LMFDB - Embedded newform 684.6.a.e.1.4

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			1.4
Root			$4.80512$ 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)

$f(q)$	$=$	$q+109.245 q^{5} -177.299 q^{7} +O(q^{10})$	$q+109.245 q^{5} -177.299 q^{7} -228.019 q^{11} -476.072 q^{13} +1868.17 q^{17} -361.000 q^{19} -1966.09 q^{23} +8809.50 q^{25} +4201.04 q^{29} -3842.27 q^{31} -19369.1 q^{35} -15414.3 q^{37} -1927.41 q^{41} -7330.22 q^{43} -5080.98 q^{47} +14628.0 q^{49} +12616.1 q^{53} -24910.0 q^{55} -7718.28 q^{59} -17993.0 q^{61} -52008.5 q^{65} +10746.2 q^{67} -50642.8 q^{71} -58036.2 q^{73} +40427.6 q^{77} +33913.4 q^{79} -30761.3 q^{83} +204088. q^{85} -97641.2 q^{89} +84407.1 q^{91} -39437.5 q^{95} +105569. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	109.245	1.95424	0.977118	−	0.212696i	$-0.0682244\pi$
$5$	109.245	1.95424	0.977118	+	0.212696i	$0.0682244\pi$
$6$	0	0
$7$	−177.299	−1.36761	−0.683804	−	0.729666i	$-0.739676\pi$
$7$	−177.299	−1.36761	−0.683804	+	0.729666i	$0.739676\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−228.019	−0.568184	−0.284092	−	0.958797i	$-0.591692\pi$
$11$	−228.019	−0.568184	−0.284092	+	0.958797i	$0.591692\pi$
$12$	0	0
$13$	−476.072	−0.781293	−0.390647	−	0.920541i	$-0.627749\pi$
$13$	−476.072	−0.781293	−0.390647	+	0.920541i	$0.627749\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1868.17	1.56781	0.783905	−	0.620880i	$-0.213225\pi$
$17$	1868.17	1.56781	0.783905	+	0.620880i	$0.213225\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1966.09	−0.774967	−0.387483	−	0.921877i	$-0.626656\pi$
$23$	−1966.09	−0.774967	−0.387483	+	0.921877i	$0.626656\pi$
$24$	0	0
$25$	8809.50	2.81904
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4201.04	0.927602	0.463801	−	0.885939i	$-0.346485\pi$
$29$	4201.04	0.927602	0.463801	+	0.885939i	$0.346485\pi$
$30$	0	0
$31$	−3842.27	−0.718097	−0.359048	−	0.933319i	$-0.616899\pi$
$31$	−3842.27	−0.718097	−0.359048	+	0.933319i	$0.616899\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−19369.1	−2.67263
$36$	0	0
$37$	−15414.3	−1.85106	−0.925528	−	0.378678i	$-0.876379\pi$
$37$	−15414.3	−1.85106	−0.925528	+	0.378678i	$0.876379\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1927.41	−0.179067	−0.0895334	−	0.995984i	$-0.528538\pi$
$41$	−1927.41	−0.179067	−0.0895334	+	0.995984i	$0.528538\pi$
$42$	0	0
$43$	−7330.22	−0.604569	−0.302284	−	0.953218i	$-0.597749\pi$
$43$	−7330.22	−0.604569	−0.302284	+	0.953218i	$0.597749\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5080.98	−0.335508	−0.167754	−	0.985829i	$-0.553651\pi$
$47$	−5080.98	−0.335508	−0.167754	+	0.985829i	$0.553651\pi$
$48$	0	0
$49$	14628.0	0.870351
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12616.1	0.616929	0.308465	−	0.951236i	$-0.400185\pi$
$53$	12616.1	0.616929	0.308465	+	0.951236i	$0.400185\pi$
$54$	0	0
$55$	−24910.0	−1.11037
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7718.28	−0.288663	−0.144331	−	0.989529i	$-0.546103\pi$
$59$	−7718.28	−0.288663	−0.144331	+	0.989529i	$0.546103\pi$
$60$	0	0
$61$	−17993.0	−0.619127	−0.309564	−	0.950879i	$-0.600183\pi$
$61$	−17993.0	−0.619127	−0.309564	+	0.950879i	$0.600183\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−52008.5	−1.52683
$66$	0	0
$67$	10746.2	0.292460	0.146230	−	0.989251i	$-0.453286\pi$
$67$	10746.2	0.292460	0.146230	+	0.989251i	$0.453286\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−50642.8	−1.19226	−0.596132	−	0.802887i	$-0.703296\pi$
$71$	−50642.8	−1.19226	−0.596132	+	0.802887i	$0.703296\pi$
$72$	0	0
$73$	−58036.2	−1.27465	−0.637326	−	0.770594i	$-0.719960\pi$
$73$	−58036.2	−1.27465	−0.637326	+	0.770594i	$0.719960\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	40427.6	0.777054
$78$	0	0
$79$	33913.4	0.611369	0.305685	−	0.952133i	$-0.401115\pi$
$79$	33913.4	0.611369	0.305685	+	0.952133i	$0.401115\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−30761.3	−0.490128	−0.245064	−	0.969507i	$-0.578809\pi$
$83$	−30761.3	−0.490128	−0.245064	+	0.969507i	$0.578809\pi$
$84$	0	0
$85$	204088.	3.06387
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−97641.2	−1.30665	−0.653323	−	0.757079i	$-0.726626\pi$
$89$	−97641.2	−1.30665	−0.653323	+	0.757079i	$0.726626\pi$
$90$	0	0
$91$	84407.1	1.06850
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−39437.5	−0.448333
$96$	0	0
$97$	105569.	1.13922	0.569609	−	0.821916i	$-0.307095\pi$
