LMFDB - Embedded newform 684.5.h.f.37.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 2574 x^{12} - 7948 x^{11} + 5136095 x^{10} - 4313010 x^{9} + 3526383758 x^{8} + \cdots + 13\!\cdots\!00 $ x^14 - 2x^13 + 2574x^12 - 7948x^11 + 5136095x^10 - 4313010x^9 + 3526383758x^8 + 42678039416x^7 + 1723219853217x^6 + 14955695567410x^5 + 301377432088480x^4 + 2343588994038264x^3 + 36239668281491904x^2 + 197066822628281760*x + 1359248846310158400
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.14
Root			$-22.3263 - 38.6702i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.f.37.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+43.6525 q^{5} -50.6192 q^{7} +O(q^{10})$	$q+43.6525 q^{5} -50.6192 q^{7} +31.7924 q^{11} +150.658i q^{13} -148.637 q^{17} +(130.750 - 336.490i) q^{19} -135.922 q^{23} +1280.54 q^{25} +1238.37i q^{29} +1672.71i q^{31} -2209.66 q^{35} +109.089i q^{37} +887.790i q^{41} +695.944 q^{43} +1410.32 q^{47} +161.307 q^{49} -1179.54i q^{53} +1387.82 q^{55} +1301.37i q^{59} +6469.01 q^{61} +6576.61i q^{65} +1102.98i q^{67} +2831.99i q^{71} -2110.00 q^{73} -1609.31 q^{77} -11348.2i q^{79} -11958.5 q^{83} -6488.39 q^{85} +7732.88i q^{89} -7626.20i q^{91} +(5707.57 - 14688.6i) q^{95} +6825.33i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) $ 14 * q - 18 * q^5 - 86 * q^7	$ 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) $ 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	43.6525			1.74610			0.873050	−	0.487631i	$-0.162139\pi$
$5$	43.6525			1.74610			0.873050	+	0.487631i	$0.162139\pi$
$6$		0			0
$7$	−50.6192			−1.03305			−0.516523	−	0.856273i	$-0.672774\pi$
$7$	−50.6192			−1.03305			−0.516523	+	0.856273i	$0.672774\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	31.7924			0.262747			0.131374	−	0.991333i	$-0.458061\pi$
$11$	31.7924			0.262747			0.131374	+	0.991333i	$0.458061\pi$
$12$		0			0
$13$			150.658i			0.891469i	0.895165	+	0.445734i	$0.147057\pi$
$13$			150.658i			0.891469i	−0.895165	+	0.445734i	$0.852943\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−148.637			−0.514316			−0.257158	−	0.966369i	$-0.582786\pi$
$17$	−148.637			−0.514316			−0.257158	+	0.966369i	$0.582786\pi$
$18$		0			0
$19$	130.750	−	336.490i	0.362189	−	0.932105i
$19$	130.750	−	336.490i	0.362189	−	0.932105i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−135.922			−0.256941			−0.128471	−	0.991713i	$-0.541007\pi$
$23$	−135.922			−0.256941			−0.128471	+	0.991713i	$0.541007\pi$
$24$		0			0
$25$	1280.54			2.04887
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1238.37i			1.47250i	0.676712	+	0.736248i	$0.263404\pi$
$29$			1238.37i			1.47250i	−0.676712	+	0.736248i	$0.736596\pi$
$30$		0			0
$31$			1672.71i			1.74060i	0.492524	+	0.870299i	$0.336074\pi$
$31$			1672.71i			1.74060i	−0.492524	+	0.870299i	$0.663926\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2209.66			−1.80380
$36$		0			0
$37$			109.089i			0.0796848i	0.999206	+	0.0398424i	$0.0126856\pi$
$37$			109.089i			0.0796848i	−0.999206	+	0.0398424i	$0.987314\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			887.790i			0.528132i	0.964505	+	0.264066i	$0.0850637\pi$
$41$			887.790i			0.528132i	−0.964505	+	0.264066i	$0.914936\pi$
$42$		0			0
$43$	695.944			0.376389			0.188195	−	0.982132i	$-0.439736\pi$
$43$	695.944			0.376389			0.188195	+	0.982132i	$0.439736\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1410.32			0.638445			0.319222	−	0.947680i	$-0.396578\pi$
$47$	1410.32			0.638445			0.319222	+	0.947680i	$0.396578\pi$
$48$		0			0
$49$	161.307			0.0671833
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1179.54i		−	0.419916i	−0.977710	−	0.209958i	$-0.932667\pi$
$53$		−	1179.54i		−	0.419916i	0.977710	−	0.209958i	$-0.0673327\pi$
$54$		0			0
$55$	1387.82			0.458783
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1301.37i			0.373849i	0.982374	+	0.186925i	$0.0598520\pi$
$59$			1301.37i			0.373849i	−0.982374	+	0.186925i	$0.940148\pi$
$60$		0			0
$61$	6469.01			1.73851			0.869257	−	0.494360i	$-0.164598\pi$
$61$	6469.01			1.73851			0.869257	+	0.494360i	$0.164598\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6576.61i			1.55659i
$66$		0			0
$67$			1102.98i			0.245707i	0.992425	+	0.122853i	$0.0392045\pi$
$67$			1102.98i			0.245707i	−0.992425	+	0.122853i	$0.960796\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2831.99i			0.561791i	0.959738	+	0.280896i	$0.0906315\pi$
$71$			2831.99i			0.561791i	−0.959738	+	0.280896i	$0.909369\pi$
$72$		0			0
$73$	−2110.00			−0.395946			−0.197973	−	0.980207i	$-0.563436\pi$
$73$	−2110.00			−0.395946			−0.197973	+	0.980207i	$0.563436\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1609.31			−0.271430
$78$		0			0
$79$		−	11348.2i		−	1.81834i	−0.416427	−	0.909169i	$-0.636718\pi$
$79$		−	11348.2i		−	1.81834i	0.416427	−	0.909169i	$-0.363282\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11958.5			−1.73589			−0.867943	−	0.496664i	$-0.834558\pi$
