LMFDB - Embedded newform 8112.2.a.cv.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.920510$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.90568 q^{5} +0.0794899 q^{7} +1.00000 q^{9} -1.50475 q^{11} +3.90568 q^{15} -6.52695 q^{17} +0.786340 q^{19} +0.0794899 q^{21} +7.03409 q^{23} +10.2543 q^{25} +1.00000 q^{27} +6.15266 q^{29} -1.43887 q^{31} -1.50475 q^{33} +0.310462 q^{35} +9.23571 q^{37} +7.75379 q^{41} -1.06562 q^{43} +3.90568 q^{45} -12.7564 q^{47} -6.99368 q^{49} -6.52695 q^{51} +2.73092 q^{53} -5.87709 q^{55} +0.786340 q^{57} -10.3775 q^{59} +7.92745 q^{61} +0.0794899 q^{63} +7.80912 q^{67} +7.03409 q^{69} +15.1722 q^{71} +4.16507 q^{73} +10.2543 q^{75} -0.119613 q^{77} -3.83731 q^{79} +1.00000 q^{81} -9.04179 q^{83} -25.4922 q^{85} +6.15266 q^{87} -4.75082 q^{89} -1.43887 q^{93} +3.07119 q^{95} +6.94297 q^{97} -1.50475 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} + 7 q^{21} + 6 q^{23} + 17 q^{25} + 6 q^{27} + 3 q^{29} + 9 q^{31} + 8 q^{33} - 4 q^{35} - 6 q^{37} + 15 q^{41} - 11 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 + 7 * q^21 + 6 * q^23 + 17 * q^25 + 6 * q^27 + 3 * q^29 + 9 * q^31 + 8 * q^33 - 4 * q^35 - 6 * q^37 + 15 * q^41 - 11 * q^43 - q^45 + 5 * q^47 + 5 * q^49 - 5 * q^51 + 12 * q^53 - 7 * q^55 + 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 + 49 * q^73 + 17 * q^75 - 25 * q^77 - 14 * q^79 + 6 * q^81 - 3 * q^83 + 19 * q^85 + 3 * q^87 - 13 * q^89 + 9 * q^93 - 43 * q^95 + 14 * q^97 + 8 * q^99

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
$5$	3.90568	1.74667	0.873337	−	0.487117i	$-0.161951\pi$
$5$	3.90568	1.74667	0.873337	+	0.487117i	$0.161951\pi$
$6$	0	0
$7$	0.0794899	0.0300444	0.0150222	−	0.999887i	$-0.495218\pi$
$7$	0.0794899	0.0300444	0.0150222	+	0.999887i	$0.495218\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.50475	−0.453700	−0.226850	−	0.973930i	$-0.572843\pi$
$11$	−1.50475	−0.453700	−0.226850	+	0.973930i	$0.572843\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.90568	1.00844
$16$	0	0
$17$	−6.52695	−1.58302	−0.791509	−	0.611158i	$-0.790704\pi$
$17$	−6.52695	−1.58302	−0.791509	+	0.611158i	$0.790704\pi$
$18$	0	0
$19$	0.786340	0.180399	0.0901994	−	0.995924i	$-0.471250\pi$
$19$	0.786340	0.180399	0.0901994	+	0.995924i	$0.471250\pi$
$20$	0	0
$21$	0.0794899	0.0173461
$22$	0	0
$23$	7.03409	1.46671	0.733354	−	0.679846i	$-0.237954\pi$
$23$	7.03409	1.46671	0.733354	+	0.679846i	$0.237954\pi$
$24$	0	0
$25$	10.2543	2.05087
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.15266	1.14252	0.571260	−	0.820769i	$-0.306455\pi$
$29$	6.15266	1.14252	0.571260	+	0.820769i	$0.306455\pi$
$30$	0	0
$31$	−1.43887	−0.258429	−0.129214	−	0.991617i	$-0.541246\pi$
$31$	−1.43887	−0.258429	−0.129214	+	0.991617i	$0.541246\pi$
$32$	0	0
$33$	−1.50475	−0.261944
$34$	0	0
$35$	0.310462	0.0524777
$36$	0	0
$37$	9.23571	1.51834	0.759171	−	0.650891i	$-0.225605\pi$
$37$	9.23571	1.51834	0.759171	+	0.650891i	$0.225605\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.75379	1.21094	0.605469	−	0.795869i	$-0.292985\pi$
$41$	7.75379	1.21094	0.605469	+	0.795869i	$0.292985\pi$
$42$	0	0
$43$	−1.06562	−0.162506	−0.0812528	−	0.996694i	$-0.525892\pi$
$43$	−1.06562	−0.162506	−0.0812528	+	0.996694i	$0.525892\pi$
$44$	0	0
$45$	3.90568	0.582225
$46$	0	0
$47$	−12.7564	−1.86071	−0.930355	−	0.366659i	$-0.880502\pi$
$47$	−12.7564	−1.86071	−0.930355	+	0.366659i	$0.880502\pi$
$48$	0	0
$49$	−6.99368	−0.999097
$50$	0	0
$51$	−6.52695	−0.913956
$52$	0	0
$53$	2.73092	0.375121	0.187560	−	0.982253i	$-0.439942\pi$
$53$	2.73092	0.375121	0.187560	+	0.982253i	$0.439942\pi$
$54$	0	0
$55$	−5.87709	−0.792466
$56$	0	0
$57$	0.786340	0.104153
$58$	0	0
$59$	−10.3775	−1.35104	−0.675520	−	0.737341i	$-0.736081\pi$
$59$	−10.3775	−1.35104	−0.675520	+	0.737341i	$0.736081\pi$
$60$	0	0
$61$	7.92745	1.01501	0.507503	−	0.861650i	$-0.330569\pi$
$61$	7.92745	1.01501	0.507503	+	0.861650i	$0.330569\pi$
$62$	0	0
$63$	0.0794899	0.0100148
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.80912	0.954036	0.477018	−	0.878893i	$-0.341718\pi$
$67$	7.80912	0.954036	0.477018	+	0.878893i	$0.341718\pi$
$68$	0	0
$69$	7.03409	0.846805
$70$	0	0
$71$	15.1722	1.80061	0.900306	−	0.435257i	$-0.143343\pi$
$71$	15.1722	1.80061	0.900306	+	0.435257i	$0.143343\pi$
$72$	0	0
$73$	4.16507	0.487485	0.243742	−	0.969840i	$-0.421625\pi$
$73$	4.16507	0.487485	0.243742	+	0.969840i	$0.421625\pi$
$74$	0	0
$75$	10.2543	1.18407
$76$	0	0
$77$	−0.119613	−0.0136311
$78$	0	0
$79$	−3.83731	−0.431732	−0.215866	−	0.976423i	$-0.569257\pi$
