LMFDB - Embedded newform 1040.6.a.e.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ 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)

$f(q)$	$=$	$q-13.8114 q^{3} +25.0000 q^{5} -213.246 q^{7} -52.2456 q^{9} -587.285 q^{11} -169.000 q^{13} -345.285 q^{15} +162.605 q^{17} +81.3914 q^{19} +2945.22 q^{21} +2948.84 q^{23} +625.000 q^{25} +4077.75 q^{27} +6254.43 q^{29} +4034.62 q^{31} +8111.22 q^{33} -5331.14 q^{35} +7617.22 q^{37} +2334.12 q^{39} +958.121 q^{41} -169.655 q^{43} -1306.14 q^{45} -21612.6 q^{47} +28666.7 q^{49} -2245.80 q^{51} +24789.1 q^{53} -14682.1 q^{55} -1124.13 q^{57} -40109.2 q^{59} -16343.0 q^{61} +11141.1 q^{63} -4225.00 q^{65} -18362.1 q^{67} -40727.6 q^{69} +77846.0 q^{71} -62130.2 q^{73} -8632.12 q^{75} +125236. q^{77} +60599.1 q^{79} -43623.7 q^{81} +2654.46 q^{83} +4065.12 q^{85} -86382.4 q^{87} +8229.77 q^{89} +36038.5 q^{91} -55723.7 q^{93} +2034.79 q^{95} -53844.0 q^{97} +30683.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 50 q^{5} - 300 q^{7} + 22 q^{9} - 384 q^{11} - 338 q^{13} + 100 q^{15} - 244 q^{17} + 2408 q^{19} + 1400 q^{21} - 332 q^{23} + 1250 q^{25} + 1072 q^{27} + 3528 q^{29} - 880 q^{31} + 11732 q^{33}+ \cdots + 45776 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 50 * q^5 - 300 * q^7 + 22 * q^9 - 384 * q^11 - 338 * q^13 + 100 * q^15 - 244 * q^17 + 2408 * q^19 + 1400 * q^21 - 332 * q^23 + 1250 * q^25 + 1072 * q^27 + 3528 * q^29 - 880 * q^31 + 11732 * q^33 - 7500 * q^35 + 10744 * q^37 - 676 * q^39 + 16020 * q^41 - 2964 * q^43 + 550 * q^45 - 38292 * q^47 + 19386 * q^49 - 9488 * q^51 + 42052 * q^53 - 9600 * q^55 + 40316 * q^57 - 42872 * q^59 - 33192 * q^61 + 4700 * q^63 - 8450 * q^65 - 37420 * q^67 - 99164 * q^69 + 69520 * q^71 - 12632 * q^73 + 2500 * q^75 + 107600 * q^77 - 35208 * q^79 - 115202 * q^81 - 31500 * q^83 - 6100 * q^85 - 134944 * q^87 - 18452 * q^89 + 50700 * q^91 - 143260 * q^93 + 60200 * q^95 - 28884 * q^97 + 45776 * 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
$3$	−13.8114	−0.886001	−0.443000	−	0.896521i	$-0.646086\pi$
$3$	−13.8114	−0.886001	−0.443000	+	0.896521i	$0.646086\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−213.246	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$7$	−213.246	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$8$	0	0
$9$	−52.2456	−0.215002
$10$	0	0
$11$	−587.285	−1.46341	−0.731707	−	0.681620i	$-0.761276\pi$
$11$	−587.285	−1.46341	−0.731707	+	0.681620i	$0.761276\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	−345.285	−0.396232
$16$	0	0
$17$	162.605	0.136462	0.0682310	−	0.997670i	$-0.478265\pi$
$17$	162.605	0.136462	0.0682310	+	0.997670i	$0.478265\pi$
$18$	0	0
$19$	81.3914	0.0517243	0.0258622	−	0.999666i	$-0.491767\pi$
$19$	81.3914	0.0517243	0.0258622	+	0.999666i	$0.491767\pi$
$20$	0	0
$21$	2945.22	1.45737
$22$	0	0
$23$	2948.84	1.16234	0.581169	−	0.813783i	$-0.302596\pi$
$23$	2948.84	1.16234	0.581169	+	0.813783i	$0.302596\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4077.75	1.07649
$28$	0	0
$29$	6254.43	1.38100	0.690499	−	0.723333i	$-0.257391\pi$
$29$	6254.43	1.38100	0.690499	+	0.723333i	$0.257391\pi$
$30$	0	0
$31$	4034.62	0.754048	0.377024	−	0.926204i	$-0.376948\pi$
$31$	4034.62	0.754048	0.377024	+	0.926204i	$0.376948\pi$
$32$	0	0
$33$	8111.22	1.29659
$34$	0	0
$35$	−5331.14	−0.735614
$36$	0	0
$37$	7617.22	0.914728	0.457364	−	0.889280i	$-0.348794\pi$
$37$	7617.22	0.914728	0.457364	+	0.889280i	$0.348794\pi$
$38$	0	0
$39$	2334.12	0.245732
$40$	0	0
$41$	958.121	0.0890145	0.0445072	−	0.999009i	$-0.485828\pi$
$41$	958.121	0.0890145	0.0445072	+	0.999009i	$0.485828\pi$
$42$	0	0
$43$	−169.655	−0.0139925	−0.00699624	−	0.999976i	$-0.502227\pi$
$43$	−169.655	−0.0139925	−0.00699624	+	0.999976i	$0.502227\pi$
$44$	0	0
$45$	−1306.14	−0.0961519
$46$	0	0
$47$	−21612.6	−1.42712	−0.713562	−	0.700592i	$-0.752919\pi$
$47$	−21612.6	−1.42712	−0.713562	+	0.700592i	$0.752919\pi$
$48$	0	0
$49$	28666.7	1.70564
$50$	0	0
$51$	−2245.80	−0.120905
$52$	0	0
$53$	24789.1	1.21219	0.606096	−	0.795392i	$-0.292735\pi$
$53$	24789.1	1.21219	0.606096	+	0.795392i	$0.292735\pi$
$54$	0	0
$55$	−14682.1	−0.654458
$56$	0	0
$57$	−1124.13	−0.0458278
$58$	0	0
$59$	−40109.2	−1.50008	−0.750040	−	0.661392i	$-0.769966\pi$
$59$	−40109.2	−1.50008	−0.750040	+	0.661392i	$0.769966\pi$
$60$	0	0
$61$	−16343.0	−0.562351	−0.281176	−	0.959656i	$-0.590724\pi$
$61$	−16343.0	−0.562351	−0.281176	+	0.959656i	$0.590724\pi$
$62$	0	0
$63$	11141.1	0.353653
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−18362.1	−0.499731	−0.249866	−	0.968281i	$-0.580386\pi$
$67$	−18362.1	−0.499731	−0.249866	+	0.968281i	$0.580386\pi$
$68$	0	0
$69$	−40727.6	−1.02983
$70$	0	0
$71$	77846.0	1.83270	0.916348	−	0.400382i	$-0.131123\pi$
$71$	77846.0	1.83270	0.916348	+	0.400382i	$0.131123\pi$
