LMFDB - Embedded newform 1040.6.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1458804.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 361x - 1139 $ x^3 - x^2 - 361*x - 1139
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$20.8905$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.8905 q^{3} +25.0000 q^{5} +240.426 q^{7} +534.880 q^{9} -544.151 q^{11} +169.000 q^{13} +697.262 q^{15} +1629.30 q^{17} +805.920 q^{19} +6705.59 q^{21} -373.152 q^{23} +625.000 q^{25} +8140.67 q^{27} +1503.62 q^{29} +2200.08 q^{31} -15176.7 q^{33} +6010.64 q^{35} -13109.1 q^{37} +4713.49 q^{39} -17099.9 q^{41} +8935.58 q^{43} +13372.0 q^{45} -15749.7 q^{47} +40997.5 q^{49} +45442.0 q^{51} +40379.7 q^{53} -13603.8 q^{55} +22477.5 q^{57} +47562.1 q^{59} -30280.0 q^{61} +128599. q^{63} +4225.00 q^{65} -38769.4 q^{67} -10407.4 q^{69} +10519.7 q^{71} +1582.12 q^{73} +17431.6 q^{75} -130828. q^{77} +6191.23 q^{79} +97071.7 q^{81} -37849.2 q^{83} +40732.5 q^{85} +41936.6 q^{87} +49151.0 q^{89} +40631.9 q^{91} +61361.3 q^{93} +20148.0 q^{95} -15654.2 q^{97} -291056. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 22 q^{3} + 75 q^{5} + 234 q^{7} + 155 q^{9} - 48 q^{11} + 507 q^{13} + 550 q^{15} + 1506 q^{17} + 360 q^{19} + 6904 q^{21} + 2370 q^{23} + 1875 q^{25} + 10168 q^{27} - 3078 q^{29} + 5388 q^{31} - 22572 q^{33}+ \cdots - 350340 q^{99}+O(q^{100}) $ 3 * q + 22 * q^3 + 75 * q^5 + 234 * q^7 + 155 * q^9 - 48 * q^11 + 507 * q^13 + 550 * q^15 + 1506 * q^17 + 360 * q^19 + 6904 * q^21 + 2370 * q^23 + 1875 * q^25 + 10168 * q^27 - 3078 * q^29 + 5388 * q^31 - 22572 * q^33 + 5850 * q^35 - 25362 * q^37 + 3718 * q^39 - 15906 * q^41 + 39306 * q^43 + 3875 * q^45 + 17778 * q^47 + 7767 * q^49 + 33972 * q^51 + 9246 * q^53 - 1200 * q^55 - 2996 * q^57 + 77760 * q^59 + 17982 * q^61 + 128762 * q^63 + 12675 * q^65 + 2922 * q^67 - 31860 * q^69 + 4944 * q^71 + 43278 * q^73 + 13750 * q^75 - 144420 * q^77 - 42120 * q^79 + 146519 * q^81 - 58098 * q^83 + 37650 * q^85 - 2016 * q^87 + 19614 * q^89 + 39546 * q^91 + 162796 * q^93 + 9000 * q^95 - 87078 * q^97 - 350340 * 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
$3$	27.8905	1.78918	0.894588	−	0.446892i	$-0.147469\pi$
$3$	27.8905	1.78918	0.894588	+	0.446892i	$0.147469\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	240.426	1.85454	0.927269	−	0.374397i	$-0.122150\pi$
$7$	240.426	1.85454	0.927269	+	0.374397i	$0.122150\pi$
$8$	0	0
$9$	534.880	2.20115
$10$	0	0
$11$	−544.151	−1.35593	−0.677966	−	0.735093i	$-0.737138\pi$
$11$	−544.151	−1.35593	−0.677966	+	0.735093i	$0.737138\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	697.262	0.800144
$16$	0	0
$17$	1629.30	1.36735	0.683673	−	0.729788i	$-0.260381\pi$
$17$	1629.30	1.36735	0.683673	+	0.729788i	$0.260381\pi$
$18$	0	0
$19$	805.920	0.512162	0.256081	−	0.966655i	$-0.417569\pi$
$19$	805.920	0.512162	0.256081	+	0.966655i	$0.417569\pi$
$20$	0	0
$21$	6705.59	3.31809
$22$	0	0
$23$	−373.152	−0.147084	−0.0735421	−	0.997292i	$-0.523430\pi$
$23$	−373.152	−0.147084	−0.0735421	+	0.997292i	$0.523430\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	8140.67	2.14907
$28$	0	0
$29$	1503.62	0.332003	0.166001	−	0.986126i	$-0.446914\pi$
$29$	1503.62	0.332003	0.166001	+	0.986126i	$0.446914\pi$
$30$	0	0
$31$	2200.08	0.411182	0.205591	−	0.978638i	$-0.434088\pi$
$31$	2200.08	0.411182	0.205591	+	0.978638i	$0.434088\pi$
$32$	0	0
$33$	−15176.7	−2.42600
$34$	0	0
$35$	6010.64	0.829374
$36$	0	0
$37$	−13109.1	−1.57423	−0.787117	−	0.616804i	$-0.788427\pi$
$37$	−13109.1	−1.57423	−0.787117	+	0.616804i	$0.788427\pi$
$38$	0	0
$39$	4713.49	0.496228
$40$	0	0
$41$	−17099.9	−1.58867	−0.794337	−	0.607477i	$-0.792182\pi$
$41$	−17099.9	−1.58867	−0.794337	+	0.607477i	$0.792182\pi$
$42$	0	0
$43$	8935.58	0.736973	0.368487	−	0.929633i	$-0.379876\pi$
$43$	8935.58	0.736973	0.368487	+	0.929633i	$0.379876\pi$
$44$	0	0
$45$	13372.0	0.984385
$46$	0	0
$47$	−15749.7	−1.03999	−0.519995	−	0.854170i	$-0.674066\pi$
$47$	−15749.7	−1.03999	−0.519995	+	0.854170i	$0.674066\pi$
$48$	0	0
$49$	40997.5	2.43931
$50$	0	0
$51$	45442.0	2.44642
$52$	0	0
$53$	40379.7	1.97457	0.987286	−	0.158951i	$-0.0508112\pi$
$53$	40379.7	1.97457	0.987286	+	0.158951i	$0.0508112\pi$
$54$	0	0
$55$	−13603.8	−0.606391
$56$	0	0
$57$	22477.5	0.916349
$58$	0	0
$59$	47562.1	1.77881	0.889407	−	0.457116i	$-0.151118\pi$
$59$	47562.1	1.77881	0.889407	+	0.457116i	$0.151118\pi$
$60$	0	0
$61$	−30280.0	−1.04191	−0.520956	−	0.853584i	$-0.674424\pi$
$61$	−30280.0	−1.04191	−0.520956	+	0.853584i	$0.674424\pi$
$62$	0	0
$63$	128599.	4.08212
$64$	0	0
$65$	4225.00	0.124035
$66$	0	0
$67$	−38769.4	−1.05512	−0.527561	−	0.849517i	$-0.676893\pi$
$67$	−38769.4	−1.05512	−0.527561	+	0.849517i	$0.676893\pi$
$68$	0	0
$69$	−10407.4	−0.263160
$70$	0	0
