	$-0.172978\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			280.238i			1.14383i
$246$		0			0
$247$			111.576i			0.451726i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			41.4671i			0.165208i	0.996582	+	0.0826038i	$0.0263236\pi$
$251$			41.4671i			0.165208i	−0.996582	+	0.0826038i	$0.973676\pi$
$252$		0			0
$253$	59.1893	+	250.974i	0.233950	+	0.991993i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	220.328			0.857309			0.428654	−	0.903469i	$-0.358988\pi$
$257$	220.328			0.857309			0.428654	+	0.903469i	$0.358988\pi$
$258$		0			0
$259$	442.965			1.71029
$260$		0			0
$261$		0			0
$262$		0			0
$263$			319.837i			1.21611i	0.793895	+	0.608054i	$0.208050\pi$
$263$			319.837i			1.21611i	−0.793895	+	0.608054i	$0.791950\pi$
$264$		0			0
$265$	3.72705			0.0140643
$266$		0			0
$267$		0			0
$268$		0			0
$269$	355.239			1.32059			0.660295	−	0.751006i	$-0.270431\pi$
$269$	355.239			1.32059			0.660295	+	0.751006i	$0.270431\pi$
$270$		0			0
$271$	−140.163			−0.517208			−0.258604	−	0.965983i	$-0.583263\pi$
$271$	−140.163			−0.517208			−0.258604	+	0.965983i	$0.583263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			56.0564i			0.203841i
$276$		0			0
$277$	−349.667			−1.26234			−0.631169	−	0.775646i	$-0.717424\pi$
$277$	−349.667			−1.26234			−0.631169	+	0.775646i	$0.717424\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	155.570i		−	0.553631i	−0.960923	−	0.276815i	$-0.910721\pi$
$281$		−	155.570i		−	0.553631i	0.960923	−	0.276815i	$-0.0892790\pi$
$282$		0			0
$283$		−	430.828i		−	1.52236i	−0.648540	−	0.761181i	$-0.724620\pi$
$283$		−	430.828i		−	1.52236i	0.648540	−	0.761181i	$-0.275380\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			124.107i			0.432427i
$288$		0			0
$289$	−199.217			−0.689331
$290$		0			0
$291$		0			0
$292$		0			0
$293$			30.1358i			0.102853i	0.998677	+	0.0514263i	$0.0163767\pi$
$293$			30.1358i			0.102853i	−0.998677	+	0.0514263i	$0.983623\pi$
$294$		0			0
$295$		−	28.6359i		−	0.0970707i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	390.978	−	92.2075i	1.30762	−	0.308386i
$300$		0			0
$301$	−455.569			−1.51352
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−167.488			−0.549142
$306$		0			0
$307$	−170.383			−0.554994			−0.277497	−	0.960726i	$-0.589505\pi$
$307$	−170.383			−0.554994			−0.277497	+	0.960726i	$0.589505\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−300.372			−0.965826			−0.482913	−	0.875668i	$-0.660421\pi$
$311$	−300.372			−0.965826			−0.482913	+	0.875668i	$0.660421\pi$
$312$		0			0
$313$		−	320.778i		−	1.02485i	−0.858732	−	0.512424i	$-0.828747\pi$
$313$		−	320.778i		−	1.02485i	0.858732	−	0.512424i	$-0.171253\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−488.337			−1.54050			−0.770248	−	0.637744i	$-0.779868\pi$
$317$	−488.337			−1.54050			−0.770248	+	0.637744i	$0.779868\pi$
$318$		0			0
$319$		−	298.962i		−	0.937186i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	141.156			0.437017
$324$		0			0
$325$	87.3270			0.268698
$326$		0			0
$327$		0			0
$328$		0			0
$329$			6.93620i			0.0210827i
$330$		0			0
$331$	−117.006			−0.353492			−0.176746	−	0.984257i	$-0.556557\pi$
$331$	−117.006			−0.353492			−0.176746	+	0.984257i	$0.556557\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	95.1071			0.283902
$336$		0			0
$337$		−	190.806i		−	0.566190i	−0.959092	−	0.283095i	$-0.908639\pi$
$337$		−	190.806i		−	0.566190i	0.959092	−	0.283095i	$-0.0913612\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			193.299i			0.566860i