$83$	−11958.5			−1.73589			−0.867943	+	0.496664i	$0.834558\pi$
$84$		0			0
$85$	−6488.39			−0.898047
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7732.88i			0.976251i	0.872774	+	0.488125i	$0.162319\pi$
$89$			7732.88i			0.976251i	−0.872774	+	0.488125i	$0.837681\pi$
$90$		0			0
$91$		−	7626.20i		−	0.920928i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5707.57	−	14688.6i	0.632418	−	1.62755i
$96$		0			0
$97$			6825.33i			0.725405i	0.931905	+	0.362702i	$0.118146\pi$
$97$			6825.33i			0.725405i	−0.931905	+	0.362702i	$0.881854\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15483.4			1.51784			0.758918	−	0.651186i	$-0.225728\pi$
$101$	15483.4			1.51784			0.758918	+	0.651186i	$0.225728\pi$
$102$		0			0
$103$		−	5392.54i		−	0.508299i	−0.967165	−	0.254149i	$-0.918204\pi$
$103$		−	5392.54i		−	0.508299i	0.967165	−	0.254149i	$-0.0817955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3885.03i		−	0.339334i	−0.985501	−	0.169667i	$-0.945731\pi$
$107$		−	3885.03i		−	0.339334i	0.985501	−	0.169667i	$-0.0542691\pi$
$108$		0			0
$109$			19873.7i			1.67273i	0.548176	+	0.836363i	$0.315322\pi$
$109$			19873.7i			1.67273i	−0.548176	+	0.836363i	$0.684678\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			21170.3i			1.65795i	0.559287	+	0.828974i	$0.311075\pi$
$113$			21170.3i			1.65795i	−0.559287	+	0.828974i	$0.688925\pi$
$114$		0			0
$115$	−5933.33			−0.448645
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7523.90			0.531312
$120$		0			0
$121$	−13630.2			−0.930964
$122$		0			0
$123$		0			0
$124$		0			0
$125$	28616.0			1.83142
$126$		0			0
$127$			11543.6i			0.715705i	0.933778	+	0.357852i	$0.116491\pi$
$127$			11543.6i			0.715705i	−0.933778	+	0.357852i	$0.883509\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11020.2			−0.642165			−0.321083	−	0.947051i	$-0.604047\pi$
$131$	−11020.2			−0.642165			−0.321083	+	0.947051i	$0.604047\pi$
$132$		0			0
$133$	−6618.47	+	17032.9i	−0.374158	+	0.962907i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20273.6			1.08016			0.540082	−	0.841612i	$-0.318393\pi$
$137$	20273.6			1.08016			0.540082	+	0.841612i	$0.318393\pi$
$138$		0			0
$139$	24502.5			1.26818			0.634090	−	0.773259i	$-0.281375\pi$
$139$	24502.5			1.26818			0.634090	+	0.773259i	$0.281375\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4789.79i			0.234231i
$144$		0			0
$145$			54057.9i			2.57112i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6735.72			0.303397			0.151699	−	0.988427i	$-0.451526\pi$
$149$	6735.72			0.303397			0.151699	+	0.988427i	$0.451526\pi$
$150$		0			0
$151$			3712.50i			0.162822i	0.996681	+	0.0814109i	$0.0259426\pi$
$151$			3712.50i			0.162822i	−0.996681	+	0.0814109i	$0.974057\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			73018.2i			3.03926i
$156$		0			0
$157$	3514.28			0.142573			0.0712865	−	0.997456i	$-0.477290\pi$
$157$	3514.28			0.142573			0.0712865	+	0.997456i	$0.477290\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6880.26			0.265432
$162$		0			0
$163$	39081.0			1.47092			0.735462	−	0.677566i	$-0.236965\pi$
$163$	39081.0			1.47092			0.735462	+	0.677566i	$0.236965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11688.3i		−	0.419101i	−0.977798	−	0.209550i	$-0.932800\pi$
$167$		−	11688.3i		−	0.419101i	0.977798	−	0.209550i	$-0.0672001\pi$
$168$		0			0
$169$	5863.10			0.205283
$170$		0			0
$171$		0			0
$172$		0			0
$173$			41377.4i			1.38252i	0.722607	+	0.691259i	$0.242944\pi$
$173$			41377.4i			1.38252i	−0.722607	+	0.691259i	$0.757056\pi$
$174$		0			0
$175$	−64820.0			−2.11657
$176$		0			0
$177$		0			0
$178$		0			0
$179$			49458.3i			1.54359i	0.635869	+	0.771797i	$0.280642\pi$
$179$			49458.3i			1.54359i	−0.635869	+	0.771797i	$0.719358\pi$
$180$		0			0
$181$			39604.1i			1.20888i	0.796651	+	0.604440i	$0.206603\pi$
$181$			39604.1i			1.20888i	−0.796651	+	0.604440i	$0.793397\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4761.99i			0.139138i
$186$		0			0
$187$	−4725.54			−0.135135
$188$		0			0
$189$		0			0
$190$		0			0
$191$	23461.3			0.643110			0.321555	−	0.946891i	$-0.395794\pi$
$191$	23461.3			0.643110			0.321555	+	0.946891i	$0.395794\pi$
$192$		0			0
$193$		−	46221.6i		−	1.24088i	−0.784253	−	0.620441i	$-0.786953\pi$
$193$		−	46221.6i		−	1.24088i	0.784253	−	0.620441i	$-0.213047\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12258.7			0.315874			0.157937	−	0.987449i	$-0.449516\pi$
$197$	12258.7			0.315874			0.157937	+	0.987449i	$0.449516\pi$
$198$		0			0
$199$	11457.9			0.289334			0.144667	−	0.989480i	$-0.453789\pi$
$199$	11457.9			0.289334			0.144667	+	0.989480i	$0.453789\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	62685.3i		−	1.52115i
$204$		0			0
$205$			38754.3i			0.922172i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4156.87	−	10697.8i	0.0951642	−	0.244908i
$210$		0			0