$79$	−3.83731	−0.431732	−0.215866	+	0.976423i	$0.569257\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.04179	−0.992465	−0.496232	−	0.868190i	$-0.665284\pi$
$83$	−9.04179	−0.992465	−0.496232	+	0.868190i	$0.665284\pi$
$84$	0	0
$85$	−25.4922	−2.76502
$86$	0	0
$87$	6.15266	0.659635
$88$	0	0
$89$	−4.75082	−0.503586	−0.251793	−	0.967781i	$-0.581020\pi$
$89$	−4.75082	−0.503586	−0.251793	+	0.967781i	$0.581020\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.43887	−0.149204
$94$	0	0
$95$	3.07119	0.315098
$96$	0	0
$97$	6.94297	0.704952	0.352476	−	0.935821i	$-0.385340\pi$
$97$	6.94297	0.704952	0.352476	+	0.935821i	$0.385340\pi$
$98$	0	0
$99$	−1.50475	−0.151233
$100$	0	0
$101$	16.9785	1.68943	0.844713	−	0.535219i	$-0.179771\pi$
$101$	16.9785	1.68943	0.844713	+	0.535219i	$0.179771\pi$
$102$	0	0
$103$	0.534038	0.0526203	0.0263102	−	0.999654i	$-0.491624\pi$
$103$	0.534038	0.0526203	0.0263102	+	0.999654i	$0.491624\pi$
$104$	0	0
$105$	0.310462	0.0302980
$106$	0	0
$107$	−10.9729	−1.06079	−0.530396	−	0.847750i	$-0.677957\pi$
$107$	−10.9729	−1.06079	−0.530396	+	0.847750i	$0.677957\pi$
$108$	0	0
$109$	8.24364	0.789597	0.394799	−	0.918768i	$-0.370814\pi$
$109$	8.24364	0.789597	0.394799	+	0.918768i	$0.370814\pi$
$110$	0	0
$111$	9.23571	0.876615
$112$	0	0
$113$	1.35771	0.127723	0.0638613	−	0.997959i	$-0.479658\pi$
$113$	1.35771	0.127723	0.0638613	+	0.997959i	$0.479658\pi$
$114$	0	0
$115$	27.4729	2.56186
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.518827	−0.0475607
$120$	0	0
$121$	−8.73572	−0.794156
$122$	0	0
$123$	7.75379	0.699136
$124$	0	0
$125$	20.5218	1.83553
$126$	0	0
$127$	11.0111	0.977081	0.488541	−	0.872541i	$-0.337529\pi$
$127$	11.0111	0.977081	0.488541	+	0.872541i	$0.337529\pi$
$128$	0	0
$129$	−1.06562	−0.0938226
$130$	0	0
$131$	19.8552	1.73476	0.867378	−	0.497650i	$-0.165804\pi$
$131$	19.8552	1.73476	0.867378	+	0.497650i	$0.165804\pi$
$132$	0	0
$133$	0.0625061	0.00541996
$134$	0	0
$135$	3.90568	0.336148
$136$	0	0
$137$	4.80881	0.410844	0.205422	−	0.978673i	$-0.434143\pi$
$137$	4.80881	0.410844	0.205422	+	0.978673i	$0.434143\pi$
$138$	0	0
$139$	−20.7637	−1.76115	−0.880575	−	0.473906i	$-0.842844\pi$
$139$	−20.7637	−1.76115	−0.880575	+	0.473906i	$0.842844\pi$
$140$	0	0
$141$	−12.7564	−1.07428
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0303	1.99561
$146$	0	0
$147$	−6.99368	−0.576829
$148$	0	0
$149$	4.10785	0.336528	0.168264	−	0.985742i	$-0.446184\pi$
$149$	4.10785	0.336528	0.168264	+	0.985742i	$0.446184\pi$
$150$	0	0
$151$	12.6178	1.02683	0.513413	−	0.858142i	$-0.328381\pi$
$151$	12.6178	1.02683	0.513413	+	0.858142i	$0.328381\pi$
$152$	0	0
$153$	−6.52695	−0.527673
$154$	0	0
$155$	−5.61977	−0.451390
$156$	0	0
$157$	−10.1958	−0.813711	−0.406855	−	0.913493i	$-0.633375\pi$
$157$	−10.1958	−0.813711	−0.406855	+	0.913493i	$0.633375\pi$
$158$	0	0
$159$	2.73092	0.216576
$160$	0	0
$161$	0.559139	0.0440663
$162$	0	0
$163$	15.8095	1.23829	0.619147	−	0.785275i	$-0.287478\pi$
$163$	15.8095	1.23829	0.619147	+	0.785275i	$0.287478\pi$
$164$	0	0
$165$	−5.87709	−0.457531
$166$	0	0
$167$	24.8487	1.92285	0.961425	−	0.275066i	$-0.0886997\pi$
$167$	24.8487	1.92285	0.961425	+	0.275066i	$0.0886997\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.786340	0.0601329
$172$	0	0
$173$	12.6367	0.960752	0.480376	−	0.877063i	$-0.340500\pi$
$173$	12.6367	0.960752	0.480376	+	0.877063i	$0.340500\pi$
$174$	0	0
$175$	0.815117	0.0616170
$176$	0	0
$177$	−10.3775	−0.780024
$178$	0	0
$179$	−2.34535	−0.175299	−0.0876497	−	0.996151i	$-0.527936\pi$
$179$	−2.34535	−0.175299	−0.0876497	+	0.996151i	$0.527936\pi$
$180$	0	0
$181$	10.8755	0.808370	0.404185	−	0.914677i	$-0.367555\pi$
$181$	10.8755	0.808370	0.404185	+	0.914677i	$0.367555\pi$
$182$	0	0
$183$	7.92745	0.586014
$184$	0	0
$185$	36.0717	2.65205
$186$	0	0
$187$	9.82145	0.718216
$188$	0	0
$189$	0.0794899	0.00578204
$190$	0	0
$191$	−25.0561	−1.81300	−0.906499	−	0.422207i	$-0.861255\pi$
$191$	−25.0561	−1.81300	−0.906499	+	0.422207i	$0.861255\pi$
$192$	0	0
$193$	2.03469	0.146460	0.0732300	−	0.997315i	$-0.476669\pi$
$193$	2.03469	0.146460	0.0732300	+	0.997315i	$0.476669\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.93816	−0.209336	−0.104668	−	0.994507i	$-0.533378\pi$
$197$	−2.93816	−0.209336	−0.104668	+	0.994507i	$0.533378\pi$
$198$	0	0
$199$	−21.3170	−1.51112	−0.755561	−	0.655078i	$-0.772636\pi$
$199$	−21.3170	−1.51112	−0.755561	+	0.655078i	$0.772636\pi$
$200$	0	0
$201$	7.80912	0.550813
$202$	0	0
$203$	0.489074	0.0343263
$204$	0	0
$205$	30.2838	2.11511
$206$	0	0
$207$	7.03409	0.488903
$208$	0	0
$209$	−1.18325	−0.0818469
$210$	0	0
$211$	−9.26059	−0.637525	−0.318763	−	0.947835i	$-0.603267\pi$
