LMFDB - Embedded newform 684.5.h.f.37.6

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			37.6
Root			$6.18424 - 10.7114i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.f.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.3685 q^{5} -81.9550 q^{7} +O(q^{10})$	$q-13.3685 q^{5} -81.9550 q^{7} -108.151 q^{11} +204.956i q^{13} +24.5332 q^{17} +(224.291 + 282.869i) q^{19} -832.333 q^{23} -446.283 q^{25} +895.912i q^{29} +1371.67i q^{31} +1095.61 q^{35} -1282.50i q^{37} -2045.46i q^{41} +2302.62 q^{43} -3566.48 q^{47} +4315.62 q^{49} -4859.83i q^{53} +1445.82 q^{55} -4699.94i q^{59} -4471.99 q^{61} -2739.95i q^{65} +2466.44i q^{67} +7309.42i q^{71} +5081.66 q^{73} +8863.53 q^{77} -949.794i q^{79} +4532.53 q^{83} -327.971 q^{85} -14417.2i q^{89} -16797.2i q^{91} +(-2998.43 - 3781.53i) q^{95} +12086.2i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−13.3685			−0.534740			−0.267370	−	0.963594i	$-0.586155\pi$
$5$	−13.3685			−0.534740			−0.267370	+	0.963594i	$0.586155\pi$
$6$		0			0
$7$	−81.9550			−1.67255			−0.836275	−	0.548310i	$-0.815272\pi$
$7$	−81.9550			−1.67255			−0.836275	+	0.548310i	$0.815272\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−108.151			−0.893812			−0.446906	−	0.894581i	$-0.647474\pi$
$11$	−108.151			−0.893812			−0.446906	+	0.894581i	$0.647474\pi$
$12$		0			0
$13$			204.956i			1.21276i	0.795176	+	0.606378i	$0.207378\pi$
$13$			204.956i			1.21276i	−0.795176	+	0.606378i	$0.792622\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	24.5332			0.0848898			0.0424449	−	0.999099i	$-0.486485\pi$
$17$	24.5332			0.0848898			0.0424449	+	0.999099i	$0.486485\pi$
$18$		0			0
$19$	224.291	+	282.869i	0.621304	+	0.783570i
$19$	224.291	+	282.869i	0.621304	+	0.783570i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−832.333			−1.57341			−0.786704	−	0.617331i	$-0.788214\pi$
$23$	−832.333			−1.57341			−0.786704	+	0.617331i	$0.788214\pi$
$24$		0			0
$25$	−446.283			−0.714054
$26$		0			0
$27$		0			0
$28$		0			0
$29$			895.912i			1.06529i	0.846338	+	0.532647i	$0.178803\pi$
$29$			895.912i			1.06529i	−0.846338	+	0.532647i	$0.821197\pi$
$30$		0			0
$31$			1371.67i			1.42733i	0.700486	+	0.713667i	$0.252967\pi$
$31$			1371.67i			1.42733i	−0.700486	+	0.713667i	$0.747033\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1095.61			0.894379
$36$		0			0
$37$		−	1282.50i		−	0.936813i	−0.883513	−	0.468407i	$-0.844828\pi$
$37$		−	1282.50i		−	0.936813i	0.883513	−	0.468407i	$-0.155172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2045.46i		−	1.21681i	−0.793625	−	0.608407i	$-0.791809\pi$
$41$		−	2045.46i		−	1.21681i	0.793625	−	0.608407i	$-0.208191\pi$
$42$		0			0
$43$	2302.62			1.24533			0.622667	−	0.782487i	$-0.286049\pi$
$43$	2302.62			1.24533			0.622667	+	0.782487i	$0.286049\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3566.48			−1.61452			−0.807260	−	0.590195i	$-0.799051\pi$
$47$	−3566.48			−1.61452			−0.807260	+	0.590195i	$0.799051\pi$
$48$		0			0
$49$	4315.62			1.79743
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4859.83i		−	1.73009i	−0.501691	−	0.865047i	$-0.667289\pi$
$53$		−	4859.83i		−	1.73009i	0.501691	−	0.865047i	$-0.332711\pi$
$54$		0			0
$55$	1445.82			0.477957
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4699.94i		−	1.35017i	−0.737740	−	0.675085i	$-0.764107\pi$
$59$		−	4699.94i		−	1.35017i	0.737740	−	0.675085i	$-0.235893\pi$
$60$		0			0
$61$	−4471.99			−1.20182			−0.600912	−	0.799315i	$-0.705196\pi$
$61$	−4471.99			−1.20182			−0.600912	+	0.799315i	$0.705196\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2739.95i		−	0.648509i
$66$		0			0
$67$			2466.44i			0.549440i	0.961524	+	0.274720i	$0.0885851\pi$
$67$			2466.44i			0.549440i	−0.961524	+	0.274720i	$0.911415\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7309.42i			1.44999i	0.688752	+	0.724997i	$0.258159\pi$
$71$			7309.42i			1.44999i	−0.688752	+	0.724997i	$0.741841\pi$
$72$		0			0
$73$	5081.66			0.953587			0.476793	−	0.879015i	$-0.341799\pi$
$73$	5081.66			0.953587			0.476793	+	0.879015i	$0.341799\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8863.53			1.49495
$78$		0			0
$79$		−	949.794i		−	0.152186i	−0.997101	−	0.0760931i	$-0.975755\pi$
$79$		−	949.794i		−	0.152186i	0.997101	−	0.0760931i	$-0.0242446\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4532.53			0.657937			0.328968	−	0.944341i	$-0.393299\pi$
$83$	4532.53			0.657937			0.328968	+	0.944341i	$0.393299\pi$
$84$		0			0
$85$	−327.971			−0.0453939
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14417.2i		−	1.82012i	−0.414471	−	0.910062i	$-0.636033\pi$
$89$		−	14417.2i		−	1.82012i	0.414471	−	0.910062i	$-0.363967\pi$
$90$		0			0
$91$		−	16797.2i		−	2.02840i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2998.43	−	3781.53i	−0.332236	−	0.419006i
$96$		0			0
