LMFDB - Embedded newform 960.6.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,6,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	960.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:	$153.968467020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.61187$ of defining polynomial
Character	$\chi$	$=$	960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -25.0000 q^{5} +217.790 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -25.0000 q^{5} +217.790 q^{7} +81.0000 q^{9} -199.580 q^{11} -599.790 q^{13} +225.000 q^{15} +209.050 q^{17} +2835.27 q^{19} -1960.11 q^{21} +2093.37 q^{23} +625.000 q^{25} -729.000 q^{27} -326.840 q^{29} -3115.69 q^{31} +1796.22 q^{33} -5444.75 q^{35} +3917.29 q^{37} +5398.11 q^{39} -9314.86 q^{41} -7572.84 q^{43} -2025.00 q^{45} +22137.8 q^{47} +30625.5 q^{49} -1881.45 q^{51} -15796.2 q^{53} +4989.50 q^{55} -25517.4 q^{57} +32229.6 q^{59} +5745.42 q^{61} +17641.0 q^{63} +14994.7 q^{65} +6491.20 q^{67} -18840.3 q^{69} -11870.0 q^{71} +23414.0 q^{73} -5625.00 q^{75} -43466.5 q^{77} -15779.4 q^{79} +6561.00 q^{81} -101815. q^{83} -5226.25 q^{85} +2941.56 q^{87} -22865.0 q^{89} -130628. q^{91} +28041.2 q^{93} -70881.7 q^{95} +113814. q^{97} -16166.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 50 q^{5} + 112 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 - 50 * q^5 + 112 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} - 50 q^{5} + 112 q^{7} + 162 q^{9} + 248 q^{11} - 876 q^{13} + 450 q^{15} + 2036 q^{17} + 1464 q^{19} - 1008 q^{21} + 3216 q^{23} + 1250 q^{25} - 1458 q^{27} - 1948 q^{29} - 2672 q^{31} - 2232 q^{33} - 2800 q^{35} - 8668 q^{37} + 7884 q^{39} - 7628 q^{41} - 16440 q^{43} - 4050 q^{45} + 19360 q^{47} + 25010 q^{49} - 18324 q^{51} + 14356 q^{53} - 6200 q^{55} - 13176 q^{57} - 904 q^{59} - 20220 q^{61} + 9072 q^{63} + 21900 q^{65} - 12904 q^{67} - 28944 q^{69} + 40976 q^{71} + 59124 q^{73} - 11250 q^{75} - 90816 q^{77} - 107600 q^{79} + 13122 q^{81} - 122088 q^{83} - 50900 q^{85} + 17532 q^{87} + 103764 q^{89} - 101408 q^{91} + 24048 q^{93} - 36600 q^{95} - 24764 q^{97} + 20088 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 50 * q^5 + 112 * q^7 + 162 * q^9 + 248 * q^11 - 876 * q^13 + 450 * q^15 + 2036 * q^17 + 1464 * q^19 - 1008 * q^21 + 3216 * q^23 + 1250 * q^25 - 1458 * q^27 - 1948 * q^29 - 2672 * q^31 - 2232 * q^33 - 2800 * q^35 - 8668 * q^37 + 7884 * q^39 - 7628 * q^41 - 16440 * q^43 - 4050 * q^45 + 19360 * q^47 + 25010 * q^49 - 18324 * q^51 + 14356 * q^53 - 6200 * q^55 - 13176 * q^57 - 904 * q^59 - 20220 * q^61 + 9072 * q^63 + 21900 * q^65 - 12904 * q^67 - 28944 * q^69 + 40976 * q^71 + 59124 * q^73 - 11250 * q^75 - 90816 * q^77 - 107600 * q^79 + 13122 * q^81 - 122088 * q^83 - 50900 * q^85 + 17532 * q^87 + 103764 * q^89 - 101408 * q^91 + 24048 * q^93 - 36600 * q^95 - 24764 * q^97 + 20088 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	217.790	1.67994	0.839968	−	0.542636i	$-0.182574\pi$
$7$	217.790	1.67994	0.839968	+	0.542636i	$0.182574\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−199.580	−0.497319	−0.248660	−	0.968591i	$-0.579990\pi$
$11$	−199.580	−0.497319	−0.248660	+	0.968591i	$0.579990\pi$
$12$	0	0
$13$	−599.790	−0.984330	−0.492165	−	0.870502i	$-0.663794\pi$
$13$	−599.790	−0.984330	−0.492165	+	0.870502i	$0.663794\pi$
$14$	0	0
$15$	225.000	0.258199
$16$	0	0
$17$	209.050	0.175440	0.0877199	−	0.996145i	$-0.472042\pi$
$17$	209.050	0.175440	0.0877199	+	0.996145i	$0.472042\pi$
$18$	0	0
$19$	2835.27	1.80182	0.900908	−	0.434010i	$-0.142902\pi$
$19$	2835.27	1.80182	0.900908	+	0.434010i	$0.142902\pi$
$20$	0	0
$21$	−1960.11	−0.969912
$22$	0	0
$23$	2093.37	0.825138	0.412569	−	0.910926i	$-0.364632\pi$
$23$	2093.37	0.825138	0.412569	+	0.910926i	$0.364632\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−326.840	−0.0721673	−0.0360836	−	0.999349i	$-0.511488\pi$
$29$	−326.840	−0.0721673	−0.0360836	+	0.999349i	$0.511488\pi$
$30$	0	0
$31$	−3115.69	−0.582304	−0.291152	−	0.956677i	$-0.594039\pi$
$31$	−3115.69	−0.582304	−0.291152	+	0.956677i	$0.594039\pi$
$32$	0	0
$33$	1796.22	0.287127
$34$	0	0
$35$	−5444.75	−0.751290
$36$	0	0
$37$	3917.29	0.470415	0.235208	−	0.971945i	$-0.424423\pi$
$37$	3917.29	0.470415	0.235208	+	0.971945i	$0.424423\pi$
$38$	0	0
$39$	5398.11	0.568303
$40$	0	0
$41$	−9314.86	−0.865400	−0.432700	−	0.901538i	$-0.642439\pi$
$41$	−9314.86	−0.865400	−0.432700	+	0.901538i	$0.642439\pi$
$42$	0	0
$43$	−7572.84	−0.624579	−0.312290	−	0.949987i	$-0.601096\pi$
$43$	−7572.84	−0.624579	−0.312290	+	0.949987i	$0.601096\pi$
$44$	0	0
$45$	−2025.00	−0.149071
$46$	0	0
$47$	22137.8	1.46181	0.730904	−	0.682480i	$-0.239099\pi$
$47$	22137.8	1.46181	0.730904	+	0.682480i	$0.239099\pi$
$48$	0	0
$49$	30625.5	1.82219
$50$	0	0
$51$	−1881.45	−0.101290
$52$	0	0
$53$	−15796.2	−0.772436	−0.386218	−	0.922408i	$-0.626219\pi$
$53$	−15796.2	−0.772436	−0.386218	+	0.922408i	$0.626219\pi$
$54$	0	0
$55$	4989.50	0.222408
$56$	0	0
$57$	−25517.4	−1.04028
$58$	0	0
$59$	32229.6	1.20538	0.602691	−	0.797975i	$-0.294095\pi$
