LMFDB - Embedded newform 912.6.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,6,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, 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("912.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	912.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:	$146.270043669$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4327x^{2} + 78705x - 258666 $ x^4 - x^3 - 4327x^2 + 78705x - 258666
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.29948$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -31.7936 q^{5} +136.490 q^{7} +81.0000 q^{9} +691.738 q^{11} -261.653 q^{13} -286.142 q^{15} +1408.97 q^{17} +361.000 q^{19} +1228.41 q^{21} +2844.99 q^{23} -2114.17 q^{25} +729.000 q^{27} +1469.54 q^{29} -863.414 q^{31} +6225.64 q^{33} -4339.50 q^{35} +3162.80 q^{37} -2354.87 q^{39} +7469.23 q^{41} -10854.2 q^{43} -2575.28 q^{45} +12364.1 q^{47} +1822.47 q^{49} +12680.8 q^{51} +19022.9 q^{53} -21992.8 q^{55} +3249.00 q^{57} -17232.8 q^{59} +9423.41 q^{61} +11055.7 q^{63} +8318.88 q^{65} -35827.5 q^{67} +25604.9 q^{69} -78965.5 q^{71} -21729.2 q^{73} -19027.5 q^{75} +94415.2 q^{77} +43428.1 q^{79} +6561.00 q^{81} -18576.8 q^{83} -44796.3 q^{85} +13225.9 q^{87} -27875.0 q^{89} -35712.9 q^{91} -7770.72 q^{93} -11477.5 q^{95} +19004.2 q^{97} +56030.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 117 q^{5} + 33 q^{7} + 324 q^{9} + 129 q^{11} + 638 q^{13} + 1053 q^{15} + 285 q^{17} + 1444 q^{19} + 297 q^{21} + 2340 q^{23} + 3629 q^{25} + 2916 q^{27} + 438 q^{29} - 160 q^{31} + 1161 q^{33}+ \cdots + 10449 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 117 * q^5 + 33 * q^7 + 324 * q^9 + 129 * q^11 + 638 * q^13 + 1053 * q^15 + 285 * q^17 + 1444 * q^19 + 297 * q^21 + 2340 * q^23 + 3629 * q^25 + 2916 * q^27 + 438 * q^29 - 160 * q^31 + 1161 * q^33 + 2409 * q^35 + 15520 * q^37 + 5742 * q^39 + 14664 * q^41 - 22519 * q^43 + 9477 * q^45 - 16395 * q^47 + 64925 * q^49 + 2565 * q^51 + 61266 * q^53 - 49921 * q^55 + 12996 * q^57 - 67512 * q^59 + 104285 * q^61 + 2673 * q^63 + 134466 * q^65 - 40584 * q^67 + 21060 * q^69 - 98856 * q^71 + 46323 * q^73 + 32661 * q^75 + 164715 * q^77 + 25126 * q^79 + 26244 * q^81 - 34026 * q^83 - 21861 * q^85 + 3942 * q^87 + 132606 * q^89 + 150138 * q^91 - 1440 * q^93 + 42237 * q^95 - 31892 * q^97 + 10449 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−31.7936	−0.568741	−0.284371	−	0.958714i	$-0.591785\pi$
$5$	−31.7936	−0.568741	−0.284371	+	0.958714i	$0.591785\pi$
$6$	0	0
$7$	136.490	1.05282	0.526411	−	0.850230i	$-0.323537\pi$
$7$	136.490	1.05282	0.526411	+	0.850230i	$0.323537\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	691.738	1.72369	0.861846	−	0.507169i	$-0.169308\pi$
$11$	691.738	1.72369	0.861846	+	0.507169i	$0.169308\pi$
$12$	0	0
$13$	−261.653	−0.429405	−0.214702	−	0.976680i	$-0.568878\pi$
$13$	−261.653	−0.429405	−0.214702	+	0.976680i	$0.568878\pi$
$14$	0	0
$15$	−286.142	−0.328363
$16$	0	0
$17$	1408.97	1.18244	0.591222	−	0.806509i	$-0.298646\pi$
$17$	1408.97	1.18244	0.591222	+	0.806509i	$0.298646\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	1228.41	0.607847
$22$	0	0
$23$	2844.99	1.12140	0.560700	−	0.828019i	$-0.310532\pi$
$23$	2844.99	1.12140	0.560700	+	0.828019i	$0.310532\pi$
$24$	0	0
$25$	−2114.17	−0.676533
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	1469.54	0.324479	0.162240	−	0.986751i	$-0.448128\pi$
$29$	1469.54	0.324479	0.162240	+	0.986751i	$0.448128\pi$
$30$	0	0
$31$	−863.414	−0.161367	−0.0806835	−	0.996740i	$-0.525710\pi$
$31$	−863.414	−0.161367	−0.0806835	+	0.996740i	$0.525710\pi$
$32$	0	0
$33$	6225.64	0.995175
$34$	0	0
$35$	−4339.50	−0.598784
$36$	0	0
$37$	3162.80	0.379811	0.189905	−	0.981802i	$-0.439182\pi$
$37$	3162.80	0.379811	0.189905	+	0.981802i	$0.439182\pi$
$38$	0	0
$39$	−2354.87	−0.247917
$40$	0	0
$41$	7469.23	0.693931	0.346966	−	0.937878i	$-0.387212\pi$
$41$	7469.23	0.693931	0.346966	+	0.937878i	$0.387212\pi$
$42$	0	0
$43$	−10854.2	−0.895212	−0.447606	−	0.894231i	$-0.647723\pi$
$43$	−10854.2	−0.895212	−0.447606	+	0.894231i	$0.647723\pi$
$44$	0	0
$45$	−2575.28	−0.189580
$46$	0	0
$47$	12364.1	0.816431	0.408216	−	0.912886i	$-0.366151\pi$
$47$	12364.1	0.816431	0.408216	+	0.912886i	$0.366151\pi$
$48$	0	0
$49$	1822.47	0.108435
$50$	0	0
$51$	12680.8	0.682684
$52$	0	0
$53$	19022.9	0.930225	0.465112	−	0.885252i	$-0.346014\pi$
$53$	19022.9	0.930225	0.465112	+	0.885252i	$0.346014\pi$
$54$	0	0
$55$	−21992.8	−0.980335
$56$	0	0
$57$	3249.00	0.132453
$58$	0	0
$59$	−17232.8	−0.644504	−0.322252	−	0.946654i	$-0.604440\pi$
$59$	−17232.8	−0.644504	−0.322252	+	0.946654i	$0.604440\pi$
$60$	0	0
$61$	9423.41	0.324252	0.162126	−	0.986770i	$-0.448165\pi$
$61$	9423.41	0.324252	0.162126	+	0.986770i	$0.448165\pi$
$62$	0	0
$63$	11055.7	0.350941
$64$	0	0
