LMFDB - Embedded newform 684.7.h.c.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,7,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, 7, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
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:	$157.356993196$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 $ x^8 + 5090x^6 + 8905881x^4 + 5831691048*x^2 + 827887219200
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$-42.5965i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.7.h.c.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-103.439 q^{5} -619.422 q^{7} +O(q^{10})$	$q-103.439 q^{5} -619.422 q^{7} +57.6072 q^{11} +481.215i q^{13} +2339.07 q^{17} +(4359.73 - 5295.16i) q^{19} +8694.38 q^{23} -4925.35 q^{25} +46018.1i q^{29} +36728.1i q^{31} +64072.4 q^{35} +40458.4i q^{37} +61518.9i q^{41} -64375.2 q^{43} +93996.5 q^{47} +266034. q^{49} +97805.1i q^{53} -5958.83 q^{55} -132484. i q^{59} -1921.32 q^{61} -49776.4i q^{65} +378794. i q^{67} -451799. i q^{71} -669896. q^{73} -35683.1 q^{77} +747688. i q^{79} +495260. q^{83} -241951. q^{85} -1.35076e6i q^{89} -298075. i q^{91} +(-450966. + 547726. i) q^{95} -75990.3i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} + 362 q^{7}+O(q^{10}) $ 8 * q - 2 * q^5 + 362 * q^7	$ 8 q - 2 q^{5} + 362 q^{7} - 902 q^{11} - 1550 q^{17} + 6232 q^{19} + 18820 q^{23} - 12158 q^{25} + 101762 q^{35} - 335042 q^{43} + 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} - 632014 q^{61} - 852938 q^{73} - 1850530 q^{77} - 441200 q^{83} - 1828374 q^{85} - 627950 q^{95}+O(q^{100}) $ 8 * q - 2 * q^5 + 362 * q^7 - 902 * q^11 - 1550 * q^17 + 6232 * q^19 + 18820 * q^23 - 12158 * q^25 + 101762 * q^35 - 335042 * q^43 + 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 - 632014 * q^61 - 852938 * q^73 - 1850530 * q^77 - 441200 * q^83 - 1828374 * q^85 - 627950 * 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
$5$	−103.439			−0.827513			−0.413756	−	0.910388i	$-0.635783\pi$
$5$	−103.439			−0.827513			−0.413756	+	0.910388i	$0.635783\pi$
$6$		0			0
$7$	−619.422			−1.80589			−0.902947	−	0.429752i	$-0.858601\pi$
$7$	−619.422			−1.80589			−0.902947	+	0.429752i	$0.858601\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	57.6072			0.0432811			0.0216406	−	0.999766i	$-0.493111\pi$
$11$	57.6072			0.0432811			0.0216406	+	0.999766i	$0.493111\pi$
$12$		0			0
$13$			481.215i			0.219033i	0.993985	+	0.109516i	$0.0349302\pi$
$13$			481.215i			0.219033i	−0.993985	+	0.109516i	$0.965070\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2339.07			0.476098			0.238049	−	0.971253i	$-0.423492\pi$
$17$	2339.07			0.476098			0.238049	+	0.971253i	$0.423492\pi$
$18$		0			0
$19$	4359.73	−	5295.16i	0.635621	−	0.772001i
$19$	4359.73	−	5295.16i	0.635621	−	0.772001i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8694.38			0.714587			0.357293	−	0.933992i	$-0.383700\pi$
$23$	8694.38			0.714587			0.357293	+	0.933992i	$0.383700\pi$
$24$		0			0
$25$	−4925.35			−0.315222
$26$		0			0
$27$		0			0
$28$		0			0
$29$			46018.1i			1.88684i	0.331603	+	0.943419i	$0.392410\pi$
$29$			46018.1i			1.88684i	−0.331603	+	0.943419i	$0.607590\pi$
$30$		0			0
$31$			36728.1i			1.23286i	0.787410	+	0.616430i	$0.211421\pi$
$31$			36728.1i			1.23286i	−0.787410	+	0.616430i	$0.788579\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	64072.4			1.49440
$36$		0			0
$37$			40458.4i			0.798737i	0.916791	+	0.399368i	$0.130771\pi$
$37$			40458.4i			0.798737i	−0.916791	+	0.399368i	$0.869229\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			61518.9i			0.892600i	0.894883	+	0.446300i	$0.147259\pi$
$41$			61518.9i			0.892600i	−0.894883	+	0.446300i	$0.852741\pi$
$42$		0			0
$43$	−64375.2			−0.809680			−0.404840	−	0.914388i	$-0.632673\pi$
$43$	−64375.2			−0.809680			−0.404840	+	0.914388i	$0.632673\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	93996.5			0.905353			0.452676	−	0.891675i	$-0.350469\pi$
$47$	93996.5			0.905353			0.452676	+	0.891675i	$0.350469\pi$
$48$		0			0
$49$	266034.			2.26125
$50$		0			0
$51$		0			0
$52$		0			0
$53$			97805.1i			0.656952i	0.944512	+	0.328476i	$0.106535\pi$
$53$			97805.1i			0.656952i	−0.944512	+	0.328476i	$0.893465\pi$
$54$		0			0
$55$	−5958.83			−0.0358157
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	132484.i		−	0.645070i	−0.946558	−	0.322535i	$-0.895465\pi$
$59$		−	132484.i		−	0.645070i	0.946558	−	0.322535i	$-0.104535\pi$
$60$		0			0
$61$	−1921.32			−0.00846469			−0.00423235	−	0.999991i	$-0.501347\pi$
$61$	−1921.32			−0.00846469			−0.00423235	+	0.999991i	$0.501347\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	49776.4i		−	0.181252i
$66$		0			0
$67$			378794.i			1.25944i	0.776821	+	0.629721i	$0.216831\pi$
$67$			378794.i			1.25944i	−0.776821	+	0.629721i	$0.783169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	451799.i		−	1.26232i	−0.775652	−	0.631161i	$-0.782579\pi$
$71$		−	451799.i		−	1.26232i	0.775652	−	0.631161i	$-0.217421\pi$
$72$		0			0
$73$	−669896.			−1.72202			−0.861011	−	0.508587i	$-0.830168\pi$
$73$	−669896.			−1.72202			−0.861011	+	0.508587i	$0.830168\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−35683.1			−0.0781611
