LMFDB - Embedded newform 1028.6.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.91744 q^{3} +8.20543 q^{5} +107.816 q^{7} -144.644 q^{9} +O(q^{10})$	$q-9.91744 q^{3} +8.20543 q^{5} +107.816 q^{7} -144.644 q^{9} +167.608 q^{11} -534.101 q^{13} -81.3768 q^{15} +2016.65 q^{17} -1857.37 q^{19} -1069.26 q^{21} +1096.31 q^{23} -3057.67 q^{25} +3844.44 q^{27} -609.863 q^{29} +3021.39 q^{31} -1662.24 q^{33} +884.675 q^{35} -4670.67 q^{37} +5296.91 q^{39} +526.091 q^{41} +6569.57 q^{43} -1186.87 q^{45} +11555.1 q^{47} -5182.74 q^{49} -20000.0 q^{51} -15738.2 q^{53} +1375.29 q^{55} +18420.4 q^{57} +28544.2 q^{59} +23990.6 q^{61} -15594.9 q^{63} -4382.52 q^{65} -38620.2 q^{67} -10872.6 q^{69} -17100.1 q^{71} -4402.18 q^{73} +30324.3 q^{75} +18070.8 q^{77} -28673.8 q^{79} -2978.46 q^{81} +97401.8 q^{83} +16547.5 q^{85} +6048.28 q^{87} -39455.8 q^{89} -57584.5 q^{91} -29964.5 q^{93} -15240.6 q^{95} +73609.7 q^{97} -24243.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.91744	−0.636204	−0.318102	−	0.948056i	$-0.603046\pi$
$3$	−9.91744	−0.636204	−0.318102	+	0.948056i	$0.603046\pi$
$4$	0	0
$5$	8.20543	0.146783	0.0733916	−	0.997303i	$-0.476618\pi$
$5$	8.20543	0.146783	0.0733916	+	0.997303i	$0.476618\pi$
$6$	0	0
$7$	107.816	0.831644	0.415822	−	0.909446i	$-0.363494\pi$
$7$	107.816	0.831644	0.415822	+	0.909446i	$0.363494\pi$
$8$	0	0
$9$	−144.644	−0.595244
$10$	0	0
$11$	167.608	0.417650	0.208825	−	0.977953i	$-0.433036\pi$
$11$	167.608	0.417650	0.208825	+	0.977953i	$0.433036\pi$
$12$	0	0
$13$	−534.101	−0.876526	−0.438263	−	0.898847i	$-0.644406\pi$
$13$	−534.101	−0.876526	−0.438263	+	0.898847i	$0.644406\pi$
$14$	0	0
$15$	−81.3768	−0.0933840
$16$	0	0
$17$	2016.65	1.69242	0.846209	−	0.532851i	$-0.178879\pi$
$17$	2016.65	1.69242	0.846209	+	0.532851i	$0.178879\pi$
$18$	0	0
$19$	−1857.37	−1.18036	−0.590181	−	0.807271i	$-0.700944\pi$
$19$	−1857.37	−1.18036	−0.590181	+	0.807271i	$0.700944\pi$
$20$	0	0
$21$	−1069.26	−0.529096
$22$	0	0
$23$	1096.31	0.432131	0.216066	−	0.976379i	$-0.430677\pi$
$23$	1096.31	0.432131	0.216066	+	0.976379i	$0.430677\pi$
$24$	0	0
$25$	−3057.67	−0.978455
$26$	0	0
$27$	3844.44	1.01490
$28$	0	0
$29$	−609.863	−0.134660	−0.0673298	−	0.997731i	$-0.521448\pi$
$29$	−609.863	−0.134660	−0.0673298	+	0.997731i	$0.521448\pi$
$30$	0	0
$31$	3021.39	0.564681	0.282340	−	0.959314i	$-0.408889\pi$
$31$	3021.39	0.564681	0.282340	+	0.959314i	$0.408889\pi$
$32$	0	0
$33$	−1662.24	−0.265711
$34$	0	0
$35$	884.675	0.122071
$36$	0	0
$37$	−4670.67	−0.560886	−0.280443	−	0.959871i	$-0.590481\pi$
$37$	−4670.67	−0.560886	−0.280443	+	0.959871i	$0.590481\pi$
$38$	0	0
$39$	5296.91	0.557650
$40$	0	0
$41$	526.091	0.0488767	0.0244383	−	0.999701i	$-0.492220\pi$
$41$	526.091	0.0488767	0.0244383	+	0.999701i	$0.492220\pi$
$42$	0	0
$43$	6569.57	0.541833	0.270917	−	0.962603i	$-0.412673\pi$
$43$	6569.57	0.541833	0.270917	+	0.962603i	$0.412673\pi$
$44$	0	0
$45$	−1186.87	−0.0873718
$46$	0	0
$47$	11555.1	0.763011	0.381506	−	0.924367i	$-0.375406\pi$
$47$	11555.1	0.763011	0.381506	+	0.924367i	$0.375406\pi$
$48$	0	0
$49$	−5182.74	−0.308368
$50$	0	0
$51$	−20000.0	−1.07672
$52$	0	0
$53$	−15738.2	−0.769599	−0.384799	−	0.923000i	$-0.625729\pi$
$53$	−15738.2	−0.769599	−0.384799	+	0.923000i	$0.625729\pi$
$54$	0	0
$55$	1375.29	0.0613040
$56$	0	0
$57$	18420.4	0.750952
$58$	0	0
$59$	28544.2	1.06755	0.533774	−	0.845627i	$-0.320773\pi$
$59$	28544.2	1.06755	0.533774	+	0.845627i	$0.320773\pi$
$60$	0	0
$61$	23990.6	0.825501	0.412750	−	0.910844i	$-0.364568\pi$
$61$	23990.6	0.825501	0.412750	+	0.910844i	$0.364568\pi$
$62$	0	0
$63$	−15594.9	−0.495031
$64$	0	0
$65$	−4382.52	−0.128659
$66$	0	0
$67$	−38620.2	−1.05106	−0.525530	−	0.850775i	$-0.676133\pi$
$67$	−38620.2	−1.05106	−0.525530	+	0.850775i	$0.676133\pi$
$68$	0	0
$69$	−10872.6	−0.274924
$70$	0	0
$71$	−17100.1	−0.402581	−0.201290	−	0.979532i	$-0.564514\pi$
$71$	−17100.1	−0.402581	−0.201290	+	0.979532i	$0.564514\pi$
$72$	0	0
$73$	−4402.18	−0.0966853	−0.0483427	−	0.998831i	$-0.515394\pi$
$73$	−4402.18	−0.0966853	−0.0483427	+	0.998831i	$0.515394\pi$
$74$	0	0
$75$	30324.3	0.622497
$76$	0	0
$77$	18070.8	0.347336
$78$	0	0
$79$	−28673.8	−0.516914	−0.258457	−	0.966023i	$-0.583214\pi$
$79$	−28673.8	−0.516914	−0.258457	+	0.966023i	$0.583214\pi$
$80$	0	0
$81$	−2978.46	−0.0504404
$82$	0	0
$83$	97401.8	1.55193	0.775964	−	0.630777i	$-0.217264\pi$
$83$	97401.8	1.55193	0.775964	+	0.630777i	$0.217264\pi$
$84$	0	0
$85$	16547.5	0.248418
$86$	0	0
$87$	6048.28	0.0856710
$88$	0	0