$59$	32229.6	1.20538	0.602691	+	0.797975i	$0.294095\pi$
$60$	0	0
$61$	5745.42	0.197696	0.0988478	−	0.995103i	$-0.468484\pi$
$61$	5745.42	0.197696	0.0988478	+	0.995103i	$0.468484\pi$
$62$	0	0
$63$	17641.0	0.559979
$64$	0	0
$65$	14994.7	0.440206
$66$	0	0
$67$	6491.20	0.176660	0.0883299	−	0.996091i	$-0.471847\pi$
$67$	6491.20	0.176660	0.0883299	+	0.996091i	$0.471847\pi$
$68$	0	0
$69$	−18840.3	−0.476393
$70$	0	0
$71$	−11870.0	−0.279450	−0.139725	−	0.990190i	$-0.544622\pi$
$71$	−11870.0	−0.279450	−0.139725	+	0.990190i	$0.544622\pi$
$72$	0	0
$73$	23414.0	0.514243	0.257121	−	0.966379i	$-0.417226\pi$
$73$	23414.0	0.514243	0.257121	+	0.966379i	$0.417226\pi$
$74$	0	0
$75$	−5625.00	−0.115470
$76$	0	0
$77$	−43466.5	−0.835465
$78$	0	0
$79$	−15779.4	−0.284460	−0.142230	−	0.989834i	$-0.545427\pi$
$79$	−15779.4	−0.284460	−0.142230	+	0.989834i	$0.545427\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−101815.	−1.62225	−0.811123	−	0.584875i	$-0.801143\pi$
$83$	−101815.	−1.62225	−0.811123	+	0.584875i	$0.801143\pi$
$84$	0	0
$85$	−5226.25	−0.0784590
$86$	0	0
$87$	2941.56	0.0416658
$88$	0	0
$89$	−22865.0	−0.305982	−0.152991	−	0.988228i	$-0.548891\pi$
$89$	−22865.0	−0.305982	−0.152991	+	0.988228i	$0.548891\pi$
$90$	0	0
$91$	−130628.	−1.65361
$92$	0	0
$93$	28041.2	0.336194
$94$	0	0
$95$	−70881.7	−0.805797
$96$	0	0
$97$	113814.	1.22819	0.614097	−	0.789230i	$-0.289520\pi$
$97$	113814.	1.22819	0.614097	+	0.789230i	$0.289520\pi$
$98$	0	0
$99$	−16166.0	−0.165773
$100$	0	0
$101$	−88036.9	−0.858739	−0.429370	−	0.903129i	$-0.641264\pi$
$101$	−88036.9	−0.858739	−0.429370	+	0.903129i	$0.641264\pi$
$102$	0	0
$103$	−52836.4	−0.490727	−0.245364	−	0.969431i	$-0.578907\pi$
$103$	−52836.4	−0.490727	−0.245364	+	0.969431i	$0.578907\pi$
$104$	0	0
$105$	49002.7	0.433758
$106$	0	0
$107$	−141097.	−1.19141	−0.595703	−	0.803205i	$-0.703126\pi$
$107$	−141097.	−1.19141	−0.595703	+	0.803205i	$0.703126\pi$
$108$	0	0
$109$	14517.3	0.117036	0.0585179	−	0.998286i	$-0.481363\pi$
$109$	14517.3	0.117036	0.0585179	+	0.998286i	$0.481363\pi$
$110$	0	0
$111$	−35255.6	−0.271594
$112$	0	0
$113$	−29499.2	−0.217327	−0.108664	−	0.994079i	$-0.534657\pi$
$113$	−29499.2	−0.217327	−0.108664	+	0.994079i	$0.534657\pi$
$114$	0	0
$115$	−52334.2	−0.369013
$116$	0	0
$117$	−48583.0	−0.328110
$118$	0	0
$119$	45529.0	0.294728
$120$	0	0
$121$	−121219.	−0.752674
$122$	0	0
$123$	83833.7	0.499639
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−197789.	−1.08816	−0.544081	−	0.839033i	$-0.683121\pi$
$127$	−197789.	−1.08816	−0.544081	+	0.839033i	$0.683121\pi$
$128$	0	0
$129$	68155.6	0.360601
$130$	0	0
$131$	−141216.	−0.718959	−0.359480	−	0.933153i	$-0.617046\pi$
$131$	−141216.	−0.718959	−0.359480	+	0.933153i	$0.617046\pi$
$132$	0	0
$133$	617493.	3.02694
$134$	0	0
$135$	18225.0	0.0860663
$136$	0	0
$137$	159983.	0.728235	0.364117	−	0.931353i	$-0.381371\pi$
$137$	159983.	0.728235	0.364117	+	0.931353i	$0.381371\pi$
$138$	0	0
$139$	319007.	1.40044	0.700219	−	0.713928i	$-0.253086\pi$
$139$	319007.	1.40044	0.700219	+	0.713928i	$0.253086\pi$
$140$	0	0
$141$	−199240.	−0.843975
$142$	0	0
$143$	119706.	0.489526
$144$	0	0
$145$	8171.00	0.0322742
$146$	0	0
$147$	−275629.	−1.05204
$148$	0	0
$149$	398964.	1.47220	0.736102	−	0.676870i	$-0.236664\pi$
$149$	398964.	1.47220	0.736102	+	0.676870i	$0.236664\pi$
$150$	0	0
$151$	166591.	0.594577	0.297289	−	0.954788i	$-0.403918\pi$
$151$	166591.	0.594577	0.297289	+	0.954788i	$0.403918\pi$
$152$	0	0
$153$	16933.1	0.0584799
$154$	0	0
$155$	77892.2	0.260414
$156$	0	0
$157$	256463.	0.830378	0.415189	−	0.909735i	$-0.363715\pi$
$157$	256463.	0.830378	0.415189	+	0.909735i	$0.363715\pi$
$158$	0	0
$159$	142166.	0.445966
$160$	0	0
$161$	455915.	1.38618
$162$	0	0
$163$	654000.	1.92801	0.964004	−	0.265887i	$-0.0856650\pi$
$163$	654000.	1.92801	0.964004	+	0.265887i	$0.0856650\pi$
$164$	0	0
$165$	−44905.5	−0.128407
$166$	0	0
$167$	−667196.	−1.85124	−0.925619	−	0.378456i	$-0.876455\pi$
$167$	−667196.	−1.85124	−0.925619	+	0.378456i	$0.876455\pi$
$168$	0	0
$169$	−11545.0	−0.0310940
$170$	0	0
$171$	229657.	0.600605
$172$	0	0
$173$	−381680.	−0.969582	−0.484791	−	0.874630i	$-0.661104\pi$
$173$	−381680.	−0.969582	−0.484791	+	0.874630i	$0.661104\pi$
$174$	0	0
$175$	136119.	0.335987
$176$	0	0
$177$	−290066.	−0.695927
$178$	0	0
$179$	614936.	1.43449	0.717244	−	0.696822i	$-0.245403\pi$
$179$	614936.	1.43449	0.717244	+	0.696822i	$0.245403\pi$
$180$	0	0
$181$	611099.	1.38648	0.693242	−	0.720705i	$-0.256182\pi$
$181$	611099.	1.38648	0.693242	+	0.720705i	$0.256182\pi$
$182$	0	0
$183$	−51708.8	−0.114140
$184$	0	0
$185$	−97932.2	−0.210376
$186$	0	0
$187$	−41722.2	−0.0872496
$188$	0	0
$189$	−158769.	−0.323304
$190$	0	0
$191$	151563.	0.300615	0.150307	−	0.988639i	$-0.451974\pi$
$191$	151563.	0.300615	0.150307	+	0.988639i	$0.451974\pi$
$192$	0	0
$193$	669868.	1.29448	0.647241	−	0.762285i	$-0.275923\pi$
$193$	669868.	1.29448	0.647241	+	0.762285i	$0.275923\pi$
$194$	0	0
$195$	−134953.	−0.254153
$196$	0	0
$197$	843520.	1.54857	0.774283	−	0.632839i	$-0.218111\pi$
