LMFDB - Embedded newform 7800.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.461844.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} - 12x + 6 $ x^4 - x^3 - 14x^2 - 12x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.60110$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} -5.61488 q^{11} -1.00000 q^{13} -5.77950 q^{17} -0.234280 q^{19} +2.60110 q^{21} -7.01378 q^{23} -1.00000 q^{27} +4.20220 q^{29} +4.41269 q^{31} +5.61488 q^{33} +2.96792 q^{37} +1.00000 q^{39} -10.9955 q^{41} -0.188415 q^{43} -7.84916 q^{47} -0.234280 q^{49} +5.77950 q^{51} -11.9496 q^{53} +0.234280 q^{57} -12.1601 q^{59} -2.98170 q^{61} -2.60110 q^{63} -3.56902 q^{67} +7.01378 q^{69} +3.76572 q^{71} +14.5728 q^{73} +14.6049 q^{77} -4.96792 q^{79} +1.00000 q^{81} -3.39890 q^{83} -4.20220 q^{87} +17.2298 q^{89} +2.60110 q^{91} -4.41269 q^{93} +5.20220 q^{97} -5.61488 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + 5 q^{11} - 4 q^{13} - 7 q^{17} + 3 q^{19} + 3 q^{21} - 8 q^{23} - 4 q^{27} + 2 q^{29} + 5 q^{31} - 5 q^{33} + q^{37} + 4 q^{39} + 7 q^{41} - 6 q^{43} + 3 q^{49} + 7 q^{51} - 6 q^{53} - 3 q^{57} - 9 q^{59} + 19 q^{61} - 3 q^{63} + 4 q^{67} + 8 q^{69} + 19 q^{71} + 6 q^{73} + 17 q^{77} - 9 q^{79} + 4 q^{81} - 21 q^{83} - 2 q^{87} + 14 q^{89} + 3 q^{91} - 5 q^{93} + 6 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9 + 5 * q^11 - 4 * q^13 - 7 * q^17 + 3 * q^19 + 3 * q^21 - 8 * q^23 - 4 * q^27 + 2 * q^29 + 5 * q^31 - 5 * q^33 + q^37 + 4 * q^39 + 7 * q^41 - 6 * q^43 + 3 * q^49 + 7 * q^51 - 6 * q^53 - 3 * q^57 - 9 * q^59 + 19 * q^61 - 3 * q^63 + 4 * q^67 + 8 * q^69 + 19 * q^71 + 6 * q^73 + 17 * q^77 - 9 * q^79 + 4 * q^81 - 21 * q^83 - 2 * q^87 + 14 * q^89 + 3 * q^91 - 5 * q^93 + 6 * q^97 + 5 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.60110	−0.983123	−0.491562	−	0.870843i	$-0.663574\pi$
$7$	−2.60110	−0.983123	−0.491562	+	0.870843i	$0.663574\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.61488	−1.69295	−0.846476	−	0.532427i	$-0.821280\pi$
$11$	−5.61488	−1.69295	−0.846476	+	0.532427i	$0.821280\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.77950	−1.40174	−0.700868	−	0.713291i	$-0.747204\pi$
$17$	−5.77950	−1.40174	−0.700868	+	0.713291i	$0.747204\pi$
$18$	0	0
$19$	−0.234280	−0.0537475	−0.0268738	−	0.999639i	$-0.508555\pi$
$19$	−0.234280	−0.0537475	−0.0268738	+	0.999639i	$0.508555\pi$
$20$	0	0
$21$	2.60110	0.567607
$22$	0	0
$23$	−7.01378	−1.46248	−0.731238	−	0.682123i	$-0.761057\pi$
$23$	−7.01378	−1.46248	−0.731238	+	0.682123i	$0.761057\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.20220	0.780329	0.390164	−	0.920745i	$-0.372418\pi$
$29$	4.20220	0.780329	0.390164	+	0.920745i	$0.372418\pi$
$30$	0	0
$31$	4.41269	0.792542	0.396271	−	0.918134i	$-0.370304\pi$
$31$	4.41269	0.792542	0.396271	+	0.918134i	$0.370304\pi$
$32$	0	0
$33$	5.61488	0.977426
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.96792	0.487923	0.243961	−	0.969785i	$-0.421553\pi$
$37$	2.96792	0.487923	0.243961	+	0.969785i	$0.421553\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.9955	−1.71721	−0.858603	−	0.512640i	$-0.828667\pi$
$41$	−10.9955	−1.71721	−0.858603	+	0.512640i	$0.828667\pi$
$42$	0	0
$43$	−0.188415	−0.0287330	−0.0143665	−	0.999897i	$-0.504573\pi$
$43$	−0.188415	−0.0287330	−0.0143665	+	0.999897i	$0.504573\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.84916	−1.14492	−0.572459	−	0.819933i	$-0.694010\pi$
$47$	−7.84916	−1.14492	−0.572459	+	0.819933i	$0.694010\pi$
$48$	0	0
$49$	−0.234280	−0.0334686
$50$	0	0
$51$	5.77950	0.809293
$52$	0	0
$53$	−11.9496	−1.64141	−0.820704	−	0.571354i	$-0.806418\pi$
$53$	−11.9496	−1.64141	−0.820704	+	0.571354i	$0.806418\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.234280	0.0310312
$58$	0	0
$59$	−12.1601	−1.58311	−0.791556	−	0.611097i	$-0.790729\pi$
$59$	−12.1601	−1.58311	−0.791556	+	0.611097i	$0.790729\pi$
$60$	0	0
$61$	−2.98170	−0.381768	−0.190884	−	0.981613i	$-0.561135\pi$
$61$	−2.98170	−0.381768	−0.190884	+	0.981613i	$0.561135\pi$
$62$	0	0
$63$	−2.60110	−0.327708
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.56902	−0.436025	−0.218013	−	0.975946i	$-0.569957\pi$
$67$	−3.56902	−0.436025	−0.218013	+	0.975946i	$0.569957\pi$
$68$	0	0
$69$	7.01378	0.844360
$70$	0	0
$71$	3.76572	0.446909	0.223454	−	0.974714i	$-0.428267\pi$
$71$	3.76572	0.446909	0.223454	+	0.974714i	$0.428267\pi$
$72$	0	0
$73$	14.5728	1.70562	0.852808	−	0.522224i	$-0.174898\pi$
$73$	14.5728	1.70562	0.852808	+	0.522224i	$0.174898\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.6049	1.66438