$211$			7800.02i			0.175199i	0.996156	+	0.0875994i	$0.0279195\pi$
$211$			7800.02i			0.175199i	−0.996156	+	0.0875994i	$0.972080\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	30379.7			0.657214
$216$		0			0
$217$		−	84671.5i		−	1.79812i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	22393.4i		−	0.458496i
$222$		0			0
$223$		−	69599.3i		−	1.39957i	−0.714353	−	0.699786i	$-0.753279\pi$
$223$		−	69599.3i		−	1.39957i	0.714353	−	0.699786i	$-0.246721\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			45789.5i			0.888615i	0.895874	+	0.444308i	$0.146550\pi$
$227$			45789.5i			0.888615i	−0.895874	+	0.444308i	$0.853450\pi$
$228$		0			0
$229$	19460.5			0.371093			0.185547	−	0.982635i	$-0.440594\pi$
$229$	19460.5			0.371093			0.185547	+	0.982635i	$0.440594\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−95276.3			−1.75498			−0.877491	−	0.479592i	$-0.840785\pi$
$233$	−95276.3			−1.75498			−0.877491	+	0.479592i	$0.840785\pi$
$234$		0			0
$235$	61564.2			1.11479
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−77773.2			−1.36155			−0.680776	−	0.732492i	$-0.738357\pi$
$239$	−77773.2			−1.36155			−0.680776	+	0.732492i	$0.738357\pi$
$240$		0			0
$241$		−	22190.7i		−	0.382065i	−0.981584	−	0.191033i	$-0.938816\pi$
$241$		−	22190.7i		−	0.382065i	0.981584	−	0.191033i	$-0.0611837\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7041.46			0.117309
$246$		0			0
$247$	50695.0	+	19698.6i	0.830942	+	0.322880i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−113978.			−1.80915			−0.904576	−	0.426313i	$-0.859812\pi$
$251$	−113978.			−1.80915			−0.904576	+	0.426313i	$0.859812\pi$
$252$		0			0
$253$	−4321.29			−0.0675107
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	103119.i		−	1.56125i	−0.625001	−	0.780624i	$-0.714901\pi$
$257$		−	103119.i		−	1.56125i	0.625001	−	0.780624i	$-0.285099\pi$
$258$		0			0
$259$		−	5521.98i		−	0.0823181i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−43454.0			−0.628229			−0.314114	−	0.949385i	$-0.601708\pi$
$263$	−43454.0			−0.628229			−0.314114	+	0.949385i	$0.601708\pi$
$264$		0			0
$265$		−	51490.0i		−	0.733215i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			101870.i			1.40781i	0.710296	+	0.703903i	$0.248561\pi$
$269$			101870.i			1.40781i	−0.710296	+	0.703903i	$0.751439\pi$
$270$		0			0
$271$	−53480.5			−0.728211			−0.364106	−	0.931358i	$-0.618625\pi$
$271$	−53480.5			−0.728211			−0.364106	+	0.931358i	$0.618625\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	40711.5			0.538334
$276$		0			0
$277$	51740.6			0.674329			0.337164	−	0.941446i	$-0.390532\pi$
$277$	51740.6			0.674329			0.337164	+	0.941446i	$0.390532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			85690.2i			1.08522i	0.839984	+	0.542611i	$0.182564\pi$
$281$			85690.2i			1.08522i	−0.839984	+	0.542611i	$0.817436\pi$
$282$		0			0
$283$	123827.			1.54612			0.773060	−	0.634333i	$-0.218725\pi$
$283$	123827.			1.54612			0.773060	+	0.634333i	$0.218725\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	44939.3i		−	0.545585i
$288$		0			0
$289$	−61428.0			−0.735479
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	95949.1i		−	1.11765i	−0.829286	−	0.558825i	$-0.811252\pi$
$293$		−	95949.1i		−	1.11765i	0.829286	−	0.558825i	$-0.188748\pi$
$294$		0			0
$295$			56808.0i			0.652778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	20477.8i		−	0.229055i
$300$		0			0
$301$	−35228.2			−0.388827
$302$		0			0
$303$		0			0
$304$		0			0
$305$	282389.			3.03562
$306$		0			0
$307$		−	651.880i		−	0.00691658i	−0.999994	−	0.00345829i	$-0.998899\pi$
$307$		−	651.880i		−	0.00691658i	0.999994	−	0.00345829i	$-0.00110081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−77619.4			−0.802509			−0.401254	−	0.915967i	$-0.631426\pi$
$311$	−77619.4			−0.802509			−0.401254	+	0.915967i	$0.631426\pi$
$312$		0			0
$313$	−169389.			−1.72901			−0.864506	−	0.502623i	$-0.832368\pi$
$313$	−169389.			−1.72901			−0.864506	+	0.502623i	$0.832368\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27880.5i		−	0.277449i	−0.990331	−	0.138724i	$-0.955700\pi$
$317$		−	27880.5i		−	0.277449i	0.990331	−	0.138724i	$-0.0443002\pi$
$318$		0			0
$319$			39370.8i			0.386894i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−19434.3	+	50014.9i	−0.186279	+	0.479396i
$324$		0			0
$325$			192924.i			1.82650i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−71389.6			−0.659543
$330$		0			0
$331$		−	85855.5i		−	0.783632i	−0.920044	−	0.391816i	$-0.871847\pi$
$331$		−	85855.5i		−	0.783632i	0.920044	−	0.391816i	$-0.128153\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			48147.7i			0.429029i
$336$		0			0
$337$			20954.8i			0.184511i	0.995735	+	0.0922557i	$0.0294077\pi$
$337$			20954.8i			0.184511i	−0.995735	+	0.0922557i	$0.970592\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			53179.7i			0.457338i
$342$		0			0
$343$	113372.			0.963642