$78$		0			0
$79$			747688.i			1.51649i	0.651971	+	0.758244i	$0.273942\pi$
$79$			747688.i			1.51649i	−0.651971	+	0.758244i	$0.726058\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	495260.			0.866161			0.433081	−	0.901355i	$-0.357427\pi$
$83$	495260.			0.866161			0.433081	+	0.901355i	$0.357427\pi$
$84$		0			0
$85$	−241951.			−0.393977
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.35076e6i		−	1.91606i	−0.286672	−	0.958029i	$-0.592549\pi$
$89$		−	1.35076e6i		−	1.91606i	0.286672	−	0.958029i	$-0.407451\pi$
$90$		0			0
$91$		−	298075.i		−	0.395550i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−450966.	+	547726.i	−0.525985	+	0.638841i
$96$		0			0
$97$		−	75990.3i		−	0.0832612i	−0.999133	−	0.0416306i	$-0.986745\pi$
$97$		−	75990.3i		−	0.0832612i	0.999133	−	0.0416306i	$-0.0132553\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.08825e6			−1.05624			−0.528121	−	0.849169i	$-0.677103\pi$
$101$	−1.08825e6			−1.05624			−0.528121	+	0.849169i	$0.677103\pi$
$102$		0			0
$103$		−	1.69273e6i		−	1.54909i	−0.632518	−	0.774545i	$-0.717979\pi$
$103$		−	1.69273e6i		−	1.54909i	0.632518	−	0.774545i	$-0.282021\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	624544.i		−	0.509814i	−0.966966	−	0.254907i	$-0.917955\pi$
$107$		−	624544.i		−	0.509814i	0.966966	−	0.254907i	$-0.0820448\pi$
$108$		0			0
$109$		−	1.40732e6i		−	1.08671i	−0.839503	−	0.543356i	$-0.817154\pi$
$109$		−	1.40732e6i		−	1.08671i	0.839503	−	0.543356i	$-0.182846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.11788e6i		−	0.774749i	−0.921922	−	0.387375i	$-0.873382\pi$
$113$		−	1.11788e6i		−	0.774749i	0.921922	−	0.387375i	$-0.126618\pi$
$114$		0			0
$115$	−899339.			−0.591330
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.44887e6			−0.859782
$120$		0			0
$121$	−1.76824e6			−0.998127
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.12571e6			1.08836
$126$		0			0
$127$			2.06130e6i			1.00631i	0.864198	+	0.503153i	$0.167827\pi$
$127$			2.06130e6i			1.00631i	−0.864198	+	0.503153i	$0.832173\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.33564e6			1.03894			0.519471	−	0.854488i	$-0.326129\pi$
$131$	2.33564e6			1.03894			0.519471	+	0.854488i	$0.326129\pi$
$132$		0			0
$133$	−2.70051e6	+	3.27993e6i	−1.14786	+	1.39415i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.26330e6			1.26910			0.634550	−	0.772882i	$-0.281185\pi$
$137$	3.26330e6			1.26910			0.634550	+	0.772882i	$0.281185\pi$
$138$		0			0
$139$	114357.			0.0425813			0.0212907	−	0.999773i	$-0.493222\pi$
$139$	114357.			0.0425813			0.0212907	+	0.999773i	$0.493222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			27721.4i			0.00947998i
$144$		0			0
$145$		−	4.76007e6i		−	1.56138i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.12978e6			−1.24844			−0.624221	−	0.781248i	$-0.714583\pi$
$149$	−4.12978e6			−1.24844			−0.624221	+	0.781248i	$0.714583\pi$
$150$		0			0
$151$		−	504349.i		−	0.146487i	−0.997314	−	0.0732437i	$-0.976665\pi$
$151$		−	504349.i		−	0.146487i	0.997314	−	0.0732437i	$-0.0233351\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.79912e6i		−	1.02021i
$156$		0			0
$157$	3.09258e6			0.799138			0.399569	−	0.916703i	$-0.369160\pi$
$157$	3.09258e6			0.799138			0.399569	+	0.916703i	$0.369160\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.38549e6			−1.29047
$162$		0			0
$163$	−587796.			−0.135726			−0.0678632	−	0.997695i	$-0.521618\pi$
$163$	−587796.			−0.135726			−0.0678632	+	0.997695i	$0.521618\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.10037e6i			1.73922i	0.493736	+	0.869612i	$0.335631\pi$
$167$			8.10037e6i			1.73922i	−0.493736	+	0.869612i	$0.664369\pi$
$168$		0			0
$169$	4.59524e6			0.952025
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.92729e6i		−	0.372229i	−0.982528	−	0.186114i	$-0.940411\pi$
$173$		−	1.92729e6i		−	0.372229i	0.982528	−	0.186114i	$-0.0595895\pi$
$174$		0			0
$175$	3.05087e6			0.569258
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.82927e6i			0.842020i	0.907056	+	0.421010i	$0.138324\pi$
$179$			4.82927e6i			0.842020i	−0.907056	+	0.421010i	$0.861676\pi$
$180$		0			0
$181$		−	3.36727e6i		−	0.567862i	−0.958845	−	0.283931i	$-0.908361\pi$
$181$		−	3.36727e6i		−	0.567862i	0.958845	−	0.283931i	$-0.0916387\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	4.18498e6i		−	0.660965i
$186$		0			0
$187$	134747.			0.0206061
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.65880e6			−0.812128			−0.406064	−	0.913845i	$-0.633099\pi$
$191$	−5.65880e6			−0.812128			−0.406064	+	0.913845i	$0.633099\pi$
$192$		0			0
$193$			1.02483e7i			1.42554i	0.701400	+	0.712768i	$0.252559\pi$
$193$			1.02483e7i			1.42554i	−0.701400	+	0.712768i	$0.747441\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.47915e7			−1.93470			−0.967351	−	0.253441i	$-0.918438\pi$
$197$	−1.47915e7			−1.93470			−0.967351	+	0.253441i	$0.918438\pi$
$198$		0			0
$199$	−4.92464e6			−0.624907			−0.312453	−	0.949933i	$-0.601151\pi$
$199$	−4.92464e6			−0.624907			−0.312453	+	0.949933i	$0.601151\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.85046e7i		−	3.40743i