$71$	10519.7	0.247660	0.123830	−	0.992303i	$-0.460482\pi$
$71$	10519.7	0.247660	0.123830	+	0.992303i	$0.460482\pi$
$72$	0	0
$73$	1582.12	0.0347483	0.0173742	−	0.999849i	$-0.494469\pi$
$73$	1582.12	0.0347483	0.0173742	+	0.999849i	$0.494469\pi$
$74$	0	0
$75$	17431.6	0.357835
$76$	0	0
$77$	−130828.	−2.51463
$78$	0	0
$79$	6191.23	0.111612	0.0558058	−	0.998442i	$-0.482227\pi$
$79$	6191.23	0.111612	0.0558058	+	0.998442i	$0.482227\pi$
$80$	0	0
$81$	97071.7	1.64392
$82$	0	0
$83$	−37849.2	−0.603062	−0.301531	−	0.953456i	$-0.597498\pi$
$83$	−37849.2	−0.603062	−0.301531	+	0.953456i	$0.597498\pi$
$84$	0	0
$85$	40732.5	0.611496
$86$	0	0
$87$	41936.6	0.594011
$88$	0	0
$89$	49151.0	0.657744	0.328872	−	0.944374i	$-0.393331\pi$
$89$	49151.0	0.657744	0.328872	+	0.944374i	$0.393331\pi$
$90$	0	0
$91$	40631.9	0.514356
$92$	0	0
$93$	61361.3	0.735677
$94$	0	0
$95$	20148.0	0.229046
$96$	0	0
$97$	−15654.2	−0.168928	−0.0844641	−	0.996427i	$-0.526918\pi$
$97$	−15654.2	−0.168928	−0.0844641	+	0.996427i	$0.526918\pi$
$98$	0	0
$99$	−291056.	−2.98461
$100$	0	0
$101$	−138108.	−1.34714	−0.673572	−	0.739122i	$-0.735241\pi$
$101$	−138108.	−1.34714	−0.673572	+	0.739122i	$0.735241\pi$
$102$	0	0
$103$	47642.1	0.442484	0.221242	−	0.975219i	$-0.428989\pi$
$103$	47642.1	0.442484	0.221242	+	0.975219i	$0.428989\pi$
$104$	0	0
$105$	167640.	1.48390
$106$	0	0
$107$	186527.	1.57500	0.787501	−	0.616313i	$-0.211374\pi$
$107$	186527.	1.57500	0.787501	+	0.616313i	$0.211374\pi$
$108$	0	0
$109$	14407.1	0.116147	0.0580736	−	0.998312i	$-0.481504\pi$
$109$	14407.1	0.116147	0.0580736	+	0.998312i	$0.481504\pi$
$110$	0	0
$111$	−365620.	−2.81658
$112$	0	0
$113$	−50802.0	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$113$	−50802.0	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$114$	0	0
$115$	−9328.80	−0.0657781
$116$	0	0
$117$	90394.7	0.610490
$118$	0	0
$119$	391725.	2.53580
$120$	0	0
$121$	135050.	0.838552
$122$	0	0
$123$	−476926.	−2.84242
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	182278.	1.00282	0.501411	−	0.865209i	$-0.332814\pi$
$127$	182278.	1.00282	0.501411	+	0.865209i	$0.332814\pi$
$128$	0	0
$129$	249218.	1.31858
$130$	0	0
$131$	−199714.	−1.01679	−0.508394	−	0.861124i	$-0.669761\pi$
$131$	−199714.	−1.01679	−0.508394	+	0.861124i	$0.669761\pi$
$132$	0	0
$133$	193764.	0.949824
$134$	0	0
$135$	203517.	0.961094
$136$	0	0
$137$	−345418.	−1.57233	−0.786165	−	0.618017i	$-0.787936\pi$
$137$	−345418.	−1.57233	−0.786165	+	0.618017i	$0.787936\pi$
$138$	0	0
$139$	−183088.	−0.803754	−0.401877	−	0.915694i	$-0.631642\pi$
$139$	−183088.	−0.803754	−0.401877	+	0.915694i	$0.631642\pi$
$140$	0	0
$141$	−439268.	−1.86072
$142$	0	0
$143$	−91961.6	−0.376068
$144$	0	0
$145$	37590.4	0.148476
$146$	0	0
$147$	1.14344e6	4.36435
$148$	0	0
$149$	−187082.	−0.690344	−0.345172	−	0.938540i	$-0.612179\pi$
$149$	−187082.	−0.690344	−0.345172	+	0.938540i	$0.612179\pi$
$150$	0	0
$151$	10616.3	0.0378907	0.0189454	−	0.999821i	$-0.493969\pi$
$151$	10616.3	0.0378907	0.0189454	+	0.999821i	$0.493969\pi$
$152$	0	0
$153$	871480.	3.00974
$154$	0	0
$155$	55002.0	0.183886
$156$	0	0
$157$	−335884.	−1.08753	−0.543764	−	0.839238i	$-0.683001\pi$
$157$	−335884.	−1.08753	−0.543764	+	0.839238i	$0.683001\pi$
$158$	0	0
$159$	1.12621e6	3.53286
$160$	0	0
$161$	−89715.3	−0.272773
$162$	0	0
$163$	−201881.	−0.595150	−0.297575	−	0.954698i	$-0.596178\pi$
$163$	−201881.	−0.595150	−0.297575	+	0.954698i	$0.596178\pi$
$164$	0	0
$165$	−379416.	−1.08494
$166$	0	0
$167$	446050.	1.23764	0.618818	−	0.785535i	$-0.287612\pi$
$167$	446050.	1.23764	0.618818	+	0.785535i	$0.287612\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	431070.	1.12735
$172$	0	0
$173$	57408.8	0.145835	0.0729177	−	0.997338i	$-0.476769\pi$
$173$	57408.8	0.145835	0.0729177	+	0.997338i	$0.476769\pi$
$174$	0	0
$175$	150266.	0.370907
$176$	0	0
$177$	1.32653e6	3.18261
$178$	0	0
$179$	−87439.4	−0.203974	−0.101987	−	0.994786i	$-0.532520\pi$
$179$	−87439.4	−0.203974	−0.101987	+	0.994786i	$0.532520\pi$
$180$	0	0
$181$	21341.6	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$181$	21341.6	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$182$	0	0
$183$	−844523.	−1.86416
$184$	0	0
$185$	−327728.	−0.704019
$186$	0	0
$187$	−886586.	−1.85403
$188$	0	0
$189$	1.95723e6	3.98553
$190$	0	0
$191$	−307517.	−0.609938	−0.304969	−	0.952362i	$-0.598646\pi$
$191$	−307517.	−0.609938	−0.304969	+	0.952362i	$0.598646\pi$
$192$	0	0
$193$	182409.	0.352496	0.176248	−	0.984346i	$-0.443604\pi$
$193$	182409.	0.352496	0.176248	+	0.984346i	$0.443604\pi$
$194$	0	0
$195$	117837.	0.221920
$196$	0	0
$197$	−152508.	−0.279980	−0.139990	−	0.990153i	$-0.544707\pi$
$197$	−152508.	−0.279980	−0.139990	+	0.990153i	$0.544707\pi$
$198$	0	0
$199$	586606.	1.05006	0.525029	−	0.851084i	$-0.324054\pi$
$199$	586606.	1.05006	0.525029	+	0.851084i	$0.324054\pi$