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19445.9			−0.161499			−0.0807494	−	0.996734i	$-0.525731\pi$
$347$	−19445.9			−0.161499			−0.0807494	+	0.996734i	$0.525731\pi$
$348$		0			0
$349$	−11070.5			−0.0908903			−0.0454451	−	0.998967i	$-0.514471\pi$
$349$	−11070.5			−0.0908903			−0.0454451	+	0.998967i	$0.514471\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	136172.			1.09279			0.546397	−	0.837526i	$-0.315999\pi$
$353$	136172.			1.09279			0.546397	+	0.837526i	$0.315999\pi$
$354$		0			0
$355$			123623.i			0.980943i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	117143.			0.908927			0.454464	−	0.890765i	$-0.349831\pi$
$359$	117143.			0.908927			0.454464	+	0.890765i	$0.349831\pi$
$360$		0			0
$361$	−96129.8	−	87992.2i	−0.737639	−	0.675196i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−92106.6			−0.691361
$366$		0			0
$367$	128966.			0.957513			0.478756	−	0.877948i	$-0.341088\pi$
$367$	128966.			0.957513			0.478756	+	0.877948i	$0.341088\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			59707.6i			0.433792i
$372$		0			0
$373$		−	216583.i		−	1.55671i	−0.627826	−	0.778354i	$-0.716055\pi$
$373$		−	216583.i		−	1.55671i	0.627826	−	0.778354i	$-0.283945\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−186570.			−1.31268
$378$		0			0
$379$			96159.3i			0.669442i	0.942317	+	0.334721i	$0.108642\pi$
$379$			96159.3i			0.669442i	−0.942317	+	0.334721i	$0.891358\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	277514.i		−	1.89185i	−0.324384	−	0.945926i	$-0.605157\pi$
$383$		−	277514.i		−	1.89185i	0.324384	−	0.945926i	$-0.394843\pi$
$384$		0			0
$385$	−70250.4			−0.473944
$386$		0			0
$387$		0			0
$388$		0			0
$389$	102821.			0.679489			0.339744	−	0.940518i	$-0.389659\pi$
$389$	102821.			0.679489			0.339744	+	0.940518i	$0.389659\pi$
$390$		0			0
$391$	20203.1			0.132149
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	495379.i		−	3.17500i
$396$		0			0
$397$	195341.			1.23940			0.619700	−	0.784839i	$-0.287254\pi$
$397$	195341.			1.23940			0.619700	+	0.784839i	$0.287254\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	241903.i		−	1.50436i	−0.658955	−	0.752182i	$-0.729001\pi$
$401$		−	241903.i		−	1.50436i	0.658955	−	0.752182i	$-0.270999\pi$
$402$		0			0
$403$	−252008.			−1.55169
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3468.19i			0.0209370i
$408$		0			0
$409$		−	241532.i		−	1.44387i	−0.691960	−	0.721936i	$-0.743253\pi$
$409$		−	241532.i		−	1.44387i	0.691960	−	0.721936i	$-0.256747\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	65874.3i		−	0.386203i
$414$		0			0
$415$	−522019.			−3.03103
$416$		0			0
$417$		0			0
$418$		0			0
$419$	118451.			0.674700			0.337350	−	0.941379i	$-0.390469\pi$
$419$	118451.			0.674700			0.337350	+	0.941379i	$0.390469\pi$
$420$		0			0
$421$			22262.7i			0.125607i	0.998026	+	0.0628034i	$0.0200041\pi$
$421$			22262.7i			0.125607i	−0.998026	+	0.0628034i	$0.979996\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−190336.			−1.05376
$426$		0			0
$427$	−327456.			−1.79596
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	295793.i		−	1.59233i	−0.605077	−	0.796167i	$-0.706858\pi$
$431$		−	295793.i		−	1.59233i	0.605077	−	0.796167i	$-0.293142\pi$
$432$		0			0
$433$		−	36812.8i		−	0.196347i	−0.995169	−	0.0981733i	$-0.968700\pi$
$433$		−	36812.8i		−	0.196347i	0.995169	−	0.0981733i	$-0.0313000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17771.8	+	45736.3i	−0.0930613	+	0.239496i
$438$		0			0
$439$		−	300989.i		−	1.56178i	−0.624666	−	0.780892i	$-0.714765\pi$
$439$		−	300989.i		−	1.56178i	0.624666	−	0.780892i	$-0.285235\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	246797.			1.25757			0.628786	−	0.777578i	$-0.283552\pi$
$443$	246797.			1.25757			0.628786	+	0.777578i	$0.283552\pi$
$444$		0			0
$445$			337560.i			1.70463i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			232912.i			1.15531i	0.816280	+	0.577657i	$0.196033\pi$
$449$			232912.i			1.15531i	−0.816280	+	0.577657i	$0.803967\pi$
$450$		0			0
$451$			28225.0i			0.138765i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	332903.i		−	1.60803i
$456$		0			0
$457$	225698.			1.08068			0.540338	−	0.841448i	$-0.318296\pi$
$457$	225698.			1.08068			0.540338	+	0.841448i	$0.318296\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−163252.			−0.768171			−0.384085	−	0.923298i	$-0.625483\pi$
$461$	−163252.			−0.768171			−0.384085	+	0.923298i	$0.625483\pi$
$462$		0			0
$463$	−87602.3			−0.408652			−0.204326	−	0.978903i	$-0.565500\pi$
$463$	−87602.3			−0.408652			−0.204326	+	0.978903i	$0.565500\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−56036.2			−0.256942			−0.128471	−	0.991713i	$-0.541007\pi$
$467$	−56036.2			−0.256942			−0.128471	+	0.991713i	$0.541007\pi$
$468$		0			0
$469$		−	55831.9i		−	0.253826i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22125.8			0.0988954
$474$		0			0