$204$		0			0
$205$		−	6.36346e6i		−	0.738638i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	251152.	−	305039.i	0.0275104	−	0.0334131i
$210$		0			0
$211$			997042.i			0.106137i	0.998591	+	0.0530684i	$0.0169001\pi$
$211$			997042.i			0.106137i	−0.998591	+	0.0530684i	$0.983100\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.65892e6			0.670021
$216$		0			0
$217$		−	2.27502e7i		−	2.22641i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.12559e6i			0.104281i
$222$		0			0
$223$		−	1.24527e7i		−	1.12292i	−0.827502	−	0.561462i	$-0.810239\pi$
$223$		−	1.24527e7i		−	1.12292i	0.827502	−	0.561462i	$-0.189761\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			7.22930e6i			0.618043i	0.951055	+	0.309022i	$0.100002\pi$
$227$			7.22930e6i			0.618043i	−0.951055	+	0.309022i	$0.899998\pi$
$228$		0			0
$229$	4.79865e6			0.399588			0.199794	−	0.979838i	$-0.435973\pi$
$229$	4.79865e6			0.399588			0.199794	+	0.979838i	$0.435973\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.55746e6			0.202181			0.101091	−	0.994877i	$-0.467767\pi$
$233$	2.55746e6			0.202181			0.101091	+	0.994877i	$0.467767\pi$
$234$		0			0
$235$	−9.72291e6			−0.749191
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.36214e7			−1.73026			−0.865132	−	0.501544i	$-0.832765\pi$
$239$	−2.36214e7			−1.73026			−0.865132	+	0.501544i	$0.832765\pi$
$240$		0			0
$241$			1.86092e7i			1.32946i	0.747082	+	0.664732i	$0.231454\pi$
$241$			1.86092e7i			1.32946i	−0.747082	+	0.664732i	$0.768546\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.75183e7			−1.87122
$246$		0			0
$247$	2.54811e6	+	2.09796e6i	0.169093	+	0.139222i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.87813e7			−1.18769			−0.593846	−	0.804578i	$-0.702391\pi$
$251$	−1.87813e7			−1.18769			−0.593846	+	0.804578i	$0.702391\pi$
$252$		0			0
$253$	500859.			0.0309281
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.43001e7i		−	0.842441i	−0.906958	−	0.421220i	$-0.861602\pi$
$257$		−	1.43001e7i		−	0.842441i	0.906958	−	0.421220i	$-0.138398\pi$
$258$		0			0
$259$		−	2.50608e7i		−	1.44243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.68886e7			1.47809			0.739044	−	0.673657i	$-0.235278\pi$
$263$	2.68886e7			1.47809			0.739044	+	0.673657i	$0.235278\pi$
$264$		0			0
$265$		−	1.01169e7i		−	0.543636i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.30414e7i		−	0.669986i	−0.942221	−	0.334993i	$-0.891266\pi$
$269$		−	1.30414e7i		−	0.669986i	0.942221	−	0.334993i	$-0.108734\pi$
$270$		0			0
$271$	−1.69825e7			−0.853284			−0.426642	−	0.904421i	$-0.640303\pi$
$271$	−1.69825e7			−0.853284			−0.426642	+	0.904421i	$0.640303\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−283736.			−0.0136432
$276$		0			0
$277$	6.95844e6			0.327395			0.163698	−	0.986511i	$-0.447658\pi$
$277$	6.95844e6			0.327395			0.163698	+	0.986511i	$0.447658\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.61455e6i			0.343183i	0.985168	+	0.171591i	$0.0548908\pi$
$281$			7.61455e6i			0.343183i	−0.985168	+	0.171591i	$0.945109\pi$
$282$		0			0
$283$	−1.19767e7			−0.528418			−0.264209	−	0.964465i	$-0.585111\pi$
$283$	−1.19767e7			−0.528418			−0.264209	+	0.964465i	$0.585111\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3.81061e7i		−	1.61194i
$288$		0			0
$289$	−1.86663e7			−0.773331
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.84558e6i		−	0.272149i	−0.990699	−	0.136075i	$-0.956551\pi$
$293$		−	6.84558e6i		−	0.272149i	0.990699	−	0.136075i	$-0.0434487\pi$
$294$		0			0
$295$			1.37040e7i			0.533804i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			4.18386e6i			0.156518i
$300$		0			0
$301$	3.98754e7			1.46220
$302$		0			0
$303$		0			0
$304$		0			0
$305$	198740.			0.00700464
$306$		0			0
$307$		−	1.30979e7i		−	0.452675i	−0.974049	−	0.226338i	$-0.927325\pi$
$307$		−	1.30979e7i		−	0.452675i	0.974049	−	0.226338i	$-0.0726753\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.13128e6			0.104097			0.0520487	−	0.998645i	$-0.483425\pi$
$311$	3.13128e6			0.104097			0.0520487	+	0.998645i	$0.483425\pi$
$312$		0			0
$313$	−2.85162e7			−0.929948			−0.464974	−	0.885324i	$-0.653936\pi$
$313$	−2.85162e7			−0.929948			−0.464974	+	0.885324i	$0.653936\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.08011e6i		−	0.0652993i	−0.999467	−	0.0326497i	$-0.989605\pi$
$317$		−	2.08011e6i		−	0.0652993i	0.999467	−	0.0326497i	$-0.0103946\pi$
$318$		0			0
$319$			2.65097e6i			0.0816645i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.01977e7	−	1.23857e7i	0.302618	−	0.367548i
$324$		0			0
$325$		−	2.37015e6i		−	0.0690440i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.82234e7			−1.63497
$330$		0			0
$331$			8.68996e6i			0.239626i	0.992796	+	0.119813i	$0.0382295\pi$
$331$			8.68996e6i			0.239626i	−0.992796	+	0.119813i	$0.961770\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	3.91821e7i		−	1.04220i
$336$		0			0
$337$		−	1.62302e7i		−	0.424066i	−0.977262	−	0.212033i	$-0.931992\pi$
$337$		−	1.62302e7i		−	0.424066i	0.977262	−	0.212033i	$-0.0680085\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.11580e6i			0.0533595i