$97$	105569.	1.13922	0.569609	+	0.821916i	$0.307095\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	70055.6	0.683344	0.341672	−	0.939819i	$-0.389007\pi$
$101$	70055.6	0.683344	0.341672	+	0.939819i	$0.389007\pi$
$102$	0	0
$103$	29196.8	0.271171	0.135585	−	0.990766i	$-0.456709\pi$
$103$	29196.8	0.271171	0.135585	+	0.990766i	$0.456709\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	83750.0	0.707173	0.353586	−	0.935402i	$-0.384962\pi$
$107$	83750.0	0.707173	0.353586	+	0.935402i	$0.384962\pi$
$108$	0	0
$109$	−51998.8	−0.419206	−0.209603	−	0.977787i	$-0.567217\pi$
$109$	−51998.8	−0.419206	−0.209603	+	0.977787i	$0.567217\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−174906.	−1.28857	−0.644287	−	0.764784i	$-0.722846\pi$
$113$	−174906.	−1.28857	−0.644287	+	0.764784i	$0.722846\pi$
$114$	0	0
$115$	−214785.	−1.51447
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−331225.	−2.14415
$120$	0	0
$121$	−109058.	−0.677166
$122$	0	0
$123$	0	0
$124$	0	0
$125$	621005.	3.55484
$126$	0	0
$127$	−10574.2	−0.0581755	−0.0290878	−	0.999577i	$-0.509260\pi$
$127$	−10574.2	−0.0581755	−0.0290878	+	0.999577i	$0.509260\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−38671.2	−0.196884	−0.0984418	−	0.995143i	$-0.531386\pi$
$131$	−38671.2	−0.196884	−0.0984418	+	0.995143i	$0.531386\pi$
$132$	0	0
$133$	64005.0	0.313751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−85272.3	−0.388156	−0.194078	−	0.980986i	$-0.562172\pi$
$137$	−85272.3	−0.388156	−0.194078	+	0.980986i	$0.562172\pi$
$138$	0	0
$139$	−198686.	−0.872230	−0.436115	−	0.899891i	$-0.643646\pi$
$139$	−198686.	−0.872230	−0.436115	+	0.899891i	$0.643646\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	108553.	0.443919
$144$	0	0
$145$	458943.	1.81275
$146$	0	0
$147$	0	0
$148$	0	0
$149$	81459.5	0.300591	0.150295	−	0.988641i	$-0.451977\pi$
$149$	81459.5	0.300591	0.150295	+	0.988641i	$0.451977\pi$
$150$	0	0
$151$	−517065.	−1.84545	−0.922727	−	0.385455i	$-0.874045\pi$
$151$	−517065.	−1.84545	−0.922727	+	0.385455i	$0.874045\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−419749.	−1.40333
$156$	0	0
$157$	169444.	0.548627	0.274314	−	0.961640i	$-0.411549\pi$
$157$	169444.	0.548627	0.274314	+	0.961640i	$0.411549\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	348585.	1.05985
$162$	0	0
$163$	−116429.	−0.343234	−0.171617	−	0.985164i	$-0.554899\pi$
$163$	−116429.	−0.343234	−0.171617	+	0.985164i	$0.554899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9810.91	0.0272219	0.0136109	−	0.999907i	$-0.495667\pi$
$167$	9810.91	0.0272219	0.0136109	+	0.999907i	$0.495667\pi$
$168$	0	0
$169$	−144649.	−0.389581
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−191280.	−0.485908	−0.242954	−	0.970038i	$-0.578116\pi$
$173$	−191280.	−0.485908	−0.242954	+	0.970038i	$0.578116\pi$
$174$	0	0
$175$	−1.56192e6	−3.85534
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−631191.	−1.47241	−0.736204	−	0.676759i	$-0.763384\pi$
$179$	−631191.	−1.47241	−0.736204	+	0.676759i	$0.763384\pi$
$180$	0	0
$181$	−119770.	−0.271739	−0.135870	−	0.990727i	$-0.543383\pi$
$181$	−119770.	−0.271739	−0.135870	+	0.990727i	$0.543383\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.68394e6	−3.61740
$186$	0	0
$187$	−425978.	−0.890806
$188$	0	0
$189$	0	0
$190$	0	0
$191$	942206.	1.86880	0.934400	−	0.356226i	$-0.115937\pi$
$191$	942206.	1.86880	0.934400	+	0.356226i	$0.115937\pi$
$192$	0	0
$193$	−560559.	−1.08325	−0.541624	−	0.840621i	$-0.682190\pi$
$193$	−560559.	−1.08325	−0.541624	+	0.840621i	$0.682190\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−792520.	−1.45494	−0.727469	−	0.686140i	$-0.759304\pi$
$197$	−792520.	−1.45494	−0.727469	+	0.686140i	$0.759304\pi$
$198$	0	0
$199$	477931.	0.855524	0.427762	−	0.903891i	$-0.359302\pi$
$199$	477931.	0.855524	0.427762	+	0.903891i	$0.359302\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−744840.	−1.26860
$204$	0	0
$205$	−210561.	−0.349939
$206$	0	0
$207$	0	0
$208$	0	0
$209$	82314.9	0.130350
$210$	0	0
$211$	−1.09610e6	−1.69490	−0.847451	−	0.530874i	$-0.821864\pi$
$211$	−1.09610e6	−1.69490	−0.847451	+	0.530874i	$0.821864\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−800791.	−1.18147
$216$	0	0
$217$	681230.	0.982075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−889382.	−1.22492
$222$	0	0
$223$	−78676.4	−0.105945	−0.0529727	−	0.998596i	$-0.516870\pi$
