LMFDB - Embedded newform 684.6.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,6,Mod(1,684)]

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, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("684.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	684.a (trivial)

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:	yes
Analytic conductor:	$109.702532752$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 140x^{2} - 84x + 3103 $ x^4 - 140x^2 - 84x + 3103
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.0437$ of defining polynomial
Character	$\chi$	$=$	684.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+64.0791 q^{5} +65.8093 q^{7} +O(q^{10})$	$q+64.0791 q^{5} +65.8093 q^{7} -635.790 q^{11} +467.894 q^{13} -522.359 q^{17} -361.000 q^{19} +3220.28 q^{23} +981.128 q^{25} -6979.01 q^{29} -3388.33 q^{31} +4217.00 q^{35} -13423.9 q^{37} -7104.69 q^{41} -14035.1 q^{43} +1444.37 q^{47} -12476.1 q^{49} +37171.0 q^{53} -40740.8 q^{55} -37865.1 q^{59} +39660.1 q^{61} +29982.2 q^{65} -11022.0 q^{67} -9099.27 q^{71} +29926.9 q^{73} -41840.9 q^{77} -33256.4 q^{79} +50768.6 q^{83} -33472.3 q^{85} +128742. q^{89} +30791.8 q^{91} -23132.5 q^{95} -111792. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 110 q^{5} + 30 q^{7}+O(q^{10}) $ 4 * q + 110 * q^5 + 30 * q^7	$ 4 q + 110 q^{5} + 30 q^{7} - 706 q^{11} + 788 q^{13} - 240 q^{17} - 1444 q^{19} - 5884 q^{23} + 11774 q^{25} - 5240 q^{29} - 860 q^{31} - 20322 q^{35} - 20732 q^{37} + 10204 q^{41} - 12554 q^{43} + 4826 q^{47} - 21376 q^{49} + 76484 q^{53} - 72914 q^{55} - 23898 q^{59} - 32482 q^{61} + 6076 q^{65} + 5022 q^{67} - 121300 q^{71} - 104700 q^{73} + 10002 q^{77} + 117128 q^{79} - 92832 q^{83} + 80322 q^{85} - 5988 q^{89} + 165618 q^{91} - 39710 q^{95} + 22972 q^{97}+O(q^{100}) $ 4 * q + 110 * q^5 + 30 * q^7 - 706 * q^11 + 788 * q^13 - 240 * q^17 - 1444 * q^19 - 5884 * q^23 + 11774 * q^25 - 5240 * q^29 - 860 * q^31 - 20322 * q^35 - 20732 * q^37 + 10204 * q^41 - 12554 * q^43 + 4826 * q^47 - 21376 * q^49 + 76484 * q^53 - 72914 * q^55 - 23898 * q^59 - 32482 * q^61 + 6076 * q^65 + 5022 * q^67 - 121300 * q^71 - 104700 * q^73 + 10002 * q^77 + 117128 * q^79 - 92832 * q^83 + 80322 * q^85 - 5988 * q^89 + 165618 * q^91 - 39710 * q^95 + 22972 * q^97

Display 100 coefficients 10 coefficients

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
$5$	64.0791	1.14628	0.573141	−	0.819457i	$-0.305725\pi$
$5$	64.0791	1.14628	0.573141	+	0.819457i	$0.305725\pi$
$6$	0	0
$7$	65.8093	0.507624	0.253812	−	0.967254i	$-0.418316\pi$
$7$	65.8093	0.507624	0.253812	+	0.967254i	$0.418316\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−635.790	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$11$	−635.790	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$12$	0	0
$13$	467.894	0.767873	0.383936	−	0.923360i	$-0.374568\pi$
$13$	467.894	0.767873	0.383936	+	0.923360i	$0.374568\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−522.359	−0.438376	−0.219188	−	0.975683i	$-0.570341\pi$
$17$	−522.359	−0.438376	−0.219188	+	0.975683i	$0.570341\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3220.28	1.26933	0.634664	−	0.772788i	$-0.281139\pi$
$23$	3220.28	1.26933	0.634664	+	0.772788i	$0.281139\pi$
$24$	0	0
$25$	981.128	0.313961
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6979.01	−1.54099	−0.770493	−	0.637448i	$-0.779990\pi$
$29$	−6979.01	−1.54099	−0.770493	+	0.637448i	$0.779990\pi$
$30$	0	0
$31$	−3388.33	−0.633259	−0.316630	−	0.948549i	$-0.602551\pi$
$31$	−3388.33	−0.633259	−0.316630	+	0.948549i	$0.602551\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4217.00	0.581880
$36$	0	0
$37$	−13423.9	−1.61204	−0.806020	−	0.591888i	$-0.798383\pi$
$37$	−13423.9	−1.61204	−0.806020	+	0.591888i	$0.798383\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7104.69	−0.660063	−0.330032	−	0.943970i	$-0.607059\pi$
$41$	−7104.69	−0.660063	−0.330032	+	0.943970i	$0.607059\pi$
$42$	0	0
$43$	−14035.1	−1.15756	−0.578781	−	0.815483i	$-0.696471\pi$
$43$	−14035.1	−1.15756	−0.578781	+	0.815483i	$0.696471\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1444.37	0.0953745	0.0476873	−	0.998862i	$-0.484815\pi$
$47$	1444.37	0.0953745	0.0476873	+	0.998862i	$0.484815\pi$
$48$	0	0
$49$	−12476.1	−0.742318
$50$	0	0
$51$	0	0
$52$	0	0
$53$	37171.0	1.81767	0.908834	−	0.417158i	$-0.136974\pi$
$53$	37171.0	1.81767	0.908834	+	0.417158i	$0.136974\pi$
$54$	0	0
$55$	−40740.8	−1.81603
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−37865.1	−1.41615	−0.708074	−	0.706138i	$-0.750436\pi$
$59$	−37865.1	−1.41615	−0.708074	+	0.706138i	$0.750436\pi$
$60$	0	0
$61$	39660.1	1.36468	0.682338	−	0.731037i	$-0.260963\pi$