$342$		0			0
$343$		−	1007.76i		−	2.93807i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	633.222			1.82485			0.912424	−	0.409246i	$-0.134208\pi$
$347$	633.222			1.82485			0.912424	+	0.409246i	$0.134208\pi$
$348$		0			0
$349$	353.303			1.01233			0.506165	−	0.862437i	$-0.331063\pi$
$349$	353.303			1.01233			0.506165	+	0.862437i	$0.331063\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−615.486			−1.74359			−0.871793	−	0.489875i	$-0.837042\pi$
$353$	−615.486			−1.74359			−0.871793	+	0.489875i	$0.837042\pi$
$354$		0			0
$355$		−	69.2852i		−	0.195169i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			503.984i			1.40385i	0.712249	+	0.701927i	$0.247677\pi$
$359$			503.984i			1.40385i	−0.712249	+	0.701927i	$0.752323\pi$
$360$		0			0
$361$	320.188			0.886947
$362$		0			0
$363$		0			0
$364$		0			0
$365$			47.2013i			0.129319i
$366$		0			0
$367$		−	491.044i		−	1.33800i	−0.743264	−	0.668998i	$-0.766724\pi$
$367$		−	491.044i		−	1.33800i	0.743264	−	0.668998i	$-0.233276\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.0070			−0.0593181
$372$		0			0
$373$			251.072i			0.673116i	0.941663	+	0.336558i	$0.109263\pi$
$373$			251.072i			0.673116i	−0.941663	+	0.336558i	$0.890737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−465.736			−1.23537
$378$		0			0
$379$		−	680.011i		−	1.79422i	−0.441804	−	0.897112i	$-0.645661\pi$
$379$		−	680.011i		−	1.79422i	0.441804	−	0.897112i	$-0.354339\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	70.9926i		−	0.185359i	−0.995696	−	0.0926796i	$-0.970457\pi$
$383$		−	70.9926i		−	0.185359i	0.995696	−	0.0926796i	$-0.0295432\pi$
$384$		0			0
$385$		−	330.995i		−	0.859728i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	161.082i		−	0.414092i	−0.978331	−	0.207046i	$-0.933615\pi$
$389$		−	161.082i		−	0.414092i	0.978331	−	0.207046i	$-0.0663850\pi$
$390$		0			0
$391$	−116.653	−	494.630i	−0.298344	−	1.26504i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−57.1713			−0.144738
$396$		0			0
$397$	−644.249			−1.62279			−0.811397	−	0.584496i	$-0.801292\pi$
$397$	−644.249			−1.62279			−0.811397	+	0.584496i	$0.801292\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	341.135i		−	0.850711i	−0.905026	−	0.425356i	$-0.860149\pi$
$401$		−	341.135i		−	0.850711i	0.905026	−	0.425356i	$-0.139851\pi$
$402$		0			0
$403$	301.130			0.747220
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−376.134			−0.924163
$408$		0			0
$409$	−234.203			−0.572623			−0.286312	−	0.958137i	$-0.592429\pi$
$409$	−234.203			−0.572623			−0.286312	+	0.958137i	$0.592429\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			169.086i			0.409409i
$414$		0			0
$415$	−71.0508			−0.171207
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	381.051i		−	0.909429i	−0.890637	−	0.454714i	$-0.849741\pi$
$419$		−	381.051i		−	0.909429i	0.890637	−	0.454714i	$-0.150259\pi$
$420$		0			0
$421$			524.072i			1.24483i	0.782689	+	0.622413i	$0.213848\pi$
$421$			524.072i			1.24483i	−0.782689	+	0.622413i	$0.786152\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	110.478i		−	0.259949i
$426$		0			0
$427$	988.966			2.31608
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	589.078i		−	1.36677i	−0.730058	−	0.683385i	$-0.760507\pi$
$431$		−	589.078i		−	1.36677i	0.730058	−	0.683385i	$-0.239493\pi$
$432$		0			0
$433$		−	433.074i		−	1.00017i	−0.865976	−	0.500085i	$-0.833302\pi$