$89$	−39455.8	−0.528003	−0.264001	−	0.964522i	$-0.585042\pi$
$89$	−39455.8	−0.528003	−0.264001	+	0.964522i	$0.585042\pi$
$90$	0	0
$91$	−57584.5	−0.728958
$92$	0	0
$93$	−29964.5	−0.359252
$94$	0	0
$95$	−15240.6	−0.173257
$96$	0	0
$97$	73609.7	0.794339	0.397169	−	0.917745i	$-0.369993\pi$
$97$	73609.7	0.794339	0.397169	+	0.917745i	$0.369993\pi$
$98$	0	0
$99$	−24243.5	−0.248604
$100$	0	0
$101$	113153.	1.10373	0.551865	−	0.833934i	$-0.313917\pi$
$101$	113153.	1.10373	0.551865	+	0.833934i	$0.313917\pi$
$102$	0	0
$103$	−148166.	−1.37612	−0.688061	−	0.725653i	$-0.741538\pi$
$103$	−148166.	−1.37612	−0.688061	+	0.725653i	$0.741538\pi$
$104$	0	0
$105$	−8773.71	−0.0776623
$106$	0	0
$107$	−113163.	−0.955531	−0.477766	−	0.878487i	$-0.658553\pi$
$107$	−113163.	−0.955531	−0.477766	+	0.878487i	$0.658553\pi$
$108$	0	0
$109$	−101629.	−0.819312	−0.409656	−	0.912240i	$-0.634351\pi$
$109$	−101629.	−0.819312	−0.409656	+	0.912240i	$0.634351\pi$
$110$	0	0
$111$	46321.1	0.356838
$112$	0	0
$113$	−78476.3	−0.578153	−0.289076	−	0.957306i	$-0.593348\pi$
$113$	−78476.3	−0.578153	−0.289076	+	0.957306i	$0.593348\pi$
$114$	0	0
$115$	8995.73	0.0634296
$116$	0	0
$117$	77254.6	0.521747
$118$	0	0
$119$	217427.	1.40749
$120$	0	0
$121$	−132959.	−0.825568
$122$	0	0
$123$	−5217.48	−0.0310955
$124$	0	0
$125$	−50731.4	−0.290404
$126$	0	0
$127$	−132363.	−0.728213	−0.364107	−	0.931357i	$-0.618626\pi$
$127$	−132363.	−0.728213	−0.364107	+	0.931357i	$0.618626\pi$
$128$	0	0
$129$	−65153.3	−0.344717
$130$	0	0
$131$	182551.	0.929407	0.464703	−	0.885466i	$-0.346161\pi$
$131$	182551.	0.929407	0.464703	+	0.885466i	$0.346161\pi$
$132$	0	0
$133$	−200254.	−0.981642
$134$	0	0
$135$	31545.3	0.148970
$136$	0	0
$137$	43942.8	0.200026	0.100013	−	0.994986i	$-0.468112\pi$
$137$	43942.8	0.200026	0.100013	+	0.994986i	$0.468112\pi$
$138$	0	0
$139$	−81634.5	−0.358374	−0.179187	−	0.983815i	$-0.557347\pi$
$139$	−81634.5	−0.358374	−0.179187	+	0.983815i	$0.557347\pi$
$140$	0	0
$141$	−114598.	−0.485431
$142$	0	0
$143$	−89519.5	−0.366081
$144$	0	0
$145$	−5004.19	−0.0197658
$146$	0	0
$147$	51399.6	0.196185
$148$	0	0
$149$	2062.29	0.00760997	0.00380499	−	0.999993i	$-0.498789\pi$
$149$	2062.29	0.00760997	0.00380499	+	0.999993i	$0.498789\pi$
$150$	0	0
$151$	21041.6	0.0750995	0.0375498	−	0.999295i	$-0.488045\pi$
$151$	21041.6	0.0750995	0.0375498	+	0.999295i	$0.488045\pi$
$152$	0	0
$153$	−291697.	−1.00740
$154$	0	0
$155$	24791.8	0.0828856
$156$	0	0
$157$	−243388.	−0.788044	−0.394022	−	0.919101i	$-0.628917\pi$
$157$	−243388.	−0.788044	−0.394022	+	0.919101i	$0.628917\pi$
$158$	0	0
$159$	156082.	0.489622
$160$	0	0
$161$	118200.	0.359379
$162$	0	0
$163$	574292.	1.69303	0.846513	−	0.532368i	$-0.178698\pi$
$163$	574292.	1.69303	0.846513	+	0.532368i	$0.178698\pi$
$164$	0	0
$165$	−13639.4	−0.0390019
$166$	0	0
$167$	514837.	1.42849	0.714247	−	0.699893i	$-0.246769\pi$
$167$	514837.	1.42849	0.714247	+	0.699893i	$0.246769\pi$
$168$	0	0
$169$	−86029.3	−0.231702
$170$	0	0
$171$	268659.	0.702604
$172$	0	0
$173$	769386.	1.95447	0.977235	−	0.212159i	$-0.0680496\pi$
$173$	769386.	1.95447	0.977235	+	0.212159i	$0.0680496\pi$
$174$	0	0
$175$	−329665.	−0.813726
$176$	0	0
$177$	−283085.	−0.679178
$178$	0	0
$179$	622271.	1.45160	0.725800	−	0.687906i	$-0.241470\pi$
$179$	622271.	1.45160	0.725800	+	0.687906i	$0.241470\pi$
$180$	0	0
$181$	710892.	1.61290	0.806449	−	0.591303i	$-0.201386\pi$
$181$	710892.	1.61290	0.806449	+	0.591303i	$0.201386\pi$
$182$	0	0
$183$	−237926.	−0.525187
$184$	0	0
$185$	−38324.8	−0.0823286
$186$	0	0
$187$	338006.	0.706839
$188$	0	0
$189$	414492.	0.844037
$190$	0	0
$191$	491721.	0.975294	0.487647	−	0.873041i	$-0.337855\pi$
$191$	491721.	0.975294	0.487647	+	0.873041i	$0.337855\pi$
$192$	0	0
$193$	−567502.	−1.09667	−0.548333	−	0.836260i	$-0.684737\pi$
$193$	−567502.	−1.09667	−0.548333	+	0.836260i	$0.684737\pi$
$194$	0	0
$195$	43463.4	0.0818536
$196$	0	0
$197$	−187277.	−0.343810	−0.171905	−	0.985114i	$-0.554992\pi$
$197$	−187277.	−0.343810	−0.171905	+	0.985114i	$0.554992\pi$
$198$	0	0
$199$	−352209.	−0.630475	−0.315237	−	0.949013i	$-0.602084\pi$
$199$	−352209.	−0.630475	−0.315237	+	0.949013i	$0.602084\pi$
$200$	0	0
$201$	383013.	0.668688
$202$	0	0
$203$	−65752.9	−0.111989
$204$	0	0
$205$	4316.80	0.00717427
$206$	0	0
$207$	−158576.	−0.257224
$208$	0	0
$209$	−311311.	−0.492979
$210$	0	0
$211$	−662909.	−1.02506	−0.512528	−	0.858670i	$-0.671291\pi$
$211$	−662909.	−1.02506	−0.512528	+	0.858670i	$0.671291\pi$
$212$	0	0
$213$	169589.	0.256124
$214$	0	0
$215$	53906.1	0.0795320
$216$	0	0