$97$			12086.2i			1.28453i	0.766481	+	0.642267i	$0.222006\pi$
$97$			12086.2i			1.28453i	−0.766481	+	0.642267i	$0.777994\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3696.36			0.362353			0.181177	−	0.983451i	$-0.442009\pi$
$101$	3696.36			0.362353			0.181177	+	0.983451i	$0.442009\pi$
$102$		0			0
$103$			7703.79i			0.726156i	0.931759	+	0.363078i	$0.118274\pi$
$103$			7703.79i			0.726156i	−0.931759	+	0.363078i	$0.881726\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			836.583i			0.0730704i	0.999332	+	0.0365352i	$0.0116321\pi$
$107$			836.583i			0.0730704i	−0.999332	+	0.0365352i	$0.988368\pi$
$108$		0			0
$109$		−	23242.7i		−	1.95629i	−0.207921	−	0.978146i	$-0.566670\pi$
$109$		−	23242.7i		−	1.95629i	0.207921	−	0.978146i	$-0.433330\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17946.7i			1.40549i	0.711443	+	0.702743i	$0.248042\pi$
$113$			17946.7i			1.40549i	−0.711443	+	0.702743i	$0.751958\pi$
$114$		0			0
$115$	11127.0			0.841363
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2010.61			−0.141982
$120$		0			0
$121$	−2944.31			−0.201101
$122$		0			0
$123$		0			0
$124$		0			0
$125$	14321.4			0.916572
$126$		0			0
$127$			15263.8i			0.946358i	0.880966	+	0.473179i	$0.156894\pi$
$127$			15263.8i			0.946358i	−0.880966	+	0.473179i	$0.843106\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2872.42			0.167380			0.0836902	−	0.996492i	$-0.473329\pi$
$131$	2872.42			0.167380			0.0836902	+	0.996492i	$0.473329\pi$
$132$		0			0
$133$	−18381.7	−	23182.5i	−1.03916	−	1.31056i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13524.7			0.720589			0.360295	−	0.932839i	$-0.382676\pi$
$137$	13524.7			0.720589			0.360295	+	0.932839i	$0.382676\pi$
$138$		0			0
$139$	−25274.6			−1.30814			−0.654071	−	0.756433i	$-0.726940\pi$
$139$	−25274.6			−1.30814			−0.654071	+	0.756433i	$0.726940\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	22166.2i		−	1.08398i
$144$		0			0
$145$		−	11977.0i		−	0.569654i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10578.3			0.476479			0.238240	−	0.971206i	$-0.423430\pi$
$149$	10578.3			0.476479			0.238240	+	0.971206i	$0.423430\pi$
$150$		0			0
$151$		−	31302.3i		−	1.37285i	−0.727203	−	0.686423i	$-0.759180\pi$
$151$		−	31302.3i		−	1.37285i	0.727203	−	0.686423i	$-0.240820\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	18337.1i		−	0.763252i
$156$		0			0
$157$	−3605.28			−0.146265			−0.0731324	−	0.997322i	$-0.523300\pi$
$157$	−3605.28			−0.146265			−0.0731324	+	0.997322i	$0.523300\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	68213.8			2.63160
$162$		0			0
$163$	30685.7			1.15494			0.577472	−	0.816410i	$-0.304039\pi$
$163$	30685.7			1.15494			0.577472	+	0.816410i	$0.304039\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12295.3i		−	0.440867i	−0.975402	−	0.220433i	$-0.929253\pi$
$167$		−	12295.3i		−	0.440867i	0.975402	−	0.220433i	$-0.0707471\pi$
$168$		0			0
$169$	−13445.9			−0.470779
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3234.07i		−	0.108058i	−0.998539	−	0.0540290i	$-0.982794\pi$
$173$		−	3234.07i		−	0.108058i	0.998539	−	0.0540290i	$-0.0172063\pi$
$174$		0			0
$175$	36575.2			1.19429
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	34471.8i		−	1.07586i	−0.842988	−	0.537932i	$-0.819206\pi$
$179$		−	34471.8i		−	1.07586i	0.842988	−	0.537932i	$-0.180794\pi$
$180$		0			0
$181$			6994.30i			0.213495i	0.994286	+	0.106747i	$0.0340436\pi$
$181$			6994.30i			0.213495i	−0.994286	+	0.106747i	$0.965956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17145.1i			0.500951i
$186$		0			0
$187$	−2653.29			−0.0758755
$188$		0			0
$189$		0			0
$190$		0			0
$191$	59531.9			1.63186			0.815930	−	0.578151i	$-0.196225\pi$
$191$	59531.9			1.63186			0.815930	+	0.578151i	$0.196225\pi$
$192$		0			0
$193$			30306.9i			0.813631i	0.913510	+	0.406816i	$0.133361\pi$
$193$			30306.9i			0.813631i	−0.913510	+	0.406816i	$0.866639\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−40475.9			−1.04295			−0.521476	−	0.853266i	$-0.674618\pi$
$197$	−40475.9			−1.04295			−0.521476	+	0.853266i	$0.674618\pi$
$198$		0			0
$199$	27389.1			0.691627			0.345814	−	0.938303i	$-0.387603\pi$
$199$	27389.1			0.691627			0.345814	+	0.938303i	$0.387603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	73424.4i		−	1.78176i
$204$		0			0
$205$			27344.8i			0.650678i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24257.3	−	30592.6i	−0.555329	−	0.700364i
$210$		0			0
$211$			8121.05i			0.182409i	0.995832	+	0.0912047i	$0.0290718\pi$
$211$			8121.05i			0.182409i	−0.995832	+	0.0912047i	$0.970928\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−30782.6			−0.665929
$216$		0			0
$217$		−	112415.i		−	2.38729i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5028.21i			0.102951i
$222$		0			0
$223$		−	730.403i		−	0.0146877i	−0.999973	−	0.00734384i	$-0.997662\pi$