$433$		−	433.074i		−	1.00017i	0.865976	−	0.500085i	$-0.166698\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	33.7273	+	143.011i	0.0771792	+	0.327255i
$438$		0			0
$439$	−843.550			−1.92153			−0.960763	−	0.277371i	$-0.910537\pi$
$439$	−843.550			−1.92153			−0.960763	+	0.277371i	$0.910537\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	765.775			1.72861			0.864305	−	0.502967i	$-0.167758\pi$
$443$	765.775			1.72861			0.864305	+	0.502967i	$0.167758\pi$
$444$		0			0
$445$	−202.942			−0.456049
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−773.319			−1.72231			−0.861157	−	0.508339i	$-0.830260\pi$
$449$	−773.319			−1.72231			−0.861157	+	0.508339i	$0.830260\pi$
$450$		0			0
$451$		−	105.383i		−	0.233664i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−515.638			−1.13327
$456$		0			0
$457$		−	598.517i		−	1.30966i	−0.755774	−	0.654832i	$-0.772739\pi$
$457$		−	598.517i		−	1.30966i	0.755774	−	0.654832i	$-0.227261\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−58.1169			−0.126067			−0.0630335	−	0.998011i	$-0.520077\pi$
$461$	−58.1169			−0.126067			−0.0630335	+	0.998011i	$0.520077\pi$
$462$		0			0
$463$	−583.571			−1.26041			−0.630207	−	0.776427i	$-0.717030\pi$
$463$	−583.571			−1.26041			−0.630207	+	0.776427i	$0.717030\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	671.076i		−	1.43699i	−0.695531	−	0.718496i	$-0.744831\pi$
$467$		−	671.076i		−	1.43699i	0.695531	−	0.718496i	$-0.255169\pi$
$468$		0			0
$469$	−561.577			−1.19739
$470$		0			0
$471$		0			0
$472$		0			0
$473$	386.837			0.817836
$474$		0			0
$475$			31.9421i			0.0672466i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	162.207i		−	0.338637i	−0.985561	−	0.169318i	$-0.945843\pi$
$479$		−	162.207i		−	0.338637i	0.985561	−	0.169318i	$-0.0541566\pi$
$480$		0			0
$481$			585.958i			1.21821i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−115.324			−0.237782
$486$		0			0
$487$	−428.312			−0.879490			−0.439745	−	0.898123i	$-0.644931\pi$
$487$	−428.312			−0.879490			−0.439745	+	0.898123i	$0.644931\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−324.673			−0.661248			−0.330624	−	0.943763i	$-0.607259\pi$
$491$	−324.673			−0.661248			−0.330624	+	0.943763i	$0.607259\pi$
$492$		0			0
$493$			589.207i			1.19515i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			409.107i			0.823153i
$498$		0			0
$499$	332.157			0.665646			0.332823	−	0.942989i	$-0.391999\pi$
$499$	332.157			0.665646			0.332823	+	0.942989i	$0.391999\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			135.734i			0.269849i	0.990856	+	0.134924i	$0.0430792\pi$
$503$			135.734i			0.269849i	−0.990856	+	0.134924i	$0.956921\pi$
$504$		0			0
$505$			34.5491i			0.0684141i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	822.660			1.61623			0.808114	−	0.589026i	$-0.200488\pi$
$509$	822.660			1.61623			0.808114	+	0.589026i	$0.200488\pi$
$510$		0			0
$511$		−	278.709i		−	0.545419i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−20.8268			−0.0404403
$516$		0			0
$517$		−	5.88973i		−	0.0113921i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			707.690i			1.35833i	0.733985	+	0.679165i	$0.237658\pi$
$521$			707.690i			1.35833i	−0.733985	+	0.679165i	$0.762342\pi$
$522$		0			0
$523$		−	889.599i		−	1.70095i	−0.526013	−	0.850477i	$-0.676314\pi$
$523$		−	889.599i		−	1.70095i	0.526013	−	0.850477i	$-0.323686\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	380.962i		−	0.722888i
$528$		0			0