$211$	−9.26059	−0.637525	−0.318763	+	0.947835i	$0.603267\pi$
$212$	0	0
$213$	15.1722	1.03958
$214$	0	0
$215$	−4.16197	−0.283844
$216$	0	0
$217$	−0.114376	−0.00776432
$218$	0	0
$219$	4.16507	0.281449
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−19.1700	−1.28372	−0.641861	−	0.766821i	$-0.721837\pi$
$223$	−19.1700	−1.28372	−0.641861	+	0.766821i	$0.721837\pi$
$224$	0	0
$225$	10.2543	0.683623
$226$	0	0
$227$	20.6095	1.36790	0.683951	−	0.729528i	$-0.260260\pi$
$227$	20.6095	1.36790	0.683951	+	0.729528i	$0.260260\pi$
$228$	0	0
$229$	11.2937	0.746309	0.373154	−	0.927769i	$-0.378276\pi$
$229$	11.2937	0.746309	0.373154	+	0.927769i	$0.378276\pi$
$230$	0	0
$231$	−0.119613	−0.00786993
$232$	0	0
$233$	−13.2668	−0.869137	−0.434569	−	0.900639i	$-0.643099\pi$
$233$	−13.2668	−0.869137	−0.434569	+	0.900639i	$0.643099\pi$
$234$	0	0
$235$	−49.8224	−3.25005
$236$	0	0
$237$	−3.83731	−0.249260
$238$	0	0
$239$	−0.412559	−0.0266862	−0.0133431	−	0.999911i	$-0.504247\pi$
$239$	−0.412559	−0.0266862	−0.0133431	+	0.999911i	$0.504247\pi$
$240$	0	0
$241$	−20.8992	−1.34624	−0.673118	−	0.739535i	$-0.735045\pi$
$241$	−20.8992	−1.34624	−0.673118	+	0.739535i	$0.735045\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−27.3151	−1.74510
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−9.04179	−0.573000
$250$	0	0
$251$	−5.10303	−0.322100	−0.161050	−	0.986946i	$-0.551488\pi$
$251$	−5.10303	−0.322100	−0.161050	+	0.986946i	$0.551488\pi$
$252$	0	0
$253$	−10.5846	−0.665446
$254$	0	0
$255$	−25.4922	−1.59638
$256$	0	0
$257$	−26.7241	−1.66700	−0.833501	−	0.552518i	$-0.813667\pi$
$257$	−26.7241	−1.66700	−0.833501	+	0.552518i	$0.813667\pi$
$258$	0	0
$259$	0.734146	0.0456176
$260$	0	0
$261$	6.15266	0.380840
$262$	0	0
$263$	9.89772	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$263$	9.89772	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$264$	0	0
$265$	10.6661	0.655214
$266$	0	0
$267$	−4.75082	−0.290745
$268$	0	0
$269$	−18.9609	−1.15607	−0.578035	−	0.816012i	$-0.696180\pi$
$269$	−18.9609	−1.15607	−0.578035	+	0.816012i	$0.696180\pi$
$270$	0	0
$271$	−23.7343	−1.44176	−0.720878	−	0.693062i	$-0.756261\pi$
$271$	−23.7343	−1.44176	−0.720878	+	0.693062i	$0.756261\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.4303	−0.930480
$276$	0	0
$277$	16.1577	0.970822	0.485411	−	0.874286i	$-0.338670\pi$
$277$	16.1577	0.970822	0.485411	+	0.874286i	$0.338670\pi$
$278$	0	0
$279$	−1.43887	−0.0861429
$280$	0	0
$281$	4.20881	0.251077	0.125538	−	0.992089i	$-0.459934\pi$
$281$	4.20881	0.251077	0.125538	+	0.992089i	$0.459934\pi$
$282$	0	0
$283$	−3.57052	−0.212245	−0.106123	−	0.994353i	$-0.533844\pi$
$283$	−3.57052	−0.212245	−0.106123	+	0.994353i	$0.533844\pi$
$284$	0	0
$285$	3.07119	0.181922
$286$	0	0
$287$	0.616348	0.0363819
$288$	0	0
$289$	25.6011	1.50595
$290$	0	0
$291$	6.94297	0.407004
$292$	0	0
$293$	−15.0734	−0.880599	−0.440300	−	0.897851i	$-0.645128\pi$
$293$	−15.0734	−0.880599	−0.440300	+	0.897851i	$0.645128\pi$
$294$	0	0
$295$	−40.5314	−2.35983
$296$	0	0
$297$	−1.50475	−0.0873146
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.0847060	−0.00488238
$302$	0	0
$303$	16.9785	0.975391
$304$	0	0
$305$	30.9621	1.77288
$306$	0	0
$307$	22.3150	1.27359	0.636793	−	0.771035i	$-0.280261\pi$
$307$	22.3150	1.27359	0.636793	+	0.771035i	$0.280261\pi$
$308$	0	0
$309$	0.534038	0.0303804
$310$	0	0
$311$	−2.15132	−0.121990	−0.0609951	−	0.998138i	$-0.519427\pi$
$311$	−2.15132	−0.121990	−0.0609951	+	0.998138i	$0.519427\pi$
$312$	0	0
$313$	−26.0404	−1.47189	−0.735945	−	0.677042i	$-0.763262\pi$
$313$	−26.0404	−1.47189	−0.735945	+	0.677042i	$0.763262\pi$
$314$	0	0
$315$	0.310462	0.0174926
$316$	0	0
$317$	13.3249	0.748403	0.374201	−	0.927347i	$-0.377917\pi$
$317$	13.3249	0.748403	0.374201	+	0.927347i	$0.377917\pi$
$318$	0	0
$319$	−9.25824	−0.518362
$320$	0	0
$321$	−10.9729	−0.612448
$322$	0	0
$323$	−5.13240	−0.285574
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.24364	0.455874
$328$	0	0
$329$	−1.01400	−0.0559038
$330$	0	0
$331$	−9.11986	−0.501273	−0.250637	−	0.968081i	$-0.580640\pi$
$331$	−9.11986	−0.501273	−0.250637	+	0.968081i	$0.580640\pi$
$332$	0	0
$333$	9.23571	0.506114
$334$	0	0
$335$	30.5000	1.66639
$336$	0	0
$337$	11.8690	0.646546	0.323273	−	0.946306i	$-0.395217\pi$
$337$	11.8690	0.646546	0.323273	+	0.946306i	$0.395217\pi$
$338$	0	0
$339$	1.35771	0.0737406
$340$	0	0
$341$	2.16514	0.117249
$342$	0	0
$343$	−1.11236	−0.0600616
$344$	0	0
$345$	27.4729	1.47909
$346$	0	0
$347$	1.20818	0.0648584	0.0324292	−	0.999474i	$-0.489676\pi$
$347$	1.20818	0.0648584	0.0324292	+	0.999474i	$0.489676\pi$