$217$	325754.	0.469614
$218$	0	0
$219$	43658.4	0.0615116
$220$	0	0
$221$	−1.07709e6	−1.48345
$222$	0	0
$223$	609498.	0.820749	0.410375	−	0.911917i	$-0.365398\pi$
$223$	609498.	0.820749	0.410375	+	0.911917i	$0.365398\pi$
$224$	0	0
$225$	442275.	0.582419
$226$	0	0
$227$	1.35964e6	1.75130	0.875651	−	0.482945i	$-0.160433\pi$
$227$	1.35964e6	1.75130	0.875651	+	0.482945i	$0.160433\pi$
$228$	0	0
$229$	175149.	0.220708	0.110354	−	0.993892i	$-0.464801\pi$
$229$	175149.	0.220708	0.110354	+	0.993892i	$0.464801\pi$
$230$	0	0
$231$	−179216.	−0.220977
$232$	0	0
$233$	857290.	1.03452	0.517259	−	0.855829i	$-0.326953\pi$
$233$	857290.	1.03452	0.517259	+	0.855829i	$0.326953\pi$
$234$	0	0
$235$	94814.9	0.111997
$236$	0	0
$237$	284371.	0.328863
$238$	0	0
$239$	1.28575e6	1.45600	0.728002	−	0.685576i	$-0.240449\pi$
$239$	1.28575e6	1.45600	0.728002	+	0.685576i	$0.240449\pi$
$240$	0	0
$241$	1.03927e6	1.15262	0.576311	−	0.817231i	$-0.304492\pi$
$241$	1.03927e6	1.15262	0.576311	+	0.817231i	$0.304492\pi$
$242$	0	0
$243$	−904660.	−0.982811
$244$	0	0
$245$	−42526.6	−0.0452632
$246$	0	0
$247$	992025.	1.03462
$248$	0	0
$249$	−965977.	−0.987344
$250$	0	0
$251$	1.27619e6	1.27859	0.639294	−	0.768962i	$-0.279227\pi$
$251$	1.27619e6	1.27859	0.639294	+	0.768962i	$0.279227\pi$
$252$	0	0
$253$	183751.	0.180480
$254$	0	0
$255$	−164108.	−0.158045
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−503572.	−0.466457
$260$	0	0
$261$	88213.2	0.0801554
$262$	0	0
$263$	−1.89519e6	−1.68952	−0.844762	−	0.535142i	$-0.820258\pi$
$263$	−1.89519e6	−1.68952	−0.844762	+	0.535142i	$0.820258\pi$
$264$	0	0
$265$	−129138.	−0.112964
$266$	0	0
$267$	391301.	0.335918
$268$	0	0
$269$	1.58513e6	1.33562	0.667811	−	0.744331i	$-0.267231\pi$
$269$	1.58513e6	1.33562	0.667811	+	0.744331i	$0.267231\pi$
$270$	0	0
$271$	704076.	0.582367	0.291183	−	0.956667i	$-0.405951\pi$
$271$	704076.	0.582367	0.291183	+	0.956667i	$0.405951\pi$
$272$	0	0
$273$	571091.	0.463766
$274$	0	0
$275$	−512490.	−0.408652
$276$	0	0
$277$	105092.	0.0822944	0.0411472	−	0.999153i	$-0.486899\pi$
$277$	105092.	0.0822944	0.0411472	+	0.999153i	$0.486899\pi$
$278$	0	0
$279$	−437027.	−0.336123
$280$	0	0
$281$	2.55493e6	1.93025	0.965125	−	0.261789i	$-0.0843126\pi$
$281$	2.55493e6	1.93025	0.965125	+	0.261789i	$0.0843126\pi$
$282$	0	0
$283$	−25775.4	−0.0191311	−0.00956555	−	0.999954i	$-0.503045\pi$
$283$	−25775.4	−0.0191311	−0.00956555	+	0.999954i	$0.503045\pi$
$284$	0	0
$285$	151147.	0.110227
$286$	0	0
$287$	56721.0	0.0406480
$288$	0	0
$289$	2.64701e6	1.86428
$290$	0	0
$291$	−730020.	−0.505362
$292$	0	0
$293$	849448.	0.578053	0.289027	−	0.957321i	$-0.406668\pi$
$293$	849448.	0.578053	0.289027	+	0.957321i	$0.406668\pi$
$294$	0	0
$295$	234217.	0.156698
$296$	0	0
$297$	644358.	0.423874
$298$	0	0
$299$	−585543.	−0.378774
$300$	0	0
$301$	708304.	0.450612
$302$	0	0
$303$	−1.12219e6	−0.702197
$304$	0	0
$305$	196853.	0.121170
$306$	0	0
$307$	2.30954e6	1.39855	0.699276	−	0.714852i	$-0.253506\pi$
$307$	2.30954e6	1.39855	0.699276	+	0.714852i	$0.253506\pi$
$308$	0	0
$309$	1.46943e6	0.875494
$310$	0	0
$311$	−2.56536e6	−1.50400	−0.751999	−	0.659164i	$-0.770910\pi$
$311$	−2.56536e6	−1.50400	−0.751999	+	0.659164i	$0.770910\pi$
$312$	0	0
$313$	776557.	0.448035	0.224018	−	0.974585i	$-0.428083\pi$
$313$	776557.	0.448035	0.224018	+	0.974585i	$0.428083\pi$
$314$	0	0
$315$	−127963.	−0.0726622
$316$	0	0
$317$	1.62954e6	0.910790	0.455395	−	0.890290i	$-0.349498\pi$
$317$	1.62954e6	0.910790	0.455395	+	0.890290i	$0.349498\pi$
$318$	0	0
$319$	−102218.	−0.0562406
$320$	0	0
$321$	1.12229e6	0.607913
$322$	0	0
$323$	−3.74567e6	−1.99767
$324$	0	0
$325$	1.63310e6	0.857641
$326$	0	0
$327$	1.00790e6	0.521250
$328$	0	0
$329$	1.24583e6	0.634554
$330$	0	0
$331$	2.95068e6	1.48031	0.740154	−	0.672437i	$-0.234753\pi$
$331$	2.95068e6	1.48031	0.740154	+	0.672437i	$0.234753\pi$
$332$	0	0
$333$	675585.	0.333864
$334$	0	0
$335$	−316895.	−0.154278
$336$	0	0
$337$	−3.94188e6	−1.89073	−0.945364	−	0.326017i	$-0.894293\pi$
$337$	−3.94188e6	−1.89073	−0.945364	+	0.326017i	$0.894293\pi$
$338$	0	0
$339$	778284.	0.367823
$340$	0	0
$341$	506409.	0.235839
$342$	0	0
$343$	−2.37084e6	−1.08810
$344$	0	0
$345$	−89214.6	−0.0403542
$346$	0	0
$347$	1.90748e6	0.850427	0.425213	−	0.905093i	$-0.360199\pi$
$347$	1.90748e6	0.850427	0.425213	+	0.905093i	$0.360199\pi$
$348$	0	0
$349$	−567403.	−0.249361	−0.124680	−	0.992197i	$-0.539791\pi$
$349$	−567403.	−0.249361	−0.124680	+	0.992197i	$0.539791\pi$
$350$	0	0
$351$	−2.05332e6	−0.889587
$352$	0	0
