LMFDB - Embedded newform 688.6.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,6,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("688.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	688.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:	$110.344068031$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.70631$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.3440 q^{3} -72.1865 q^{5} +96.4803 q^{7} +93.5034 q^{9} +O(q^{10})$	$q-18.3440 q^{3} -72.1865 q^{5} +96.4803 q^{7} +93.5034 q^{9} +684.136 q^{11} +344.655 q^{13} +1324.19 q^{15} +1319.32 q^{17} -739.016 q^{19} -1769.84 q^{21} -3165.85 q^{23} +2085.89 q^{25} +2742.37 q^{27} +7073.21 q^{29} +3791.93 q^{31} -12549.8 q^{33} -6964.58 q^{35} -12682.3 q^{37} -6322.37 q^{39} -10882.5 q^{41} -1849.00 q^{43} -6749.68 q^{45} -3873.59 q^{47} -7498.54 q^{49} -24201.7 q^{51} +6479.38 q^{53} -49385.4 q^{55} +13556.5 q^{57} -34750.3 q^{59} +26452.9 q^{61} +9021.24 q^{63} -24879.4 q^{65} +58007.3 q^{67} +58074.5 q^{69} +23477.7 q^{71} +45184.4 q^{73} -38263.6 q^{75} +66005.7 q^{77} -17895.9 q^{79} -73027.4 q^{81} -39799.2 q^{83} -95237.3 q^{85} -129751. q^{87} -30802.0 q^{89} +33252.5 q^{91} -69559.2 q^{93} +53346.9 q^{95} +22524.5 q^{97} +63969.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * 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
$3$	−18.3440	−1.17677	−0.588385	−	0.808581i	$-0.700236\pi$
$3$	−18.3440	−1.17677	−0.588385	+	0.808581i	$0.700236\pi$
$4$	0	0
$5$	−72.1865	−1.29131	−0.645656	−	0.763629i	$-0.723416\pi$
$5$	−72.1865	−1.29131	−0.645656	+	0.763629i	$0.723416\pi$
$6$	0	0
$7$	96.4803	0.744207	0.372103	−	0.928191i	$-0.378637\pi$
$7$	96.4803	0.744207	0.372103	+	0.928191i	$0.378637\pi$
$8$	0	0
$9$	93.5034	0.384788
$10$	0	0
$11$	684.136	1.70475	0.852376	−	0.522930i	$-0.175161\pi$
$11$	684.136	1.70475	0.852376	+	0.522930i	$0.175161\pi$
$12$	0	0
$13$	344.655	0.565622	0.282811	−	0.959176i	$-0.408733\pi$
$13$	344.655	0.565622	0.282811	+	0.959176i	$0.408733\pi$
$14$	0	0
$15$	1324.19	1.51958
$16$	0	0
$17$	1319.32	1.10721	0.553604	−	0.832780i	$-0.313252\pi$
$17$	1319.32	1.10721	0.553604	+	0.832780i	$0.313252\pi$
$18$	0	0
$19$	−739.016	−0.469645	−0.234823	−	0.972038i	$-0.575451\pi$
$19$	−739.016	−0.469645	−0.234823	+	0.972038i	$0.575451\pi$
$20$	0	0
$21$	−1769.84	−0.875760
$22$	0	0
$23$	−3165.85	−1.24787	−0.623937	−	0.781474i	$-0.714468\pi$
$23$	−3165.85	−1.24787	−0.623937	+	0.781474i	$0.714468\pi$
$24$	0	0
$25$	2085.89	0.667484
$26$	0	0
$27$	2742.37	0.723963
$28$	0	0
$29$	7073.21	1.56179	0.780893	−	0.624665i	$-0.214765\pi$
$29$	7073.21	1.56179	0.780893	+	0.624665i	$0.214765\pi$
$30$	0	0
$31$	3791.93	0.708689	0.354345	−	0.935115i	$-0.384704\pi$
$31$	3791.93	0.708689	0.354345	+	0.935115i	$0.384704\pi$
$32$	0	0
$33$	−12549.8	−2.00610
$34$	0	0
$35$	−6964.58	−0.961003
$36$	0	0
$37$	−12682.3	−1.52298	−0.761489	−	0.648177i	$-0.775531\pi$
$37$	−12682.3	−1.52298	−0.761489	+	0.648177i	$0.775531\pi$
$38$	0	0
$39$	−6322.37	−0.665607
$40$	0	0
$41$	−10882.5	−1.01105	−0.505523	−	0.862813i	$-0.668700\pi$
$41$	−10882.5	−1.01105	−0.505523	+	0.862813i	$0.668700\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	−6749.68	−0.496881