$61$	39660.1	1.36468	0.682338	+	0.731037i	$0.260963\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	29982.2	0.880198
$66$	0	0
$67$	−11022.0	−0.299968	−0.149984	−	0.988688i	$-0.547922\pi$
$67$	−11022.0	−0.299968	−0.149984	+	0.988688i	$0.547922\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9099.27	−0.214220	−0.107110	−	0.994247i	$-0.534160\pi$
$71$	−9099.27	−0.214220	−0.107110	+	0.994247i	$0.534160\pi$
$72$	0	0
$73$	29926.9	0.657286	0.328643	−	0.944454i	$-0.393409\pi$
$73$	29926.9	0.657286	0.328643	+	0.944454i	$0.393409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−41840.9	−0.804218
$78$	0	0
$79$	−33256.4	−0.599525	−0.299763	−	0.954014i	$-0.596907\pi$
$79$	−33256.4	−0.599525	−0.299763	+	0.954014i	$0.596907\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	50768.6	0.808910	0.404455	−	0.914558i	$-0.367461\pi$
$83$	50768.6	0.808910	0.404455	+	0.914558i	$0.367461\pi$
$84$	0	0
$85$	−33472.3	−0.502503
$86$	0	0
$87$	0	0
$88$	0	0
$89$	128742.	1.72284	0.861421	−	0.507892i	$-0.169575\pi$
$89$	128742.	1.72284	0.861421	+	0.507892i	$0.169575\pi$
$90$	0	0
$91$	30791.8	0.389791
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−23132.5	−0.262975
$96$	0	0
$97$	−111792.	−1.20638	−0.603189	−	0.797598i	$-0.706103\pi$
$97$	−111792.	−1.20638	−0.603189	+	0.797598i	$0.706103\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	106980.	1.04352	0.521758	−	0.853093i	$-0.325276\pi$
$101$	106980.	1.04352	0.521758	+	0.853093i	$0.325276\pi$
$102$	0	0
$103$	−2453.53	−0.0227876	−0.0113938	−	0.999935i	$-0.503627\pi$
$103$	−2453.53	−0.0227876	−0.0113938	+	0.999935i	$0.503627\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−25637.7	−0.216481	−0.108241	−	0.994125i	$-0.534522\pi$
$107$	−25637.7	−0.216481	−0.108241	+	0.994125i	$0.534522\pi$
$108$	0	0
$109$	97941.0	0.789584	0.394792	−	0.918771i	$-0.370817\pi$
$109$	97941.0	0.789584	0.394792	+	0.918771i	$0.370817\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−221831.	−1.63428	−0.817138	−	0.576442i	$-0.804441\pi$
$113$	−221831.	−1.63428	−0.817138	+	0.576442i	$0.804441\pi$
$114$	0	0
$115$	206352.	1.45501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−34376.1	−0.222530
$120$	0	0
$121$	243178.	1.50994
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−137377.	−0.786394
$126$	0	0
$127$	−308453.	−1.69699	−0.848496	−	0.529202i	$-0.822491\pi$
$127$	−308453.	−1.69699	−0.848496	+	0.529202i	$0.822491\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−262055.	−1.33418	−0.667090	−	0.744977i	$-0.732461\pi$
$131$	−262055.	−1.33418	−0.667090	+	0.744977i	$0.732461\pi$
$132$	0	0
$133$	−23757.2	−0.116457
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−191566.	−0.872001	−0.436001	−	0.899946i	$-0.643605\pi$
$137$	−191566.	−0.872001	−0.436001	+	0.899946i	$0.643605\pi$
$138$	0	0
$139$	−398392.	−1.74893	−0.874467	−	0.485085i	$-0.838789\pi$
$139$	−398392.	−1.74893	−0.874467	+	0.485085i	$0.838789\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−297482.	−1.21653
$144$	0	0
$145$	−447209.	−1.76640
$146$	0	0
$147$	0	0
$148$	0	0
$149$	91457.4	0.337484	0.168742	−	0.985660i	$-0.446030\pi$
$149$	91457.4	0.337484	0.168742	+	0.985660i	$0.446030\pi$
$150$	0	0
$151$	168957.	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$151$	168957.	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−217121.	−0.725893
$156$	0	0
$157$	−210494.	−0.681539	−0.340770	−	0.940147i	$-0.610688\pi$
$157$	−210494.	−0.681539	−0.340770	+	0.940147i	$0.610688\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	211924.	0.644341
$162$	0	0
$163$	619993.	1.82775	0.913876	−	0.405992i	$-0.133074\pi$
$163$	619993.	1.82775	0.913876	+	0.405992i	$0.133074\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−49657.8	−0.137783	−0.0688916	−	0.997624i	$-0.521946\pi$
$167$	−49657.8	−0.137783	−0.0688916	+	0.997624i	$0.521946\pi$
$168$	0	0
$169$	−152368.	−0.410372
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−268048.	−0.680923	−0.340461	−	0.940259i	$-0.610583\pi$
$173$	−268048.	−0.680923	−0.340461	+	0.940259i	$0.610583\pi$
$174$	0	0
$175$	64567.4	0.159374
$176$	0	0
$177$	0	0
$178$	0	0
$179$	266354.	0.621337	0.310668	−	0.950518i	$-0.399447\pi$
$179$	266354.	0.621337	0.310668	+	0.950518i	$0.399447\pi$
$180$	0	0
$181$	276641.	0.627654	0.313827	−	0.949480i	$-0.398389\pi$
$181$	276641.	0.627654	0.313827	+	0.949480i	$0.398389\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−860194.	−1.84785
$186$	0	0
$187$	332111.	0.694511
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−643635.	−1.27660	−0.638302	−	0.769786i	$-0.720363\pi$
$191$	−643635.	−1.27660	−0.638302	+	0.769786i	$0.720363\pi$
$192$	0	0
$193$	59352.1	0.114695	0.0573473	−	0.998354i	$-0.481736\pi$