$78$	0	0
$79$	−4.96792	−0.558935	−0.279467	−	0.960155i	$-0.590158\pi$
$79$	−4.96792	−0.558935	−0.279467	+	0.960155i	$0.590158\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.39890	−0.373078	−0.186539	−	0.982448i	$-0.559727\pi$
$83$	−3.39890	−0.373078	−0.186539	+	0.982448i	$0.559727\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.20220	−0.450523
$88$	0	0
$89$	17.2298	1.82635	0.913176	−	0.407566i	$-0.133622\pi$
$89$	17.2298	1.82635	0.913176	+	0.407566i	$0.133622\pi$
$90$	0	0
$91$	2.60110	0.272669
$92$	0	0
$93$	−4.41269	−0.457574
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.20220	0.528203	0.264102	−	0.964495i	$-0.414925\pi$
$97$	5.20220	0.528203	0.264102	+	0.964495i	$0.414925\pi$
$98$	0	0
$99$	−5.61488	−0.564317
$100$	0	0
$101$	5.96792	0.593830	0.296915	−	0.954904i	$-0.404042\pi$
$101$	5.96792	0.593830	0.296915	+	0.954904i	$0.404042\pi$
$102$	0	0
$103$	−15.8392	−1.56068	−0.780339	−	0.625357i	$-0.784954\pi$
$103$	−15.8392	−1.56068	−0.780339	+	0.625357i	$0.784954\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.04587	−0.391129	−0.195564	−	0.980691i	$-0.562654\pi$
$107$	−4.04587	−0.391129	−0.195564	+	0.980691i	$0.562654\pi$
$108$	0	0
$109$	−17.7291	−1.69814	−0.849071	−	0.528278i	$-0.822838\pi$
$109$	−17.7291	−1.69814	−0.849071	+	0.528278i	$0.822838\pi$
$110$	0	0
$111$	−2.96792	−0.281702
$112$	0	0
$113$	0.733639	0.0690150	0.0345075	−	0.999404i	$-0.489014\pi$
$113$	0.733639	0.0690150	0.0345075	+	0.999404i	$0.489014\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	15.0331	1.37808
$120$	0	0
$121$	20.5269	1.86608
$122$	0	0
$123$	10.9955	0.991430
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.17012	0.370038	0.185019	−	0.982735i	$-0.440765\pi$
$127$	4.17012	0.370038	0.185019	+	0.982735i	$0.440765\pi$
$128$	0	0
$129$	0.188415	0.0165890
$130$	0	0
$131$	18.2618	1.59555	0.797773	−	0.602958i	$-0.206012\pi$
$131$	18.2618	1.59555	0.797773	+	0.602958i	$0.206012\pi$
$132$	0	0
$133$	0.609386	0.0528405
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0093	1.53864	0.769318	−	0.638866i	$-0.220596\pi$
$137$	18.0093	1.53864	0.769318	+	0.638866i	$0.220596\pi$
$138$	0	0
$139$	3.55901	0.301871	0.150936	−	0.988544i	$-0.451771\pi$
$139$	3.55901	0.301871	0.150936	+	0.988544i	$0.451771\pi$
$140$	0	0
$141$	7.84916	0.661019
$142$	0	0
$143$	5.61488	0.469540
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.234280	0.0193231
$148$	0	0
$149$	9.20220	0.753874	0.376937	−	0.926239i	$-0.376977\pi$
$149$	9.20220	0.753874	0.376937	+	0.926239i	$0.376977\pi$
$150$	0	0
$151$	3.86746	0.314729	0.157365	−	0.987541i	$-0.449700\pi$
$151$	3.86746	0.314729	0.157365	+	0.987541i	$0.449700\pi$
$152$	0	0
$153$	−5.77950	−0.467245
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.2343	0.896593	0.448297	−	0.893885i	$-0.352031\pi$
$157$	11.2343	0.896593	0.448297	+	0.893885i	$0.352031\pi$
$158$	0	0
$159$	11.9496	0.947667
$160$	0	0
$161$	18.2436	1.43779
$162$	0	0
$163$	1.48234	0.116106	0.0580531	−	0.998313i	$-0.481511\pi$
$163$	1.48234	0.116106	0.0580531	+	0.998313i	$0.481511\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.37232	−0.725252	−0.362626	−	0.931935i	$-0.618120\pi$
$167$	−9.37232	−0.725252	−0.362626	+	0.931935i	$0.618120\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.234280	−0.0179158
$172$	0	0
$173$	−21.7121	−1.65074	−0.825371	−	0.564591i	$-0.809034\pi$
$173$	−21.7121	−1.65074	−0.825371	+	0.564591i	$0.809034\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.1601	0.914010
$178$	0	0
$179$	−18.6204	−1.39175	−0.695876	−	0.718162i	$-0.744984\pi$
$179$	−18.6204	−1.39175	−0.695876	+	0.718162i	$0.744984\pi$
$180$	0	0
$181$	14.4365	1.07306	0.536528	−	0.843883i	$-0.319736\pi$
$181$	14.4365	1.07306	0.536528	+	0.843883i	$0.319736\pi$
$182$	0	0
$183$	2.98170	0.220414
$184$	0	0
$185$	0	0
$186$	0	0
$187$	32.4513	2.37307
$188$	0	0
$189$	2.60110	0.189202
$190$	0	0
$191$	20.4044	1.47641	0.738205	−	0.674576i	$-0.235674\pi$
$191$	20.4044	1.47641	0.738205	+	0.674576i	$0.235674\pi$
$192$	0	0
$193$	−26.1318	−1.88101	−0.940504	−	0.339782i	$-0.889647\pi$
$193$	−26.1318	−1.88101	−0.940504	+	0.339782i	$0.889647\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8026	−1.55337	−0.776684	−	0.629890i	$-0.783100\pi$
$197$	−21.8026	−1.55337	−0.776684	+	0.629890i	$0.783100\pi$
$198$	0	0
$199$	−5.22050	−0.370071	−0.185036	−	0.982732i	$-0.559240\pi$
$199$	−5.22050	−0.370071	−0.185036	+	0.982732i	$0.559240\pi$
$200$	0	0
$201$	3.56902	0.251739
$202$	0	0
$203$	−10.9303	−0.767159