$342$		0			0
$343$	−9.19130e7			−2.27769
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.91081e7			−1.65402			−0.827010	−	0.562188i	$-0.809960\pi$
$347$	−6.91081e7			−1.65402			−0.827010	+	0.562188i	$0.809960\pi$
$348$		0			0
$349$	1.74773e7			0.411148			0.205574	−	0.978642i	$-0.434094\pi$
$349$	1.74773e7			0.411148			0.205574	+	0.978642i	$0.434094\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.02764e7			0.460964			0.230482	−	0.973077i	$-0.425970\pi$
$353$	2.02764e7			0.460964			0.230482	+	0.973077i	$0.425970\pi$
$354$		0			0
$355$			4.67337e7i			1.04459i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.66182e7			1.43982			0.719912	−	0.694065i	$-0.244182\pi$
$359$	6.66182e7			1.43982			0.719912	+	0.694065i	$0.244182\pi$
$360$		0			0
$361$	−9.03146e6	−	4.61709e7i	−0.191971	−	0.981401i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.92934e7			1.42499
$366$		0			0
$367$	−2.30816e7			−0.466948			−0.233474	−	0.972363i	$-0.575009\pi$
$367$	−2.30816e7			−0.466948			−0.233474	+	0.972363i	$0.575009\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	6.05826e7i		−	1.18639i
$372$		0			0
$373$			8.39871e7i			1.61840i	0.587532	+	0.809201i	$0.300100\pi$
$373$			8.39871e7i			1.61840i	−0.587532	+	0.809201i	$0.699900\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.21446e7			−0.413279
$378$		0			0
$379$		−	9.57993e7i		−	1.75973i	−0.475228	−	0.879863i	$-0.657634\pi$
$379$		−	9.57993e7i		−	1.75973i	0.475228	−	0.879863i	$-0.342366\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5.84087e7i		−	1.03964i	−0.854277	−	0.519818i	$-0.826000\pi$
$383$		−	5.84087e7i		−	1.03964i	0.854277	−	0.519818i	$-0.174000\pi$
$384$		0			0
$385$	3.69103e6			0.0646793
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.66058e7			−0.791755			−0.395878	−	0.918303i	$-0.629560\pi$
$389$	−4.66058e7			−0.791755			−0.395878	+	0.918303i	$0.629560\pi$
$390$		0			0
$391$	2.03368e7			0.340213
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.73401e7i		−	1.25491i
$396$		0			0
$397$	−3.42232e6			−0.0546952			−0.0273476	−	0.999626i	$-0.508706\pi$
$397$	−3.42232e6			−0.0546952			−0.0273476	+	0.999626i	$0.508706\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	3.36234e7i		−	0.521444i	−0.965414	−	0.260722i	$-0.916039\pi$
$401$		−	3.36234e7i		−	0.521444i	0.965414	−	0.260722i	$-0.0839607\pi$
$402$		0			0
$403$	−1.76741e7			−0.270036
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.33069e6i			0.0345702i
$408$		0			0
$409$			3.29912e7i			0.482201i	0.970500	+	0.241100i	$0.0775083\pi$
$409$			3.29912e7i			0.482201i	−0.970500	+	0.241100i	$0.922492\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.20633e7i			1.16493i
$414$		0			0
$415$	−5.12292e7			−0.716760
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.94603e7			−1.08021			−0.540105	−	0.841598i	$-0.681615\pi$
$419$	−7.94603e7			−1.08021			−0.540105	+	0.841598i	$0.681615\pi$
$420$		0			0
$421$		−	1.20635e8i		−	1.61669i	−0.588706	−	0.808347i	$-0.700362\pi$
$421$		−	1.20635e8i		−	1.61669i	0.588706	−	0.808347i	$-0.299638\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.15207e7			−0.150077
$426$		0			0
$427$	1.19011e6			0.0152863
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1.08169e8i			1.35105i	0.737337	+	0.675525i	$0.236083\pi$
$431$			1.08169e8i			1.35105i	−0.737337	+	0.675525i	$0.763917\pi$
$432$		0			0
$433$		−	5.91296e7i		−	0.728352i	−0.931330	−	0.364176i	$-0.881351\pi$
$433$		−	5.91296e7i		−	0.728352i	0.931330	−	0.364176i	$-0.118649\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.79051e7	−	4.60381e7i	0.454207	−	0.551662i
$438$		0			0
$439$			1.06497e8i			1.25877i	0.777094	+	0.629384i	$0.216693\pi$
$439$			1.06497e8i			1.25877i	−0.777094	+	0.629384i	$0.783307\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.08889e7			−0.125248			−0.0626242	−	0.998037i	$-0.519947\pi$
$443$	−1.08889e7			−0.125248			−0.0626242	+	0.998037i	$0.519947\pi$
$444$		0			0
$445$			1.39722e8i			1.58556i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.25704e7i		−	0.138870i	−0.997586	−	0.0694351i	$-0.977880\pi$
$449$		−	1.25704e7i		−	0.138870i	0.997586	−	0.0694351i	$-0.0221197\pi$
$450$		0			0
$451$			3.54393e6i			0.0386327i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.08326e7i			0.327323i
$456$		0			0
$457$	3.27105e7			0.342719			0.171359	−	0.985209i	$-0.445184\pi$
$457$	3.27105e7			0.342719			0.171359	+	0.985209i	$0.445184\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.30874e7			−0.643932			−0.321966	−	0.946751i	$-0.604344\pi$
$461$	−6.30874e7			−0.643932			−0.321966	+	0.946751i	$0.604344\pi$
$462$		0			0
$463$	1.48175e8			1.49290			0.746450	−	0.665441i	$-0.231757\pi$
$463$	1.48175e8			1.49290			0.746450	+	0.665441i	$0.231757\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.45801e7			−0.634086			−0.317043	−	0.948411i	$-0.602690\pi$
$467$	−6.45801e7			−0.634086			−0.317043	+	0.948411i	$0.602690\pi$
$468$		0			0
$469$		−	2.34633e8i		−	2.27442i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.70847e6			−0.0350439
$474$		0			0