$353$	572621.	0.244585	0.122293	−	0.992494i	$-0.460975\pi$
$353$	572621.	0.244585	0.122293	+	0.992494i	$0.460975\pi$
$354$	0	0
$355$	−140314.	−0.0590921
$356$	0	0
$357$	−2.15632e6	−0.895451
$358$	0	0
$359$	−2.01058e6	−0.823350	−0.411675	−	0.911331i	$-0.635056\pi$
$359$	−2.01058e6	−0.823350	−0.411675	+	0.911331i	$0.635056\pi$
$360$	0	0
$361$	973743.	0.393257
$362$	0	0
$363$	1.31861e6	0.525230
$364$	0	0
$365$	−36121.8	−0.0141918
$366$	0	0
$367$	−1.27750e6	−0.495104	−0.247552	−	0.968875i	$-0.579626\pi$
$367$	−1.27750e6	−0.495104	−0.247552	+	0.968875i	$0.579626\pi$
$368$	0	0
$369$	−76096.1	−0.0290935
$370$	0	0
$371$	−1.69682e6	−0.640032
$372$	0	0
$373$	−2.14049e6	−0.796602	−0.398301	−	0.917255i	$-0.630400\pi$
$373$	−2.14049e6	−0.796602	−0.398301	+	0.917255i	$0.630400\pi$
$374$	0	0
$375$	503126.	0.184756
$376$	0	0
$377$	325728.	0.118033
$378$	0	0
$379$	4.56086e6	1.63098	0.815491	−	0.578770i	$-0.196467\pi$
$379$	4.56086e6	1.63098	0.815491	+	0.578770i	$0.196467\pi$
$380$	0	0
$381$	1.31271e6	0.463292
$382$	0	0
$383$	−1.61402e6	−0.562227	−0.281114	−	0.959674i	$-0.590704\pi$
$383$	−1.61402e6	−0.562227	−0.281114	+	0.959674i	$0.590704\pi$
$384$	0	0
$385$	148278.	0.0509831
$386$	0	0
$387$	−950251.	−0.322523
$388$	0	0
$389$	3.95178e6	1.32409	0.662047	−	0.749462i	$-0.269688\pi$
$389$	3.95178e6	1.32409	0.662047	+	0.749462i	$0.269688\pi$
$390$	0	0
$391$	2.21088e6	0.731347
$392$	0	0
$393$	−1.81044e6	−0.591293
$394$	0	0
$395$	−235281.	−0.0758742
$396$	0	0
$397$	5.64531e6	1.79768	0.898838	−	0.438281i	$-0.144413\pi$
$397$	5.64531e6	1.79768	0.898838	+	0.438281i	$0.144413\pi$
$398$	0	0
$399$	1.98601e6	0.624525
$400$	0	0
$401$	−3.32043e6	−1.03118	−0.515588	−	0.856836i	$-0.672427\pi$
$401$	−3.32043e6	−1.03118	−0.515588	+	0.856836i	$0.672427\pi$
$402$	0	0
$403$	−1.61373e6	−0.494958
$404$	0	0
$405$	−24439.5	−0.00740380
$406$	0	0
$407$	−782840.	−0.234254
$408$	0	0
$409$	4.96320e6	1.46708	0.733539	−	0.679648i	$-0.237867\pi$
$409$	4.96320e6	1.46708	0.733539	+	0.679648i	$0.237867\pi$
$410$	0	0
$411$	−435800.	−0.127257
$412$	0	0
$413$	3.07751e6	0.887820
$414$	0	0
$415$	799223.	0.227797
$416$	0	0
$417$	809606.	0.227999
$418$	0	0
$419$	−3.58056e6	−0.996358	−0.498179	−	0.867074i	$-0.665998\pi$
$419$	−3.58056e6	−0.996358	−0.498179	+	0.867074i	$0.665998\pi$
$420$	0	0
$421$	1.48247e6	0.407644	0.203822	−	0.979008i	$-0.434664\pi$
$421$	1.48247e6	0.407644	0.203822	+	0.979008i	$0.434664\pi$
$422$	0	0
$423$	−1.67139e6	−0.454178
$424$	0	0
$425$	−6.16624e6	−1.65595
$426$	0	0
$427$	2.58657e6	0.686523
$428$	0	0
$429$	887804.	0.232902
$430$	0	0
$431$	4.72311e6	1.22471	0.612357	−	0.790581i	$-0.290221\pi$
$431$	4.72311e6	1.22471	0.612357	+	0.790581i	$0.290221\pi$
$432$	0	0
$433$	−494863.	−0.126843	−0.0634214	−	0.997987i	$-0.520201\pi$
$433$	−494863.	−0.126843	−0.0634214	+	0.997987i	$0.520201\pi$
$434$	0	0
$435$	49628.7	0.0125751
$436$	0	0
$437$	−2.03627e6	−0.510072
$438$	0	0
$439$	1.73585e6	0.429883	0.214942	−	0.976627i	$-0.431044\pi$
$439$	1.73585e6	0.429883	0.214942	+	0.976627i	$0.431044\pi$
$440$	0	0
$441$	749654.	0.183554
$442$	0	0
$443$	−2.74728e6	−0.665111	−0.332555	−	0.943084i	$-0.607911\pi$
$443$	−2.74728e6	−0.665111	−0.332555	+	0.943084i	$0.607911\pi$
$444$	0	0
$445$	−323752.	−0.0775019
$446$	0	0
$447$	−20452.6	−0.00484150
$448$	0	0
$449$	2.00897e6	0.470281	0.235141	−	0.971961i	$-0.424445\pi$
$449$	2.00897e6	0.470281	0.235141	+	0.971961i	$0.424445\pi$
$450$	0	0
$451$	88177.0	0.0204133
$452$	0	0
$453$	−208679.	−0.0477786
$454$	0	0
$455$	−472506.	−0.106999
$456$	0	0
$457$	4.22720e6	0.946810	0.473405	−	0.880845i	$-0.343025\pi$
$457$	4.22720e6	0.946810	0.473405	+	0.880845i	$0.343025\pi$
$458$	0	0
$459$	7.75288e6	1.71764
$460$	0	0
$461$	1.24293e6	0.272392	0.136196	−	0.990682i	$-0.456512\pi$
$461$	1.24293e6	0.272392	0.136196	+	0.990682i	$0.456512\pi$
$462$	0	0
$463$	6.21687e6	1.34778	0.673891	−	0.738831i	$-0.264622\pi$
$463$	6.21687e6	1.34778	0.673891	+	0.738831i	$0.264622\pi$
$464$	0	0
$465$	−245872.	−0.0527322
$466$	0	0
$467$	4.36147e6	0.925423	0.462712	−	0.886509i	$-0.346877\pi$
$467$	4.36147e6	0.925423	0.462712	+	0.886509i	$0.346877\pi$
$468$	0	0
$469$	−4.16387e6	−0.874107
$470$	0	0
$471$	2.41379e6	0.501357
$472$	0	0
$473$	1.10111e6	0.226297
$474$	0	0
$475$	5.67924e6	1.15493
$476$	0	0
$477$	2.27644e6	0.458099
$478$	0	0
$479$	3.57408e6	0.711748	0.355874	−	0.934534i	$-0.384183\pi$
$479$	3.57408e6	0.711748	0.355874	+	0.934534i	$0.384183\pi$
$480$	0	0
$481$	2.49461e6	0.491631
$482$	0	0
$483$	−1.17224e6	−0.228639
$484$	0	0