$223$		−	730.403i		−	0.0146877i	0.999973	−	0.00734384i	$-0.00233764\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			58508.5i			1.13545i	0.823219	+	0.567724i	$0.192176\pi$
$227$			58508.5i			1.13545i	−0.823219	+	0.567724i	$0.807824\pi$
$228$		0			0
$229$	−5274.04			−0.100571			−0.0502855	−	0.998735i	$-0.516013\pi$
$229$	−5274.04			−0.100571			−0.0502855	+	0.998735i	$0.516013\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12816.2			0.236075			0.118037	−	0.993009i	$-0.462340\pi$
$233$	12816.2			0.236075			0.118037	+	0.993009i	$0.462340\pi$
$234$		0			0
$235$	47678.4			0.863348
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11814.8			0.206837			0.103419	−	0.994638i	$-0.467022\pi$
$239$	11814.8			0.206837			0.103419	+	0.994638i	$0.467022\pi$
$240$		0			0
$241$		−	36326.7i		−	0.625449i	−0.949844	−	0.312725i	$-0.898758\pi$
$241$		−	36326.7i		−	0.625449i	0.949844	−	0.312725i	$-0.101242\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−57693.3			−0.961155
$246$		0			0
$247$	−57975.6	+	45969.7i	−0.950280	+	0.753490i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−52902.0			−0.839700			−0.419850	−	0.907593i	$-0.637917\pi$
$251$	−52902.0			−0.839700			−0.419850	+	0.907593i	$0.637917\pi$
$252$		0			0
$253$	90017.8			1.40633
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	98763.5i		−	1.49531i	−0.664090	−	0.747653i	$-0.731181\pi$
$257$		−	98763.5i		−	1.49531i	0.664090	−	0.747653i	$-0.268819\pi$
$258$		0			0
$259$			105107.i			1.56687i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−114821.			−1.66001			−0.830006	−	0.557754i	$-0.811663\pi$
$263$	−114821.			−1.66001			−0.830006	+	0.557754i	$0.811663\pi$
$264$		0			0
$265$			64968.6i			0.925150i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			35137.2i			0.485582i	0.970079	+	0.242791i	$0.0780629\pi$
$269$			35137.2i			0.485582i	−0.970079	+	0.242791i	$0.921937\pi$
$270$		0			0
$271$	88780.2			1.20886			0.604432	−	0.796657i	$-0.293400\pi$
$271$	88780.2			1.20886			0.604432	+	0.796657i	$0.293400\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	48266.1			0.638229
$276$		0			0
$277$	52251.1			0.680983			0.340491	−	0.940248i	$-0.389407\pi$
$277$	52251.1			0.680983			0.340491	+	0.940248i	$0.389407\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			71845.0i			0.909879i	0.890522	+	0.454939i	$0.150339\pi$
$281$			71845.0i			0.909879i	−0.890522	+	0.454939i	$0.849661\pi$
$282$		0			0
$283$	−1186.28			−0.0148121			−0.00740603	−	0.999973i	$-0.502357\pi$
$283$	−1186.28			−0.0148121			−0.00740603	+	0.999973i	$0.502357\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			167636.i			2.03518i
$288$		0			0
$289$	−82919.1			−0.992794
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	47328.0i		−	0.551294i	−0.961259	−	0.275647i	$-0.911108\pi$
$293$		−	47328.0i		−	0.551294i	0.961259	−	0.275647i	$-0.0888920\pi$
$294$		0			0
$295$			62831.1i			0.721990i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	170591.i		−	1.90816i
$300$		0			0
$301$	−188711.			−2.08288
$302$		0			0
$303$		0			0
$304$		0			0
$305$	59783.7			0.642663
$306$		0			0
$307$		−	42985.4i		−	0.456083i	−0.973651	−	0.228041i	$-0.926768\pi$
$307$		−	42985.4i		−	0.456083i	0.973651	−	0.228041i	$-0.0732322\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−39527.9			−0.408679			−0.204340	−	0.978900i	$-0.565505\pi$
$311$	−39527.9			−0.408679			−0.204340	+	0.978900i	$0.565505\pi$
$312$		0			0
$313$	107827.			1.10062			0.550309	−	0.834961i	$-0.314510\pi$
$313$	107827.			1.10062			0.550309	+	0.834961i	$0.314510\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			92174.7i			0.917262i	0.888627	+	0.458631i	$0.151660\pi$
$317$			92174.7i			0.917262i	−0.888627	+	0.458631i	$0.848340\pi$
$318$		0			0
$319$		−	96893.9i		−	0.952172i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5502.56	+	6939.66i	0.0527423	+	0.0665171i
$324$		0			0
$325$		−	91468.4i		−	0.865973i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	292291.			2.70037
$330$		0			0
$331$			132917.i			1.21318i	0.795016	+	0.606589i	$0.207463\pi$
$331$			132917.i			1.21318i	−0.795016	+	0.606589i	$0.792537\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	32972.5i		−	0.293807i
$336$		0			0
$337$		−	33846.3i		−	0.298024i	−0.988835	−	0.149012i	$-0.952391\pi$
$337$		−	33846.3i		−	0.298024i	0.988835	−	0.149012i	$-0.0476094\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	148347.i		−	1.27577i
$342$		0			0
$343$	−156913.			−1.33374
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−159566.			−1.32520			−0.662602	−	0.748972i	$-0.730548\pi$
$347$	−159566.			−1.32520			−0.662602	+	0.748972i	$0.730548\pi$
$348$		0			0
$349$	131195.			1.07712			0.538562	−	0.842586i	$-0.318968\pi$