$46$	0	0
$47$	−3873.59	−0.255782	−0.127891	−	0.991788i	$-0.540821\pi$
$47$	−3873.59	−0.255782	−0.127891	+	0.991788i	$0.540821\pi$
$48$	0	0
$49$	−7498.54	−0.446156
$50$	0	0
$51$	−24201.7	−1.30293
$52$	0	0
$53$	6479.38	0.316843	0.158421	−	0.987372i	$-0.449360\pi$
$53$	6479.38	0.316843	0.158421	+	0.987372i	$0.449360\pi$
$54$	0	0
$55$	−49385.4	−2.20136
$56$	0	0
$57$	13556.5	0.552664
$58$	0	0
$59$	−34750.3	−1.29966	−0.649829	−	0.760081i	$-0.725159\pi$
$59$	−34750.3	−1.29966	−0.649829	+	0.760081i	$0.725159\pi$
$60$	0	0
$61$	26452.9	0.910224	0.455112	−	0.890434i	$-0.349599\pi$
$61$	26452.9	0.910224	0.455112	+	0.890434i	$0.349599\pi$
$62$	0	0
$63$	9021.24	0.286362
$64$	0	0
$65$	−24879.4	−0.730394
$66$	0	0
$67$	58007.3	1.57868	0.789342	−	0.613953i	$-0.210422\pi$
$67$	58007.3	1.57868	0.789342	+	0.613953i	$0.210422\pi$
$68$	0	0
$69$	58074.5	1.46846
$70$	0	0
$71$	23477.7	0.552726	0.276363	−	0.961053i	$-0.410871\pi$
$71$	23477.7	0.552726	0.276363	+	0.961053i	$0.410871\pi$
$72$	0	0
$73$	45184.4	0.992388	0.496194	−	0.868212i	$-0.334731\pi$
$73$	45184.4	0.992388	0.496194	+	0.868212i	$0.334731\pi$
$74$	0	0
$75$	−38263.6	−0.785475
$76$	0	0
$77$	66005.7	1.26869
$78$	0	0
$79$	−17895.9	−0.322617	−0.161308	−	0.986904i	$-0.551571\pi$
$79$	−17895.9	−0.322617	−0.161308	+	0.986904i	$0.551571\pi$
$80$	0	0
$81$	−73027.4	−1.23673
$82$	0	0
$83$	−39799.2	−0.634132	−0.317066	−	0.948403i	$-0.602698\pi$
$83$	−39799.2	−0.634132	−0.317066	+	0.948403i	$0.602698\pi$
$84$	0	0
$85$	−95237.3	−1.42975
$86$	0	0
$87$	−129751.	−1.83786
$88$	0	0
$89$	−30802.0	−0.412197	−0.206098	−	0.978531i	$-0.566077\pi$
$89$	−30802.0	−0.412197	−0.206098	+	0.978531i	$0.566077\pi$
$90$	0	0
$91$	33252.5	0.420940
$92$	0	0
$93$	−69559.2	−0.833964
$94$	0	0
$95$	53346.9	0.606458
$96$	0	0
$97$	22524.5	0.243067	0.121533	−	0.992587i	$-0.461219\pi$
$97$	22524.5	0.243067	0.121533	+	0.992587i	$0.461219\pi$
$98$	0	0
$99$	63969.1	0.655968
$100$	0	0
$101$	70592.5	0.688581	0.344290	−	0.938863i	$-0.388120\pi$
$101$	70592.5	0.688581	0.344290	+	0.938863i	$0.388120\pi$
$102$	0	0
$103$	36628.5	0.340193	0.170097	−	0.985427i	$-0.445592\pi$
$103$	36628.5	0.340193	0.170097	+	0.985427i	$0.445592\pi$
$104$	0	0
$105$	127758.	1.13088
$106$	0	0
$107$	48388.9	0.408588	0.204294	−	0.978910i	$-0.434510\pi$
$107$	48388.9	0.408588	0.204294	+	0.978910i	$0.434510\pi$
$108$	0	0
$109$	−75896.0	−0.611861	−0.305930	−	0.952054i	$-0.598967\pi$
$109$	−75896.0	−0.611861	−0.305930	+	0.952054i	$0.598967\pi$
$110$	0	0
$111$	232645.	1.79220
$112$	0	0
$113$	−212156.	−1.56300	−0.781500	−	0.623905i	$-0.785545\pi$
$113$	−212156.	−1.56300	−0.781500	+	0.623905i	$0.785545\pi$
$114$	0	0
$115$	228532.	1.61139
$116$	0	0
$117$	32226.4	0.217645
$118$	0	0
$119$	127289.	0.823992
$120$	0	0
$121$	306992.	1.90618
$122$	0	0
$123$	199630.	1.18977
$124$	0	0
$125$	75009.8	0.429381
$126$	0	0
$127$	74547.8	0.410134	0.205067	−	0.978748i	$-0.434259\pi$
$127$	74547.8	0.410134	0.205067	+	0.978748i	$0.434259\pi$
$128$	0	0
$129$	33918.1	0.179456
$130$	0	0
$131$	57691.8	0.293722	0.146861	−	0.989157i	$-0.453083\pi$
$131$	57691.8	0.293722	0.146861	+	0.989157i	$0.453083\pi$