$485$	603999.	0.116596
$486$	0	0
$487$	−8.44952e6	−1.61440	−0.807198	−	0.590281i	$-0.799017\pi$
$487$	−8.44952e6	−1.61440	−0.807198	+	0.590281i	$0.799017\pi$
$488$	0	0
$489$	−5.69551e6	−1.07711
$490$	0	0
$491$	−4.39999e6	−0.823661	−0.411830	−	0.911261i	$-0.635110\pi$
$491$	−4.39999e6	−0.823661	−0.411830	+	0.911261i	$0.635110\pi$
$492$	0	0
$493$	−1.22988e6	−0.227900
$494$	0	0
$495$	−198928.	−0.0364908
$496$	0	0
$497$	−1.84366e6	−0.334804
$498$	0	0
$499$	4.46027e6	0.801881	0.400940	−	0.916104i	$-0.368683\pi$
$499$	4.46027e6	0.801881	0.400940	+	0.916104i	$0.368683\pi$
$500$	0	0
$501$	−5.10587e6	−0.908815
$502$	0	0
$503$	−8.42679e6	−1.48505	−0.742527	−	0.669817i	$-0.766373\pi$
$503$	−8.42679e6	−1.48505	−0.742527	+	0.669817i	$0.766373\pi$
$504$	0	0
$505$	928468.	0.162009
$506$	0	0
$507$	853191.	0.147410
$508$	0	0
$509$	−3.43877e6	−0.588313	−0.294157	−	0.955757i	$-0.595039\pi$
$509$	−3.43877e6	−0.588313	−0.294157	+	0.955757i	$0.595039\pi$
$510$	0	0
$511$	−474625.	−0.0804078
$512$	0	0
$513$	−7.14057e6	−1.19795
$514$	0	0
$515$	−1.21577e6	−0.201991
$516$	0	0
$517$	1.93673e6	0.318672
$518$	0	0
$519$	−7.63034e6	−1.24344
$520$	0	0
$521$	4.62064e6	0.745775	0.372887	−	0.927877i	$-0.378368\pi$
$521$	4.62064e6	0.745775	0.372887	+	0.927877i	$0.378368\pi$
$522$	0	0
$523$	−7.93237e6	−1.26809	−0.634043	−	0.773298i	$-0.718606\pi$
$523$	−7.93237e6	−1.26809	−0.634043	+	0.773298i	$0.718606\pi$
$524$	0	0
$525$	3.26944e6	0.517696
$526$	0	0
$527$	6.09309e6	0.955676
$528$	0	0
$529$	−5.23444e6	−0.813263
$530$	0	0
$531$	−4.12875e6	−0.635451
$532$	0	0
$533$	−280986.	−0.0428417
$534$	0	0
$535$	−928550.	−0.140256
$536$	0	0
$537$	−6.17134e6	−0.923514
$538$	0	0
$539$	−868669.	−0.128790
$540$	0	0
$541$	7.32108e6	1.07543	0.537715	−	0.843127i	$-0.319288\pi$
$541$	7.32108e6	1.07543	0.537715	+	0.843127i	$0.319288\pi$
$542$	0	0
$543$	−7.05023e6	−1.02613
$544$	0	0
$545$	−833905.	−0.120261
$546$	0	0
$547$	−3.14464e6	−0.449369	−0.224684	−	0.974432i	$-0.572135\pi$
$547$	−3.14464e6	−0.449369	−0.224684	+	0.974432i	$0.572135\pi$
$548$	0	0
$549$	−3.47011e6	−0.491374
$550$	0	0
$551$	1.13274e6	0.158947
$552$	0	0
$553$	−3.09149e6	−0.429888
$554$	0	0
$555$	380084.	0.0523778
$556$	0	0
$557$	−3.80355e6	−0.519458	−0.259729	−	0.965682i	$-0.583633\pi$
$557$	−3.80355e6	−0.519458	−0.259729	+	0.965682i	$0.583633\pi$
$558$	0	0
$559$	−3.50881e6	−0.474931
$560$	0	0
$561$	−3.35215e6	−0.449694
$562$	0	0
$563$	−6.92420e6	−0.920659	−0.460329	−	0.887748i	$-0.652269\pi$
$563$	−6.92420e6	−0.920659	−0.460329	+	0.887748i	$0.652269\pi$
$564$	0	0
$565$	−643932.	−0.0848630
$566$	0	0
$567$	−321125.	−0.0419485
$568$	0	0
$569$	1.01645e7	1.31614	0.658072	−	0.752955i	$-0.271372\pi$
$569$	1.01645e7	1.31614	0.658072	+	0.752955i	$0.271372\pi$
$570$	0	0
$571$	−7.02455e6	−0.901629	−0.450815	−	0.892618i	$-0.648866\pi$
$571$	−7.02455e6	−0.901629	−0.450815	+	0.892618i	$0.648866\pi$
$572$	0	0
$573$	−4.87662e6	−0.620486
$574$	0	0
$575$	−3.35217e6	−0.422821
$576$	0	0
$577$	1.54189e6	0.192804	0.0964018	−	0.995343i	$-0.469267\pi$
$577$	1.54189e6	0.192804	0.0964018	+	0.995343i	$0.469267\pi$
$578$	0	0
$579$	5.62817e6	0.697704
$580$	0	0
$581$	1.05015e7	1.29065
$582$	0	0
$583$	−2.63784e6	−0.321423
$584$	0	0
$585$	633907.	0.0765836
$586$	0	0
$587$	−3.89326e6	−0.466357	−0.233179	−	0.972434i	$-0.574913\pi$
$587$	−3.89326e6	−0.466357	−0.233179	+	0.972434i	$0.574913\pi$
$588$	0	0
$589$	−5.61186e6	−0.666528
$590$	0	0
$591$	1.85731e6	0.218733
$592$	0	0
$593$	6.00124e6	0.700816	0.350408	−	0.936597i	$-0.386043\pi$
$593$	6.00124e6	0.700816	0.350408	+	0.936597i	$0.386043\pi$
$594$	0	0
$595$	1.78408e6	0.206596
$596$	0	0
$597$	3.49301e6	0.401111
$598$	0	0
$599$	−1.07431e7	−1.22338	−0.611689	−	0.791098i	$-0.709510\pi$
$599$	−1.07431e7	−1.22338	−0.611689	+	0.791098i	$0.709510\pi$
$600$	0	0
$601$	−1.70233e6	−0.192246	−0.0961229	−	0.995369i	$-0.530644\pi$
$601$	−1.70233e6	−0.192246	−0.0961229	+	0.995369i	$0.530644\pi$
$602$	0	0
$603$	5.58619e6	0.625637
$604$	0	0
$605$	−1.09098e6	−0.121179
$606$	0	0
$607$	7.02771e6	0.774181	0.387090	−	0.922042i	$-0.373480\pi$
$607$	7.02771e6	0.774181	0.387090	+	0.922042i	$0.373480\pi$
$608$	0	0
$609$	652101.	0.0712478
$610$	0	0
$611$	−6.17161e6	−0.668799
$612$	0	0
$613$	−1.64467e6	−0.176778	−0.0883888	−	0.996086i	$-0.528172\pi$
$613$	−1.64467e6	−0.176778	−0.0883888	+	0.996086i	$0.528172\pi$
$614$	0	0
$615$	−42811.7	−0.00456430
$616$	0	0
$617$	1.62959e7	1.72332	0.861658	−	0.507490i	$-0.169427\pi$
$617$	1.62959e7	1.72332	0.861658	+	0.507490i	$0.169427\pi$
$618$	0	0
