LMFDB - Embedded newform 63.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("63.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.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:	$10.1041806482$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.29150 q^{2} -4.00000 q^{4} +37.0405 q^{5} -49.0000 q^{7} +190.494 q^{8} +O(q^{10})$	\(q-5.29150 q^{2} -4.00000 q^{4} +37.0405 q^{5} -49.0000 q^{7} +190.494 q^{8} -196.000 q^{10} +227.535 q^{11} -518.000 q^{13} +259.284 q^{14} -880.000 q^{16} -777.851 q^{17} -1484.00 q^{19} -148.162 q^{20} -1204.00 q^{22} -3825.76 q^{23} -1753.00 q^{25} +2741.00 q^{26} +196.000 q^{28} -105.830 q^{29} -2604.00 q^{31} -1439.29 q^{32} +4116.00 q^{34} -1814.99 q^{35} +402.000 q^{37} +7852.59 q^{38} +7056.00 q^{40} +629.689 q^{41} +6956.00 q^{43} -910.138 q^{44} +20244.0 q^{46} +27335.9 q^{47} +2401.00 q^{49} +9276.00 q^{50} +2072.00 q^{52} -30510.8 q^{53} +8428.00 q^{55} -9334.21 q^{56} +560.000 q^{58} -45115.4 q^{59} -22610.0 q^{61} +13779.1 q^{62} +35776.0 q^{64} -19187.0 q^{65} -13124.0 q^{67} +3111.40 q^{68} +9604.00 q^{70} +48210.9 q^{71} -82866.0 q^{73} -2127.18 q^{74} +5936.00 q^{76} -11149.2 q^{77} -81112.0 q^{79} -32595.7 q^{80} -3332.00 q^{82} +66672.9 q^{83} -28812.0 q^{85} -36807.7 q^{86} +43344.0 q^{88} +126716. q^{89} +25382.0 q^{91} +15303.0 q^{92} -144648. q^{94} -54968.1 q^{95} -10626.0 q^{97} -12704.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 98 q^{7}+O(q^{10}) $ 2 * q - 8 * q^4 - 98 * q^7	$ 2 q - 8 q^{4} - 98 q^{7} - 392 q^{10} - 1036 q^{13} - 1760 q^{16} - 2968 q^{19} - 2408 q^{22} - 3506 q^{25} + 392 q^{28} - 5208 q^{31} + 8232 q^{34} + 804 q^{37} + 14112 q^{40} + 13912 q^{43} + 40488 q^{46} + 4802 q^{49} + 4144 q^{52} + 16856 q^{55} + 1120 q^{58} - 45220 q^{61} + 71552 q^{64} - 26248 q^{67} + 19208 q^{70} - 165732 q^{73} + 11872 q^{76} - 162224 q^{79} - 6664 q^{82} - 57624 q^{85} + 86688 q^{88} + 50764 q^{91} - 289296 q^{94} - 21252 q^{97}+O(q^{100}) $ 2 * q - 8 * q^4 - 98 * q^7 - 392 * q^10 - 1036 * q^13 - 1760 * q^16 - 2968 * q^19 - 2408 * q^22 - 3506 * q^25 + 392 * q^28 - 5208 * q^31 + 8232 * q^34 + 804 * q^37 + 14112 * q^40 + 13912 * q^43 + 40488 * q^46 + 4802 * q^49 + 4144 * q^52 + 16856 * q^55 + 1120 * q^58 - 45220 * q^61 + 71552 * q^64 - 26248 * q^67 + 19208 * q^70 - 165732 * q^73 + 11872 * q^76 - 162224 * q^79 - 6664 * q^82 - 57624 * q^85 + 86688 * q^88 + 50764 * q^91 - 289296 * q^94 - 21252 * 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 $

$2$	−5.29150	−0.935414	−0.467707	−	0.883883i	$-0.654920\pi$
$2$	−5.29150	−0.935414	−0.467707	+	0.883883i	$0.654920\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.125000
$5$	37.0405	0.662601	0.331300	−	0.943525i	$-0.392513\pi$
$5$	37.0405	0.662601	0.331300	+	0.943525i	$0.392513\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	190.494	1.05234
$9$	0	0
$10$	−196.000	−0.619806
$11$	227.535	0.566977	0.283489	−	0.958976i	$-0.408508\pi$
$11$	227.535	0.566977	0.283489	+	0.958976i	$0.408508\pi$
$12$	0	0
$13$	−518.000	−0.850103	−0.425051	−	0.905169i	$-0.639744\pi$
$13$	−518.000	−0.850103	−0.425051	+	0.905169i	$0.639744\pi$
$14$	259.284	0.353553
$15$	0	0
$16$	−880.000	−0.859375
$17$	−777.851	−0.652791	−0.326395	−	0.945233i	$-0.605834\pi$
$17$	−777.851	−0.652791	−0.326395	+	0.945233i	$0.605834\pi$
$18$	0	0
$19$	−1484.00	−0.943083	−0.471541	−	0.881844i	$-0.656302\pi$
$19$	−1484.00	−0.943083	−0.471541	+	0.881844i	$0.656302\pi$
$20$	−148.162	−0.0828251
$21$	0	0
$22$	−1204.00	−0.530359
$23$	−3825.76	−1.50799	−0.753994	−	0.656882i	$-0.771875\pi$
$23$	−3825.76	−1.50799	−0.753994	+	0.656882i	$0.771875\pi$
$24$	0	0
$25$	−1753.00	−0.560960
$26$	2741.00	0.795198
$27$	0	0
$28$	196.000	0.0472456
$29$	−105.830	−0.0233676	−0.0116838	−	0.999932i	$-0.503719\pi$
$29$	−105.830	−0.0233676	−0.0116838	+	0.999932i	$0.503719\pi$
$30$	0	0
$31$	−2604.00	−0.486672	−0.243336	−	0.969942i	$-0.578242\pi$
$31$	−2604.00	−0.486672	−0.243336	+	0.969942i	$0.578242\pi$
$32$	−1439.29	−0.248469
$33$	0	0
$34$	4116.00	0.610630
$35$	−1814.99	−0.250440
$36$	0	0
$37$	402.000	0.0482749	0.0241375	−	0.999709i	$-0.492316\pi$
$37$	402.000	0.0482749	0.0241375	+	0.999709i	$0.492316\pi$
$38$	7852.59	0.882173
$39$	0	0
$40$	7056.00	0.697282
$41$	629.689	0.0585014	0.0292507	−	0.999572i	$-0.490688\pi$
$41$	629.689	0.0585014	0.0292507	+	0.999572i	$0.490688\pi$
$42$	0	0
$43$	6956.00	0.573705	0.286852	−	0.957975i	$-0.407391\pi$
$43$	6956.00	0.573705	0.286852	+	0.957975i	$0.407391\pi$
$44$	−910.138	−0.0708722
$45$	0	0
$46$	20244.0	1.41059
$47$	27335.9	1.80505	0.902524	−	0.430639i	$-0.141712\pi$
$47$	27335.9	1.80505	0.902524	+	0.430639i	$0.141712\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	9276.00	0.524730
$51$	0	0
$52$	2072.00	0.106263
$53$	−30510.8	−1.49198	−0.745992	−	0.665955i	$-0.768024\pi$
$53$	−30510.8	−1.49198	−0.745992	+	0.665955i	$0.768024\pi$
$54$	0	0
$55$	8428.00	0.375680
$56$	−9334.21	−0.397748
$57$	0	0
$58$	560.000	0.0218584
$59$	−45115.4	−1.68731	−0.843654	−	0.536887i	$-0.819600\pi$
$59$	−45115.4	−1.68731	−0.843654	+	0.536887i	$0.819600\pi$
$60$	0	0
$61$	−22610.0	−0.777994	−0.388997	−	0.921239i	$-0.627178\pi$
$61$	−22610.0	−0.777994	−0.388997	+	0.921239i	$0.627178\pi$
$62$	13779.1	0.455240