$132$	0	0
$133$	−71300.5	−0.349513
$134$	0	0
$135$	−197962.	−0.934862
$136$	0	0
$137$	−23674.0	−0.107763	−0.0538815	−	0.998547i	$-0.517159\pi$
$137$	−23674.0	−0.107763	−0.0538815	+	0.998547i	$0.517159\pi$
$138$	0	0
$139$	−234996.	−1.03163	−0.515815	−	0.856700i	$-0.672511\pi$
$139$	−234996.	−1.03163	−0.515815	+	0.856700i	$0.672511\pi$
$140$	0	0
$141$	71057.3	0.300996
$142$	0	0
$143$	235791.	0.964245
$144$	0	0
$145$	−510590.	−2.01675
$146$	0	0
$147$	137554.	0.525023
$148$	0	0
$149$	236660.	0.873292	0.436646	−	0.899633i	$-0.356166\pi$
$149$	236660.	0.873292	0.436646	+	0.899633i	$0.356166\pi$
$150$	0	0
$151$	106852.	0.381363	0.190681	−	0.981652i	$-0.438930\pi$
$151$	106852.	0.381363	0.190681	+	0.981652i	$0.438930\pi$
$152$	0	0
$153$	123361.	0.426040
$154$	0	0
$155$	−273726.	−0.915138
$156$	0	0
$157$	316009.	1.02318	0.511588	−	0.859231i	$-0.329057\pi$
$157$	316009.	1.02318	0.511588	+	0.859231i	$0.329057\pi$
$158$	0	0
$159$	−118858.	−0.372851
$160$	0	0
$161$	−305442.	−0.928677
$162$	0	0
$163$	467943.	1.37951	0.689753	−	0.724045i	$-0.257719\pi$
$163$	467943.	1.37951	0.689753	+	0.724045i	$0.257719\pi$
$164$	0	0
$165$	905927.	2.59050
$166$	0	0
$167$	−329143.	−0.913258	−0.456629	−	0.889657i	$-0.650943\pi$
$167$	−329143.	−0.913258	−0.456629	+	0.889657i	$0.650943\pi$
$168$	0	0
$169$	−252506.	−0.680072
$170$	0	0
$171$	−69100.5	−0.180714
$172$	0	0
$173$	197867.	0.502641	0.251321	−	0.967904i	$-0.419135\pi$
$173$	197867.	0.502641	0.251321	+	0.967904i	$0.419135\pi$
$174$	0	0
$175$	201247.	0.496746
$176$	0	0
$177$	637461.	1.52940
$178$	0	0
$179$	−71110.2	−0.165882	−0.0829410	−	0.996554i	$-0.526431\pi$
$179$	−71110.2	−0.165882	−0.0829410	+	0.996554i	$0.526431\pi$
$180$	0	0
$181$	111059.	0.251975	0.125987	−	0.992032i	$-0.459790\pi$
$181$	111059.	0.251975	0.125987	+	0.992032i	$0.459790\pi$
$182$	0	0
$183$	−485252.	−1.07112
$184$	0	0
$185$	915491.	1.96664
$186$	0	0
$187$	902597.	1.88751
$188$	0	0
$189$	264585.	0.538779
$190$	0	0
$191$	−620980.	−1.23167	−0.615835	−	0.787875i	$-0.711181\pi$
$191$	−620980.	−1.23167	−0.615835	+	0.787875i	$0.711181\pi$
$192$	0	0
$193$	−725314.	−1.40163	−0.700814	−	0.713344i	$-0.747180\pi$
$193$	−725314.	−1.40163	−0.700814	+	0.713344i	$0.747180\pi$
$194$	0	0
$195$	456389.	0.859506
$196$	0	0
$197$	273782.	0.502619	0.251310	−	0.967907i	$-0.419139\pi$
$197$	273782.	0.502619	0.251310	+	0.967907i	$0.419139\pi$
$198$	0	0
$199$	314223.	0.562477	0.281239	−	0.959638i	$-0.409255\pi$
$199$	314223.	0.562477	0.281239	+	0.959638i	$0.409255\pi$
$200$	0	0
$201$	−1.06409e6	−1.85775
$202$	0	0
$203$	682425.	1.16229
$204$	0	0
$205$	785573.	1.30557
$206$	0	0
$207$	−296018.	−0.480167
$208$	0	0
$209$	−505588.	−0.800628
$210$	0	0
$211$	540561.	0.835869	0.417935	−	0.908477i	$-0.362754\pi$
$211$	540561.	0.835869	0.417935	+	0.908477i	$0.362754\pi$
$212$	0	0
$213$	−430676.	−0.650432
$214$	0	0
$215$	133473.	0.196923
$216$	0	0
$217$	365846.	0.527411
$218$	0	0
$219$	−828864.	−1.16781
$220$	0	0
$221$	454712.	0.626261
$222$	0	0
$223$	231856.	0.312216	0.156108	−	0.987740i	$-0.450105\pi$
$223$	231856.	0.312216	0.156108	+	0.987740i	$0.450105\pi$
$224$	0	0
$225$	195038.	0.256840