$619$	−3.79462e6	−0.398054	−0.199027	−	0.979994i	$-0.563778\pi$
$619$	−3.79462e6	−0.398054	−0.199027	+	0.979994i	$0.563778\pi$
$620$	0	0
$621$	4.21472e6	0.438571
$622$	0	0
$623$	−4.25396e6	−0.439110
$624$	0	0
$625$	9.13895e6	0.935828
$626$	0	0
$627$	3.08741e6	0.313635
$628$	0	0
$629$	−9.41909e6	−0.949253
$630$	0	0
$631$	8.06660e6	0.806523	0.403262	−	0.915085i	$-0.367876\pi$
$631$	8.06660e6	0.806523	0.403262	+	0.915085i	$0.367876\pi$
$632$	0	0
$633$	6.57436e6	0.652145
$634$	0	0
$635$	−1.08610e6	−0.106889
$636$	0	0
$637$	2.76811e6	0.270293
$638$	0	0
$639$	2.47343e6	0.239634
$640$	0	0
$641$	978823.	0.0940934	0.0470467	−	0.998893i	$-0.485019\pi$
$641$	978823.	0.0940934	0.0470467	+	0.998893i	$0.485019\pi$
$642$	0	0
$643$	−1.28919e7	−1.22967	−0.614834	−	0.788656i	$-0.710777\pi$
$643$	−1.28919e7	−1.22967	−0.614834	+	0.788656i	$0.710777\pi$
$644$	0	0
$645$	−534611.	−0.0505986
$646$	0	0
$647$	1.41170e7	1.32582	0.662908	−	0.748701i	$-0.269322\pi$
$647$	1.41170e7	1.32582	0.662908	+	0.748701i	$0.269322\pi$
$648$	0	0
$649$	4.78422e6	0.445861
$650$	0	0
$651$	−3.23065e6	−0.298770
$652$	0	0
$653$	5.17261e6	0.474709	0.237354	−	0.971423i	$-0.423720\pi$
$653$	5.17261e6	0.474709	0.237354	+	0.971423i	$0.423720\pi$
$654$	0	0
$655$	1.49791e6	0.136421
$656$	0	0
$657$	636750.	0.0575514
$658$	0	0
$659$	−9.73336e6	−0.873071	−0.436535	−	0.899687i	$-0.643795\pi$
$659$	−9.73336e6	−0.873071	−0.436535	+	0.899687i	$0.643795\pi$
$660$	0	0
$661$	1.44416e7	1.28562	0.642808	−	0.766027i	$-0.277769\pi$
$661$	1.44416e7	1.28562	0.642808	+	0.766027i	$0.277769\pi$
$662$	0	0
$663$	1.06820e7	0.943777
$664$	0	0
$665$	−1.64317e6	−0.144088
$666$	0	0
$667$	−668602.	−0.0581906
$668$	0	0
$669$	−6.04467e6	−0.522164
$670$	0	0
$671$	4.02102e6	0.344770
$672$	0	0
$673$	−2.92385e6	−0.248838	−0.124419	−	0.992230i	$-0.539707\pi$
$673$	−2.92385e6	−0.248838	−0.124419	+	0.992230i	$0.539707\pi$
$674$	0	0
$675$	−1.17550e7	−0.993035
$676$	0	0
$677$	2.24130e7	1.87944	0.939718	−	0.341950i	$-0.111087\pi$
$677$	2.24130e7	1.87944	0.939718	+	0.341950i	$0.111087\pi$
$678$	0	0
$679$	7.93629e6	0.660607
$680$	0	0
$681$	−1.34842e7	−1.11419
$682$	0	0
$683$	−1.31681e7	−1.08012	−0.540058	−	0.841627i	$-0.681598\pi$
$683$	−1.31681e7	−1.08012	−0.540058	+	0.841627i	$0.681598\pi$
$684$	0	0
$685$	360569.	0.0293604
$686$	0	0
$687$	−1.73703e6	−0.140416
$688$	0	0
$689$	8.40576e6	0.674573
$690$	0	0
$691$	−9.69458e6	−0.772385	−0.386193	−	0.922418i	$-0.626210\pi$
$691$	−9.69458e6	−0.772385	−0.386193	+	0.922418i	$0.626210\pi$
$692$	0	0
$693$	−2.61384e6	−0.206750
$694$	0	0
$695$	−669846.	−0.0526033
$696$	0	0
$697$	1.06094e6	0.0827198
$698$	0	0
$699$	−8.50212e6	−0.658164
$700$	0	0
$701$	−1.08116e7	−0.830985	−0.415492	−	0.909597i	$-0.636391\pi$
$701$	−1.08116e7	−0.830985	−0.415492	+	0.909597i	$0.636391\pi$
$702$	0	0
$703$	8.67518e6	0.662049
$704$	0	0
$705$	−940322.	−0.0712531
$706$	0	0
$707$	1.21997e7	0.917910
$708$	0	0
$709$	4.48732e6	0.335252	0.167626	−	0.985851i	$-0.446390\pi$
$709$	4.48732e6	0.335252	0.167626	+	0.985851i	$0.446390\pi$
$710$	0	0
$711$	4.14751e6	0.307690
$712$	0	0
$713$	3.31240e6	0.244016
$714$	0	0
$715$	−734545.	−0.0537345
$716$	0	0
$717$	−1.27514e7	−0.926315
$718$	0	0
$719$	1.69383e7	1.22193	0.610967	−	0.791656i	$-0.290781\pi$
$719$	1.69383e7	1.22193	0.610967	+	0.791656i	$0.290781\pi$
$720$	0	0
$721$	−1.59747e7	−1.14444
$722$	0	0
$723$	−1.03069e7	−0.733302
$724$	0	0
$725$	1.86476e6	0.131758
$726$	0	0
$727$	−1.38320e7	−0.970622	−0.485311	−	0.874342i	$-0.661294\pi$
$727$	−1.38320e7	−0.970622	−0.485311	+	0.874342i	$0.661294\pi$
$728$	0	0
$729$	9.69568e6	0.675709
$730$	0	0
$731$	1.32485e7	0.917009
$732$	0	0
$733$	3.26058e6	0.224148	0.112074	−	0.993700i	$-0.464251\pi$
$733$	3.26058e6	0.224148	0.112074	+	0.993700i	$0.464251\pi$
$734$	0	0
$735$	421755.	0.0287967
$736$	0	0
$737$	−6.47304e6	−0.438975
$738$	0	0
$739$	−9.94296e6	−0.669738	−0.334869	−	0.942265i	$-0.608692\pi$
$739$	−9.94296e6	−0.669738	−0.334869	+	0.942265i	$0.608692\pi$
$740$	0	0
$741$	−9.83836e6	−0.658229
$742$	0	0
$743$	−1.40357e7	−0.932743	−0.466371	−	0.884589i	$-0.654439\pi$
$743$	−1.40357e7	−0.932743	−0.466371	+	0.884589i	$0.654439\pi$
$744$	0	0
$745$	16921.9	0.00111702
$746$	0	0
$747$	−1.40886e7	−0.923776
$748$	0	0
$749$	−1.22008e7	−0.794662
$750$	0	0
$751$	−1.38916e7	−0.898778	−0.449389	−	0.893336i	$-0.648358\pi$
$751$	−1.38916e7	−0.898778	−0.449389	+	0.893336i	$0.648358\pi$
$752$	0	0
$753$	−1.26565e7	−0.813443
$754$	0	0
$755$	172656.	0.0110233
$756$	0	0