$72$	0	0
$73$	−62130.2	−1.36457	−0.682285	−	0.731087i	$-0.739014\pi$
$73$	−62130.2	−1.36457	−0.682285	+	0.731087i	$0.739014\pi$
$74$	0	0
$75$	−8632.12	−0.177200
$76$	0	0
$77$	125236.	2.40714
$78$	0	0
$79$	60599.1	1.09244	0.546221	−	0.837641i	$-0.316066\pi$
$79$	60599.1	1.09244	0.546221	+	0.837641i	$0.316066\pi$
$80$	0	0
$81$	−43623.7	−0.738772
$82$	0	0
$83$	2654.46	0.0422941	0.0211471	−	0.999776i	$-0.493268\pi$
$83$	2654.46	0.0422941	0.0211471	+	0.999776i	$0.493268\pi$
$84$	0	0
$85$	4065.12	0.0610276
$86$	0	0
$87$	−86382.4	−1.22357
$88$	0	0
$89$	8229.77	0.110132	0.0550659	−	0.998483i	$-0.482463\pi$
$89$	8229.77	0.110132	0.0550659	+	0.998483i	$0.482463\pi$
$90$	0	0
$91$	36038.5	0.456208
$92$	0	0
$93$	−55723.7	−0.668087
$94$	0	0
$95$	2034.79	0.0231318
$96$	0	0
$97$	−53844.0	−0.581042	−0.290521	−	0.956869i	$-0.593829\pi$
$97$	−53844.0	−0.581042	−0.290521	+	0.956869i	$0.593829\pi$
$98$	0	0
$99$	30683.0	0.314637
$100$	0	0
$101$	−167556.	−1.63439	−0.817196	−	0.576360i	$-0.804473\pi$
$101$	−167556.	−1.63439	−0.817196	+	0.576360i	$0.804473\pi$
$102$	0	0
$103$	−30973.8	−0.287675	−0.143837	−	0.989601i	$-0.545944\pi$
$103$	−30973.8	−0.287675	−0.143837	+	0.989601i	$0.545944\pi$
$104$	0	0
$105$	73630.4	0.651755
$106$	0	0
$107$	97343.8	0.821956	0.410978	−	0.911645i	$-0.365187\pi$
$107$	97343.8	0.821956	0.410978	+	0.911645i	$0.365187\pi$
$108$	0	0
$109$	16696.8	0.134607	0.0673035	−	0.997733i	$-0.478560\pi$
$109$	16696.8	0.134607	0.0673035	+	0.997733i	$0.478560\pi$
$110$	0	0
$111$	−105204.	−0.810450
$112$	0	0
$113$	219357.	1.61605	0.808027	−	0.589145i	$-0.200535\pi$
$113$	219357.	1.61605	0.808027	+	0.589145i	$0.200535\pi$
$114$	0	0
$115$	73721.1	0.519813
$116$	0	0
$117$	8829.50	0.0596309
$118$	0	0
$119$	−34674.8	−0.224464
$120$	0	0
$121$	183852.	1.14158
$122$	0	0
$123$	−13233.0	−0.0788669
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−158753.	−0.873399	−0.436700	−	0.899607i	$-0.643853\pi$
$127$	−158753.	−0.873399	−0.436700	+	0.899607i	$0.643853\pi$
$128$	0	0
$129$	2343.17	0.0123974
$130$	0	0
$131$	−160566.	−0.817477	−0.408739	−	0.912651i	$-0.634031\pi$
$131$	−160566.	−0.817477	−0.408739	+	0.912651i	$0.634031\pi$
$132$	0	0
$133$	−17356.4	−0.0850804
$134$	0	0
$135$	101944.	0.481422
$136$	0	0
$137$	−295668.	−1.34587	−0.672934	−	0.739702i	$-0.734966\pi$
$137$	−295668.	−1.34587	−0.672934	+	0.739702i	$0.734966\pi$
$138$	0	0
$139$	−6197.57	−0.0272072	−0.0136036	−	0.999907i	$-0.504330\pi$
$139$	−6197.57	−0.0272072	−0.0136036	+	0.999907i	$0.504330\pi$
$140$	0	0
$141$	298500.	1.26443
$142$	0	0
$143$	99251.1	0.405878
$144$	0	0
$145$	156361.	0.617601
$146$	0	0
$147$	−395926.	−1.51120
$148$	0	0
$149$	−371033.	−1.36914	−0.684569	−	0.728948i	$-0.740010\pi$
$149$	−371033.	−1.36914	−0.684569	+	0.728948i	$0.740010\pi$
$150$	0	0
$151$	508361.	1.81439	0.907193	−	0.420714i	$-0.138220\pi$
$151$	508361.	1.81439	0.907193	+	0.420714i	$0.138220\pi$
$152$	0	0
$153$	−8495.39	−0.0293396
$154$	0	0
$155$	100866.	0.337220
$156$	0	0
$157$	60014.9	0.194317	0.0971583	−	0.995269i	$-0.469025\pi$
$157$	60014.9	0.194317	0.0971583	+	0.995269i	$0.469025\pi$
$158$	0	0
$159$	−342372.	−1.07400
$160$	0	0
$161$	−628828.	−1.91191
$162$	0	0
$163$	365913.	1.07872	0.539360	−	0.842075i	$-0.318666\pi$
$163$	365913.	1.07872	0.539360	+	0.842075i	$0.318666\pi$
$164$	0	0
$165$	202780.	0.579851
$166$	0	0
$167$	−67392.0	−0.186990	−0.0934948	−	0.995620i	$-0.529804\pi$
$167$	−67392.0	−0.186990	−0.0934948	+	0.995620i	$0.529804\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−4252.34	−0.0111208
$172$	0	0
$173$	518921.	1.31821	0.659107	−	0.752049i	$-0.270934\pi$
$173$	518921.	1.31821	0.659107	+	0.752049i	$0.270934\pi$
$174$	0	0
$175$	−133278.	−0.328977
$176$	0	0
$177$	553964.	1.32907
$178$	0	0
$179$	89728.6	0.209314	0.104657	−	0.994508i	$-0.466626\pi$
$179$	89728.6	0.209314	0.104657	+	0.994508i	$0.466626\pi$
$180$	0	0
$181$	−376272.	−0.853700	−0.426850	−	0.904322i	$-0.640377\pi$
$181$	−376272.	−0.853700	−0.426850	+	0.904322i	$0.640377\pi$
$182$	0	0
$183$	225720.	0.498244
$184$	0	0
$185$	190430.	0.409079
$186$	0	0
$187$	−95495.4	−0.199700
$188$	0	0
$189$	−869562.	−1.77070
$190$	0	0
$191$	−206748.	−0.410070	−0.205035	−	0.978755i	$-0.565731\pi$
$191$	−206748.	−0.410070	−0.205035	+	0.978755i	$0.565731\pi$
$192$	0	0
$193$	−413088.	−0.798269	−0.399135	−	0.916892i	$-0.630689\pi$
$193$	−413088.	−0.798269	−0.399135	+	0.916892i	$0.630689\pi$
$194$	0	0
$195$	58353.1	0.109895
$196$	0	0
$197$	538825.	0.989196	0.494598	−	0.869122i	$-0.335315\pi$
$197$	538825.	0.989196	0.494598	+	0.869122i	$0.335315\pi$
$198$	0	0
$199$	147336.	0.263740	0.131870	−	0.991267i	$-0.457902\pi$
$199$	147336.	0.263740	0.131870	+	0.991267i	$0.457902\pi$
$200$	0	0
$201$	253607.	0.442762
$202$	0	0
$203$	−1.33373e6	−2.27158