$226$	0	0
$227$	1.39293e6	1.79418	0.897090	−	0.441848i	$-0.145677\pi$
$227$	1.39293e6	1.79418	0.897090	+	0.441848i	$0.145677\pi$
$228$	0	0
$229$	−565105.	−0.712099	−0.356050	−	0.934467i	$-0.615877\pi$
$229$	−565105.	−0.712099	−0.356050	+	0.934467i	$0.615877\pi$
$230$	0	0
$231$	−1.21081e6	−1.49295
$232$	0	0
$233$	237550.	0.286658	0.143329	−	0.989675i	$-0.454219\pi$
$233$	237550.	0.286658	0.143329	+	0.989675i	$0.454219\pi$
$234$	0	0
$235$	279621.	0.330294
$236$	0	0
$237$	328284.	0.379645
$238$	0	0
$239$	−1.31472e6	−1.48881	−0.744403	−	0.667731i	$-0.767266\pi$
$239$	−1.31472e6	−1.48881	−0.744403	+	0.667731i	$0.767266\pi$
$240$	0	0
$241$	1.03644e6	1.14948	0.574739	−	0.818337i	$-0.305104\pi$
$241$	1.03644e6	1.14948	0.574739	+	0.818337i	$0.305104\pi$
$242$	0	0
$243$	673222.	0.731379
$244$	0	0
$245$	541293.	0.576126
$246$	0	0
$247$	−254706.	−0.265642
$248$	0	0
$249$	730079.	0.746227
$250$	0	0
$251$	1.67030e6	1.67344	0.836720	−	0.547632i	$-0.184470\pi$
$251$	1.67030e6	1.67344	0.836720	+	0.547632i	$0.184470\pi$
$252$	0	0
$253$	−2.16587e6	−2.12732
$254$	0	0
$255$	1.74704e6	1.68249
$256$	0	0
$257$	311852.	0.294521	0.147261	−	0.989098i	$-0.452954\pi$
$257$	311852.	0.294521	0.147261	+	0.989098i	$0.452954\pi$
$258$	0	0
$259$	−1.22359e6	−1.13341
$260$	0	0
$261$	661369.	0.600956
$262$	0	0
$263$	341600.	0.304529	0.152264	−	0.988340i	$-0.451343\pi$
$263$	341600.	0.304529	0.152264	+	0.988340i	$0.451343\pi$
$264$	0	0
$265$	−467723.	−0.409142
$266$	0	0
$267$	565034.	0.485061
$268$	0	0
$269$	865544.	0.729304	0.364652	−	0.931144i	$-0.381188\pi$
$269$	865544.	0.729304	0.364652	+	0.931144i	$0.381188\pi$
$270$	0	0
$271$	−1.49244e6	−1.23445	−0.617226	−	0.786786i	$-0.711744\pi$
$271$	−1.49244e6	−1.23445	−0.617226	+	0.786786i	$0.711744\pi$
$272$	0	0
$273$	−609984.	−0.495350
$274$	0	0
$275$	1.42703e6	1.13789
$276$	0	0
$277$	−395961.	−0.310065	−0.155033	−	0.987909i	$-0.549548\pi$
$277$	−395961.	−0.310065	−0.155033	+	0.987909i	$0.549548\pi$
$278$	0	0
$279$	354558.	0.272695
$280$	0	0
$281$	2.19634e6	1.65934	0.829668	−	0.558257i	$-0.188530\pi$
$281$	2.19634e6	1.65934	0.829668	+	0.558257i	$0.188530\pi$
$282$	0	0
$283$	−562659.	−0.417618	−0.208809	−	0.977956i	$-0.566959\pi$
$283$	−562659.	−0.417618	−0.208809	+	0.977956i	$0.566959\pi$
$284$	0	0
$285$	−978598.	−0.713661
$286$	0	0
$287$	−1.04995e6	−0.752427
$288$	0	0
$289$	320758.	0.225909
$290$	0	0
$291$	−413190.	−0.286034
$292$	0	0
$293$	2.24866e6	1.53022	0.765111	−	0.643898i	$-0.222684\pi$
$293$	2.24866e6	1.53022	0.765111	+	0.643898i	$0.222684\pi$
$294$	0	0
$295$	2.50850e6	1.67826
$296$	0	0
$297$	1.87615e6	1.23418
$298$	0	0
$299$	−1.09113e6	−0.705826
$300$	0	0
$301$	−178392.	−0.113490
$302$	0	0
$303$	−1.29495e6	−0.810301
$304$	0	0
$305$	−1.90954e6	−1.17538
$306$	0	0
$307$	−2.14795e6	−1.30070	−0.650352	−	0.759633i	$-0.725379\pi$
$307$	−2.14795e6	−1.30070	−0.650352	+	0.759633i	$0.725379\pi$
$308$	0	0
$309$	−671914.	−0.400329
$310$	0	0
$311$	2.61384e6	1.53242	0.766210	−	0.642590i	$-0.222140\pi$
$311$	2.61384e6	1.53242	0.766210	+	0.642590i	$0.222140\pi$
$312$	0	0
$313$	968697.	0.558891	0.279445	−	0.960162i	$-0.409849\pi$
$313$	968697.	0.558891	0.279445	+	0.960162i	$0.409849\pi$