$200$	0	0
$201$	−1.08130e6	−1.88780
$202$	0	0
$203$	361508.	0.615711
$204$	0	0
$205$	−427498.	−0.710477
$206$	0	0
$207$	−199592.	−0.323755
$208$	0	0
$209$	−438542.	−0.694458
$210$	0	0
$211$	207041.	0.320147	0.160073	−	0.987105i	$-0.448827\pi$
$211$	207041.	0.320147	0.160073	+	0.987105i	$0.448827\pi$
$212$	0	0
$213$	293399.	0.443108
$214$	0	0
$215$	223390.	0.329585
$216$	0	0
$217$	528955.	0.762552
$218$	0	0
$219$	44126.3	0.0621708
$220$	0	0
$221$	275352.	0.379234
$222$	0	0
$223$	1.04889e6	1.41243	0.706214	−	0.707998i	$-0.250401\pi$
$223$	1.04889e6	1.41243	0.706214	+	0.707998i	$0.250401\pi$
$224$	0	0
$225$	334300.	0.440230
$226$	0	0
$227$	100709.	0.129719	0.0648593	−	0.997894i	$-0.479340\pi$
$227$	100709.	0.129719	0.0648593	+	0.997894i	$0.479340\pi$
$228$	0	0
$229$	487174.	0.613896	0.306948	−	0.951726i	$-0.400692\pi$
$229$	487174.	0.613896	0.306948	+	0.951726i	$0.400692\pi$
$230$	0	0
$231$	−3.64886e6	−4.49911
$232$	0	0
$233$	−832844.	−1.00502	−0.502509	−	0.864572i	$-0.667590\pi$
$233$	−832844.	−1.00502	−0.502509	+	0.864572i	$0.667590\pi$
$234$	0	0
$235$	−393744.	−0.465097
$236$	0	0
$237$	172677.	0.199693
$238$	0	0
$239$	−1.34919e6	−1.52785	−0.763923	−	0.645308i	$-0.776729\pi$
$239$	−1.34919e6	−1.52785	−0.763923	+	0.645308i	$0.776729\pi$
$240$	0	0
$241$	−1.66283e6	−1.84419	−0.922096	−	0.386962i	$-0.873524\pi$
$241$	−1.66283e6	−1.84419	−0.922096	+	0.386962i	$0.873524\pi$
$242$	0	0
$243$	729193.	0.792185
$244$	0	0
$245$	1.02494e6	1.09089
$246$	0	0
$247$	136200.	0.142048
$248$	0	0
$249$	−1.05563e6	−1.07898
$250$	0	0
$251$	291273.	0.291820	0.145910	−	0.989298i	$-0.453389\pi$
$251$	291273.	0.291820	0.145910	+	0.989298i	$0.453389\pi$
$252$	0	0
$253$	203051.	0.199436
$254$	0	0
$255$	1.13605e6	1.09407
$256$	0	0
$257$	−548204.	−0.517737	−0.258868	−	0.965913i	$-0.583350\pi$
$257$	−548204.	−0.517737	−0.258868	+	0.965913i	$0.583350\pi$
$258$	0	0
$259$	−3.15177e6	−2.91947
$260$	0	0
$261$	804253.	0.730788
$262$	0	0
$263$	−818412.	−0.729596	−0.364798	−	0.931087i	$-0.618862\pi$
$263$	−818412.	−0.729596	−0.364798	+	0.931087i	$0.618862\pi$
$264$	0	0
$265$	1.00949e6	0.883056
$266$	0	0
$267$	1.37085e6	1.17682
$268$	0	0
$269$	−939455.	−0.791581	−0.395790	−	0.918341i	$-0.629529\pi$
$269$	−939455.	−0.791581	−0.395790	+	0.918341i	$0.629529\pi$
$270$	0	0
$271$	1.52416e6	1.26069	0.630345	−	0.776315i	$-0.282913\pi$
$271$	1.52416e6	1.26069	0.630345	+	0.776315i	$0.282913\pi$
$272$	0	0
$273$	1.13324e6	0.920274
$274$	0	0
$275$	−340095.	−0.271186
$276$	0	0
$277$	−439704.	−0.344319	−0.172159	−	0.985069i	$-0.555074\pi$
$277$	−439704.	−0.344319	−0.172159	+	0.985069i	$0.555074\pi$
$278$	0	0
$279$	1.17678e6	0.905074
$280$	0	0
$281$	−592033.	−0.447280	−0.223640	−	0.974672i	$-0.571794\pi$
$281$	−592033.	−0.447280	−0.223640	+	0.974672i	$0.571794\pi$
$282$	0	0
$283$	777226.	0.576875	0.288437	−	0.957499i	$-0.406864\pi$
$283$	777226.	0.576875	0.288437	+	0.957499i	$0.406864\pi$
$284$	0	0
$285$	561938.	0.409804
$286$	0	0
$287$	−4.11126e6	−2.94626
$288$	0	0
$289$	1.23476e6	0.869637
$290$	0	0
$291$	−436604.	−0.302242
$292$	0	0
$293$	609025.	0.414444	0.207222	−	0.978294i	$-0.433558\pi$
$293$	609025.	0.414444	0.207222	+	0.978294i	$0.433558\pi$
$294$	0	0
$295$	1.18905e6	0.795510
$296$	0	0
$297$	−4.42976e6	−2.91400
$298$	0	0
$299$	−63062.7	−0.0407938
$300$	0	0
$301$	2.14834e6	1.36674
$302$	0	0
$303$	−3.85189e6	−2.41028
$304$	0	0
$305$	−756999.	−0.465957
$306$	0	0
$307$	2.66761e6	1.61538	0.807692	−	0.589604i	$-0.200716\pi$
$307$	2.66761e6	1.61538	0.807692	+	0.589604i	$0.200716\pi$
$308$	0	0
$309$	1.32876e6	0.791682
$310$	0	0
$311$	1.62320e6	0.951634	0.475817	−	0.879544i	$-0.342153\pi$
$311$	1.62320e6	0.951634	0.475817	+	0.879544i	$0.342153\pi$
$312$	0	0
$313$	592115.	0.341622	0.170811	−	0.985304i	$-0.445361\pi$
$313$	592115.	0.341622	0.170811	+	0.985304i	$0.445361\pi$
$314$	0	0
$315$	3.21497e6	1.82558
$316$	0	0
$317$	−2.61036e6	−1.45899	−0.729494	−	0.683987i	$-0.760244\pi$
$317$	−2.61036e6	−1.45899	−0.729494	+	0.683987i	$0.760244\pi$
$318$	0	0
$319$	−818194.	−0.450173
$320$	0	0
$321$	5.20232e6	2.81796
$322$	0	0
$323$	1.31308e6	0.700304
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	401820.	0.207808
$328$	0	0
$329$	−3.78664e6	−1.92870
$330$	0	0
$331$	−818785.	−0.410771	−0.205386	−	0.978681i	$-0.565845\pi$
$331$	−818785.	−0.410771	−0.205386	+	0.978681i	$0.565845\pi$
$332$	0	0
$333$	−7.01180e6	−3.46513
$334$	0	0
$335$	−969236.	−0.471864
$336$	0	0
$337$	3.28910e6	1.57762	0.788809	−	0.614638i	$-0.210698\pi$
$337$	3.28910e6	1.57762	0.788809	+	0.614638i	$0.210698\pi$
$338$	0	0
$339$	−1.41689e6	−0.669634
$340$	0	0
$341$	−1.19718e6	−0.557535
$342$	0	0
$343$	5.81600e6	2.66925
$344$	0	0
$345$	−260185.	−0.117689
$346$	0	0
$347$	−139108.	−0.0620195	−0.0310098	−	0.999519i	$-0.509872\pi$