$204$	0	0
$205$	23953.0	0.0398085
$206$	0	0
$207$	−154064.	−0.249905
$208$	0	0
$209$	−47799.9	−0.0756940
$210$	0	0
$211$	1.14921e6	1.77703	0.888515	−	0.458847i	$-0.151738\pi$
$211$	1.14921e6	1.77703	0.888515	+	0.458847i	$0.151738\pi$
$212$	0	0
$213$	−1.07516e6	−1.62377
$214$	0	0
$215$	−4241.37	−0.00625763
$216$	0	0
$217$	−860365.	−1.24032
$218$	0	0
$219$	858104.	1.20901
$220$	0	0
$221$	−27480.2	−0.0378477
$222$	0	0
$223$	1.39430e6	1.87756	0.938782	−	0.344513i	$-0.111956\pi$
$223$	1.39430e6	1.87756	0.938782	+	0.344513i	$0.111956\pi$
$224$	0	0
$225$	−32653.5	−0.0430005
$226$	0	0
$227$	−1.40291e6	−1.80703	−0.903515	−	0.428556i	$-0.859023\pi$
$227$	−1.40291e6	−1.80703	−0.903515	+	0.428556i	$0.859023\pi$
$228$	0	0
$229$	−796626.	−1.00384	−0.501922	−	0.864913i	$-0.667373\pi$
$229$	−796626.	−1.00384	−0.501922	+	0.864913i	$0.667373\pi$
$230$	0	0
$231$	−1.72968e6	−2.13273
$232$	0	0
$233$	1.15975e6	1.39951	0.699753	−	0.714385i	$-0.253293\pi$
$233$	1.15975e6	1.39951	0.699753	+	0.714385i	$0.253293\pi$
$234$	0	0
$235$	−540314.	−0.638230
$236$	0	0
$237$	−836958.	−0.967905
$238$	0	0
$239$	687529.	0.778568	0.389284	−	0.921118i	$-0.372722\pi$
$239$	687529.	0.778568	0.389284	+	0.921118i	$0.372722\pi$
$240$	0	0
$241$	−391345.	−0.434028	−0.217014	−	0.976169i	$-0.569632\pi$
$241$	−391345.	−0.434028	−0.217014	+	0.976169i	$0.569632\pi$
$242$	0	0
$243$	−388389.	−0.421941
$244$	0	0
$245$	716667.	0.762785
$246$	0	0
$247$	−13755.2	−0.0143457
$248$	0	0
$249$	−36661.7	−0.0374727
$250$	0	0
$251$	−944723.	−0.946499	−0.473249	−	0.880928i	$-0.656919\pi$
$251$	−944723.	−0.946499	−0.473249	+	0.880928i	$0.656919\pi$
$252$	0	0
$253$	−1.73181e6	−1.70098
$254$	0	0
$255$	−56145.0	−0.0540705
$256$	0	0
$257$	297219.	0.280701	0.140351	−	0.990102i	$-0.455177\pi$
$257$	297219.	0.280701	0.140351	+	0.990102i	$0.455177\pi$
$258$	0	0
$259$	−1.62434e6	−1.50462
$260$	0	0
$261$	−326766.	−0.296918
$262$	0	0
$263$	1.53097e6	1.36483	0.682414	−	0.730966i	$-0.260930\pi$
$263$	1.53097e6	1.36483	0.682414	+	0.730966i	$0.260930\pi$
$264$	0	0
$265$	619728.	0.542109
$266$	0	0
$267$	−113665.	−0.0975769
$268$	0	0
$269$	−1.00878e6	−0.849994	−0.424997	−	0.905195i	$-0.639725\pi$
$269$	−1.00878e6	−0.849994	−0.424997	+	0.905195i	$0.639725\pi$
$270$	0	0
$271$	−2.19122e6	−1.81244	−0.906220	−	0.422806i	$-0.861045\pi$
$271$	−2.19122e6	−1.81244	−0.906220	+	0.422806i	$0.861045\pi$
$272$	0	0
$273$	−497742.	−0.404201
$274$	0	0
$275$	−367053.	−0.292683
$276$	0	0
$277$	−17924.0	−0.0140357	−0.00701786	−	0.999975i	$-0.502234\pi$
$277$	−17924.0	−0.0140357	−0.00701786	+	0.999975i	$0.502234\pi$
$278$	0	0
$279$	−210791.	−0.162122
$280$	0	0
$281$	−2.40663e6	−1.81821	−0.909106	−	0.416566i	$-0.863234\pi$
$281$	−2.40663e6	−1.81821	−0.909106	+	0.416566i	$0.863234\pi$
$282$	0	0
$283$	1.36921e6	1.01626	0.508128	−	0.861281i	$-0.330338\pi$
$283$	1.36921e6	1.01626	0.508128	+	0.861281i	$0.330338\pi$
$284$	0	0
$285$	−28103.2	−0.0204948
$286$	0	0
$287$	−204315.	−0.146418
$288$	0	0
$289$	−1.39342e6	−0.981378
$290$	0	0
$291$	743660.	0.514804
$292$	0	0
$293$	−390649.	−0.265838	−0.132919	−	0.991127i	$-0.542435\pi$
$293$	−390649.	−0.265838	−0.132919	+	0.991127i	$0.542435\pi$
$294$	0	0
$295$	−1.00273e6	−0.670856
$296$	0	0
$297$	−2.39480e6	−1.57535
$298$	0	0
$299$	−498355.	−0.322374
$300$	0	0
$301$	36178.1	0.0230160
$302$	0	0
$303$	2.31418e6	1.44807
$304$	0	0
$305$	−408575.	−0.251491
$306$	0	0
$307$	−566546.	−0.343075	−0.171537	−	0.985178i	$-0.554873\pi$
$307$	−566546.	−0.343075	−0.171537	+	0.985178i	$0.554873\pi$
$308$	0	0
$309$	427791.	0.254880
$310$	0	0
$311$	−881239.	−0.516646	−0.258323	−	0.966059i	$-0.583170\pi$
$311$	−881239.	−0.516646	−0.258323	+	0.966059i	$0.583170\pi$
$312$	0	0
$313$	−916421.	−0.528731	−0.264365	−	0.964423i	$-0.585162\pi$
$313$	−916421.	−0.528731	−0.264365	+	0.964423i	$0.585162\pi$
$314$	0	0
$315$	278528.	0.158159
$316$	0	0
$317$	2.01084e6	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$317$	2.01084e6	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$318$	0	0
$319$	−3.67313e6	−2.02097
$320$	0	0
$321$	−1.34445e6	−0.728254
$322$	0	0
$323$	13234.7	0.00705840
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	−230606.	−0.119262
$328$	0	0
$329$	4.60879e6	2.34745
$330$	0	0
$331$	−2.47715e6	−1.24275	−0.621373	−	0.783515i	$-0.713425\pi$
$331$	−2.47715e6	−1.24275	−0.621373	+	0.783515i	$0.713425\pi$
$332$	0	0
$333$	−397966.	−0.196669
$334$	0	0
$335$	−459054.	−0.223487
$336$	0	0
$337$	2.55550e6	1.22575	0.612873	−	0.790181i	$-0.290014\pi$
$337$	2.55550e6	1.22575	0.612873	+	0.790181i	$0.290014\pi$
$338$	0	0
$339$	−3.02963e6	−1.43183
$340$	0	0
$341$	−2.36947e6	−1.10348
$342$	0	0
$343$	−2.52902e6	−1.16069
$344$	0	0
$345$	−1.01819e6	−0.460555
$346$	0	0
$347$	475000.	0.211773	0.105886	−	0.994378i	$-0.466232\pi$