$475$	167431.	−	430889.i	0.742076	−	1.90976i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−111163.			−0.484496			−0.242248	−	0.970214i	$-0.577885\pi$
$479$	−111163.			−0.484496			−0.242248	+	0.970214i	$0.577885\pi$
$480$		0			0
$481$	−16435.1			−0.0710365
$482$		0			0
$483$		0			0
$484$		0			0
$485$			297943.i			1.26663i
$486$		0			0
$487$		−	235462.i		−	0.992801i	−0.868094	−	0.496400i	$-0.834655\pi$
$487$		−	235462.i		−	0.992801i	0.868094	−	0.496400i	$-0.165345\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−200535.			−0.831818			−0.415909	−	0.909406i	$-0.636536\pi$
$491$	−200535.			−0.831818			−0.415909	+	0.909406i	$0.636536\pi$
$492$		0			0
$493$		−	184068.i		−	0.757327i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	143353.i		−	0.580356i
$498$		0			0
$499$	−61537.1			−0.247136			−0.123568	−	0.992336i	$-0.539434\pi$
$499$	−61537.1			−0.247136			−0.123568	+	0.992336i	$0.539434\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−293935.			−1.16176			−0.580878	−	0.813991i	$-0.697291\pi$
$503$	−293935.			−1.16176			−0.580878	+	0.813991i	$0.697291\pi$
$504$		0			0
$505$	675891.			2.65029
$506$		0			0
$507$		0			0
$508$		0			0
$509$			166965.i			0.644451i	0.946663	+	0.322225i	$0.104431\pi$
$509$			166965.i			0.644451i	−0.946663	+	0.322225i	$0.895569\pi$
$510$		0			0
$511$	106806.			0.409030
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	235398.i		−	0.887541i
$516$		0			0
$517$	44837.7			0.167750
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	384.781i		−	0.00141755i	−1.00000		0.000708774i	$-0.999774\pi$
$521$		−	384.781i		−	0.00141755i	1.00000		0.000708774i	$-0.000225610\pi$
$522$		0			0
$523$		−	271182.i		−	0.991421i	−0.868488	−	0.495710i	$-0.834908\pi$
$523$		−	271182.i		−	0.991421i	0.868488	−	0.495710i	$-0.165092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	248628.i		−	0.895217i
$528$		0			0
$529$	−261366.			−0.933981
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−133753.			−0.470813
$534$		0			0
$535$		−	169591.i		−	0.592510i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5128.35			0.0176523
$540$		0			0
$541$	126674.			0.432807			0.216404	−	0.976304i	$-0.430567\pi$
$541$	126674.			0.432807			0.216404	+	0.976304i	$0.430567\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			867535.i			2.92075i
$546$		0			0
$547$			186930.i			0.624747i	0.949959	+	0.312374i	$0.101124\pi$
$547$			186930.i			0.624747i	−0.949959	+	0.312374i	$0.898876\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	416698.	+	161917.i	1.37252	+	0.533321i
$552$		0			0
$553$			574440.i			1.87843i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24277.5			0.0782516			0.0391258	−	0.999234i	$-0.487543\pi$
$557$	24277.5			0.0782516			0.0391258	+	0.999234i	$0.487543\pi$
$558$		0			0
$559$			104850.i			0.335539i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			342537.i			1.08066i	0.841452	+	0.540331i	$0.181701\pi$
$563$			342537.i			1.08066i	−0.841452	+	0.540331i	$0.818299\pi$
$564$		0			0
$565$			924138.i			2.89494i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	110667.i		−	0.341816i	−0.985287	−	0.170908i	$-0.945330\pi$
$569$		−	110667.i		−	0.341816i	0.985287	−	0.170908i	$-0.0546700\pi$
$570$		0			0
$571$	−51343.2			−0.157475			−0.0787373	−	0.996895i	$-0.525089\pi$
$571$	−51343.2			−0.157475			−0.0787373	+	0.996895i	$0.525089\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−174054.			−0.526438
$576$		0			0
$577$	19162.2			0.0575565			0.0287782	−	0.999586i	$-0.490838\pi$
$577$	19162.2			0.0575565			0.0287782	+	0.999586i	$0.490838\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	605331.			1.79325
$582$		0			0
$583$		−	37500.6i		−	0.110332i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−572610.			−1.66182			−0.830908	−	0.556410i	$-0.812178\pi$
$587$	−572610.			−1.66182			−0.830908	+	0.556410i	$0.812178\pi$
$588$		0			0
$589$	562851.	+	218708.i	1.62242	+	0.630425i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	34217.3			0.0973053			0.0486526	−	0.998816i	$-0.484507\pi$
$593$	34217.3			0.0973053			0.0486526	+	0.998816i	$0.484507\pi$
$594$		0			0
$595$	328437.			0.927723
$596$		0			0
$597$		0			0
$598$		0			0
$599$			274960.i			0.766329i	0.923680	+	0.383164i	$0.125166\pi$
$599$			274960.i			0.766329i	−0.923680	+	0.383164i	$0.874834\pi$
$600$		0			0
$601$		−	566883.i		−	1.56944i	−0.619852	−	0.784719i	$-0.712807\pi$
$601$		−	566883.i		−	1.56944i	0.619852	−	0.784719i	$-0.287193\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−594994.			−1.62556
$606$		0			0
$607$			498944.i			1.35417i	0.735903	+	0.677087i	$0.236758\pi$
$607$			498944.i			1.35417i	−0.735903	+	0.677087i	$0.763242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			212477.i			0.569154i
$612$		0			0
$613$	−618533.			−1.64604			−0.823022	−	0.568009i	$-0.807714\pi$