$63$	0	0
$64$	35776.0	1.09180
$65$	−19187.0	−0.563279
$66$	0	0
$67$	−13124.0	−0.357173	−0.178587	−	0.983924i	$-0.557153\pi$
$67$	−13124.0	−0.357173	−0.178587	+	0.983924i	$0.557153\pi$
$68$	3111.40	0.0815989
$69$	0	0
$70$	9604.00	0.234265
$71$	48210.9	1.13501	0.567504	−	0.823370i	$-0.307909\pi$
$71$	48210.9	1.13501	0.567504	+	0.823370i	$0.307909\pi$
$72$	0	0
$73$	−82866.0	−1.81999	−0.909995	−	0.414618i	$-0.863915\pi$
$73$	−82866.0	−1.81999	−0.909995	+	0.414618i	$0.863915\pi$
$74$	−2127.18	−0.0451571
$75$	0	0
$76$	5936.00	0.117885
$77$	−11149.2	−0.214297
$78$	0	0
$79$	−81112.0	−1.46224	−0.731118	−	0.682251i	$-0.761001\pi$
$79$	−81112.0	−1.46224	−0.731118	+	0.682251i	$0.761001\pi$
$80$	−32595.7	−0.569423
$81$	0	0
$82$	−3332.00	−0.0547231
$83$	66672.9	1.06232	0.531159	−	0.847272i	$-0.321757\pi$
$83$	66672.9	1.06232	0.531159	+	0.847272i	$0.321757\pi$
$84$	0	0
$85$	−28812.0	−0.432540
$86$	−36807.7	−0.536652
$87$	0	0
$88$	43344.0	0.596654
$89$	126716.	1.69572	0.847862	−	0.530217i	$-0.177890\pi$
$89$	126716.	1.69572	0.847862	+	0.530217i	$0.177890\pi$
$90$	0	0
$91$	25382.0	0.321309
$92$	15303.0	0.188498
$93$	0	0
$94$	−144648.	−1.68847
$95$	−54968.1	−0.624888
$96$	0	0
$97$	−10626.0	−0.114668	−0.0573338	−	0.998355i	$-0.518260\pi$
$97$	−10626.0	−0.114668	−0.0573338	+	0.998355i	$0.518260\pi$
$98$	−12704.9	−0.133631
$99$	0	0
$100$	7012.00	0.0701200
$101$	71080.8	0.693344	0.346672	−	0.937986i	$-0.387312\pi$
$101$	71080.8	0.693344	0.346672	+	0.937986i	$0.387312\pi$
$102$	0	0
$103$	−75964.0	−0.705529	−0.352764	−	0.935712i	$-0.614758\pi$
$103$	−75964.0	−0.705529	−0.352764	+	0.935712i	$0.614758\pi$
$104$	−98675.9	−0.894598
$105$	0	0
$106$	161448.	1.39562
$107$	24420.3	0.206201	0.103101	−	0.994671i	$-0.467124\pi$
$107$	24420.3	0.206201	0.103101	+	0.994671i	$0.467124\pi$
$108$	0	0
$109$	162882.	1.31313	0.656564	−	0.754271i	$-0.272009\pi$
$109$	162882.	1.31313	0.656564	+	0.754271i	$0.272009\pi$
$110$	−44596.8	−0.351416
$111$	0	0
$112$	43120.0	0.324813
$113$	−133134.	−0.980830	−0.490415	−	0.871489i	$-0.663155\pi$
$113$	−133134.	−0.980830	−0.490415	+	0.871489i	$0.663155\pi$
$114$	0	0
$115$	−141708.	−0.999194
$116$	423.320	0.00292095
$117$	0	0
$118$	238728.	1.57833
$119$	38114.7	0.246732
$120$	0	0
$121$	−109279.	−0.678537
$122$	119641.	0.727746
$123$	0	0
$124$	10416.0	0.0608341
$125$	−180684.	−1.03429
$126$	0	0
$127$	353208.	1.94322	0.971608	−	0.236595i	$-0.0760315\pi$
$127$	353208.	1.94322	0.971608	+	0.236595i	$0.0760315\pi$
$128$	−143252.	−0.772813
$129$	0	0
$130$	101528.	0.526899
$131$	−79859.4	−0.406581	−0.203291	−	0.979118i	$-0.565164\pi$
$131$	−79859.4	−0.406581	−0.203291	+	0.979118i	$0.565164\pi$
$132$	0	0
$133$	72716.0	0.356452
$134$	69445.7	0.334105
$135$	0	0
$136$	−148176.	−0.686959
$137$	154586.	0.703669	0.351835	−	0.936062i	$-0.385558\pi$
$137$	154586.	0.703669	0.351835	+	0.936062i	$0.385558\pi$
$138$	0	0
$139$	118944.	0.522162	0.261081	−	0.965317i	$-0.415921\pi$
$139$	118944.	0.522162	0.261081	+	0.965317i	$0.415921\pi$
$140$	7259.94	0.0313050
$141$	0	0
$142$	−255108.	−1.06170
$143$	−117863.	−0.481989
$144$	0	0
$145$	−3920.00	−0.0154834
$146$	438486.	1.70245
$147$	0	0
$148$	−1608.00	−0.00603437
$149$	358161.	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$149$	358161.	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$150$	0	0
$151$	327368.	1.16841	0.584203	−	0.811607i	$-0.301407\pi$
$151$	327368.	1.16841	0.584203	+	0.811607i	$0.301407\pi$
$152$	−282693.	−0.992445
$153$	0	0
$154$	58996.0	0.200457
$155$	−96453.5	−0.322470
$156$	0	0
$157$	−612290.	−1.98248	−0.991238	−	0.132086i	$-0.957832\pi$
$157$	−612290.	−1.98248	−0.991238	+	0.132086i	$0.957832\pi$
$158$	429204.	1.36780
$159$	0	0
$160$	−53312.0	−0.164636
$161$	187462.	0.569966
$162$	0	0
$163$	131380.	0.387311	0.193656	−	0.981070i	$-0.437966\pi$
$163$	131380.	0.387311	0.193656	+	0.981070i	$0.437966\pi$
$164$	−2518.76	−0.00731268
$165$	0	0
$166$	−352800.	−0.993707
$167$	−338328.	−0.938743	−0.469372	−	0.883001i	$-0.655520\pi$
$167$	−338328.	−0.938743	−0.469372	+	0.883001i	$0.655520\pi$
$168$	0	0
$169$	−102969.	−0.277325
$170$	152459.	0.404604
$171$	0	0
$172$	−27824.0	−0.0717131
$173$	−718771.	−1.82589	−0.912947	−	0.408079i	$-0.866199\pi$
$173$	−718771.	−1.82589	−0.912947	+	0.408079i	$0.866199\pi$
$174$	0	0
$175$	85897.0	0.212023
$176$	−200230.	−0.487246
$177$	0	0
$178$	−670516.	−1.58620
$179$	−173006.	−0.403578	−0.201789	−	0.979429i	$-0.564676\pi$
$179$	−173006.	−0.403578	−0.201789	+	0.979429i	$0.564676\pi$
$180$	0	0
$181$	234682.	0.532456	0.266228	−	0.963910i	$-0.414223\pi$
$181$	234682.	0.532456	0.266228	+	0.963910i	$0.414223\pi$
$182$	−134309.	−0.300557
$183$	0	0
$184$	−728784.	−1.58692
$185$	14890.3	0.0319870
$186$	0	0
$187$	−176988.	−0.370118
$188$	−109344.	−0.225631
$189$	0	0
$190$	290864.	0.584529
$191$	394423.	0.782311	0.391155	−	0.920325i	$-0.372076\pi$
$191$	394423.	0.782311	0.391155	+	0.920325i	$0.372076\pi$
$192$	0	0
$193$	−69826.0	−0.134935	−0.0674674	−	0.997721i	$-0.521492\pi$
$193$	−69826.0	−0.134935	−0.0674674	+	0.997721i	$0.521492\pi$
$194$	56227.5	0.107262
$195$	0	0
$196$	−9604.00	−0.0178571
$197$	145082.	0.266348	0.133174	−	0.991093i	$-0.457483\pi$
$197$	145082.	0.266348	0.133174	+	0.991093i	$0.457483\pi$
$198$	0	0