$349$	131195.			1.07712			0.538562	+	0.842586i	$0.318968\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11866.1			−0.0952264			−0.0476132	−	0.998866i	$-0.515161\pi$
$353$	−11866.1			−0.0952264			−0.0476132	+	0.998866i	$0.515161\pi$
$354$		0			0
$355$		−	97715.8i		−	0.775369i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−173900.			−1.34931			−0.674654	−	0.738134i	$-0.735707\pi$
$359$	−173900.			−1.34931			−0.674654	+	0.738134i	$0.735707\pi$
$360$		0			0
$361$	−29708.4	+	126890.i	−0.227964	+	0.973670i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−67934.2			−0.509921
$366$		0			0
$367$	−190202.			−1.41216			−0.706078	−	0.708134i	$-0.749537\pi$
$367$	−190202.			−1.41216			−0.706078	+	0.708134i	$0.749537\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			398288.i			2.89367i
$372$		0			0
$373$		−	23159.5i		−	0.166461i	−0.996530	−	0.0832304i	$-0.973476\pi$
$373$		−	23159.5i		−	0.166461i	0.996530	−	0.0832304i	$-0.0265237\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−183622.			−1.29194
$378$		0			0
$379$			108736.i			0.757001i	0.925601	+	0.378501i	$0.123560\pi$
$379$			108736.i			0.757001i	−0.925601	+	0.378501i	$0.876440\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			193237.i			1.31732i	0.752440	+	0.658661i	$0.228877\pi$
$383$			193237.i			1.31732i	−0.752440	+	0.658661i	$0.771123\pi$
$384$		0			0
$385$	−118492.			−0.799407
$386$		0			0
$387$		0			0
$388$		0			0
$389$	257431.			1.70122			0.850611	−	0.525796i	$-0.176232\pi$
$389$	257431.			1.70122			0.850611	+	0.525796i	$0.176232\pi$
$390$		0			0
$391$	−20419.7			−0.133566
$392$		0			0
$393$		0			0
$394$		0			0
$395$			12697.3i			0.0813800i
$396$		0			0
$397$	−60954.1			−0.386742			−0.193371	−	0.981126i	$-0.561942\pi$
$397$	−60954.1			−0.386742			−0.193371	+	0.981126i	$0.561942\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			119175.i			0.741132i	0.928806	+	0.370566i	$0.120836\pi$
$401$			119175.i			0.741132i	−0.928806	+	0.370566i	$0.879164\pi$
$402$		0			0
$403$	−281131.			−1.73101
$404$		0			0
$405$		0			0
$406$		0			0
$407$			138704.i			0.837335i
$408$		0			0
$409$		−	149621.i		−	0.894427i	−0.894427	−	0.447213i	$-0.852416\pi$
$409$		−	149621.i		−	0.894427i	0.894427	−	0.447213i	$-0.147584\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			385184.i			2.25823i
$414$		0			0
$415$	−60593.0			−0.351825
$416$		0			0
$417$		0			0
$418$		0			0
$419$	174064.			0.991474			0.495737	−	0.868473i	$-0.334898\pi$
$419$	174064.			0.991474			0.495737	+	0.868473i	$0.334898\pi$
$420$		0			0
$421$			182840.i			1.03159i	0.856711	+	0.515796i	$0.172504\pi$
$421$			182840.i			1.03159i	−0.856711	+	0.515796i	$0.827496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10948.7			−0.0606159
$426$		0			0
$427$	366502.			2.01011
$428$		0			0
$429$		0			0
$430$		0			0
$431$			302356.i			1.62766i	0.581103	+	0.813830i	$0.302621\pi$
$431$			302356.i			1.62766i	−0.581103	+	0.813830i	$0.697379\pi$
$432$		0			0
$433$		−	82990.9i		−	0.442644i	−0.975201	−	0.221322i	$-0.928963\pi$
$433$		−	82990.9i		−	0.442644i	0.975201	−	0.221322i	$-0.0710372\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−186684.	−	235441.i	−0.977564	−	1.23287i
$438$		0			0
$439$		−	180064.i		−	0.934325i	−0.884171	−	0.467162i	$-0.845276\pi$
$439$		−	180064.i		−	0.934325i	0.884171	−	0.467162i	$-0.154724\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−282685.			−1.44044			−0.720219	−	0.693747i	$-0.755959\pi$
$443$	−282685.			−1.44044			−0.720219	+	0.693747i	$0.755959\pi$
$444$		0			0
$445$			192736.i			0.973293i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	193737.i		−	0.960991i	−0.876997	−	0.480495i	$-0.840457\pi$
$449$		−	193737.i		−	0.960991i	0.876997	−	0.480495i	$-0.159543\pi$
$450$		0			0
$451$			221219.i			1.08760i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			224553.i			1.08466i
$456$		0			0
$457$	37883.9			0.181394			0.0906968	−	0.995879i	$-0.471091\pi$
$457$	37883.9			0.181394			0.0906968	+	0.995879i	$0.471091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−23481.9			−0.110492			−0.0552461	−	0.998473i	$-0.517594\pi$
$461$	−23481.9			−0.110492			−0.0552461	+	0.998473i	$0.517594\pi$
$462$		0			0
$463$	−270053.			−1.25976			−0.629878	−	0.776694i	$-0.716895\pi$
$463$	−270053.			−1.25976			−0.629878	+	0.776694i	$0.716895\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	108467.			0.497354			0.248677	−	0.968586i	$-0.420004\pi$
$467$	108467.			0.497354			0.248677	+	0.968586i	$0.420004\pi$
$468$		0			0
$469$		−	202137.i		−	0.918966i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−249031.			−1.11309
$474$		0			0
$475$	−100097.	−	126240.i	−0.443644	−	0.559511i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−257055.			−1.12035			−0.560176	−	0.828374i	$-0.689267\pi$
$479$	−257055.			−1.12035			−0.560176	+	0.828374i	$0.689267\pi$
$480$		0			0
$481$	262855.			1.13613