$757$	2.40650e7	1.52632	0.763160	−	0.646210i	$-0.223647\pi$
$757$	2.40650e7	1.52632	0.763160	+	0.646210i	$0.223647\pi$
$758$	0	0
$759$	−1.82234e6	−0.114822
$760$	0	0
$761$	7.06989e6	0.442538	0.221269	−	0.975213i	$-0.428980\pi$
$761$	7.06989e6	0.442538	0.221269	+	0.975213i	$0.428980\pi$
$762$	0	0
$763$	−1.09572e7	−0.681376
$764$	0	0
$765$	−2.39349e6	−0.147870
$766$	0	0
$767$	−1.52455e7	−0.935733
$768$	0	0
$769$	−2.71632e7	−1.65640	−0.828200	−	0.560433i	$-0.810635\pi$
$769$	−2.71632e7	−1.65640	−0.828200	+	0.560433i	$0.810635\pi$
$770$	0	0
$771$	655037.	0.0396853
$772$	0	0
$773$	−3.09178e6	−0.186106	−0.0930530	−	0.995661i	$-0.529663\pi$
$773$	−3.09178e6	−0.186106	−0.0930530	+	0.995661i	$0.529663\pi$
$774$	0	0
$775$	−9.23843e6	−0.552515
$776$	0	0
$777$	4.99414e6	0.296762
$778$	0	0
$779$	−977149.	−0.0576922
$780$	0	0
$781$	−2.86611e6	−0.168138
$782$	0	0
$783$	−2.34458e6	−0.136666
$784$	0	0
$785$	−1.99710e6	−0.115672
$786$	0	0
$787$	5.45062e6	0.313696	0.156848	−	0.987623i	$-0.449867\pi$
$787$	5.45062e6	0.313696	0.156848	+	0.987623i	$0.449867\pi$
$788$	0	0
$789$	1.87955e7	1.07488
$790$	0	0
$791$	−8.46099e6	−0.480817
$792$	0	0
$793$	−1.28134e7	−0.723573
$794$	0	0
$795$	1.28072e6	0.0718682
$796$	0	0
$797$	−2.80881e7	−1.56631	−0.783154	−	0.621828i	$-0.786390\pi$
$797$	−2.80881e7	−1.56631	−0.783154	+	0.621828i	$0.786390\pi$
$798$	0	0
$799$	2.33027e7	1.29133
$800$	0	0
$801$	5.70706e6	0.314290
$802$	0	0
$803$	−737840.	−0.0403806
$804$	0	0
$805$	969882.	0.0527508
$806$	0	0
$807$	−1.57204e7	−0.849729
$808$	0	0
$809$	748621.	0.0402153	0.0201076	−	0.999798i	$-0.493599\pi$
$809$	748621.	0.0402153	0.0201076	+	0.999798i	$0.493599\pi$
$810$	0	0
$811$	2.30243e7	1.22923	0.614617	−	0.788825i	$-0.289310\pi$
$811$	2.30243e7	1.22923	0.614617	+	0.788825i	$0.289310\pi$
$812$	0	0
$813$	−6.98264e6	−0.370504
$814$	0	0
$815$	4.71231e6	0.248508
$816$	0	0
$817$	−1.22022e7	−0.639560
$818$	0	0
$819$	8.32927e6	0.433908
$820$	0	0
$821$	−380799.	−0.0197169	−0.00985844	−	0.999951i	$-0.503138\pi$
$821$	−380799.	−0.0197169	−0.00985844	+	0.999951i	$0.503138\pi$
$822$	0	0
$823$	2.84051e7	1.46183	0.730915	−	0.682468i	$-0.239094\pi$
$823$	2.84051e7	1.46183	0.730915	+	0.682468i	$0.239094\pi$
$824$	0	0
$825$	5.08259e6	0.259986
$826$	0	0
$827$	2.38871e7	1.21450	0.607252	−	0.794509i	$-0.292272\pi$
$827$	2.38871e7	1.21450	0.607252	+	0.794509i	$0.292272\pi$
$828$	0	0
$829$	−1.74243e7	−0.880582	−0.440291	−	0.897855i	$-0.645125\pi$
$829$	−1.74243e7	−0.880582	−0.440291	+	0.897855i	$0.645125\pi$
$830$	0	0
$831$	−1.04224e6	−0.0523560
$832$	0	0
$833$	−1.04518e7	−0.521888
$834$	0	0
$835$	4.22446e6	0.209679
$836$	0	0
$837$	1.16156e7	0.573095
$838$	0	0
$839$	864802.	0.0424142	0.0212071	−	0.999775i	$-0.493249\pi$
$839$	864802.	0.0424142	0.0212071	+	0.999775i	$0.493249\pi$
$840$	0	0
$841$	−2.01392e7	−0.981867
$842$	0	0
$843$	−2.53384e7	−1.22803
$844$	0	0
$845$	−705907.	−0.0340099
$846$	0	0
$847$	−1.43350e7	−0.686579
$848$	0	0
$849$	255626.	0.0121713
$850$	0	0
$851$	−5.12052e6	−0.242376
$852$	0	0
$853$	−3.63403e6	−0.171008	−0.0855039	−	0.996338i	$-0.527250\pi$
$853$	−3.63403e6	−0.171008	−0.0855039	+	0.996338i	$0.527250\pi$
$854$	0	0
$855$	2.20446e6	0.103130
$856$	0	0
$857$	−3.62619e7	−1.68655	−0.843274	−	0.537485i	$-0.819375\pi$
$857$	−3.62619e7	−1.68655	−0.843274	+	0.537485i	$0.819375\pi$
$858$	0	0
$859$	−1.66674e7	−0.770698	−0.385349	−	0.922771i	$-0.625919\pi$
$859$	−1.66674e7	−0.770698	−0.385349	+	0.922771i	$0.625919\pi$
$860$	0	0
$861$	−562527.	−0.0258604
$862$	0	0
$863$	1.64866e7	0.753538	0.376769	−	0.926307i	$-0.377035\pi$
$863$	1.64866e7	0.753538	0.376769	+	0.926307i	$0.377035\pi$
$864$	0	0
$865$	6.31314e6	0.286883
$866$	0	0
$867$	−2.62516e7	−1.18606
$868$	0	0
$869$	−4.80596e6	−0.215889
$870$	0	0
$871$	2.06271e7	0.921281
$872$	0	0
$873$	−1.06472e7	−0.472826
$874$	0	0
$875$	−5.46965e6	−0.241513
$876$	0	0
$877$	3.78014e7	1.65962	0.829811	−	0.558045i	$-0.188448\pi$
$877$	3.78014e7	1.65962	0.829811	+	0.558045i	$0.188448\pi$
$878$	0	0
$879$	−8.42435e6	−0.367760
$880$	0	0
$881$	3.67410e7	1.59482	0.797410	−	0.603438i	$-0.206203\pi$
$881$	3.67410e7	1.59482	0.797410	+	0.603438i	$0.206203\pi$
$882$	0	0
$883$	−4.50800e7	−1.94573	−0.972864	−	0.231377i	$-0.925677\pi$
$883$	−4.50800e7	−1.94573	−0.972864	+	0.231377i	$0.925677\pi$
$884$	0	0
$885$	−2.32283e6	−0.0996919
$886$	0	0
$887$	−4.34701e6	−0.185516	−0.0927580	−	0.995689i	$-0.529568\pi$
$887$	−4.34701e6	−0.185516	−0.0927580	+	0.995689i	$0.529568\pi$
$888$	0	0
$889$	−1.42709e7	−0.605614
$890$	0	0
$891$	−499213.	−0.0210665