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.01378	−0.487492
$208$	0	0
$209$	1.31546	0.0909920
$210$	0	0
$211$	11.0138	0.758220	0.379110	−	0.925352i	$-0.376230\pi$
$211$	11.0138	0.758220	0.379110	+	0.925352i	$0.376230\pi$
$212$	0	0
$213$	−3.76572	−0.258023
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.4778	−0.779166
$218$	0	0
$219$	−14.5728	−0.984738
$220$	0	0
$221$	5.77950	0.388772
$222$	0	0
$223$	−27.4733	−1.83975	−0.919875	−	0.392212i	$-0.871710\pi$
$223$	−27.4733	−1.83975	−0.919875	+	0.392212i	$0.871710\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.47556	−0.0979367	−0.0489683	−	0.998800i	$-0.515593\pi$
$227$	−1.47556	−0.0979367	−0.0489683	+	0.998800i	$0.515593\pi$
$228$	0	0
$229$	−9.65246	−0.637853	−0.318926	−	0.947779i	$-0.603322\pi$
$229$	−9.65246	−0.637853	−0.318926	+	0.947779i	$0.603322\pi$
$230$	0	0
$231$	−14.6049	−0.960930
$232$	0	0
$233$	−19.2298	−1.25978	−0.629892	−	0.776683i	$-0.716901\pi$
$233$	−19.2298	−1.25978	−0.629892	+	0.776683i	$0.716901\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.96792	0.322701
$238$	0	0
$239$	−0.244290	−0.0158018	−0.00790089	−	0.999969i	$-0.502515\pi$
$239$	−0.244290	−0.0158018	−0.00790089	+	0.999969i	$0.502515\pi$
$240$	0	0
$241$	0.986215	0.0635277	0.0317639	−	0.999495i	$-0.489888\pi$
$241$	0.986215	0.0635277	0.0317639	+	0.999495i	$0.489888\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.234280	0.0149069
$248$	0	0
$249$	3.39890	0.215397
$250$	0	0
$251$	−0.995489	−0.0628347	−0.0314174	−	0.999506i	$-0.510002\pi$
$251$	−0.995489	−0.0628347	−0.0314174	+	0.999506i	$0.510002\pi$
$252$	0	0
$253$	39.3816	2.47590
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.3523	−1.26954	−0.634771	−	0.772701i	$-0.718905\pi$
$257$	−20.3523	−1.26954	−0.634771	+	0.772701i	$0.718905\pi$
$258$	0	0
$259$	−7.71985	−0.479688
$260$	0	0
$261$	4.20220	0.260110
$262$	0	0
$263$	−0.252576	−0.0155745	−0.00778724	−	0.999970i	$-0.502479\pi$
$263$	−0.252576	−0.0155745	−0.00778724	+	0.999970i	$0.502479\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−17.2298	−1.05444
$268$	0	0
$269$	14.9358	0.910654	0.455327	−	0.890324i	$-0.349522\pi$
$269$	14.9358	0.910654	0.455327	+	0.890324i	$0.349522\pi$
$270$	0	0
$271$	−3.09723	−0.188143	−0.0940716	−	0.995565i	$-0.529988\pi$
$271$	−3.09723	−0.188143	−0.0940716	+	0.995565i	$0.529988\pi$
$272$	0	0
$273$	−2.60110	−0.157426
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.2939	1.27943	0.639714	−	0.768613i	$-0.279053\pi$
$277$	21.2939	1.27943	0.639714	+	0.768613i	$0.279053\pi$
$278$	0	0
$279$	4.41269	0.264181
$280$	0	0
$281$	13.1839	0.786486	0.393243	−	0.919435i	$-0.371353\pi$
$281$	13.1839	0.786486	0.393243	+	0.919435i	$0.371353\pi$
$282$	0	0
$283$	7.18218	0.426936	0.213468	−	0.976950i	$-0.431524\pi$
$283$	7.18218	0.426936	0.213468	+	0.976950i	$0.431524\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.6004	1.68823
$288$	0	0
$289$	16.4027	0.964863
$290$	0	0
$291$	−5.20220	−0.304958
$292$	0	0
$293$	18.9005	1.10418	0.552090	−	0.833784i	$-0.313830\pi$
$293$	18.9005	1.10418	0.552090	+	0.833784i	$0.313830\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.61488	0.325809
$298$	0	0
$299$	7.01378	0.405618
$300$	0	0
$301$	0.490085	0.0282481
$302$	0	0
$303$	−5.96792	−0.342848
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.56352	0.260454	0.130227	−	0.991484i	$-0.458429\pi$
$307$	4.56352	0.260454	0.130227	+	0.991484i	$0.458429\pi$
$308$	0	0
$309$	15.8392	0.901058
$310$	0	0
$311$	−25.5500	−1.44881	−0.724403	−	0.689376i	$-0.757885\pi$
$311$	−25.5500	−1.44881	−0.724403	+	0.689376i	$0.757885\pi$
$312$	0	0
$313$	30.4092	1.71883	0.859414	−	0.511281i	$-0.170829\pi$
$313$	30.4092	1.71883	0.859414	+	0.511281i	$0.170829\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0401	1.79955	0.899775	−	0.436354i	$-0.143731\pi$
$317$	32.0401	1.79955	0.899775	+	0.436354i	$0.143731\pi$
$318$	0	0
$319$	−23.5949	−1.32106
$320$	0	0
$321$	4.04587	0.225818
$322$	0	0
$323$	1.35402	0.0753398
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.7291	0.980423
$328$	0	0
$329$	20.4165	1.12560
$330$	0	0
$331$	7.45478	0.409752	0.204876	−	0.978788i	$-0.434321\pi$
$331$	7.45478	0.409752	0.204876	+	0.978788i	$0.434321\pi$
$332$	0	0
$333$	2.96792	0.162641
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.4916	−0.680462	−0.340231	−	0.940342i	$-0.610505\pi$
$337$	−12.4916	−0.680462	−0.340231	+	0.940342i	$0.610505\pi$
$338$	0	0
$339$	−0.733639	−0.0398458
$340$	0	0
$341$	−24.7767	−1.34173
$342$	0	0
$343$	18.8171	1.01603