$347$	−139108.	−0.0620195	−0.0310098	+	0.999519i	$0.509872\pi$
$348$	0	0
$349$	1.41781e6	0.623097	0.311549	−	0.950230i	$-0.399152\pi$
$349$	1.41781e6	0.623097	0.311549	+	0.950230i	$0.399152\pi$
$350$	0	0
$351$	1.37577e6	0.596045
$352$	0	0
$353$	2.45675e6	1.04936	0.524679	−	0.851300i	$-0.324185\pi$
$353$	2.45675e6	1.04936	0.524679	+	0.851300i	$0.324185\pi$
$354$	0	0
$355$	262992.	0.110757
$356$	0	0
$357$	1.09254e7	4.53698
$358$	0	0
$359$	2.88601e6	1.18185	0.590924	−	0.806727i	$-0.298763\pi$
$359$	2.88601e6	1.18185	0.590924	+	0.806727i	$0.298763\pi$
$360$	0	0
$361$	−1.82659e6	−0.737690
$362$	0	0
$363$	3.76660e6	1.50032
$364$	0	0
$365$	39553.1	0.0155399
$366$	0	0
$367$	2.83587e6	1.09906	0.549530	−	0.835474i	$-0.314807\pi$
$367$	2.83587e6	1.09906	0.549530	+	0.835474i	$0.314807\pi$
$368$	0	0
$369$	−9.14641e6	−3.49691
$370$	0	0
$371$	9.70831e6	3.66192
$372$	0	0
$373$	5.02691e6	1.87081	0.935403	−	0.353583i	$-0.115037\pi$
$373$	5.02691e6	1.87081	0.935403	+	0.353583i	$0.115037\pi$
$374$	0	0
$375$	435789.	0.160029
$376$	0	0
$377$	254111.	0.0920810
$378$	0	0
$379$	−2.14892e6	−0.768462	−0.384231	−	0.923237i	$-0.625533\pi$
$379$	−2.14892e6	−0.768462	−0.384231	+	0.923237i	$0.625533\pi$
$380$	0	0
$381$	5.08381e6	1.79423
$382$	0	0
$383$	−832971.	−0.290157	−0.145078	−	0.989420i	$-0.546343\pi$
$383$	−832971.	−0.290157	−0.145078	+	0.989420i	$0.546343\pi$
$384$	0	0
$385$	−3.27070e6	−1.12458
$386$	0	0
$387$	4.77946e6	1.62219
$388$	0	0
$389$	−725243.	−0.243002	−0.121501	−	0.992591i	$-0.538771\pi$
$389$	−725243.	−0.243002	−0.121501	+	0.992591i	$0.538771\pi$
$390$	0	0
$391$	−607976.	−0.201115
$392$	0	0
$393$	−5.57013e6	−1.81921
$394$	0	0
$395$	154781.	0.0499142
$396$	0	0
$397$	2.42654e6	0.772701	0.386351	−	0.922352i	$-0.373735\pi$
$397$	2.42654e6	0.772701	0.386351	+	0.922352i	$0.373735\pi$
$398$	0	0
$399$	5.40417e6	1.69940
$400$	0	0
$401$	−2.14376e6	−0.665758	−0.332879	−	0.942970i	$-0.608020\pi$
$401$	−2.14376e6	−0.665758	−0.332879	+	0.942970i	$0.608020\pi$
$402$	0	0
$403$	371813.	0.114041
$404$	0	0
$405$	2.42679e6	0.735182
$406$	0	0
$407$	7.13335e6	2.13455
$408$	0	0
$409$	−3.66144e6	−1.08229	−0.541145	−	0.840929i	$-0.682009\pi$
$409$	−3.66144e6	−1.08229	−0.541145	+	0.840929i	$0.682009\pi$
$410$	0	0
$411$	−9.63388e6	−2.81317
$412$	0	0
$413$	1.14351e7	3.29888
$414$	0	0
$415$	−946231.	−0.269698
$416$	0	0
$417$	−5.10642e6	−1.43806
$418$	0	0
$419$	−3.03786e6	−0.845341	−0.422671	−	0.906283i	$-0.638907\pi$
$419$	−3.03786e6	−0.845341	−0.422671	+	0.906283i	$0.638907\pi$
$420$	0	0
$421$	−1.88663e6	−0.518777	−0.259389	−	0.965773i	$-0.583521\pi$
$421$	−1.88663e6	−0.518777	−0.259389	+	0.965773i	$0.583521\pi$
$422$	0	0
$423$	−8.42422e6	−2.28917
$424$	0	0
$425$	1.01831e6	0.273469
$426$	0	0
$427$	−7.28008e6	−1.93226
$428$	0	0
$429$	−2.56485e6	−0.672852
$430$	0	0
$431$	−717193.	−0.185970	−0.0929850	−	0.995668i	$-0.529641\pi$
$431$	−717193.	−0.185970	−0.0929850	+	0.995668i	$0.529641\pi$
$432$	0	0
$433$	2.64569e6	0.678140	0.339070	−	0.940761i	$-0.389888\pi$
$433$	2.64569e6	0.678140	0.339070	+	0.940761i	$0.389888\pi$
$434$	0	0
$435$	1.04841e6	0.265650
$436$	0	0
$437$	−300731.	−0.0753310
$438$	0	0
$439$	−4.47973e6	−1.10941	−0.554704	−	0.832048i	$-0.687168\pi$
$439$	−4.47973e6	−1.10941	−0.554704	+	0.832048i	$0.687168\pi$
$440$	0	0
$441$	2.19287e7	5.36929
$442$	0	0
$443$	−4.30731e6	−1.04279	−0.521395	−	0.853315i	$-0.674588\pi$
$443$	−4.30731e6	−1.04279	−0.521395	+	0.853315i	$0.674588\pi$
$444$	0	0
$445$	1.22877e6	0.294152
$446$	0	0
$447$	−5.21780e6	−1.23515
$448$	0	0
$449$	−25272.9	−0.00591615	−0.00295808	−	0.999996i	$-0.500942\pi$
$449$	−25272.9	−0.00591615	−0.00295808	+	0.999996i	$0.500942\pi$
$450$	0	0
$451$	9.30495e6	2.15413
$452$	0	0
$453$	296095.	0.0677932
$454$	0	0
$455$	1.01580e6	0.230027
$456$	0	0
$457$	1.14360e6	0.256143	0.128072	−	0.991765i	$-0.459121\pi$
$457$	1.14360e6	0.256143	0.128072	+	0.991765i	$0.459121\pi$
$458$	0	0
$459$	1.32636e7	2.93853
$460$	0	0
$461$	−8.34911e6	−1.82973	−0.914867	−	0.403755i	$-0.867705\pi$
$461$	−8.34911e6	−1.82973	−0.914867	+	0.403755i	$0.867705\pi$
$462$	0	0
$463$	3.41596e6	0.740561	0.370280	−	0.928920i	$-0.379262\pi$
$463$	3.41596e6	0.740561	0.370280	+	0.928920i	$0.379262\pi$
$464$	0	0
$465$	1.53403e6	0.329005
$466$	0	0
$467$	−5.67105e6	−1.20329	−0.601646	−	0.798763i	$-0.705488\pi$
$467$	−5.67105e6	−1.20329	−0.601646	+	0.798763i	$0.705488\pi$
$468$	0	0
$469$	−9.32116e6	−1.95676
$470$	0	0
$471$	−9.36797e6	−1.94578
$472$	0	0
$473$	−4.86231e6	−0.999286
$474$	0	0
$475$	503700.	0.102432
$476$	0	0
$477$	2.15983e7	4.34633
$478$	0	0
$479$	6.16340e6	1.22739	0.613694	−	0.789544i	$-0.289683\pi$
$479$	6.16340e6	1.22739	0.613694	+	0.789544i	$0.289683\pi$
$480$	0	0
$481$	−2.21544e6	−0.436614
$482$	0	0
$483$	−2.50220e6	−0.488039
$484$	0	0
$485$	−391356.	−0.0755470