$197$	843520.	1.54857	0.774283	+	0.632839i	$0.218111\pi$
$198$	0	0
$199$	99215.8	0.177602	0.0888010	−	0.996049i	$-0.471696\pi$
$199$	99215.8	0.177602	0.0888010	+	0.996049i	$0.471696\pi$
$200$	0	0
$201$	−58420.8	−0.101995
$202$	0	0
$203$	−71182.5	−0.121236
$204$	0	0
$205$	232871.	0.387018
$206$	0	0
$207$	169563.	0.275046
$208$	0	0
$209$	−565863.	−0.896078
$210$	0	0
$211$	954803.	1.47641	0.738206	−	0.674575i	$-0.235673\pi$
$211$	954803.	1.47641	0.738206	+	0.674575i	$0.235673\pi$
$212$	0	0
$213$	106830.	0.161341
$214$	0	0
$215$	189321.	0.279320
$216$	0	0
$217$	−678566.	−0.978234
$218$	0	0
$219$	−210726.	−0.296898
$220$	0	0
$221$	−125386.	−0.172691
$222$	0	0
$223$	99120.0	0.133475	0.0667374	−	0.997771i	$-0.478741\pi$
$223$	99120.0	0.133475	0.0667374	+	0.997771i	$0.478741\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	−489116.	−0.630009	−0.315005	−	0.949090i	$-0.602006\pi$
$227$	−489116.	−0.630009	−0.315005	+	0.949090i	$0.602006\pi$
$228$	0	0
$229$	77461.9	0.0976111	0.0488056	−	0.998808i	$-0.484459\pi$
$229$	77461.9	0.0976111	0.0488056	+	0.998808i	$0.484459\pi$
$230$	0	0
$231$	391199.	0.482356
$232$	0	0
$233$	−128093.	−0.154574	−0.0772868	−	0.997009i	$-0.524626\pi$
$233$	−128093.	−0.154574	−0.0772868	+	0.997009i	$0.524626\pi$
$234$	0	0
$235$	−553446.	−0.653740
$236$	0	0
$237$	142014.	0.164233
$238$	0	0
$239$	1.03342e6	1.17026	0.585129	−	0.810940i	$-0.301044\pi$
$239$	1.03342e6	1.17026	0.585129	+	0.810940i	$0.301044\pi$
$240$	0	0
$241$	491546.	0.545157	0.272578	−	0.962134i	$-0.412124\pi$
$241$	491546.	0.545157	0.272578	+	0.962134i	$0.412124\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−765637.	−0.814906
$246$	0	0
$247$	−1.70057e6	−1.77358
$248$	0	0
$249$	916336.	0.936604
$250$	0	0
$251$	−1.23517e6	−1.23750	−0.618748	−	0.785590i	$-0.712360\pi$
$251$	−1.23517e6	−1.23750	−0.618748	+	0.785590i	$0.712360\pi$
$252$	0	0
$253$	−417795.	−0.410357
$254$	0	0
$255$	47036.3	0.0452983
$256$	0	0
$257$	1.90909e6	1.80299	0.901495	−	0.432789i	$-0.142470\pi$
$257$	1.90909e6	1.80299	0.901495	+	0.432789i	$0.142470\pi$
$258$	0	0
$259$	853146.	0.790268
$260$	0	0
$261$	−26474.0	−0.0240558
$262$	0	0
$263$	614059.	0.547420	0.273710	−	0.961812i	$-0.411749\pi$
$263$	614059.	0.547420	0.273710	+	0.961812i	$0.411749\pi$
$264$	0	0
$265$	394904.	0.345444
$266$	0	0
$267$	205785.	0.176659
$268$	0	0
$269$	2.05806e6	1.73411	0.867055	−	0.498213i	$-0.166010\pi$
$269$	2.05806e6	1.73411	0.867055	+	0.498213i	$0.166010\pi$
$270$	0	0
$271$	−178987.	−0.148046	−0.0740231	−	0.997257i	$-0.523584\pi$
$271$	−178987.	−0.148046	−0.0740231	+	0.997257i	$0.523584\pi$
$272$	0	0
$273$	1.17565e6	0.954713
$274$	0	0
$275$	−124737.	−0.0994638
$276$	0	0
$277$	1.89158e6	1.48124	0.740619	−	0.671925i	$-0.234532\pi$
$277$	1.89158e6	1.48124	0.740619	+	0.671925i	$0.234532\pi$
$278$	0	0
$279$	−252371.	−0.194101
$280$	0	0
$281$	2.19927e6	1.66154	0.830772	−	0.556613i	$-0.187899\pi$
$281$	2.19927e6	1.66154	0.830772	+	0.556613i	$0.187899\pi$
$282$	0	0
$283$	804296.	0.596967	0.298483	−	0.954415i	$-0.403519\pi$
$283$	804296.	0.596967	0.298483	+	0.954415i	$0.403519\pi$
$284$	0	0
$285$	637936.	0.465227
$286$	0	0
$287$	−2.02868e6	−1.45382
$288$	0	0
$289$	−1.37616e6	−0.969221
$290$	0	0
$291$	−1.02433e6	−0.709098
$292$	0	0
$293$	435069.	0.296067	0.148033	−	0.988982i	$-0.452706\pi$
$293$	435069.	0.296067	0.148033	+	0.988982i	$0.452706\pi$
$294$	0	0
$295$	−805739.	−0.539063
$296$	0	0
$297$	145494.	0.0957091
$298$	0	0
$299$	−1.25558e6	−0.812208
$300$	0	0
$301$	−1.64929e6	−1.04925
$302$	0	0
$303$	792332.	0.495793
$304$	0	0
$305$	−143635.	−0.0884122
$306$	0	0
$307$	379288.	0.229680	0.114840	−	0.993384i	$-0.463364\pi$
$307$	379288.	0.229680	0.114840	+	0.993384i	$0.463364\pi$
$308$	0	0
$309$	475528.	0.283322
$310$	0	0
$311$	1.71677e6	1.00649	0.503246	−	0.864143i	$-0.332139\pi$
$311$	1.71677e6	1.00649	0.503246	+	0.864143i	$0.332139\pi$
$312$	0	0
$313$	−1.46409e6	−0.844711	−0.422356	−	0.906430i	$-0.638797\pi$
$313$	−1.46409e6	−0.844711	−0.422356	+	0.906430i	$0.638797\pi$
$314$	0	0
$315$	−441025.	−0.250430
$316$	0	0
$317$	−90285.6	−0.0504627	−0.0252313	−	0.999682i	$-0.508032\pi$
$317$	−90285.6	−0.0504627	−0.0252313	+	0.999682i	$0.508032\pi$
$318$	0	0
$319$	65230.7	0.0358902
$320$	0	0
$321$	1.26988e6	0.687858
$322$	0	0
$323$	592713.	0.316110
$324$	0	0
$325$	−374869.	−0.196866
$326$	0	0
$327$	−130656.	−0.0675707
$328$	0	0
$329$	4.82140e6	2.45574
$330$	0	0
$331$	−3.37162e6	−1.69148	−0.845742	−	0.533591i	$-0.820842\pi$
$331$	−3.37162e6	−1.69148	−0.845742	+	0.533591i	$0.820842\pi$
$332$	0	0
$333$	317300.	0.156805
$334$	0	0
$335$	−162280.	−0.0790047
$336$	0	0
$337$	−2.09461e6	−1.00468	−0.502341	−	0.864669i	$-0.667528\pi$
$337$	−2.09461e6	−1.00468	−0.502341	+	0.864669i	$0.667528\pi$
$338$	0	0
$339$	265493.	0.125474
$340$	0	0
$341$	621829.	0.289591
$342$	0	0
$343$	3.00953e6	1.38122
$344$	0	0
$345$	471008.	0.213050
$346$	0	0
$347$	3.42706e6	1.52791	0.763955	−	0.645269i	$-0.223255\pi$
$347$	3.42706e6	1.52791	0.763955	+	0.645269i	$0.223255\pi$
$348$	0	0