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	161574.i		−	0.686892i
$486$		0			0
$487$			303267.i			1.27870i	0.768917	+	0.639348i	$0.220796\pi$
$487$			303267.i			1.27870i	−0.768917	+	0.639348i	$0.779204\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−23463.3			−0.0973253			−0.0486626	−	0.998815i	$-0.515496\pi$
$491$	−23463.3			−0.0973253			−0.0486626	+	0.998815i	$0.515496\pi$
$492$		0			0
$493$			21979.5i			0.0904325i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	599043.i		−	2.42519i
$498$		0			0
$499$	192267.			0.772152			0.386076	−	0.922467i	$-0.373830\pi$
$499$	192267.			0.772152			0.386076	+	0.922467i	$0.373830\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	489238.			1.93368			0.966840	−	0.255384i	$-0.0822020\pi$
$503$	489238.			1.93368			0.966840	+	0.255384i	$0.0822020\pi$
$504$		0			0
$505$	−49414.8			−0.193765
$506$		0			0
$507$		0			0
$508$		0			0
$509$			49146.1i			0.189694i	0.995492	+	0.0948469i	$0.0302362\pi$
$509$			49146.1i			0.189694i	−0.995492	+	0.0948469i	$0.969764\pi$
$510$		0			0
$511$	−416468.			−1.59492
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	102988.i		−	0.388304i
$516$		0			0
$517$	385719.			1.44308
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	202398.i		−	0.745642i	−0.927903	−	0.372821i	$-0.878391\pi$
$521$		−	202398.i		−	0.745642i	0.927903	−	0.372821i	$-0.121609\pi$
$522$		0			0
$523$		−	107037.i		−	0.391317i	−0.980672	−	0.195659i	$-0.937316\pi$
$523$		−	107037.i		−	0.391317i	0.980672	−	0.195659i	$-0.0626845\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			33651.3i			0.121166i
$528$		0			0
$529$	412937.			1.47561
$530$		0			0
$531$		0			0
$532$		0			0
$533$	419230.			1.47570
$534$		0			0
$535$		−	11183.8i		−	0.0390736i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−466740.			−1.60656
$540$		0			0
$541$	−306226.			−1.04628			−0.523139	−	0.852248i	$-0.675239\pi$
$541$	−306226.			−1.04628			−0.523139	+	0.852248i	$0.675239\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			310720.i			1.04611i
$546$		0			0
$547$		−	114944.i		−	0.384161i	−0.981379	−	0.192081i	$-0.938477\pi$
$547$		−	114944.i		−	0.384161i	0.981379	−	0.192081i	$-0.0615235\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−253425.	+	200945.i	−0.834732	+	0.661871i
$552$		0			0
$553$			77840.4i			0.254539i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	45325.2			0.146093			0.0730465	−	0.997329i	$-0.476728\pi$
$557$	45325.2			0.146093			0.0730465	+	0.997329i	$0.476728\pi$
$558$		0			0
$559$			471936.i			1.51029i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			2010.56i			0.00634309i	0.999995	+	0.00317155i	$0.00100954\pi$
$563$			2010.56i			0.00634309i	−0.999995	+	0.00317155i	$0.998990\pi$
$564$		0			0
$565$		−	239920.i		−	0.751569i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	485775.i		−	1.50041i	−0.661203	−	0.750207i	$-0.729954\pi$
$569$		−	485775.i		−	1.50041i	0.661203	−	0.750207i	$-0.270046\pi$
$570$		0			0
$571$	547633.			1.67964			0.839822	−	0.542862i	$-0.182659\pi$
$571$	547633.			1.67964			0.839822	+	0.542862i	$0.182659\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	371456.			1.12350
$576$		0			0
$577$	162881.			0.489235			0.244618	−	0.969620i	$-0.421338\pi$
$577$	162881.			0.489235			0.244618	+	0.969620i	$0.421338\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−371463.			−1.10043
$582$		0			0
$583$			525597.i			1.54638i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	106188.			0.308177			0.154089	−	0.988057i	$-0.450756\pi$
$587$	106188.			0.308177			0.154089	+	0.988057i	$0.450756\pi$
$588$		0			0
$589$	−388002.	+	307652.i	−1.11842	+	0.886807i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−361718.			−1.02863			−0.514316	−	0.857601i	$-0.671954\pi$
$593$	−361718.			−1.02863			−0.514316	+	0.857601i	$0.671954\pi$
$594$		0			0
$595$	26878.9			0.0759237
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	261061.i		−	0.727592i	−0.931479	−	0.363796i	$-0.881480\pi$
$599$		−	261061.i		−	0.727592i	0.931479	−	0.363796i	$-0.118520\pi$
$600$		0			0
$601$		−	247417.i		−	0.684983i	−0.939521	−	0.342491i	$-0.888729\pi$
$601$		−	247417.i		−	0.684983i	0.939521	−	0.342491i	$-0.111271\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	39361.0			0.107536
$606$		0			0
$607$		−	435069.i		−	1.18081i	−0.807106	−	0.590406i	$-0.798968\pi$
$607$		−	435069.i		−	1.18081i	0.807106	−	0.590406i	$-0.201032\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	730970.i		−	1.95802i
$612$		0			0
$613$	664904.			1.76945			0.884725	−	0.466113i	$-0.154346\pi$
$613$	664904.			1.76945			0.884725	+	0.466113i	$0.154346\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−330735.			−0.868779			−0.434389	−	0.900725i	$-0.643036\pi$
$617$	−330735.			−0.868779			−0.434389	+	0.900725i	$0.643036\pi$