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.2115	1.24606	0.623029	−	0.782199i	$-0.285902\pi$
$347$	23.2115	1.24606	0.623029	+	0.782199i	$0.285902\pi$
$348$	0	0
$349$	−17.3982	−0.931302	−0.465651	−	0.884968i	$-0.654180\pi$
$349$	−17.3982	−0.931302	−0.465651	+	0.884968i	$0.654180\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	6.65525	0.354223	0.177112	−	0.984191i	$-0.443325\pi$
$353$	6.65525	0.354223	0.177112	+	0.984191i	$0.443325\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−15.0331	−0.795634
$358$	0	0
$359$	8.55523	0.451528	0.225764	−	0.974182i	$-0.427512\pi$
$359$	8.55523	0.451528	0.225764	+	0.974182i	$0.427512\pi$
$360$	0	0
$361$	−18.9451	−0.997111
$362$	0	0
$363$	−20.5269	−1.07738
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.7153	−0.768135	−0.384067	−	0.923305i	$-0.625477\pi$
$367$	−14.7153	−0.768135	−0.384067	+	0.923305i	$0.625477\pi$
$368$	0	0
$369$	−10.9955	−0.572402
$370$	0	0
$371$	31.0822	1.61371
$372$	0	0
$373$	8.87747	0.459658	0.229829	−	0.973231i	$-0.426183\pi$
$373$	8.87747	0.459658	0.229829	+	0.973231i	$0.426183\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.20220	−0.216424
$378$	0	0
$379$	−1.11875	−0.0574666	−0.0287333	−	0.999587i	$-0.509147\pi$
$379$	−1.11875	−0.0574666	−0.0287333	+	0.999587i	$0.509147\pi$
$380$	0	0
$381$	−4.17012	−0.213642
$382$	0	0
$383$	−17.3247	−0.885252	−0.442626	−	0.896706i	$-0.645953\pi$
$383$	−17.3247	−0.885252	−0.442626	+	0.896706i	$0.645953\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.188415	−0.00957766
$388$	0	0
$389$	0.459286	0.0232867	0.0116434	−	0.999932i	$-0.496294\pi$
$389$	0.459286	0.0232867	0.0116434	+	0.999932i	$0.496294\pi$
$390$	0	0
$391$	40.5362	2.05000
$392$	0	0
$393$	−18.2618	−0.921188
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.5144	1.02959	0.514795	−	0.857313i	$-0.327868\pi$
$397$	20.5144	1.02959	0.514795	+	0.857313i	$0.327868\pi$
$398$	0	0
$399$	−0.609386	−0.0305074
$400$	0	0
$401$	−7.31546	−0.365316	−0.182658	−	0.983176i	$-0.558470\pi$
$401$	−7.31546	−0.365316	−0.182658	+	0.983176i	$0.558470\pi$
$402$	0	0
$403$	−4.41269	−0.219812
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.6645	−0.826030
$408$	0	0
$409$	29.0889	1.43836	0.719178	−	0.694826i	$-0.244519\pi$
$409$	29.0889	1.43836	0.719178	+	0.694826i	$0.244519\pi$
$410$	0	0
$411$	−18.0093	−0.888332
$412$	0	0
$413$	31.6297	1.55639
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.55901	−0.174285
$418$	0	0
$419$	−3.90504	−0.190774	−0.0953868	−	0.995440i	$-0.530409\pi$
$419$	−3.90504	−0.190774	−0.0953868	+	0.995440i	$0.530409\pi$
$420$	0	0
$421$	2.06416	0.100601	0.0503005	−	0.998734i	$-0.483982\pi$
$421$	2.06416	0.100601	0.0503005	+	0.998734i	$0.483982\pi$
$422$	0	0
$423$	−7.84916	−0.381639
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.75571	0.375325
$428$	0	0
$429$	−5.61488	−0.271089
$430$	0	0
$431$	33.7567	1.62600	0.813001	−	0.582262i	$-0.197832\pi$
$431$	33.7567	1.62600	0.813001	+	0.582262i	$0.197832\pi$
$432$	0	0
$433$	−22.4137	−1.07713	−0.538566	−	0.842583i	$-0.681034\pi$
$433$	−22.4137	−1.07713	−0.538566	+	0.842583i	$0.681034\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.64319	0.0786044
$438$	0	0
$439$	10.6662	0.509072	0.254536	−	0.967063i	$-0.418077\pi$
$439$	10.6662	0.509072	0.254536	+	0.967063i	$0.418077\pi$
$440$	0	0
$441$	−0.234280	−0.0111562
$442$	0	0
$443$	14.5821	0.692815	0.346407	−	0.938084i	$-0.387401\pi$
$443$	14.5821	0.692815	0.346407	+	0.938084i	$0.387401\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.20220	−0.435249
$448$	0	0
$449$	−22.3019	−1.05249	−0.526246	−	0.850332i	$-0.676401\pi$
$449$	−22.3019	−1.05249	−0.526246	+	0.850332i	$0.676401\pi$
$450$	0	0
$451$	61.7384	2.90715
$452$	0	0
$453$	−3.86746	−0.181709
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.3706	0.999674	0.499837	−	0.866119i	$-0.333393\pi$
$457$	21.3706	0.999674	0.499837	+	0.866119i	$0.333393\pi$
$458$	0	0
$459$	5.77950	0.269764
$460$	0	0
$461$	−18.4686	−0.860167	−0.430083	−	0.902789i	$-0.641516\pi$
$461$	−18.4686	−0.860167	−0.430083	+	0.902789i	$0.641516\pi$
$462$	0	0
$463$	−1.52315	−0.0707870	−0.0353935	−	0.999373i	$-0.511268\pi$
$463$	−1.52315	−0.0707870	−0.0353935	+	0.999373i	$0.511268\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.0446	0.696180	0.348090	−	0.937461i	$-0.386830\pi$
$467$	15.0446	0.696180	0.348090	+	0.937461i	$0.386830\pi$
$468$	0	0
$469$	9.28337	0.428666
$470$	0	0
$471$	−11.2343	−0.517648
$472$	0	0
$473$	1.05793	0.0486435
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.9496	−0.547136
$478$	0	0