$349$	−101956.	−0.0448074	−0.0224037	−	0.999749i	$-0.507132\pi$
$349$	−101956.	−0.0448074	−0.0224037	+	0.999749i	$0.507132\pi$
$350$	0	0
$351$	437247.	0.189434
$352$	0	0
$353$	−704751.	−0.301022	−0.150511	−	0.988608i	$-0.548092\pi$
$353$	−704751.	−0.301022	−0.150511	+	0.988608i	$0.548092\pi$
$354$	0	0
$355$	296750.	0.124974
$356$	0	0
$357$	−409761.	−0.170161
$358$	0	0
$359$	2.50896e6	1.02744	0.513721	−	0.857957i	$-0.328267\pi$
$359$	2.50896e6	1.02744	0.513721	+	0.857957i	$0.328267\pi$
$360$	0	0
$361$	5.56266e6	2.24654
$362$	0	0
$363$	1.09097e6	0.434556
$364$	0	0
$365$	−585350.	−0.229976
$366$	0	0
$367$	4.06900e6	1.57697	0.788483	−	0.615057i	$-0.210867\pi$
$367$	4.06900e6	1.57697	0.788483	+	0.615057i	$0.210867\pi$
$368$	0	0
$369$	−754504.	−0.288467
$370$	0	0
$371$	−3.44025e6	−1.29764
$372$	0	0
$373$	957594.	0.356377	0.178188	−	0.983996i	$-0.442976\pi$
$373$	957594.	0.356377	0.178188	+	0.983996i	$0.442976\pi$
$374$	0	0
$375$	140625.	0.0516398
$376$	0	0
$377$	196035.	0.0710364
$378$	0	0
$379$	328048.	0.117311	0.0586556	−	0.998278i	$-0.481319\pi$
$379$	328048.	0.117311	0.0586556	+	0.998278i	$0.481319\pi$
$380$	0	0
$381$	1.78010e6	0.628250
$382$	0	0
$383$	−3.22117e6	−1.12206	−0.561031	−	0.827795i	$-0.689595\pi$
$383$	−3.22117e6	−1.12206	−0.561031	+	0.827795i	$0.689595\pi$
$384$	0	0
$385$	1.08666e6	0.373631
$386$	0	0
$387$	−613400.	−0.208193
$388$	0	0
$389$	−2.38697e6	−0.799785	−0.399893	−	0.916562i	$-0.630953\pi$
$389$	−2.38697e6	−0.799785	−0.399893	+	0.916562i	$0.630953\pi$
$390$	0	0
$391$	437619.	0.144762
$392$	0	0
$393$	1.27094e6	0.415091
$394$	0	0
$395$	394484.	0.127214
$396$	0	0
$397$	3.13956e6	0.999751	0.499876	−	0.866097i	$-0.333379\pi$
$397$	3.13956e6	0.999751	0.499876	+	0.866097i	$0.333379\pi$
$398$	0	0
$399$	−5.55744e6	−1.74760
$400$	0	0
$401$	−2.49379e6	−0.774459	−0.387229	−	0.921983i	$-0.626568\pi$
$401$	−2.49379e6	−0.774459	−0.387229	+	0.921983i	$0.626568\pi$
$402$	0	0
$403$	1.86876e6	0.573180
$404$	0	0
$405$	−164025.	−0.0496904
$406$	0	0
$407$	−781813.	−0.233947
$408$	0	0
$409$	114720.	0.0339103	0.0169552	−	0.999856i	$-0.494603\pi$
$409$	114720.	0.0339103	0.0169552	+	0.999856i	$0.494603\pi$
$410$	0	0
$411$	−1.43984e6	−0.420446
$412$	0	0
$413$	7.01928e6	2.02496
$414$	0	0
$415$	2.54538e6	0.725491
$416$	0	0
$417$	−2.87107e6	−0.808543
$418$	0	0
$419$	2.25805e6	0.628344	0.314172	−	0.949366i	$-0.398273\pi$
$419$	2.25805e6	0.628344	0.314172	+	0.949366i	$0.398273\pi$
$420$	0	0
$421$	352348.	0.0968873	0.0484436	−	0.998826i	$-0.484574\pi$
$421$	352348.	0.0968873	0.0484436	+	0.998826i	$0.484574\pi$
$422$	0	0
$423$	1.79316e6	0.487269
$424$	0	0
$425$	130656.	0.0350880
$426$	0	0
$427$	1.25129e6	0.332116
$428$	0	0
$429$	−1.07735e6	−0.282628
$430$	0	0
$431$	−2.43169e6	−0.630543	−0.315272	−	0.949001i	$-0.602096\pi$
$431$	−2.43169e6	−0.630543	−0.315272	+	0.949001i	$0.602096\pi$
$432$	0	0
$433$	5.37990e6	1.37897	0.689485	−	0.724300i	$-0.257837\pi$
$433$	5.37990e6	1.37897	0.689485	+	0.724300i	$0.257837\pi$
$434$	0	0
$435$	−73539.0	−0.0186335
$436$	0	0
$437$	5.93527e6	1.48675
$438$	0	0
$439$	−6.85919e6	−1.69868	−0.849340	−	0.527847i	$-0.823000\pi$
$439$	−6.85919e6	−1.69868	−0.849340	+	0.527847i	$0.823000\pi$
$440$	0	0
$441$	2.48066e6	0.607395
$442$	0	0
$443$	−568384.	−0.137605	−0.0688023	−	0.997630i	$-0.521918\pi$
$443$	−568384.	−0.137605	−0.0688023	+	0.997630i	$0.521918\pi$
$444$	0	0
$445$	571624.	0.136839
$446$	0	0
$447$	−3.59068e6	−0.849978
$448$	0	0
$449$	−4.12355e6	−0.965285	−0.482642	−	0.875818i	$-0.660323\pi$
$449$	−4.12355e6	−0.965285	−0.482642	+	0.875818i	$0.660323\pi$
$450$	0	0
$451$	1.85906e6	0.430380
$452$	0	0
$453$	−1.49932e6	−0.343279
$454$	0	0
$455$	3.26571e6	0.739518
$456$	0	0
$457$	1.48425e6	0.332443	0.166221	−	0.986088i	$-0.446843\pi$
$457$	1.48425e6	0.332443	0.166221	+	0.986088i	$0.446843\pi$
$458$	0	0
$459$	−152397.	−0.0337634
$460$	0	0
$461$	−6.49347e6	−1.42306	−0.711532	−	0.702654i	$-0.751998\pi$
$461$	−6.49347e6	−1.42306	−0.711532	+	0.702654i	$0.751998\pi$
$462$	0	0
$463$	1.85239e6	0.401586	0.200793	−	0.979634i	$-0.435648\pi$
$463$	1.85239e6	0.401586	0.200793	+	0.979634i	$0.435648\pi$
$464$	0	0
$465$	−701030.	−0.150350
$466$	0	0
$467$	−1.64352e6	−0.348724	−0.174362	−	0.984682i	$-0.555786\pi$
$467$	−1.64352e6	−0.348724	−0.174362	+	0.984682i	$0.555786\pi$
$468$	0	0
$469$	1.41372e6	0.296777
$470$	0	0
$471$	−2.30817e6	−0.479419
$472$	0	0
$473$	1.51139e6	0.310615
$474$	0	0
$475$	1.77204e6	0.360363
$476$	0	0
$477$	−1.27949e6	−0.257479
$478$	0	0
$479$	−7.72002e6	−1.53737	−0.768687	−	0.639625i	$-0.779090\pi$
$479$	−7.72002e6	−1.53737	−0.768687	+	0.639625i	$0.779090\pi$
$480$	0	0
$481$	−2.34955e6	−0.463044
$482$	0	0
$483$	−4.10324e6	−0.800311
$484$	0	0
$485$	−2.84535e6	−0.549265
$486$	0	0
$487$	−4.10435e6	−0.784191	−0.392096	−	0.919924i	$-0.628250\pi$
$487$	−4.10435e6	−0.784191	−0.392096	+	0.919924i	$0.628250\pi$
$488$	0	0
$489$	−5.88600e6	−1.11314
$490$	0	0
$491$	−450180.	−0.0842719	−0.0421360	−	0.999112i	$-0.513416\pi$
$491$	−450180.	−0.0842719	−0.0421360	+	0.999112i	$0.513416\pi$