$65$	8318.88	0.244220
$66$	0	0
$67$	−35827.5	−0.975056	−0.487528	−	0.873107i	$-0.662101\pi$
$67$	−35827.5	−0.975056	−0.487528	+	0.873107i	$0.662101\pi$
$68$	0	0
$69$	25604.9	0.647441
$70$	0	0
$71$	−78965.5	−1.85905	−0.929526	−	0.368757i	$-0.879783\pi$
$71$	−78965.5	−1.85905	−0.929526	+	0.368757i	$0.879783\pi$
$72$	0	0
$73$	−21729.2	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$73$	−21729.2	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$74$	0	0
$75$	−19027.5	−0.390597
$76$	0	0
$77$	94415.2	1.81474
$78$	0	0
$79$	43428.1	0.782893	0.391447	−	0.920201i	$-0.371975\pi$
$79$	43428.1	0.782893	0.391447	+	0.920201i	$0.371975\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−18576.8	−0.295990	−0.147995	−	0.988988i	$-0.547282\pi$
$83$	−18576.8	−0.295990	−0.147995	+	0.988988i	$0.547282\pi$
$84$	0	0
$85$	−44796.3	−0.672504
$86$	0	0
$87$	13225.9	0.187338
$88$	0	0
$89$	−27875.0	−0.373027	−0.186513	−	0.982452i	$-0.559719\pi$
$89$	−27875.0	−0.373027	−0.186513	+	0.982452i	$0.559719\pi$
$90$	0	0
$91$	−35712.9	−0.452087
$92$	0	0
$93$	−7770.72	−0.0931653
$94$	0	0
$95$	−11477.5	−0.130478
$96$	0	0
$97$	19004.2	0.205078	0.102539	−	0.994729i	$-0.467303\pi$
$97$	19004.2	0.205078	0.102539	+	0.994729i	$0.467303\pi$
$98$	0	0
$99$	56030.8	0.574564
$100$	0	0
$101$	−37006.5	−0.360973	−0.180486	−	0.983577i	$-0.557767\pi$
$101$	−37006.5	−0.360973	−0.180486	+	0.983577i	$0.557767\pi$
$102$	0	0
$103$	132725.	1.23270	0.616351	−	0.787471i	$-0.288610\pi$
$103$	132725.	1.23270	0.616351	+	0.787471i	$0.288610\pi$
$104$	0	0
$105$	−39055.5	−0.345708
$106$	0	0
$107$	−127034.	−1.07266	−0.536329	−	0.844009i	$-0.680189\pi$
$107$	−127034.	−1.07266	−0.536329	+	0.844009i	$0.680189\pi$
$108$	0	0
$109$	183607.	1.48021	0.740104	−	0.672492i	$-0.234776\pi$
$109$	183607.	1.48021	0.740104	+	0.672492i	$0.234776\pi$
$110$	0	0
$111$	28465.2	0.219284
$112$	0	0
$113$	−38263.8	−0.281898	−0.140949	−	0.990017i	$-0.545015\pi$
$113$	−38263.8	−0.281898	−0.140949	+	0.990017i	$0.545015\pi$
$114$	0	0
$115$	−90452.4	−0.637787
$116$	0	0
$117$	−21193.9	−0.143135
$118$	0	0
$119$	192310.	1.24490
$120$	0	0
$121$	317450.	1.97112
$122$	0	0
$123$	67223.1	0.400641
$124$	0	0
$125$	166572.	0.953514
$126$	0	0
$127$	171357.	0.942742	0.471371	−	0.881935i	$-0.343759\pi$
$127$	171357.	0.942742	0.471371	+	0.881935i	$0.343759\pi$
$128$	0	0
$129$	−97687.7	−0.516851
$130$	0	0
$131$	92853.5	0.472737	0.236369	−	0.971663i	$-0.424043\pi$
$131$	92853.5	0.472737	0.236369	+	0.971663i	$0.424043\pi$
$132$	0	0
$133$	49272.8	0.241534
$134$	0	0
$135$	−23177.5	−0.109454
$136$	0	0
$137$	−97281.4	−0.442821	−0.221410	−	0.975181i	$-0.571066\pi$
$137$	−97281.4	−0.442821	−0.221410	+	0.975181i	$0.571066\pi$
$138$	0	0
$139$	323196.	1.41883	0.709413	−	0.704793i	$-0.248960\pi$
$139$	323196.	1.41883	0.709413	+	0.704793i	$0.248960\pi$
$140$	0	0
$141$	111277.	0.471367
$142$	0	0
$143$	−180995.	−0.740162
$144$	0	0
$145$	−46722.0	−0.184545
$146$	0	0
$147$	16402.3	0.0626052
$148$	0	0
$149$	−66353.4	−0.244848	−0.122424	−	0.992478i	$-0.539067\pi$
$149$	−66353.4	−0.244848	−0.122424	+	0.992478i	$0.539067\pi$
$150$	0	0
$151$	372223.	1.32850	0.664249	−	0.747512i	$-0.268752\pi$
$151$	372223.	1.32850	0.664249	+	0.747512i	$0.268752\pi$
$152$	0	0
$153$	114127.	0.394148
$154$	0	0
$155$	27451.0	0.0917760
$156$	0	0
$157$	−493158.	−1.59675	−0.798376	−	0.602160i	$-0.794307\pi$
$157$	−493158.	−1.59675	−0.798376	+	0.602160i	$0.794307\pi$
$158$	0	0
$159$	171206.	0.537065
$160$	0	0
$161$	388312.	1.18064
$162$	0	0
$163$	211574.	0.623724	0.311862	−	0.950127i	$-0.399047\pi$
$163$	211574.	0.623724	0.311862	+	0.950127i	$0.399047\pi$
$164$	0	0
$165$	−197936.	−0.565997
$166$	0	0
$167$	138869.	0.385312	0.192656	−	0.981266i	$-0.438290\pi$
$167$	138869.	0.385312	0.192656	+	0.981266i	$0.438290\pi$
$168$	0	0
$169$	−302831.	−0.815612
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	159542.	0.405284	0.202642	−	0.979253i	$-0.435047\pi$
$173$	159542.	0.405284	0.202642	+	0.979253i	$0.435047\pi$
$174$	0	0
$175$	−288562.	−0.712270
$176$	0	0
$177$	−155095.	−0.372104
$178$	0	0
$179$	−820067.	−1.91301	−0.956504	−	0.291719i	$-0.905773\pi$
$179$	−820067.	−1.91301	−0.956504	+	0.291719i	$0.905773\pi$
$180$	0	0
$181$	144944.	0.328854	0.164427	−	0.986389i	$-0.447422\pi$
$181$	144944.	0.328854	0.164427	+	0.986389i	$0.447422\pi$
$182$	0	0
$183$	84810.7	0.187207
$184$	0	0
$185$	−100557.	−0.216014
$186$	0	0
$187$	974640.	2.03817
$188$	0	0
$189$	99501.1	0.202616
$190$	0	0
$191$	−602801.	−1.19561	−0.597806	−	0.801641i	$-0.703961\pi$
$191$	−602801.	−1.19561	−0.597806	+	0.801641i	$0.703961\pi$
$192$	0	0
$193$	625582.	1.20890	0.604451	−	0.796643i	$-0.293393\pi$
$193$	625582.	1.20890	0.604451	+	0.796643i	$0.293393\pi$
$194$	0	0
$195$	74869.9	0.141001
$196$	0	0
$197$	−27843.8	−0.0511167	−0.0255584	−	0.999673i	$-0.508136\pi$