$223$	−78676.4	−0.105945	−0.0529727	+	0.998596i	$0.516870\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	965468.	1.24358	0.621789	−	0.783185i	$-0.286406\pi$
$227$	965468.	1.24358	0.621789	+	0.783185i	$0.286406\pi$
$228$	0	0
$229$	837329.	1.05513	0.527566	−	0.849514i	$-0.323105\pi$
$229$	837329.	1.05513	0.527566	+	0.849514i	$0.323105\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	881429.	1.06365	0.531824	−	0.846855i	$-0.321507\pi$
$233$	881429.	1.06365	0.531824	+	0.846855i	$0.321507\pi$
$234$	0	0
$235$	−555072.	−0.655662
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−298682.	−0.338232	−0.169116	−	0.985596i	$-0.554091\pi$
$239$	−298682.	−0.338232	−0.169116	+	0.985596i	$0.554091\pi$
$240$	0	0
$241$	−1.04452e6	−1.15844	−0.579219	−	0.815172i	$-0.696643\pi$
$241$	−1.04452e6	−1.15844	−0.579219	+	0.815172i	$0.696643\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.59804e6	1.70087
$246$	0	0
$247$	171862.	0.179241
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.10873e6	−1.11082	−0.555408	−	0.831578i	$-0.687438\pi$
$251$	−1.10873e6	−1.11082	−0.555408	+	0.831578i	$0.687438\pi$
$252$	0	0
$253$	448305.	0.440324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	83101.8	0.0784834	0.0392417	−	0.999230i	$-0.487506\pi$
$257$	83101.8	0.0784834	0.0392417	+	0.999230i	$0.487506\pi$
$258$	0	0
$259$	2.73294e6	2.53152
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.75925e6	1.56833	0.784167	−	0.620550i	$-0.213091\pi$
$263$	1.75925e6	1.56833	0.784167	+	0.620550i	$0.213091\pi$
$264$	0	0
$265$	1.37825e6	1.20563
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−524521.	−0.441959	−0.220980	−	0.975278i	$-0.570925\pi$
$269$	−524521.	−0.441959	−0.220980	+	0.975278i	$0.570925\pi$
$270$	0	0
$271$	761057.	0.629498	0.314749	−	0.949175i	$-0.398080\pi$
$271$	761057.	0.629498	0.314749	+	0.949175i	$0.398080\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00873e6	−1.60174
$276$	0	0
$277$	−349747.	−0.273877	−0.136938	−	0.990580i	$-0.543726\pi$
$277$	−349747.	−0.273877	−0.136938	+	0.990580i	$0.543726\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.71890e6	−1.29863	−0.649314	−	0.760520i	$-0.724944\pi$
$281$	−1.71890e6	−1.29863	−0.649314	+	0.760520i	$0.724944\pi$
$282$	0	0
$283$	−654088.	−0.485479	−0.242739	−	0.970092i	$-0.578046\pi$
$283$	−654088.	−0.485479	−0.242739	+	0.970092i	$0.578046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	341729.	0.244893
$288$	0	0
$289$	2.07020e6	1.45803
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.93430e6	−1.31630	−0.658152	−	0.752885i	$-0.728661\pi$
$293$	−1.93430e6	−1.31630	−0.658152	+	0.752885i	$0.728661\pi$
$294$	0	0
$295$	−843185.	−0.564115
$296$	0	0
$297$	0	0
$298$	0	0
$299$	935998.	0.605476
$300$	0	0
$301$	1.29964e6	0.826813
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.96565e6	−1.20992
$306$	0	0
$307$	1.65795e6	1.00398	0.501992	−	0.864872i	$-0.332601\pi$
$307$	1.65795e6	1.00398	0.501992	+	0.864872i	$0.332601\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	794581.	0.465840	0.232920	−	0.972496i	$-0.425172\pi$
$311$	794581.	0.465840	0.232920	+	0.972496i	$0.425172\pi$
$312$	0	0
$313$	−2.21582e6	−1.27842	−0.639211	−	0.769032i	$-0.720739\pi$
$313$	−2.21582e6	−1.27842	−0.639211	+	0.769032i	$0.720739\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.59381e6	0.890815	0.445408	−	0.895328i	$-0.353059\pi$
$317$	1.59381e6	0.890815	0.445408	+	0.895328i	$0.353059\pi$
$318$	0	0
$319$	−957916.	−0.527049
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−674409.	−0.359681
$324$	0	0
$325$	−4.19396e6	−2.20250
$326$	0	0
$327$	0	0
$328$	0	0
$329$	900853.	0.458843
$330$	0	0
$331$	3.22458e6	1.61772	0.808860	−	0.588001i	$-0.200085\pi$
$331$	3.22458e6	1.61772	0.808860	+	0.588001i	$0.200085\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.17397e6	0.571536
$336$	0	0
$337$	1.24462e6	0.596982	0.298491	−	0.954412i	$-0.403517\pi$
$337$	1.24462e6	0.596982	0.298491	+	0.954412i	$0.403517\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	876109.	0.408012
$342$	0	0
$343$	386336.	0.177309
$344$	0	0
$345$	0	0
$346$	0	0
$347$	324910.	0.144857	0.0724284	−	0.997374i	$-0.476925\pi$
$347$	324910.	0.144857	0.0724284	+	0.997374i	$0.476925\pi$
$348$	0	0
$349$	337444.	0.148299	0.0741495	−	0.997247i	$-0.476376\pi$
$349$	337444.	0.148299	0.0741495	+	0.997247i	$0.476376\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.80025e6	−1.19608	−0.598039	−	0.801467i	$-0.704053\pi$