$618$		0			0
$619$	−78389.9			−0.204587			−0.102294	−	0.994754i	$-0.532618\pi$
$619$	−78389.9			−0.204587			−0.102294	+	0.994754i	$0.532618\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.18156e6i			3.04425i
$624$		0			0
$625$	87471.1			0.223926
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	31463.7i		−	0.0795259i
$630$		0			0
$631$	−472556.			−1.18685			−0.593423	−	0.804891i	$-0.702224\pi$
$631$	−472556.			−1.18685			−0.593423	+	0.804891i	$0.702224\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	204054.i		−	0.506055i
$636$		0			0
$637$			884512.i			2.17984i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			295589.i			0.719402i	0.933068	+	0.359701i	$0.117121\pi$
$641$			295589.i			0.719402i	−0.933068	+	0.359701i	$0.882879\pi$
$642$		0			0
$643$	−69120.6			−0.167180			−0.0835902	−	0.996500i	$-0.526639\pi$
$643$	−69120.6			−0.167180			−0.0835902	+	0.996500i	$0.526639\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−381989.			−0.912520			−0.456260	−	0.889847i	$-0.650811\pi$
$647$	−381989.			−0.912520			−0.456260	+	0.889847i	$0.650811\pi$
$648$		0			0
$649$			508305.i			1.20680i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−71764.6			−0.168300			−0.0841500	−	0.996453i	$-0.526817\pi$
$653$	−71764.6			−0.168300			−0.0841500	+	0.996453i	$0.526817\pi$
$654$		0			0
$655$	−38399.9			−0.0895050
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	336370.i		−	0.774544i	−0.921965	−	0.387272i	$-0.873417\pi$
$659$		−	336370.i		−	0.774544i	0.921965	−	0.387272i	$-0.126583\pi$
$660$		0			0
$661$		−	325195.i		−	0.744289i	−0.928175	−	0.372144i	$-0.878623\pi$
$661$		−	325195.i		−	0.744289i	0.928175	−	0.372144i	$-0.121377\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	245736.	+	309915.i	0.555681	+	0.700809i
$666$		0			0
$667$		−	745697.i		−	1.67614i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	483651.			1.07420
$672$		0			0
$673$		−	376184.i		−	0.830558i	−0.909694	−	0.415279i	$-0.863684\pi$
$673$		−	376184.i		−	0.830558i	0.909694	−	0.415279i	$-0.136316\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	177663.i		−	0.387633i	−0.981038	−	0.193816i	$-0.937913\pi$
$677$		−	177663.i		−	0.387633i	0.981038	−	0.193816i	$-0.0620866\pi$
$678$		0			0
$679$		−	990523.i		−	2.14845i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			92119.7i			0.197475i	0.995114	+	0.0987373i	$0.0314804\pi$
$683$			92119.7i			0.197475i	−0.995114	+	0.0987373i	$0.968520\pi$
$684$		0			0
$685$	−180805.			−0.385328
$686$		0			0
$687$		0			0
$688$		0			0
$689$	996051.			2.09818
$690$		0			0
$691$	−503706.			−1.05492			−0.527462	−	0.849579i	$-0.676856\pi$
$691$	−503706.			−1.05492			−0.527462	+	0.849579i	$0.676856\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	337883.			0.699515
$696$		0			0
$697$		−	50181.7i		−	0.103295i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	603516.			1.22815			0.614077	−	0.789246i	$-0.289528\pi$
$701$	603516.			1.22815			0.614077	+	0.789246i	$0.289528\pi$
$702$		0			0
$703$	362778.	−	287652.i	0.734059	−	0.582046i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−302936.			−0.606054
$708$		0			0
$709$	59077.3			0.117524			0.0587622	−	0.998272i	$-0.481285\pi$
$709$	59077.3			0.117524			0.0587622	+	0.998272i	$0.481285\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.14168e6i		−	2.24578i
$714$		0			0
$715$			296329.i			0.579645i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	355935.			0.688513			0.344257	−	0.938876i	$-0.388131\pi$
$719$	355935.			0.688513			0.344257	+	0.938876i	$0.388131\pi$
$720$		0			0
$721$		−	631364.i		−	1.21453i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	399831.i		−	0.760676i
$726$		0			0
$727$	4884.59			0.00924186			0.00462093	−	0.999989i	$-0.498529\pi$
$727$	4884.59			0.00924186			0.00462093	+	0.999989i	$0.498529\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	56490.6			0.105716
$732$		0			0
$733$	−624774.			−1.16283			−0.581414	−	0.813608i	$-0.697500\pi$
$733$	−624774.			−1.16283			−0.581414	+	0.813608i	$0.697500\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	266748.i		−	0.491096i
$738$		0			0
$739$	392348.			0.718426			0.359213	−	0.933256i	$-0.383045\pi$
$739$	392348.			0.718426			0.359213	+	0.933256i	$0.383045\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			287157.i			0.520166i	0.965586	+	0.260083i	$0.0837499\pi$
$743$			287157.i			0.520166i	−0.965586	+	0.260083i	$0.916250\pi$
$744$		0			0
$745$	−141416.			−0.254792
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	68562.1i		−	0.122214i
$750$		0			0
$751$		−	668681.i		−	1.18560i	−0.805349	−	0.592801i	$-0.798022\pi$
$751$		−	668681.i		−	1.18560i	0.805349	−	0.592801i	$-0.201978\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			418464.i			0.734115i
$756$		0			0
$757$	740640.			1.29245			0.646227	−	0.763145i	$-0.276346\pi$