$193$	59352.1	0.114695	0.0573473	+	0.998354i	$0.481736\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−60559.8	−0.111178	−0.0555890	−	0.998454i	$-0.517704\pi$
$197$	−60559.8	−0.111178	−0.0555890	+	0.998454i	$0.517704\pi$
$198$	0	0
$199$	−260073.	−0.465546	−0.232773	−	0.972531i	$-0.574780\pi$
$199$	−260073.	−0.465546	−0.232773	+	0.972531i	$0.574780\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−459284.	−0.782242
$204$	0	0
$205$	−455262.	−0.756618
$206$	0	0
$207$	0	0
$208$	0	0
$209$	229520.	0.363459
$210$	0	0
$211$	−501653.	−0.775706	−0.387853	−	0.921721i	$-0.626783\pi$
$211$	−501653.	−0.775706	−0.387853	+	0.921721i	$0.626783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−899356.	−1.32689
$216$	0	0
$217$	−222984.	−0.321458
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−244409.	−0.336617
$222$	0	0
$223$	−625533.	−0.842341	−0.421170	−	0.906981i	$-0.638381\pi$
$223$	−625533.	−0.842341	−0.421170	+	0.906981i	$0.638381\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	128165.	0.165084	0.0825422	−	0.996588i	$-0.473696\pi$
$227$	128165.	0.165084	0.0825422	+	0.996588i	$0.473696\pi$
$228$	0	0
$229$	−400845.	−0.505112	−0.252556	−	0.967582i	$-0.581271\pi$
$229$	−400845.	−0.505112	−0.252556	+	0.967582i	$0.581271\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	233608.	0.281902	0.140951	−	0.990017i	$-0.454984\pi$
$233$	233608.	0.281902	0.140951	+	0.990017i	$0.454984\pi$
$234$	0	0
$235$	92553.6	0.109326
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.47603e6	−1.67147	−0.835737	−	0.549130i	$-0.814959\pi$
$239$	−1.47603e6	−1.67147	−0.835737	+	0.549130i	$0.814959\pi$
$240$	0	0
$241$	1.50349e6	1.66747	0.833735	−	0.552164i	$-0.186198\pi$
$241$	1.50349e6	1.66747	0.833735	+	0.552164i	$0.186198\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−799459.	−0.850905
$246$	0	0
$247$	−168910.	−0.176162
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.24638e6	1.24872	0.624361	−	0.781136i	$-0.285359\pi$
$251$	1.24638e6	1.24872	0.624361	+	0.781136i	$0.285359\pi$
$252$	0	0
$253$	−2.04742e6	−2.01097
$254$	0	0
$255$	0	0
$256$	0	0
$257$	185409.	0.175105	0.0875525	−	0.996160i	$-0.472095\pi$
$257$	185409.	0.175105	0.0875525	+	0.996160i	$0.472095\pi$
$258$	0	0
$259$	−883421.	−0.818311
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−685160.	−0.610805	−0.305403	−	0.952223i	$-0.598791\pi$
$263$	−685160.	−0.610805	−0.305403	+	0.952223i	$0.598791\pi$
$264$	0	0
$265$	2.38188e6	2.08356
$266$	0	0
$267$	0	0
$268$	0	0
$269$	46236.6	0.0389587	0.0194794	−	0.999810i	$-0.493799\pi$
$269$	46236.6	0.0389587	0.0194794	+	0.999810i	$0.493799\pi$
$270$	0	0
$271$	635296.	0.525476	0.262738	−	0.964867i	$-0.415375\pi$
$271$	635296.	0.525476	0.262738	+	0.964867i	$0.415375\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−623791.	−0.497402
$276$	0	0
$277$	−912448.	−0.714511	−0.357255	−	0.934007i	$-0.616287\pi$
$277$	−912448.	−0.714511	−0.357255	+	0.934007i	$0.616287\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.37085e6	1.79118	0.895589	−	0.444883i	$-0.146755\pi$
$281$	2.37085e6	1.79118	0.895589	+	0.444883i	$0.146755\pi$
$282$	0	0
$283$	−87826.9	−0.0651871	−0.0325936	−	0.999469i	$-0.510377\pi$
$283$	−87826.9	−0.0651871	−0.0325936	+	0.999469i	$0.510377\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−467555.	−0.335064
$288$	0	0
$289$	−1.14700e6	−0.807826
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−368613.	−0.250842	−0.125421	−	0.992104i	$-0.540028\pi$
$293$	−368613.	−0.250842	−0.125421	+	0.992104i	$0.540028\pi$
$294$	0	0
$295$	−2.42636e6	−1.62330
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.50675e6	0.974682
$300$	0	0
$301$	−923640.	−0.587607
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.54138e6	1.56430
$306$	0	0
$307$	−915441.	−0.554351	−0.277175	−	0.960819i	$-0.589398\pi$
$307$	−915441.	−0.554351	−0.277175	+	0.960819i	$0.589398\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.65178e6	−0.968390	−0.484195	−	0.874960i	$-0.660887\pi$
$311$	−1.65178e6	−0.968390	−0.484195	+	0.874960i	$0.660887\pi$
$312$	0	0
$313$	2.74796e6	1.58544	0.792720	−	0.609586i	$-0.208664\pi$
$313$	2.74796e6	1.58544	0.792720	+	0.609586i	$0.208664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20962.0	0.0117161	0.00585807	−	0.999983i	$-0.498135\pi$
$317$	20962.0	0.0117161	0.00585807	+	0.999983i	$0.498135\pi$
$318$	0	0
$319$	4.43718e6	2.44135
$320$	0	0
$321$	0	0
$322$	0	0
$323$	188572.	0.100570
$324$	0	0
$325$	459064.	0.241082
$326$	0	0
$327$	0	0
$328$	0	0
$329$	95052.7	0.0484144
$330$	0	0
$331$	1.41911e6	0.711947	0.355973	−	0.934496i	$-0.384149\pi$
$331$	1.41911e6	0.711947	0.355973	+	0.934496i	$0.384149\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−706282.	−0.343848
$336$	0	0
$337$	−2.30351e6	−1.10488	−0.552441	−	0.833552i	$-0.686303\pi$