$479$	32.4695	1.48357	0.741786	−	0.670637i	$-0.233979\pi$
$479$	32.4695	1.48357	0.741786	+	0.670637i	$0.233979\pi$
$480$	0	0
$481$	−2.96792	−0.135325
$482$	0	0
$483$	−18.2436	−0.830110
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−28.0469	−1.27092	−0.635462	−	0.772132i	$-0.719190\pi$
$487$	−28.0469	−1.27092	−0.635462	+	0.772132i	$0.719190\pi$
$488$	0	0
$489$	−1.48234	−0.0670340
$490$	0	0
$491$	29.2097	1.31822	0.659109	−	0.752048i	$-0.270934\pi$
$491$	29.2097	1.31822	0.659109	+	0.752048i	$0.270934\pi$
$492$	0	0
$493$	−24.2866	−1.09381
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.79501	−0.439366
$498$	0	0
$499$	−34.7497	−1.55561	−0.777805	−	0.628506i	$-0.783667\pi$
$499$	−34.7497	−1.55561	−0.777805	+	0.628506i	$0.783667\pi$
$500$	0	0
$501$	9.37232	0.418724
$502$	0	0
$503$	22.5728	1.00647	0.503236	−	0.864149i	$-0.332143\pi$
$503$	22.5728	1.00647	0.503236	+	0.864149i	$0.332143\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	25.8502	1.14579	0.572894	−	0.819630i	$-0.305821\pi$
$509$	25.8502	1.14579	0.572894	+	0.819630i	$0.305821\pi$
$510$	0	0
$511$	−37.9053	−1.67683
$512$	0	0
$513$	0.234280	0.0103437
$514$	0	0
$515$	0	0
$516$	0	0
$517$	44.0722	1.93829
$518$	0	0
$519$	21.7121	0.953056
$520$	0	0
$521$	−25.9634	−1.13748	−0.568739	−	0.822518i	$-0.692568\pi$
$521$	−25.9634	−1.13748	−0.568739	+	0.822518i	$0.692568\pi$
$522$	0	0
$523$	−38.1579	−1.66853	−0.834264	−	0.551366i	$-0.814107\pi$
$523$	−38.1579	−1.66853	−0.834264	+	0.551366i	$0.814107\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.5031	−1.11093
$528$	0	0
$529$	26.1932	1.13883
$530$	0	0
$531$	−12.1601	−0.527704
$532$	0	0
$533$	10.9955	0.476268
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.6204	0.803529
$538$	0	0
$539$	1.31546	0.0566607
$540$	0	0
$541$	27.0414	1.16260	0.581299	−	0.813690i	$-0.302545\pi$
$541$	27.0414	1.16260	0.581299	+	0.813690i	$0.302545\pi$
$542$	0	0
$543$	−14.4365	−0.619529
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.5287	0.621200	0.310600	−	0.950541i	$-0.399470\pi$
$547$	14.5287	0.621200	0.310600	+	0.950541i	$0.399470\pi$
$548$	0	0
$549$	−2.98170	−0.127256
$550$	0	0
$551$	−0.984492	−0.0419408
$552$	0	0
$553$	12.9221	0.549502
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.1684	−0.600334	−0.300167	−	0.953887i	$-0.597042\pi$
$557$	−14.1684	−0.600334	−0.300167	+	0.953887i	$0.597042\pi$
$558$	0	0
$559$	0.188415	0.00796909
$560$	0	0
$561$	−32.4513	−1.37009
$562$	0	0
$563$	−28.1701	−1.18723	−0.593614	−	0.804750i	$-0.702300\pi$
$563$	−28.1701	−1.18723	−0.593614	+	0.804750i	$0.702300\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.60110	−0.109236
$568$	0	0
$569$	−36.0140	−1.50979	−0.754893	−	0.655847i	$-0.772311\pi$
$569$	−36.0140	−1.50979	−0.754893	+	0.655847i	$0.772311\pi$
$570$	0	0
$571$	13.9233	0.582673	0.291337	−	0.956621i	$-0.405900\pi$
$571$	13.9233	0.582673	0.291337	+	0.956621i	$0.405900\pi$
$572$	0	0
$573$	−20.4044	−0.852406
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.32022	−0.179853	−0.0899265	−	0.995948i	$-0.528663\pi$
$577$	−4.32022	−0.179853	−0.0899265	+	0.995948i	$0.528663\pi$
$578$	0	0
$579$	26.1318	1.08600
$580$	0	0
$581$	8.84088	0.366781
$582$	0	0
$583$	67.0958	2.77882
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.4878	1.75366	0.876830	−	0.480800i	$-0.159654\pi$
$587$	42.4878	1.75366	0.876830	+	0.480800i	$0.159654\pi$
$588$	0	0
$589$	−1.03380	−0.0425972
$590$	0	0
$591$	21.8026	0.896838
$592$	0	0
$593$	−18.8712	−0.774949	−0.387474	−	0.921880i	$-0.626652\pi$
$593$	−18.8712	−0.774949	−0.387474	+	0.921880i	$0.626652\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.22050	0.213661
$598$	0	0
$599$	−43.4367	−1.77478	−0.887388	−	0.461023i	$-0.847483\pi$
$599$	−43.4367	−1.77478	−0.887388	+	0.461023i	$0.847483\pi$
$600$	0	0
$601$	−27.6645	−1.12846	−0.564230	−	0.825618i	$-0.690827\pi$
$601$	−27.6645	−1.12846	−0.564230	+	0.825618i	$0.690827\pi$
$602$	0	0
$603$	−3.56902	−0.145342
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.9188	−0.443181	−0.221591	−	0.975140i	$-0.571125\pi$
$607$	−10.9188	−0.443181	−0.221591	+	0.975140i	$0.571125\pi$
$608$	0	0
$609$	10.9303	0.442920
$610$	0	0
$611$	7.84916	0.317543
$612$	0	0
$613$	−0.459538	−0.0185606	−0.00928029	−	0.999957i	$-0.502954\pi$
$613$	−0.459538	−0.0185606	−0.00928029	+	0.999957i	$0.502954\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.2280	0.532541	0.266271	−	0.963898i	$-0.414209\pi$
$617$	13.2280	0.532541	0.266271	+	0.963898i	$0.414209\pi$
$618$	0	0
$619$	36.6480	1.47301	0.736503	−	0.676434i	$-0.236476\pi$