$199$	448224.	0.802347	0.401174	−	0.916002i	$-0.368602\pi$
$199$	448224.	0.802347	0.401174	+	0.916002i	$0.368602\pi$
$200$	−333936.	−0.590321
$201$	0	0
$202$	−376124.	−0.648564
$203$	5185.67	0.00883212
$204$	0	0
$205$	23324.0	0.0387631
$206$	401964.	0.659962
$207$	0	0
$208$	455840.	0.730557
$209$	−337661.	−0.534707
$210$	0	0
$211$	572500.	0.885257	0.442628	−	0.896705i	$-0.354046\pi$
$211$	572500.	0.885257	0.442628	+	0.896705i	$0.354046\pi$
$212$	122043.	0.186498
$213$	0	0
$214$	−129220.	−0.192884
$215$	257654.	0.380137
$216$	0	0
$217$	127596.	0.183945
$218$	−861891.	−1.22832
$219$	0	0
$220$	−33712.0	−0.0469600
$221$	402927.	0.554939
$222$	0	0
$223$	515368.	0.693993	0.346997	−	0.937866i	$-0.387202\pi$
$223$	515368.	0.693993	0.346997	+	0.937866i	$0.387202\pi$
$224$	70525.1	0.0939126
$225$	0	0
$226$	704480.	0.917482
$227$	501603.	0.646093	0.323047	−	0.946383i	$-0.395293\pi$
$227$	501603.	0.646093	0.323047	+	0.946383i	$0.395293\pi$
$228$	0	0
$229$	−413182.	−0.520658	−0.260329	−	0.965520i	$-0.583831\pi$
$229$	−413182.	−0.520658	−0.260329	+	0.965520i	$0.583831\pi$
$230$	749848.	0.934660
$231$	0	0
$232$	−20160.0	−0.0245907
$233$	−392069.	−0.473121	−0.236561	−	0.971617i	$-0.576020\pi$
$233$	−392069.	−0.473121	−0.236561	+	0.971617i	$0.576020\pi$
$234$	0	0
$235$	1.01254e6	1.19603
$236$	180461.	0.210913
$237$	0	0
$238$	−201684.	−0.230796
$239$	990617.	1.12179	0.560894	−	0.827887i	$-0.310457\pi$
$239$	990617.	1.12179	0.560894	+	0.827887i	$0.310457\pi$
$240$	0	0
$241$	659750.	0.731706	0.365853	−	0.930673i	$-0.380777\pi$
$241$	659750.	0.731706	0.365853	+	0.930673i	$0.380777\pi$
$242$	578250.	0.634713
$243$	0	0
$244$	90440.0	0.0972492
$245$	88934.3	0.0946573
$246$	0	0
$247$	768712.	0.801717
$248$	−496047.	−0.512145
$249$	0	0
$250$	956088.	0.967493
$251$	−280989.	−0.281518	−0.140759	−	0.990044i	$-0.544954\pi$
$251$	−280989.	−0.281518	−0.140759	+	0.990044i	$0.544954\pi$
$252$	0	0
$253$	−870492.	−0.854995
$254$	−1.86900e6	−1.81771
$255$	0	0
$256$	−386816.	−0.368896
$257$	1.70612e6	1.61130	0.805652	−	0.592389i	$-0.201815\pi$
$257$	1.70612e6	1.61130	0.805652	+	0.592389i	$0.201815\pi$
$258$	0	0
$259$	−19698.0	−0.0182462
$260$	76748.0	0.0704099
$261$	0	0
$262$	422576.	0.380322
$263$	−1.48763e6	−1.32619	−0.663093	−	0.748537i	$-0.730757\pi$
$263$	−1.48763e6	−1.32619	−0.663093	+	0.748537i	$0.730757\pi$
$264$	0	0
$265$	−1.13014e6	−0.988590
$266$	−384777.	−0.333430
$267$	0	0
$268$	52496.0	0.0446467
$269$	−1.79406e6	−1.51167	−0.755833	−	0.654765i	$-0.772768\pi$
$269$	−1.79406e6	−1.51167	−0.755833	+	0.654765i	$0.772768\pi$
$270$	0	0
$271$	298060.	0.246536	0.123268	−	0.992373i	$-0.460663\pi$
$271$	298060.	0.246536	0.123268	+	0.992373i	$0.460663\pi$
$272$	684509.	0.560992
$273$	0	0
$274$	−817992.	−0.658222
$275$	−398868.	−0.318052
$276$	0	0
$277$	−1.80497e6	−1.41342	−0.706709	−	0.707504i	$-0.749821\pi$
$277$	−1.80497e6	−1.41342	−0.706709	+	0.707504i	$0.749821\pi$
$278$	−629392.	−0.488438
$279$	0	0
$280$	−345744.	−0.263548
$281$	1.87130e6	1.41376	0.706882	−	0.707331i	$-0.250101\pi$
$281$	1.87130e6	1.41376	0.706882	+	0.707331i	$0.250101\pi$
$282$	0	0
$283$	1.52228e6	1.12987	0.564933	−	0.825136i	$-0.308902\pi$
$283$	1.52228e6	1.12987	0.564933	+	0.825136i	$0.308902\pi$
$284$	−192844.	−0.141876
$285$	0	0
$286$	623672.	0.450859
$287$	−30854.8	−0.0221115
$288$	0	0
$289$	−814805.	−0.573864
$290$	20742.7	0.0144834
$291$	0	0
$292$	331464.	0.227499
$293$	−1.85488e6	−1.26225	−0.631126	−	0.775680i	$-0.717407\pi$
$293$	−1.85488e6	−1.26225	−0.631126	+	0.775680i	$0.717407\pi$
$294$	0	0
$295$	−1.67110e6	−1.11801
$296$	76578.6	0.0508017
$297$	0	0
$298$	−1.89521e6	−1.23628
$299$	1.98174e6	1.28194
$300$	0	0
$301$	−340844.	−0.216840
$302$	−1.73227e6	−1.09294
$303$	0	0
$304$	1.30592e6	0.810462
$305$	−837486.	−0.515499
$306$	0	0
$307$	−2.29855e6	−1.39190	−0.695949	−	0.718091i	$-0.745016\pi$
$307$	−2.29855e6	−1.39190	−0.695949	+	0.718091i	$0.745016\pi$
$308$	44596.8	0.0267872
$309$	0	0
$310$	510384.	0.301643
$311$	927865.	0.543981	0.271991	−	0.962300i	$-0.412318\pi$
$311$	927865.	0.543981	0.271991	+	0.962300i	$0.412318\pi$
$312$	0	0
$313$	−2.73864e6	−1.58006	−0.790030	−	0.613068i	$-0.789935\pi$
$313$	−2.73864e6	−1.58006	−0.790030	+	0.613068i	$0.789935\pi$
$314$	3.23993e6	1.85444
$315$	0	0
$316$	324448.	0.182779
$317$	1.93228e6	1.07999	0.539997	−	0.841667i	$-0.318425\pi$
$317$	1.93228e6	1.07999	0.539997	+	0.841667i	$0.318425\pi$
$318$	0	0
$319$	−24080.0	−0.0132489
$320$	1.32516e6	0.723426
$321$	0	0
$322$	−991956.	−0.533154
$323$	1.15433e6	0.615636
$324$	0	0
$325$	908054.	0.476874
$326$	−695198.	−0.362297
$327$	0	0
$328$	119952.	0.0615634
$329$	−1.33946e6	−0.682244
$330$	0	0
$331$	2.64146e6	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$331$	2.64146e6	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$332$	−266692.	−0.132790
$333$	0	0
$334$	1.79026e6	0.878114
$335$	−486120.	−0.236663
$336$	0	0
$337$	−3.13119e6	−1.50188	−0.750938	−	0.660373i	$-0.770398\pi$
$337$	−3.13119e6	−1.50188	−0.750938	+	0.660373i	$0.770398\pi$
$338$	544861.	0.259414
$339$	0	0
$340$	115248.	0.0540675
$341$	−592500.	−0.275932
$342$	0	0
$343$	−117649.	−0.0539949
$344$	1.32508e6	0.603733
$345$	0	0
$346$	3.80338e6	1.70797
$347$	−2.09446e6	−0.933786	−0.466893	−	0.884314i	$-0.654627\pi$