$348$	0	0
$349$	−5.45450	−0.291973	−0.145986	−	0.989287i	$-0.546636\pi$
$349$	−5.45450	−0.291973	−0.145986	+	0.989287i	$0.546636\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.5083	−1.88991	−0.944957	−	0.327195i	$-0.893897\pi$
$353$	−35.5083	−1.88991	−0.944957	+	0.327195i	$0.893897\pi$
$354$	0	0
$355$	59.2579	3.14508
$356$	0	0
$357$	−0.518827	−0.0274592
$358$	0	0
$359$	18.4695	0.974782	0.487391	−	0.873184i	$-0.337949\pi$
$359$	18.4695	0.974782	0.487391	+	0.873184i	$0.337949\pi$
$360$	0	0
$361$	−18.3817	−0.967456
$362$	0	0
$363$	−8.73572	−0.458506
$364$	0	0
$365$	16.2674	0.851477
$366$	0	0
$367$	−18.7044	−0.976362	−0.488181	−	0.872742i	$-0.662339\pi$
$367$	−18.7044	−0.976362	−0.488181	+	0.872742i	$0.662339\pi$
$368$	0	0
$369$	7.75379	0.403646
$370$	0	0
$371$	0.217081	0.0112703
$372$	0	0
$373$	12.3213	0.637973	0.318987	−	0.947759i	$-0.396657\pi$
$373$	12.3213	0.637973	0.318987	+	0.947759i	$0.396657\pi$
$374$	0	0
$375$	20.5218	1.05974
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.9123	0.765991	0.382996	−	0.923750i	$-0.374892\pi$
$379$	14.9123	0.765991	0.382996	+	0.923750i	$0.374892\pi$
$380$	0	0
$381$	11.0111	0.564118
$382$	0	0
$383$	−5.27199	−0.269386	−0.134693	−	0.990887i	$-0.543005\pi$
$383$	−5.27199	−0.269386	−0.134693	+	0.990887i	$0.543005\pi$
$384$	0	0
$385$	−0.467169	−0.0238091
$386$	0	0
$387$	−1.06562	−0.0541685
$388$	0	0
$389$	−2.66665	−0.135205	−0.0676023	−	0.997712i	$-0.521535\pi$
$389$	−2.66665	−0.135205	−0.0676023	+	0.997712i	$0.521535\pi$
$390$	0	0
$391$	−45.9111	−2.32183
$392$	0	0
$393$	19.8552	1.00156
$394$	0	0
$395$	−14.9873	−0.754094
$396$	0	0
$397$	12.7244	0.638618	0.319309	−	0.947651i	$-0.396549\pi$
$397$	12.7244	0.638618	0.319309	+	0.947651i	$0.396549\pi$
$398$	0	0
$399$	0.0625061	0.00312922
$400$	0	0
$401$	23.5633	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$401$	23.5633	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.90568	0.194075
$406$	0	0
$407$	−13.8975	−0.688872
$408$	0	0
$409$	29.2475	1.44620	0.723098	−	0.690745i	$-0.242717\pi$
$409$	29.2475	1.44620	0.723098	+	0.690745i	$0.242717\pi$
$410$	0	0
$411$	4.80881	0.237201
$412$	0	0
$413$	−0.824910	−0.0405911
$414$	0	0
$415$	−35.3143	−1.73351
$416$	0	0
$417$	−20.7637	−1.01680
$418$	0	0
$419$	23.3488	1.14066	0.570331	−	0.821415i	$-0.306815\pi$
$419$	23.3488	1.14066	0.570331	+	0.821415i	$0.306815\pi$
$420$	0	0
$421$	6.92772	0.337636	0.168818	−	0.985647i	$-0.446005\pi$
$421$	6.92772	0.337636	0.168818	+	0.985647i	$0.446005\pi$
$422$	0	0
$423$	−12.7564	−0.620237
$424$	0	0
$425$	−66.9296	−3.24656
$426$	0	0
$427$	0.630152	0.0304952
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.3178	0.978675	0.489337	−	0.872095i	$-0.337239\pi$
$431$	20.3178	0.978675	0.489337	+	0.872095i	$0.337239\pi$
$432$	0	0
$433$	−35.6153	−1.71156	−0.855780	−	0.517340i	$-0.826922\pi$
$433$	−35.6153	−1.71156	−0.855780	+	0.517340i	$0.826922\pi$
$434$	0	0
$435$	24.0303	1.15217
$436$	0	0
$437$	5.53118	0.264592
$438$	0	0
$439$	−28.4092	−1.35590	−0.677948	−	0.735110i	$-0.737131\pi$
$439$	−28.4092	−1.35590	−0.677948	+	0.735110i	$0.737131\pi$
$440$	0	0
$441$	−6.99368	−0.333032
$442$	0	0
$443$	−24.8982	−1.18295	−0.591474	−	0.806324i	$-0.701454\pi$
$443$	−24.8982	−1.18295	−0.591474	+	0.806324i	$0.701454\pi$
$444$	0	0
$445$	−18.5552	−0.879600
$446$	0	0
$447$	4.10785	0.194295
$448$	0	0
$449$	7.33361	0.346094	0.173047	−	0.984914i	$-0.444639\pi$
$449$	7.33361	0.346094	0.173047	+	0.984914i	$0.444639\pi$
$450$	0	0
$451$	−11.6675	−0.549403
$452$	0	0
$453$	12.6178	0.592838
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.8511	0.741481	0.370741	−	0.928736i	$-0.379104\pi$
$457$	15.8511	0.741481	0.370741	+	0.928736i	$0.379104\pi$
$458$	0	0
$459$	−6.52695	−0.304652
$460$	0	0
$461$	−10.5739	−0.492474	−0.246237	−	0.969210i	$-0.579194\pi$
$461$	−10.5739	−0.492474	−0.246237	+	0.969210i	$0.579194\pi$
$462$	0	0
$463$	−5.34050	−0.248194	−0.124097	−	0.992270i	$-0.539603\pi$
$463$	−5.34050	−0.248194	−0.124097	+	0.992270i	$0.539603\pi$
$464$	0	0
$465$	−5.61977	−0.260610
$466$	0	0
$467$	−25.7751	−1.19273	−0.596364	−	0.802714i	$-0.703389\pi$
$467$	−25.7751	−1.19273	−0.596364	+	0.802714i	$0.703389\pi$
$468$	0	0
$469$	0.620746	0.0286634
$470$	0	0
$471$	−10.1958	−0.469796
$472$	0	0
$473$	1.60350	0.0737288
$474$	0	0
$475$	8.06340	0.369974
$476$	0	0
$477$	2.73092	0.125040
$478$	0	0
$479$	−17.2136	−0.786511	−0.393255	−	0.919429i	$-0.628651\pi$
$479$	−17.2136	−0.786511	−0.393255	+	0.919429i	$0.628651\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.559139	0.0254417
$484$	0	0
$485$	27.1170	1.23132