$353$	−2.80025e6	−1.19608	−0.598039	+	0.801467i	$0.704053\pi$
$354$	0	0
$355$	−5.53249e6	−2.32997
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.26867e6	1.33855	0.669275	−	0.743015i	$-0.266605\pi$
$359$	3.26867e6	1.33855	0.669275	+	0.743015i	$0.266605\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.34017e6	−2.49097
$366$	0	0
$367$	−4.37614e6	−1.69600	−0.848001	−	0.529995i	$-0.822194\pi$
$367$	−4.37614e6	−1.69600	−0.848001	+	0.529995i	$0.822194\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.23682e6	−0.843717
$372$	0	0
$373$	385783.	0.143572	0.0717862	−	0.997420i	$-0.477130\pi$
$373$	385783.	0.143572	0.0717862	+	0.997420i	$0.477130\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000e6	−0.724729
$378$	0	0
$379$	2.75351e6	0.984665	0.492333	−	0.870407i	$-0.336144\pi$
$379$	2.75351e6	0.984665	0.492333	+	0.870407i	$0.336144\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.67232e6	−1.97589	−0.987947	−	0.154792i	$-0.950529\pi$
$383$	−5.67232e6	−1.97589	−0.987947	+	0.154792i	$0.950529\pi$
$384$	0	0
$385$	4.41652e6	1.51855
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.22486e6	−1.41559	−0.707797	−	0.706416i	$-0.750311\pi$
$389$	−4.22486e6	−1.41559	−0.707797	+	0.706416i	$0.750311\pi$
$390$	0	0
$391$	−3.67298e6	−1.21500
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.70487e6	1.19476
$396$	0	0
$397$	−1.60192e6	−0.510110	−0.255055	−	0.966927i	$-0.582094\pi$
$397$	−1.60192e6	−0.510110	−0.255055	+	0.966927i	$0.582094\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	584352.	0.181474	0.0907369	−	0.995875i	$-0.471078\pi$
$401$	584352.	0.181474	0.0907369	+	0.995875i	$0.471078\pi$
$402$	0	0
$403$	1.82919e6	0.561044
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.51476e6	1.05174
$408$	0	0
$409$	−1.70400e6	−0.503687	−0.251844	−	0.967768i	$-0.581037\pi$
$409$	−1.70400e6	−0.503687	−0.251844	+	0.967768i	$0.581037\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.36845e6	0.394777
$414$	0	0
$415$	−3.36052e6	−0.957825
$416$	0	0
$417$	0	0
$418$	0	0
$419$	42813.6	0.0119137	0.00595684	−	0.999982i	$-0.498104\pi$
$419$	42813.6	0.0119137	0.00595684	+	0.999982i	$0.498104\pi$
$420$	0	0
$421$	−4.45775e6	−1.22577	−0.612887	−	0.790170i	$-0.709992\pi$
$421$	−4.45775e6	−1.22577	−0.612887	+	0.790170i	$0.709992\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.64576e7	4.41972
$426$	0	0
$427$	3.19015e6	0.846724
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−127317.	−0.0330136	−0.0165068	−	0.999864i	$-0.505255\pi$
$431$	−127317.	−0.0330136	−0.0165068	+	0.999864i	$0.505255\pi$
$432$	0	0
$433$	7.00930e6	1.79661	0.898307	−	0.439369i	$-0.144798\pi$
$433$	7.00930e6	1.79661	0.898307	+	0.439369i	$0.144798\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	709757.	0.177790
$438$	0	0
$439$	4.17533e6	1.03402	0.517011	−	0.855979i	$-0.327045\pi$
$439$	4.17533e6	1.03402	0.517011	+	0.855979i	$0.327045\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−529129.	−0.128101	−0.0640504	−	0.997947i	$-0.520402\pi$
$443$	−529129.	−0.128101	−0.0640504	+	0.997947i	$0.520402\pi$
$444$	0	0
$445$	−1.06668e7	−2.55350
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.14414e6	0.501924	0.250962	−	0.967997i	$-0.419253\pi$
$449$	2.14414e6	0.501924	0.250962	+	0.967997i	$0.419253\pi$
$450$	0	0
$451$	439487.	0.101743
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.22107e6	2.08811
$456$	0	0
$457$	−6.90778e6	−1.54721	−0.773603	−	0.633670i	$-0.781548\pi$
$457$	−6.90778e6	−1.54721	−0.773603	+	0.633670i	$0.781548\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.58026e6	1.44208	0.721042	−	0.692891i	$-0.243664\pi$
$461$	6.58026e6	1.44208	0.721042	+	0.692891i	$0.243664\pi$
$462$	0	0
$463$	6.91768e6	1.49971	0.749856	−	0.661601i	$-0.230123\pi$
$463$	6.91768e6	1.49971	0.749856	+	0.661601i	$0.230123\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.45033e6	−0.307734	−0.153867	−	0.988092i	$-0.549173\pi$
$467$	−1.45033e6	−0.307734	−0.153867	+	0.988092i	$0.549173\pi$
$468$	0	0
$469$	−1.90528e6	−0.399970
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.67143e6	0.343507
$474$	0	0
$475$	−3.18023e6	−0.646732
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.81138e6	−0.360720	−0.180360	−	0.983601i	$-0.557726\pi$
$479$	−1.81138e6	−0.360720	−0.180360	+	0.983601i	$0.557726\pi$
$480$	0	0
$481$	7.33832e6	1.44622
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.15329e7	2.22630