$347$	−2.09446e6	−0.933786	−0.466893	+	0.884314i	$0.654627\pi$
$348$	0	0
$349$	−1.45515e6	−0.639504	−0.319752	−	0.947501i	$-0.603600\pi$
$349$	−1.45515e6	−0.639504	−0.319752	+	0.947501i	$0.603600\pi$
$350$	−454524.	−0.198329
$351$	0	0
$352$	−327488.	−0.140877
$353$	−88193.5	−0.0376704	−0.0188352	−	0.999823i	$-0.505996\pi$
$353$	−88193.5	−0.0376704	−0.0188352	+	0.999823i	$0.505996\pi$
$354$	0	0
$355$	1.78576e6	0.752058
$356$	−506862.	−0.211965
$357$	0	0
$358$	915460.	0.377513
$359$	−2.05956e6	−0.843411	−0.421705	−	0.906733i	$-0.638568\pi$
$359$	−2.05956e6	−0.843411	−0.421705	+	0.906733i	$0.638568\pi$
$360$	0	0
$361$	−273843.	−0.110595
$362$	−1.24182e6	−0.498067
$363$	0	0
$364$	−101528.	−0.0401636
$365$	−3.06940e6	−1.20593
$366$	0	0
$367$	2.69606e6	1.04488	0.522438	−	0.852677i	$-0.325022\pi$
$367$	2.69606e6	1.04488	0.522438	+	0.852677i	$0.325022\pi$
$368$	3.36667e6	1.29593
$369$	0	0
$370$	−78792.0	−0.0299211
$371$	1.49503e6	0.563917
$372$	0	0
$373$	3.09959e6	1.15354	0.576769	−	0.816907i	$-0.304313\pi$
$373$	3.09959e6	1.15354	0.576769	+	0.816907i	$0.304313\pi$
$374$	936532.	0.346213
$375$	0	0
$376$	5.20733e6	1.89953
$377$	54820.0	0.0198649
$378$	0	0
$379$	−1.86899e6	−0.668357	−0.334178	−	0.942510i	$-0.608459\pi$
$379$	−1.86899e6	−0.668357	−0.334178	+	0.942510i	$0.608459\pi$
$380$	219873.	0.0781110
$381$	0	0
$382$	−2.08709e6	−0.731784
$383$	−3.55278e6	−1.23757	−0.618787	−	0.785559i	$-0.712376\pi$
$383$	−3.55278e6	−1.23757	−0.618787	+	0.785559i	$0.712376\pi$
$384$	0	0
$385$	−412972.	−0.141994
$386$	369484.	0.126220
$387$	0	0
$388$	42504.0	0.0143334
$389$	1.65133e6	0.553299	0.276649	−	0.960971i	$-0.410776\pi$
$389$	1.65133e6	0.553299	0.276649	+	0.960971i	$0.410776\pi$
$390$	0	0
$391$	2.97587e6	0.984400
$392$	457376.	0.150334
$393$	0	0
$394$	−767704.	−0.249146
$395$	−3.00443e6	−0.968879
$396$	0	0
$397$	−4.66120e6	−1.48430	−0.742150	−	0.670234i	$-0.766194\pi$
$397$	−4.66120e6	−1.48430	−0.742150	+	0.670234i	$0.766194\pi$
$398$	−2.37178e6	−0.750527
$399$	0	0
$400$	1.54264e6	0.482075
$401$	1.83798e6	0.570795	0.285398	−	0.958409i	$-0.407874\pi$
$401$	1.83798e6	0.570795	0.285398	+	0.958409i	$0.407874\pi$
$402$	0	0
$403$	1.34887e6	0.413722
$404$	−284323.	−0.0866680
$405$	0	0
$406$	−27440.0	−0.00826169
$407$	91468.9	0.0273708
$408$	0	0
$409$	1.80265e6	0.532849	0.266424	−	0.963856i	$-0.414158\pi$
$409$	1.80265e6	0.532849	0.266424	+	0.963856i	$0.414158\pi$
$410$	−123419.	−0.0362595
$411$	0	0
$412$	303856.	0.0881911
$413$	2.21065e6	0.637742
$414$	0	0
$415$	2.46960e6	0.703893
$416$	745552.	0.211225
$417$	0	0
$418$	1.78674e6	0.500172
$419$	−69117.6	−0.0192333	−0.00961665	−	0.999954i	$-0.503061\pi$
$419$	−69117.6	−0.0192333	−0.00961665	+	0.999954i	$0.503061\pi$
$420$	0	0
$421$	−872170.	−0.239826	−0.119913	−	0.992784i	$-0.538262\pi$
$421$	−872170.	−0.239826	−0.119913	+	0.992784i	$0.538262\pi$
$422$	−3.02939e6	−0.828082
$423$	0	0
$424$	−5.81213e6	−1.57008
$425$	1.36357e6	0.366190
$426$	0	0
$427$	1.10789e6	0.294054
$428$	−97681.1	−0.0257752
$429$	0	0
$430$	−1.36338e6	−0.355586
$431$	4.76618e6	1.23588	0.617941	−	0.786224i	$-0.287967\pi$
$431$	4.76618e6	1.23588	0.617941	+	0.786224i	$0.287967\pi$
$432$	0	0
$433$	−5.53813e6	−1.41953	−0.709764	−	0.704440i	$-0.751198\pi$
$433$	−5.53813e6	−1.41953	−0.709764	+	0.704440i	$0.751198\pi$
$434$	−675175.	−0.172065
$435$	0	0
$436$	−651528.	−0.164141
$437$	5.67742e6	1.42216
$438$	0	0
$439$	−4.71778e6	−1.16836	−0.584179	−	0.811625i	$-0.698583\pi$
$439$	−4.71778e6	−1.16836	−0.584179	+	0.811625i	$0.698583\pi$
$440$	1.60548e6	0.395343
$441$	0	0
$442$	−2.13209e6	−0.519098
$443$	5.04236e6	1.22074	0.610372	−	0.792115i	$-0.291020\pi$
$443$	5.04236e6	1.22074	0.610372	+	0.792115i	$0.291020\pi$
$444$	0	0
$445$	4.69361e6	1.12359
$446$	−2.72707e6	−0.649171
$447$	0	0
$448$	−1.75302e6	−0.412660
$449$	−709083.	−0.165990	−0.0829948	−	0.996550i	$-0.526448\pi$
$449$	−709083.	−0.165990	−0.0829948	+	0.996550i	$0.526448\pi$
$450$	0	0
$451$	143276.	0.0331690
$452$	532537.	0.122604
$453$	0	0
$454$	−2.65423e6	−0.604365
$455$	940162.	0.212899
$456$	0	0
$457$	−5.26385e6	−1.17900	−0.589499	−	0.807769i	$-0.700675\pi$
$457$	−5.26385e6	−1.17900	−0.589499	+	0.807769i	$0.700675\pi$
$458$	2.18635e6	0.487031
$459$	0	0
$460$	566832.	0.124899
$461$	−3.51326e6	−0.769941	−0.384971	−	0.922929i	$-0.625788\pi$
$461$	−3.51326e6	−0.769941	−0.384971	+	0.922929i	$0.625788\pi$
$462$	0	0
$463$	2.24315e6	0.486302	0.243151	−	0.969988i	$-0.421819\pi$
$463$	2.24315e6	0.486302	0.243151	+	0.969988i	$0.421819\pi$
$464$	93130.4	0.0200815
$465$	0	0
$466$	2.07463e6	0.442564
$467$	−5.87100e6	−1.24572	−0.622859	−	0.782334i	$-0.714029\pi$
$467$	−5.87100e6	−1.24572	−0.622859	+	0.782334i	$0.714029\pi$
$468$	0	0
$469$	643076.	0.134999
$470$	−5.35784e6	−1.11878
$471$	0	0
$472$	−8.59421e6	−1.77562
$473$	1.58273e6	0.325278
$474$	0	0
$475$	2.60145e6	0.529032
$476$	−152459.	−0.0308415
$477$	0	0
$478$	−5.24185e6	−1.04934
$479$	2.28696e6	0.455427	0.227714	−	0.973728i	$-0.426875\pi$
$479$	2.28696e6	0.455427	0.227714	+	0.973728i	$0.426875\pi$
$480$	0	0
$481$	−208236.	−0.0410387
$482$	−3.49107e6	−0.684449
$483$	0	0
$484$	437116.	0.0848171
$485$	−393593.	−0.0759788
$486$	0	0
$487$	−269480.	−0.0514878	−0.0257439	−	0.999669i	$-0.508195\pi$