$486$	0	0
$487$	17.0725	0.773628	0.386814	−	0.922158i	$-0.373575\pi$
$487$	17.0725	0.773628	0.386814	+	0.922158i	$0.373575\pi$
$488$	0	0
$489$	15.8095	0.714930
$490$	0	0
$491$	24.7127	1.11527	0.557634	−	0.830087i	$-0.311709\pi$
$491$	24.7127	1.11527	0.557634	+	0.830087i	$0.311709\pi$
$492$	0	0
$493$	−40.1581	−1.80863
$494$	0	0
$495$	−5.87709	−0.264155
$496$	0	0
$497$	1.20604	0.0540982
$498$	0	0
$499$	−0.887509	−0.0397303	−0.0198652	−	0.999803i	$-0.506324\pi$
$499$	−0.887509	−0.0397303	−0.0198652	+	0.999803i	$0.506324\pi$
$500$	0	0
$501$	24.8487	1.11016
$502$	0	0
$503$	−7.24671	−0.323115	−0.161557	−	0.986863i	$-0.551652\pi$
$503$	−7.24671	−0.323115	−0.161557	+	0.986863i	$0.551652\pi$
$504$	0	0
$505$	66.3127	2.95088
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.6937	−1.58210	−0.791048	−	0.611754i	$-0.790464\pi$
$509$	−35.6937	−1.58210	−0.791048	+	0.611754i	$0.790464\pi$
$510$	0	0
$511$	0.331081	0.0146462
$512$	0	0
$513$	0.786340	0.0347178
$514$	0	0
$515$	2.08578	0.0919106
$516$	0	0
$517$	19.1952	0.844205
$518$	0	0
$519$	12.6367	0.554690
$520$	0	0
$521$	−3.56585	−0.156223	−0.0781113	−	0.996945i	$-0.524889\pi$
$521$	−3.56585	−0.156223	−0.0781113	+	0.996945i	$0.524889\pi$
$522$	0	0
$523$	8.08344	0.353464	0.176732	−	0.984259i	$-0.443447\pi$
$523$	8.08344	0.353464	0.176732	+	0.984259i	$0.443447\pi$
$524$	0	0
$525$	0.815117	0.0355746
$526$	0	0
$527$	9.39143	0.409097
$528$	0	0
$529$	26.4784	1.15123
$530$	0	0
$531$	−10.3775	−0.450347
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−42.8567	−1.85286
$536$	0	0
$537$	−2.34535	−0.101209
$538$	0	0
$539$	10.5238	0.453291
$540$	0	0
$541$	−14.2925	−0.614483	−0.307241	−	0.951632i	$-0.599406\pi$
$541$	−14.2925	−0.614483	−0.307241	+	0.951632i	$0.599406\pi$
$542$	0	0
$543$	10.8755	0.466712
$544$	0	0
$545$	32.1970	1.37917
$546$	0	0
$547$	25.0004	1.06894	0.534471	−	0.845187i	$-0.320511\pi$
$547$	25.0004	1.06894	0.534471	+	0.845187i	$0.320511\pi$
$548$	0	0
$549$	7.92745	0.338335
$550$	0	0
$551$	4.83808	0.206109
$552$	0	0
$553$	−0.305028	−0.0129711
$554$	0	0
$555$	36.0717	1.53116
$556$	0	0
$557$	−2.39071	−0.101298	−0.0506488	−	0.998717i	$-0.516129\pi$
$557$	−2.39071	−0.101298	−0.0506488	+	0.998717i	$0.516129\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.82145	0.414662
$562$	0	0
$563$	−37.8778	−1.59636	−0.798178	−	0.602421i	$-0.794203\pi$
$563$	−37.8778	−1.59636	−0.798178	+	0.602421i	$0.794203\pi$
$564$	0	0
$565$	5.30278	0.223090
$566$	0	0
$567$	0.0794899	0.00333826
$568$	0	0
$569$	−6.58037	−0.275863	−0.137932	−	0.990442i	$-0.544045\pi$
$569$	−6.58037	−0.275863	−0.137932	+	0.990442i	$0.544045\pi$
$570$	0	0
$571$	21.5937	0.903670	0.451835	−	0.892102i	$-0.350770\pi$
$571$	21.5937	0.903670	0.451835	+	0.892102i	$0.350770\pi$
$572$	0	0
$573$	−25.0561	−1.04674
$574$	0	0
$575$	72.1300	3.00803
$576$	0	0
$577$	20.0143	0.833208	0.416604	−	0.909088i	$-0.363220\pi$
$577$	20.0143	0.833208	0.416604	+	0.909088i	$0.363220\pi$
$578$	0	0
$579$	2.03469	0.0845588
$580$	0	0
$581$	−0.718730	−0.0298180
$582$	0	0
$583$	−4.10936	−0.170192
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.9326	−0.657610	−0.328805	−	0.944398i	$-0.606646\pi$
$587$	−15.9326	−0.657610	−0.328805	+	0.944398i	$0.606646\pi$
$588$	0	0
$589$	−1.13144	−0.0466202
$590$	0	0
$591$	−2.93816	−0.120860
$592$	0	0
$593$	−12.7746	−0.524592	−0.262296	−	0.964987i	$-0.584480\pi$
$593$	−12.7746	−0.524592	−0.262296	+	0.964987i	$0.584480\pi$
$594$	0	0
$595$	−2.02637	−0.0830731
$596$	0	0
$597$	−21.3170	−0.872446
$598$	0	0
$599$	20.4940	0.837362	0.418681	−	0.908133i	$-0.362493\pi$
$599$	20.4940	0.837362	0.418681	+	0.908133i	$0.362493\pi$
$600$	0	0
$601$	−3.59245	−0.146539	−0.0732695	−	0.997312i	$-0.523343\pi$
$601$	−3.59245	−0.146539	−0.0732695	+	0.997312i	$0.523343\pi$
$602$	0	0
$603$	7.80912	0.318012
$604$	0	0
$605$	−34.1189	−1.38713
$606$	0	0
$607$	13.0627	0.530197	0.265099	−	0.964221i	$-0.414596\pi$
$607$	13.0627	0.530197	0.265099	+	0.964221i	$0.414596\pi$
$608$	0	0
$609$	0.489074	0.0198183
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.56139	0.345791	0.172896	−	0.984940i	$-0.444688\pi$
$613$	8.56139	0.345791	0.172896	+	0.984940i	$0.444688\pi$
$614$	0	0
$615$	30.2838	1.22116
$616$	0	0
$617$	9.50428	0.382628	0.191314	−	0.981529i	$-0.438725\pi$
$617$	9.50428	0.382628	0.191314	+	0.981529i	$0.438725\pi$
$618$	0	0
$619$	−0.765606	−0.0307723	−0.0153861	−	0.999882i	$-0.504898\pi$
$619$	−0.765606	−0.0307723	−0.0153861	+	0.999882i	$0.504898\pi$
$620$	0	0
$621$	7.03409	0.282268
$622$	0	0
$623$	−0.377642	−0.0151299
$624$	0	0
$625$	28.8799	1.15520
$626$	0	0