$613$	−618533.			−1.64604			−0.823022	+	0.568009i	$0.807714\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	90695.6			0.238241			0.119120	−	0.992880i	$-0.461993\pi$
$617$	90695.6			0.238241			0.119120	+	0.992880i	$0.461993\pi$
$618$		0			0
$619$	605838.			1.58116			0.790578	−	0.612361i	$-0.209780\pi$
$619$	605838.			1.58116			0.790578	+	0.612361i	$0.209780\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	391433.i		−	1.00851i
$624$		0			0
$625$	448822.			1.14898
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16214.6i		−	0.0409832i
$630$		0			0
$631$	−670169.			−1.68316			−0.841581	−	0.540131i	$-0.818375\pi$
$631$	−670169.			−1.68316			−0.841581	+	0.540131i	$0.818375\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			503907.i			1.24969i
$636$		0			0
$637$			24302.3i			0.0598919i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	561901.i		−	1.36755i	−0.729692	−	0.683776i	$-0.760337\pi$
$641$		−	561901.i		−	1.36755i	0.729692	−	0.683776i	$-0.239663\pi$
$642$		0			0
$643$	306343.			0.740944			0.370472	−	0.928844i	$-0.379196\pi$
$643$	306343.			0.740944			0.370472	+	0.928844i	$0.379196\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	200604.			0.479215			0.239608	−	0.970870i	$-0.422981\pi$
$647$	200604.			0.479215			0.239608	+	0.970870i	$0.422981\pi$
$648$		0			0
$649$			41373.7i			0.0982279i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−73458.7			−0.172273			−0.0861364	−	0.996283i	$-0.527452\pi$
$653$	−73458.7			−0.172273			−0.0861364	+	0.996283i	$0.527452\pi$
$654$		0			0
$655$	−481059.			−1.12128
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	352876.i		−	0.812553i	−0.913750	−	0.406276i	$-0.866827\pi$
$659$		−	352876.i		−	0.812553i	0.913750	−	0.406276i	$-0.133173\pi$
$660$		0			0
$661$			451848.i			1.03416i	0.855936	+	0.517082i	$0.172982\pi$
$661$			451848.i			1.03416i	−0.855936	+	0.517082i	$0.827018\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−288913.	+	743527.i	−0.653317	+	1.68133i
$666$		0			0
$667$		−	168321.i		−	0.378345i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	205666.			0.456790
$672$		0			0
$673$		−	565185.i		−	1.24784i	−0.781486	−	0.623922i	$-0.785538\pi$
$673$		−	565185.i		−	1.24784i	0.781486	−	0.623922i	$-0.214462\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			537021.i			1.17169i	0.810422	+	0.585846i	$0.199238\pi$
$677$			537021.i			1.17169i	−0.810422	+	0.585846i	$0.800762\pi$
$678$		0			0
$679$		−	345493.i		−	0.749376i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	843141.i		−	1.80742i	−0.428146	−	0.903710i	$-0.640833\pi$
$683$		−	843141.i		−	1.80742i	0.428146	−	0.903710i	$-0.359167\pi$
$684$		0			0
$685$	884993.			1.88607
$686$		0			0
$687$		0			0
$688$		0			0
$689$	177708.			0.374342
$690$		0			0
$691$	−472040.			−0.988604			−0.494302	−	0.869290i	$-0.664576\pi$
$691$	−472040.			−0.988604			−0.494302	+	0.869290i	$0.664576\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.06960e6			2.21437
$696$		0			0
$697$		−	131959.i		−	0.271627i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	854584.			1.73908			0.869539	−	0.493865i	$-0.164416\pi$
$701$	854584.			1.73908			0.869539	+	0.493865i	$0.164416\pi$
$702$		0			0
$703$	36707.2	+	14263.3i	0.0742746	+	0.0288610i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−783760.			−1.56799
$708$		0			0
$709$	−489624.			−0.974026			−0.487013	−	0.873395i	$-0.661914\pi$
$709$	−489624.			−0.974026			−0.487013	+	0.873395i	$0.661914\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	227359.i		−	0.447231i
$714$		0			0
$715$			209086.i			0.408991i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	661267.			1.27914			0.639572	−	0.768732i	$-0.279112\pi$
$719$	661267.			1.27914			0.639572	+	0.768732i	$0.279112\pi$
$720$		0			0
$721$			272966.i			0.525096i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1.58578e6i			3.01694i
$726$		0			0
$727$	−822942.			−1.55704			−0.778521	−	0.627619i	$-0.784030\pi$
$727$	−822942.			−1.55704			−0.778521	+	0.627619i	$0.784030\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−103443.			−0.193583
$732$		0			0
$733$	−386539.			−0.719424			−0.359712	−	0.933063i	$-0.617125\pi$
$733$	−386539.			−0.719424			−0.359712	+	0.933063i	$0.617125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			35066.4i			0.0645588i
$738$		0			0
$739$	95863.7			0.175536			0.0877679	−	0.996141i	$-0.472027\pi$
$739$	95863.7			0.175536			0.0877679	+	0.996141i	$0.472027\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	90530.4i		−	0.163990i	−0.996633	−	0.0819949i	$-0.973871\pi$
$743$		−	90530.4i		−	0.163990i	0.996633	−	0.0819949i	$-0.0261291\pi$
$744$		0			0
$745$	294031.			0.529762
$746$		0			0
$747$		0			0
$748$		0			0
$749$			196657.i			0.350547i
$750$		0			0
$751$		−	614966.i		−	1.09036i	−0.838318	−	0.545182i	$-0.816461\pi$
$751$		−	614966.i		−	1.09036i	0.838318	−	0.545182i	$-0.183539\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			162060.i			0.284303i