$487$	−269480.	−0.0514878	−0.0257439	+	0.999669i	$0.508195\pi$
$488$	−4.30707e6	−0.818715
$489$	0	0
$490$	−470596.	−0.0885438
$491$	−5.42798e6	−1.01609	−0.508047	−	0.861329i	$-0.669633\pi$
$491$	−5.42798e6	−1.01609	−0.508047	+	0.861329i	$0.669633\pi$
$492$	0	0
$493$	82320.0	0.0152542
$494$	−4.06764e6	−0.749938
$495$	0	0
$496$	2.29152e6	0.418234
$497$	−2.36233e6	−0.428993
$498$	0	0
$499$	4.19720e6	0.754585	0.377292	−	0.926094i	$-0.376855\pi$
$499$	4.19720e6	0.754585	0.377292	+	0.926094i	$0.376855\pi$
$500$	722735.	0.129287
$501$	0	0
$502$	1.48686e6	0.263336
$503$	5.43607e6	0.957998	0.478999	−	0.877815i	$-0.341000\pi$
$503$	5.43607e6	0.957998	0.478999	+	0.877815i	$0.341000\pi$
$504$	0	0
$505$	2.63287e6	0.459410
$506$	4.60621e6	0.799774
$507$	0	0
$508$	−1.41283e6	−0.242902
$509$	3.22849e6	0.552338	0.276169	−	0.961109i	$-0.410935\pi$
$509$	3.22849e6	0.552338	0.276169	+	0.961109i	$0.410935\pi$
$510$	0	0
$511$	4.06043e6	0.687892
$512$	6.63089e6	1.11788
$513$	0	0
$514$	−9.02796e6	−1.50724
$515$	−2.81375e6	−0.467484
$516$	0	0
$517$	6.21986e6	1.02342
$518$	104232.	0.0170678
$519$	0	0
$520$	−3.65501e6	−0.592761
$521$	−1.17530e7	−1.89694	−0.948471	−	0.316863i	$-0.897370\pi$
$521$	−1.17530e7	−1.89694	−0.948471	+	0.316863i	$0.897370\pi$
$522$	0	0
$523$	−3.22482e6	−0.515526	−0.257763	−	0.966208i	$-0.582985\pi$
$523$	−3.22482e6	−0.515526	−0.257763	+	0.966208i	$0.582985\pi$
$524$	319437.	0.0508227
$525$	0	0
$526$	7.87178e6	1.24053
$527$	2.02552e6	0.317695
$528$	0	0
$529$	8.20007e6	1.27403
$530$	5.98012e6	0.924741
$531$	0	0
$532$	−290864.	−0.0445565
$533$	−326179.	−0.0497322
$534$	0	0
$535$	904540.	0.136629
$536$	−2.50004e6	−0.375868
$537$	0	0
$538$	9.49326e6	1.41403
$539$	546311.	0.0809968
$540$	0	0
$541$	7.85617e6	1.15403	0.577016	−	0.816733i	$-0.304217\pi$
$541$	7.85617e6	1.15403	0.577016	+	0.816733i	$0.304217\pi$
$542$	−1.57719e6	−0.230613
$543$	0	0
$544$	1.11955e6	0.162199
$545$	6.03323e6	0.870079
$546$	0	0
$547$	−1.81480e6	−0.259335	−0.129668	−	0.991558i	$-0.541391\pi$
$547$	−1.81480e6	−0.259335	−0.129668	+	0.991558i	$0.541391\pi$
$548$	−618344.	−0.0879587
$549$	0	0
$550$	2.11061e6	0.297510
$551$	157052.	0.0220376
$552$	0	0
$553$	3.97449e6	0.552673
$554$	9.55100e6	1.32213
$555$	0	0
$556$	−475776.	−0.0652703
$557$	5.85343e6	0.799415	0.399708	−	0.916643i	$-0.369112\pi$
$557$	5.85343e6	0.799415	0.399708	+	0.916643i	$0.369112\pi$
$558$	0	0
$559$	−3.60321e6	−0.487708
$560$	1.59719e6	0.215222
$561$	0	0
$562$	−9.90198e6	−1.32246
$563$	9.35236e6	1.24351	0.621756	−	0.783211i	$-0.286419\pi$
$563$	9.35236e6	1.24351	0.621756	+	0.783211i	$0.286419\pi$
$564$	0	0
$565$	−4.93136e6	−0.649899
$566$	−8.05513e6	−1.05689
$567$	0	0
$568$	9.18389e6	1.19442
$569$	−1.07712e7	−1.39471	−0.697357	−	0.716724i	$-0.745641\pi$
$569$	−1.07712e7	−1.39471	−0.697357	+	0.716724i	$0.745641\pi$
$570$	0	0
$571$	1.16786e7	1.49900	0.749499	−	0.662006i	$-0.230295\pi$
$571$	1.16786e7	1.49900	0.749499	+	0.662006i	$0.230295\pi$
$572$	471452.	0.0602486
$573$	0	0
$574$	163268.	0.0206834
$575$	6.70655e6	0.845921
$576$	0	0
$577$	−7.94272e6	−0.993184	−0.496592	−	0.867984i	$-0.665415\pi$
$577$	−7.94272e6	−0.993184	−0.496592	+	0.867984i	$0.665415\pi$
$578$	4.31154e6	0.536801
$579$	0	0
$580$	15680.0	0.00193542
$581$	−3.26697e6	−0.401518
$582$	0	0
$583$	−6.94226e6	−0.845921
$584$	−1.57855e7	−1.91525
$585$	0	0
$586$	9.81509e6	1.18073
$587$	7.14682e6	0.856086	0.428043	−	0.903758i	$-0.359203\pi$
$587$	7.14682e6	0.856086	0.428043	+	0.903758i	$0.359203\pi$
$588$	0	0
$589$	3.86434e6	0.458972
$590$	8.84261e6	1.04580
$591$	0	0
$592$	−353760.	−0.0414863
$593$	4.17028e6	0.487000	0.243500	−	0.969901i	$-0.421704\pi$
$593$	4.17028e6	0.487000	0.243500	+	0.969901i	$0.421704\pi$
$594$	0	0
$595$	1.41179e6	0.163485
$596$	−1.43264e6	−0.165205
$597$	0	0
$598$	−1.04864e7	−1.19915
$599$	−1.65906e7	−1.88927	−0.944635	−	0.328123i	$-0.893584\pi$
$599$	−1.65906e7	−1.88927	−0.944635	+	0.328123i	$0.893584\pi$
$600$	0	0
$601$	7.22270e6	0.815668	0.407834	−	0.913056i	$-0.366284\pi$
$601$	7.22270e6	0.815668	0.407834	+	0.913056i	$0.366284\pi$
$602$	1.80358e6	0.202835
$603$	0	0
$604$	−1.30947e6	−0.146051
$605$	−4.04775e6	−0.449599
$606$	0	0
$607$	−1.31990e7	−1.45402	−0.727010	−	0.686627i	$-0.759091\pi$
$607$	−1.31990e7	−1.45402	−0.727010	+	0.686627i	$0.759091\pi$
$608$	2.13590e6	0.234327
$609$	0	0
$610$	4.43156e6	0.482205
$611$	−1.41600e7	−1.53448
$612$	0	0
$613$	7.26132e6	0.780484	0.390242	−	0.920712i	$-0.372391\pi$
$613$	7.26132e6	0.780484	0.390242	+	0.920712i	$0.372391\pi$
$614$	1.21628e7	1.30200
$615$	0	0
$616$	−2.12386e6	−0.225514
$617$	−6.18512e6	−0.654087	−0.327043	−	0.945009i	$-0.606052\pi$
$617$	−6.18512e6	−0.654087	−0.327043	+	0.945009i	$0.606052\pi$
$618$	0	0
$619$	−1.46063e7	−1.53219	−0.766097	−	0.642725i	$-0.777804\pi$
$619$	−1.46063e7	−1.53219	−0.766097	+	0.642725i	$0.777804\pi$
$620$	385814.	0.0403087
$621$	0	0
$622$	−4.90980e6	−0.508848
$623$	−6.20907e6	−0.640923
$624$	0	0
$625$	−1.21449e6	−0.124364
$626$	1.44915e7	1.47801
$627$	0	0
$628$	2.44916e6	0.247810
$629$	−312696.	−0.0315134
$630$	0	0
$631$	4.52521e6	0.452444	0.226222	−	0.974076i	$-0.427362\pi$
$631$	4.52521e6	0.452444	0.226222	+	0.974076i	$0.427362\pi$
$632$	−1.54514e7	−1.53877
$633$	0	0