$492$	0	0
$493$	−68325.9	−0.0126610
$494$	0	0
$495$	404149.	0.0741360
$496$	0	0
$497$	−2.58517e6	−0.469459
$498$	0	0
$499$	−2.01251e6	−0.361815	−0.180907	−	0.983500i	$-0.557903\pi$
$499$	−2.01251e6	−0.361815	−0.180907	+	0.983500i	$0.557903\pi$
$500$	0	0
$501$	6.00477e6	1.06881
$502$	0	0
$503$	−2.74624e6	−0.483970	−0.241985	−	0.970280i	$-0.577798\pi$
$503$	−2.74624e6	−0.483970	−0.241985	+	0.970280i	$0.577798\pi$
$504$	0	0
$505$	2.20092e6	0.384040
$506$	0	0
$507$	103905.	0.0179521
$508$	0	0
$509$	4.47345e6	0.765328	0.382664	−	0.923888i	$-0.375007\pi$
$509$	4.47345e6	0.765328	0.382664	+	0.923888i	$0.375007\pi$
$510$	0	0
$511$	5.09933e6	0.863895
$512$	0	0
$513$	−2.06691e6	−0.346760
$514$	0	0
$515$	1.32091e6	0.219460
$516$	0	0
$517$	−4.41827e6	−0.726985
$518$	0	0
$519$	3.43512e6	0.559789
$520$	0	0
$521$	−4.20576e6	−0.678814	−0.339407	−	0.940640i	$-0.610226\pi$
$521$	−4.20576e6	−0.678814	−0.339407	+	0.940640i	$0.610226\pi$
$522$	0	0
$523$	−3.27485e6	−0.523524	−0.261762	−	0.965132i	$-0.584304\pi$
$523$	−3.27485e6	−0.523524	−0.261762	+	0.965132i	$0.584304\pi$
$524$	0	0
$525$	−1.22507e6	−0.193982
$526$	0	0
$527$	−651335.	−0.102159
$528$	0	0
$529$	−2.05415e6	−0.319148
$530$	0	0
$531$	2.61060e6	0.401794
$532$	0	0
$533$	5.58696e6	0.851839
$534$	0	0
$535$	3.52743e6	0.532813
$536$	0	0
$537$	−5.53442e6	−0.828202
$538$	0	0
$539$	−6.11223e6	−0.906208
$540$	0	0
$541$	4.55984e6	0.669817	0.334909	−	0.942251i	$-0.391295\pi$
$541$	4.55984e6	0.669817	0.334909	+	0.942251i	$0.391295\pi$
$542$	0	0
$543$	−5.49989e6	−0.800487
$544$	0	0
$545$	−362932.	−0.0523400
$546$	0	0
$547$	1.12208e7	1.60345	0.801723	−	0.597695i	$-0.203917\pi$
$547$	1.12208e7	1.60345	0.801723	+	0.597695i	$0.203917\pi$
$548$	0	0
$549$	465379.	0.0658985
$550$	0	0
$551$	−926680.	−0.130032
$552$	0	0
$553$	−3.43659e6	−0.477875
$554$	0	0
$555$	881390.	0.121461
$556$	0	0
$557$	−1.90478e6	−0.260140	−0.130070	−	0.991505i	$-0.541520\pi$
$557$	−1.90478e6	−0.260140	−0.130070	+	0.991505i	$0.541520\pi$
$558$	0	0
$559$	4.54211e6	0.614792
$560$	0	0
$561$	375500.	0.0503736
$562$	0	0
$563$	8.81706e6	1.17234	0.586169	−	0.810189i	$-0.300635\pi$
$563$	8.81706e6	1.17234	0.586169	+	0.810189i	$0.300635\pi$
$564$	0	0
$565$	737481.	0.0971918
$566$	0	0
$567$	1.42892e6	0.186660
$568$	0	0
$569$	−8.58116e6	−1.11113	−0.555565	−	0.831473i	$-0.687498\pi$
$569$	−8.58116e6	−1.11113	−0.555565	+	0.831473i	$0.687498\pi$
$570$	0	0
$571$	−4.01627e6	−0.515504	−0.257752	−	0.966211i	$-0.582982\pi$
$571$	−4.01627e6	−0.515504	−0.257752	+	0.966211i	$0.582982\pi$
$572$	0	0
$573$	−1.36407e6	−0.173560
$574$	0	0
$575$	1.30836e6	0.165028
$576$	0	0
$577$	2.73670e6	0.342206	0.171103	−	0.985253i	$-0.445267\pi$
$577$	2.73670e6	0.342206	0.171103	+	0.985253i	$0.445267\pi$
$578$	0	0
$579$	−6.02881e6	−0.747370
$580$	0	0
$581$	−2.21743e7	−2.72527
$582$	0	0
$583$	3.15260e6	0.384147
$584$	0	0
$585$	1.21457e6	0.146735
$586$	0	0
$587$	1.12479e6	0.134734	0.0673670	−	0.997728i	$-0.478540\pi$
$587$	1.12479e6	0.134734	0.0673670	+	0.997728i	$0.478540\pi$
$588$	0	0
$589$	−8.83382e6	−1.04921
$590$	0	0
$591$	−7.59168e6	−0.894065
$592$	0	0
$593$	6.61107e6	0.772032	0.386016	−	0.922492i	$-0.373851\pi$
$593$	6.61107e6	0.772032	0.386016	+	0.922492i	$0.373851\pi$
$594$	0	0
$595$	−1.13823e6	−0.131806
$596$	0	0
$597$	−892942.	−0.102539
$598$	0	0
$599$	−3.78429e6	−0.430941	−0.215470	−	0.976510i	$-0.569128\pi$
$599$	−3.78429e6	−0.430941	−0.215470	+	0.976510i	$0.569128\pi$
$600$	0	0
$601$	1.36699e7	1.54376	0.771878	−	0.635771i	$-0.219318\pi$
$601$	1.36699e7	1.54376	0.771878	+	0.635771i	$0.219318\pi$
$602$	0	0
$603$	525787.	0.0588866
$604$	0	0
$605$	3.03047e6	0.336606
$606$	0	0
$607$	1.05825e7	1.16578	0.582890	−	0.812551i	$-0.301922\pi$
$607$	1.05825e7	1.16578	0.582890	+	0.812551i	$0.301922\pi$
$608$	0	0
$609$	640642.	0.0699959
$610$	0	0
$611$	−1.32780e7	−1.43890
$612$	0	0
$613$	−1.49493e7	−1.60683	−0.803414	−	0.595421i	$-0.796985\pi$
$613$	−1.49493e7	−1.60683	−0.803414	+	0.595421i	$0.796985\pi$
$614$	0	0
$615$	−2.09584e6	−0.223445
$616$	0	0
$617$	8.44272e6	0.892832	0.446416	−	0.894826i	$-0.352700\pi$
$617$	8.44272e6	0.892832	0.446416	+	0.894826i	$0.352700\pi$
$618$	0	0
$619$	−7.49113e6	−0.785815	−0.392908	−	0.919578i	$-0.628531\pi$
$619$	−7.49113e6	−0.785815	−0.392908	+	0.919578i	$0.628531\pi$
$620$	0	0
$621$	−1.52607e6	−0.158798
$622$	0	0
$623$	−4.97976e6	−0.514030
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	5.09277e6	0.517351
$628$	0	0
$629$	818910.	0.0825295
$630$	0	0
$631$	9.74348e6	0.974184	0.487092	−	0.873351i	$-0.338058\pi$
$631$	9.74348e6	0.974184	0.487092	+	0.873351i	$0.338058\pi$
$632$	0	0
$633$	−8.59323e6	−0.852407
$634$	0	0
$635$	4.94473e6	0.486641
$636$	0	0
$637$	−1.83689e7	−1.79363
$638$	0	0
$639$	−961470.	−0.0931501
$640$	0	0
$641$	−1.30814e7	−1.25750	−0.628750	−	0.777608i	$-0.716433\pi$
$641$	−1.30814e7	−1.25750	−0.628750	+	0.777608i	$0.716433\pi$
$642$	0	0
$643$	6.93389e6	0.661377	0.330689	−	0.943740i	$-0.392719\pi$
$643$	6.93389e6	0.661377	0.330689	+	0.943740i	$0.392719\pi$