$756$		0			0
$757$	−557166.			−0.972284			−0.486142	−	0.873880i	$-0.661596\pi$
$757$	−557166.			−0.972284			−0.486142	+	0.873880i	$0.661596\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	522882.			0.902889			0.451445	−	0.892299i	$-0.350909\pi$
$761$	522882.			0.902889			0.451445	+	0.892299i	$0.350909\pi$
$762$		0			0
$763$		−	1.00599e6i		−	1.72800i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−196062.			−0.333275
$768$		0			0
$769$	−871976.			−1.47452			−0.737262	−	0.675607i	$-0.763882\pi$
$769$	−871976.			−1.47452			−0.737262	+	0.675607i	$0.763882\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	122858.i		−	0.205611i	−0.994701	−	0.102805i	$-0.967218\pi$
$773$		−	122858.i		−	0.205611i	0.994701	−	0.102805i	$-0.0327819\pi$
$774$		0			0
$775$			2.14198e6i			3.56625i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	298732.	+	116079.i	0.492275	+	0.191284i
$780$		0			0
$781$			90035.9i			0.147609i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	153407.			0.248947
$786$		0			0
$787$		−	573825.i		−	0.926468i	−0.886236	−	0.463234i	$-0.846689\pi$
$787$		−	573825.i		−	0.926468i	0.886236	−	0.463234i	$-0.153311\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.07163e6i		−	1.71274i
$792$		0			0
$793$			974610.i			1.54983i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.07399e6i			1.69076i	0.534162	+	0.845382i	$0.320627\pi$
$797$			1.07399e6i			1.69076i	−0.534162	+	0.845382i	$0.679373\pi$
$798$		0			0
$799$	−209627.			−0.328362
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−67081.9			−0.104034
$804$		0			0
$805$	300341.			0.463471
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−916296.			−1.40004			−0.700018	−	0.714126i	$-0.746825\pi$
$809$	−916296.			−1.40004			−0.700018	+	0.714126i	$0.746825\pi$
$810$		0			0
$811$			1.00638e6i			1.53010i	0.643973	+	0.765048i	$0.277285\pi$
$811$			1.00638e6i			1.53010i	−0.643973	+	0.765048i	$0.722715\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.70598e6			2.56838
$816$		0			0
$817$	90994.8	−	234178.i	0.136324	−	0.350834i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	249844.			0.370667			0.185333	−	0.982676i	$-0.440664\pi$
$821$	249844.			0.370667			0.185333	+	0.982676i	$0.440664\pi$
$822$		0			0
$823$	−67236.8			−0.0992676			−0.0496338	−	0.998767i	$-0.515805\pi$
$823$	−67236.8			−0.0992676			−0.0496338	+	0.998767i	$0.515805\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			477032.i			0.697487i	0.937218	+	0.348743i	$0.113392\pi$
$827$			477032.i			0.697487i	−0.937218	+	0.348743i	$0.886608\pi$
$828$		0			0
$829$		−	353974.i		−	0.515065i	−0.966270	−	0.257532i	$-0.917091\pi$
$829$		−	353974.i		−	0.515065i	0.966270	−	0.257532i	$-0.0829094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23976.3			−0.0345534
$834$		0			0
$835$		−	510224.i		−	0.731792i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.01183e6i			1.43742i	0.695308	+	0.718712i	$0.255268\pi$
$839$			1.01183e6i			1.43742i	−0.695308	+	0.718712i	$0.744732\pi$
$840$		0			0
$841$	−826276.			−1.16824
$842$		0			0
$843$		0			0
$844$		0			0
$845$	255939.			0.358445
$846$		0			0
$847$	689952.			0.961728
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	14827.5i		−	0.0204743i
$852$		0			0
$853$	1.40206e6			1.92694			0.963470	−	0.267818i	$-0.0863025\pi$
$853$	1.40206e6			1.92694			0.963470	+	0.267818i	$0.0863025\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.01397e6i		−	1.38058i	−0.723532	−	0.690291i	$-0.757483\pi$
$857$		−	1.01397e6i		−	1.38058i	0.723532	−	0.690291i	$-0.242517\pi$
$858$		0			0
$859$	1.25699e6			1.70352			0.851759	−	0.523934i	$-0.175536\pi$
$859$	1.25699e6			1.70352			0.851759	+	0.523934i	$0.175536\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			934732.i			1.25506i	0.778591	+	0.627532i	$0.215935\pi$
$863$			934732.i			1.25506i	−0.778591	+	0.627532i	$0.784065\pi$
$864$		0			0
$865$			1.80623e6i			2.41402i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	360789.i		−	0.477764i
$870$		0			0
$871$	−166173.			−0.219040
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.44852e6			−1.89194
$876$		0			0
$877$		−	1.33841e6i		−	1.74016i	−0.492909	−	0.870081i	$-0.664066\pi$
$877$		−	1.33841e6i		−	1.74016i	0.492909	−	0.870081i	$-0.335934\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	754436.			0.972010			0.486005	−	0.873956i	$-0.338454\pi$
$881$	754436.			0.972010			0.486005	+	0.873956i	$0.338454\pi$
$882$		0			0
$883$	914747.			1.17322			0.586610	−	0.809869i	$-0.300462\pi$
$883$	914747.			1.17322			0.586610	+	0.809869i	$0.300462\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	850013.i		−	1.08039i	−0.841541	−	0.540193i	$-0.818351\pi$
$887$		−	850013.i		−	1.08039i	0.841541	−	0.540193i	$-0.181649\pi$
$888$		0			0
$889$		−	584328.i		−	0.739356i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	184400.	−	474560.i	0.231238	−	0.595097i