$634$	−1.02246e7	−1.01024
$635$	1.30830e7	1.28758
$636$	0	0
$637$	−1.24372e6	−0.121443
$638$	127419.	0.0123932
$639$	0	0
$640$	−5.30611e6	−0.512067
$641$	−1.61783e6	−0.155521	−0.0777605	−	0.996972i	$-0.524777\pi$
$641$	−1.61783e6	−0.155521	−0.0777605	+	0.996972i	$0.524777\pi$
$642$	0	0
$643$	−1.38733e7	−1.32328	−0.661641	−	0.749821i	$-0.730140\pi$
$643$	−1.38733e7	−1.32328	−0.661641	+	0.749821i	$0.730140\pi$
$644$	−749848.	−0.0712457
$645$	0	0
$646$	−6.10814e6	−0.575875
$647$	−2.05567e6	−0.193061	−0.0965303	−	0.995330i	$-0.530774\pi$
$647$	−2.05567e6	−0.193061	−0.0965303	+	0.995330i	$0.530774\pi$
$648$	0	0
$649$	−1.02653e7	−0.956665
$650$	−4.80497e6	−0.446074
$651$	0	0
$652$	−525520.	−0.0484139
$653$	−3.97072e6	−0.364407	−0.182203	−	0.983261i	$-0.558323\pi$
$653$	−3.97072e6	−0.364407	−0.182203	+	0.983261i	$0.558323\pi$
$654$	0	0
$655$	−2.95803e6	−0.269401
$656$	−554126.	−0.0502746
$657$	0	0
$658$	7.08775e6	0.638181
$659$	−4.35538e6	−0.390672	−0.195336	−	0.980736i	$-0.562580\pi$
$659$	−4.35538e6	−0.390672	−0.195336	+	0.980736i	$0.562580\pi$
$660$	0	0
$661$	1.71451e6	0.152629	0.0763144	−	0.997084i	$-0.475685\pi$
$661$	1.71451e6	0.152629	0.0763144	+	0.997084i	$0.475685\pi$
$662$	−1.39773e7	−1.23959
$663$	0	0
$664$	1.27008e7	1.11792
$665$	2.69344e6	0.236185
$666$	0	0
$667$	404880.	0.0352380
$668$	1.35331e6	0.117343
$669$	0	0
$670$	2.57230e6	0.221378
$671$	−5.14456e6	−0.441105
$672$	0	0
$673$	−1.73630e7	−1.47770	−0.738850	−	0.673870i	$-0.764631\pi$
$673$	−1.73630e7	−1.47770	−0.738850	+	0.673870i	$0.764631\pi$
$674$	1.65687e7	1.40488
$675$	0	0
$676$	411876.	0.0346657
$677$	−2.27699e6	−0.190937	−0.0954684	−	0.995432i	$-0.530435\pi$
$677$	−2.27699e6	−0.190937	−0.0954684	+	0.995432i	$0.530435\pi$
$678$	0	0
$679$	520674.	0.0433403
$680$	−5.48852e6	−0.455179
$681$	0	0
$682$	3.13522e6	0.258111
$683$	1.14583e7	0.939874	0.469937	−	0.882700i	$-0.344277\pi$
$683$	1.14583e7	0.939874	0.469937	+	0.882700i	$0.344277\pi$
$684$	0	0
$685$	5.72594e6	0.466252
$686$	622540.	0.0505076
$687$	0	0
$688$	−6.12128e6	−0.493028
$689$	1.58046e7	1.26834
$690$	0	0
$691$	−1.38106e7	−1.10031	−0.550156	−	0.835062i	$-0.685432\pi$
$691$	−1.38106e7	−1.10031	−0.550156	+	0.835062i	$0.685432\pi$
$692$	2.87509e6	0.228237
$693$	0	0
$694$	1.10828e7	0.873477
$695$	4.40575e6	0.345985
$696$	0	0
$697$	−489804.	−0.0381892
$698$	7.69991e6	0.598201
$699$	0	0
$700$	−343588.	−0.0265029
$701$	2.17478e7	1.67155	0.835775	−	0.549072i	$-0.185019\pi$
$701$	2.17478e7	1.67155	0.835775	+	0.549072i	$0.185019\pi$
$702$	0	0
$703$	−596568.	−0.0455273
$704$	8.14028e6	0.619024
$705$	0	0
$706$	466676.	0.0352374
$707$	−3.48296e6	−0.262059
$708$	0	0
$709$	3.12067e6	0.233148	0.116574	−	0.993182i	$-0.462809\pi$
$709$	3.12067e6	0.233148	0.116574	+	0.993182i	$0.462809\pi$
$710$	−9.44933e6	−0.703486
$711$	0	0
$712$	2.41386e7	1.78448
$713$	9.96227e6	0.733896
$714$	0	0
$715$	−4.36570e6	−0.319366
$716$	692023.	0.0504473
$717$	0	0
$718$	1.08982e7	0.788939
$719$	−1.15140e7	−0.830621	−0.415311	−	0.909680i	$-0.636327\pi$
$719$	−1.15140e7	−0.830621	−0.415311	+	0.909680i	$0.636327\pi$
$720$	0	0
$721$	3.72224e6	0.266665
$722$	1.44904e6	0.103452
$723$	0	0
$724$	−938728.	−0.0665569
$725$	185520.	0.0131083
$726$	0	0
$727$	−7.30668e6	−0.512725	−0.256362	−	0.966581i	$-0.582524\pi$
$727$	−7.30668e6	−0.512725	−0.256362	+	0.966581i	$0.582524\pi$
$728$	4.83512e6	0.338126
$729$	0	0
$730$	1.62417e7	1.12804
$731$	−5.41073e6	−0.374509
$732$	0	0
$733$	8.23826e6	0.566338	0.283169	−	0.959070i	$-0.408614\pi$
$733$	8.23826e6	0.566338	0.283169	+	0.959070i	$0.408614\pi$
$734$	−1.42662e7	−0.977393
$735$	0	0
$736$	5.50637e6	0.374689
$737$	−2.98616e6	−0.202509
$738$	0	0
$739$	228180.	0.0153697	0.00768487	−	0.999970i	$-0.497554\pi$
$739$	228180.	0.0153697	0.00768487	+	0.999970i	$0.497554\pi$
$740$	−59561.2	−0.00399838
$741$	0	0
$742$	−7.91095e6	−0.527496
$743$	−2.43456e6	−0.161789	−0.0808945	−	0.996723i	$-0.525778\pi$
$743$	−2.43456e6	−0.161789	−0.0808945	+	0.996723i	$0.525778\pi$
$744$	0	0
$745$	1.32665e7	0.875718
$746$	−1.64015e7	−1.07904
$747$	0	0
$748$	707952.	0.0462647
$749$	−1.19659e6	−0.0779367
$750$	0	0
$751$	6.19161e6	0.400593	0.200297	−	0.979735i	$-0.435809\pi$
$751$	6.19161e6	0.400593	0.200297	+	0.979735i	$0.435809\pi$
$752$	−2.40556e7	−1.55121
$753$	0	0
$754$	−290080.	−0.0185819
$755$	1.21259e7	0.774187
$756$	0	0
$757$	1.01573e7	0.644227	0.322113	−	0.946701i	$-0.395607\pi$
$757$	1.01573e7	0.644227	0.322113	+	0.946701i	$0.395607\pi$
$758$	9.88975e6	0.625191
$759$	0	0
$760$	−1.04711e7	−0.657595
$761$	−2.11305e7	−1.32266	−0.661331	−	0.750094i	$-0.730008\pi$
$761$	−2.11305e7	−1.32266	−0.661331	+	0.750094i	$0.730008\pi$
$762$	0	0
$763$	−7.98122e6	−0.496315
$764$	−1.57769e6	−0.0977888
$765$	0	0
$766$	1.87995e7	1.15764
$767$	2.33698e7	1.43438
$768$	0	0
$769$	1.27606e6	0.0778134	0.0389067	−	0.999243i	$-0.487612\pi$
$769$	1.27606e6	0.0778134	0.0389067	+	0.999243i	$0.487612\pi$
$770$	2.18524e6	0.132823
$771$	0	0
$772$	279304.	0.0168668
$773$	−6.79223e6	−0.408850	−0.204425	−	0.978882i	$-0.565532\pi$
$773$	−6.79223e6	−0.408850	−0.204425	+	0.978882i	$0.565532\pi$
$774$	0	0
$775$	4.56481e6	0.273004
$776$	−2.02419e6	−0.120669
$777$	0	0
$778$	−8.73802e6	−0.517564
$779$	−934458.	−0.0551717
$780$	0	0