$529$	473.255	−	236.370i	0.894622	−	0.446824i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−164.169			−0.308010
$534$		0			0
$535$	−351.363			−0.656754
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1405.07i			2.60681i
$540$		0			0
$541$	−217.745			−0.402486			−0.201243	−	0.979541i	$-0.564498\pi$
$541$	−217.745			−0.402486			−0.201243	+	0.979541i	$0.564498\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−263.391			−0.483286
$546$		0			0
$547$	523.617			0.957252			0.478626	−	0.878019i	$-0.341135\pi$
$547$	523.617			0.957252			0.478626	+	0.878019i	$0.341135\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	170.355i		−	0.309175i
$552$		0			0
$553$	337.579			0.610449
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	609.072i		−	1.09349i	−0.837300	−	0.546743i	$-0.815867\pi$
$557$		−	609.072i		−	1.09349i	0.837300	−	0.546743i	$-0.184133\pi$
$558$		0			0
$559$		−	602.630i		−	1.07805i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			547.226i			0.971982i	0.873964	+	0.485991i	$0.161541\pi$
$563$			547.226i			0.971982i	−0.873964	+	0.485991i	$0.838459\pi$
$564$		0			0
$565$	494.512			0.875243
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	960.993i		−	1.68892i	−0.535621	−	0.844458i	$-0.679923\pi$
$569$		−	960.993i		−	1.68892i	0.535621	−	0.844458i	$-0.320077\pi$
$570$		0			0
$571$			28.4911i			0.0498969i	0.999689	+	0.0249484i	$0.00794216\pi$
$571$			28.4911i			0.0498969i	−0.999689	+	0.0249484i	$0.992058\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	111.929	−	26.3972i	0.194660	−	0.0459082i
$576$		0			0
$577$	324.840			0.562980			0.281490	−	0.959564i	$-0.409171\pi$
$577$	324.840			0.562980			0.281490	+	0.959564i	$0.409171\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	419.532			0.722086
$582$		0			0
$583$	18.6868			0.0320528
$584$		0			0
$585$		0			0
$586$		0			0
$587$	399.193			0.680055			0.340028	−	0.940415i	$-0.389564\pi$
$587$	399.193			0.680055			0.340028	+	0.940415i	$0.389564\pi$
$588$		0			0
$589$			110.146i			0.187005i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	475.610			0.802041			0.401021	−	0.916069i	$-0.368656\pi$
$593$	475.610			0.802041			0.401021	+	0.916069i	$0.368656\pi$
$594$		0			0
$595$			652.338i			1.09637i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	327.989			0.547561			0.273780	−	0.961792i	$-0.411726\pi$
$599$	327.989			0.547561			0.273780	+	0.961792i	$0.411726\pi$
$600$		0			0
$601$	396.256			0.659328			0.329664	−	0.944098i	$-0.393065\pi$
$601$	396.256			0.659328			0.329664	+	0.944098i	$0.393065\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			10.4933i			0.0173444i
$606$		0			0
$607$	−811.377			−1.33670			−0.668350	−	0.743847i	$-0.732999\pi$
$607$	−811.377			−1.33670			−0.668350	+	0.743847i	$0.732999\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.17526			−0.0150168
$612$		0			0
$613$		−	133.195i		−	0.217284i	−0.994081	−	0.108642i	$-0.965350\pi$
$613$		−	133.195i		−	0.217284i	0.994081	−	0.108642i	$-0.0346502\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			960.566i			1.55683i	0.627748	+	0.778417i	$0.283977\pi$
$617$			960.566i			1.55683i	−0.627748	+	0.778417i	$0.716023\pi$
$618$		0			0
$619$			213.414i			0.344771i	0.985030	+	0.172386i	$0.0551475\pi$
$619$			213.414i			0.344771i	−0.985030	+	0.172386i	$0.944852\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1198.31			1.92345
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	741.301			1.17854
$630$		0			0
$631$			263.928i			0.418269i	0.977887	+	0.209134i	$0.0670646\pi$