$347$	475000.	0.211773	0.105886	+	0.994378i	$0.466232\pi$
$348$	0	0
$349$	615277.	0.270400	0.135200	−	0.990818i	$-0.456832\pi$
$349$	615277.	0.270400	0.135200	+	0.990818i	$0.456832\pi$
$350$	0	0
$351$	−689140.	−0.298565
$352$	0	0
$353$	−4.37400e6	−1.86828	−0.934140	−	0.356908i	$-0.883831\pi$
$353$	−4.37400e6	−1.86828	−0.934140	+	0.356908i	$0.883831\pi$
$354$	0	0
$355$	1.94615e6	0.819607
$356$	0	0
$357$	478907.	0.198875
$358$	0	0
$359$	−4.59562e6	−1.88195	−0.940975	−	0.338476i	$-0.890089\pi$
$359$	−4.59562e6	−1.88195	−0.940975	+	0.338476i	$0.890089\pi$
$360$	0	0
$361$	−2.46947e6	−0.997325
$362$	0	0
$363$	−2.53926e6	−1.01144
$364$	0	0
$365$	−1.55326e6	−0.610254
$366$	0	0
$367$	4.59362e6	1.78029	0.890143	−	0.455681i	$-0.150604\pi$
$367$	4.59362e6	1.78029	0.890143	+	0.455681i	$0.150604\pi$
$368$	0	0
$369$	−50057.6	−0.0191383
$370$	0	0
$371$	−5.28617e6	−1.99391
$372$	0	0
$373$	1.74314e6	0.648724	0.324362	−	0.945933i	$-0.394850\pi$
$373$	1.74314e6	0.648724	0.324362	+	0.945933i	$0.394850\pi$
$374$	0	0
$375$	−215803.	−0.0792463
$376$	0	0
$377$	−1.05700e6	−0.383020
$378$	0	0
$379$	4.42529e6	1.58250	0.791250	−	0.611493i	$-0.209431\pi$
$379$	4.42529e6	1.58250	0.791250	+	0.611493i	$0.209431\pi$
$380$	0	0
$381$	2.19260e6	0.773833
$382$	0	0
$383$	−1.39773e6	−0.486884	−0.243442	−	0.969915i	$-0.578277\pi$
$383$	−1.39773e6	−0.486884	−0.243442	+	0.969915i	$0.578277\pi$
$384$	0	0
$385$	3.13090e6	1.07651
$386$	0	0
$387$	8863.71	0.00300842
$388$	0	0
$389$	1.63535e6	0.547946	0.273973	−	0.961737i	$-0.411662\pi$
$389$	1.63535e6	0.547946	0.273973	+	0.961737i	$0.411662\pi$
$390$	0	0
$391$	479497.	0.158615
$392$	0	0
$393$	2.21764e6	0.724286
$394$	0	0
$395$	1.51498e6	0.488555
$396$	0	0
$397$	4.90653e6	1.56242	0.781210	−	0.624268i	$-0.214603\pi$
$397$	4.90653e6	1.56242	0.781210	+	0.624268i	$0.214603\pi$
$398$	0	0
$399$	239715.	0.0753813
$400$	0	0
$401$	−951184.	−0.295395	−0.147698	−	0.989033i	$-0.547186\pi$
$401$	−951184.	−0.295395	−0.147698	+	0.989033i	$0.547186\pi$
$402$	0	0
$403$	−681851.	−0.209135
$404$	0	0
$405$	−1.09059e6	−0.330389
$406$	0	0
$407$	−4.47348e6	−1.33863
$408$	0	0
$409$	4.49464e6	1.32858	0.664289	−	0.747476i	$-0.268735\pi$
$409$	4.49464e6	1.32858	0.664289	+	0.747476i	$0.268735\pi$
$410$	0	0
$411$	4.08358e6	1.19244
$412$	0	0
$413$	8.55312e6	2.46746
$414$	0	0
$415$	66361.4	0.0189145
$416$	0	0
$417$	85597.0	0.0241056
$418$	0	0
$419$	828906.	0.230659	0.115329	−	0.993327i	$-0.463208\pi$
$419$	828906.	0.230659	0.115329	+	0.993327i	$0.463208\pi$
$420$	0	0
$421$	744813.	0.204806	0.102403	−	0.994743i	$-0.467347\pi$
$421$	744813.	0.204806	0.102403	+	0.994743i	$0.467347\pi$
$422$	0	0
$423$	1.12916e6	0.306835
$424$	0	0
$425$	101628.	0.0272924
$426$	0	0
$427$	3.48508e6	0.925002
$428$	0	0
$429$	−1.37080e6	−0.359608
$430$	0	0
$431$	3.89529e6	1.01006	0.505030	−	0.863102i	$-0.331481\pi$
$431$	3.89529e6	1.01006	0.505030	+	0.863102i	$0.331481\pi$
$432$	0	0
$433$	−5.88297e6	−1.50792	−0.753958	−	0.656923i	$-0.771858\pi$
$433$	−5.88297e6	−1.50792	−0.753958	+	0.656923i	$0.771858\pi$
$434$	0	0
$435$	−2.15956e6	−0.547195
$436$	0	0
$437$	240011.	0.0601211
$438$	0	0
$439$	−1.73024e6	−0.428495	−0.214248	−	0.976779i	$-0.568730\pi$
$439$	−1.73024e6	−0.428495	−0.214248	+	0.976779i	$0.568730\pi$
$440$	0	0
$441$	−1.49771e6	−0.366716
$442$	0	0
$443$	106432.	0.0257670	0.0128835	−	0.999917i	$-0.495899\pi$
$443$	106432.	0.0257670	0.0128835	+	0.999917i	$0.495899\pi$
$444$	0	0
$445$	205744.	0.0492524
$446$	0	0
$447$	5.12449e6	1.21306
$448$	0	0
$449$	−4.89220e6	−1.14522	−0.572609	−	0.819828i	$-0.694069\pi$
$449$	−4.89220e6	−1.14522	−0.572609	+	0.819828i	$0.694069\pi$
$450$	0	0
$451$	−562690.	−0.130265
$452$	0	0
$453$	−7.02117e6	−1.60755
$454$	0	0
$455$	900962.	0.204023
$456$	0	0
$457$	177252.	0.0397008	0.0198504	−	0.999803i	$-0.493681\pi$
$457$	177252.	0.0397008	0.0198504	+	0.999803i	$0.493681\pi$
$458$	0	0
$459$	663063.	0.146900
$460$	0	0
$461$	7.15676e6	1.56843	0.784213	−	0.620491i	$-0.213067\pi$
$461$	7.15676e6	1.56843	0.784213	+	0.620491i	$0.213067\pi$
$462$	0	0
$463$	−3.44228e6	−0.746267	−0.373133	−	0.927778i	$-0.621717\pi$
$463$	−3.44228e6	−0.746267	−0.373133	+	0.927778i	$0.621717\pi$
$464$	0	0
$465$	−1.39309e6	−0.298778
$466$	0	0
$467$	9.22486e6	1.95735	0.978673	−	0.205423i	$-0.0658571\pi$
$467$	9.22486e6	1.95735	0.978673	+	0.205423i	$0.0658571\pi$
$468$	0	0
$469$	3.91565e6	0.821999
$470$	0	0
$471$	−828889.	−0.172165
$472$	0	0
$473$	99635.7	0.0204768
$474$	0	0
$475$	50869.6	0.0103449
$476$	0	0
$477$	−1.29512e6	−0.260624
$478$	0	0
$479$	−1.23331e6	−0.245602	−0.122801	−	0.992431i	$-0.539188\pi$
$479$	−1.23331e6	−0.245602	−0.122801	+	0.992431i	$0.539188\pi$
$480$	0	0
$481$	−1.28731e6	−0.253700
$482$	0	0
$483$	8.68498e6	1.69395
$484$	0	0
$485$	−1.34610e6	−0.259850
$486$	0	0