$892$	0	0
$893$	−2.14622e7	−0.900630
$894$	0	0
$895$	5.10600e6	0.213070
$896$	0	0
$897$	5.80709e6	0.240978
$898$	0	0
$899$	−1.84264e6	−0.0760397
$900$	0	0
$901$	−3.17383e7	−1.30248
$902$	0	0
$903$	−7.02456e6	−0.286682
$904$	0	0
$905$	5.83317e6	0.236746
$906$	0	0
$907$	−4.67354e7	−1.88637	−0.943186	−	0.332265i	$-0.892187\pi$
$907$	−4.67354e7	−1.88637	−0.943186	+	0.332265i	$0.892187\pi$
$908$	0	0
$909$	−1.63669e7	−0.656988
$910$	0	0
$911$	−3.19626e7	−1.27598	−0.637992	−	0.770043i	$-0.720235\pi$
$911$	−3.19626e7	−1.27598	−0.637992	+	0.770043i	$0.720235\pi$
$912$	0	0
$913$	1.63253e7	0.648163
$914$	0	0
$915$	−1.95228e6	−0.0770886
$916$	0	0
$917$	1.96819e7	0.772936
$918$	0	0
$919$	9.58201e6	0.374255	0.187128	−	0.982336i	$-0.440082\pi$
$919$	9.58201e6	0.374255	0.187128	+	0.982336i	$0.440082\pi$
$920$	0	0
$921$	−2.29047e7	−0.889765
$922$	0	0
$923$	9.13319e6	0.352873
$924$	0	0
$925$	1.42814e7	0.548801
$926$	0	0
$927$	2.14314e7	0.819128
$928$	0	0
$929$	−2.47656e7	−0.941477	−0.470739	−	0.882273i	$-0.656012\pi$
$929$	−2.47656e7	−0.941477	−0.470739	+	0.882273i	$0.656012\pi$
$930$	0	0
$931$	9.62630e6	0.363986
$932$	0	0
$933$	2.54418e7	0.956850
$934$	0	0
$935$	2.77348e6	0.103752
$936$	0	0
$937$	−3.07347e7	−1.14362	−0.571808	−	0.820387i	$-0.693758\pi$
$937$	−3.07347e7	−1.14362	−0.571808	+	0.820387i	$0.693758\pi$
$938$	0	0
$939$	−7.70146e6	−0.285042
$940$	0	0
$941$	−3.18988e7	−1.17436	−0.587179	−	0.809457i	$-0.699761\pi$
$941$	−3.18988e7	−1.17436	−0.587179	+	0.809457i	$0.699761\pi$
$942$	0	0
$943$	576762.	0.0211211
$944$	0	0
$945$	3.40108e6	0.123890
$946$	0	0
$947$	−2.01256e7	−0.729246	−0.364623	−	0.931155i	$-0.618802\pi$
$947$	−2.01256e7	−0.729246	−0.364623	+	0.931155i	$0.618802\pi$
$948$	0	0
$949$	2.35121e6	0.0847472
$950$	0	0
$951$	−1.61609e7	−0.579448
$952$	0	0
$953$	1.32424e7	0.472320	0.236160	−	0.971714i	$-0.424111\pi$
$953$	1.32424e7	0.472320	0.236160	+	0.971714i	$0.424111\pi$
$954$	0	0
$955$	4.03478e6	0.143157
$956$	0	0
$957$	1.01374e6	0.0357805
$958$	0	0
$959$	4.73773e6	0.166350
$960$	0	0
$961$	−1.95003e7	−0.681135
$962$	0	0
$963$	1.63684e7	0.568774
$964$	0	0
$965$	−4.65660e6	−0.160972
$966$	0	0
$967$	1.71866e7	0.591049	0.295525	−	0.955335i	$-0.404506\pi$
$967$	1.71866e7	0.591049	0.295525	+	0.955335i	$0.404506\pi$
$968$	0	0
$969$	3.71475e7	1.27092
$970$	0	0
$971$	−1.86541e7	−0.634930	−0.317465	−	0.948270i	$-0.602832\pi$
$971$	−1.86541e7	−0.634930	−0.317465	+	0.948270i	$0.602832\pi$
$972$	0	0
$973$	−8.80150e6	−0.298040
$974$	0	0
$975$	−1.61962e7	−0.545635
$976$	0	0
$977$	−9.58458e6	−0.321245	−0.160623	−	0.987016i	$-0.551350\pi$
$977$	−9.58458e6	−0.321245	−0.160623	+	0.987016i	$0.551350\pi$
$978$	0	0
$979$	−6.61311e6	−0.220520
$980$	0	0
$981$	1.47000e7	0.487691
$982$	0	0
$983$	2.64870e7	0.874276	0.437138	−	0.899394i	$-0.355992\pi$
$983$	2.64870e7	0.874276	0.437138	+	0.899394i	$0.355992\pi$
$984$	0	0
$985$	−1.53668e6	−0.0504655
$986$	0	0
$987$	−1.23554e7	−0.403706
$988$	0	0
$989$	7.20232e6	0.234143
$990$	0	0
$991$	−2.30457e7	−0.745427	−0.372714	−	0.927946i	$-0.621573\pi$
$991$	−2.30457e7	−0.745427	−0.372714	+	0.927946i	$0.621573\pi$
$992$	0	0
$993$	−2.92632e7	−0.941778
$994$	0	0
$995$	−2.89002e6	−0.0925430
$996$	0	0
$997$	5.00174e7	1.59362	0.796808	−	0.604233i	$-0.206520\pi$
$997$	5.00174e7	1.59362	0.796808	+	0.604233i	$0.206520\pi$
$998$	0	0
$999$	−1.79561e7	−0.569244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.20	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.20	✓	57	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-9.91744 q^{3} +8.20543 q^{5} +107.816 q^{7} -144.644 q^{9} +O(q^{10})\)	\(q-9.91744 q^{3} +8.20543 q^{5} +107.816 q^{7} -144.644 q^{9} +167.608 q^{11} -534.101 q^{13} -81.3768 q^{15} +2016.65 q^{17} -1857.37 q^{19} -1069.26 q^{21} +1096.31 q^{23} -3057.67 q^{25} +3844.44 q^{27} -609.863 q^{29} +3021.39 q^{31} -1662.24 q^{33} +884.675 q^{35} -4670.67 q^{37} +5296.91 q^{39} +526.091 q^{41} +6569.57 q^{43} -1186.87 q^{45} +11555.1 q^{47} -5182.74 q^{49} -20000.0 q^{51} -15738.2 q^{53} +1375.29 q^{55} +18420.4 q^{57} +28544.2 q^{59} +23990.6 q^{61} -15594.9 q^{63} -4382.52 q^{65} -38620.2 q^{67} -10872.6 q^{69} -17100.1 q^{71} -4402.18 q^{73} +30324.3 q^{75} +18070.8 q^{77} -28673.8 q^{79} -2978.46 q^{81} +97401.8 q^{83} +16547.5 q^{85} +6048.28 q^{87} -39455.8 q^{89} -57584.5 q^{91} -29964.5 q^{93} -15240.6 q^{95} +73609.7 q^{97} -24243.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

Varieties

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