$631$			263.928i			0.418269i	−0.977887	+	0.209134i	$0.932935\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	79.7908i		−	0.125655i
$636$		0			0
$637$	2188.88			3.43622
$638$		0			0
$639$		0			0
$640$		0			0
$641$			185.770i			0.289813i	0.989445	+	0.144906i	$0.0462881\pi$
$641$			185.770i			0.289813i	−0.989445	+	0.144906i	$0.953712\pi$
$642$		0			0
$643$			50.0241i			0.0777979i	0.999243	+	0.0388990i	$0.0123850\pi$
$643$			50.0241i			0.0777979i	−0.999243	+	0.0388990i	$0.987615\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	686.937			1.06173			0.530863	−	0.847458i	$-0.321868\pi$
$647$	686.937			1.06173			0.530863	+	0.847458i	$0.321868\pi$
$648$		0			0
$649$		−	143.576i		−	0.221226i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1218.00			−1.86524			−0.932620	−	0.360859i	$-0.882484\pi$
$653$	−1218.00			−1.86524			−0.932620	+	0.360859i	$0.882484\pi$
$654$		0			0
$655$		−	34.7360i		−	0.0530321i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			716.541i			1.08732i	0.839307	+	0.543658i	$0.182961\pi$
$659$			716.541i			1.08732i	−0.839307	+	0.543658i	$0.817039\pi$
$660$		0			0
$661$			786.925i			1.19051i	0.803538	+	0.595253i	$0.202948\pi$
$661$			786.925i			1.19051i	−0.803538	+	0.595253i	$0.797052\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	188.608i		−	0.283621i
$666$		0			0
$667$	−596.946	+	140.783i	−0.894972	+	0.211068i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−839.759			−1.25150
$672$		0			0
$673$	794.006			1.17980			0.589900	−	0.807476i	$-0.299167\pi$
$673$	794.006			1.17980			0.589900	+	0.807476i	$0.299167\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			689.897i			1.01905i	0.860456	+	0.509525i	$0.170179\pi$
$677$			689.897i			1.01905i	−0.860456	+	0.509525i	$0.829821\pi$
$678$		0			0
$679$	680.953			1.00288
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−172.107			−0.251987			−0.125993	−	0.992031i	$-0.540212\pi$
$683$	−172.107			−0.251987			−0.125993	+	0.992031i	$0.540212\pi$
$684$		0			0
$685$	233.211			0.340454
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	29.1111i		−	0.0422512i
$690$		0			0
$691$	117.871			0.170580			0.0852900	−	0.996356i	$-0.472818\pi$
$691$	117.871			0.170580			0.0852900	+	0.996356i	$0.472818\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	244.314i		−	0.351531i
$696$		0			0
$697$			207.692i			0.297980i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	48.6907i		−	0.0694590i	−0.999397	−	0.0347295i	$-0.988943\pi$
$701$		−	48.6907i		−	0.0694590i	0.999397	−	0.0347295i	$-0.0110570\pi$
$702$		0			0
$703$	−214.330			−0.304878
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	204.002i		−	0.288546i
$708$		0			0
$709$			356.209i			0.502411i	0.967934	+	0.251205i	$0.0808269\pi$
$709$			356.209i			0.502411i	−0.967934	+	0.251205i	$0.919173\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	385.966	−	91.0255i	0.541327	−	0.127665i
$714$		0			0
$715$	437.843			0.612368
$716$		0			0
$717$		0			0
$718$		0			0
$719$	654.975			0.910953			0.455477	−	0.890248i	$-0.349469\pi$
$719$	654.975			0.910953			0.455477	+	0.890248i	$0.349469\pi$
$720$		0			0
$721$	122.976			0.170562
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−133.331			−0.183905
$726$		0			0
$727$		−	498.965i		−	0.686334i	−0.939274	−	0.343167i	$-0.888500\pi$
$727$		−	498.965i		−	0.686334i	0.939274	−	0.343167i	$-0.111500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−762.393			−1.04294