$781$	1.09696e7	0.643524
$782$	−1.57468e7	−0.920822
$783$	0	0
$784$	−2.11288e6	−0.122768
$785$	−2.26795e7	−1.31359
$786$	0	0
$787$	8.29595e6	0.477452	0.238726	−	0.971087i	$-0.423270\pi$
$787$	8.29595e6	0.477452	0.238726	+	0.971087i	$0.423270\pi$
$788$	−580330.	−0.0332935
$789$	0	0
$790$	1.58980e7	0.906303
$791$	6.52358e6	0.370719
$792$	0	0
$793$	1.17120e7	0.661374
$794$	2.46648e7	1.38844
$795$	0	0
$796$	−1.79290e6	−0.100293
$797$	7.91300e6	0.441261	0.220630	−	0.975357i	$-0.429189\pi$
$797$	7.91300e6	0.441261	0.220630	+	0.975357i	$0.429189\pi$
$798$	0	0
$799$	−2.12633e7	−1.17832
$800$	2.52307e6	0.139381
$801$	0	0
$802$	−9.72569e6	−0.533930
$803$	−1.88549e7	−1.03189
$804$	0	0
$805$	6.94369e6	0.377660
$806$	−7.13756e6	−0.387001
$807$	0	0
$808$	1.35405e7	0.729634
$809$	2.48124e7	1.33290	0.666449	−	0.745551i	$-0.267813\pi$
$809$	2.48124e7	1.33290	0.666449	+	0.745551i	$0.267813\pi$
$810$	0	0
$811$	−3.28455e7	−1.75357	−0.876785	−	0.480882i	$-0.840316\pi$
$811$	−3.28455e7	−1.75357	−0.876785	+	0.480882i	$0.840316\pi$
$812$	−20742.7	−0.00110402
$813$	0	0
$814$	−484008.	−0.0256030
$815$	4.86638e6	0.256633
$816$	0	0
$817$	−1.03227e7	−0.541051
$818$	−9.53875e6	−0.498435
$819$	0	0
$820$	−93296.0	−0.00484539
$821$	3.14948e7	1.63073	0.815364	−	0.578949i	$-0.196537\pi$
$821$	3.14948e7	1.63073	0.815364	+	0.578949i	$0.196537\pi$
$822$	0	0
$823$	−2.31689e7	−1.19236	−0.596179	−	0.802852i	$-0.703315\pi$
$823$	−2.31689e7	−1.19236	−0.596179	+	0.802852i	$0.703315\pi$
$824$	−1.44707e7	−0.742457
$825$	0	0
$826$	−1.16977e7	−0.596553
$827$	−5.32817e6	−0.270903	−0.135452	−	0.990784i	$-0.543249\pi$
$827$	−5.32817e6	−0.270903	−0.135452	+	0.990784i	$0.543249\pi$
$828$	0	0
$829$	3.48465e7	1.76105	0.880527	−	0.473996i	$-0.157189\pi$
$829$	3.48465e7	1.76105	0.880527	+	0.473996i	$0.157189\pi$
$830$	−1.30679e7	−0.658431
$831$	0	0
$832$	−1.85320e7	−0.928139
$833$	−1.86762e6	−0.0932558
$834$	0	0
$835$	−1.25318e7	−0.622012
$836$	1.35065e6	0.0668383
$837$	0	0
$838$	365736.	0.0179911
$839$	−7.12608e6	−0.349499	−0.174749	−	0.984613i	$-0.555912\pi$
$839$	−7.12608e6	−0.349499	−0.174749	+	0.984613i	$0.555912\pi$
$840$	0	0
$841$	−2.04999e7	−0.999454
$842$	4.61509e6	0.224336
$843$	0	0
$844$	−2.29000e6	−0.110657
$845$	−3.81403e6	−0.183756
$846$	0	0
$847$	5.35467e6	0.256463
$848$	2.68495e7	1.28217
$849$	0	0
$850$	−7.21535e6	−0.342539
$851$	−1.53795e6	−0.0727980
$852$	0	0
$853$	2.67997e7	1.26112	0.630562	−	0.776139i	$-0.282825\pi$
$853$	2.67997e7	1.26112	0.630562	+	0.776139i	$0.282825\pi$
$854$	−5.86240e6	−0.275062
$855$	0	0
$856$	4.65192e6	0.216994
$857$	2.84036e7	1.32106	0.660528	−	0.750801i	$-0.270332\pi$
$857$	2.84036e7	1.32106	0.660528	+	0.750801i	$0.270332\pi$
$858$	0	0
$859$	4.52556e6	0.209261	0.104631	−	0.994511i	$-0.466634\pi$
$859$	4.52556e6	0.209261	0.104631	+	0.994511i	$0.466634\pi$
$860$	−1.03062e6	−0.0475172
$861$	0	0
$862$	−2.52202e7	−1.15606
$863$	−1.64572e7	−0.752194	−0.376097	−	0.926580i	$-0.622734\pi$
$863$	−1.64572e7	−0.752194	−0.376097	+	0.926580i	$0.622734\pi$
$864$	0	0
$865$	−2.66237e7	−1.20984
$866$	2.93051e7	1.32785
$867$	0	0
$868$	−510384.	−0.0229931
$869$	−1.84558e7	−0.829055
$870$	0	0
$871$	6.79823e6	0.303634
$872$	3.10281e7	1.38186
$873$	0	0
$874$	−3.00421e7	−1.33031
$875$	8.85350e6	0.390926
$876$	0	0
$877$	1.67759e7	0.736522	0.368261	−	0.929722i	$-0.379953\pi$
$877$	1.67759e7	0.736522	0.368261	+	0.929722i	$0.379953\pi$
$878$	2.49641e7	1.09290
$879$	0	0
$880$	−7.41664e6	−0.322850
$881$	−1.84212e7	−0.799609	−0.399804	−	0.916600i	$-0.630922\pi$
$881$	−1.84212e7	−0.799609	−0.399804	+	0.916600i	$0.630922\pi$
$882$	0	0
$883$	−4.36205e6	−0.188273	−0.0941367	−	0.995559i	$-0.530009\pi$
$883$	−4.36205e6	−0.188273	−0.0941367	+	0.995559i	$0.530009\pi$
$884$	−1.61171e6	−0.0693674
$885$	0	0
$886$	−2.66817e7	−1.14190
$887$	1.20003e7	0.512134	0.256067	−	0.966659i	$-0.417573\pi$
$887$	1.20003e7	0.512134	0.256067	+	0.966659i	$0.417573\pi$
$888$	0	0
$889$	−1.73072e7	−0.734467
$890$	−2.48363e7	−1.05102
$891$	0	0
$892$	−2.06147e6	−0.0867492
$893$	−4.05665e7	−1.70231
$894$	0	0
$895$	−6.40822e6	−0.267411
$896$	7.01933e6	0.292096
$897$	0	0
$898$	3.75211e6	0.155269
$899$	275581.	0.0113724
$900$	0	0
$901$	2.37329e7	0.973953
$902$	−758145.	−0.0310267
$903$	0	0
$904$	−2.53613e7	−1.03217
$905$	8.69274e6	0.352806
$906$	0	0
$907$	−3.39963e6	−0.137219	−0.0686093	−	0.997644i	$-0.521856\pi$
$907$	−3.39963e6	−0.137219	−0.0686093	+	0.997644i	$0.521856\pi$
$908$	−2.00641e6	−0.0807617
$909$	0	0
$910$	−4.97487e6	−0.199149
$911$	6.49345e6	0.259227	0.129613	−	0.991565i	$-0.458626\pi$
$911$	6.49345e6	0.259227	0.129613	+	0.991565i	$0.458626\pi$
$912$	0	0
$913$	1.51704e7	0.602310
$914$	2.78537e7	1.10285
$915$	0	0
$916$	1.65273e6	0.0650823
$917$	3.91311e6	0.153673
$918$	0	0
$919$	3.11184e7	1.21543	0.607713	−	0.794157i	$-0.292087\pi$
$919$	3.11184e7	1.21543	0.607713	+	0.794157i	$0.292087\pi$
$920$	−2.69945e7	−1.05149
$921$	0	0
$922$	1.85904e7	0.720214
$923$	−2.49732e7	−0.964874
$924$	0	0
$925$	−704706.	−0.0270803
$926$	−1.18696e7	−0.454894
$927$	0	0
$928$	152320.	0.00580613
$929$	−3.33767e7	−1.26883	−0.634417	−	0.772991i	$-0.718760\pi$
$929$	−3.33767e7	−1.26883	−0.634417	+	0.772991i	$0.718760\pi$
$930$	0	0
$931$	−3.56308e6	−0.134726