$197$	−27843.8	−0.0511167	−0.0255584	+	0.999673i	$0.508136\pi$
$198$	0	0
$199$	715911.	1.28152	0.640761	−	0.767740i	$-0.278619\pi$
$199$	715911.	1.28152	0.640761	+	0.767740i	$0.278619\pi$
$200$	0	0
$201$	−322447.	−0.562949
$202$	0	0
$203$	200578.	0.341619
$204$	0	0
$205$	−237474.	−0.394667
$206$	0	0
$207$	230444.	0.373800
$208$	0	0
$209$	249717.	0.395442
$210$	0	0
$211$	1.19468e6	1.84734	0.923668	−	0.383193i	$-0.125176\pi$
$211$	1.19468e6	1.84734	0.923668	+	0.383193i	$0.125176\pi$
$212$	0	0
$213$	−710689.	−1.07332
$214$	0	0
$215$	345094.	0.509144
$216$	0	0
$217$	−117847.	−0.169891
$218$	0	0
$219$	−195563.	−0.275535
$220$	0	0
$221$	−368661.	−0.507747
$222$	0	0
$223$	−437325.	−0.588901	−0.294450	−	0.955667i	$-0.595137\pi$
$223$	−437325.	−0.588901	−0.294450	+	0.955667i	$0.595137\pi$
$224$	0	0
$225$	−171248.	−0.225511
$226$	0	0
$227$	−370776.	−0.477582	−0.238791	−	0.971071i	$-0.576751\pi$
$227$	−370776.	−0.477582	−0.238791	+	0.971071i	$0.576751\pi$
$228$	0	0
$229$	−1.27852e6	−1.61108	−0.805540	−	0.592541i	$-0.798125\pi$
$229$	−1.27852e6	−1.61108	−0.805540	+	0.592541i	$0.798125\pi$
$230$	0	0
$231$	849737.	1.04774
$232$	0	0
$233$	1.00010e6	1.20685	0.603423	−	0.797421i	$-0.293803\pi$
$233$	1.00010e6	1.20685	0.603423	+	0.797421i	$0.293803\pi$
$234$	0	0
$235$	−393101.	−0.464338
$236$	0	0
$237$	390853.	0.452004
$238$	0	0
$239$	145864.	0.165179	0.0825893	−	0.996584i	$-0.473681\pi$
$239$	145864.	0.165179	0.0825893	+	0.996584i	$0.473681\pi$
$240$	0	0
$241$	1.41435e6	1.56861	0.784304	−	0.620376i	$-0.213020\pi$
$241$	1.41435e6	1.56861	0.784304	+	0.620376i	$0.213020\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−57943.0	−0.0616717
$246$	0	0
$247$	−94456.6	−0.0985122
$248$	0	0
$249$	−167192.	−0.170890
$250$	0	0
$251$	−216775.	−0.217183	−0.108591	−	0.994086i	$-0.534634\pi$
$251$	−216775.	−0.217183	−0.108591	+	0.994086i	$0.534634\pi$
$252$	0	0
$253$	1.96799e6	1.93295
$254$	0	0
$255$	−403167.	−0.388271
$256$	0	0
$257$	853978.	0.806518	0.403259	−	0.915086i	$-0.367877\pi$
$257$	853978.	0.806518	0.403259	+	0.915086i	$0.367877\pi$
$258$	0	0
$259$	431690.	0.399874
$260$	0	0
$261$	119033.	0.108160
$262$	0	0
$263$	467645.	0.416896	0.208448	−	0.978033i	$-0.433159\pi$
$263$	467645.	0.416896	0.208448	+	0.978033i	$0.433159\pi$
$264$	0	0
$265$	−604808.	−0.529057
$266$	0	0
$267$	−250875.	−0.215367
$268$	0	0
$269$	449741.	0.378950	0.189475	−	0.981886i	$-0.439321\pi$
$269$	449741.	0.378950	0.189475	+	0.981886i	$0.439321\pi$
$270$	0	0
$271$	−1.01480e6	−0.839376	−0.419688	−	0.907668i	$-0.637861\pi$
$271$	−1.01480e6	−0.839376	−0.419688	+	0.907668i	$0.637861\pi$
$272$	0	0
$273$	−321416.	−0.261012
$274$	0	0
$275$	−1.46245e6	−1.16614
$276$	0	0
$277$	−202928.	−0.158906	−0.0794532	−	0.996839i	$-0.525317\pi$
$277$	−202928.	−0.158906	−0.0794532	+	0.996839i	$0.525317\pi$
$278$	0	0
$279$	−69936.5	−0.0537890
$280$	0	0
$281$	589539.	0.445397	0.222698	−	0.974887i	$-0.428514\pi$
$281$	589539.	0.445397	0.222698	+	0.974887i	$0.428514\pi$
$282$	0	0
$283$	−2.55801e6	−1.89861	−0.949307	−	0.314350i	$-0.898213\pi$
$283$	−2.55801e6	−1.89861	−0.949307	+	0.314350i	$0.898213\pi$
$284$	0	0
$285$	−103297.	−0.0753316
$286$	0	0
$287$	1.01947e6	0.730586
$288$	0	0
$289$	565348.	0.398172
$290$	0	0
$291$	171037.	0.118402
$292$	0	0
$293$	2.71564e6	1.84801	0.924004	−	0.382382i	$-0.124896\pi$
$293$	2.71564e6	1.84801	0.924004	+	0.382382i	$0.124896\pi$
$294$	0	0
$295$	547892.	0.366556
$296$	0	0
$297$	504277.	0.331725
$298$	0	0
$299$	−744398.	−0.481535
$300$	0	0
$301$	−1.48149e6	−0.942500
$302$	0	0
$303$	−333058.	−0.208408
$304$	0	0
$305$	−299604.	−0.184416
$306$	0	0
$307$	−851449.	−0.515600	−0.257800	−	0.966198i	$-0.582998\pi$
$307$	−851449.	−0.515600	−0.257800	+	0.966198i	$0.582998\pi$
$308$	0	0
$309$	1.19452e6	0.711701
$310$	0	0
$311$	533143.	0.312567	0.156283	−	0.987712i	$-0.450049\pi$
$311$	533143.	0.312567	0.156283	+	0.987712i	$0.450049\pi$
$312$	0	0
$313$	−109835.	−0.0633694	−0.0316847	−	0.999498i	$-0.510087\pi$
$313$	−109835.	−0.0633694	−0.0316847	+	0.999498i	$0.510087\pi$
$314$	0	0
$315$	−351500.	−0.199595
$316$	0	0
$317$	2.77656e6	1.55188	0.775942	−	0.630805i	$-0.217275\pi$
$317$	2.77656e6	1.55188	0.775942	+	0.630805i	$0.217275\pi$
$318$	0	0
$319$	1.01654e6	0.559303
$320$	0	0
$321$	−1.14331e6	−0.619300
$322$	0	0
$323$	508639.	0.271271
$324$	0	0
$325$	553177.	0.290507
$326$	0	0
$327$	1.65246e6	0.854598
$328$	0	0
$329$	1.68758e6	0.859557
$330$	0	0
$331$	2.79226e6	1.40083	0.700415	−	0.713736i	$-0.252998\pi$
$331$	2.79226e6	1.40083	0.700415	+	0.713736i	$0.252998\pi$
$332$	0	0
$333$	256187.	0.126604
$334$	0	0
$335$	1.13909e6	0.554554
$336$	0	0
$337$	−1.65336e6	−0.793037	−0.396519	−	0.918027i	$-0.629782\pi$
$337$	−1.65336e6	−0.793037	−0.396519	+	0.918027i	$0.629782\pi$
$338$	0	0
$339$	−344374.	−0.162754