$732$		0			0
$733$			612.320i			0.835362i	0.908594	+	0.417681i	$0.137157\pi$
$733$			612.320i			0.835362i	−0.908594	+	0.417681i	$0.862843\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	476.851			0.647017
$738$		0			0
$739$	−772.436			−1.04524			−0.522622	−	0.852564i	$-0.675046\pi$
$739$	−772.436			−1.04524			−0.522622	+	0.852564i	$0.675046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	919.192i		−	1.23714i	−0.785731	−	0.618568i	$-0.787713\pi$
$743$		−	919.192i		−	1.23714i	0.785731	−	0.618568i	$-0.212287\pi$
$744$		0			0
$745$	−63.2187			−0.0848573
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2074.69			2.76995
$750$		0			0
$751$		−	910.779i		−	1.21276i	−0.795177	−	0.606378i	$-0.792622\pi$
$751$		−	910.779i		−	1.21276i	0.795177	−	0.606378i	$-0.207378\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	432.530i		−	0.572888i
$756$		0			0
$757$		−	303.316i		−	0.400681i	−0.979726	−	0.200341i	$-0.935795\pi$
$757$		−	303.316i		−	0.400681i	0.979726	−	0.200341i	$-0.0642049\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	722.666			0.949627			0.474814	−	0.880086i	$-0.342515\pi$
$761$	722.666			0.949627			0.474814	+	0.880086i	$0.342515\pi$
$762$		0			0
$763$	1555.24			2.03832
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−223.668			−0.291614
$768$		0			0
$769$		−	1281.01i		−	1.66582i	−0.553411	−	0.832909i	$-0.686674\pi$
$769$		−	1281.01i		−	1.66582i	0.553411	−	0.832909i	$-0.313326\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	713.668i		−	0.923245i	−0.887076	−	0.461623i	$-0.847267\pi$
$773$		−	713.668i		−	0.923245i	0.887076	−	0.461623i	$-0.152733\pi$
$774$		0			0
$775$	86.2075			0.111236
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	60.0492i		−	0.0770850i
$780$		0			0
$781$		−	347.384i		−	0.444794i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	154.718			0.197093
$786$		0			0
$787$		−	788.604i		−	1.00204i	−0.865436	−	0.501019i	$-0.832959\pi$
$787$		−	788.604i		−	1.00204i	0.865436	−	0.501019i	$-0.167041\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2919.94			−3.69145
$792$		0			0
$793$			1308.21i			1.64970i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1090.39i			1.36812i	0.729426	+	0.684060i	$0.239787\pi$
$797$			1090.39i			1.36812i	−0.729426	+	0.684060i	$0.760213\pi$
$798$		0			0
$799$			11.6077i			0.0145278i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			236.660i			0.294720i
$804$		0			0
$805$	−660.908	+	155.867i	−0.821003	+	0.193624i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−986.070			−1.21887			−0.609437	−	0.792834i	$-0.708605\pi$
$809$	−986.070			−1.21887			−0.609437	+	0.792834i	$0.708605\pi$
$810$		0			0
$811$	751.844			0.927057			0.463529	−	0.886082i	$-0.346583\pi$
$811$	751.844			0.927057			0.463529	+	0.886082i	$0.346583\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			83.4703i			0.102418i
$816$		0			0
$817$	220.428			0.269802
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1298.45			−1.58155			−0.790773	−	0.612110i	$-0.790321\pi$
$821$	−1298.45			−1.58155			−0.790773	+	0.612110i	$0.790321\pi$
$822$		0			0
$823$	−664.003			−0.806808			−0.403404	−	0.915022i	$-0.632173\pi$
$823$	−664.003			−0.806808			−0.403404	+	0.915022i	$0.632173\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			204.075i			0.246766i	0.992359	+	0.123383i	$0.0393743\pi$
$827$			204.075i			0.246766i	−0.992359	+	0.123383i	$0.960626\pi$
$828$		0			0
$829$	1234.50			1.48914			0.744570	−	0.667544i	$-0.232655\pi$