$475$	−2.14732e7	+	2.60805e7i	−0.200362	+	0.243352i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.70533e8			1.55168			0.775838	−	0.630932i	$-0.217328\pi$
$479$	1.70533e8			1.55168			0.775838	+	0.630932i	$0.217328\pi$
$480$		0			0
$481$	−1.94692e7			−0.174949
$482$		0			0
$483$		0			0
$484$		0			0
$485$			7.86037e6i			0.0688997i
$486$		0			0
$487$			1.79460e8i			1.55375i	0.629657	+	0.776873i	$0.283195\pi$
$487$			1.79460e8i			1.55375i	−0.629657	+	0.776873i	$0.716805\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.53625e8			1.29783			0.648913	−	0.760863i	$-0.275224\pi$
$491$	1.53625e8			1.29783			0.648913	+	0.760863i	$0.275224\pi$
$492$		0			0
$493$			1.07639e8i			0.898320i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.79854e8i			2.27962i
$498$		0			0
$499$	−1.34585e7			−0.108317			−0.0541584	−	0.998532i	$-0.517248\pi$
$499$	−1.34585e7			−0.108317			−0.0541584	+	0.998532i	$0.517248\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.76561e7			−0.453045			−0.226523	−	0.974006i	$-0.572736\pi$
$503$	−5.76561e7			−0.453045			−0.226523	+	0.974006i	$0.572736\pi$
$504$		0			0
$505$	1.12567e8			0.874054
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	2.15748e8i		−	1.63604i	−0.575193	−	0.818018i	$-0.695073\pi$
$509$		−	2.15748e8i		−	1.63604i	0.575193	−	0.818018i	$-0.304927\pi$
$510$		0			0
$511$	4.14948e8			3.10979
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.75095e8i			1.28189i
$516$		0			0
$517$	5.41487e6			0.0391847
$518$		0			0
$519$		0			0
$520$		0			0
$521$			8.48064e7i			0.599674i	0.953990	+	0.299837i	$0.0969323\pi$
$521$			8.48064e7i			0.599674i	−0.953990	+	0.299837i	$0.903068\pi$
$522$		0			0
$523$		−	2.36947e7i		−	0.165633i	−0.996565	−	0.0828165i	$-0.973608\pi$
$523$		−	2.36947e7i		−	0.165633i	0.996565	−	0.0828165i	$-0.0263915\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.59096e7i			0.586962i
$528$		0			0
$529$	−7.24437e7			−0.489366
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.96038e7			−0.195509
$534$		0			0
$535$			6.46022e7i			0.421877i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.53255e7			0.0978696
$540$		0			0
$541$	−2.29774e8			−1.45114			−0.725569	−	0.688149i	$-0.758423\pi$
$541$	−2.29774e8			−1.45114			−0.725569	+	0.688149i	$0.758423\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.45572e8i			0.899268i
$546$		0			0
$547$			1.04309e8i			0.637326i	0.947868	+	0.318663i	$0.103234\pi$
$547$			1.04309e8i			0.637326i	−0.947868	+	0.318663i	$0.896766\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.43673e8	+	2.00626e8i	1.45664	+	1.19931i
$552$		0			0
$553$		−	4.63134e8i		−	2.73862i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.31503e8			−0.760973			−0.380486	−	0.924787i	$-0.624243\pi$
$557$	−1.31503e8			−0.760973			−0.380486	+	0.924787i	$0.624243\pi$
$558$		0			0
$559$		−	3.09783e7i		−	0.177346i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	7.60307e7i		−	0.426053i	−0.977046	−	0.213027i	$-0.931668\pi$
$563$		−	7.60307e7i		−	0.426053i	0.977046	−	0.213027i	$-0.0683321\pi$
$564$		0			0
$565$			1.15633e8i			0.641115i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1.84138e8i		−	0.999553i	−0.866155	−	0.499776i	$-0.833416\pi$
$569$		−	1.84138e8i		−	0.999553i	0.866155	−	0.499776i	$-0.166584\pi$
$570$		0			0
$571$	−1.94513e8			−1.04482			−0.522409	−	0.852695i	$-0.674967\pi$
$571$	−1.94513e8			−1.04482			−0.522409	+	0.852695i	$0.674967\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.28229e7			−0.225254
$576$		0			0
$577$	230318.			0.00119895			0.000599475	−	1.00000i	$-0.499809\pi$
$577$	230318.			0.00119895			0.000599475		1.00000i	$0.499809\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.06775e8			−1.56420
$582$		0			0
$583$			5.63427e6i			0.0284336i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.41690e8			1.19493			0.597466	−	0.801894i	$-0.296174\pi$
$587$	2.41690e8			1.19493			0.597466	+	0.801894i	$0.296174\pi$
$588$		0			0
$589$	1.94481e8	+	1.60124e8i	0.951768	+	0.783631i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.16180e8			1.03669			0.518347	−	0.855170i	$-0.326547\pi$
$593$	2.16180e8			1.03669			0.518347	+	0.855170i	$0.326547\pi$
$594$		0			0
$595$	1.49870e8			0.711481
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.80059e8i			0.837788i	0.908035	+	0.418894i	$0.137582\pi$
$599$			1.80059e8i			0.837788i	−0.908035	+	0.418894i	$0.862418\pi$
$600$		0			0
$601$		−	2.65025e8i		−	1.22085i	−0.792072	−	0.610427i	$-0.790998\pi$
$601$		−	2.65025e8i		−	1.22085i	0.792072	−	0.610427i	$-0.209002\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.82905e8			0.825963
$606$		0			0
$607$		−	7.33490e7i		−	0.327965i	−0.986463	−	0.163983i	$-0.947566\pi$
$607$		−	7.33490e7i		−	0.327965i	0.986463	−	0.163983i	$-0.0524341\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.52325e7i			0.198302i
$612$		0			0
$613$	2.14007e8			0.929064			0.464532	−	0.885556i	$-0.346223\pi$
$613$	2.14007e8			0.929064			0.464532	+	0.885556i	$0.346223\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.01185e8			−1.28226			−0.641132	−	0.767430i	$-0.721535\pi$