$644$	0	0
$645$	−1.70389e6	−0.161266
$646$	0	0
$647$	1.98447e7	1.86373	0.931866	−	0.362801i	$-0.118180\pi$
$647$	1.98447e7	1.86373	0.931866	+	0.362801i	$0.118180\pi$
$648$	0	0
$649$	−6.43238e6	−0.599459
$650$	0	0
$651$	6.10709e6	0.564784
$652$	0	0
$653$	5.44251e6	0.499478	0.249739	−	0.968313i	$-0.419655\pi$
$653$	5.44251e6	0.499478	0.249739	+	0.968313i	$0.419655\pi$
$654$	0	0
$655$	3.53039e6	0.321528
$656$	0	0
$657$	1.89653e6	0.171414
$658$	0	0
$659$	1.32110e7	1.18501	0.592505	−	0.805567i	$-0.298139\pi$
$659$	1.32110e7	1.18501	0.592505	+	0.805567i	$0.298139\pi$
$660$	0	0
$661$	1.82594e7	1.62549	0.812744	−	0.582621i	$-0.197973\pi$
$661$	1.82594e7	1.62549	0.812744	+	0.582621i	$0.197973\pi$
$662$	0	0
$663$	1.12848e6	0.0997030
$664$	0	0
$665$	−1.54373e7	−1.35369
$666$	0	0
$667$	−684197.	−0.0595479
$668$	0	0
$669$	−892080.	−0.0770617
$670$	0	0
$671$	−1.14667e6	−0.0983178
$672$	0	0
$673$	1.96595e6	0.167315	0.0836576	−	0.996495i	$-0.473340\pi$
$673$	1.96595e6	0.167315	0.0836576	+	0.996495i	$0.473340\pi$
$674$	0	0
$675$	−455625.	−0.0384900
$676$	0	0
$677$	1.56212e7	1.30991	0.654955	−	0.755667i	$-0.272687\pi$
$677$	1.56212e7	1.30991	0.654955	+	0.755667i	$0.272687\pi$
$678$	0	0
$679$	2.47876e7	2.06329
$680$	0	0
$681$	4.40204e6	0.363736
$682$	0	0
$683$	3.86400e6	0.316946	0.158473	−	0.987363i	$-0.449343\pi$
$683$	3.86400e6	0.316946	0.158473	+	0.987363i	$0.449343\pi$
$684$	0	0
$685$	−3.99956e6	−0.325676
$686$	0	0
$687$	−697157.	−0.0563558
$688$	0	0
$689$	9.47439e6	0.760332
$690$	0	0
$691$	8.51624e6	0.678504	0.339252	−	0.940695i	$-0.389826\pi$
$691$	8.51624e6	0.678504	0.339252	+	0.940695i	$0.389826\pi$
$692$	0	0
$693$	−3.52079e6	−0.278488
$694$	0	0
$695$	−7.97518e6	−0.626295
$696$	0	0
$697$	−1.94727e6	−0.151825
$698$	0	0
$699$	1.15284e6	0.0892431
$700$	0	0
$701$	−2.36126e7	−1.81489	−0.907443	−	0.420176i	$-0.861968\pi$
$701$	−2.36126e7	−1.81489	−0.907443	+	0.420176i	$0.861968\pi$
$702$	0	0
$703$	1.11066e7	0.847602
$704$	0	0
$705$	4.98101e6	0.377437
$706$	0	0
$707$	−1.91736e7	−1.44263
$708$	0	0
$709$	−1.03409e7	−0.772582	−0.386291	−	0.922377i	$-0.626244\pi$
$709$	−1.03409e7	−0.772582	−0.386291	+	0.922377i	$0.626244\pi$
$710$	0	0
$711$	−1.27813e6	−0.0948201
$712$	0	0
$713$	−6.52229e6	−0.480481
$714$	0	0
$715$	−2.99265e6	−0.218923
$716$	0	0
$717$	−9.30076e6	−0.675648
$718$	0	0
$719$	−3.72694e6	−0.268862	−0.134431	−	0.990923i	$-0.542921\pi$
$719$	−3.72694e6	−0.268862	−0.134431	+	0.990923i	$0.542921\pi$
$720$	0	0
$721$	−1.15072e7	−0.824391
$722$	0	0
$723$	−4.42391e6	−0.314746
$724$	0	0
$725$	−204275.	−0.0144335
$726$	0	0
$727$	6.39102e6	0.448471	0.224235	−	0.974535i	$-0.428012\pi$
$727$	6.39102e6	0.448471	0.224235	+	0.974535i	$0.428012\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.58310e6	−0.109576
$732$	0	0
$733$	1.04155e7	0.716011	0.358006	−	0.933719i	$-0.383457\pi$
$733$	1.04155e7	0.716011	0.358006	+	0.933719i	$0.383457\pi$
$734$	0	0
$735$	6.89073e6	0.470486
$736$	0	0
$737$	−1.29551e6	−0.0878564
$738$	0	0
$739$	2.76262e7	1.86084	0.930421	−	0.366492i	$-0.119441\pi$
$739$	2.76262e7	1.86084	0.930421	+	0.366492i	$0.119441\pi$
$740$	0	0
$741$	1.53051e7	1.02398
$742$	0	0
$743$	7.79121e6	0.517765	0.258882	−	0.965909i	$-0.416646\pi$
$743$	7.79121e6	0.517765	0.258882	+	0.965909i	$0.416646\pi$
$744$	0	0
$745$	−9.97410e6	−0.658390
$746$	0	0
$747$	−8.24702e6	−0.540749
$748$	0	0
$749$	−3.07296e7	−2.00148
$750$	0	0
$751$	−2.52335e7	−1.63259	−0.816296	−	0.577633i	$-0.803976\pi$
$751$	−2.52335e7	−1.63259	−0.816296	+	0.577633i	$0.803976\pi$
$752$	0	0
$753$	1.11166e7	0.714469
$754$	0	0
$755$	−4.16477e6	−0.265903
$756$	0	0
$757$	1.13968e7	0.722844	0.361422	−	0.932402i	$-0.382291\pi$
$757$	1.13968e7	0.722844	0.361422	+	0.932402i	$0.382291\pi$
$758$	0	0
$759$	3.76015e6	0.236920
$760$	0	0
$761$	5.00333e6	0.313182	0.156591	−	0.987664i	$-0.449950\pi$
$761$	5.00333e6	0.313182	0.156591	+	0.987664i	$0.449950\pi$
$762$	0	0
$763$	3.16172e6	0.196613
$764$	0	0
$765$	−423326.	−0.0261530
$766$	0	0
$767$	−1.93310e7	−1.18649
$768$	0	0
$769$	−4.48204e6	−0.273313	−0.136656	−	0.990619i	$-0.543636\pi$
$769$	−4.48204e6	−0.273313	−0.136656	+	0.990619i	$0.543636\pi$
$770$	0	0
$771$	−1.71818e7	−1.04096
$772$	0	0
$773$	−2.29031e7	−1.37862	−0.689311	−	0.724465i	$-0.742087\pi$
$773$	−2.29031e7	−1.37862	−0.689311	+	0.724465i	$0.742087\pi$
$774$	0	0
$775$	−1.94731e6	−0.116461
$776$	0	0
$777$	−7.67832e6	−0.456261
$778$	0	0
$779$	−2.64101e7	−1.55929
$780$	0	0
$781$	2.36901e6	0.138976
$782$	0	0
$783$	238266.	0.0138886
$784$	0	0
$785$	−6.41158e6	−0.371356
$786$	0	0
$787$	6.83480e6	0.393359	0.196679	−	0.980468i	$-0.436984\pi$
$787$	6.83480e6	0.393359	0.196679	+	0.980468i	$0.436984\pi$
$788$	0	0
$789$	−5.52653e6	−0.316053
$790$	0	0
$791$	−6.42464e6	−0.365096
$792$	0	0
$793$	−3.44604e6	−0.194598
$794$	0	0
$795$	−3.55414e6	−0.199442
$796$	0	0
$797$	−1.23658e6	−0.0689564	−0.0344782	−	0.999405i	$-0.510977\pi$
$797$	−1.23658e6	−0.0689564	−0.0344782	+	0.999405i	$0.510977\pi$
$798$	0	0
$799$	4.62791e6	0.256459
$800$	0	0