$337$	−2.30351e6	−1.10488	−0.552441	+	0.833552i	$0.686303\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.15427e6	1.00326
$342$	0	0
$343$	−1.92710e6	−0.884442
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.93674e6	−1.30931	−0.654653	−	0.755929i	$-0.727185\pi$
$347$	−2.93674e6	−1.30931	−0.654653	+	0.755929i	$0.727185\pi$
$348$	0	0
$349$	−3.80733e6	−1.67323	−0.836617	−	0.547788i	$-0.815470\pi$
$349$	−3.80733e6	−1.67323	−0.836617	+	0.547788i	$0.815470\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−679178.	−0.290100	−0.145050	−	0.989424i	$-0.546334\pi$
$353$	−679178.	−0.290100	−0.145050	+	0.989424i	$0.546334\pi$
$354$	0	0
$355$	−583073.	−0.245557
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.29854e6	−1.35078	−0.675392	−	0.737459i	$-0.736026\pi$
$359$	−3.29854e6	−1.35078	−0.675392	+	0.737459i	$0.736026\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.91769e6	0.753435
$366$	0	0
$367$	−1.36424e6	−0.528720	−0.264360	−	0.964424i	$-0.585161\pi$
$367$	−1.36424e6	−0.528720	−0.264360	+	0.964424i	$0.585161\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.44620e6	0.922692
$372$	0	0
$373$	−359656.	−0.133849	−0.0669246	−	0.997758i	$-0.521319\pi$
$373$	−359656.	−0.133849	−0.0669246	+	0.997758i	$0.521319\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.26544e6	−1.18328
$378$	0	0
$379$	2.90865e6	1.04014	0.520072	−	0.854123i	$-0.325905\pi$
$379$	2.90865e6	1.04014	0.520072	+	0.854123i	$0.325905\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.04074e6	−0.710871	−0.355436	−	0.934701i	$-0.615667\pi$
$383$	−2.04074e6	−0.710871	−0.355436	+	0.934701i	$0.615667\pi$
$384$	0	0
$385$	−2.68112e6	−0.921861
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.37257e6	1.13002	0.565011	−	0.825083i	$-0.308872\pi$
$389$	3.37257e6	1.13002	0.565011	+	0.825083i	$0.308872\pi$
$390$	0	0
$391$	−1.68214e6	−0.556443
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.13104e6	−0.687225
$396$	0	0
$397$	850462.	0.270819	0.135409	−	0.990790i	$-0.456765\pi$
$397$	850462.	0.270819	0.135409	+	0.990790i	$0.456765\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	805071.	0.250019	0.125010	−	0.992156i	$-0.460104\pi$
$401$	805071.	0.250019	0.125010	+	0.992156i	$0.460104\pi$
$402$	0	0
$403$	−1.58538e6	−0.486262
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.53481e6	2.55392
$408$	0	0
$409$	−1.41286e6	−0.417630	−0.208815	−	0.977955i	$-0.566961\pi$
$409$	−1.41286e6	−0.417630	−0.208815	+	0.977955i	$0.566961\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.49187e6	−0.718871
$414$	0	0
$415$	3.25321e6	0.927238
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.55942e6	0.433938	0.216969	−	0.976179i	$-0.430383\pi$
$419$	1.55942e6	0.433938	0.216969	+	0.976179i	$0.430383\pi$
$420$	0	0
$421$	1.59717e6	0.439183	0.219592	−	0.975592i	$-0.429528\pi$
$421$	1.59717e6	0.439183	0.219592	+	0.975592i	$0.429528\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−512502.	−0.137633
$426$	0	0
$427$	2.61000e6	0.692742
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.28243e6	−1.11045	−0.555223	−	0.831701i	$-0.687367\pi$
$431$	−4.28243e6	−1.11045	−0.555223	+	0.831701i	$0.687367\pi$
$432$	0	0
$433$	−3.66079e6	−0.938328	−0.469164	−	0.883111i	$-0.655445\pi$
$433$	−3.66079e6	−0.938328	−0.469164	+	0.883111i	$0.655445\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.16252e6	−0.291204
$438$	0	0
$439$	5.49744e6	1.36144	0.680722	−	0.732542i	$-0.261666\pi$
$439$	5.49744e6	1.36144	0.680722	+	0.732542i	$0.261666\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.82338e6	−0.441436	−0.220718	−	0.975338i	$-0.570840\pi$
$443$	−1.82338e6	−0.441436	−0.220718	+	0.975338i	$0.570840\pi$
$444$	0	0
$445$	8.24967e6	1.97486
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.53215e6	−0.358662	−0.179331	−	0.983789i	$-0.557393\pi$
$449$	−1.53215e6	−0.358662	−0.179331	+	0.983789i	$0.557393\pi$
$450$	0	0
$451$	4.51709e6	1.04572
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.97311e6	0.446810
$456$	0	0
$457$	5.28539e6	1.18382	0.591912	−	0.806003i	$-0.298373\pi$
$457$	5.28539e6	1.18382	0.591912	+	0.806003i	$0.298373\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.69418e6	1.46705	0.733525	−	0.679662i	$-0.237874\pi$
$461$	6.69418e6	1.46705	0.733525	+	0.679662i	$0.237874\pi$
$462$	0	0
$463$	812541.	0.176154	0.0880770	−	0.996114i	$-0.471928\pi$
$463$	812541.	0.176154	0.0880770	+	0.996114i	$0.471928\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.46112e6	0.734387	0.367193	−	0.930145i	$-0.380319\pi$
$467$	3.46112e6	0.734387	0.367193	+	0.930145i	$0.380319\pi$
$468$	0	0
$469$	−725353.	−0.152271
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.92337e6	1.83390
$474$	0	0
$475$	−354187.	−0.0720276