$757$	740640.			1.29245			0.646227	+	0.763145i	$0.276346\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	210422.			0.363348			0.181674	−	0.983359i	$-0.441849\pi$
$761$	210422.			0.363348			0.181674	+	0.983359i	$0.441849\pi$
$762$		0			0
$763$			1.90485e6i			3.27200i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	963281.			1.63743
$768$		0			0
$769$	498675.			0.843267			0.421633	−	0.906766i	$-0.361457\pi$
$769$	498675.			0.843267			0.421633	+	0.906766i	$0.361457\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	457240.i		−	0.765218i	−0.923910	−	0.382609i	$-0.875026\pi$
$773$		−	457240.i		−	0.765218i	0.923910	−	0.382609i	$-0.124974\pi$
$774$		0			0
$775$		−	612152.i		−	1.01919i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	578598.	−	458778.i	0.953459	−	0.756011i
$780$		0			0
$781$		−	790522.i		−	1.29602i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	48197.1			0.0782135
$786$		0			0
$787$			414714.i			0.669575i	0.942294	+	0.334787i	$0.108664\pi$
$787$			414714.i			0.669575i	−0.942294	+	0.334787i	$0.891336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.47082e6i		−	2.35075i
$792$		0			0
$793$		−	916560.i		−	1.45752i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			548863.i			0.864067i	0.901858	+	0.432033i	$0.142204\pi$
$797$			548863.i			0.864067i	−0.901858	+	0.432033i	$0.857796\pi$
$798$		0			0
$799$	−87496.9			−0.137056
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−549588.			−0.852327
$804$		0			0
$805$	−911916.			−1.40722
$806$		0			0
$807$		0			0
$808$		0			0
$809$	971849.			1.48492			0.742458	−	0.669893i	$-0.233660\pi$
$809$	971849.			1.48492			0.742458	+	0.669893i	$0.233660\pi$
$810$		0			0
$811$		−	1.14831e6i		−	1.74590i	−0.487810	−	0.872950i	$-0.662204\pi$
$811$		−	1.14831e6i		−	1.74590i	0.487810	−	0.872950i	$-0.337796\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−410222.			−0.617595
$816$		0			0
$817$	516456.	+	651340.i	0.773730	+	0.975806i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.06981e6			1.58716			0.793578	−	0.608468i	$-0.208216\pi$
$821$	1.06981e6			1.58716			0.793578	+	0.608468i	$0.208216\pi$
$822$		0			0
$823$	−171985.			−0.253917			−0.126959	−	0.991908i	$-0.540522\pi$
$823$	−171985.			−0.253917			−0.126959	+	0.991908i	$0.540522\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.00942e6i			1.47591i	0.674851	+	0.737954i	$0.264208\pi$
$827$			1.00942e6i			1.47591i	−0.674851	+	0.737954i	$0.735792\pi$
$828$		0			0
$829$		−	165838.i		−	0.241310i	−0.992694	−	0.120655i	$-0.961501\pi$
$829$		−	165838.i		−	0.241310i	0.992694	−	0.120655i	$-0.0384995\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	105876.			0.152583
$834$		0			0
$835$			164370.i			0.235749i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	439623.i		−	0.624535i	−0.949994	−	0.312267i	$-0.898912\pi$
$839$		−	439623.i		−	0.624535i	0.949994	−	0.312267i	$-0.101088\pi$
$840$		0			0
$841$	−95376.7			−0.134850
$842$		0			0
$843$		0			0
$844$		0			0
$845$	179752.			0.251744
$846$		0			0
$847$	241301.			0.336351
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.06746e6i			1.47399i
$852$		0			0
$853$	−513622.			−0.705904			−0.352952	−	0.935642i	$-0.614822\pi$
$853$	−513622.			−0.705904			−0.352952	+	0.935642i	$0.614822\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			295407.i			0.402215i	0.979569	+	0.201108i	$0.0644541\pi$
$857$			295407.i			0.402215i	−0.979569	+	0.201108i	$0.935546\pi$
$858$		0			0
$859$	191144.			0.259045			0.129522	−	0.991576i	$-0.458656\pi$
$859$	191144.			0.259045			0.129522	+	0.991576i	$0.458656\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	82441.3i		−	0.110694i	−0.998467	−	0.0553469i	$-0.982374\pi$
$863$		−	82441.3i		−	0.110694i	0.998467	−	0.0553469i	$-0.0176265\pi$
$864$		0			0
$865$			43234.6i			0.0577829i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			102721.i			0.136026i
$870$		0			0
$871$	−505510.			−0.666337
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.17371e6			−1.53301
$876$		0			0
$877$		−	360547.i		−	0.468773i	−0.972144	−	0.234386i	$-0.924692\pi$
$877$		−	360547.i		−	0.468773i	0.972144	−	0.234386i	$-0.0753081\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−87383.1			−0.112584			−0.0562919	−	0.998414i	$-0.517928\pi$
$881$	−87383.1			−0.112584			−0.0562919	+	0.998414i	$0.517928\pi$
$882$		0			0
$883$	551825.			0.707751			0.353875	−	0.935293i	$-0.384864\pi$
$883$	551825.			0.707751			0.353875	+	0.935293i	$0.384864\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22357.0i		−	0.0284162i	−0.999899	−	0.0142081i	$-0.995477\pi$
$887$		−	22357.0i		−	0.0284162i	0.999899	−	0.0142081i	$-0.00452273\pi$
$888$		0			0
$889$		−	1.25094e6i		−	1.58283i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−799927.	−	1.00884e6i	−1.00311	−	1.26509i