$801$	−1.85206e6	−0.101994
$802$	0	0
$803$	−4.67296e6	−0.255743
$804$	0	0
$805$	−1.13979e7	−0.619918
$806$	0	0
$807$	−1.85225e7	−1.00119
$808$	0	0
$809$	−9.63877e6	−0.517786	−0.258893	−	0.965906i	$-0.583358\pi$
$809$	−9.63877e6	−0.517786	−0.258893	+	0.965906i	$0.583358\pi$
$810$	0	0
$811$	1.11387e7	0.594679	0.297339	−	0.954772i	$-0.403901\pi$
$811$	1.11387e7	0.594679	0.297339	+	0.954772i	$0.403901\pi$
$812$	0	0
$813$	1.61088e6	0.0854745
$814$	0	0
$815$	−1.63500e7	−0.862231
$816$	0	0
$817$	−2.14710e7	−1.12538
$818$	0	0
$819$	−1.05809e7	−0.551204
$820$	0	0
$821$	2.76945e7	1.43396	0.716978	−	0.697096i	$-0.245525\pi$
$821$	2.76945e7	1.43396	0.716978	+	0.697096i	$0.245525\pi$
$822$	0	0
$823$	3.29068e7	1.69351	0.846753	−	0.531987i	$-0.178554\pi$
$823$	3.29068e7	1.69351	0.846753	+	0.531987i	$0.178554\pi$
$824$	0	0
$825$	1.12264e6	0.0574255
$826$	0	0
$827$	−2.46622e7	−1.25392	−0.626958	−	0.779053i	$-0.715700\pi$
$827$	−2.46622e7	−1.25392	−0.626958	+	0.779053i	$0.715700\pi$
$828$	0	0
$829$	−2.63681e7	−1.33258	−0.666288	−	0.745694i	$-0.732118\pi$
$829$	−2.63681e7	−1.33258	−0.666288	+	0.745694i	$0.732118\pi$
$830$	0	0
$831$	−1.70242e7	−0.855193
$832$	0	0
$833$	6.40226e6	0.319684
$834$	0	0
$835$	1.66799e7	0.827899
$836$	0	0
$837$	2.27134e6	0.112065
$838$	0	0
$839$	−2.53962e7	−1.24556	−0.622778	−	0.782399i	$-0.713996\pi$
$839$	−2.53962e7	−1.24556	−0.622778	+	0.782399i	$0.713996\pi$
$840$	0	0
$841$	−2.04043e7	−0.994792
$842$	0	0
$843$	−1.97934e7	−0.959293
$844$	0	0
$845$	288624.	0.0139056
$846$	0	0
$847$	−2.64002e7	−1.26444
$848$	0	0
$849$	−7.23867e6	−0.344659
$850$	0	0
$851$	8.20034e6	0.388157
$852$	0	0
$853$	−1.46587e7	−0.689799	−0.344900	−	0.938640i	$-0.612087\pi$
$853$	−1.46587e7	−0.689799	−0.344900	+	0.938640i	$0.612087\pi$
$854$	0	0
$855$	−5.74142e6	−0.268599
$856$	0	0
$857$	3.44223e7	1.60099	0.800493	−	0.599342i	$-0.204571\pi$
$857$	3.44223e7	1.60099	0.800493	+	0.599342i	$0.204571\pi$
$858$	0	0
$859$	8.53480e6	0.394648	0.197324	−	0.980338i	$-0.436775\pi$
$859$	8.53480e6	0.394648	0.197324	+	0.980338i	$0.436775\pi$
$860$	0	0
$861$	1.82581e7	0.839361
$862$	0	0
$863$	−1.40872e7	−0.643870	−0.321935	−	0.946762i	$-0.604333\pi$
$863$	−1.40872e7	−0.643870	−0.321935	+	0.946762i	$0.604333\pi$
$864$	0	0
$865$	9.54201e6	0.433610
$866$	0	0
$867$	1.23854e7	0.559580
$868$	0	0
$869$	3.14924e6	0.141468
$870$	0	0
$871$	−3.89336e6	−0.173892
$872$	0	0
$873$	9.21895e6	0.409398
$874$	0	0
$875$	−3.40297e6	−0.150258
$876$	0	0
$877$	−2.18156e7	−0.957786	−0.478893	−	0.877873i	$-0.658962\pi$
$877$	−2.18156e7	−0.957786	−0.478893	+	0.877873i	$0.658962\pi$
$878$	0	0
$879$	−3.91562e6	−0.170934
$880$	0	0
$881$	−1.92711e7	−0.836501	−0.418250	−	0.908332i	$-0.637357\pi$
$881$	−1.92711e7	−0.836501	−0.418250	+	0.908332i	$0.637357\pi$
$882$	0	0
$883$	5.79768e6	0.250238	0.125119	−	0.992142i	$-0.460069\pi$
$883$	5.79768e6	0.250238	0.125119	+	0.992142i	$0.460069\pi$
$884$	0	0
$885$	7.25165e6	0.311228
$886$	0	0
$887$	5.72121e6	0.244163	0.122081	−	0.992520i	$-0.461043\pi$
$887$	5.72121e6	0.244163	0.122081	+	0.992520i	$0.461043\pi$
$888$	0	0
$889$	−4.30765e7	−1.82804
$890$	0	0
$891$	−1.30944e6	−0.0552577
$892$	0	0
$893$	6.27667e7	2.63391
$894$	0	0
$895$	−1.53734e7	−0.641523
$896$	0	0
$897$	1.13002e7	0.468928
$898$	0	0
$899$	1.01833e6	0.0420233
$900$	0	0
$901$	−3.30219e6	−0.135516
$902$	0	0
$903$	1.48436e7	0.605787
$904$	0	0
$905$	−1.52775e7	−0.620055
$906$	0	0
$907$	−1.43236e7	−0.578139	−0.289070	−	0.957308i	$-0.593346\pi$
$907$	−1.43236e7	−0.578139	−0.289070	+	0.957308i	$0.593346\pi$
$908$	0	0
$909$	−7.13099e6	−0.286246
$910$	0	0
$911$	2.60230e7	1.03887	0.519434	−	0.854510i	$-0.326143\pi$
$911$	2.60230e7	1.03887	0.519434	+	0.854510i	$0.326143\pi$
$912$	0	0
$913$	2.03203e7	0.806774
$914$	0	0
$915$	1.29272e6	0.0510448
$916$	0	0
$917$	−3.07553e7	−1.20781
$918$	0	0
$919$	1.51344e7	0.591121	0.295560	−	0.955324i	$-0.404494\pi$
$919$	1.51344e7	0.591121	0.295560	+	0.955324i	$0.404494\pi$
$920$	0	0
$921$	−3.41359e6	−0.132606
$922$	0	0
$923$	7.11951e6	0.275071
$924$	0	0
$925$	2.44831e6	0.0940830
$926$	0	0
$927$	−4.27975e6	−0.163576
$928$	0	0
$929$	1.28489e7	0.488459	0.244230	−	0.969717i	$-0.421465\pi$
$929$	1.28489e7	0.488459	0.244230	+	0.969717i	$0.421465\pi$
$930$	0	0
$931$	8.68315e7	3.28324
$932$	0	0
$933$	−1.54509e7	−0.581099
$934$	0	0
$935$	1.04306e6	0.0390192
$936$	0	0
$937$	−3.08935e7	−1.14952	−0.574762	−	0.818320i	$-0.694905\pi$
$937$	−3.08935e7	−1.14952	−0.574762	+	0.818320i	$0.694905\pi$
$938$	0	0
$939$	1.31768e7	0.487694
$940$	0	0
$941$	2.48240e7	0.913897	0.456948	−	0.889493i	$-0.348942\pi$
$941$	2.48240e7	0.913897	0.456948	+	0.889493i	$0.348942\pi$
$942$	0	0
$943$	−1.94994e7	−0.714074
$944$	0	0
$945$	3.96922e6	0.144586
$946$	0	0
$947$	−2.75170e7	−0.997071	−0.498535	−	0.866869i	$-0.666129\pi$
$947$	−2.75170e7	−0.997071	−0.498535	+	0.866869i	$0.666129\pi$
$948$	0	0
$949$	−1.40435e7	−0.506185
$950$	0	0
$951$	812570.	0.0291346
$952$	0	0
$953$	−5.04677e7	−1.80004	−0.900019	−	0.435851i	$-0.856447\pi$
$953$	−5.04677e7	−1.80004	−0.900019	+	0.435851i	$0.856447\pi$