$314$	0	0
$315$	−651212.	−0.369782
$316$	0	0
$317$	289378.	0.161740	0.0808700	−	0.996725i	$-0.474230\pi$
$317$	289378.	0.161740	0.0808700	+	0.996725i	$0.474230\pi$
$318$	0	0
$319$	4.83904e6	2.66246
$320$	0	0
$321$	−887647.	−0.480815
$322$	0	0
$323$	−975001.	−0.519995
$324$	0	0
$325$	718912.	0.377544
$326$	0	0
$327$	1.39224e6	0.720019
$328$	0	0
$329$	−373725.	−0.190354
$330$	0	0
$331$	−2.74453e6	−1.37689	−0.688443	−	0.725291i	$-0.741705\pi$
$331$	−2.74453e6	−1.37689	−0.688443	+	0.725291i	$0.741705\pi$
$332$	0	0
$333$	−1.18584e6	−0.586024
$334$	0	0
$335$	−4.18734e6	−2.03857
$336$	0	0
$337$	−1.03816e6	−0.497956	−0.248978	−	0.968509i	$-0.580095\pi$
$337$	−1.03816e6	−0.497956	−0.248978	+	0.968509i	$0.580095\pi$
$338$	0	0
$339$	3.89180e6	1.83929
$340$	0	0
$341$	2.59420e6	1.20814
$342$	0	0
$343$	−2.34501e6	−1.07624
$344$	0	0
$345$	−4.19219e6	−1.89624
$346$	0	0
$347$	1.87377e6	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$347$	1.87377e6	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$348$	0	0
$349$	46712.6	0.0205291	0.0102646	−	0.999947i	$-0.496733\pi$
$349$	46712.6	0.0205291	0.0102646	+	0.999947i	$0.496733\pi$
$350$	0	0
$351$	945172.	0.409490
$352$	0	0
$353$	643022.	0.274656	0.137328	−	0.990526i	$-0.456149\pi$
$353$	643022.	0.274656	0.137328	+	0.990526i	$0.456149\pi$
$354$	0	0
$355$	−1.69477e6	−0.713742
$356$	0	0
$357$	−2.33499e6	−0.969649
$358$	0	0
$359$	−361451.	−0.148017	−0.0740087	−	0.997258i	$-0.523579\pi$
$359$	−361451.	−0.148017	−0.0740087	+	0.997258i	$0.523579\pi$
$360$	0	0
$361$	−1.92995e6	−0.779434
$362$	0	0
$363$	−5.63146e6	−2.24313
$364$	0	0
$365$	−3.26170e6	−1.28148
$366$	0	0
$367$	1.12671e6	0.436663	0.218332	−	0.975875i	$-0.429939\pi$
$367$	1.12671e6	0.436663	0.218332	+	0.975875i	$0.429939\pi$
$368$	0	0
$369$	−1.01756e6	−0.389038
$370$	0	0
$371$	625132.	0.235796
$372$	0	0
$373$	−1.98041e6	−0.737027	−0.368514	−	0.929622i	$-0.620133\pi$
$373$	−1.98041e6	−0.737027	−0.368514	+	0.929622i	$0.620133\pi$
$374$	0	0
$375$	−1.37598e6	−0.505283
$376$	0	0
$377$	2.43782e6	0.883380
$378$	0	0
$379$	870212.	0.311191	0.155596	−	0.987821i	$-0.450270\pi$
$379$	870212.	0.311191	0.155596	+	0.987821i	$0.450270\pi$
$380$	0	0
$381$	−1.36751e6	−0.482633
$382$	0	0
$383$	2.78192e6	0.969055	0.484527	−	0.874776i	$-0.338992\pi$
$383$	2.78192e6	0.969055	0.484527	+	0.874776i	$0.338992\pi$
$384$	0	0
$385$	−4.76472e6	−1.63827
$386$	0	0
$387$	−172888.	−0.0586796
$388$	0	0
$389$	5.01979e6	1.68195	0.840973	−	0.541077i	$-0.181983\pi$
$389$	5.01979e6	1.68195	0.840973	+	0.541077i	$0.181983\pi$
$390$	0	0
$391$	−4.17678e6	−1.38166
$392$	0	0
$393$	−1.05830e6	−0.345643
$394$	0	0
$395$	1.29184e6	0.416598
$396$	0	0
$397$	−3.26819e6	−1.04071	−0.520357	−	0.853949i	$-0.674201\pi$
$397$	−3.26819e6	−1.04071	−0.520357	+	0.853949i	$0.674201\pi$
$398$	0	0
$399$	1.30794e6	0.411297
$400$	0	0
$401$	3.63270e6	1.12815	0.564077	−	0.825722i	$-0.309232\pi$
$401$	3.63270e6	1.12815	0.564077	+	0.825722i	$0.309232\pi$
$402$	0	0
$403$	1.30691e6	0.400850
$404$	0	0
$405$	5.27159e6	1.59700
$406$	0	0
$407$	−8.67643e6	−2.59630
$408$	0	0
$409$	−4.93318e6	−1.45821	−0.729103	−	0.684404i	$-0.760063\pi$