$487$	6.21276e6	1.18703	0.593516	−	0.804823i	$-0.297740\pi$
$487$	6.21276e6	1.18703	0.593516	+	0.804823i	$0.297740\pi$
$488$	0	0
$489$	−5.05376e6	−0.955747
$490$	0	0
$491$	7.05221e6	1.32014	0.660072	−	0.751202i	$-0.270526\pi$
$491$	7.05221e6	1.32014	0.660072	+	0.751202i	$0.270526\pi$
$492$	0	0
$493$	1.01700e6	0.188454
$494$	0	0
$495$	767075.	0.140710
$496$	0	0
$497$	−1.66003e7	−3.01457
$498$	0	0
$499$	−5.63009e6	−1.01220	−0.506098	−	0.862476i	$-0.668912\pi$
$499$	−5.63009e6	−1.01220	−0.506098	+	0.862476i	$0.668912\pi$
$500$	0	0
$501$	930778.	0.165673
$502$	0	0
$503$	−9.65803e6	−1.70204	−0.851018	−	0.525137i	$-0.824014\pi$
$503$	−9.65803e6	−1.70204	−0.851018	+	0.525137i	$0.824014\pi$
$504$	0	0
$505$	−4.18890e6	−0.730922
$506$	0	0
$507$	−394467.	−0.0681539
$508$	0	0
$509$	−4.39209e6	−0.751409	−0.375705	−	0.926739i	$-0.622599\pi$
$509$	−4.39209e6	−0.751409	−0.375705	+	0.926739i	$0.622599\pi$
$510$	0	0
$511$	1.32490e7	2.24456
$512$	0	0
$513$	331894.	0.0556809
$514$	0	0
$515$	−774345.	−0.128652
$516$	0	0
$517$	1.26927e7	2.08847
$518$	0	0
$519$	−7.16702e6	−1.16794
$520$	0	0
$521$	−8.38443e6	−1.35325	−0.676627	−	0.736326i	$-0.736559\pi$
$521$	−8.38443e6	−1.35325	−0.676627	+	0.736326i	$0.736559\pi$
$522$	0	0
$523$	−2.54489e6	−0.406831	−0.203416	−	0.979092i	$-0.565204\pi$
$523$	−2.54489e6	−0.406831	−0.203416	+	0.979092i	$0.565204\pi$
$524$	0	0
$525$	1.84076e6	0.291474
$526$	0	0
$527$	656050.	0.102899
$528$	0	0
$529$	2.25933e6	0.351028
$530$	0	0
$531$	2.09553e6	0.322521
$532$	0	0
$533$	−161922.	−0.0246882
$534$	0	0
$535$	2.43359e6	0.367590
$536$	0	0
$537$	−1.23928e6	−0.185452
$538$	0	0
$539$	−1.68355e7	−2.49605
$540$	0	0
$541$	−3.83465e6	−0.563290	−0.281645	−	0.959519i	$-0.590880\pi$
$541$	−3.83465e6	−0.563290	−0.281645	+	0.959519i	$0.590880\pi$
$542$	0	0
$543$	5.19684e6	0.756379
$544$	0	0
$545$	417421.	0.0601981
$546$	0	0
$547$	2.61787e6	0.374093	0.187047	−	0.982351i	$-0.440108\pi$
$547$	2.61787e6	0.374093	0.187047	+	0.982351i	$0.440108\pi$
$548$	0	0
$549$	853850.	0.120907
$550$	0	0
$551$	509057.	0.0714312
$552$	0	0
$553$	−1.29225e7	−1.79694
$554$	0	0
$555$	−2.63011e6	−0.362444
$556$	0	0
$557$	5.40419e6	0.738061	0.369031	−	0.929417i	$-0.379690\pi$
$557$	5.40419e6	0.738061	0.369031	+	0.929417i	$0.379690\pi$
$558$	0	0
$559$	28671.7	0.00388082
$560$	0	0
$561$	1.31892e6	0.176935
$562$	0	0
$563$	−904604.	−0.120278	−0.0601392	−	0.998190i	$-0.519154\pi$
$563$	−904604.	−0.120278	−0.0601392	+	0.998190i	$0.519154\pi$
$564$	0	0
$565$	5.48393e6	0.722722
$566$	0	0
$567$	9.30257e6	1.21519
$568$	0	0
$569$	−6.69355e6	−0.866714	−0.433357	−	0.901222i	$-0.642671\pi$
$569$	−6.69355e6	−0.866714	−0.433357	+	0.901222i	$0.642671\pi$
$570$	0	0
$571$	−8.22981e6	−1.05633	−0.528165	−	0.849142i	$-0.677120\pi$
$571$	−8.22981e6	−1.05633	−0.528165	+	0.849142i	$0.677120\pi$
$572$	0	0
$573$	2.85548e6	0.363322
$574$	0	0
$575$	1.84303e6	0.232467
$576$	0	0
$577$	3.07720e6	0.384783	0.192391	−	0.981318i	$-0.438376\pi$
$577$	3.07720e6	0.384783	0.192391	+	0.981318i	$0.438376\pi$
$578$	0	0
$579$	5.70532e6	0.707267
$580$	0	0
$581$	−566051.	−0.0695689
$582$	0	0
$583$	−1.45583e7	−1.77394
$584$	0	0
$585$	220737.	0.0266678
$586$	0	0
$587$	−2.64175e6	−0.316443	−0.158222	−	0.987404i	$-0.550576\pi$
$587$	−2.64175e6	−0.316443	−0.158222	+	0.987404i	$0.550576\pi$
$588$	0	0
$589$	328384.	0.0390026
$590$	0	0
$591$	−7.44193e6	−0.876429
$592$	0	0
$593$	−5.06641e6	−0.591649	−0.295824	−	0.955242i	$-0.595594\pi$
$593$	−5.06641e6	−0.591649	−0.295824	+	0.955242i	$0.595594\pi$
$594$	0	0
$595$	−866870.	−0.100383
$596$	0	0
$597$	−2.03492e6	−0.233674
$598$	0	0
$599$	−1.31136e7	−1.49332	−0.746662	−	0.665204i	$-0.768345\pi$
$599$	−1.31136e7	−1.49332	−0.746662	+	0.665204i	$0.768345\pi$
$600$	0	0
$601$	7.53425e6	0.850852	0.425426	−	0.904993i	$-0.360124\pi$
$601$	7.53425e6	0.850852	0.425426	+	0.904993i	$0.360124\pi$
$602$	0	0
$603$	959341.	0.107443
$604$	0	0
$605$	4.59631e6	0.510529
$606$	0	0
$607$	−8.76101e6	−0.965123	−0.482561	−	0.875862i	$-0.660293\pi$
$607$	−8.76101e6	−0.965123	−0.482561	+	0.875862i	$0.660293\pi$
$608$	0	0
$609$	1.84207e7	2.01262
$610$	0	0
$611$	3.65253e6	0.395813
$612$	0	0
$613$	−469316.	−0.0504445	−0.0252223	−	0.999682i	$-0.508029\pi$
$613$	−469316.	−0.0504445	−0.0252223	+	0.999682i	$0.508029\pi$
$614$	0	0
$615$	−330824.	−0.0352704
$616$	0	0
$617$	−4.14114e6	−0.437933	−0.218966	−	0.975732i	$-0.570268\pi$
$617$	−4.14114e6	−0.437933	−0.218966	+	0.975732i	$0.570268\pi$
$618$	0	0
$619$	1.31422e7	1.37861	0.689304	−	0.724472i	$-0.257916\pi$
$619$	1.31422e7	1.37861	0.689304	+	0.724472i	$0.257916\pi$
$620$	0	0
$621$	1.20246e7	1.25125
$622$	0	0
$623$	−1.75496e6	−0.181154
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	660184.	0.0670650
$628$	0	0
$629$	1.23860e6	0.124826
$630$	0	0