$617$	−3.01185e8			−1.28226			−0.641132	+	0.767430i	$0.721535\pi$
$618$		0			0
$619$	−3.15883e8			−1.33185			−0.665924	−	0.746019i	$-0.731963\pi$
$619$	−3.15883e8			−1.33185			−0.665924	+	0.746019i	$0.731963\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			8.36691e8i			3.46020i
$624$		0			0
$625$	−1.42923e8			−0.585412
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.46350e7i			0.380277i
$630$		0			0
$631$	1.94706e8			0.774980			0.387490	−	0.921874i	$-0.373342\pi$
$631$	1.94706e8			0.774980			0.387490	+	0.921874i	$0.373342\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	2.13219e8i		−	0.832731i
$636$		0			0
$637$			1.28020e8i			0.495288i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	4.53818e8i		−	1.72309i	−0.507684	−	0.861543i	$-0.669498\pi$
$641$		−	4.53818e8i		−	1.72309i	0.507684	−	0.861543i	$-0.330502\pi$
$642$		0			0
$643$	1.37859e8			0.518564			0.259282	−	0.965802i	$-0.416514\pi$
$643$	1.37859e8			0.518564			0.259282	+	0.965802i	$0.416514\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.74047e7			0.359639			0.179820	−	0.983700i	$-0.442449\pi$
$647$	9.74047e7			0.359639			0.179820	+	0.983700i	$0.442449\pi$
$648$		0			0
$649$		−	7.63202e6i		−	0.0279193i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.31439e7			0.0831184			0.0415592	−	0.999136i	$-0.486767\pi$
$653$	2.31439e7			0.0831184			0.0415592	+	0.999136i	$0.486767\pi$
$654$		0			0
$655$	−2.41596e8			−0.859738
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.21467e8i		−	0.773844i	−0.922112	−	0.386922i	$-0.873538\pi$
$659$		−	2.21467e8i		−	0.773844i	0.922112	−	0.386922i	$-0.126462\pi$
$660$		0			0
$661$			2.13083e8i			0.737811i	0.929467	+	0.368906i	$0.120267\pi$
$661$			2.13083e8i			0.737811i	−0.929467	+	0.368906i	$0.879733\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.79338e8	−	3.39273e8i	0.949873	−	1.15368i
$666$		0			0
$667$			4.00099e8i			1.34831i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−110682.			−0.000366361
$672$		0			0
$673$		−	1.52414e8i		−	0.500011i	−0.968244	−	0.250006i	$-0.919568\pi$
$673$		−	1.52414e8i		−	0.500011i	0.968244	−	0.250006i	$-0.0804325\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	9.86309e7i		−	0.317868i	−0.987289	−	0.158934i	$-0.949194\pi$
$677$		−	9.86309e7i		−	0.317868i	0.987289	−	0.158934i	$-0.0508057\pi$
$678$		0			0
$679$			4.70700e7i			0.150361i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.83544e8i			0.576074i	0.957619	+	0.288037i	$0.0930026\pi$
$683$			1.83544e8i			0.576074i	−0.957619	+	0.288037i	$0.906997\pi$
$684$		0			0
$685$	−3.37553e8			−1.05020
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.70653e7			−0.143894
$690$		0			0
$691$	9.11147e7			0.276156			0.138078	−	0.990421i	$-0.455908\pi$
$691$	9.11147e7			0.276156			0.138078	+	0.990421i	$0.455908\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.18290e7			−0.0352366
$696$		0			0
$697$			1.43897e8i			0.424965i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.14272e8			1.20263			0.601315	−	0.799012i	$-0.294644\pi$
$701$	4.14272e8			1.20263			0.601315	+	0.799012i	$0.294644\pi$
$702$		0			0
$703$	2.14234e8	+	1.76388e8i	0.616625	+	0.507694i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.74084e8			1.90746
$708$		0			0
$709$	−3.21198e8			−0.901226			−0.450613	−	0.892719i	$-0.648795\pi$
$709$	−3.21198e8			−0.901226			−0.450613	+	0.892719i	$0.648795\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.19328e8i			0.880985i
$714$		0			0
$715$		−	2.86748e6i		−	0.00784481i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.98125e8			1.07111			0.535553	−	0.844502i	$-0.320103\pi$
$719$	3.98125e8			1.07111			0.535553	+	0.844502i	$0.320103\pi$
$720$		0			0
$721$			1.04852e9i			2.79749i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2.26655e8i		−	0.594774i
$726$		0			0
$727$	2.93307e8			0.763343			0.381672	−	0.924298i	$-0.375349\pi$
$727$	2.93307e8			0.763343			0.381672	+	0.924298i	$0.375349\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.50578e8			−0.385487
$732$		0			0
$733$	4.75660e8			1.20777			0.603886	−	0.797071i	$-0.293618\pi$
$733$	4.75660e8			1.20777			0.603886	+	0.797071i	$0.293618\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2.18212e7i			0.0545101i
$738$		0			0
$739$	2.86348e8			0.709515			0.354757	−	0.934958i	$-0.384563\pi$
$739$	2.86348e8			0.709515			0.354757	+	0.934958i	$0.384563\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			6.29617e8i			1.53501i	0.641045	+	0.767503i	$0.278501\pi$
$743$			6.29617e8i			1.53501i	−0.641045	+	0.767503i	$0.721499\pi$
$744$		0			0
$745$	4.27181e8			1.03310
$746$		0			0
$747$		0			0
$748$		0			0
$749$			3.86856e8i			0.920669i
$750$		0			0
$751$		−	5.36049e8i		−	1.26557i	−0.774329	−	0.632783i	$-0.781913\pi$
$751$		−	5.36049e8i		−	1.26557i	0.774329	−	0.632783i	$-0.218087\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.21694e7i			0.121220i
$756$		0			0
$757$	−4.72181e7			−0.108848			−0.0544240	−	0.998518i	$-0.517332\pi$
$757$	−4.72181e7			−0.108848			−0.0544240	+	0.998518i	$0.517332\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.01530e8			−0.684189			−0.342095	−	0.939666i	$-0.611136\pi$