$627$	−1.18325	−0.0472544
$628$	0	0
$629$	−60.2810	−2.40356
$630$	0	0
$631$	−15.2955	−0.608902	−0.304451	−	0.952528i	$-0.598473\pi$
$631$	−15.2955	−0.608902	−0.304451	+	0.952528i	$0.598473\pi$
$632$	0	0
$633$	−9.26059	−0.368075
$634$	0	0
$635$	43.0060	1.70664
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.1722	0.600204
$640$	0	0
$641$	−16.7020	−0.659690	−0.329845	−	0.944035i	$-0.606997\pi$
$641$	−16.7020	−0.659690	−0.329845	+	0.944035i	$0.606997\pi$
$642$	0	0
$643$	12.3219	0.485927	0.242963	−	0.970035i	$-0.421881\pi$
$643$	12.3219	0.485927	0.242963	+	0.970035i	$0.421881\pi$
$644$	0	0
$645$	−4.16197	−0.163878
$646$	0	0
$647$	6.92612	0.272294	0.136147	−	0.990689i	$-0.456528\pi$
$647$	6.92612	0.272294	0.136147	+	0.990689i	$0.456528\pi$
$648$	0	0
$649$	15.6156	0.612967
$650$	0	0
$651$	−0.114376	−0.00448273
$652$	0	0
$653$	−6.42407	−0.251393	−0.125697	−	0.992069i	$-0.540117\pi$
$653$	−6.42407	−0.251393	−0.125697	+	0.992069i	$0.540117\pi$
$654$	0	0
$655$	77.5480	3.03005
$656$	0	0
$657$	4.16507	0.162495
$658$	0	0
$659$	−14.0188	−0.546096	−0.273048	−	0.962000i	$-0.588032\pi$
$659$	−14.0188	−0.546096	−0.273048	+	0.962000i	$0.588032\pi$
$660$	0	0
$661$	17.4663	0.679359	0.339680	−	0.940541i	$-0.389681\pi$
$661$	17.4663	0.679359	0.339680	+	0.940541i	$0.389681\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.244129	0.00946691
$666$	0	0
$667$	43.2784	1.67574
$668$	0	0
$669$	−19.1700	−0.741157
$670$	0	0
$671$	−11.9289	−0.460508
$672$	0	0
$673$	32.6688	1.25929	0.629644	−	0.776883i	$-0.283201\pi$
$673$	32.6688	1.25929	0.629644	+	0.776883i	$0.283201\pi$
$674$	0	0
$675$	10.2543	0.394690
$676$	0	0
$677$	−49.3618	−1.89713	−0.948565	−	0.316583i	$-0.897465\pi$
$677$	−49.3618	−1.89713	−0.948565	+	0.316583i	$0.897465\pi$
$678$	0	0
$679$	0.551896	0.0211798
$680$	0	0
$681$	20.6095	0.789759
$682$	0	0
$683$	−44.9159	−1.71866	−0.859330	−	0.511422i	$-0.829119\pi$
$683$	−44.9159	−1.71866	−0.859330	+	0.511422i	$0.829119\pi$
$684$	0	0
$685$	18.7817	0.717611
$686$	0	0
$687$	11.2937	0.430882
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−21.5931	−0.821440	−0.410720	−	0.911762i	$-0.634723\pi$
$691$	−21.5931	−0.821440	−0.410720	+	0.911762i	$0.634723\pi$
$692$	0	0
$693$	−0.119613	−0.00454371
$694$	0	0
$695$	−81.0962	−3.07616
$696$	0	0
$697$	−50.6086	−1.91694
$698$	0	0
$699$	−13.2668	−0.501797
$700$	0	0
$701$	−31.8311	−1.20224	−0.601121	−	0.799158i	$-0.705279\pi$
$701$	−31.8311	−1.20224	−0.601121	+	0.799158i	$0.705279\pi$
$702$	0	0
$703$	7.26241	0.273907
$704$	0	0
$705$	−49.8224	−1.87642
$706$	0	0
$707$	1.34962	0.0507577
$708$	0	0
$709$	−48.5378	−1.82288	−0.911439	−	0.411436i	$-0.865027\pi$
$709$	−48.5378	−1.82288	−0.911439	+	0.411436i	$0.865027\pi$
$710$	0	0
$711$	−3.83731	−0.143911
$712$	0	0
$713$	−10.1211	−0.379039
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.412559	−0.0154073
$718$	0	0
$719$	−5.13015	−0.191322	−0.0956611	−	0.995414i	$-0.530497\pi$
$719$	−5.13015	−0.191322	−0.0956611	+	0.995414i	$0.530497\pi$
$720$	0	0
$721$	0.0424506	0.00158094
$722$	0	0
$723$	−20.8992	−0.777249
$724$	0	0
$725$	63.0915	2.34316
$726$	0	0
$727$	45.8944	1.70213	0.851065	−	0.525061i	$-0.175957\pi$
$727$	45.8944	1.70213	0.851065	+	0.525061i	$0.175957\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.95525	0.257249
$732$	0	0
$733$	−19.2958	−0.712706	−0.356353	−	0.934351i	$-0.615980\pi$
$733$	−19.2958	−0.712706	−0.356353	+	0.934351i	$0.615980\pi$
$734$	0	0
$735$	−27.3151	−1.00753
$736$	0	0
$737$	−11.7508	−0.432846
$738$	0	0
$739$	−24.7702	−0.911187	−0.455594	−	0.890188i	$-0.650573\pi$
$739$	−24.7702	−0.911187	−0.455594	+	0.890188i	$0.650573\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.1083	1.06788	0.533939	−	0.845523i	$-0.320711\pi$
$743$	29.1083	1.06788	0.533939	+	0.845523i	$0.320711\pi$
$744$	0	0
$745$	16.0440	0.587805
$746$	0	0
$747$	−9.04179	−0.330822
$748$	0	0
$749$	−0.872235	−0.0318708
$750$	0	0
$751$	−27.5195	−1.00420	−0.502101	−	0.864809i	$-0.667439\pi$
$751$	−27.5195	−1.00420	−0.502101	+	0.864809i	$0.667439\pi$
$752$	0	0
$753$	−5.10303	−0.185965
$754$	0	0
$755$	49.2813	1.79353
$756$	0	0
$757$	5.71110	0.207574	0.103787	−	0.994600i	$-0.466904\pi$
$757$	5.71110	0.207574	0.103787	+	0.994600i	$0.466904\pi$
$758$	0	0
$759$	−10.5846	−0.384195
$760$	0	0
$761$	−40.1705	−1.45618	−0.728089	−	0.685482i	$-0.759591\pi$
$761$	−40.1705	−1.45618	−0.728089	+	0.685482i	$0.759591\pi$
$762$	0	0
$763$	0.655286	0.0237229
$764$	0	0
$765$	−25.4922	−0.921672
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−15.8754	−0.572483	−0.286241	−	0.958158i	$-0.592406\pi$