$631$	−9.79229e6	−0.979063	−0.489532	−	0.871986i	$-0.662832\pi$
$631$	−9.79229e6	−0.979063	−0.489532	+	0.871986i	$0.662832\pi$
$632$	0	0
$633$	−1.58722e7	−1.57445
$634$	0	0
$635$	−3.96883e6	−0.390596
$636$	0	0
$637$	−4.84467e6	−0.473059
$638$	0	0
$639$	−4.06711e6	−0.394034
$640$	0	0
$641$	3.38154e6	0.325064	0.162532	−	0.986703i	$-0.448034\pi$
$641$	3.38154e6	0.325064	0.162532	+	0.986703i	$0.448034\pi$
$642$	0	0
$643$	3.04366e6	0.290314	0.145157	−	0.989409i	$-0.453631\pi$
$643$	3.04366e6	0.290314	0.145157	+	0.989409i	$0.453631\pi$
$644$	0	0
$645$	58579.2	0.00554427
$646$	0	0
$647$	−8.51735e6	−0.799915	−0.399957	−	0.916534i	$-0.630975\pi$
$647$	−8.51735e6	−0.799915	−0.399957	+	0.916534i	$0.630975\pi$
$648$	0	0
$649$	2.35555e7	2.19524
$650$	0	0
$651$	1.18828e7	1.09892
$652$	0	0
$653$	−8.83998e6	−0.811275	−0.405638	−	0.914034i	$-0.632950\pi$
$653$	−8.83998e6	−0.811275	−0.405638	+	0.914034i	$0.632950\pi$
$654$	0	0
$655$	−4.01415e6	−0.365587
$656$	0	0
$657$	3.24603e6	0.293386
$658$	0	0
$659$	2.97807e6	0.267129	0.133564	−	0.991040i	$-0.457358\pi$
$659$	2.97807e6	0.267129	0.133564	+	0.991040i	$0.457358\pi$
$660$	0	0
$661$	−7.90309e6	−0.703548	−0.351774	−	0.936085i	$-0.614421\pi$
$661$	−7.90309e6	−0.703548	−0.351774	+	0.936085i	$0.614421\pi$
$662$	0	0
$663$	379540.	0.0335331
$664$	0	0
$665$	−433909.	−0.0380491
$666$	0	0
$667$	1.84433e7	1.60519
$668$	0	0
$669$	−1.92572e7	−1.66352
$670$	0	0
$671$	9.59800e6	0.822952
$672$	0	0
$673$	1.82522e7	1.55338	0.776688	−	0.629885i	$-0.216898\pi$
$673$	1.82522e7	1.55338	0.776688	+	0.629885i	$0.216898\pi$
$674$	0	0
$675$	2.54859e6	0.215299
$676$	0	0
$677$	5.39380e6	0.452296	0.226148	−	0.974093i	$-0.427387\pi$
$677$	5.39380e6	0.452296	0.226148	+	0.974093i	$0.427387\pi$
$678$	0	0
$679$	1.14820e7	0.955746
$680$	0	0
$681$	1.93762e7	1.60103
$682$	0	0
$683$	−2.30171e7	−1.88799	−0.943994	−	0.329963i	$-0.892964\pi$
$683$	−2.30171e7	−1.88799	−0.943994	+	0.329963i	$0.892964\pi$
$684$	0	0
$685$	−7.39169e6	−0.601891
$686$	0	0
$687$	1.10025e7	0.889406
$688$	0	0
$689$	−4.18936e6	−0.336201
$690$	0	0
$691$	−4.87910e6	−0.388727	−0.194364	−	0.980930i	$-0.562264\pi$
$691$	−4.87910e6	−0.388727	−0.194364	+	0.980930i	$0.562264\pi$
$692$	0	0
$693$	−6.54302e6	−0.517541
$694$	0	0
$695$	−154939.	−0.0121674
$696$	0	0
$697$	155795.	0.0121471
$698$	0	0
$699$	−1.60178e7	−1.23996
$700$	0	0
$701$	1.47820e7	1.13615	0.568077	−	0.822976i	$-0.307688\pi$
$701$	1.47820e7	1.13615	0.568077	+	0.822976i	$0.307688\pi$
$702$	0	0
$703$	619976.	0.0473137
$704$	0	0
$705$	7.46249e6	0.565472
$706$	0	0
$707$	3.57305e7	2.68838
$708$	0	0
$709$	−2.75364e6	−0.205727	−0.102863	−	0.994695i	$-0.532800\pi$
$709$	−2.75364e6	−0.205727	−0.102863	+	0.994695i	$0.532800\pi$
$710$	0	0
$711$	−3.16603e6	−0.234878
$712$	0	0
$713$	1.18975e7	0.876457
$714$	0	0
$715$	2.48128e6	0.181514
$716$	0	0
$717$	−9.49574e6	−0.689812
$718$	0	0
$719$	2.10250e7	1.51675	0.758373	−	0.651820i	$-0.225994\pi$
$719$	2.10250e7	1.51675	0.758373	+	0.651820i	$0.225994\pi$
$720$	0	0
$721$	6.60502e6	0.473191
$722$	0	0
$723$	5.40502e6	0.384549
$724$	0	0
$725$	3.90902e6	0.276200
$726$	0	0
$727$	−2.21381e7	−1.55347	−0.776737	−	0.629825i	$-0.783127\pi$
$727$	−2.21381e7	−1.55347	−0.776737	+	0.629825i	$0.783127\pi$
$728$	0	0
$729$	1.59648e7	1.11261
$730$	0	0
$731$	−27586.7	−0.00190944
$732$	0	0
$733$	1.23177e7	0.846775	0.423388	−	0.905949i	$-0.360841\pi$
$733$	1.23177e7	0.846775	0.423388	+	0.905949i	$0.360841\pi$
$734$	0	0
$735$	−9.89816e6	−0.675828
$736$	0	0
$737$	1.07838e7	0.731313
$738$	0	0
$739$	1.79039e7	1.20597	0.602986	−	0.797752i	$-0.293977\pi$
$739$	1.79039e7	1.20597	0.602986	+	0.797752i	$0.293977\pi$
$740$	0	0
$741$	189978.	0.0127103
$742$	0	0
$743$	9.82085e6	0.652645	0.326323	−	0.945258i	$-0.394190\pi$
$743$	9.82085e6	0.652645	0.326323	+	0.945258i	$0.394190\pi$
$744$	0	0
$745$	−9.27584e6	−0.612297
$746$	0	0
$747$	−138684.	−0.00909334
$748$	0	0
$749$	−2.07581e7	−1.35202
$750$	0	0
$751$	−1.04747e7	−0.677705	−0.338852	−	0.940840i	$-0.610039\pi$
$751$	−1.04747e7	−0.677705	−0.338852	+	0.940840i	$0.610039\pi$
$752$	0	0
$753$	1.30479e7	0.838599
$754$	0	0
$755$	1.27090e7	0.811418
$756$	0	0
$757$	2.20487e7	1.39844	0.699219	−	0.714908i	$-0.253531\pi$
$757$	2.20487e7	1.39844	0.699219	+	0.714908i	$0.253531\pi$
$758$	0	0
$759$	2.39187e7	1.50707
$760$	0	0
$761$	−2.75293e7	−1.72319	−0.861596	−	0.507594i	$-0.830535\pi$
$761$	−2.75293e7	−1.72319	−0.861596	+	0.507594i	$0.830535\pi$
$762$	0	0
$763$	−3.56052e6	−0.221413
$764$	0	0
$765$	−212385.	−0.0131211
$766$	0	0
$767$	6.77846e6	0.416047
$768$	0	0
$769$	1.50464e7	0.917525	0.458763	−	0.888559i	$-0.348293\pi$
$769$	1.50464e7	0.917525	0.458763	+	0.888559i	$0.348293\pi$
$770$	0	0
$771$	−4.10501e6	−0.248701
$772$	0	0
$773$	2.28606e6	0.137606	0.0688031	−	0.997630i	$-0.478082\pi$