$761$	−3.01530e8			−0.684189			−0.342095	+	0.939666i	$0.611136\pi$
$762$		0			0
$763$			8.71726e8i			1.96249i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.37532e7			0.141291
$768$		0			0
$769$	−7.09285e8			−1.55970			−0.779851	−	0.625965i	$-0.784705\pi$
$769$	−7.09285e8			−1.55970			−0.779851	+	0.625965i	$0.784705\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.38166e8i			0.948638i	0.880353	+	0.474319i	$0.157306\pi$
$773$			4.38166e8i			0.948638i	−0.880353	+	0.474319i	$0.842694\pi$
$774$		0			0
$775$		−	1.80899e8i		−	0.388625i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.25752e8	+	2.68206e8i	0.689088	+	0.567356i
$780$		0			0
$781$		−	2.60269e7i		−	0.0546347i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.19893e8			−0.661297
$786$		0			0
$787$		−	4.08238e8i		−	0.837509i	−0.908099	−	0.418755i	$-0.862467\pi$
$787$		−	4.08238e8i		−	0.837509i	0.908099	−	0.418755i	$-0.137533\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			6.92441e8i			1.39911i
$792$		0			0
$793$		−	924569.i		−	0.00185404i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.93088e8i		−	0.776452i	−0.921564	−	0.388226i	$-0.873088\pi$
$797$		−	3.93088e8i		−	0.776452i	0.921564	−	0.388226i	$-0.126912\pi$
$798$		0			0
$799$	2.19864e8			0.431037
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.85908e7			−0.0745310
$804$		0			0
$805$	5.57070e8			1.06788
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.89585e8			0.358062			0.179031	−	0.983843i	$-0.442704\pi$
$809$	1.89585e8			0.358062			0.179031	+	0.983843i	$0.442704\pi$
$810$		0			0
$811$			3.61235e8i			0.677217i	0.940927	+	0.338608i	$0.109956\pi$
$811$			3.61235e8i			0.677217i	−0.940927	+	0.338608i	$0.890044\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.08011e7			0.112315
$816$		0			0
$817$	−2.80658e8	+	3.40877e8i	−0.514650	+	0.625074i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.28100e8			1.67712			0.838562	−	0.544806i	$-0.183397\pi$
$821$	9.28100e8			1.67712			0.838562	+	0.544806i	$0.183397\pi$
$822$		0			0
$823$	−5.88701e8			−1.05608			−0.528038	−	0.849221i	$-0.677072\pi$
$823$	−5.88701e8			−1.05608			−0.528038	+	0.849221i	$0.677072\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.62734e8i		−	0.641316i	−0.947195	−	0.320658i	$-0.896096\pi$
$827$		−	3.62734e8i		−	0.641316i	0.947195	−	0.320658i	$-0.103904\pi$
$828$		0			0
$829$		−	2.99074e8i		−	0.524946i	−0.964939	−	0.262473i	$-0.915462\pi$
$829$		−	2.99074e8i		−	0.524946i	0.964939	−	0.262473i	$-0.0845381\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.22272e8			1.07658
$834$		0			0
$835$		−	8.37895e8i		−	1.43923i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	9.24397e8i		−	1.56521i	−0.622518	−	0.782605i	$-0.713890\pi$
$839$		−	9.24397e8i		−	1.56521i	0.622518	−	0.782605i	$-0.286110\pi$
$840$		0			0
$841$	−1.52284e9			−2.56016
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.75328e8			−0.787813
$846$		0			0
$847$	1.09529e9			1.80251
$848$		0			0
$849$		0			0
$850$		0			0
$851$			3.51761e8i			0.570767i
$852$		0			0
$853$	9.56120e8			1.54051			0.770256	−	0.637734i	$-0.220128\pi$
$853$	9.56120e8			1.54051			0.770256	+	0.637734i	$0.220128\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7.58039e8i		−	1.20434i	−0.798368	−	0.602170i	$-0.794303\pi$
$857$		−	7.58039e8i		−	1.20434i	0.798368	−	0.602170i	$-0.205697\pi$
$858$		0			0
$859$	−1.10887e9			−1.74945			−0.874724	−	0.484621i	$-0.838957\pi$
$859$	−1.10887e9			−1.74945			−0.874724	+	0.484621i	$0.838957\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	3.02345e8i		−	0.470403i	−0.971947	−	0.235201i	$-0.924425\pi$
$863$		−	3.02345e8i		−	0.470403i	0.971947	−	0.235201i	$-0.0755750\pi$
$864$		0			0
$865$			1.99358e8i			0.308024i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.30722e7i			0.0656353i
$870$		0			0
$871$	−1.82281e8			−0.275859
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.31671e9			−1.96547
$876$		0			0
$877$			7.86910e7i			0.116661i	0.998297	+	0.0583306i	$0.0185777\pi$
$877$			7.86910e7i			0.116661i	−0.998297	+	0.0583306i	$0.981422\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.85796e7			0.0710438			0.0355219	−	0.999369i	$-0.488691\pi$
$881$	4.85796e7			0.0710438			0.0355219	+	0.999369i	$0.488691\pi$
$882$		0			0
$883$	−6.43174e8			−0.934214			−0.467107	−	0.884201i	$-0.654704\pi$
$883$	−6.43174e8			−0.934214			−0.467107	+	0.884201i	$0.654704\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	8.45733e8i		−	1.21189i	−0.795507	−	0.605944i	$-0.792796\pi$
$887$		−	8.45733e8i		−	1.21189i	0.795507	−	0.605944i	$-0.207204\pi$
$888$		0			0
$889$		−	1.27681e9i		−	1.81728i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.09799e8	−	4.97726e8i	0.575462	−	0.698933i
$894$		0			0
$895$		−	4.99536e8i		−	0.696783i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.69016e9			−2.32621
$900$		0			0
$901$			2.28773e8i			0.312774i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			3.48308e8i			0.469913i
$906$		0			0
$907$		−	6.28543e8i		−	0.842390i	−0.906970	−	0.421195i	$-0.861611\pi$