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.38975e6	−0.475898	−0.237949	−	0.971278i	$-0.576475\pi$
$479$	−2.38975e6	−0.475898	−0.237949	+	0.971278i	$0.576475\pi$
$480$	0	0
$481$	−6.28099e6	−1.23784
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.16356e6	−1.38285
$486$	0	0
$487$	1.58853e6	0.303511	0.151755	−	0.988418i	$-0.451507\pi$
$487$	1.58853e6	0.303511	0.151755	+	0.988418i	$0.451507\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	949072.	0.177662	0.0888312	−	0.996047i	$-0.471687\pi$
$491$	949072.	0.177662	0.0888312	+	0.996047i	$0.471687\pi$
$492$	0	0
$493$	3.64555e6	0.675532
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−598816.	−0.108743
$498$	0	0
$499$	5.93440e6	1.06690	0.533452	−	0.845830i	$-0.320894\pi$
$499$	5.93440e6	1.06690	0.533452	+	0.845830i	$0.320894\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.26101e6	1.10338	0.551689	−	0.834050i	$-0.313984\pi$
$503$	6.26101e6	1.10338	0.551689	+	0.834050i	$0.313984\pi$
$504$	0	0
$505$	6.85518e6	1.19616
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.36180e6	−0.575145	−0.287572	−	0.957759i	$-0.592848\pi$
$509$	−3.36180e6	−0.575145	−0.287572	+	0.957759i	$0.592848\pi$
$510$	0	0
$511$	1.96947e6	0.333654
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−157220.	−0.0261210
$516$	0	0
$517$	−918313.	−0.151100
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.54384e6	0.733380	0.366690	−	0.930343i	$-0.380491\pi$
$521$	4.54384e6	0.733380	0.366690	+	0.930343i	$0.380491\pi$
$522$	0	0
$523$	−9.28544e6	−1.48439	−0.742195	−	0.670184i	$-0.766215\pi$
$523$	−9.28544e6	−1.48439	−0.742195	+	0.670184i	$0.766215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.76993e6	0.277606
$528$	0	0
$529$	3.93385e6	0.611193
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.32424e6	−0.506845
$534$	0	0
$535$	−1.64284e6	−0.248148
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7.93220e6	1.17604
$540$	0	0
$541$	−875434.	−0.128597	−0.0642984	−	0.997931i	$-0.520481\pi$
$541$	−875434.	−0.128597	−0.0642984	+	0.997931i	$0.520481\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.27597e6	0.905085
$546$	0	0
$547$	1.16991e7	1.67180	0.835901	−	0.548881i	$-0.184946\pi$
$547$	1.16991e7	1.67180	0.835901	+	0.548881i	$0.184946\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.51942e6	0.353527
$552$	0	0
$553$	−2.18858e6	−0.304333
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.93911e6	0.537972	0.268986	−	0.963144i	$-0.413311\pi$
$557$	3.93911e6	0.537972	0.268986	+	0.963144i	$0.413311\pi$
$558$	0	0
$559$	−6.56694e6	−0.888861
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.82008e6	−1.30570	−0.652851	−	0.757486i	$-0.726427\pi$
$563$	−9.82008e6	−1.30570	−0.652851	+	0.757486i	$0.726427\pi$
$564$	0	0
$565$	−1.42147e7	−1.87334
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.83108e6	−1.14349	−0.571746	−	0.820431i	$-0.693734\pi$
$569$	−8.83108e6	−1.14349	−0.571746	+	0.820431i	$0.693734\pi$
$570$	0	0
$571$	7.13743e6	0.916118	0.458059	−	0.888922i	$-0.348545\pi$
$571$	7.13743e6	0.916118	0.458059	+	0.888922i	$0.348545\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.15951e6	0.398519
$576$	0	0
$577$	1.30687e7	1.63415	0.817077	−	0.576528i	$-0.195593\pi$
$577$	1.30687e7	1.63415	0.817077	+	0.576528i	$0.195593\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.34105e6	0.410622
$582$	0	0
$583$	−2.36329e7	−2.87969
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.11604e6	1.09197	0.545985	−	0.837795i	$-0.316155\pi$
$587$	9.11604e6	1.09197	0.545985	+	0.837795i	$0.316155\pi$
$588$	0	0
$589$	1.22319e6	0.145280
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.76736e6	0.790283	0.395141	−	0.918620i	$-0.370696\pi$
$593$	6.76736e6	0.790283	0.395141	+	0.918620i	$0.370696\pi$
$594$	0	0
$595$	−2.20279e6	−0.255082
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.98188e6	0.453441	0.226720	−	0.973960i	$-0.427200\pi$
$599$	3.98188e6	0.453441	0.226720	+	0.973960i	$0.427200\pi$
$600$	0	0
$601$	1.55761e7	1.75903	0.879516	−	0.475870i	$-0.157867\pi$
$601$	1.55761e7	1.75903	0.879516	+	0.475870i	$0.157867\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.55826e7	1.73082
$606$	0	0
$607$	3.12960e6	0.344760	0.172380	−	0.985031i	$-0.444854\pi$
$607$	3.12960e6	0.344760	0.172380	+	0.985031i	$0.444854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	675810.	0.0732355
$612$	0	0
$613$	1.40524e7	1.51043	0.755215	−	0.655477i	$-0.227532\pi$
$613$	1.40524e7	1.51043	0.755215	+	0.655477i	$0.227532\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.27324e6	0.557654	0.278827	−	0.960341i	$-0.410054\pi$
$617$	5.27324e6	0.557654	0.278827	+	0.960341i	$0.410054\pi$
$618$	0	0
$619$	−7.91873e6	−0.830670	−0.415335	−	0.909668i	$-0.636336\pi$