$486$	0	0
$487$	2.19793e6	0.419944	0.209972	−	0.977707i	$-0.432663\pi$
$487$	2.19793e6	0.419944	0.209972	+	0.977707i	$0.432663\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.64236e6	−1.43062	−0.715310	−	0.698808i	$-0.753714\pi$
$491$	−7.64236e6	−1.43062	−0.715310	+	0.698808i	$0.753714\pi$
$492$	0	0
$493$	7.84825e6	1.45430
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.97893e6	1.63055
$498$	0	0
$499$	−1.96363e6	−0.353028	−0.176514	−	0.984298i	$-0.556482\pi$
$499$	−1.96363e6	−0.353028	−0.176514	+	0.984298i	$0.556482\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.51813e6	0.972460	0.486230	−	0.873831i	$-0.338372\pi$
$503$	5.51813e6	0.972460	0.486230	+	0.873831i	$0.338372\pi$
$504$	0	0
$505$	7.65323e6	1.33542
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.59852e6	0.444562	0.222281	−	0.974983i	$-0.428650\pi$
$509$	2.59852e6	0.444562	0.222281	+	0.974983i	$0.428650\pi$
$510$	0	0
$511$	1.02898e7	1.74322
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.18961e6	0.529932
$516$	0	0
$517$	1.15856e6	0.190630
$518$	0	0
$519$	0	0
$520$	0	0
$521$	665007.	0.107333	0.0536664	−	0.998559i	$-0.482909\pi$
$521$	665007.	0.107333	0.0536664	+	0.998559i	$0.482909\pi$
$522$	0	0
$523$	−2.70303e6	−0.432113	−0.216057	−	0.976381i	$-0.569320\pi$
$523$	−2.70303e6	−0.432113	−0.216057	+	0.976381i	$0.569320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.17800e6	−1.12584
$528$	0	0
$529$	−2.57085e6	−0.399427
$530$	0	0
$531$	0	0
$532$	0	0
$533$	917587.	0.139904
$534$	0	0
$535$	9.14929e6	1.38198
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.33546e6	−0.494520
$540$	0	0
$541$	9.58822e6	1.40846	0.704230	−	0.709972i	$-0.251292\pi$
$541$	9.58822e6	1.40846	0.704230	+	0.709972i	$0.251292\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.68062e6	−0.819227
$546$	0	0
$547$	1.05846e7	1.51254	0.756268	−	0.654261i	$-0.227020\pi$
$547$	1.05846e7	1.51254	0.756268	+	0.654261i	$0.227020\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.51657e6	−0.212806
$552$	0	0
$553$	−6.01282e6	−0.836113
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.71496e6	0.643932	0.321966	−	0.946751i	$-0.395656\pi$
$557$	4.71496e6	0.643932	0.321966	+	0.946751i	$0.395656\pi$
$558$	0	0
$559$	3.48971e6	0.472346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.60502e6	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$563$	9.60502e6	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$564$	0	0
$565$	−1.91077e7	−2.51818
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.97008e6	0.902521	0.451260	−	0.892392i	$-0.350975\pi$
$569$	6.97008e6	0.902521	0.451260	+	0.892392i	$0.350975\pi$
$570$	0	0
$571$	−5.27949e6	−0.677644	−0.338822	−	0.940851i	$-0.610028\pi$
$571$	−5.27949e6	−0.677644	−0.338822	+	0.940851i	$0.610028\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.73202e7	−2.18466
$576$	0	0
$577$	1.01323e7	1.26697	0.633486	−	0.773754i	$-0.281623\pi$
$577$	1.01323e7	1.26697	0.633486	+	0.773754i	$0.281623\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.45395e6	0.670302
$582$	0	0
$583$	−2.87671e6	−0.350530
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.91129e6	−0.947659	−0.473830	−	0.880617i	$-0.657129\pi$
$587$	−7.91129e6	−0.947659	−0.473830	+	0.880617i	$0.657129\pi$
$588$	0	0
$589$	1.38706e6	0.164743
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.98912e6	1.04974	0.524869	−	0.851183i	$-0.324114\pi$
$593$	8.98912e6	1.04974	0.524869	+	0.851183i	$0.324114\pi$
$594$	0	0
$595$	−3.61847e7	−4.19018
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.04333e6	−1.02982	−0.514910	−	0.857244i	$-0.672175\pi$
$599$	−9.04333e6	−1.02982	−0.514910	+	0.857244i	$0.672175\pi$
$600$	0	0
$601$	8.48341e6	0.958042	0.479021	−	0.877803i	$-0.340992\pi$
$601$	8.48341e6	0.958042	0.479021	+	0.877803i	$0.340992\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.19141e7	−1.32334
$606$	0	0
$607$	1.37916e7	1.51930	0.759650	−	0.650332i	$-0.225370\pi$
$607$	1.37916e7	1.51930	0.759650	+	0.650332i	$0.225370\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.41891e6	0.262130
$612$	0	0
$613$	−8.69314e6	−0.934384	−0.467192	−	0.884156i	$-0.654734\pi$
$613$	−8.69314e6	−0.934384	−0.467192	+	0.884156i	$0.654734\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.63565e6	−0.278725	−0.139362	−	0.990241i	$-0.544505\pi$
$617$	−2.63565e6	−0.278725	−0.139362	+	0.990241i	$0.544505\pi$
$618$	0	0