$907$		−	6.28543e8i		−	0.842390i	0.906970	−	0.421195i	$-0.138389\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.01606e9i			1.34389i	0.740601	+	0.671945i	$0.234541\pi$
$911$			1.01606e9i			1.34389i	−0.740601	+	0.671945i	$0.765459\pi$
$912$		0			0
$913$	2.85305e7			0.0374884
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.44674e9			−1.87622
$918$		0			0
$919$	2.64340e8			0.340578			0.170289	−	0.985394i	$-0.445530\pi$
$919$	2.64340e8			0.340578			0.170289	+	0.985394i	$0.445530\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.17412e8			0.276490
$924$		0			0
$925$		−	1.99272e8i		−	0.251780i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.45301e8			−0.680126			−0.340063	−	0.940403i	$-0.610448\pi$
$929$	−5.45301e8			−0.680126			−0.340063	+	0.940403i	$0.610448\pi$
$930$		0			0
$931$	1.15984e9	−	1.40869e9i	1.43730	−	1.74569i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.39381e7			−0.0170518
$936$		0			0
$937$	−4.24987e7			−0.0516602			−0.0258301	−	0.999666i	$-0.508223\pi$
$937$	−4.24987e7			−0.0516602			−0.0258301	+	0.999666i	$0.508223\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			2.88776e8i			0.346570i	0.984872	+	0.173285i	$0.0554382\pi$
$941$			2.88776e8i			0.346570i	−0.984872	+	0.173285i	$0.944562\pi$
$942$		0			0
$943$			5.34869e8i			0.637840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.44768e8			−0.641449			−0.320724	−	0.947173i	$-0.603926\pi$
$947$	−5.44768e8			−0.641449			−0.320724	+	0.947173i	$0.603926\pi$
$948$		0			0
$949$		−	3.22364e8i		−	0.377179i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	8.40684e8i		−	0.971302i	−0.874153	−	0.485651i	$-0.838583\pi$
$953$		−	8.40684e8i		−	0.971302i	0.874153	−	0.485651i	$-0.161417\pi$
$954$		0			0
$955$	5.85341e8			0.672046
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.02136e9			−2.29186
$960$		0			0
$961$	−4.61450e8			−0.519941
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.06007e9i		−	1.17965i
$966$		0			0
$967$	−5.07252e8			−0.560976			−0.280488	−	0.959857i	$-0.590496\pi$
$967$	−5.07252e8			−0.560976			−0.280488	+	0.959857i	$0.590496\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	8.79559e8i		−	0.960743i	−0.877065	−	0.480371i	$-0.840502\pi$
$971$		−	8.79559e8i		−	0.960743i	0.877065	−	0.480371i	$-0.159498\pi$
$972$		0			0
$973$	−7.08353e7			−0.0768973
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.00687e9i		−	1.07967i	−0.841771	−	0.539835i	$-0.818487\pi$
$977$		−	1.00687e9i		−	1.07967i	0.841771	−	0.539835i	$-0.181513\pi$
$978$		0			0
$979$		−	7.78135e7i		−	0.0829291i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.63063e9i			1.71670i	0.513066	+	0.858349i	$0.328509\pi$
$983$			1.63063e9i			1.71670i	−0.513066	+	0.858349i	$0.671491\pi$
$984$		0			0
$985$	1.53002e9			1.60099
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.59703e8			−0.578587
$990$		0			0
$991$			4.15200e8i			0.426615i	0.976985	+	0.213307i	$0.0684236\pi$
$991$			4.15200e8i			0.426615i	−0.976985	+	0.213307i	$0.931576\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.09400e8			0.517118
$996$		0			0
$997$	−1.10642e9			−1.11644			−0.558218	−	0.829695i	$-0.688515\pi$
$997$	−1.10642e9			−1.11644			−0.558218	+	0.829695i	$0.688515\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.7.h.c.37.2		8
3.2	odd	2		76.7.c.b.37.8	yes	8
12.11	even	2		304.7.e.c.113.1		8
19.18	odd	2	inner	684.7.h.c.37.1		8
57.56	even	2		76.7.c.b.37.1	✓	8
228.227	odd	2		304.7.e.c.113.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.c.b.37.1	✓	8	57.56	even	2
76.7.c.b.37.8	yes	8	3.2	odd	2
304.7.e.c.113.1		8	12.11	even	2
304.7.e.c.113.8		8	228.227	odd	2
684.7.h.c.37.1		8	19.18	odd	2	inner
684.7.h.c.37.2		8	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-103.439 q^{5} -619.422 q^{7} +O(q^{10})\)	\(q-103.439 q^{5} -619.422 q^{7} +57.6072 q^{11} +481.215i q^{13} +2339.07 q^{17} +(4359.73 - 5295.16i) q^{19} +8694.38 q^{23} -4925.35 q^{25} +46018.1i q^{29} +36728.1i q^{31} +64072.4 q^{35} +40458.4i q^{37} +61518.9i q^{41} -64375.2 q^{43} +93996.5 q^{47} +266034. q^{49} +97805.1i q^{53} -5958.83 q^{55} -132484. i q^{59} -1921.32 q^{61} -49776.4i q^{65} +378794. i q^{67} -451799. i q^{71} -669896. q^{73} -35683.1 q^{77} +747688. i q^{79} +495260. q^{83} -241951. q^{85} -1.35076e6i q^{89} -298075. i q^{91} +(-450966. + 547726. i) q^{95} -75990.3i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} + 362 q^{7}+O(q^{10}) \) 8 * q - 2 * q^5 + 362 * q^7	\( 8 q - 2 q^{5} + 362 q^{7} - 902 q^{11} - 1550 q^{17} + 6232 q^{19} + 18820 q^{23} - 12158 q^{25} + 101762 q^{35} - 335042 q^{43} + 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} - 632014 q^{61} - 852938 q^{73} - 1850530 q^{77} - 441200 q^{83} - 1828374 q^{85} - 627950 q^{95}+O(q^{100}) \) 8 * q - 2 * q^5 + 362 * q^7 - 902 * q^11 - 1550 * q^17 + 6232 * q^19 + 18820 * q^23 - 12158 * q^25 + 101762 * q^35 - 335042 * q^43 + 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 - 632014 * q^61 - 852938 * q^73 - 1850530 * q^77 - 441200 * q^83 - 1828374 * q^85 - 627950 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more