$619$	−7.91873e6	−0.830670	−0.415335	+	0.909668i	$0.636336\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.47242e6	0.874556
$624$	0	0
$625$	−1.18690e7	−1.21539
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.01213e6	0.706680
$630$	0	0
$631$	1.62979e7	1.62951	0.814755	−	0.579805i	$-0.196871\pi$
$631$	1.62979e7	1.62951	0.814755	+	0.579805i	$0.196871\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.97654e7	−1.94523
$636$	0	0
$637$	−5.83751e6	−0.570006
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.00004e7	−0.961330	−0.480665	−	0.876904i	$-0.659605\pi$
$641$	−1.00004e7	−0.961330	−0.480665	+	0.876904i	$0.659605\pi$
$642$	0	0
$643$	−1.66096e7	−1.58428	−0.792138	−	0.610342i	$-0.791032\pi$
$643$	−1.66096e7	−1.58428	−0.792138	+	0.610342i	$0.791032\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.48062e6	0.326886	0.163443	−	0.986553i	$-0.447740\pi$
$647$	3.48062e6	0.326886	0.163443	+	0.986553i	$0.447740\pi$
$648$	0	0
$649$	2.40742e7	2.24358
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.11534e6	0.652999	0.326500	−	0.945197i	$-0.394131\pi$
$653$	7.11534e6	0.652999	0.326500	+	0.945197i	$0.394131\pi$
$654$	0	0
$655$	−1.67923e7	−1.52935
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.01255e6	0.539318	0.269659	−	0.962956i	$-0.413089\pi$
$659$	6.01255e6	0.539318	0.269659	+	0.962956i	$0.413089\pi$
$660$	0	0
$661$	−1.66126e6	−0.147889	−0.0739444	−	0.997262i	$-0.523559\pi$
$661$	−1.66126e6	−0.147889	−0.0739444	+	0.997262i	$0.523559\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.52234e6	−0.133492
$666$	0	0
$667$	−2.24744e7	−1.95602
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.52155e7	−2.16203
$672$	0	0
$673$	2.09343e6	0.178164	0.0890821	−	0.996024i	$-0.471607\pi$
$673$	2.09343e6	0.178164	0.0890821	+	0.996024i	$0.471607\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.41602e7	1.18740	0.593700	−	0.804686i	$-0.297667\pi$
$677$	1.41602e7	1.18740	0.593700	+	0.804686i	$0.297667\pi$
$678$	0	0
$679$	−7.35698e6	−0.612386
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.94105e6	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$683$	4.94105e6	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$684$	0	0
$685$	−1.22754e7	−0.999559
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.73921e7	1.39574
$690$	0	0
$691$	−9.44784e6	−0.752727	−0.376363	−	0.926472i	$-0.622826\pi$
$691$	−9.44784e6	−0.752727	−0.376363	+	0.926472i	$0.622826\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.55286e7	−2.00477
$696$	0	0
$697$	3.71120e6	0.289356
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.83585e6	0.525409	0.262705	−	0.964876i	$-0.415386\pi$
$701$	6.83585e6	0.525409	0.262705	+	0.964876i	$0.415386\pi$
$702$	0	0
$703$	4.84605e6	0.369827
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.04028e6	0.529714
$708$	0	0
$709$	2.04148e6	0.152521	0.0762604	−	0.997088i	$-0.475702\pi$
$709$	2.04148e6	0.152521	0.0762604	+	0.997088i	$0.475702\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.09114e7	−0.803814
$714$	0	0
$715$	−1.90624e7	−1.39448
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.41812e7	−1.02303	−0.511516	−	0.859274i	$-0.670916\pi$
$719$	−1.41812e7	−1.02303	−0.511516	+	0.859274i	$0.670916\pi$
$720$	0	0
$721$	−161465.	−0.0115675
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.84730e6	−0.483810
$726$	0	0
$727$	7.23601e6	0.507766	0.253883	−	0.967235i	$-0.418292\pi$
$727$	7.23601e6	0.507766	0.253883	+	0.967235i	$0.418292\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.33137e6	0.507448
$732$	0	0
$733$	1.34837e7	0.926933	0.463467	−	0.886114i	$-0.346605\pi$
$733$	1.34837e7	0.926933	0.463467	+	0.886114i	$0.346605\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.00770e6	0.475233
$738$	0	0
$739$	1.52779e6	0.102909	0.0514544	−	0.998675i	$-0.483614\pi$
$739$	1.52779e6	0.102909	0.0514544	+	0.998675i	$0.483614\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.01918e6	−0.400005	−0.200003	−	0.979795i	$-0.564095\pi$
$743$	−6.01918e6	−0.400005	−0.200003	+	0.979795i	$0.564095\pi$
$744$	0	0
$745$	5.86050e6	0.386852
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.68720e6	−0.109891
$750$	0	0
$751$	−2.06310e6	−0.133481	−0.0667407	−	0.997770i	$-0.521260\pi$
$751$	−2.06310e6	−0.133481	−0.0667407	+	0.997770i	$0.521260\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.08266e7	0.691234
$756$	0	0
$757$	−2.73849e7	−1.73689	−0.868444	−	0.495787i	$-0.834880\pi$
$757$	−2.73849e7	−1.73689	−0.868444	+	0.495787i	$0.834880\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.55235e6	−0.222359	−0.111179	−	0.993800i	$-0.535463\pi$
$761$	−3.55235e6	−0.222359	−0.111179	+	0.993800i	$0.535463\pi$
$762$	0	0
$763$	6.44543e6	0.400812