$773$	2.28606e6	0.137606	0.0688031	+	0.997630i	$0.478082\pi$
$774$	0	0
$775$	2.52164e6	0.150810
$776$	0	0
$777$	2.24344e7	1.33310
$778$	0	0
$779$	77982.8	0.00460421
$780$	0	0
$781$	−4.57178e7	−2.68199
$782$	0	0
$783$	2.55040e7	1.48663
$784$	0	0
$785$	1.50037e6	0.0869011
$786$	0	0
$787$	−9.89052e6	−0.569223	−0.284611	−	0.958643i	$-0.591865\pi$
$787$	−9.89052e6	−0.569223	−0.284611	+	0.958643i	$0.591865\pi$
$788$	0	0
$789$	−2.11449e7	−1.20924
$790$	0	0
$791$	−4.67770e7	−2.65822
$792$	0	0
$793$	2.76197e6	0.155968
$794$	0	0
$795$	−8.55930e6	−0.480309
$796$	0	0
$797$	2.93049e7	1.63416	0.817080	−	0.576525i	$-0.195592\pi$
$797$	2.93049e7	1.63416	0.817080	+	0.576525i	$0.195592\pi$
$798$	0	0
$799$	−3.51431e6	−0.194748
$800$	0	0
$801$	−429969.	−0.0236786
$802$	0	0
$803$	3.64881e7	1.99693
$804$	0	0
$805$	−1.57207e7	−0.855031
$806$	0	0
$807$	1.39327e7	0.753095
$808$	0	0
$809$	1.02027e7	0.548078	0.274039	−	0.961719i	$-0.411640\pi$
$809$	1.02027e7	0.548078	0.274039	+	0.961719i	$0.411640\pi$
$810$	0	0
$811$	1.26432e7	0.675001	0.337501	−	0.941325i	$-0.390418\pi$
$811$	1.26432e7	0.675001	0.337501	+	0.941325i	$0.390418\pi$
$812$	0	0
$813$	3.02639e7	1.60582
$814$	0	0
$815$	9.14782e6	0.482418
$816$	0	0
$817$	−13808.4	−0.000723752 0
$818$	0	0
$819$	−1.88285e6	−0.0980858
$820$	0	0
$821$	−1.56881e7	−0.812294	−0.406147	−	0.913808i	$-0.633128\pi$
$821$	−1.56881e7	−0.812294	−0.406147	+	0.913808i	$0.633128\pi$
$822$	0	0
$823$	−2.45799e7	−1.26497	−0.632487	−	0.774571i	$-0.717966\pi$
$823$	−2.45799e7	−1.26497	−0.632487	+	0.774571i	$0.717966\pi$
$824$	0	0
$825$	5.06951e6	0.259317
$826$	0	0
$827$	−3.62365e7	−1.84239	−0.921197	−	0.389098i	$-0.872787\pi$
$827$	−3.62365e7	−1.84239	−0.921197	+	0.389098i	$0.872787\pi$
$828$	0	0
$829$	−2.81815e7	−1.42422	−0.712110	−	0.702068i	$-0.752260\pi$
$829$	−2.81815e7	−1.42422	−0.712110	+	0.702068i	$0.752260\pi$
$830$	0	0
$831$	247555.	0.0124357
$832$	0	0
$833$	4.66134e6	0.232755
$834$	0	0
$835$	−1.68480e6	−0.0836243
$836$	0	0
$837$	1.64522e7	0.811727
$838$	0	0
$839$	8.42806e6	0.413354	0.206677	−	0.978409i	$-0.433735\pi$
$839$	8.42806e6	0.413354	0.206677	+	0.978409i	$0.433735\pi$
$840$	0	0
$841$	1.86068e7	0.907155
$842$	0	0
$843$	3.32390e7	1.61094
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	−3.92057e7	−1.87776
$848$	0	0
$849$	−1.89107e7	−0.900404
$850$	0	0
$851$	2.24620e7	1.06322
$852$	0	0
$853$	−3.10763e7	−1.46237	−0.731184	−	0.682180i	$-0.761032\pi$
$853$	−3.10763e7	−1.46237	−0.731184	+	0.682180i	$0.761032\pi$
$854$	0	0
$855$	−106309.	−0.00497339
$856$	0	0
$857$	−5.72064e6	−0.266068	−0.133034	−	0.991111i	$-0.542472\pi$
$857$	−5.72064e6	−0.266068	−0.133034	+	0.991111i	$0.542472\pi$
$858$	0	0
$859$	6.04140e6	0.279354	0.139677	−	0.990197i	$-0.455394\pi$
$859$	6.04140e6	0.279354	0.139677	+	0.990197i	$0.455394\pi$
$860$	0	0
$861$	2.82187e6	0.129727
$862$	0	0
$863$	−2.09227e7	−0.956292	−0.478146	−	0.878280i	$-0.658691\pi$
$863$	−2.09227e7	−0.956292	−0.478146	+	0.878280i	$0.658691\pi$
$864$	0	0
$865$	1.29730e7	0.589523
$866$	0	0
$867$	1.92450e7	0.869502
$868$	0	0
$869$	−3.55889e7	−1.59869
$870$	0	0
$871$	3.10320e6	0.138601
$872$	0	0
$873$	2.81311e6	0.124925
$874$	0	0
$875$	−3.33196e6	−0.147123
$876$	0	0
$877$	−5.01539e6	−0.220194	−0.110097	−	0.993921i	$-0.535116\pi$
$877$	−5.01539e6	−0.220194	−0.110097	+	0.993921i	$0.535116\pi$
$878$	0	0
$879$	5.39541e6	0.235533
$880$	0	0
$881$	−2.14791e7	−0.932346	−0.466173	−	0.884694i	$-0.654368\pi$
$881$	−2.14791e7	−0.932346	−0.466173	+	0.884694i	$0.654368\pi$
$882$	0	0
$883$	−996549.	−0.0430127	−0.0215064	−	0.999769i	$-0.506846\pi$
$883$	−996549.	−0.0430127	−0.0215064	+	0.999769i	$0.506846\pi$
$884$	0	0
$885$	1.38491e7	0.594379
$886$	0	0
$887$	−4.67336e6	−0.199444	−0.0997219	−	0.995015i	$-0.531795\pi$
$887$	−4.67336e6	−0.199444	−0.0997219	+	0.995015i	$0.531795\pi$
$888$	0	0
$889$	3.38534e7	1.43664
$890$	0	0
$891$	2.56196e7	1.08113
$892$	0	0
$893$	−1.75908e6	−0.0738170
$894$	0	0
$895$	2.24321e6	0.0936080
$896$	0	0
$897$	6.88297e6	0.285624
$898$	0	0
$899$	2.52343e7	1.04134
$900$	0	0
$901$	4.03083e6	0.165418
$902$	0	0
$903$	−499670.	−0.0203922
$904$	0	0
$905$	−9.40680e6	−0.381786
$906$	0	0
$907$	−1.18130e7	−0.476807	−0.238403	−	0.971166i	$-0.576624\pi$
$907$	−1.18130e7	−0.476807	−0.238403	+	0.971166i	$0.576624\pi$
$908$	0	0
$909$	8.75405e6	0.351398
$910$	0	0
$911$	−3.55560e7	−1.41944	−0.709721	−	0.704483i	$-0.751179\pi$
$911$	−3.55560e7	−1.41944	−0.709721	+	0.704483i	$0.751179\pi$
$912$	0	0
$913$	−1.55892e6	−0.0618938
$914$	0	0
$915$	5.64299e6	0.222821
$916$	0	0
$917$	3.42400e7	1.34465
$918$	0	0
$919$	−2.66902e7	−1.04247	−0.521235	−	0.853413i	$-0.674528\pi$
$919$	−2.66902e7	−1.04247	−0.521235	+	0.853413i	$0.674528\pi$
$920$	0	0
$921$	7.82478e6	0.303965
$922$	0	0
$923$	−1.31560e7	−0.508299
$924$	0	0