$409$	−4.93318e6	−1.45821	−0.729103	+	0.684404i	$0.760063\pi$
$410$	0	0
$411$	434276.	0.126812
$412$	0	0
$413$	−3.35272e6	−0.967214
$414$	0	0
$415$	2.87297e6	0.818861
$416$	0	0
$417$	4.31078e6	1.21399
$418$	0	0
$419$	−412864.	−0.114887	−0.0574437	−	0.998349i	$-0.518295\pi$
$419$	−412864.	−0.114887	−0.0574437	+	0.998349i	$0.518295\pi$
$420$	0	0
$421$	−2.27168e6	−0.624656	−0.312328	−	0.949974i	$-0.601109\pi$
$421$	−2.27168e6	−0.624656	−0.312328	+	0.949974i	$0.601109\pi$
$422$	0	0
$423$	−362194.	−0.0984216
$424$	0	0
$425$	2.75196e6	0.739044
$426$	0	0
$427$	2.55218e6	0.677395
$428$	0	0
$429$	−4.32536e6	−1.13469
$430$	0	0
$431$	1.37855e6	0.357461	0.178731	−	0.983898i	$-0.442801\pi$
$431$	1.37855e6	0.357461	0.178731	+	0.983898i	$0.442801\pi$
$432$	0	0
$433$	−4.50720e6	−1.15528	−0.577639	−	0.816292i	$-0.696026\pi$
$433$	−4.50720e6	−1.15528	−0.577639	+	0.816292i	$0.696026\pi$
$434$	0	0
$435$	9.36628e6	2.37325
$436$	0	0
$437$	2.33961e6	0.586058
$438$	0	0
$439$	1.97588e6	0.489327	0.244663	−	0.969608i	$-0.421323\pi$
$439$	1.97588e6	0.489327	0.244663	+	0.969608i	$0.421323\pi$
$440$	0	0
$441$	−701140.	−0.171675
$442$	0	0
$443$	2.75707e6	0.667481	0.333740	−	0.942665i	$-0.391689\pi$
$443$	2.75707e6	0.667481	0.333740	+	0.942665i	$0.391689\pi$
$444$	0	0
$445$	2.22349e6	0.532274
$446$	0	0
$447$	−4.34130e6	−1.02766
$448$	0	0
$449$	5.19837e6	1.21689	0.608445	−	0.793596i	$-0.291794\pi$
$449$	5.19837e6	1.21689	0.608445	+	0.793596i	$0.291794\pi$
$450$	0	0
$451$	−7.44515e6	−1.72358
$452$	0	0
$453$	−1.96009e6	−0.448777
$454$	0	0
$455$	−2.40038e6	−0.543564
$456$	0	0
$457$	7.85140e6	1.75856	0.879279	−	0.476307i	$-0.158025\pi$
$457$	7.85140e6	1.75856	0.879279	+	0.476307i	$0.158025\pi$
$458$	0	0
$459$	3.61807e6	0.801578
$460$	0	0
$461$	528436.	0.115808	0.0579042	−	0.998322i	$-0.481558\pi$
$461$	528436.	0.115808	0.0579042	+	0.998322i	$0.481558\pi$
$462$	0	0
$463$	−825321.	−0.178925	−0.0894624	−	0.995990i	$-0.528515\pi$
$463$	−825321.	−0.178925	−0.0894624	+	0.995990i	$0.528515\pi$
$464$	0	0
$465$	5.02124e6	1.07691
$466$	0	0
$467$	−4.03618e6	−0.856404	−0.428202	−	0.903683i	$-0.640853\pi$
$467$	−4.03618e6	−0.856404	−0.428202	+	0.903683i	$0.640853\pi$
$468$	0	0
$469$	5.59656e6	1.17487
$470$	0	0
$471$	−5.79688e6	−1.20404
$472$	0	0
$473$	−1.26497e6	−0.259972
$474$	0	0
$475$	−1.54150e6	−0.313481
$476$	0	0
$477$	605844.	0.121917
$478$	0	0
$479$	6.39933e6	1.27437	0.637186	−	0.770710i	$-0.280098\pi$
$479$	6.39933e6	1.27437	0.637186	+	0.770710i	$0.280098\pi$
$480$	0	0
$481$	−4.37102e6	−0.861431
$482$	0	0
$483$	5.60305e6	1.09284
$484$	0	0
$485$	−1.62596e6	−0.313875
$486$	0	0
$487$	−4.00561e6	−0.765326	−0.382663	−	0.923888i	$-0.624993\pi$
$487$	−4.00561e6	−0.765326	−0.382663	+	0.923888i	$0.624993\pi$
$488$	0	0
$489$	−8.58395e6	−1.62336
$490$	0	0
$491$	5.31835e6	0.995573	0.497786	−	0.867300i	$-0.334146\pi$
$491$	5.31835e6	0.995573	0.497786	+	0.867300i	$0.334146\pi$
$492$	0	0
$493$	9.33185e6	1.72922
$494$	0	0
$495$	−4.61770e6	−0.847058
$496$	0	0
$497$	2.26514e6	0.411343
$498$	0	0
$499$	−172551.	−0.0310217	−0.0155108	−	0.999880i	$-0.504937\pi$
$499$	−172551.	−0.0310217	−0.0155108	+	0.999880i	$0.504937\pi$