$340$	0	0
$341$	−597256.	−0.278147
$342$	0	0
$343$	−2.04524e6	−0.938659
$344$	0	0
$345$	−814072.	−0.368226
$346$	0	0
$347$	−3.49138e6	−1.55659	−0.778293	−	0.627901i	$-0.783914\pi$
$347$	−3.49138e6	−1.55659	−0.778293	+	0.627901i	$0.783914\pi$
$348$	0	0
$349$	2.19114e6	0.962958	0.481479	−	0.876458i	$-0.340100\pi$
$349$	2.19114e6	0.962958	0.481479	+	0.876458i	$0.340100\pi$
$350$	0	0
$351$	−190745.	−0.0826389
$352$	0	0
$353$	−1.93036e6	−0.824521	−0.412261	−	0.911066i	$-0.635261\pi$
$353$	−1.93036e6	−0.824521	−0.412261	+	0.911066i	$0.635261\pi$
$354$	0	0
$355$	2.51060e6	1.05732
$356$	0	0
$357$	1.73079e6	0.718745
$358$	0	0
$359$	1.97492e6	0.808750	0.404375	−	0.914593i	$-0.367489\pi$
$359$	1.97492e6	0.808750	0.404375	+	0.914593i	$0.367489\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	2.85705e6	1.13803
$364$	0	0
$365$	690850.	0.271426
$366$	0	0
$367$	−4.82988e6	−1.87185	−0.935925	−	0.352198i	$-0.885434\pi$
$367$	−4.82988e6	−1.87185	−0.935925	+	0.352198i	$0.885434\pi$
$368$	0	0
$369$	605008.	0.231310
$370$	0	0
$371$	2.59644e6	0.979361
$372$	0	0
$373$	3.03190e6	1.12835	0.564173	−	0.825657i	$-0.309195\pi$
$373$	3.03190e6	1.12835	0.564173	+	0.825657i	$0.309195\pi$
$374$	0	0
$375$	1.49915e6	0.550511
$376$	0	0
$377$	−384510.	−0.139333
$378$	0	0
$379$	5.24333e6	1.87503	0.937517	−	0.347940i	$-0.113119\pi$
$379$	5.24333e6	1.87503	0.937517	+	0.347940i	$0.113119\pi$
$380$	0	0
$381$	1.54221e6	0.544292
$382$	0	0
$383$	4.63538e6	1.61469	0.807343	−	0.590082i	$-0.200905\pi$
$383$	4.63538e6	1.61469	0.807343	+	0.590082i	$0.200905\pi$
$384$	0	0
$385$	−3.00180e6	−1.03212
$386$	0	0
$387$	−879189.	−0.298404
$388$	0	0
$389$	2.19168e6	0.734350	0.367175	−	0.930152i	$-0.380325\pi$
$389$	2.19168e6	0.734350	0.367175	+	0.930152i	$0.380325\pi$
$390$	0	0
$391$	4.00851e6	1.32599
$392$	0	0
$393$	835681.	0.272935
$394$	0	0
$395$	−1.38073e6	−0.445264
$396$	0	0
$397$	5.22159e6	1.66275	0.831375	−	0.555712i	$-0.187554\pi$
$397$	5.22159e6	1.66275	0.831375	+	0.555712i	$0.187554\pi$
$398$	0	0
$399$	443455.	0.139450
$400$	0	0
$401$	2.26293e6	0.702764	0.351382	−	0.936232i	$-0.385712\pi$
$401$	2.26293e6	0.702764	0.351382	+	0.936232i	$0.385712\pi$
$402$	0	0
$403$	225914.	0.0692917
$404$	0	0
$405$	−208598.	−0.0631935
$406$	0	0
$407$	2.18783e6	0.654678
$408$	0	0
$409$	3.33165e6	0.984807	0.492403	−	0.870367i	$-0.336118\pi$
$409$	3.33165e6	0.984807	0.492403	+	0.870367i	$0.336118\pi$
$410$	0	0
$411$	−875532.	−0.255663
$412$	0	0
$413$	−2.35210e6	−0.678548
$414$	0	0
$415$	590624.	0.168341
$416$	0	0
$417$	2.90876e6	0.819159
$418$	0	0
$419$	−3.58378e6	−0.997256	−0.498628	−	0.866816i	$-0.666163\pi$
$419$	−3.58378e6	−0.997256	−0.498628	+	0.866816i	$0.666163\pi$
$420$	0	0
$421$	−1.61256e6	−0.443416	−0.221708	−	0.975113i	$-0.571163\pi$
$421$	−1.61256e6	−0.443416	−0.221708	+	0.975113i	$0.571163\pi$
$422$	0	0
$423$	1.00150e6	0.272144
$424$	0	0
$425$	−2.97880e6	−0.799962
$426$	0	0
$427$	1.28620e6	0.341380
$428$	0	0
$429$	−1.62896e6	−0.427332
$430$	0	0
$431$	981411.	0.254482	0.127241	−	0.991872i	$-0.459388\pi$
$431$	981411.	0.254482	0.127241	+	0.991872i	$0.459388\pi$
$432$	0	0
$433$	−255387.	−0.0654606	−0.0327303	−	0.999464i	$-0.510420\pi$
$433$	−255387.	−0.0654606	−0.0327303	+	0.999464i	$0.510420\pi$
$434$	0	0
$435$	−420498.	−0.106547
$436$	0	0
$437$	1.02704e6	0.257267
$438$	0	0
$439$	−2.60776e6	−0.645813	−0.322907	−	0.946431i	$-0.604660\pi$
$439$	−2.60776e6	−0.645813	−0.322907	+	0.946431i	$0.604660\pi$
$440$	0	0
$441$	147620.	0.0361451
$442$	0	0
$443$	4.41189e6	1.06811	0.534055	−	0.845450i	$-0.320668\pi$
$443$	4.41189e6	1.06811	0.534055	+	0.845450i	$0.320668\pi$
$444$	0	0
$445$	886247.	0.212156
$446$	0	0
$447$	−597180.	−0.141363
$448$	0	0
$449$	−8.12094e6	−1.90104	−0.950518	−	0.310669i	$-0.899447\pi$
$449$	−8.12094e6	−1.90104	−0.950518	+	0.310669i	$0.899447\pi$
$450$	0	0
$451$	5.16675e6	1.19612
$452$	0	0
$453$	3.35001e6	0.767008
$454$	0	0
$455$	1.13544e6	0.257120
$456$	0	0
$457$	−5.29918e6	−1.18691	−0.593455	−	0.804867i	$-0.702237\pi$
$457$	−5.29918e6	−1.18691	−0.593455	+	0.804867i	$0.702237\pi$
$458$	0	0
$459$	1.02714e6	0.227561
$460$	0	0
$461$	−7.84132e6	−1.71845	−0.859225	−	0.511598i	$-0.829054\pi$
$461$	−7.84132e6	−1.71845	−0.859225	+	0.511598i	$0.829054\pi$
$462$	0	0
$463$	936253.	0.202974	0.101487	−	0.994837i	$-0.467640\pi$
$463$	936253.	0.202974	0.101487	+	0.994837i	$0.467640\pi$
$464$	0	0
$465$	247059.	0.0529869
$466$	0	0
$467$	1.72974e6	0.367018	0.183509	−	0.983018i	$-0.441254\pi$
$467$	1.72974e6	0.367018	0.183509	+	0.983018i	$0.441254\pi$
$468$	0	0
$469$	−4.89009e6	−1.02656
$470$	0	0
$471$	−4.43843e6	−0.921885
$472$	0	0
$473$	−7.50825e6	−1.54307
$474$	0	0
$475$	−763214.	−0.155207
$476$	0	0
$477$	1.54086e6	0.310075
$478$	0	0
$479$	−6.69000e6	−1.33225	−0.666127	−	0.745838i	$-0.732049\pi$