$486$	0	0
$487$	7.32701e6	1.39992	0.699962	−	0.714180i	$-0.253200\pi$
$487$	7.32701e6	1.39992	0.699962	+	0.714180i	$0.253200\pi$
$488$	0	0
$489$	−5.63056e6	−1.06483
$490$	0	0
$491$	−8.89739e6	−1.66555	−0.832777	−	0.553608i	$-0.813250\pi$
$491$	−8.89739e6	−1.66555	−0.832777	+	0.553608i	$0.813250\pi$
$492$	0	0
$493$	2.44984e6	0.453963
$494$	0	0
$495$	−7.27639e6	−1.33476
$496$	0	0
$497$	2.52920e6	0.459295
$498$	0	0
$499$	−3.02237e6	−0.543371	−0.271686	−	0.962386i	$-0.587581\pi$
$499$	−3.02237e6	−0.543371	−0.271686	+	0.962386i	$0.587581\pi$
$500$	0	0
$501$	1.24406e7	2.21435
$502$	0	0
$503$	7.48799e6	1.31961	0.659805	−	0.751437i	$-0.270639\pi$
$503$	7.48799e6	1.31961	0.659805	+	0.751437i	$0.270639\pi$
$504$	0	0
$505$	−3.45269e6	−0.602461
$506$	0	0
$507$	796581.	0.137629
$508$	0	0
$509$	4.87997e6	0.834878	0.417439	−	0.908705i	$-0.362928\pi$
$509$	4.87997e6	0.834878	0.417439	+	0.908705i	$0.362928\pi$
$510$	0	0
$511$	380383.	0.0644420
$512$	0	0
$513$	6.56073e6	1.10067
$514$	0	0
$515$	1.19105e6	0.197885
$516$	0	0
$517$	8.57024e6	1.41015
$518$	0	0
$519$	1.60116e6	0.260925
$520$	0	0
$521$	−3.79679e6	−0.612805	−0.306403	−	0.951902i	$-0.599125\pi$
$521$	−3.79679e6	−0.612805	−0.306403	+	0.951902i	$0.599125\pi$
$522$	0	0
$523$	5.66643e6	0.905848	0.452924	−	0.891549i	$-0.350381\pi$
$523$	5.66643e6	0.905848	0.452924	+	0.891549i	$0.350381\pi$
$524$	0	0
$525$	4.19099e6	0.663619
$526$	0	0
$527$	3.58459e6	0.562228
$528$	0	0
$529$	−6.29710e6	−0.978366
$530$	0	0
$531$	2.54400e7	3.91544
$532$	0	0
$533$	−2.88989e6	−0.440619
$534$	0	0
$535$	4.66316e6	0.704362
$536$	0	0
$537$	−2.43873e6	−0.364945
$538$	0	0
$539$	−2.23088e7	−3.30754
$540$	0	0
$541$	1.15648e7	1.69881	0.849403	−	0.527745i	$-0.176962\pi$
$541$	1.15648e7	1.69881	0.849403	+	0.527745i	$0.176962\pi$
$542$	0	0
$543$	595228.	0.0866330
$544$	0	0
$545$	360176.	0.0519426
$546$	0	0
$547$	1.28104e7	1.83060	0.915299	−	0.402776i	$-0.131955\pi$
$547$	1.28104e7	1.83060	0.915299	+	0.402776i	$0.131955\pi$
$548$	0	0
$549$	−1.61961e7	−2.29340
$550$	0	0
$551$	1.21179e6	0.170039
$552$	0	0
$553$	1.48853e6	0.206988
$554$	0	0
$555$	−9.14050e6	−1.25961
$556$	0	0
$557$	3.51737e6	0.480374	0.240187	−	0.970727i	$-0.422791\pi$
$557$	3.51737e6	0.480374	0.240187	+	0.970727i	$0.422791\pi$
$558$	0	0
$559$	1.51011e6	0.204400
$560$	0	0
$561$	−2.47273e7	−3.31719
$562$	0	0
$563$	−9.45258e6	−1.25684	−0.628419	−	0.777875i	$-0.716298\pi$
$563$	−9.45258e6	−1.25684	−0.628419	+	0.777875i	$0.716298\pi$
$564$	0	0
$565$	−1.27005e6	−0.167378
$566$	0	0
$567$	2.33385e7	3.04871
$568$	0	0
$569$	1.11315e7	1.44136	0.720679	−	0.693269i	$-0.243830\pi$
$569$	1.11315e7	1.44136	0.720679	+	0.693269i	$0.243830\pi$
$570$	0	0
$571$	−4.92935e6	−0.632702	−0.316351	−	0.948642i	$-0.602458\pi$
$571$	−4.92935e6	−0.632702	−0.316351	+	0.948642i	$0.602458\pi$
$572$	0	0
$573$	−8.57680e6	−1.09129
$574$	0	0
$575$	−233220.	−0.0294169
$576$	0	0
$577$	1.09437e7	1.36843	0.684217	−	0.729279i	$-0.260144\pi$
$577$	1.09437e7	1.36843	0.684217	+	0.729279i	$0.260144\pi$
$578$	0	0
$579$	5.08749e6	0.630677
$580$	0	0
$581$	−9.09993e6	−1.11840
$582$	0	0
$583$	−2.19727e7	−2.67739
$584$	0	0
$585$	2.25987e6	0.273019
$586$	0	0
$587$	−7.96444e6	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$587$	−7.96444e6	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$588$	0	0
$589$	1.77309e6	0.210592
$590$	0	0
$591$	−4.25352e6	−0.500934
$592$	0	0
$593$	−1.19793e7	−1.39892	−0.699460	−	0.714671i	$-0.746576\pi$
$593$	−1.19793e7	−1.39892	−0.699460	+	0.714671i	$0.746576\pi$
$594$	0	0
$595$	9.79313e6	1.13404
$596$	0	0
$597$	1.63607e7	1.87874
$598$	0	0
$599$	−7.45663e6	−0.849132	−0.424566	−	0.905397i	$-0.639573\pi$
$599$	−7.45663e6	−0.849132	−0.424566	+	0.905397i	$0.639573\pi$
$600$	0	0
$601$	−1.04196e7	−1.17670	−0.588351	−	0.808606i	$-0.700223\pi$
$601$	−1.04196e7	−1.17670	−0.588351	+	0.808606i	$0.700223\pi$
$602$	0	0
$603$	−2.07370e7	−2.32248
$604$	0	0
$605$	3.37624e6	0.375012
$606$	0	0
$607$	−571939.	−0.0630054	−0.0315027	−	0.999504i	$-0.510029\pi$
$607$	−571939.	−0.0630054	−0.0315027	+	0.999504i	$0.510029\pi$
$608$	0	0
$609$	1.00826e7	1.10162
$610$	0	0
$611$	−2.66171e6	−0.288441
$612$	0	0
$613$	−1.61323e7	−1.73398	−0.866991	−	0.498323i	$-0.833949\pi$
$613$	−1.61323e7	−1.73398	−0.866991	+	0.498323i	$0.833949\pi$
$614$	0	0
$615$	−1.19231e7	−1.27117
$616$	0	0
$617$	−6.39133e6	−0.675894	−0.337947	−	0.941165i	$-0.609732\pi$
$617$	−6.39133e6	−0.675894	−0.337947	+	0.941165i	$0.609732\pi$
$618$	0	0
$619$	−1.27062e7	−1.33287	−0.666435	−	0.745563i	$-0.732181\pi$
$619$	−1.27062e7	−1.33287	−0.666435	+	0.745563i	$0.732181\pi$
$620$	0	0
$621$	−3.03771e6	−0.316095
$622$	0	0
$623$	1.18172e7	1.21981
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.22312e7	−1.24251
$628$	0	0
$629$	−2.13587e7	−2.15252