$500$	0	0
$501$	6.03781e6	1.07470
$502$	0	0
$503$	−1.54456e6	−0.272198	−0.136099	−	0.990695i	$-0.543457\pi$
$503$	−1.54456e6	−0.272198	−0.136099	+	0.990695i	$0.543457\pi$
$504$	0	0
$505$	−5.09582e6	−0.889172
$506$	0	0
$507$	4.63197e6	0.800288
$508$	0	0
$509$	3.51269e6	0.600960	0.300480	−	0.953788i	$-0.402853\pi$
$509$	3.51269e6	0.600960	0.300480	+	0.953788i	$0.402853\pi$
$510$	0	0
$511$	4.35941e6	0.738542
$512$	0	0
$513$	−2.02665e6	−0.340006
$514$	0	0
$515$	−2.64408e6	−0.439295
$516$	0	0
$517$	−2.65007e6	−0.436044
$518$	0	0
$519$	−3.62968e6	−0.591493
$520$	0	0
$521$	−1.33669e6	−0.215743	−0.107871	−	0.994165i	$-0.534403\pi$
$521$	−1.33669e6	−0.215743	−0.107871	+	0.994165i	$0.534403\pi$
$522$	0	0
$523$	1.35607e6	0.216784	0.108392	−	0.994108i	$-0.465430\pi$
$523$	1.35607e6	0.216784	0.108392	+	0.994108i	$0.465430\pi$
$524$	0	0
$525$	−3.69168e6	−0.584556
$526$	0	0
$527$	5.00278e6	0.784666
$528$	0	0
$529$	3.58627e6	0.557191
$530$	0	0
$531$	−3.24928e6	−0.500092
$532$	0	0
$533$	−3.75073e6	−0.571870
$534$	0	0
$535$	−3.49302e6	−0.527615
$536$	0	0
$537$	1.30445e6	0.195205
$538$	0	0
$539$	−5.13003e6	−0.760585
$540$	0	0
$541$	1.28068e6	0.188125	0.0940625	−	0.995566i	$-0.470015\pi$
$541$	1.28068e6	0.188125	0.0940625	+	0.995566i	$0.470015\pi$
$542$	0	0
$543$	−2.03727e6	−0.296516
$544$	0	0
$545$	5.47866e6	0.790103
$546$	0	0
$547$	−1.13222e7	−1.61793	−0.808967	−	0.587854i	$-0.799973\pi$
$547$	−1.13222e7	−1.61793	−0.808967	+	0.587854i	$0.799973\pi$
$548$	0	0
$549$	2.47343e6	0.350243
$550$	0	0
$551$	−5.22721e6	−0.733485
$552$	0	0
$553$	−1.72661e6	−0.240093
$554$	0	0
$555$	−1.67938e7	−2.31428
$556$	0	0
$557$	−1.13494e7	−1.55002	−0.775008	−	0.631952i	$-0.782254\pi$
$557$	−1.13494e7	−1.55002	−0.775008	+	0.631952i	$0.782254\pi$
$558$	0	0
$559$	−637267.	−0.0862566
$560$	0	0
$561$	−1.65573e7	−2.22117
$562$	0	0
$563$	8.01199e6	1.06529	0.532647	−	0.846338i	$-0.321198\pi$
$563$	8.01199e6	1.06529	0.532647	+	0.846338i	$0.321198\pi$
$564$	0	0
$565$	1.53148e7	2.01832
$566$	0	0
$567$	−7.04571e6	−0.920380
$568$	0	0
$569$	−6.18179e6	−0.800448	−0.400224	−	0.916417i	$-0.631068\pi$
$569$	−6.18179e6	−0.800448	−0.400224	+	0.916417i	$0.631068\pi$
$570$	0	0
$571$	−4.84960e6	−0.622466	−0.311233	−	0.950334i	$-0.600742\pi$
$571$	−4.84960e6	−0.622466	−0.311233	+	0.950334i	$0.600742\pi$
$572$	0	0
$573$	1.13913e7	1.44939
$574$	0	0
$575$	−6.60361e6	−0.832937
$576$	0	0
$577$	−1.20012e6	−0.150067	−0.0750335	−	0.997181i	$-0.523906\pi$
$577$	−1.20012e6	−0.150067	−0.0750335	+	0.997181i	$0.523906\pi$
$578$	0	0
$579$	1.33052e7	1.64939
$580$	0	0
$581$	−3.83984e6	−0.471925
$582$	0	0
$583$	4.43278e6	0.540138
$584$	0	0
$585$	−2.32631e6	−0.281047
$586$	0	0
$587$	−2.03492e6	−0.243754	−0.121877	−	0.992545i	$-0.538891\pi$
$587$	−2.03492e6	−0.243754	−0.121877	+	0.992545i	$0.538891\pi$
$588$	0	0
$589$	−2.80229e6	−0.332832
$590$	0	0
$591$	−5.02226e6	−0.591467
$592$	0	0
$593$	−1.38521e7	−1.61762	−0.808812	−	0.588067i	$-0.799889\pi$
$593$	−1.38521e7	−1.61762	−0.808812	+	0.588067i	$0.799889\pi$
$594$	0	0
$595$	−9.18853e6	−1.06403
$596$	0	0
$597$	−5.76411e6	−0.661906
$598$	0	0
$599$	1.64523e7	1.87352	0.936760	−	0.349972i	$-0.113809\pi$