$954$	0	0
$955$	−3.78908e6	−0.134439
$956$	0	0
$957$	−587077.	−0.0207212
$958$	0	0
$959$	3.48426e7	1.22339
$960$	0	0
$961$	−1.89216e7	−0.660922
$962$	0	0
$963$	−1.14289e7	−0.397135
$964$	0	0
$965$	−1.67467e7	−0.578910
$966$	0	0
$967$	2.51529e7	0.865010	0.432505	−	0.901632i	$-0.357630\pi$
$967$	2.51529e7	0.865010	0.432505	+	0.901632i	$0.357630\pi$
$968$	0	0
$969$	−5.33442e6	−0.182506
$970$	0	0
$971$	−6.68530e6	−0.227548	−0.113774	−	0.993507i	$-0.536294\pi$
$971$	−6.68530e6	−0.227548	−0.113774	+	0.993507i	$0.536294\pi$
$972$	0	0
$973$	6.94766e7	2.35265
$974$	0	0
$975$	3.37382e6	0.113661
$976$	0	0
$977$	2.48404e7	0.832573	0.416286	−	0.909234i	$-0.363331\pi$
$977$	2.48404e7	0.832573	0.416286	+	0.909234i	$0.363331\pi$
$978$	0	0
$979$	4.56339e6	0.152171
$980$	0	0
$981$	1.17590e6	0.0390120
$982$	0	0
$983$	−3.77957e7	−1.24755	−0.623776	−	0.781603i	$-0.714402\pi$
$983$	−3.77957e7	−1.24755	−0.623776	+	0.781603i	$0.714402\pi$
$984$	0	0
$985$	−2.10880e7	−0.692540
$986$	0	0
$987$	−4.33926e7	−1.41782
$988$	0	0
$989$	−1.58528e7	−0.515364
$990$	0	0
$991$	7.63130e6	0.246839	0.123420	−	0.992355i	$-0.460614\pi$
$991$	7.63130e6	0.246839	0.123420	+	0.992355i	$0.460614\pi$
$992$	0	0
$993$	3.03445e7	0.976579
$994$	0	0
$995$	−2.48039e6	−0.0794261
$996$	0	0
$997$	6.87906e6	0.219175	0.109588	−	0.993977i	$-0.465047\pi$
$997$	6.87906e6	0.219175	0.109588	+	0.993977i	$0.465047\pi$
$998$	0	0
$999$	−2.85570e6	−0.0905314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.6.a.bf.1.2		2
4.3	odd	2		960.6.a.bj.1.1		2
8.3	odd	2		15.6.a.c.1.2	✓	2
8.5	even	2		240.6.a.q.1.2		2
24.5	odd	2		720.6.a.bd.1.2		2
24.11	even	2		45.6.a.e.1.1		2
40.3	even	4		75.6.b.e.49.2		4
40.19	odd	2		75.6.a.h.1.1		2
40.27	even	4		75.6.b.e.49.3		4
56.27	even	2		735.6.a.g.1.2		2
120.59	even	2		225.6.a.m.1.2		2
120.83	odd	4		225.6.b.g.199.3		4
120.107	odd	4		225.6.b.g.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.c.1.2	✓	2	8.3	odd	2
45.6.a.e.1.1		2	24.11	even	2
75.6.a.h.1.1		2	40.19	odd	2
75.6.b.e.49.2		4	40.3	even	4
75.6.b.e.49.3		4	40.27	even	4
225.6.a.m.1.2		2	120.59	even	2
225.6.b.g.199.2		4	120.107	odd	4
225.6.b.g.199.3		4	120.83	odd	4
240.6.a.q.1.2		2	8.5	even	2
720.6.a.bd.1.2		2	24.5	odd	2
735.6.a.g.1.2		2	56.27	even	2
960.6.a.bf.1.2		2	1.1	even	1	trivial
960.6.a.bj.1.1		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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	960.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -25.0000 q^{5} +217.790 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -25.0000 q^{5} +217.790 q^{7} +81.0000 q^{9} -199.580 q^{11} -599.790 q^{13} +225.000 q^{15} +209.050 q^{17} +2835.27 q^{19} -1960.11 q^{21} +2093.37 q^{23} +625.000 q^{25} -729.000 q^{27} -326.840 q^{29} -3115.69 q^{31} +1796.22 q^{33} -5444.75 q^{35} +3917.29 q^{37} +5398.11 q^{39} -9314.86 q^{41} -7572.84 q^{43} -2025.00 q^{45} +22137.8 q^{47} +30625.5 q^{49} -1881.45 q^{51} -15796.2 q^{53} +4989.50 q^{55} -25517.4 q^{57} +32229.6 q^{59} +5745.42 q^{61} +17641.0 q^{63} +14994.7 q^{65} +6491.20 q^{67} -18840.3 q^{69} -11870.0 q^{71} +23414.0 q^{73} -5625.00 q^{75} -43466.5 q^{77} -15779.4 q^{79} +6561.00 q^{81} -101815. q^{83} -5226.25 q^{85} +2941.56 q^{87} -22865.0 q^{89} -130628. q^{91} +28041.2 q^{93} -70881.7 q^{95} +113814. q^{97} -16166.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 50 q^{5} + 112 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 - 50 * q^5 + 112 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} - 50 q^{5} + 112 q^{7} + 162 q^{9} + 248 q^{11} - 876 q^{13} + 450 q^{15} + 2036 q^{17} + 1464 q^{19} - 1008 q^{21} + 3216 q^{23} + 1250 q^{25} - 1458 q^{27} - 1948 q^{29} - 2672 q^{31} - 2232 q^{33} - 2800 q^{35} - 8668 q^{37} + 7884 q^{39} - 7628 q^{41} - 16440 q^{43} - 4050 q^{45} + 19360 q^{47} + 25010 q^{49} - 18324 q^{51} + 14356 q^{53} - 6200 q^{55} - 13176 q^{57} - 904 q^{59} - 20220 q^{61} + 9072 q^{63} + 21900 q^{65} - 12904 q^{67} - 28944 q^{69} + 40976 q^{71} + 59124 q^{73} - 11250 q^{75} - 90816 q^{77} - 107600 q^{79} + 13122 q^{81} - 122088 q^{83} - 50900 q^{85} + 17532 q^{87} + 103764 q^{89} - 101408 q^{91} + 24048 q^{93} - 36600 q^{95} - 24764 q^{97} + 20088 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 50 * q^5 + 112 * q^7 + 162 * q^9 + 248 * q^11 - 876 * q^13 + 450 * q^15 + 2036 * q^17 + 1464 * q^19 - 1008 * q^21 + 3216 * q^23 + 1250 * q^25 - 1458 * q^27 - 1948 * q^29 - 2672 * q^31 - 2232 * q^33 - 2800 * q^35 - 8668 * q^37 + 7884 * q^39 - 7628 * q^41 - 16440 * q^43 - 4050 * q^45 + 19360 * q^47 + 25010 * q^49 - 18324 * q^51 + 14356 * q^53 - 6200 * q^55 - 13176 * q^57 - 904 * q^59 - 20220 * q^61 + 9072 * q^63 + 21900 * q^65 - 12904 * q^67 - 28944 * q^69 + 40976 * q^71 + 59124 * q^73 - 11250 * q^75 - 90816 * q^77 - 107600 * q^79 + 13122 * q^81 - 122088 * q^83 - 50900 * q^85 + 17532 * q^87 + 103764 * q^89 - 101408 * q^91 + 24048 * q^93 - 36600 * q^95 - 24764 * q^97 + 20088 * q^99

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