$479$	−6.69000e6	−1.33225	−0.666127	+	0.745838i	$0.732049\pi$
$480$	0	0
$481$	−827555.	−0.163093
$482$	0	0
$483$	3.49481e6	0.681640
$484$	0	0
$485$	−604210.	−0.116636
$486$	0	0
$487$	−4.18881e6	−0.800328	−0.400164	−	0.916444i	$-0.631047\pi$
$487$	−4.18881e6	−0.800328	−0.400164	+	0.916444i	$0.631047\pi$
$488$	0	0
$489$	1.90416e6	0.360107
$490$	0	0
$491$	−5.51402e6	−1.03220	−0.516101	−	0.856528i	$-0.672617\pi$
$491$	−5.51402e6	−1.03220	−0.516101	+	0.856528i	$0.672617\pi$
$492$	0	0
$493$	2.07055e6	0.383679
$494$	0	0
$495$	−1.78142e6	−0.326778
$496$	0	0
$497$	−1.07780e7	−1.95725
$498$	0	0
$499$	−4.44154e6	−0.798513	−0.399257	−	0.916839i	$-0.630732\pi$
$499$	−4.44154e6	−0.798513	−0.399257	+	0.916839i	$0.630732\pi$
$500$	0	0
$501$	1.24982e6	0.222460
$502$	0	0
$503$	7.53545e6	1.32797	0.663987	−	0.747744i	$-0.268863\pi$
$503$	7.53545e6	1.32797	0.663987	+	0.747744i	$0.268863\pi$
$504$	0	0
$505$	1.17657e6	0.205300
$506$	0	0
$507$	−2.72548e6	−0.470894
$508$	0	0
$509$	−1.80688e6	−0.309125	−0.154563	−	0.987983i	$-0.549397\pi$
$509$	−1.80688e6	−0.309125	−0.154563	+	0.987983i	$0.549397\pi$
$510$	0	0
$511$	−2.96582e6	−0.502449
$512$	0	0
$513$	263169.	0.0441511
$514$	0	0
$515$	−4.21979e6	−0.701089
$516$	0	0
$517$	8.55275e6	1.40728
$518$	0	0
$519$	1.43588e6	0.233991
$520$	0	0
$521$	1.62195e6	0.261784	0.130892	−	0.991397i	$-0.458216\pi$
$521$	1.62195e6	0.261784	0.130892	+	0.991397i	$0.458216\pi$
$522$	0	0
$523$	−8.02004e6	−1.28210	−0.641051	−	0.767499i	$-0.721501\pi$
$523$	−8.02004e6	−1.28210	−0.641051	+	0.767499i	$0.721501\pi$
$524$	0	0
$525$	−2.59706e6	−0.411229
$526$	0	0
$527$	−1.21653e6	−0.190807
$528$	0	0
$529$	1.65761e6	0.257539
$530$	0	0
$531$	−1.39586e6	−0.214835
$532$	0	0
$533$	−1.95434e6	−0.297977
$534$	0	0
$535$	4.03888e6	0.610065
$536$	0	0
$537$	−7.38060e6	−1.10448
$538$	0	0
$539$	1.26067e6	0.186909
$540$	0	0
$541$	−91817.8	−0.0134876	−0.00674378	−	0.999977i	$-0.502147\pi$
$541$	−91817.8	−0.0134876	−0.00674378	+	0.999977i	$0.502147\pi$
$542$	0	0
$543$	1.30450e6	0.189864
$544$	0	0
$545$	−5.83752e6	−0.841855
$546$	0	0
$547$	1.56765e6	0.224018	0.112009	−	0.993707i	$-0.464272\pi$
$547$	1.56765e6	0.224018	0.112009	+	0.993707i	$0.464272\pi$
$548$	0	0
$549$	763296.	0.108084
$550$	0	0
$551$	530505.	0.0744407
$552$	0	0
$553$	5.92749e6	0.824248
$554$	0	0
$555$	−905011.	−0.124716
$556$	0	0
$557$	−6.11842e6	−0.835605	−0.417803	−	0.908538i	$-0.637200\pi$
$557$	−6.11842e6	−0.835605	−0.417803	+	0.908538i	$0.637200\pi$
$558$	0	0
$559$	2.84003e6	0.384408
$560$	0	0
$561$	8.77176e6	1.17674
$562$	0	0
$563$	6.06877e6	0.806918	0.403459	−	0.914998i	$-0.367808\pi$
$563$	6.06877e6	0.806918	0.403459	+	0.914998i	$0.367808\pi$
$564$	0	0
$565$	1.21654e6	0.160327
$566$	0	0
$567$	895510.	0.116980
$568$	0	0
$569$	−1.02217e7	−1.32355	−0.661776	−	0.749701i	$-0.730197\pi$
$569$	−1.02217e7	−1.32355	−0.661776	+	0.749701i	$0.730197\pi$
$570$	0	0
$571$	−3.55216e6	−0.455934	−0.227967	−	0.973669i	$-0.573208\pi$
$571$	−3.55216e6	−0.455934	−0.227967	+	0.973669i	$0.573208\pi$
$572$	0	0
$573$	−5.42521e6	−0.690287
$574$	0	0
$575$	−6.01478e6	−0.758665
$576$	0	0
$577$	4.96727e6	0.621125	0.310562	−	0.950553i	$-0.399483\pi$
$577$	4.96727e6	0.621125	0.310562	+	0.950553i	$0.399483\pi$
$578$	0	0
$579$	5.63024e6	0.697959
$580$	0	0
$581$	−2.53555e6	−0.311625
$582$	0	0
$583$	1.31589e7	1.60342
$584$	0	0
$585$	673829.	0.0814067
$586$	0	0
$587$	−1.47975e7	−1.77253	−0.886267	−	0.463175i	$-0.846710\pi$
$587$	−1.47975e7	−1.77253	−0.886267	+	0.463175i	$0.846710\pi$
$588$	0	0
$589$	−311692.	−0.0370201
$590$	0	0
$591$	−250594.	−0.0295123
$592$	0	0
$593$	1.09127e7	1.27437	0.637187	−	0.770709i	$-0.280098\pi$
$593$	1.09127e7	1.27437	0.637187	+	0.770709i	$0.280098\pi$
$594$	0	0
$595$	−6.11424e6	−0.708028
$596$	0	0
$597$	6.44320e6	0.739887
$598$	0	0
$599$	357302.	0.0406881	0.0203441	−	0.999793i	$-0.493524\pi$
$599$	357302.	0.0406881	0.0203441	+	0.999793i	$0.493524\pi$
$600$	0	0
$601$	1.92888e6	0.217830	0.108915	−	0.994051i	$-0.465262\pi$
$601$	1.92888e6	0.217830	0.108915	+	0.994051i	$0.465262\pi$
$602$	0	0
$603$	−2.90203e6	−0.325019
$604$	0	0
$605$	−1.00929e7	−1.12106
$606$	0	0
$607$	−4.04780e6	−0.445911	−0.222955	−	0.974829i	$-0.571570\pi$
$607$	−4.04780e6	−0.445911	−0.222955	+	0.974829i	$0.571570\pi$
$608$	0	0
$609$	1.80520e6	0.197234
$610$	0	0
$611$	−3.23511e6	−0.350579
$612$	0	0
$613$	−2.94813e6	−0.316881	−0.158440	−	0.987369i	$-0.550647\pi$
$613$	−2.94813e6	−0.316881	−0.158440	+	0.987369i	$0.550647\pi$
$614$	0	0
$615$	−2.13726e6	−0.227861
$616$	0	0
$617$	1.37469e7	1.45376	0.726878	−	0.686767i	$-0.240971\pi$
$617$	1.37469e7	1.45376	0.726878	+	0.686767i	$0.240971\pi$
$618$	0	0
$619$	−2.35692e6	−0.247239	−0.123620	−	0.992330i	$-0.539450\pi$
$619$	−2.35692e6	−0.247239	−0.123620	+	0.992330i	$0.539450\pi$
$620$	0	0
$621$	2.07400e6	0.215814