$829$	1234.50			1.48914			0.744570	+	0.667544i	$0.232655\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2769.17i		−	3.32433i
$834$		0			0
$835$		−	163.693i		−	0.196039i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	218.644i		−	0.260600i	−0.991475	−	0.130300i	$-0.958406\pi$
$839$		−	218.644i		−	0.260600i	0.991475	−	0.130300i	$-0.0415941\pi$
$840$		0			0
$841$	−129.913			−0.154475
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	304.195i		−	0.359994i
$846$		0			0
$847$		−	61.9598i		−	0.0731521i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	177.123	+	751.038i	0.208136	+	0.882536i
$852$		0			0
$853$	−1546.19			−1.81264			−0.906322	−	0.422587i	$-0.861122\pi$
$853$	−1546.19			−1.81264			−0.906322	+	0.422587i	$0.861122\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1093.05			1.27544			0.637718	−	0.770270i	$-0.279879\pi$
$857$	1093.05			1.27544			0.637718	+	0.770270i	$0.279879\pi$
$858$		0			0
$859$	784.906			0.913744			0.456872	−	0.889532i	$-0.348970\pi$
$859$	784.906			0.913744			0.456872	+	0.889532i	$0.348970\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−703.565			−0.815255			−0.407627	−	0.913148i	$-0.633644\pi$
$863$	−703.565			−0.815255			−0.407627	+	0.913148i	$0.633644\pi$
$864$		0			0
$865$			469.149i			0.542369i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−286.648			−0.329859
$870$		0			0
$871$		−	742.859i		−	0.852880i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−147.617			−0.168705
$876$		0			0
$877$	575.512			0.656228			0.328114	−	0.944638i	$-0.393587\pi$
$877$	575.512			0.656228			0.328114	+	0.944638i	$0.393587\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			362.368i			0.411314i	0.978624	+	0.205657i	$0.0659331\pi$
$881$			362.368i			0.411314i	−0.978624	+	0.205657i	$0.934067\pi$
$882$		0			0
$883$	−1312.15			−1.48602			−0.743008	−	0.669283i	$-0.766602\pi$
$883$	−1312.15			−1.48602			−0.743008	+	0.669283i	$0.766602\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	692.301			0.780498			0.390249	−	0.920709i	$-0.372389\pi$
$887$	692.301			0.780498			0.390249	+	0.920709i	$0.372389\pi$
$888$		0			0
$889$			471.139i			0.529966i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	3.35609i		−	0.00375822i
$894$		0			0
$895$			280.588i			0.313507i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−459.766			−0.511419
$900$		0			0
$901$	−36.8287			−0.0408753
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−528.819			−0.584330
$906$		0			0
$907$		−	982.356i		−	1.08308i	−0.840674	−	0.541541i	$-0.817841\pi$
$907$		−	982.356i		−	1.08308i	0.840674	−	0.541541i	$-0.182159\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1321.66i			1.45078i	0.688338	+	0.725390i	$0.258341\pi$
$911$			1321.66i			1.45078i	−0.688338	+	0.725390i	$0.741659\pi$
$912$		0			0
$913$	−356.237			−0.390183
$914$		0			0
$915$		0			0
$916$		0			0
$917$			205.105i			0.223670i
$918$		0			0
$919$			953.812i			1.03788i	0.854811	+	0.518940i	$0.173673\pi$
$919$			953.812i			1.03788i	−0.854811	+	0.518940i	$0.826327\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−541.170			−0.586316
$924$		0			0
$925$			167.748i			0.181349i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−322.827			−0.347499			−0.173749	−	0.984790i	$-0.555588\pi$
$929$	−322.827			−0.347499			−0.173749	+	0.984790i	$0.555588\pi$
$930$		0			0
$931$			800.639i			0.859977i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	553.919i		−	0.592427i