$932$	1.56827e6	0.0591401
$933$	0	0
$934$	3.10664e7	1.16526
$935$	−6.55573e6	−0.245240
$936$	0	0
$937$	−1.20340e7	−0.447775	−0.223887	−	0.974615i	$-0.571875\pi$
$937$	−1.20340e7	−0.447775	−0.223887	+	0.974615i	$0.571875\pi$
$938$	−3.40284e6	−0.126280
$939$	0	0
$940$	−4.05014e6	−0.149503
$941$	−1.30499e7	−0.480434	−0.240217	−	0.970719i	$-0.577219\pi$
$941$	−1.30499e7	−0.480434	−0.240217	+	0.970719i	$0.577219\pi$
$942$	0	0
$943$	−2.40904e6	−0.0882194
$944$	3.97015e7	1.45003
$945$	0	0
$946$	−8.37502e6	−0.304269
$947$	−1.18922e7	−0.430912	−0.215456	−	0.976514i	$-0.569124\pi$
$947$	−1.18922e7	−0.430912	−0.215456	+	0.976514i	$0.569124\pi$
$948$	0	0
$949$	4.29246e7	1.54718
$950$	−1.37656e7	−0.494864
$951$	0	0
$952$	7.26062e6	0.259646
$953$	4.76513e6	0.169958	0.0849791	−	0.996383i	$-0.472918\pi$
$953$	4.76513e6	0.169958	0.0849791	+	0.996383i	$0.472918\pi$
$954$	0	0
$955$	1.46096e7	0.518360
$956$	−3.96247e6	−0.140224
$957$	0	0
$958$	−1.21014e7	−0.426013
$959$	−7.57471e6	−0.265962
$960$	0	0
$961$	−2.18483e7	−0.763150
$962$	1.10188e6	0.0383881
$963$	0	0
$964$	−2.63900e6	−0.0914633
$965$	−2.58639e6	−0.0894079
$966$	0	0
$967$	4.42455e7	1.52161	0.760804	−	0.648981i	$-0.224805\pi$
$967$	4.42455e7	1.52161	0.760804	+	0.648981i	$0.224805\pi$
$968$	−2.08170e7	−0.714052
$969$	0	0
$970$	2.08270e6	0.0710717
$971$	4.94740e7	1.68395	0.841974	−	0.539518i	$-0.181393\pi$
$971$	4.94740e7	1.68395	0.841974	+	0.539518i	$0.181393\pi$
$972$	0	0
$973$	−5.82826e6	−0.197359
$974$	1.42595e6	0.0481624
$975$	0	0
$976$	1.98968e7	0.668588
$977$	2.59183e6	0.0868701	0.0434350	−	0.999056i	$-0.486170\pi$
$977$	2.59183e6	0.0868701	0.0434350	+	0.999056i	$0.486170\pi$
$978$	0	0
$979$	2.88322e7	0.961437
$980$	−355737.	−0.0118322
$981$	0	0
$982$	2.87221e7	0.950470
$983$	−4.28471e7	−1.41429	−0.707144	−	0.707069i	$-0.750017\pi$
$983$	−4.28471e7	−1.41429	−0.707144	+	0.707069i	$0.750017\pi$
$984$	0	0
$985$	5.37393e6	0.176482
$986$	−435596.	−0.0142690
$987$	0	0
$988$	−3.07485e6	−0.100215
$989$	−2.66120e7	−0.865140
$990$	0	0
$991$	−3.69273e7	−1.19444	−0.597219	−	0.802078i	$-0.703728\pi$
$991$	−3.69273e7	−1.19444	−0.597219	+	0.802078i	$0.703728\pi$
$992$	3.74791e6	0.120923
$993$	0	0
$994$	1.25003e7	0.401286
$995$	1.66024e7	0.531636
$996$	0	0
$997$	3.12556e7	0.995842	0.497921	−	0.867222i	$-0.334097\pi$
$997$	3.12556e7	0.995842	0.497921	+	0.867222i	$0.334097\pi$
$998$	−2.22095e7	−0.705849
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.a.g.1.1	✓	2
3.2	odd	2	inner	63.6.a.g.1.2	yes	2
4.3	odd	2		1008.6.a.bl.1.2		2
7.6	odd	2		441.6.a.p.1.1		2
12.11	even	2		1008.6.a.bl.1.1		2
21.20	even	2		441.6.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.a.g.1.1	✓	2	1.1	even	1	trivial
63.6.a.g.1.2	yes	2	3.2	odd	2	inner
441.6.a.p.1.1		2	7.6	odd	2
441.6.a.p.1.2		2	21.20	even	2
1008.6.a.bl.1.1		2	12.11	even	2
1008.6.a.bl.1.2		2	4.3	odd	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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q-5.29150 q^{2} -4.00000 q^{4} +37.0405 q^{5} -49.0000 q^{7} +190.494 q^{8} +O(q^{10})\)	\(q-5.29150 q^{2} -4.00000 q^{4} +37.0405 q^{5} -49.0000 q^{7} +190.494 q^{8} -196.000 q^{10} +227.535 q^{11} -518.000 q^{13} +259.284 q^{14} -880.000 q^{16} -777.851 q^{17} -1484.00 q^{19} -148.162 q^{20} -1204.00 q^{22} -3825.76 q^{23} -1753.00 q^{25} +2741.00 q^{26} +196.000 q^{28} -105.830 q^{29} -2604.00 q^{31} -1439.29 q^{32} +4116.00 q^{34} -1814.99 q^{35} +402.000 q^{37} +7852.59 q^{38} +7056.00 q^{40} +629.689 q^{41} +6956.00 q^{43} -910.138 q^{44} +20244.0 q^{46} +27335.9 q^{47} +2401.00 q^{49} +9276.00 q^{50} +2072.00 q^{52} -30510.8 q^{53} +8428.00 q^{55} -9334.21 q^{56} +560.000 q^{58} -45115.4 q^{59} -22610.0 q^{61} +13779.1 q^{62} +35776.0 q^{64} -19187.0 q^{65} -13124.0 q^{67} +3111.40 q^{68} +9604.00 q^{70} +48210.9 q^{71} -82866.0 q^{73} -2127.18 q^{74} +5936.00 q^{76} -11149.2 q^{77} -81112.0 q^{79} -32595.7 q^{80} -3332.00 q^{82} +66672.9 q^{83} -28812.0 q^{85} -36807.7 q^{86} +43344.0 q^{88} +126716. q^{89} +25382.0 q^{91} +15303.0 q^{92} -144648. q^{94} -54968.1 q^{95} -10626.0 q^{97} -12704.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 98 q^{7}+O(q^{10}) \) 2 * q - 8 * q^4 - 98 * q^7	\( 2 q - 8 q^{4} - 98 q^{7} - 392 q^{10} - 1036 q^{13} - 1760 q^{16} - 2968 q^{19} - 2408 q^{22} - 3506 q^{25} + 392 q^{28} - 5208 q^{31} + 8232 q^{34} + 804 q^{37} + 14112 q^{40} + 13912 q^{43} + 40488 q^{46} + 4802 q^{49} + 4144 q^{52} + 16856 q^{55} + 1120 q^{58} - 45220 q^{61} + 71552 q^{64} - 26248 q^{67} + 19208 q^{70} - 165732 q^{73} + 11872 q^{76} - 162224 q^{79} - 6664 q^{82} - 57624 q^{85} + 86688 q^{88} + 50764 q^{91} - 289296 q^{94} - 21252 q^{97}+O(q^{100}) \) 2 * q - 8 * q^4 - 98 * q^7 - 392 * q^10 - 1036 * q^13 - 1760 * q^16 - 2968 * q^19 - 2408 * q^22 - 3506 * q^25 + 392 * q^28 - 5208 * q^31 + 8232 * q^34 + 804 * q^37 + 14112 * q^40 + 13912 * q^43 + 40488 * q^46 + 4802 * q^49 + 4144 * q^52 + 16856 * q^55 + 1120 * q^58 - 45220 * q^61 + 71552 * q^64 - 26248 * q^67 + 19208 * q^70 - 165732 * q^73 + 11872 * q^76 - 162224 * q^79 - 6664 * q^82 - 57624 * q^85 + 86688 * q^88 + 50764 * q^91 - 289296 * q^94 - 21252 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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