$622$	0	0
$623$	−3.80466e6	−0.392731
$624$	0	0
$625$	1.31085e6	0.134231
$626$	0	0
$627$	2.24746e6	0.228309
$628$	0	0
$629$	4.45630e6	0.449105
$630$	0	0
$631$	9.99454e6	0.999285	0.499643	−	0.866232i	$-0.333465\pi$
$631$	9.99454e6	0.999285	0.499643	+	0.866232i	$0.333465\pi$
$632$	0	0
$633$	1.07521e7	1.06656
$634$	0	0
$635$	−5.44806e6	−0.536176
$636$	0	0
$637$	−476855.	−0.0465626
$638$	0	0
$639$	−6.39620e6	−0.619684
$640$	0	0
$641$	2.18866e6	0.210394	0.105197	−	0.994451i	$-0.466453\pi$
$641$	2.18866e6	0.210394	0.105197	+	0.994451i	$0.466453\pi$
$642$	0	0
$643$	−1.42232e7	−1.35666	−0.678329	−	0.734758i	$-0.737296\pi$
$643$	−1.42232e7	−1.35666	−0.678329	+	0.734758i	$0.737296\pi$
$644$	0	0
$645$	3.10584e6	0.293955
$646$	0	0
$647$	−1.32117e7	−1.24079	−0.620397	−	0.784288i	$-0.713028\pi$
$647$	−1.32117e7	−1.24079	−0.620397	+	0.784288i	$0.713028\pi$
$648$	0	0
$649$	−1.19206e7	−1.11093
$650$	0	0
$651$	−1.06062e6	−0.0980865
$652$	0	0
$653$	−9.55832e6	−0.877200	−0.438600	−	0.898682i	$-0.644525\pi$
$653$	−9.55832e6	−0.877200	−0.438600	+	0.898682i	$0.644525\pi$
$654$	0	0
$655$	−2.95215e6	−0.268865
$656$	0	0
$657$	−1.76007e6	−0.159080
$658$	0	0
$659$	−1.85335e7	−1.66244	−0.831218	−	0.555946i	$-0.812356\pi$
$659$	−1.85335e7	−1.66244	−0.831218	+	0.555946i	$0.812356\pi$
$660$	0	0
$661$	−5.06510e6	−0.450905	−0.225452	−	0.974254i	$-0.572386\pi$
$661$	−5.06510e6	−0.450905	−0.225452	+	0.974254i	$0.572386\pi$
$662$	0	0
$663$	−3.31795e6	−0.293148
$664$	0	0
$665$	−1.56656e6	−0.137370
$666$	0	0
$667$	4.18083e6	0.363871
$668$	0	0
$669$	−3.93593e6	−0.340002
$670$	0	0
$671$	6.51853e6	0.558912
$672$	0	0
$673$	7.33110e6	0.623923	0.311962	−	0.950095i	$-0.399014\pi$
$673$	7.33110e6	0.623923	0.311962	+	0.950095i	$0.399014\pi$
$674$	0	0
$675$	−1.54123e6	−0.130199
$676$	0	0
$677$	−7.34608e6	−0.616004	−0.308002	−	0.951386i	$-0.599660\pi$
$677$	−7.34608e6	−0.616004	−0.308002	+	0.951386i	$0.599660\pi$
$678$	0	0
$679$	2.59387e6	0.215911
$680$	0	0
$681$	−3.33699e6	−0.275732
$682$	0	0
$683$	1.54260e7	1.26532	0.632662	−	0.774428i	$-0.281962\pi$
$683$	1.54260e7	1.26532	0.632662	+	0.774428i	$0.281962\pi$
$684$	0	0
$685$	3.09292e6	0.251851
$686$	0	0
$687$	−1.15066e7	−0.930158
$688$	0	0
$689$	−4.97740e6	−0.399443
$690$	0	0
$691$	−2.01025e7	−1.60161	−0.800803	−	0.598928i	$-0.795594\pi$
$691$	−2.01025e7	−1.60161	−0.800803	+	0.598928i	$0.795594\pi$
$692$	0	0
$693$	7.64763e6	0.604914
$694$	0	0
$695$	−1.02756e7	−0.806945
$696$	0	0
$697$	1.05239e7	0.820534
$698$	0	0
$699$	9.00086e6	0.696773
$700$	0	0
$701$	1.39186e7	1.06980	0.534898	−	0.844917i	$-0.320350\pi$
$701$	1.39186e7	1.06980	0.534898	+	0.844917i	$0.320350\pi$
$702$	0	0
$703$	1.14177e6	0.0871346
$704$	0	0
$705$	−3.53791e6	−0.268086
$706$	0	0
$707$	−5.05101e6	−0.380040
$708$	0	0
$709$	−2.21375e7	−1.65391	−0.826957	−	0.562266i	$-0.809930\pi$
$709$	−2.21375e7	−1.65391	−0.826957	+	0.562266i	$0.809930\pi$
$710$	0	0
$711$	3.51767e6	0.260964
$712$	0	0
$713$	−2.45640e6	−0.180957
$714$	0	0
$715$	5.75448e6	0.420960
$716$	0	0
$717$	1.31278e6	0.0953659
$718$	0	0
$719$	2.94192e6	0.212231	0.106116	−	0.994354i	$-0.466159\pi$
$719$	2.94192e6	0.212231	0.106116	+	0.994354i	$0.466159\pi$
$720$	0	0
$721$	1.81156e7	1.29782
$722$	0	0
$723$	1.27292e7	0.905637
$724$	0	0
$725$	−3.10686e6	−0.219521
$726$	0	0
$727$	−1.79898e7	−1.26238	−0.631189	−	0.775629i	$-0.717433\pi$
$727$	−1.79898e7	−1.26238	−0.631189	+	0.775629i	$0.717433\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.52933e7	−1.05854
$732$	0	0
$733$	6.63171e6	0.455896	0.227948	−	0.973673i	$-0.426798\pi$
$733$	6.63171e6	0.455896	0.227948	+	0.973673i	$0.426798\pi$
$734$	0	0
$735$	−521487.	−0.0356061
$736$	0	0
$737$	−2.47832e7	−1.68070
$738$	0	0
$739$	−1.98503e7	−1.33708	−0.668539	−	0.743677i	$-0.733080\pi$
$739$	−1.98503e7	−1.33708	−0.668539	+	0.743677i	$0.733080\pi$
$740$	0	0
$741$	−850109.	−0.0568760
$742$	0	0
$743$	1.73059e7	1.15007	0.575033	−	0.818130i	$-0.304989\pi$
$743$	1.73059e7	1.15007	0.575033	+	0.818130i	$0.304989\pi$
$744$	0	0
$745$	2.10961e6	0.139255
$746$	0	0
$747$	−1.50472e6	−0.0986632
$748$	0	0
$749$	−1.73389e7	−1.12932
$750$	0	0
$751$	−3.05367e6	−0.197571	−0.0987854	−	0.995109i	$-0.531496\pi$
$751$	−3.05367e6	−0.197571	−0.0987854	+	0.995109i	$0.531496\pi$
$752$	0	0
$753$	−1.95098e6	−0.125390
$754$	0	0
$755$	−1.18343e7	−0.755571
$756$	0	0
$757$	−1.58229e7	−1.00357	−0.501783	−	0.864994i	$-0.667322\pi$
$757$	−1.58229e7	−1.00357	−0.501783	+	0.864994i	$0.667322\pi$
$758$	0	0
$759$	1.77119e7	1.11599
$760$	0	0
$761$	−8.84728e6	−0.553794	−0.276897	−	0.960900i	$-0.589306\pi$
$761$	−8.84728e6	−0.553794	−0.276897	+	0.960900i	$0.589306\pi$
$762$	0	0
$763$	2.50605e7	1.55840
$764$	0	0
$765$	−3.62850e6	−0.224168
$766$	0	0
$767$	4.50900e6	0.276753
$768$	0	0
$769$	2.54971e7	1.55480	0.777400	−	0.629007i	$-0.216538\pi$