$619$	−1.47325e7	−1.54543	−0.772717	−	0.634751i	$-0.781103\pi$
$619$	−1.47325e7	−1.54543	−0.772717	+	0.634751i	$0.781103\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.73117e7	1.78698
$624$	0	0
$625$	4.03120e7	4.12795
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.87965e7	−2.90211
$630$	0	0
$631$	2.04039e6	0.204005	0.102002	−	0.994784i	$-0.467475\pi$
$631$	2.04039e6	0.204005	0.102002	+	0.994784i	$0.467475\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.15519e6	−0.113689
$636$	0	0
$637$	−6.96397e6	−0.679999
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.92090e6	−0.569171	−0.284585	−	0.958651i	$-0.591856\pi$
$641$	−5.92090e6	−0.569171	−0.284585	+	0.958651i	$0.591856\pi$
$642$	0	0
$643$	8.71800e6	0.831552	0.415776	−	0.909467i	$-0.363510\pi$
$643$	8.71800e6	0.831552	0.415776	+	0.909467i	$0.363510\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.96829e7	1.84854	0.924269	−	0.381742i	$-0.124676\pi$
$647$	1.96829e7	1.84854	0.924269	+	0.381742i	$0.124676\pi$
$648$	0	0
$649$	1.75992e6	0.164014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.87843e6	0.447710	0.223855	−	0.974622i	$-0.428136\pi$
$653$	4.87843e6	0.447710	0.223855	+	0.974622i	$0.428136\pi$
$654$	0	0
$655$	−4.22464e6	−0.384757
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.08130e7	−0.969908	−0.484954	−	0.874540i	$-0.661164\pi$
$659$	−1.08130e7	−0.969908	−0.484954	+	0.874540i	$0.661164\pi$
$660$	0	0
$661$	4.22290e6	0.375930	0.187965	−	0.982176i	$-0.439811\pi$
$661$	4.22290e6	0.375930	0.187965	+	0.982176i	$0.439811\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.99224e6	0.613143
$666$	0	0
$667$	−8.25960e6	−0.718860
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.10276e6	0.351779
$672$	0	0
$673$	−4.79237e6	−0.407861	−0.203931	−	0.978985i	$-0.565372\pi$
$673$	−4.79237e6	−0.407861	−0.203931	+	0.978985i	$0.565372\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.20013e6	0.687621	0.343810	−	0.939039i	$-0.388282\pi$
$677$	8.20013e6	0.687621	0.343810	+	0.939039i	$0.388282\pi$
$678$	0	0
$679$	−1.87173e7	−1.55800
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.11848e6	0.0917438	0.0458719	−	0.998947i	$-0.485393\pi$
$683$	1.11848e6	0.0917438	0.0458719	+	0.998947i	$0.485393\pi$
$684$	0	0
$685$	−9.31559e6	−0.758549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.00617e6	−0.482003
$690$	0	0
$691$	−2.53615e6	−0.202060	−0.101030	−	0.994883i	$-0.532214\pi$
$691$	−2.53615e6	−0.202060	−0.101030	+	0.994883i	$0.532214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.17055e7	−1.70454
$696$	0	0
$697$	−3.60073e6	−0.280743
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.13234e7	1.63893	0.819466	−	0.573128i	$-0.194270\pi$
$701$	2.13234e7	1.63893	0.819466	+	0.573128i	$0.194270\pi$
$702$	0	0
$703$	5.56457e6	0.424662
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.24208e7	−0.934546
$708$	0	0
$709$	9.73372e6	0.727216	0.363608	−	0.931552i	$-0.381545\pi$
$709$	9.73372e6	0.727216	0.363608	+	0.931552i	$0.381545\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.55423e6	0.556501
$714$	0	0
$715$	1.18589e7	0.867522
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.68007e6	0.121200	0.0606002	−	0.998162i	$-0.480699\pi$
$719$	1.68007e6	0.121200	0.0606002	+	0.998162i	$0.480699\pi$
$720$	0	0
$721$	−5.17657e6	−0.370855
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.70091e7	2.61495
$726$	0	0
$727$	−1.18408e7	−0.830891	−0.415445	−	0.909618i	$-0.636374\pi$
$727$	−1.18408e7	−0.830891	−0.415445	+	0.909618i	$0.636374\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.36941e7	−0.947850
$732$	0	0
$733$	−7.38510e6	−0.507688	−0.253844	−	0.967245i	$-0.581695\pi$
$733$	−7.38510e6	−0.507688	−0.253844	+	0.967245i	$0.581695\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.45033e6	−0.166171
$738$	0	0
$739$	2.54272e7	1.71272	0.856361	−	0.516377i	$-0.172720\pi$
$739$	2.54272e7	1.71272	0.856361	+	0.516377i	$0.172720\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.96141e7	−1.30345	−0.651727	−	0.758454i	$-0.725955\pi$
$743$	−1.96141e7	−1.30345	−0.651727	+	0.758454i	$0.725955\pi$
$744$	0	0
$745$	8.89905e6	0.587426
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.48488e7	−0.967135
$750$	0	0
$751$	−1.53430e6	−0.0992685	−0.0496342	−	0.998767i	$-0.515806\pi$
$751$	−1.53430e6	−0.0992685	−0.0496342	+	0.998767i	$0.515806\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.64869e7	−3.60645
$756$	0	0