$630$	0	0
$631$	694354.	0.0694237	0.0347118	−	0.999397i	$-0.488949\pi$
$631$	694354.	0.0694237	0.0347118	+	0.999397i	$0.488949\pi$
$632$	0	0
$633$	5.77447e6	0.572799
$634$	0	0
$635$	4.55694e6	0.448476
$636$	0	0
$637$	6.92857e6	0.676542
$638$	0	0
$639$	5.62676e6	0.545138
$640$	0	0
$641$	−8.71863e6	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$641$	−8.71863e6	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$642$	0	0
$643$	−5.98737e6	−0.571096	−0.285548	−	0.958364i	$-0.592176\pi$
$643$	−5.98737e6	−0.571096	−0.285548	+	0.958364i	$0.592176\pi$
$644$	0	0
$645$	6.23045e6	0.589685
$646$	0	0
$647$	1.09581e7	1.02914	0.514571	−	0.857448i	$-0.327951\pi$
$647$	1.09581e7	1.02914	0.514571	+	0.857448i	$0.327951\pi$
$648$	0	0
$649$	−2.58810e7	−2.41195
$650$	0	0
$651$	1.47528e7	1.36434
$652$	0	0
$653$	6.47503e6	0.594236	0.297118	−	0.954841i	$-0.403975\pi$
$653$	6.47503e6	0.594236	0.297118	+	0.954841i	$0.403975\pi$
$654$	0	0
$655$	−4.99285e6	−0.454722
$656$	0	0
$657$	846247.	0.0764863
$658$	0	0
$659$	−501723.	−0.0450039	−0.0225020	−	0.999747i	$-0.507163\pi$
$659$	−501723.	−0.0450039	−0.0225020	+	0.999747i	$0.507163\pi$
$660$	0	0
$661$	4.00659e6	0.356674	0.178337	−	0.983969i	$-0.442928\pi$
$661$	4.00659e6	0.356674	0.178337	+	0.983969i	$0.442928\pi$
$662$	0	0
$663$	7.67969e6	0.678516
$664$	0	0
$665$	4.84409e6	0.424774
$666$	0	0
$667$	−561077.	−0.0488324
$668$	0	0
$669$	2.92540e7	2.52708
$670$	0	0
$671$	1.64769e7	1.41276
$672$	0	0
$673$	8.17473e6	0.695722	0.347861	−	0.937546i	$-0.386908\pi$
$673$	8.17473e6	0.695722	0.347861	+	0.937546i	$0.386908\pi$
$674$	0	0
$675$	5.08792e6	0.429814
$676$	0	0
$677$	1.00935e6	0.0846392	0.0423196	−	0.999104i	$-0.486525\pi$
$677$	1.00935e6	0.0846392	0.0423196	+	0.999104i	$0.486525\pi$
$678$	0	0
$679$	−3.76368e6	−0.313284
$680$	0	0
$681$	2.80882e6	0.232090
$682$	0	0
$683$	−1.99189e7	−1.63386	−0.816928	−	0.576739i	$-0.804325\pi$
$683$	−1.99189e7	−1.63386	−0.816928	+	0.576739i	$0.804325\pi$
$684$	0	0
$685$	−8.63545e6	−0.703167
$686$	0	0
$687$	1.35875e7	1.09837
$688$	0	0
$689$	6.82417e6	0.547648
$690$	0	0
$691$	677407.	0.0539703	0.0269851	−	0.999636i	$-0.491409\pi$
$691$	677407.	0.0539703	0.0269851	+	0.999636i	$0.491409\pi$
$692$	0	0
$693$	−6.99772e7	−5.53508
$694$	0	0
$695$	−4.57720e6	−0.359450
$696$	0	0
$697$	−2.78609e7	−2.17227
$698$	0	0
$699$	−2.32284e7	−1.79815
$700$	0	0
$701$	1.04542e7	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$701$	1.04542e7	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$702$	0	0
$703$	−1.05649e7	−0.806263
$704$	0	0
$705$	−1.09817e7	−0.832141
$706$	0	0
$707$	−3.32046e7	−2.49833
$708$	0	0
$709$	−2.25125e7	−1.68193	−0.840964	−	0.541091i	$-0.818011\pi$
$709$	−2.25125e7	−1.68193	−0.840964	+	0.541091i	$0.818011\pi$
$710$	0	0
$711$	3.31157e6	0.245674
$712$	0	0
$713$	−820964.	−0.0604784
$714$	0	0
$715$	−2.29904e6	−0.168183
$716$	0	0
$717$	−3.76297e7	−2.73358
$718$	0	0
$719$	2.77678e6	0.200317	0.100159	−	0.994971i	$-0.468065\pi$
$719$	2.77678e6	0.200317	0.100159	+	0.994971i	$0.468065\pi$
$720$	0	0
$721$	1.14544e7	0.820603
$722$	0	0
$723$	−4.63772e7	−3.29958
$724$	0	0
$725$	939759.	0.0664006
$726$	0	0
$727$	−2.76872e6	−0.194287	−0.0971434	−	0.995270i	$-0.530971\pi$
$727$	−2.76872e6	−0.194287	−0.0971434	+	0.995270i	$0.530971\pi$
$728$	0	0
$729$	−3.25086e6	−0.226558
$730$	0	0
$731$	1.45587e7	1.00770
$732$	0	0
$733$	−1.80998e7	−1.24427	−0.622133	−	0.782911i	$-0.713734\pi$
$733$	−1.80998e7	−1.24427	−0.622133	+	0.782911i	$0.713734\pi$
$734$	0	0
$735$	2.85860e7	1.95180
$736$	0	0
$737$	2.10964e7	1.43067
$738$	0	0
$739$	3.57016e6	0.240479	0.120239	−	0.992745i	$-0.461634\pi$
$739$	3.57016e6	0.240479	0.120239	+	0.992745i	$0.461634\pi$
$740$	0	0
$741$	3.79870e6	0.254149
$742$	0	0
$743$	1.73285e7	1.15157	0.575783	−	0.817603i	$-0.304697\pi$
$743$	1.73285e7	1.15157	0.575783	+	0.817603i	$0.304697\pi$
$744$	0	0
$745$	−4.67704e6	−0.308731
$746$	0	0
$747$	−2.02448e7	−1.32743
$748$	0	0
$749$	4.48457e7	2.92090
$750$	0	0
$751$	−3.04634e6	−0.197096	−0.0985480	−	0.995132i	$-0.531420\pi$
$751$	−3.04634e6	−0.197096	−0.0985480	+	0.995132i	$0.531420\pi$
$752$	0	0
$753$	8.12374e6	0.522118
$754$	0	0
$755$	265409.	0.0169452
$756$	0	0
$757$	−8.98398e6	−0.569808	−0.284904	−	0.958556i	$-0.591962\pi$
$757$	−8.98398e6	−0.569808	−0.284904	+	0.958556i	$0.591962\pi$
$758$	0	0
$759$	5.66320e6	0.356827
$760$	0	0
$761$	1.31757e7	0.824731	0.412366	−	0.911018i	$-0.364703\pi$
$761$	1.31757e7	0.824731	0.412366	+	0.911018i	$0.364703\pi$
$762$	0	0
$763$	3.46382e6	0.215399
$764$	0	0
$765$	2.17870e7	1.34600
$766$	0	0
$767$	8.03799e6	0.493354
$768$	0	0
$769$	3.53661e6	0.215661	0.107830	−	0.994169i	$-0.465610\pi$
$769$	3.53661e6	0.215661	0.107830	+	0.994169i	$0.465610\pi$
$770$	0	0
$771$	−1.52897e7	−0.926322
$772$	0	0