$769$	−15.8754	−0.572483	−0.286241	+	0.958158i	$0.592406\pi$
$770$	0	0
$771$	−26.7241	−0.962444
$772$	0	0
$773$	−1.19739	−0.0430670	−0.0215335	−	0.999768i	$-0.506855\pi$
$773$	−1.19739	−0.0430670	−0.0215335	+	0.999768i	$0.506855\pi$
$774$	0	0
$775$	−14.7547	−0.530003
$776$	0	0
$777$	0.734146	0.0263373
$778$	0	0
$779$	6.09711	0.218452
$780$	0	0
$781$	−22.8305	−0.816938
$782$	0	0
$783$	6.15266	0.219878
$784$	0	0
$785$	−39.8214	−1.42129
$786$	0	0
$787$	32.2576	1.14986	0.574929	−	0.818203i	$-0.305030\pi$
$787$	32.2576	1.14986	0.574929	+	0.818203i	$0.305030\pi$
$788$	0	0
$789$	9.89772	0.352368
$790$	0	0
$791$	0.107924	0.00383734
$792$	0	0
$793$	0	0
$794$	0	0
$795$	10.6661	0.378288
$796$	0	0
$797$	−28.2439	−1.00045	−0.500226	−	0.865895i	$-0.666750\pi$
$797$	−28.2439	−1.00045	−0.500226	+	0.865895i	$0.666750\pi$
$798$	0	0
$799$	83.2603	2.94554
$800$	0	0
$801$	−4.75082	−0.167862
$802$	0	0
$803$	−6.26740	−0.221172
$804$	0	0
$805$	2.18382	0.0769695
$806$	0	0
$807$	−18.9609	−0.667457
$808$	0	0
$809$	−37.2877	−1.31096	−0.655482	−	0.755210i	$-0.727535\pi$
$809$	−37.2877	−1.31096	−0.655482	+	0.755210i	$0.727535\pi$
$810$	0	0
$811$	−6.09538	−0.214038	−0.107019	−	0.994257i	$-0.534131\pi$
$811$	−6.09538	−0.214038	−0.107019	+	0.994257i	$0.534131\pi$
$812$	0	0
$813$	−23.7343	−0.832398
$814$	0	0
$815$	61.7468	2.16290
$816$	0	0
$817$	−0.837940	−0.0293158
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.6514	0.581138	0.290569	−	0.956854i	$-0.406155\pi$
$821$	16.6514	0.581138	0.290569	+	0.956854i	$0.406155\pi$
$822$	0	0
$823$	−32.1734	−1.12150	−0.560748	−	0.827987i	$-0.689486\pi$
$823$	−32.1734	−1.12150	−0.560748	+	0.827987i	$0.689486\pi$
$824$	0	0
$825$	−15.4303	−0.537213
$826$	0	0
$827$	53.7727	1.86986	0.934930	−	0.354831i	$-0.115462\pi$
$827$	53.7727	1.86986	0.934930	+	0.354831i	$0.115462\pi$
$828$	0	0
$829$	47.5426	1.65122	0.825611	−	0.564240i	$-0.190831\pi$
$829$	47.5426	1.65122	0.825611	+	0.564240i	$0.190831\pi$
$830$	0	0
$831$	16.1577	0.560504
$832$	0	0
$833$	45.6474	1.58159
$834$	0	0
$835$	97.0511	3.35859
$836$	0	0
$837$	−1.43887	−0.0497346
$838$	0	0
$839$	−29.9379	−1.03357	−0.516785	−	0.856115i	$-0.672871\pi$
$839$	−29.9379	−1.03357	−0.516785	+	0.856115i	$0.672871\pi$
$840$	0	0
$841$	8.85524	0.305353
$842$	0	0
$843$	4.20881	0.144959
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.694401	−0.0238599
$848$	0	0
$849$	−3.57052	−0.122540
$850$	0	0
$851$	64.9648	2.22697
$852$	0	0
$853$	−38.2677	−1.31026	−0.655130	−	0.755516i	$-0.727386\pi$
$853$	−38.2677	−1.31026	−0.655130	+	0.755516i	$0.727386\pi$
$854$	0	0
$855$	3.07119	0.105033
$856$	0	0
$857$	−10.6995	−0.365486	−0.182743	−	0.983161i	$-0.558498\pi$
$857$	−10.6995	−0.365486	−0.182743	+	0.983161i	$0.558498\pi$
$858$	0	0
$859$	22.6031	0.771207	0.385604	−	0.922665i	$-0.373993\pi$
$859$	22.6031	0.771207	0.385604	+	0.922665i	$0.373993\pi$
$860$	0	0
$861$	0.616348	0.0210051
$862$	0	0
$863$	−20.7503	−0.706349	−0.353174	−	0.935557i	$-0.614898\pi$
$863$	−20.7503	−0.706349	−0.353174	+	0.935557i	$0.614898\pi$
$864$	0	0
$865$	49.3550	1.67812
$866$	0	0
$867$	25.6011	0.869458
$868$	0	0
$869$	5.77421	0.195877
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.94297	0.234984
$874$	0	0
$875$	1.63128	0.0551472
$876$	0	0
$877$	20.8423	0.703793	0.351897	−	0.936039i	$-0.385537\pi$
$877$	20.8423	0.703793	0.351897	+	0.936039i	$0.385537\pi$
$878$	0	0
$879$	−15.0734	−0.508414
$880$	0	0
$881$	−9.03338	−0.304342	−0.152171	−	0.988354i	$-0.548627\pi$
$881$	−9.03338	−0.304342	−0.152171	+	0.988354i	$0.548627\pi$
$882$	0	0
$883$	−21.4380	−0.721447	−0.360724	−	0.932673i	$-0.617470\pi$
$883$	−21.4380	−0.721447	−0.360724	+	0.932673i	$0.617470\pi$
$884$	0	0
$885$	−40.5314	−1.36245
$886$	0	0
$887$	−5.41708	−0.181888	−0.0909438	−	0.995856i	$-0.528988\pi$
$887$	−5.41708	−0.181888	−0.0909438	+	0.995856i	$0.528988\pi$
$888$	0	0
$889$	0.875275	0.0293558
$890$	0	0
$891$	−1.50475	−0.0504111
$892$	0	0
$893$	−10.0309	−0.335670
$894$	0	0
$895$	−9.16018	−0.306191
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.85288	−0.295260
$900$	0	0
$901$	−17.8246	−0.593823
$902$	0	0
$903$	−0.0847060	−0.00281884
$904$	0	0
$905$	42.4762	1.41196
$906$	0	0
$907$	−13.5292	−0.449229	−0.224615	−	0.974448i	$-0.572112\pi$
$907$	−13.5292	−0.449229	−0.224615	+	0.974448i	$0.572112\pi$
$908$	0	0
$909$	16.9785	0.563142
$910$	0	0
$911$	25.7848	0.854290	0.427145	−	0.904183i	$-0.359519\pi$
$911$	25.7848	0.854290	0.427145	+	0.904183i	$0.359519\pi$
$912$	0	0
$913$	13.6057	0.450281
$914$	0	0
$915$	30.9621	1.02358
$916$	0	0
$917$	1.57829	0.0521196
$918$	0	0
$919$	28.3412	0.934889	0.467445	−	0.884022i	$-0.345175\pi$