$619$	36.6480	1.47301	0.736503	+	0.676434i	$0.236476\pi$
$620$	0	0
$621$	7.01378	0.281453
$622$	0	0
$623$	−44.8163	−1.79553
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.31546	−0.0525342
$628$	0	0
$629$	−17.1531	−0.683939
$630$	0	0
$631$	9.15310	0.364379	0.182190	−	0.983263i	$-0.441682\pi$
$631$	9.15310	0.364379	0.182190	+	0.983263i	$0.441682\pi$
$632$	0	0
$633$	−11.0138	−0.437759
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.234280	0.00928251
$638$	0	0
$639$	3.76572	0.148970
$640$	0	0
$641$	−32.5883	−1.28716	−0.643580	−	0.765379i	$-0.722552\pi$
$641$	−32.5883	−1.28716	−0.643580	+	0.765379i	$0.722552\pi$
$642$	0	0
$643$	−8.28942	−0.326903	−0.163451	−	0.986551i	$-0.552263\pi$
$643$	−8.28942	−0.326903	−0.163451	+	0.986551i	$0.552263\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.72590	0.303736	0.151868	−	0.988401i	$-0.451471\pi$
$647$	7.72590	0.303736	0.151868	+	0.988401i	$0.451471\pi$
$648$	0	0
$649$	68.2776	2.68013
$650$	0	0
$651$	11.4778	0.449852
$652$	0	0
$653$	15.1210	0.591731	0.295866	−	0.955230i	$-0.404392\pi$
$653$	15.1210	0.591731	0.295866	+	0.955230i	$0.404392\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.5728	0.568539
$658$	0	0
$659$	−7.65397	−0.298156	−0.149078	−	0.988825i	$-0.547631\pi$
$659$	−7.65397	−0.298156	−0.149078	+	0.988825i	$0.547631\pi$
$660$	0	0
$661$	33.3633	1.29768	0.648841	−	0.760924i	$-0.275254\pi$
$661$	33.3633	1.29768	0.648841	+	0.760924i	$0.275254\pi$
$662$	0	0
$663$	−5.77950	−0.224457
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−29.4733	−1.14121
$668$	0	0
$669$	27.4733	1.06218
$670$	0	0
$671$	16.7419	0.646315
$672$	0	0
$673$	−23.9020	−0.921356	−0.460678	−	0.887567i	$-0.652394\pi$
$673$	−23.9020	−0.921356	−0.460678	+	0.887567i	$0.652394\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.0276	1.38465	0.692326	−	0.721585i	$-0.256586\pi$
$677$	36.0276	1.38465	0.692326	+	0.721585i	$0.256586\pi$
$678$	0	0
$679$	−13.5314	−0.519289
$680$	0	0
$681$	1.47556	0.0565438
$682$	0	0
$683$	−43.2015	−1.65306	−0.826529	−	0.562894i	$-0.809688\pi$
$683$	−43.2015	−1.65306	−0.826529	+	0.562894i	$0.809688\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.65246	0.368264
$688$	0	0
$689$	11.9496	0.455244
$690$	0	0
$691$	−11.7008	−0.445120	−0.222560	−	0.974919i	$-0.571441\pi$
$691$	−11.7008	−0.445120	−0.222560	+	0.974919i	$0.571441\pi$
$692$	0	0
$693$	14.6049	0.554793
$694$	0	0
$695$	0	0
$696$	0	0
$697$	63.5485	2.40707
$698$	0	0
$699$	19.2298	0.727337
$700$	0	0
$701$	−52.7552	−1.99254	−0.996268	−	0.0863132i	$-0.972491\pi$
$701$	−52.7552	−1.99254	−0.996268	+	0.0863132i	$0.972491\pi$
$702$	0	0
$703$	−0.695324	−0.0262247
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.5232	−0.583808
$708$	0	0
$709$	30.1428	1.13204	0.566018	−	0.824393i	$-0.308483\pi$
$709$	30.1428	1.13204	0.566018	+	0.824393i	$0.308483\pi$
$710$	0	0
$711$	−4.96792	−0.186312
$712$	0	0
$713$	−30.9496	−1.15907
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.244290	0.00912317
$718$	0	0
$719$	−2.06009	−0.0768285	−0.0384142	−	0.999262i	$-0.512231\pi$
$719$	−2.06009	−0.0768285	−0.0384142	+	0.999262i	$0.512231\pi$
$720$	0	0
$721$	41.1992	1.53434
$722$	0	0
$723$	−0.986215	−0.0366777
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.25129	−0.194760	−0.0973799	−	0.995247i	$-0.531046\pi$
$727$	−5.25129	−0.194760	−0.0973799	+	0.995247i	$0.531046\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.08894	0.0402760
$732$	0	0
$733$	24.7795	0.915252	0.457626	−	0.889145i	$-0.348700\pi$
$733$	24.7795	0.915252	0.457626	+	0.889145i	$0.348700\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.0396	0.738169
$738$	0	0
$739$	−19.7635	−0.727011	−0.363506	−	0.931592i	$-0.618420\pi$
$739$	−19.7635	−0.727011	−0.363506	+	0.931592i	$0.618420\pi$
$740$	0	0
$741$	−0.234280	−0.00860649
$742$	0	0
$743$	−12.3944	−0.454706	−0.227353	−	0.973812i	$-0.573007\pi$
$743$	−12.3944	−0.454706	−0.227353	+	0.973812i	$0.573007\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.39890	−0.124359
$748$	0	0
$749$	10.5237	0.384528
$750$	0	0
$751$	23.3799	0.853144	0.426572	−	0.904454i	$-0.359721\pi$
$751$	23.3799	0.853144	0.426572	+	0.904454i	$0.359721\pi$
$752$	0	0
$753$	0.995489	0.0362776
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−14.3398	−0.521189	−0.260594	−	0.965448i	$-0.583919\pi$
$757$	−14.3398	−0.521189	−0.260594	+	0.965448i	$0.583919\pi$
$758$	0	0
$759$	−39.3816	−1.42946
$760$	0	0
$761$	11.0706	0.401311	0.200655	−	0.979662i	$-0.435693\pi$
$761$	11.0706	0.401311	0.200655	+	0.979662i	$0.435693\pi$
$762$	0	0