$773$	−3.14397e7	−1.89247	−0.946235	−	0.323479i	$-0.895147\pi$
$773$	−3.14397e7	−1.89247	−0.946235	+	0.323479i	$0.895147\pi$
$774$	0	0
$775$	1.37505e6	0.0822364
$776$	0	0
$777$	−8.79044e7	−5.22346
$778$	0	0
$779$	−1.37812e7	−0.813659
$780$	0	0
$781$	−5.72429e6	−0.335810
$782$	0	0
$783$	1.22404e7	0.713498
$784$	0	0
$785$	−8.39710e6	−0.486357
$786$	0	0
$787$	−1.27213e7	−0.732139	−0.366069	−	0.930588i	$-0.619297\pi$
$787$	−1.27213e7	−0.732139	−0.366069	+	0.930588i	$0.619297\pi$
$788$	0	0
$789$	−2.28259e7	−1.30538
$790$	0	0
$791$	−1.22141e7	−0.694097
$792$	0	0
$793$	−5.11731e6	−0.288974
$794$	0	0
$795$	2.81552e7	1.57994
$796$	0	0
$797$	1.84237e7	1.02738	0.513689	−	0.857977i	$-0.328279\pi$
$797$	1.84237e7	1.02738	0.513689	+	0.857977i	$0.328279\pi$
$798$	0	0
$799$	−2.56610e7	−1.42203
$800$	0	0
$801$	2.62899e7	1.44780
$802$	0	0
$803$	−860915.	−0.0471164
$804$	0	0
$805$	−2.24288e6	−0.121988
$806$	0	0
$807$	−2.62019e7	−1.41628
$808$	0	0
$809$	2.10360e7	1.13004	0.565018	−	0.825079i	$-0.308869\pi$
$809$	2.10360e7	1.13004	0.565018	+	0.825079i	$0.308869\pi$
$810$	0	0
$811$	−3.53571e7	−1.88767	−0.943833	−	0.330424i	$-0.892808\pi$
$811$	−3.53571e7	−1.88767	−0.943833	+	0.330424i	$0.892808\pi$
$812$	0	0
$813$	4.25097e7	2.25560
$814$	0	0
$815$	−5.04702e6	−0.266159
$816$	0	0
$817$	7.20136e6	0.377450
$818$	0	0
$819$	2.17332e7	1.13218
$820$	0	0
$821$	1.72038e7	0.890770	0.445385	−	0.895339i	$-0.353067\pi$
$821$	1.72038e7	0.890770	0.445385	+	0.895339i	$0.353067\pi$
$822$	0	0
$823$	947098.	0.0487411	0.0243705	−	0.999703i	$-0.492242\pi$
$823$	947098.	0.0487411	0.0243705	+	0.999703i	$0.492242\pi$
$824$	0	0
$825$	−9.48541e6	−0.485200
$826$	0	0
$827$	3.05385e7	1.55269	0.776345	−	0.630308i	$-0.217072\pi$
$827$	3.05385e7	1.55269	0.776345	+	0.630308i	$0.217072\pi$
$828$	0	0
$829$	−1.57415e7	−0.795538	−0.397769	−	0.917486i	$-0.630215\pi$
$829$	−1.57415e7	−0.795538	−0.397769	+	0.917486i	$0.630215\pi$
$830$	0	0
$831$	−1.22636e7	−0.616047
$832$	0	0
$833$	6.67971e7	3.33538
$834$	0	0
$835$	1.11513e7	0.553487
$836$	0	0
$837$	1.79101e7	0.883659
$838$	0	0
$839$	−2.07700e7	−1.01867	−0.509333	−	0.860570i	$-0.670108\pi$
$839$	−2.07700e7	−1.01867	−0.509333	+	0.860570i	$0.670108\pi$
$840$	0	0
$841$	−1.82503e7	−0.889774
$842$	0	0
$843$	−1.65121e7	−0.800263
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	3.24694e7	1.55513
$848$	0	0
$849$	2.16772e7	1.03213
$850$	0	0
$851$	4.89169e6	0.231545
$852$	0	0
$853$	−1.91838e7	−0.902740	−0.451370	−	0.892337i	$-0.649064\pi$
$853$	−1.91838e7	−0.902740	−0.451370	+	0.892337i	$0.649064\pi$
$854$	0	0
$855$	1.07768e7	0.504165
$856$	0	0
$857$	2.64211e7	1.22885	0.614425	−	0.788975i	$-0.289388\pi$
$857$	2.64211e7	1.22885	0.614425	+	0.788975i	$0.289388\pi$
$858$	0	0
$859$	3.30576e7	1.52858	0.764289	−	0.644873i	$-0.223090\pi$
$859$	3.30576e7	1.52858	0.764289	+	0.644873i	$0.223090\pi$
$860$	0	0
$861$	−1.14665e8	−5.27137
$862$	0	0
$863$	−1.19324e7	−0.545380	−0.272690	−	0.962102i	$-0.587913\pi$
$863$	−1.19324e7	−0.545380	−0.272690	+	0.962102i	$0.587913\pi$
$864$	0	0
$865$	1.43522e6	0.0652196
$866$	0	0
$867$	3.44381e7	1.55593
$868$	0	0
$869$	−3.36897e6	−0.151338
$870$	0	0
$871$	−6.55203e6	−0.292638
$872$	0	0
$873$	−8.37313e6	−0.371837
$874$	0	0
$875$	3.75665e6	0.165875
$876$	0	0
$877$	3.88434e6	0.170537	0.0852684	−	0.996358i	$-0.472825\pi$
$877$	3.88434e6	0.170537	0.0852684	+	0.996358i	$0.472825\pi$
$878$	0	0
$879$	1.69860e7	0.741513
$880$	0	0
$881$	−8.93397e6	−0.387797	−0.193899	−	0.981022i	$-0.562113\pi$
$881$	−8.93397e6	−0.387797	−0.193899	+	0.981022i	$0.562113\pi$
$882$	0	0
$883$	4.32552e6	0.186697	0.0933484	−	0.995634i	$-0.470243\pi$
$883$	4.32552e6	0.186697	0.0933484	+	0.995634i	$0.470243\pi$
$884$	0	0
$885$	3.31632e7	1.42331
$886$	0	0
$887$	−3.55552e7	−1.51738	−0.758689	−	0.651453i	$-0.774160\pi$
$887$	−3.55552e7	−1.51738	−0.758689	+	0.651453i	$0.774160\pi$
$888$	0	0
$889$	4.38242e7	1.85977
$890$	0	0
$891$	−5.28217e7	−2.22904
$892$	0	0
$893$	−1.26930e7	−0.532643
$894$	0	0
$895$	−2.18599e6	−0.0912199
$896$	0	0
$897$	−1.75885e6	−0.0729874
$898$	0	0
$899$	3.30807e6	0.136514
$900$	0	0
$901$	6.57906e7	2.69993
$902$	0	0
$903$	5.99184e7	2.44535
$904$	0	0
$905$	533540.	0.0216544
$906$	0	0
$907$	−2.44157e7	−0.985486	−0.492743	−	0.870175i	$-0.664006\pi$
$907$	−2.44157e7	−0.985486	−0.492743	+	0.870175i	$0.664006\pi$
$908$	0	0
$909$	−7.38709e7	−2.96527
$910$	0	0
$911$	−3.00267e7	−1.19870	−0.599352	−	0.800486i	$-0.704575\pi$
$911$	−3.00267e7	−1.19870	−0.599352	+	0.800486i	$0.704575\pi$
$912$	0	0
$913$	2.05957e7	0.817711
$914$	0	0
$915$	−2.11131e7	−0.833679
$916$	0	0
$917$	−4.80164e7	−1.88567
$918$	0	0
$919$	1.30118e7	0.508216	0.254108	−	0.967176i	$-0.418218\pi$
$919$	1.30118e7	0.508216	0.254108	+	0.967176i	$0.418218\pi$
$920$	0	0
$921$	7.44009e7	2.89021
$922$	0	0
$923$	1.77782e6	0.0686886