$919$	28.3412	0.934889	0.467445	+	0.884022i	$0.345175\pi$
$920$	0	0
$921$	22.3150	0.735305
$922$	0	0
$923$	0	0
$924$	0	0
$925$	94.7062	3.11392
$926$	0	0
$927$	0.534038	0.0175401
$928$	0	0
$929$	−27.2671	−0.894606	−0.447303	−	0.894382i	$-0.647615\pi$
$929$	−27.2671	−0.894606	−0.447303	+	0.894382i	$0.647615\pi$
$930$	0	0
$931$	−5.49941	−0.180236
$932$	0	0
$933$	−2.15132	−0.0704310
$934$	0	0
$935$	38.3595	1.25449
$936$	0	0
$937$	57.0811	1.86476	0.932379	−	0.361483i	$-0.117729\pi$
$937$	57.0811	1.86476	0.932379	+	0.361483i	$0.117729\pi$
$938$	0	0
$939$	−26.0404	−0.849796
$940$	0	0
$941$	−33.2566	−1.08413	−0.542067	−	0.840335i	$-0.682358\pi$
$941$	−33.2566	−1.08413	−0.542067	+	0.840335i	$0.682358\pi$
$942$	0	0
$943$	54.5408	1.77609
$944$	0	0
$945$	0.310462	0.0100993
$946$	0	0
$947$	−38.9304	−1.26507	−0.632534	−	0.774533i	$-0.717985\pi$
$947$	−38.9304	−1.26507	−0.632534	+	0.774533i	$0.717985\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.3249	0.432090
$952$	0	0
$953$	−27.3688	−0.886562	−0.443281	−	0.896383i	$-0.646186\pi$
$953$	−27.3688	−0.886562	−0.443281	+	0.896383i	$0.646186\pi$
$954$	0	0
$955$	−97.8613	−3.16672
$956$	0	0
$957$	−9.25824	−0.299276
$958$	0	0
$959$	0.382251	0.0123435
$960$	0	0
$961$	−28.9297	−0.933215
$962$	0	0
$963$	−10.9729	−0.353597
$964$	0	0
$965$	7.94684	0.255818
$966$	0	0
$967$	48.2047	1.55016	0.775079	−	0.631865i	$-0.217710\pi$
$967$	48.2047	1.55016	0.775079	+	0.631865i	$0.217710\pi$
$968$	0	0
$969$	−5.13240	−0.164877
$970$	0	0
$971$	40.7160	1.30664	0.653319	−	0.757083i	$-0.273376\pi$
$971$	40.7160	1.30664	0.653319	+	0.757083i	$0.273376\pi$
$972$	0	0
$973$	−1.65050	−0.0529126
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.2632	−0.552297	−0.276149	−	0.961115i	$-0.589058\pi$
$977$	−17.2632	−0.552297	−0.276149	+	0.961115i	$0.589058\pi$
$978$	0	0
$979$	7.14881	0.228477
$980$	0	0
$981$	8.24364	0.263199
$982$	0	0
$983$	50.0600	1.59667	0.798333	−	0.602216i	$-0.205715\pi$
$983$	50.0600	1.59667	0.798333	+	0.602216i	$0.205715\pi$
$984$	0	0
$985$	−11.4755	−0.365641
$986$	0	0
$987$	−1.01400	−0.0322761
$988$	0	0
$989$	−7.49567	−0.238348
$990$	0	0
$991$	−0.122255	−0.00388355	−0.00194177	−	0.999998i	$-0.500618\pi$
$991$	−0.122255	−0.00388355	−0.00194177	+	0.999998i	$0.500618\pi$
$992$	0	0
$993$	−9.11986	−0.289410
$994$	0	0
$995$	−83.2574	−2.63944
$996$	0	0
$997$	−4.95155	−0.156817	−0.0784086	−	0.996921i	$-0.524984\pi$
$997$	−4.95155	−0.156817	−0.0784086	+	0.996921i	$0.524984\pi$
$998$	0	0
$999$	9.23571	0.292205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cv.1.6		6
4.3	odd	2		4056.2.a.bf.1.6	✓	6
13.12	even	2		8112.2.a.cw.1.1		6
52.31	even	4		4056.2.c.q.337.1		12
52.47	even	4		4056.2.c.q.337.12		12
52.51	odd	2		4056.2.a.bg.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.6	✓	6	4.3	odd	2
4056.2.a.bg.1.1	yes	6	52.51	odd	2
4056.2.c.q.337.1		12	52.31	even	4
4056.2.c.q.337.12		12	52.47	even	4
8112.2.a.cv.1.6		6	1.1	even	1	trivial
8112.2.a.cw.1.1		6	13.12	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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.90568 q^{5} +0.0794899 q^{7} +1.00000 q^{9} -1.50475 q^{11} +3.90568 q^{15} -6.52695 q^{17} +0.786340 q^{19} +0.0794899 q^{21} +7.03409 q^{23} +10.2543 q^{25} +1.00000 q^{27} +6.15266 q^{29} -1.43887 q^{31} -1.50475 q^{33} +0.310462 q^{35} +9.23571 q^{37} +7.75379 q^{41} -1.06562 q^{43} +3.90568 q^{45} -12.7564 q^{47} -6.99368 q^{49} -6.52695 q^{51} +2.73092 q^{53} -5.87709 q^{55} +0.786340 q^{57} -10.3775 q^{59} +7.92745 q^{61} +0.0794899 q^{63} +7.80912 q^{67} +7.03409 q^{69} +15.1722 q^{71} +4.16507 q^{73} +10.2543 q^{75} -0.119613 q^{77} -3.83731 q^{79} +1.00000 q^{81} -9.04179 q^{83} -25.4922 q^{85} +6.15266 q^{87} -4.75082 q^{89} -1.43887 q^{93} +3.07119 q^{95} +6.94297 q^{97} -1.50475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} + 7 q^{21} + 6 q^{23} + 17 q^{25} + 6 q^{27} + 3 q^{29} + 9 q^{31} + 8 q^{33} - 4 q^{35} - 6 q^{37} + 15 q^{41} - 11 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 + 7 * q^21 + 6 * q^23 + 17 * q^25 + 6 * q^27 + 3 * q^29 + 9 * q^31 + 8 * q^33 - 4 * q^35 - 6 * q^37 + 15 * q^41 - 11 * q^43 - q^45 + 5 * q^47 + 5 * q^49 - 5 * q^51 + 12 * q^53 - 7 * q^55 + 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 + 49 * q^73 + 17 * q^75 - 25 * q^77 - 14 * q^79 + 6 * q^81 - 3 * q^83 + 19 * q^85 + 3 * q^87 - 13 * q^89 + 9 * q^93 - 43 * q^95 + 14 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

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Database

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