$763$	46.1152	1.66948
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1601	0.439076
$768$	0	0
$769$	−49.6251	−1.78953	−0.894764	−	0.446539i	$-0.852656\pi$
$769$	−49.6251	−1.78953	−0.894764	+	0.446539i	$0.852656\pi$
$770$	0	0
$771$	20.3523	0.732970
$772$	0	0
$773$	−7.66174	−0.275574	−0.137787	−	0.990462i	$-0.543999\pi$
$773$	−7.66174	−0.275574	−0.137787	+	0.990462i	$0.543999\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.71985	0.276948
$778$	0	0
$779$	2.57602	0.0922956
$780$	0	0
$781$	−21.1441	−0.756595
$782$	0	0
$783$	−4.20220	−0.150174
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.8234	1.06309	0.531544	−	0.847031i	$-0.321612\pi$
$787$	29.8234	1.06309	0.531544	+	0.847031i	$0.321612\pi$
$788$	0	0
$789$	0.252576	0.00899194
$790$	0	0
$791$	−1.90827	−0.0678502
$792$	0	0
$793$	2.98170	0.105883
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.9085	1.87412	0.937058	−	0.349174i	$-0.113538\pi$
$797$	52.9085	1.87412	0.937058	+	0.349174i	$0.113538\pi$
$798$	0	0
$799$	45.3643	1.60487
$800$	0	0
$801$	17.2298	0.608784
$802$	0	0
$803$	−81.8246	−2.88753
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.9358	−0.525766
$808$	0	0
$809$	16.6354	0.584871	0.292436	−	0.956285i	$-0.405534\pi$
$809$	16.6354	0.584871	0.292436	+	0.956285i	$0.405534\pi$
$810$	0	0
$811$	−19.0972	−0.670594	−0.335297	−	0.942112i	$-0.608837\pi$
$811$	−19.0972	−0.670594	−0.335297	+	0.942112i	$0.608837\pi$
$812$	0	0
$813$	3.09723	0.108625
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.0441418	0.00154433
$818$	0	0
$819$	2.60110	0.0908898
$820$	0	0
$821$	16.6845	0.582295	0.291147	−	0.956678i	$-0.405963\pi$
$821$	16.6845	0.582295	0.291147	+	0.956678i	$0.405963\pi$
$822$	0	0
$823$	−1.66904	−0.0581789	−0.0290895	−	0.999577i	$-0.509261\pi$
$823$	−1.66904	−0.0581789	−0.0290895	+	0.999577i	$0.509261\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.1007	−1.42921	−0.714606	−	0.699527i	$-0.753394\pi$
$827$	−41.1007	−1.42921	−0.714606	+	0.699527i	$0.753394\pi$
$828$	0	0
$829$	−20.7001	−0.718943	−0.359471	−	0.933156i	$-0.617043\pi$
$829$	−20.7001	−0.718943	−0.359471	+	0.933156i	$0.617043\pi$
$830$	0	0
$831$	−21.2939	−0.738678
$832$	0	0
$833$	1.35402	0.0469141
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.41269	−0.152525
$838$	0	0
$839$	56.4720	1.94963	0.974816	−	0.223012i	$-0.0715889\pi$
$839$	56.4720	1.94963	0.974816	+	0.223012i	$0.0715889\pi$
$840$	0	0
$841$	−11.3415	−0.391087
$842$	0	0
$843$	−13.1839	−0.454078
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−53.3926	−1.83459
$848$	0	0
$849$	−7.18218	−0.246492
$850$	0	0
$851$	−20.8163	−0.713575
$852$	0	0
$853$	45.1197	1.54487	0.772435	−	0.635093i	$-0.219038\pi$
$853$	45.1197	1.54487	0.772435	+	0.635093i	$0.219038\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.8088	0.369222	0.184611	−	0.982812i	$-0.440898\pi$
$857$	10.8088	0.369222	0.184611	+	0.982812i	$0.440898\pi$
$858$	0	0
$859$	31.0248	1.05855	0.529276	−	0.848450i	$-0.322464\pi$
$859$	31.0248	1.05855	0.529276	+	0.848450i	$0.322464\pi$
$860$	0	0
$861$	−28.6004	−0.974698
$862$	0	0
$863$	46.8264	1.59399	0.796994	−	0.603987i	$-0.206422\pi$
$863$	46.8264	1.59399	0.796994	+	0.603987i	$0.206422\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.4027	−0.557064
$868$	0	0
$869$	27.8943	0.946249
$870$	0	0
$871$	3.56902	0.120932
$872$	0	0
$873$	5.20220	0.176068
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.6049	0.932150	0.466075	−	0.884745i	$-0.345668\pi$
$877$	27.6049	0.932150	0.466075	+	0.884745i	$0.345668\pi$
$878$	0	0
$879$	−18.9005	−0.637499
$880$	0	0
$881$	−24.1348	−0.813122	−0.406561	−	0.913624i	$-0.633272\pi$
$881$	−24.1348	−0.813122	−0.406561	+	0.913624i	$0.633272\pi$
$882$	0	0
$883$	32.1579	1.08220	0.541099	−	0.840959i	$-0.318008\pi$
$883$	32.1579	1.08220	0.541099	+	0.840959i	$0.318008\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.3077	−0.749020	−0.374510	−	0.927223i	$-0.622189\pi$
$887$	−22.3077	−0.749020	−0.374510	+	0.927223i	$0.622189\pi$
$888$	0	0
$889$	−10.8469	−0.363793
$890$	0	0
$891$	−5.61488	−0.188106
$892$	0	0
$893$	1.83890	0.0615365
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.01378	−0.234183
$898$	0	0
$899$	18.5430	0.618443
$900$	0	0
$901$	69.0629	2.30082
$902$	0	0
$903$	−0.490085	−0.0163090
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−35.9634	−1.19415	−0.597073	−	0.802187i	$-0.703670\pi$
$907$	−35.9634	−1.19415	−0.597073	+	0.802187i	$0.703670\pi$
$908$	0	0
$909$	5.96792	0.197943
$910$	0	0
$911$	24.2911	0.804801	0.402401	−	0.915464i	$-0.368176\pi$