$757$	−1.20770e6	−0.0765984	−0.0382992	−	0.999266i	$-0.512194\pi$
$757$	−1.20770e6	−0.0765984	−0.0382992	+	0.999266i	$0.512194\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.52116e7	0.952165	0.476082	−	0.879401i	$-0.342056\pi$
$761$	1.52116e7	0.952165	0.476082	+	0.879401i	$0.342056\pi$
$762$	0	0
$763$	9.21934e6	0.573309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.67446e6	0.225530
$768$	0	0
$769$	−3.34827e6	−0.204176	−0.102088	−	0.994775i	$-0.532552\pi$
$769$	−3.34827e6	−0.204176	−0.102088	+	0.994775i	$0.532552\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.37017e7	−0.824758	−0.412379	−	0.911012i	$-0.635302\pi$
$773$	−1.37017e7	−0.824758	−0.412379	+	0.911012i	$0.635302\pi$
$774$	0	0
$775$	−3.38485e7	−2.02435
$776$	0	0
$777$	0	0
$778$	0	0
$779$	695796.	0.0410807
$780$	0	0
$781$	1.15475e7	0.677426
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.85110e7	1.07215
$786$	0	0
$787$	2.18608e7	1.25814	0.629070	−	0.777349i	$-0.283436\pi$
$787$	2.18608e7	1.25814	0.629070	+	0.777349i	$0.283436\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.10108e7	1.76226
$792$	0	0
$793$	8.56598e6	0.483720
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.36301e6	0.466355	0.233177	−	0.972434i	$-0.425088\pi$
$797$	8.36301e6	0.466355	0.233177	+	0.972434i	$0.425088\pi$
$798$	0	0
$799$	−9.49212e6	−0.526013
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.32334e7	0.724238
$804$	0	0
$805$	3.80813e7	2.07120
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.37307e6	0.127479	0.0637396	−	0.997967i	$-0.479697\pi$
$809$	2.37307e6	0.127479	0.0637396	+	0.997967i	$0.479697\pi$
$810$	0	0
$811$	−2.18207e7	−1.16498	−0.582489	−	0.812839i	$-0.697921\pi$
$811$	−2.18207e7	−1.16498	−0.582489	+	0.812839i	$0.697921\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.27193e7	−0.670761
$816$	0	0
$817$	2.64621e6	0.138698
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.63150e6	0.291586	0.145793	−	0.989315i	$-0.453427\pi$
$821$	5.63150e6	0.291586	0.145793	+	0.989315i	$0.453427\pi$
$822$	0	0
$823$	−2.74886e7	−1.41466	−0.707331	−	0.706882i	$-0.750101\pi$
$823$	−2.74886e7	−1.41466	−0.707331	+	0.706882i	$0.750101\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.53327e6	−0.230488	−0.115244	−	0.993337i	$-0.536765\pi$
$827$	−4.53327e6	−0.230488	−0.115244	+	0.993337i	$0.536765\pi$
$828$	0	0
$829$	−2.86843e7	−1.44963	−0.724817	−	0.688942i	$-0.758075\pi$
$829$	−2.86843e7	−1.44963	−0.724817	+	0.688942i	$0.758075\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.73276e7	1.36455
$834$	0	0
$835$	1.07179e6	0.0531980
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.47030e7	−1.21156	−0.605781	−	0.795632i	$-0.707139\pi$
$839$	−2.47030e7	−1.21156	−0.605781	+	0.795632i	$0.707139\pi$
$840$	0	0
$841$	−2.86243e6	−0.139555
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.58022e7	−0.761334
$846$	0	0
$847$	1.93360e7	0.926098
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.03059e7	1.43451
$852$	0	0
$853$	2.47224e7	1.16337	0.581686	−	0.813414i	$-0.302393\pi$
$853$	2.47224e7	1.16337	0.581686	+	0.813414i	$0.302393\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.44852e7	1.13881	0.569404	−	0.822058i	$-0.307174\pi$
$857$	2.44852e7	1.13881	0.569404	+	0.822058i	$0.307174\pi$
$858$	0	0
$859$	−6.62049e6	−0.306131	−0.153066	−	0.988216i	$-0.548915\pi$
$859$	−6.62049e6	−0.306131	−0.153066	+	0.988216i	$0.548915\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.86778e6	−0.0853687	−0.0426843	−	0.999089i	$-0.513591\pi$
$863$	−1.86778e6	−0.0853687	−0.0426843	+	0.999089i	$0.513591\pi$
$864$	0	0
$865$	−2.08964e7	−0.949579
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.73290e6	−0.347370
$870$	0	0
$871$	−5.11594e6	−0.228497
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.10104e8	−4.86162
$876$	0	0
$877$	−2.99625e7	−1.31546	−0.657732	−	0.753252i	$-0.728484\pi$
$877$	−2.99625e7	−1.31546	−0.657732	+	0.753252i	$0.728484\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.53722e7	−0.667261	−0.333631	−	0.942704i	$-0.608274\pi$
$881$	−1.53722e7	−0.667261	−0.333631	+	0.942704i	$0.608274\pi$
$882$	0	0
$883$	3.92079e7	1.69228	0.846139	−	0.532962i	$-0.178921\pi$
$883$	3.92079e7	1.69228	0.846139	+	0.532962i	$0.178921\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.26437e7	−0.539589	−0.269795	−	0.962918i	$-0.586956\pi$
$887$	−1.26437e7	−0.539589	−0.269795	+	0.962918i	$0.586956\pi$