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.77168e7	−1.08742
$768$	0	0
$769$	9.74838e6	0.594452	0.297226	−	0.954807i	$-0.403939\pi$
$769$	9.74838e6	0.594452	0.297226	+	0.954807i	$0.403939\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.11310e7	−1.87389	−0.936945	−	0.349477i	$-0.886359\pi$
$773$	−3.11310e7	−1.87389	−0.936945	+	0.349477i	$0.886359\pi$
$774$	0	0
$775$	−3.32439e6	−0.198819
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.56479e6	0.151429
$780$	0	0
$781$	5.78522e6	0.339385
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.34883e7	−0.781236
$786$	0	0
$787$	−1.96579e7	−1.13136	−0.565680	−	0.824625i	$-0.691386\pi$
$787$	−1.96579e7	−1.13136	−0.565680	+	0.824625i	$0.691386\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.45985e7	−0.829597
$792$	0	0
$793$	1.85567e7	1.04790
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.97340e7	−1.10045	−0.550224	−	0.835017i	$-0.685458\pi$
$797$	−1.97340e7	−1.10045	−0.550224	+	0.835017i	$0.685458\pi$
$798$	0	0
$799$	−754478.	−0.0418099
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.90272e7	−1.04133
$804$	0	0
$805$	1.35799e7	0.738596
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.73904e7	1.47139	0.735695	−	0.677313i	$-0.236856\pi$
$809$	2.73904e7	1.47139	0.735695	+	0.677313i	$0.236856\pi$
$810$	0	0
$811$	9.78395e6	0.522351	0.261175	−	0.965291i	$-0.415890\pi$
$811$	9.78395e6	0.522351	0.261175	+	0.965291i	$0.415890\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.97286e7	2.09512
$816$	0	0
$817$	5.06667e6	0.265563
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.07144e6	0.107254	0.0536272	−	0.998561i	$-0.482922\pi$
$821$	2.07144e6	0.107254	0.0536272	+	0.998561i	$0.482922\pi$
$822$	0	0
$823$	2.10882e7	1.08528	0.542638	−	0.839967i	$-0.317426\pi$
$823$	2.10882e7	1.08528	0.542638	+	0.839967i	$0.317426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.73194e6	0.393119	0.196560	−	0.980492i	$-0.437023\pi$
$827$	7.73194e6	0.393119	0.196560	+	0.980492i	$0.437023\pi$
$828$	0	0
$829$	2.77388e7	1.40185	0.700924	−	0.713236i	$-0.252771\pi$
$829$	2.77388e7	1.40185	0.700924	+	0.713236i	$0.252771\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.51703e6	0.325415
$834$	0	0
$835$	−3.18203e6	−0.157938
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.08334e7	−1.02177	−0.510887	−	0.859648i	$-0.670683\pi$
$839$	−2.08334e7	−1.02177	−0.510887	+	0.859648i	$0.670683\pi$
$840$	0	0
$841$	2.81954e7	1.37464
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.76361e6	−0.470401
$846$	0	0
$847$	1.60033e7	0.766483
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.32288e7	−2.04621
$852$	0	0
$853$	3.04500e7	1.43290	0.716449	−	0.697639i	$-0.245766\pi$
$853$	3.04500e7	1.43290	0.716449	+	0.697639i	$0.245766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.74993e6	0.0813895	0.0406947	−	0.999172i	$-0.487043\pi$
$857$	1.74993e6	0.0813895	0.0406947	+	0.999172i	$0.487043\pi$
$858$	0	0
$859$	−2.31559e7	−1.07073	−0.535364	−	0.844622i	$-0.679825\pi$
$859$	−2.31559e7	−1.07073	−0.535364	+	0.844622i	$0.679825\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.92908e7	−0.881704	−0.440852	−	0.897580i	$-0.645324\pi$
$863$	−1.92908e7	−0.881704	−0.440852	+	0.897580i	$0.645324\pi$
$864$	0	0
$865$	−1.71763e7	−0.780529
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.11441e7	0.949816
$870$	0	0
$871$	−5.15715e6	−0.230337
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.04071e6	−0.399192
$876$	0	0
$877$	−6.13591e6	−0.269389	−0.134695	−	0.990887i	$-0.543005\pi$
$877$	−6.13591e6	−0.269389	−0.134695	+	0.990887i	$0.543005\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.50035e6	0.108533	0.0542665	−	0.998526i	$-0.482718\pi$
$881$	2.50035e6	0.108533	0.0542665	+	0.998526i	$0.482718\pi$
$882$	0	0
$883$	8.48262e6	0.366124	0.183062	−	0.983101i	$-0.441399\pi$
$883$	8.48262e6	0.366124	0.183062	+	0.983101i	$0.441399\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.42625e7	−1.03544	−0.517722	−	0.855549i	$-0.673220\pi$
$887$	−2.42625e7	−1.03544	−0.517722	+	0.855549i	$0.673220\pi$
$888$	0	0
$889$	−2.02991e7	−0.861434
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−521416.	−0.0218804
$894$	0	0
$895$	1.70677e7	0.712227
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.36472e7	0.975844
$900$	0	0
$901$	−1.94166e7	−0.796823
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.77269e7	0.719468
$906$	0	0
$907$	−4.47763e7	−1.80730	−0.903649	−	0.428274i	$-0.859122\pi$
$907$	−4.47763e7	−1.80730	−0.903649	+	0.428274i	$0.859122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.62280e6	−0.384154	−0.192077	−	0.981380i	$-0.561522\pi$
$911$	−9.62280e6	−0.384154	−0.192077	+	0.981380i	$0.561522\pi$
$912$	0	0