$894$		0			0
$895$			460835.i			0.575307i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.22889e6			−1.52053
$900$		0			0
$901$		−	119227.i		−	0.146867i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	93503.3i		−	0.114164i
$906$		0			0
$907$			876341.i			1.06527i	0.846346	+	0.532634i	$0.178798\pi$
$907$			876341.i			1.06527i	−0.846346	+	0.532634i	$0.821202\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	643776.i		−	0.775707i	−0.921721	−	0.387854i	$-0.873217\pi$
$911$		−	643776.i		−	0.775707i	0.921721	−	0.387854i	$-0.126783\pi$
$912$		0			0
$913$	−490198.			−0.588072
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−235409.			−0.279952
$918$		0			0
$919$	1.34660e6			1.59444			0.797219	−	0.603690i	$-0.206303\pi$
$919$	1.34660e6			1.59444			0.797219	+	0.603690i	$0.206303\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.49811e6			−1.75849
$924$		0			0
$925$			572358.i			0.668935i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.24433e6			1.44180			0.720899	−	0.693040i	$-0.243729\pi$
$929$	1.24433e6			1.44180			0.720899	+	0.693040i	$0.243729\pi$
$930$		0			0
$931$	967953.	+	1.22075e6i	1.11675	+	1.40841i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	35470.5			0.0405736
$936$		0			0
$937$	−272651.			−0.310547			−0.155274	−	0.987872i	$-0.549626\pi$
$937$	−272651.			−0.310547			−0.155274	+	0.987872i	$0.549626\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	220249.i		−	0.248734i	−0.992236	−	0.124367i	$-0.960310\pi$
$941$		−	220249.i		−	0.248734i	0.992236	−	0.124367i	$-0.0396900\pi$
$942$		0			0
$943$			1.70251e6i			1.91454i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	469074.			0.523048			0.261524	−	0.965197i	$-0.415775\pi$
$947$	469074.			0.523048			0.261524	+	0.965197i	$0.415775\pi$
$948$		0			0
$949$			1.04152e6i			1.15647i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			358061.i			0.394250i	0.980378	+	0.197125i	$0.0631604\pi$
$953$			358061.i			0.394250i	−0.980378	+	0.197125i	$0.936840\pi$
$954$		0			0
$955$	−795851.			−0.872620
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.10842e6			−1.20522
$960$		0			0
$961$	−957950.			−1.03728
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	405158.i		−	0.435081i
$966$		0			0
$967$	−915995.			−0.979581			−0.489790	−	0.871840i	$-0.662927\pi$
$967$	−915995.			−0.979581			−0.489790	+	0.871840i	$0.662927\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.71250e6i			1.81632i	0.418620	+	0.908162i	$0.362514\pi$
$971$			1.71250e6i			1.81632i	−0.418620	+	0.908162i	$0.637486\pi$
$972$		0			0
$973$	2.07138e6			2.18793
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.68940e6i		−	1.76988i	−0.465709	−	0.884938i	$-0.654201\pi$
$977$		−	1.68940e6i		−	1.76988i	0.465709	−	0.884938i	$-0.345799\pi$
$978$		0			0
$979$			1.55924e6i			1.62685i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			889710.i			0.920750i	0.887725	+	0.460375i	$0.152285\pi$
$983$			889710.i			0.920750i	−0.887725	+	0.460375i	$0.847715\pi$
$984$		0			0
$985$	541102.			0.557707
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.91655e6			−1.95942
$990$		0			0
$991$		−	1.85097e6i		−	1.88474i	−0.334567	−	0.942372i	$-0.608590\pi$
$991$		−	1.85097e6i		−	1.88474i	0.334567	−	0.942372i	$-0.391410\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−366151.			−0.369841
$996$		0			0
$997$	237857.			0.239291			0.119645	−	0.992817i	$-0.461824\pi$
$997$	237857.			0.239291			0.119645	+	0.992817i	$0.461824\pi$
$998$		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.37.6		14
3.2	odd	2		228.5.h.a.37.5	✓	14
12.11	even	2		912.5.o.c.721.12		14
19.18	odd	2	inner	684.5.h.f.37.5		14
57.56	even	2		228.5.h.a.37.12	yes	14
228.227	odd	2		912.5.o.c.721.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.5.h.a.37.5	✓	14	3.2	odd	2
228.5.h.a.37.12	yes	14	57.56	even	2
684.5.h.f.37.5		14	19.18	odd	2	inner
684.5.h.f.37.6		14	1.1	even	1	trivial
912.5.o.c.721.5		14	228.227	odd	2
912.5.o.c.721.12		14	12.11	even	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)

\(f(q)\)	\(=\)	\(q-13.3685 q^{5} -81.9550 q^{7} +O(q^{10})\)	\(q-13.3685 q^{5} -81.9550 q^{7} -108.151 q^{11} +204.956i q^{13} +24.5332 q^{17} +(224.291 + 282.869i) q^{19} -832.333 q^{23} -446.283 q^{25} +895.912i q^{29} +1371.67i q^{31} +1095.61 q^{35} -1282.50i q^{37} -2045.46i q^{41} +2302.62 q^{43} -3566.48 q^{47} +4315.62 q^{49} -4859.83i q^{53} +1445.82 q^{55} -4699.94i q^{59} -4471.99 q^{61} -2739.95i q^{65} +2466.44i q^{67} +7309.42i q^{71} +5081.66 q^{73} +8863.53 q^{77} -949.794i q^{79} +4532.53 q^{83} -327.971 q^{85} -14417.2i q^{89} -16797.2i q^{91} +(-2998.43 - 3781.53i) q^{95} +12086.2i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) \) 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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