$888$	0	0
$889$	1.87481e6	0.0795613
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.83423e6	0.0769708
$894$	0	0
$895$	−6.89546e7	−2.87743
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.61415e7	−0.666108
$900$	0	0
$901$	2.35690e7	0.967228
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.30843e7	−0.531043
$906$	0	0
$907$	−2.09717e7	−0.846475	−0.423238	−	0.906019i	$-0.639106\pi$
$907$	−2.09717e7	−0.846475	−0.423238	+	0.906019i	$0.639106\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.60866e7	1.83983	0.919916	−	0.392114i	$-0.128256\pi$
$911$	4.60866e7	1.83983	0.919916	+	0.392114i	$0.128256\pi$
$912$	0	0
$913$	7.01416e6	0.278483
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.85637e6	0.269260
$918$	0	0
$919$	5.84244e6	0.228195	0.114097	−	0.993470i	$-0.463602\pi$
$919$	5.84244e6	0.228195	0.114097	+	0.993470i	$0.463602\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.41096e7	0.931507
$924$	0	0
$925$	−1.35792e8	−5.21821
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.13555e7	−0.811839	−0.405919	−	0.913909i	$-0.633049\pi$
$929$	−2.13555e7	−0.811839	−0.405919	+	0.913909i	$0.633049\pi$
$930$	0	0
$931$	−5.28071e6	−0.199672
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.65360e7	−1.74085
$936$	0	0
$937$	−1.24265e7	−0.462382	−0.231191	−	0.972908i	$-0.574262\pi$
$937$	−1.24265e7	−0.462382	−0.231191	+	0.972908i	$0.574262\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.44931e7	−1.26987	−0.634934	−	0.772566i	$-0.718973\pi$
$941$	−3.44931e7	−1.26987	−0.634934	+	0.772566i	$0.718973\pi$
$942$	0	0
$943$	3.78946e6	0.138771
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.32057e7	−0.840853	−0.420426	−	0.907327i	$-0.638119\pi$
$947$	−2.32057e7	−0.840853	−0.420426	+	0.907327i	$0.638119\pi$
$948$	0	0
$949$	2.76294e7	0.995877
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.54048e7	0.906116	0.453058	−	0.891481i	$-0.350333\pi$
$953$	2.54048e7	0.906116	0.453058	+	0.891481i	$0.350333\pi$
$954$	0	0
$955$	1.02931e8	3.65208
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.51187e7	0.530845
$960$	0	0
$961$	−1.38662e7	−0.484337
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.12384e7	−2.11692
$966$	0	0
$967$	5.24428e6	0.180352	0.0901758	−	0.995926i	$-0.471257\pi$
$967$	5.24428e6	0.180352	0.0901758	+	0.995926i	$0.471257\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.50453e7	1.53321	0.766605	−	0.642119i	$-0.221945\pi$
$971$	4.50453e7	1.53321	0.766605	+	0.642119i	$0.221945\pi$
$972$	0	0
$973$	3.52269e7	1.19287
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.97190e7	1.33126	0.665628	−	0.746284i	$-0.268164\pi$
$977$	3.97190e7	1.33126	0.665628	+	0.746284i	$0.268164\pi$
$978$	0	0
$979$	2.22641e7	0.742416
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.93498e7	1.95900	0.979502	−	0.201436i	$-0.0645609\pi$
$983$	5.93498e7	1.95900	0.979502	+	0.201436i	$0.0645609\pi$
$984$	0	0
$985$	−8.65790e7	−2.84329
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.44118e7	0.468521
$990$	0	0
$991$	9.53405e6	0.308385	0.154193	−	0.988041i	$-0.450722\pi$
$991$	9.53405e6	0.308385	0.154193	+	0.988041i	$0.450722\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.22116e7	1.67190
$996$	0	0
$997$	2.39124e7	0.761878	0.380939	−	0.924600i	$-0.375601\pi$
$997$	2.39124e7	0.761878	0.380939	+	0.924600i	$0.375601\pi$
$998$	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.1.4		4
3.2	odd	2		76.6.a.b.1.1	✓	4
12.11	even	2		304.6.a.k.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.a.b.1.1	✓	4	3.2	odd	2
304.6.a.k.1.4		4	12.11	even	2
684.6.a.e.1.4		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)

\(f(q)\)	\(=\)	\(q+109.245 q^{5} -177.299 q^{7} +O(q^{10})\)	\(q+109.245 q^{5} -177.299 q^{7} -228.019 q^{11} -476.072 q^{13} +1868.17 q^{17} -361.000 q^{19} -1966.09 q^{23} +8809.50 q^{25} +4201.04 q^{29} -3842.27 q^{31} -19369.1 q^{35} -15414.3 q^{37} -1927.41 q^{41} -7330.22 q^{43} -5080.98 q^{47} +14628.0 q^{49} +12616.1 q^{53} -24910.0 q^{55} -7718.28 q^{59} -17993.0 q^{61} -52008.5 q^{65} +10746.2 q^{67} -50642.8 q^{71} -58036.2 q^{73} +40427.6 q^{77} +33913.4 q^{79} -30761.3 q^{83} +204088. q^{85} -97641.2 q^{89} +84407.1 q^{91} -39437.5 q^{95} +105569. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 110 q^{5} + 30 q^{7}+O(q^{10}) \) 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

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