$924$	0	0
$925$	−8.19320e6	−0.314847
$926$	0	0
$927$	2.54828e7	0.973975
$928$	0	0
$929$	−5.16678e7	−1.96418	−0.982089	−	0.188419i	$-0.939664\pi$
$929$	−5.16678e7	−1.96418	−0.982089	+	0.188419i	$0.939664\pi$
$930$	0	0
$931$	3.30407e7	1.24932
$932$	0	0
$933$	4.52717e7	1.70264
$934$	0	0
$935$	−2.21646e7	−0.829147
$936$	0	0
$937$	−4.04703e6	−0.150587	−0.0752936	−	0.997161i	$-0.523989\pi$
$937$	−4.04703e6	−0.150587	−0.0752936	+	0.997161i	$0.523989\pi$
$938$	0	0
$939$	1.65144e7	0.611221
$940$	0	0
$941$	−4.43008e7	−1.63094	−0.815470	−	0.578799i	$-0.803521\pi$
$941$	−4.43008e7	−1.63094	−0.815470	+	0.578799i	$0.803521\pi$
$942$	0	0
$943$	6.38088e6	0.233669
$944$	0	0
$945$	4.89307e7	1.78239
$946$	0	0
$947$	3.60463e7	1.30613	0.653064	−	0.757303i	$-0.273483\pi$
$947$	3.60463e7	1.30613	0.653064	+	0.757303i	$0.273483\pi$
$948$	0	0
$949$	267379.	0.00963745
$950$	0	0
$951$	−7.28042e7	−2.61039
$952$	0	0
$953$	−1.31248e7	−0.468123	−0.234061	−	0.972222i	$-0.575202\pi$
$953$	−1.31248e7	−0.468123	−0.234061	+	0.972222i	$0.575202\pi$
$954$	0	0
$955$	−7.68792e6	−0.272773
$956$	0	0
$957$	−2.28198e7	−0.805439
$958$	0	0
$959$	−8.30473e7	−2.91594
$960$	0	0
$961$	−2.37888e7	−0.830929
$962$	0	0
$963$	9.97693e7	3.46682
$964$	0	0
$965$	4.56023e6	0.157641
$966$	0	0
$967$	−2.48150e7	−0.853390	−0.426695	−	0.904396i	$-0.640322\pi$
$967$	−2.48150e7	−0.853390	−0.426695	+	0.904396i	$0.640322\pi$
$968$	0	0
$969$	3.66226e7	1.25297
$970$	0	0
$971$	3.54227e6	0.120569	0.0602843	−	0.998181i	$-0.480799\pi$
$971$	3.54227e6	0.120569	0.0602843	+	0.998181i	$0.480799\pi$
$972$	0	0
$973$	−4.40191e7	−1.49059
$974$	0	0
$975$	2.94593e6	0.0992456
$976$	0	0
$977$	9.16916e6	0.307322	0.153661	−	0.988124i	$-0.450894\pi$
$977$	9.16916e6	0.307322	0.153661	+	0.988124i	$0.450894\pi$
$978$	0	0
$979$	−2.67456e7	−0.891857
$980$	0	0
$981$	7.70604e6	0.255658
$982$	0	0
$983$	3.05912e7	1.00975	0.504874	−	0.863193i	$-0.331539\pi$
$983$	3.05912e7	1.00975	0.504874	+	0.863193i	$0.331539\pi$
$984$	0	0
$985$	−3.81270e6	−0.125211
$986$	0	0
$987$	−1.05611e8	−3.45078
$988$	0	0
$989$	−3.33433e6	−0.108397
$990$	0	0
$991$	−3.47852e7	−1.12515	−0.562575	−	0.826746i	$-0.690189\pi$
$991$	−3.47852e7	−1.12515	−0.562575	+	0.826746i	$0.690189\pi$
$992$	0	0
$993$	−2.28363e7	−0.734942
$994$	0	0
$995$	1.46651e7	0.469600
$996$	0	0
$997$	−4.21812e7	−1.34394	−0.671972	−	0.740577i	$-0.734552\pi$
$997$	−4.21812e7	−1.34394	−0.671972	+	0.740577i	$0.734552\pi$
$998$	0	0
$999$	−1.06717e8	−3.38314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.l.1.3		3
4.3	odd	2		130.6.a.f.1.1	✓	3
20.3	even	4		650.6.b.j.599.4		6
20.7	even	4		650.6.b.j.599.3		6
20.19	odd	2		650.6.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.1	✓	3	4.3	odd	2
650.6.a.j.1.3		3	20.19	odd	2
650.6.b.j.599.3		6	20.7	even	4
650.6.b.j.599.4		6	20.3	even	4
1040.6.a.l.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+27.8905 q^{3} +25.0000 q^{5} +240.426 q^{7} +534.880 q^{9} -544.151 q^{11} +169.000 q^{13} +697.262 q^{15} +1629.30 q^{17} +805.920 q^{19} +6705.59 q^{21} -373.152 q^{23} +625.000 q^{25} +8140.67 q^{27} +1503.62 q^{29} +2200.08 q^{31} -15176.7 q^{33} +6010.64 q^{35} -13109.1 q^{37} +4713.49 q^{39} -17099.9 q^{41} +8935.58 q^{43} +13372.0 q^{45} -15749.7 q^{47} +40997.5 q^{49} +45442.0 q^{51} +40379.7 q^{53} -13603.8 q^{55} +22477.5 q^{57} +47562.1 q^{59} -30280.0 q^{61} +128599. q^{63} +4225.00 q^{65} -38769.4 q^{67} -10407.4 q^{69} +10519.7 q^{71} +1582.12 q^{73} +17431.6 q^{75} -130828. q^{77} +6191.23 q^{79} +97071.7 q^{81} -37849.2 q^{83} +40732.5 q^{85} +41936.6 q^{87} +49151.0 q^{89} +40631.9 q^{91} +61361.3 q^{93} +20148.0 q^{95} -15654.2 q^{97} -291056. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 22 q^{3} + 75 q^{5} + 234 q^{7} + 155 q^{9} - 48 q^{11} + 507 q^{13} + 550 q^{15} + 1506 q^{17} + 360 q^{19} + 6904 q^{21} + 2370 q^{23} + 1875 q^{25} + 10168 q^{27} - 3078 q^{29} + 5388 q^{31} - 22572 q^{33}+ \cdots - 350340 q^{99}+O(q^{100}) \) 3 * q + 22 * q^3 + 75 * q^5 + 234 * q^7 + 155 * q^9 - 48 * q^11 + 507 * q^13 + 550 * q^15 + 1506 * q^17 + 360 * q^19 + 6904 * q^21 + 2370 * q^23 + 1875 * q^25 + 10168 * q^27 - 3078 * q^29 + 5388 * q^31 - 22572 * q^33 + 5850 * q^35 - 25362 * q^37 + 3718 * q^39 - 15906 * q^41 + 39306 * q^43 + 3875 * q^45 + 17778 * q^47 + 7767 * q^49 + 33972 * q^51 + 9246 * q^53 - 1200 * q^55 - 2996 * q^57 + 77760 * q^59 + 17982 * q^61 + 128762 * q^63 + 12675 * q^65 + 2922 * q^67 - 31860 * q^69 + 4944 * q^71 + 43278 * q^73 + 13750 * q^75 - 144420 * q^77 - 42120 * q^79 + 146519 * q^81 - 58098 * q^83 + 37650 * q^85 - 2016 * q^87 + 19614 * q^89 + 39546 * q^91 + 162796 * q^93 + 9000 * q^95 - 87078 * q^97 - 350340 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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