$925$	4.76076e6	0.182946
$926$	0	0
$927$	1.61824e6	0.0618507
$928$	0	0
$929$	−3.08229e7	−1.17175	−0.585874	−	0.810402i	$-0.699249\pi$
$929$	−3.08229e7	−1.17175	−0.585874	+	0.810402i	$0.699249\pi$
$930$	0	0
$931$	2.33322e6	0.0882230
$932$	0	0
$933$	1.21711e7	0.457749
$934$	0	0
$935$	−2.38739e6	−0.0893087
$936$	0	0
$937$	2.78462e7	1.03613	0.518067	−	0.855340i	$-0.326652\pi$
$937$	2.78462e7	1.03613	0.518067	+	0.855340i	$0.326652\pi$
$938$	0	0
$939$	1.26571e7	0.468456
$940$	0	0
$941$	2.80836e7	1.03390	0.516950	−	0.856016i	$-0.327067\pi$
$941$	2.80836e7	1.03390	0.516950	+	0.856016i	$0.327067\pi$
$942$	0	0
$943$	2.82535e6	0.103465
$944$	0	0
$945$	−2.17391e7	−0.791883
$946$	0	0
$947$	4.49006e7	1.62696	0.813481	−	0.581592i	$-0.197570\pi$
$947$	4.49006e7	1.62696	0.813481	+	0.581592i	$0.197570\pi$
$948$	0	0
$949$	1.05000e7	0.378463
$950$	0	0
$951$	−2.77724e7	−0.995779
$952$	0	0
$953$	−1.13992e7	−0.406577	−0.203289	−	0.979119i	$-0.565163\pi$
$953$	−1.13992e7	−0.406577	−0.203289	+	0.979119i	$0.565163\pi$
$954$	0	0
$955$	−5.16870e6	−0.183389
$956$	0	0
$957$	5.07311e7	1.79058
$958$	0	0
$959$	6.30498e7	2.21380
$960$	0	0
$961$	−1.23510e7	−0.431412
$962$	0	0
$963$	−5.08578e6	−0.176722
$964$	0	0
$965$	−1.03272e7	−0.356997
$966$	0	0
$967$	5.54751e7	1.90780	0.953898	−	0.300130i	$-0.0970300\pi$
$967$	5.54751e7	1.90780	0.953898	+	0.300130i	$0.0970300\pi$
$968$	0	0
$969$	−182789.	−0.00625375
$970$	0	0
$971$	2.45562e6	0.0835819	0.0417910	−	0.999126i	$-0.486694\pi$
$971$	2.45562e6	0.0835819	0.0417910	+	0.999126i	$0.486694\pi$
$972$	0	0
$973$	1.32160e6	0.0447527
$974$	0	0
$975$	1.45883e6	0.0491465
$976$	0	0
$977$	3.31344e7	1.11056	0.555280	−	0.831663i	$-0.312611\pi$
$977$	3.31344e7	1.11056	0.555280	+	0.831663i	$0.312611\pi$
$978$	0	0
$979$	−4.83322e6	−0.161168
$980$	0	0
$981$	−872335.	−0.0289408
$982$	0	0
$983$	4.46461e7	1.47367	0.736834	−	0.676073i	$-0.236320\pi$
$983$	4.46461e7	1.47367	0.736834	+	0.676073i	$0.236320\pi$
$984$	0	0
$985$	1.34706e7	0.442382
$986$	0	0
$987$	−6.36537e7	−2.07985
$988$	0	0
$989$	−500285.	−0.0162640
$990$	0	0
$991$	−4.99797e7	−1.61662	−0.808312	−	0.588754i	$-0.799619\pi$
$991$	−4.99797e7	−1.61662	−0.808312	+	0.588754i	$0.799619\pi$
$992$	0	0
$993$	3.42129e7	1.10107
$994$	0	0
$995$	3.68340e6	0.117948
$996$	0	0
$997$	−1.25647e7	−0.400326	−0.200163	−	0.979763i	$-0.564147\pi$
$997$	−1.25647e7	−0.400326	−0.200163	+	0.979763i	$0.564147\pi$
$998$	0	0
$999$	3.10611e7	0.984699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.e.1.1		2
4.3	odd	2		130.6.a.a.1.2	✓	2
20.3	even	4		650.6.b.g.599.4		4
20.7	even	4		650.6.b.g.599.1		4
20.19	odd	2		650.6.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.a.1.2	✓	2	4.3	odd	2
650.6.a.h.1.1		2	20.19	odd	2
650.6.b.g.599.1		4	20.7	even	4
650.6.b.g.599.4		4	20.3	even	4
1040.6.a.e.1.1		2	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)

\(f(q)\)	\(=\)	\(q-13.8114 q^{3} +25.0000 q^{5} -213.246 q^{7} -52.2456 q^{9} -587.285 q^{11} -169.000 q^{13} -345.285 q^{15} +162.605 q^{17} +81.3914 q^{19} +2945.22 q^{21} +2948.84 q^{23} +625.000 q^{25} +4077.75 q^{27} +6254.43 q^{29} +4034.62 q^{31} +8111.22 q^{33} -5331.14 q^{35} +7617.22 q^{37} +2334.12 q^{39} +958.121 q^{41} -169.655 q^{43} -1306.14 q^{45} -21612.6 q^{47} +28666.7 q^{49} -2245.80 q^{51} +24789.1 q^{53} -14682.1 q^{55} -1124.13 q^{57} -40109.2 q^{59} -16343.0 q^{61} +11141.1 q^{63} -4225.00 q^{65} -18362.1 q^{67} -40727.6 q^{69} +77846.0 q^{71} -62130.2 q^{73} -8632.12 q^{75} +125236. q^{77} +60599.1 q^{79} -43623.7 q^{81} +2654.46 q^{83} +4065.12 q^{85} -86382.4 q^{87} +8229.77 q^{89} +36038.5 q^{91} -55723.7 q^{93} +2034.79 q^{95} -53844.0 q^{97} +30683.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 50 q^{5} - 300 q^{7} + 22 q^{9} - 384 q^{11} - 338 q^{13} + 100 q^{15} - 244 q^{17} + 2408 q^{19} + 1400 q^{21} - 332 q^{23} + 1250 q^{25} + 1072 q^{27} + 3528 q^{29} - 880 q^{31} + 11732 q^{33}+ \cdots + 45776 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 50 * q^5 - 300 * q^7 + 22 * q^9 - 384 * q^11 - 338 * q^13 + 100 * q^15 - 244 * q^17 + 2408 * q^19 + 1400 * q^21 - 332 * q^23 + 1250 * q^25 + 1072 * q^27 + 3528 * q^29 - 880 * q^31 + 11732 * q^33 - 7500 * q^35 + 10744 * q^37 - 676 * q^39 + 16020 * q^41 - 2964 * q^43 + 550 * q^45 - 38292 * q^47 + 19386 * q^49 - 9488 * q^51 + 42052 * q^53 - 9600 * q^55 + 40316 * q^57 - 42872 * q^59 - 33192 * q^61 + 4700 * q^63 - 8450 * q^65 - 37420 * q^67 - 99164 * q^69 + 69520 * q^71 - 12632 * q^73 + 2500 * q^75 + 107600 * q^77 - 35208 * q^79 - 115202 * q^81 - 31500 * q^83 - 6100 * q^85 - 134944 * q^87 - 18452 * q^89 + 50700 * q^91 - 143260 * q^93 + 60200 * q^95 - 28884 * q^97 + 45776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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