$936$		0			0
$937$			1159.90i			1.23789i	0.785434	+	0.618946i	$0.212440\pi$
$937$			1159.90i			1.23789i	−0.785434	+	0.618946i	$0.787560\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			400.394i			0.425499i	0.977107	+	0.212749i	$0.0682418\pi$
$941$			400.394i			0.425499i	−0.977107	+	0.212749i	$0.931758\pi$
$942$		0			0
$943$	−210.420	+	49.6251i	−0.223139	+	0.0526247i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.0706711			7.46263e−5			3.73132e−5	−	1.00000i	$-0.499988\pi$
$947$	0.0706711			7.46263e−5			3.73132e−5		1.00000i	$0.499988\pi$
$948$		0			0
$949$	368.678			0.388492
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	377.286i		−	0.395893i	−0.980213	−	0.197947i	$-0.936573\pi$
$953$		−	377.286i		−	0.395893i	0.980213	−	0.197947i	$-0.0634272\pi$
$954$		0			0
$955$	−490.350			−0.513455
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1377.04			−1.43591
$960$		0			0
$961$	−663.730			−0.690666
$962$		0			0
$963$		0			0
$964$		0			0
$965$			396.271i			0.410644i
$966$		0			0
$967$	−979.222			−1.01264			−0.506319	−	0.862346i	$-0.668994\pi$
$967$	−979.222			−1.01264			−0.506319	+	0.862346i	$0.668994\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	204.360i		−	0.210463i	−0.994448	−	0.105232i	$-0.966442\pi$
$971$		−	204.360i		−	0.210463i	0.994448	−	0.105232i	$-0.0335584\pi$
$972$		0			0
$973$			1442.60i			1.48263i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1474.62i			1.50934i	0.656106	+	0.754669i	$0.272202\pi$
$977$			1474.62i			1.50934i	−0.656106	+	0.754669i	$0.727798\pi$
$978$		0			0
$979$	−1017.52			−1.03934
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	462.209i		−	0.470203i	−0.971971	−	0.235101i	$-0.924458\pi$
$983$		−	462.209i		−	0.470203i	0.971971	−	0.235101i	$-0.0755422\pi$
$984$		0			0
$985$		−	724.079i		−	0.735106i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−182.163	−	772.408i	−0.184189	−	0.780999i
$990$		0			0
$991$	−1470.31			−1.48366			−0.741830	−	0.670587i	$-0.766042\pi$
$991$	−1470.31			−1.48366			−0.741830	+	0.670587i	$0.766042\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	302.643			0.304164
$996$		0			0
$997$	−1599.83			−1.60465			−0.802324	−	0.596889i	$-0.796403\pi$
$997$	−1599.83			−1.60465			−0.802324	+	0.596889i	$0.796403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.b.2161.16	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.32	yes	32
23.22	odd	2	inner	4140.3.d.b.2161.17	yes	32
69.68	even	2	inner	4140.3.d.b.2161.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.1	✓	32	69.68	even	2	inner
4140.3.d.b.2161.16	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.17	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.32	yes	32	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +13.2033i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +13.2033i q^{7} -11.2113i q^{11} -17.4654 q^{13} +22.0956i q^{17} -6.38843i q^{19} +(-22.3859 + 5.27944i) q^{23} -5.00000 q^{25} +26.6662 q^{29} -17.2415 q^{31} +29.5234 q^{35} -33.5497i q^{37} +9.39969 q^{41} +34.5042i q^{43} +0.525339 q^{47} -125.326 q^{49} +1.66679i q^{53} -25.0692 q^{55} +12.8063 q^{59} -74.9031i q^{61} +39.0538i q^{65} +42.5332i q^{67} +30.9853 q^{71} -21.1091 q^{73} +148.026 q^{77} -25.5678i q^{79} -31.7749i q^{83} +49.4073 q^{85} -90.7583i q^{89} -230.600i q^{91} -14.2850 q^{95} -51.5745i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) \) 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more