$911$	24.2911	0.804801	0.402401	+	0.915464i	$0.368176\pi$
$912$	0	0
$913$	19.0844	0.631603
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−47.5009	−1.56862
$918$	0	0
$919$	−53.0997	−1.75160	−0.875799	−	0.482676i	$-0.839665\pi$
$919$	−53.0997	−1.75160	−0.875799	+	0.482676i	$0.839665\pi$
$920$	0	0
$921$	−4.56352	−0.150373
$922$	0	0
$923$	−3.76572	−0.123950
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−15.8392	−0.520226
$928$	0	0
$929$	−51.6986	−1.69618	−0.848088	−	0.529856i	$-0.822246\pi$
$929$	−51.6986	−1.69618	−0.848088	+	0.529856i	$0.822246\pi$
$930$	0	0
$931$	0.0548871	0.00179885
$932$	0	0
$933$	25.5500	0.836469
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.3310	0.370166	0.185083	−	0.982723i	$-0.440745\pi$
$937$	11.3310	0.370166	0.185083	+	0.982723i	$0.440745\pi$
$938$	0	0
$939$	−30.4092	−0.992366
$940$	0	0
$941$	−39.3816	−1.28380	−0.641902	−	0.766787i	$-0.721854\pi$
$941$	−39.3816	−1.28380	−0.641902	+	0.766787i	$0.721854\pi$
$942$	0	0
$943$	77.1200	2.51137
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.0481	−0.748964	−0.374482	−	0.927234i	$-0.622179\pi$
$947$	−23.0481	−0.748964	−0.374482	+	0.927234i	$0.622179\pi$
$948$	0	0
$949$	−14.5728	−0.473053
$950$	0	0
$951$	−32.0401	−1.03897
$952$	0	0
$953$	−40.0231	−1.29647	−0.648237	−	0.761439i	$-0.724493\pi$
$953$	−40.0231	−1.29647	−0.648237	+	0.761439i	$0.724493\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23.5949	0.762714
$958$	0	0
$959$	−46.8439	−1.51267
$960$	0	0
$961$	−11.5282	−0.371878
$962$	0	0
$963$	−4.04587	−0.130376
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.8875	−1.21838	−0.609190	−	0.793024i	$-0.708505\pi$
$967$	−37.8875	−1.21838	−0.609190	+	0.793024i	$0.708505\pi$
$968$	0	0
$969$	−1.35402	−0.0434975
$970$	0	0
$971$	−23.5593	−0.756053	−0.378026	−	0.925795i	$-0.623397\pi$
$971$	−23.5593	−0.756053	−0.378026	+	0.925795i	$0.623397\pi$
$972$	0	0
$973$	−9.25734	−0.296777
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.0564	−0.737640	−0.368820	−	0.929501i	$-0.620238\pi$
$977$	−23.0564	−0.737640	−0.368820	+	0.929501i	$0.620238\pi$
$978$	0	0
$979$	−96.7432	−3.09192
$980$	0	0
$981$	−17.7291	−0.566048
$982$	0	0
$983$	0.844654	0.0269403	0.0134701	−	0.999909i	$-0.495712\pi$
$983$	0.844654	0.0269403	0.0134701	+	0.999909i	$0.495712\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.4165	−0.649863
$988$	0	0
$989$	1.32150	0.0420213
$990$	0	0
$991$	−14.2709	−0.453329	−0.226665	−	0.973973i	$-0.572782\pi$
$991$	−14.2709	−0.453329	−0.226665	+	0.973973i	$0.572782\pi$
$992$	0	0
$993$	−7.45478	−0.236570
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.3874	0.518994	0.259497	−	0.965744i	$-0.416443\pi$
$997$	16.3874	0.518994	0.259497	+	0.965744i	$0.416443\pi$
$998$	0	0
$999$	−2.96792	−0.0939008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bu.1.2	✓	4
5.4	even	2		7800.2.a.bx.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bu.1.2	✓	4	1.1	even	1	trivial
7800.2.a.bx.1.3	yes	4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} -5.61488 q^{11} -1.00000 q^{13} -5.77950 q^{17} -0.234280 q^{19} +2.60110 q^{21} -7.01378 q^{23} -1.00000 q^{27} +4.20220 q^{29} +4.41269 q^{31} +5.61488 q^{33} +2.96792 q^{37} +1.00000 q^{39} -10.9955 q^{41} -0.188415 q^{43} -7.84916 q^{47} -0.234280 q^{49} +5.77950 q^{51} -11.9496 q^{53} +0.234280 q^{57} -12.1601 q^{59} -2.98170 q^{61} -2.60110 q^{63} -3.56902 q^{67} +7.01378 q^{69} +3.76572 q^{71} +14.5728 q^{73} +14.6049 q^{77} -4.96792 q^{79} +1.00000 q^{81} -3.39890 q^{83} -4.20220 q^{87} +17.2298 q^{89} +2.60110 q^{91} -4.41269 q^{93} +5.20220 q^{97} -5.61488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + 5 q^{11} - 4 q^{13} - 7 q^{17} + 3 q^{19} + 3 q^{21} - 8 q^{23} - 4 q^{27} + 2 q^{29} + 5 q^{31} - 5 q^{33} + q^{37} + 4 q^{39} + 7 q^{41} - 6 q^{43} + 3 q^{49} + 7 q^{51} - 6 q^{53} - 3 q^{57} - 9 q^{59} + 19 q^{61} - 3 q^{63} + 4 q^{67} + 8 q^{69} + 19 q^{71} + 6 q^{73} + 17 q^{77} - 9 q^{79} + 4 q^{81} - 21 q^{83} - 2 q^{87} + 14 q^{89} + 3 q^{91} - 5 q^{93} + 6 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9 + 5 * q^11 - 4 * q^13 - 7 * q^17 + 3 * q^19 + 3 * q^21 - 8 * q^23 - 4 * q^27 + 2 * q^29 + 5 * q^31 - 5 * q^33 + q^37 + 4 * q^39 + 7 * q^41 - 6 * q^43 + 3 * q^49 + 7 * q^51 - 6 * q^53 - 3 * q^57 - 9 * q^59 + 19 * q^61 - 3 * q^63 + 4 * q^67 + 8 * q^69 + 19 * q^71 + 6 * q^73 + 17 * q^77 - 9 * q^79 + 4 * q^81 - 21 * q^83 - 2 * q^87 + 14 * q^89 + 3 * q^91 - 5 * q^93 + 6 * q^97 + 5 * q^99

Introduction

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