$913$	−3.22782e7	−1.28154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.72457e7	−0.677262
$918$	0	0
$919$	4.09421e7	1.59912	0.799561	−	0.600585i	$-0.205066\pi$
$919$	4.09421e7	1.59912	0.799561	+	0.600585i	$0.205066\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.25749e6	−0.164494
$924$	0	0
$925$	−1.31706e7	−0.506118
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.50754e7	−0.573099	−0.286550	−	0.958065i	$-0.592508\pi$
$929$	−1.50754e7	−0.573099	−0.286550	+	0.958065i	$0.592508\pi$
$930$	0	0
$931$	4.50389e6	0.170299
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.12814e7	0.796105
$936$	0	0
$937$	−2.25249e7	−0.838135	−0.419068	−	0.907955i	$-0.637643\pi$
$937$	−2.25249e7	−0.838135	−0.419068	+	0.907955i	$0.637643\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.04598e7	0.385078	0.192539	−	0.981289i	$-0.438328\pi$
$941$	1.04598e7	0.385078	0.192539	+	0.981289i	$0.438328\pi$
$942$	0	0
$943$	−2.28791e7	−0.837837
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.91938e7	−0.695481	−0.347741	−	0.937591i	$-0.613051\pi$
$947$	−1.91938e7	−0.695481	−0.347741	+	0.937591i	$0.613051\pi$
$948$	0	0
$949$	1.40026e7	0.504712
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.15274e7	−1.48116	−0.740581	−	0.671967i	$-0.765450\pi$
$953$	−4.15274e7	−1.48116	−0.740581	+	0.671967i	$0.765450\pi$
$954$	0	0
$955$	−4.12435e7	−1.46335
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.26068e7	−0.442649
$960$	0	0
$961$	−1.71484e7	−0.598983
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.80323e6	0.131472
$966$	0	0
$967$	5.33594e7	1.83504	0.917518	−	0.397694i	$-0.130189\pi$
$967$	5.33594e7	1.83504	0.917518	+	0.397694i	$0.130189\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.51646e7	−1.53727	−0.768635	−	0.639688i	$-0.779064\pi$
$971$	−4.51646e7	−1.53727	−0.768635	+	0.639688i	$0.779064\pi$
$972$	0	0
$973$	−2.62179e7	−0.887801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.79834e6	−0.0937916	−0.0468958	−	0.998900i	$-0.514933\pi$
$977$	−2.79834e6	−0.0937916	−0.0468958	+	0.998900i	$0.514933\pi$
$978$	0	0
$979$	−8.18529e7	−2.72946
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.61378e7	−0.862750	−0.431375	−	0.902173i	$-0.641971\pi$
$983$	−2.61378e7	−0.862750	−0.431375	+	0.902173i	$0.641971\pi$
$984$	0	0
$985$	−3.88062e6	−0.127441
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.51969e7	−1.46933
$990$	0	0
$991$	4.58748e7	1.48385	0.741925	−	0.670483i	$-0.233913\pi$
$991$	4.58748e7	1.48385	0.741925	+	0.670483i	$0.233913\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.66652e7	−0.533647
$996$	0	0
$997$	−4.97107e7	−1.58384	−0.791921	−	0.610623i	$-0.790919\pi$
$997$	−4.97107e7	−1.58384	−0.791921	+	0.610623i	$0.790919\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.6.a.e.1.3		4
3.2	odd	2		76.6.a.b.1.4	✓	4
12.11	even	2		304.6.a.k.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.a.b.1.4	✓	4	3.2	odd	2
304.6.a.k.1.1		4	12.11	even	2
684.6.a.e.1.3		4	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	684.a (trivial)

\(f(q)\)	\(=\)	\(q+64.0791 q^{5} +65.8093 q^{7} +O(q^{10})\)	\(q+64.0791 q^{5} +65.8093 q^{7} -635.790 q^{11} +467.894 q^{13} -522.359 q^{17} -361.000 q^{19} +3220.28 q^{23} +981.128 q^{25} -6979.01 q^{29} -3388.33 q^{31} +4217.00 q^{35} -13423.9 q^{37} -7104.69 q^{41} -14035.1 q^{43} +1444.37 q^{47} -12476.1 q^{49} +37171.0 q^{53} -40740.8 q^{55} -37865.1 q^{59} +39660.1 q^{61} +29982.2 q^{65} -11022.0 q^{67} -9099.27 q^{71} +29926.9 q^{73} -41840.9 q^{77} -33256.4 q^{79} +50768.6 q^{83} -33472.3 q^{85} +128742. q^{89} +30791.8 q^{91} -23132.5 q^{95} -111792. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 110 q^{5} + 30 q^{7}+O(q^{10}) \) 4 * q + 110 * q^5 + 30 * q^7	\( 4 q + 110 q^{5} + 30 q^{7} - 706 q^{11} + 788 q^{13} - 240 q^{17} - 1444 q^{19} - 5884 q^{23} + 11774 q^{25} - 5240 q^{29} - 860 q^{31} - 20322 q^{35} - 20732 q^{37} + 10204 q^{41} - 12554 q^{43} + 4826 q^{47} - 21376 q^{49} + 76484 q^{53} - 72914 q^{55} - 23898 q^{59} - 32482 q^{61} + 6076 q^{65} + 5022 q^{67} - 121300 q^{71} - 104700 q^{73} + 10002 q^{77} + 117128 q^{79} - 92832 q^{83} + 80322 q^{85} - 5988 q^{89} + 165618 q^{91} - 39710 q^{95} + 22972 q^{97}+O(q^{100}) \) 4 * q + 110 * q^5 + 30 * q^7 - 706 * q^11 + 788 * q^13 - 240 * q^17 - 1444 * q^19 - 5884 * q^23 + 11774 * q^25 - 5240 * q^29 - 860 * q^31 - 20322 * q^35 - 20732 * q^37 + 10204 * q^41 - 12554 * q^43 + 4826 * q^47 - 21376 * q^49 + 76484 * q^53 - 72914 * q^55 - 23898 * q^59 - 32482 * q^61 + 6076 * q^65 + 5022 * q^67 - 121300 * q^71 - 104700 * q^73 + 10002 * q^77 + 117128 * q^79 - 92832 * q^83 + 80322 * q^85 - 5988 * q^89 + 165618 * q^91 - 39710 * q^95 + 22972 * q^97

Introduction

L-functions

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