$769$	2.54971e7	1.55480	0.777400	+	0.629007i	$0.216538\pi$
$770$	0	0
$771$	7.68580e6	0.465643
$772$	0	0
$773$	762315.	0.0458866	0.0229433	−	0.999737i	$-0.492696\pi$
$773$	762315.	0.0458866	0.0229433	+	0.999737i	$0.492696\pi$
$774$	0	0
$775$	1.82540e6	0.109170
$776$	0	0
$777$	3.88521e6	0.230867
$778$	0	0
$779$	2.69639e6	0.159199
$780$	0	0
$781$	−5.46234e7	−3.20443
$782$	0	0
$783$	1.07130e6	0.0624461
$784$	0	0
$785$	1.56793e7	0.908138
$786$	0	0
$787$	−5.26619e6	−0.303082	−0.151541	−	0.988451i	$-0.548423\pi$
$787$	−5.26619e6	−0.303082	−0.151541	+	0.988451i	$0.548423\pi$
$788$	0	0
$789$	4.20881e6	0.240695
$790$	0	0
$791$	−5.22261e6	−0.296788
$792$	0	0
$793$	−2.46566e6	−0.139235
$794$	0	0
$795$	−5.44327e6	−0.305451
$796$	0	0
$797$	−7.81454e6	−0.435770	−0.217885	−	0.975974i	$-0.569916\pi$
$797$	−7.81454e6	−0.435770	−0.217885	+	0.975974i	$0.569916\pi$
$798$	0	0
$799$	1.74207e7	0.965384
$800$	0	0
$801$	−2.25788e6	−0.124342
$802$	0	0
$803$	−1.50309e7	−0.822616
$804$	0	0
$805$	−1.23458e7	−0.671476
$806$	0	0
$807$	4.04767e6	0.218787
$808$	0	0
$809$	8.53525e6	0.458506	0.229253	−	0.973367i	$-0.426372\pi$
$809$	8.53525e6	0.458506	0.229253	+	0.973367i	$0.426372\pi$
$810$	0	0
$811$	522103.	0.0278743	0.0139371	−	0.999903i	$-0.495564\pi$
$811$	522103.	0.0278743	0.0139371	+	0.999903i	$0.495564\pi$
$812$	0	0
$813$	−9.13319e6	−0.484614
$814$	0	0
$815$	−6.72669e6	−0.354738
$816$	0	0
$817$	−3.91836e6	−0.205376
$818$	0	0
$819$	−2.89275e6	−0.150696
$820$	0	0
$821$	−1.52943e7	−0.791902	−0.395951	−	0.918272i	$-0.629585\pi$
$821$	−1.52943e7	−0.791902	−0.395951	+	0.918272i	$0.629585\pi$
$822$	0	0
$823$	2.75392e7	1.41727	0.708635	−	0.705575i	$-0.249311\pi$
$823$	2.75392e7	1.41727	0.708635	+	0.705575i	$0.249311\pi$
$824$	0	0
$825$	−1.31620e7	−0.673269
$826$	0	0
$827$	−3.19965e7	−1.62682	−0.813409	−	0.581693i	$-0.802391\pi$
$827$	−3.19965e7	−1.62682	−0.813409	+	0.581693i	$0.802391\pi$
$828$	0	0
$829$	−6.88245e6	−0.347822	−0.173911	−	0.984761i	$-0.555640\pi$
$829$	−6.88245e6	−0.347822	−0.173911	+	0.984761i	$0.555640\pi$
$830$	0	0
$831$	−1.82635e6	−0.0917447
$832$	0	0
$833$	2.56782e6	0.128219
$834$	0	0
$835$	−4.41513e6	−0.219143
$836$	0	0
$837$	−629428.	−0.0310551
$838$	0	0
$839$	−1.74797e7	−0.857292	−0.428646	−	0.903472i	$-0.641009\pi$
$839$	−1.74797e7	−0.857292	−0.428646	+	0.903472i	$0.641009\pi$
$840$	0	0
$841$	−1.83516e7	−0.894713
$842$	0	0
$843$	5.30585e6	0.257150
$844$	0	0
$845$	9.62809e6	0.463872
$846$	0	0
$847$	4.33288e7	2.07524
$848$	0	0
$849$	−2.30221e7	−1.09617
$850$	0	0
$851$	8.99813e6	0.425920
$852$	0	0
$853$	1.12972e7	0.531615	0.265808	−	0.964026i	$-0.414361\pi$
$853$	1.12972e7	0.531615	0.265808	+	0.964026i	$0.414361\pi$
$854$	0	0
$855$	−929677.	−0.0434927
$856$	0	0
$857$	−5.02156e6	−0.233553	−0.116777	−	0.993158i	$-0.537256\pi$
$857$	−5.02156e6	−0.233553	−0.116777	+	0.993158i	$0.537256\pi$
$858$	0	0
$859$	1.53818e7	0.711254	0.355627	−	0.934628i	$-0.384267\pi$
$859$	1.53818e7	0.711254	0.355627	+	0.934628i	$0.384267\pi$
$860$	0	0
$861$	9.17527e6	0.421804
$862$	0	0
$863$	−2.34908e7	−1.07367	−0.536834	−	0.843688i	$-0.680380\pi$
$863$	−2.34908e7	−1.07367	−0.536834	+	0.843688i	$0.680380\pi$
$864$	0	0
$865$	−5.07241e6	−0.230502
$866$	0	0
$867$	5.08813e6	0.229885
$868$	0	0
$869$	3.00408e7	1.34947
$870$	0	0
$871$	9.37436e6	0.418693
$872$	0	0
$873$	1.53934e6	0.0683593
$874$	0	0
$875$	2.27354e7	1.00388
$876$	0	0
$877$	1.64937e7	0.724132	0.362066	−	0.932152i	$-0.382071\pi$
$877$	1.64937e7	0.724132	0.362066	+	0.932152i	$0.382071\pi$
$878$	0	0
$879$	2.44408e7	1.06695
$880$	0	0
$881$	−8.58613e6	−0.372699	−0.186349	−	0.982484i	$-0.559666\pi$
$881$	−8.58613e6	−0.372699	−0.186349	+	0.982484i	$0.559666\pi$
$882$	0	0
$883$	−3.36967e7	−1.45441	−0.727203	−	0.686423i	$-0.759180\pi$
$883$	−3.36967e7	−1.45441	−0.727203	+	0.686423i	$0.759180\pi$
$884$	0	0
$885$	4.93103e6	0.211631
$886$	0	0
$887$	2.88644e6	0.123184	0.0615918	−	0.998101i	$-0.480382\pi$
$887$	2.88644e6	0.123184	0.0615918	+	0.998101i	$0.480382\pi$
$888$	0	0
$889$	2.33885e7	0.992540
$890$	0	0
$891$	4.53849e6	0.191521
$892$	0	0
$893$	4.46346e6	0.187302
$894$	0	0
$895$	2.60729e7	1.08801
$896$	0	0
$897$	−6.69959e6	−0.278014
$898$	0	0
$899$	−1.26882e6	−0.0523603
$900$	0	0
$901$	2.68028e7	1.09994
$902$	0	0
$903$	−1.33334e7	−0.544153
$904$	0	0
$905$	−4.60829e6	−0.187033
$906$	0	0
$907$	−1.33506e6	−0.0538869	−0.0269434	−	0.999637i	$-0.508577\pi$
$907$	−1.33506e6	−0.0538869	−0.0269434	+	0.999637i	$0.508577\pi$
$908$	0	0
$909$	−2.99753e6	−0.120324
$910$	0	0
$911$	3.15965e7	1.26137	0.630685	−	0.776039i	$-0.282774\pi$
$911$	3.15965e7	1.26137	0.630685	+	0.776039i	$0.282774\pi$
$912$	0	0
$913$	−1.28503e7	−0.510195
$914$	0	0
$915$	−2.69644e6	−0.106472
$916$	0	0
$917$	1.26736e7	0.497709
$918$	0	0
$919$	−3.64069e6	−0.142198	−0.0710992	−	0.997469i	$-0.522651\pi$