$894$		0			0
$895$			2.15898e6i			2.69527i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.07144e6			−2.56302
$900$		0			0
$901$			175324.i			0.215969i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.72882e6i			2.11082i
$906$		0			0
$907$		−	661996.i		−	0.804713i	−0.915483	−	0.402356i	$-0.868191\pi$
$907$		−	661996.i		−	0.804713i	0.915483	−	0.402356i	$-0.131809\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			885361.i			1.06680i	0.845863	+	0.533401i	$0.179086\pi$
$911$			885361.i			1.06680i	−0.845863	+	0.533401i	$0.820914\pi$
$912$		0			0
$913$	−380191.			−0.456100
$914$		0			0
$915$		0			0
$916$		0			0
$917$	557834.			0.663386
$918$		0			0
$919$	−700462.			−0.829380			−0.414690	−	0.909963i	$-0.636110\pi$
$919$	−700462.			−0.829380			−0.414690	+	0.909963i	$0.636110\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−426662.			−0.500819
$924$		0			0
$925$			139692.i			0.163263i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.01149e6			1.17201			0.586006	−	0.810307i	$-0.300699\pi$
$929$	1.01149e6			1.17201			0.586006	+	0.810307i	$0.300699\pi$
$930$		0			0
$931$	21090.9	−	54278.2i	0.0243331	−	0.0626219i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−206282.			−0.235959
$936$		0			0
$937$	1.29435e6			1.47426			0.737129	−	0.675752i	$-0.236181\pi$
$937$	1.29435e6			1.47426			0.737129	+	0.675752i	$0.236181\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.18632e6i		−	1.33975i	−0.742475	−	0.669874i	$-0.766348\pi$
$941$		−	1.18632e6i		−	1.33975i	0.742475	−	0.669874i	$-0.233652\pi$
$942$		0			0
$943$		−	120670.i		−	0.135699i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−398078.			−0.443883			−0.221941	−	0.975060i	$-0.571239\pi$
$947$	−398078.			−0.443883			−0.221941	+	0.975060i	$0.571239\pi$
$948$		0			0
$949$		−	317888.i		−	0.352974i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			677092.i			0.745525i	0.927927	+	0.372762i	$0.121589\pi$
$953$			677092.i			0.745525i	−0.927927	+	0.372762i	$0.878411\pi$
$954$		0			0
$955$	1.02414e6			1.12294
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.02623e6			−1.11586
$960$		0			0
$961$	−1.87445e6			−2.02968
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2.01769e6i		−	2.16671i
$966$		0			0
$967$	497168.			0.531680			0.265840	−	0.964017i	$-0.414351\pi$
$967$	497168.			0.531680			0.265840	+	0.964017i	$0.414351\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			79543.5i			0.0843657i	0.999110	+	0.0421829i	$0.0134312\pi$
$971$			79543.5i			0.0843657i	−0.999110	+	0.0421829i	$0.986569\pi$
$972$		0			0
$973$	−1.24030e6			−1.31009
$974$		0			0
$975$		0			0
$976$		0			0
$977$			644874.i			0.675594i	0.941219	+	0.337797i	$0.109682\pi$
$977$			644874.i			0.675594i	−0.941219	+	0.337797i	$0.890318\pi$
$978$		0			0
$979$			245847.i			0.256507i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			127085.i			0.131519i	0.997836	+	0.0657594i	$0.0209470\pi$
$983$			127085.i			0.131519i	−0.997836	+	0.0657594i	$0.979053\pi$
$984$		0			0
$985$	535125.			0.551547
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−94594.1			−0.0967100
$990$		0			0
$991$		−	1.03477e6i		−	1.05365i	−0.849974	−	0.526824i	$-0.823383\pi$
$991$		−	1.03477e6i		−	1.05365i	0.849974	−	0.526824i	$-0.176617\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	500166.			0.505206
$996$		0			0
$997$	276282.			0.277948			0.138974	−	0.990296i	$-0.455620\pi$
$997$	276282.			0.277948			0.138974	+	0.990296i	$0.455620\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.f.37.14		14
3.2	odd	2		228.5.h.a.37.8	yes	14
12.11	even	2		912.5.o.c.721.1		14
19.18	odd	2	inner	684.5.h.f.37.13		14
57.56	even	2		228.5.h.a.37.1	✓	14
228.227	odd	2		912.5.o.c.721.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.5.h.a.37.1	✓	14	57.56	even	2
228.5.h.a.37.8	yes	14	3.2	odd	2
684.5.h.f.37.13		14	19.18	odd	2	inner
684.5.h.f.37.14		14	1.1	even	1	trivial
912.5.o.c.721.1		14	12.11	even	2
912.5.o.c.721.8		14	228.227	odd	2

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+43.6525 q^{5} -50.6192 q^{7} +O(q^{10})\)	\(q+43.6525 q^{5} -50.6192 q^{7} +31.7924 q^{11} +150.658i q^{13} -148.637 q^{17} +(130.750 - 336.490i) q^{19} -135.922 q^{23} +1280.54 q^{25} +1238.37i q^{29} +1672.71i q^{31} -2209.66 q^{35} +109.089i q^{37} +887.790i q^{41} +695.944 q^{43} +1410.32 q^{47} +161.307 q^{49} -1179.54i q^{53} +1387.82 q^{55} +1301.37i q^{59} +6469.01 q^{61} +6576.61i q^{65} +1102.98i q^{67} +2831.99i q^{71} -2110.00 q^{73} -1609.31 q^{77} -11348.2i q^{79} -11958.5 q^{83} -6488.39 q^{85} +7732.88i q^{89} -7626.20i q^{91} +(5707.57 - 14688.6i) q^{95} +6825.33i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) \) 14 * q - 18 * q^5 - 86 * q^7	\( 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) \) 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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