$919$	−3.64069e6	−0.142198	−0.0710992	+	0.997469i	$0.522651\pi$
$920$	0	0
$921$	−7.66304e6	−0.297682
$922$	0	0
$923$	2.06615e7	0.798285
$924$	0	0
$925$	−6.68669e6	−0.256955
$926$	0	0
$927$	1.07507e7	0.410901
$928$	0	0
$929$	1.50483e7	0.572068	0.286034	−	0.958219i	$-0.407663\pi$
$929$	1.50483e7	0.572068	0.286034	+	0.958219i	$0.407663\pi$
$930$	0	0
$931$	657913.	0.0248768
$932$	0	0
$933$	4.79829e6	0.180461
$934$	0	0
$935$	−3.09873e7	−1.15919
$936$	0	0
$937$	−1.49448e7	−0.556084	−0.278042	−	0.960569i	$-0.589685\pi$
$937$	−1.49448e7	−0.556084	−0.278042	+	0.960569i	$0.589685\pi$
$938$	0	0
$939$	−988515.	−0.0365864
$940$	0	0
$941$	187690.	0.00690984	0.00345492	−	0.999994i	$-0.498900\pi$
$941$	187690.	0.00690984	0.00345492	+	0.999994i	$0.498900\pi$
$942$	0	0
$943$	2.12499e7	0.778175
$944$	0	0
$945$	−3.16350e6	−0.115236
$946$	0	0
$947$	−5.36291e7	−1.94324	−0.971618	−	0.236557i	$-0.923981\pi$
$947$	−5.36291e7	−1.94324	−0.971618	+	0.236557i	$0.923981\pi$
$948$	0	0
$949$	5.68551e6	0.204929
$950$	0	0
$951$	2.49890e7	0.895980
$952$	0	0
$953$	−3.59848e7	−1.28347	−0.641737	−	0.766925i	$-0.721786\pi$
$953$	−3.59848e7	−1.28347	−0.641737	+	0.766925i	$0.721786\pi$
$954$	0	0
$955$	1.91652e7	0.679994
$956$	0	0
$957$	9.14885e6	0.322914
$958$	0	0
$959$	−1.32779e7	−0.466212
$960$	0	0
$961$	−2.78837e7	−0.973961
$962$	0	0
$963$	−1.02898e7	−0.357553
$964$	0	0
$965$	−1.98895e7	−0.687552
$966$	0	0
$967$	1.31181e7	0.451132	0.225566	−	0.974228i	$-0.427577\pi$
$967$	1.31181e7	0.451132	0.225566	+	0.974228i	$0.427577\pi$
$968$	0	0
$969$	4.57775e6	0.156618
$970$	0	0
$971$	−2.04465e7	−0.695940	−0.347970	−	0.937506i	$-0.613129\pi$
$971$	−2.04465e7	−0.695940	−0.347970	+	0.937506i	$0.613129\pi$
$972$	0	0
$973$	4.41130e7	1.49377
$974$	0	0
$975$	4.97860e6	0.167724
$976$	0	0
$977$	3.26765e7	1.09521	0.547607	−	0.836736i	$-0.315539\pi$
$977$	3.26765e7	1.09521	0.547607	+	0.836736i	$0.315539\pi$
$978$	0	0
$979$	−1.92822e7	−0.642984
$980$	0	0
$981$	1.48722e7	0.493403
$982$	0	0
$983$	−3.07073e7	−1.01358	−0.506789	−	0.862070i	$-0.669168\pi$
$983$	−3.07073e7	−1.01358	−0.506789	+	0.862070i	$0.669168\pi$
$984$	0	0
$985$	885255.	0.0290722
$986$	0	0
$987$	1.51882e7	0.496266
$988$	0	0
$989$	−3.08800e7	−1.00389
$990$	0	0
$991$	2.86036e7	0.925203	0.462601	−	0.886566i	$-0.346916\pi$
$991$	2.86036e7	0.925203	0.462601	+	0.886566i	$0.346916\pi$
$992$	0	0
$993$	2.51303e7	0.808770
$994$	0	0
$995$	−2.27614e7	−0.728855
$996$	0	0
$997$	3.62989e7	1.15653	0.578263	−	0.815850i	$-0.303731\pi$
$997$	3.62989e7	1.15653	0.578263	+	0.815850i	$0.303731\pi$
$998$	0	0
$999$	2.30568e6	0.0730947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.6.a.s.1.1		4
4.3	odd	2		228.6.a.c.1.1	✓	4
12.11	even	2		684.6.a.c.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.a.c.1.1	✓	4	4.3	odd	2
684.6.a.c.1.4		4	12.11	even	2
912.6.a.s.1.1		4	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -31.7936 q^{5} +136.490 q^{7} +81.0000 q^{9} +691.738 q^{11} -261.653 q^{13} -286.142 q^{15} +1408.97 q^{17} +361.000 q^{19} +1228.41 q^{21} +2844.99 q^{23} -2114.17 q^{25} +729.000 q^{27} +1469.54 q^{29} -863.414 q^{31} +6225.64 q^{33} -4339.50 q^{35} +3162.80 q^{37} -2354.87 q^{39} +7469.23 q^{41} -10854.2 q^{43} -2575.28 q^{45} +12364.1 q^{47} +1822.47 q^{49} +12680.8 q^{51} +19022.9 q^{53} -21992.8 q^{55} +3249.00 q^{57} -17232.8 q^{59} +9423.41 q^{61} +11055.7 q^{63} +8318.88 q^{65} -35827.5 q^{67} +25604.9 q^{69} -78965.5 q^{71} -21729.2 q^{73} -19027.5 q^{75} +94415.2 q^{77} +43428.1 q^{79} +6561.00 q^{81} -18576.8 q^{83} -44796.3 q^{85} +13225.9 q^{87} -27875.0 q^{89} -35712.9 q^{91} -7770.72 q^{93} -11477.5 q^{95} +19004.2 q^{97} +56030.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 117 q^{5} + 33 q^{7} + 324 q^{9} + 129 q^{11} + 638 q^{13} + 1053 q^{15} + 285 q^{17} + 1444 q^{19} + 297 q^{21} + 2340 q^{23} + 3629 q^{25} + 2916 q^{27} + 438 q^{29} - 160 q^{31} + 1161 q^{33}+ \cdots + 10449 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 117 * q^5 + 33 * q^7 + 324 * q^9 + 129 * q^11 + 638 * q^13 + 1053 * q^15 + 285 * q^17 + 1444 * q^19 + 297 * q^21 + 2340 * q^23 + 3629 * q^25 + 2916 * q^27 + 438 * q^29 - 160 * q^31 + 1161 * q^33 + 2409 * q^35 + 15520 * q^37 + 5742 * q^39 + 14664 * q^41 - 22519 * q^43 + 9477 * q^45 - 16395 * q^47 + 64925 * q^49 + 2565 * q^51 + 61266 * q^53 - 49921 * q^55 + 12996 * q^57 - 67512 * q^59 + 104285 * q^61 + 2673 * q^63 + 134466 * q^65 - 40584 * q^67 + 21060 * q^69 - 98856 * q^71 + 46323 * q^73 + 32661 * q^75 + 164715 * q^77 + 25126 * q^79 + 26244 * q^81 - 34026 * q^83 - 21861 * q^85 + 3942 * q^87 + 132606 * q^89 + 150138 * q^91 - 1440 * q^93 + 42237 * q^95 - 31892 * q^97 + 10449 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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