$599$	1.64523e7	1.87352	0.936760	+	0.349972i	$0.113809\pi$
$600$	0	0
$601$	1.48314e7	1.67493	0.837465	−	0.546491i	$-0.184037\pi$
$601$	1.48314e7	1.67493	0.837465	+	0.546491i	$0.184037\pi$
$602$	0	0
$603$	5.42388e6	0.607459
$604$	0	0
$605$	−2.21606e7	−2.46147
$606$	0	0
$607$	1.64197e7	1.80881	0.904404	−	0.426677i	$-0.140316\pi$
$607$	1.64197e7	1.80881	0.904404	+	0.426677i	$0.140316\pi$
$608$	0	0
$609$	−1.25184e7	−1.36775
$610$	0	0
$611$	−1.33505e6	−0.144676
$612$	0	0
$613$	8.77423e6	0.943100	0.471550	−	0.881839i	$-0.343695\pi$
$613$	8.77423e6	0.943100	0.471550	+	0.881839i	$0.343695\pi$
$614$	0	0
$615$	−1.44106e7	−1.53636
$616$	0	0
$617$	1.23910e7	1.31037	0.655183	−	0.755470i	$-0.272591\pi$
$617$	1.23910e7	1.31037	0.655183	+	0.755470i	$0.272591\pi$
$618$	0	0
$619$	1.58048e6	0.165791	0.0828957	−	0.996558i	$-0.473583\pi$
$619$	1.58048e6	0.165791	0.0828957	+	0.996558i	$0.473583\pi$
$620$	0	0
$621$	−8.68193e6	−0.903415
$622$	0	0
$623$	−2.97179e6	−0.306760
$624$	0	0
$625$	−1.19331e7	−1.22195
$626$	0	0
$627$	9.27451e6	0.942155
$628$	0	0
$629$	−1.67321e7	−1.68625
$630$	0	0
$631$	3.95881e6	0.395814	0.197907	−	0.980221i	$-0.436586\pi$
$631$	3.95881e6	0.395814	0.197907	+	0.980221i	$0.436586\pi$
$632$	0	0
$633$	−9.91606e6	−0.983626
$634$	0	0
$635$	−5.38135e6	−0.529611
$636$	0	0
$637$	−2.58441e6	−0.252356
$638$	0	0
$639$	2.19525e6	0.212682
$640$	0	0
$641$	8.19225e6	0.787514	0.393757	−	0.919215i	$-0.371175\pi$
$641$	8.19225e6	0.787514	0.393757	+	0.919215i	$0.371175\pi$
$642$	0	0
$643$	9.53703e6	0.909674	0.454837	−	0.890575i	$-0.349697\pi$
$643$	9.53703e6	0.909674	0.454837	+	0.890575i	$0.349697\pi$
$644$	0	0
$645$	−2.44843e6	−0.231733
$646$	0	0
$647$	1.17034e7	1.09913	0.549566	−	0.835450i	$-0.314793\pi$
$647$	1.17034e7	1.09913	0.549566	+	0.835450i	$0.314793\pi$
$648$	0	0
$649$	−2.37740e7	−2.21559
$650$	0	0
$651$	−6.71110e6	−0.620642
$652$	0	0
$653$	9.87401e6	0.906172	0.453086	−	0.891467i	$-0.350323\pi$
$653$	9.87401e6	0.906172	0.453086	+	0.891467i	$0.350323\pi$
$654$	0	0
$655$	−4.16457e6	−0.379286
$656$	0	0
$657$	4.22490e6	0.381859
$658$	0	0
$659$	−1.87688e7	−1.68354	−0.841770	−	0.539836i	$-0.818486\pi$
$659$	−1.87688e7	−1.68354	−0.841770	+	0.539836i	$0.818486\pi$
$660$	0	0
$661$	1.84811e6	0.164522	0.0822612	−	0.996611i	$-0.473786\pi$
$661$	1.84811e6	0.164522	0.0822612	+	0.996611i	$0.473786\pi$
$662$	0	0
$663$	−8.34125e6	−0.736966
$664$	0	0
$665$	5.14693e6	0.451330
$666$	0	0
$667$	−2.23927e7	−1.94891
$668$	0	0
$669$	−4.25317e6	−0.367407
$670$	0	0
$671$	1.80974e7	1.55171
$672$	0	0
$673$	−3.40289e6	−0.289608	−0.144804	−	0.989460i	$-0.546255\pi$
$673$	−3.40289e6	−0.289608	−0.144804	+	0.989460i	$0.546255\pi$
$674$	0	0
$675$	5.72028e6	0.483234
$676$	0	0
$677$	2.23309e7	1.87255	0.936276	−	0.351266i	$-0.114249\pi$
$677$	2.23309e7	1.87255	0.936276	+	0.351266i	$0.114249\pi$
$678$	0	0
$679$	2.17317e6	0.180892
$680$	0	0
$681$	−2.55520e7	−2.11134
$682$	0	0
$683$	−7.78181e6	−0.638306	−0.319153	−	0.947703i	$-0.603398\pi$
$683$	−7.78181e6	−0.638306	−0.319153	+	0.947703i	$0.603398\pi$
$684$	0	0
$685$	1.70894e6	0.139155
$686$	0	0
$687$	1.03663e7	0.837977
$688$	0	0
$689$	2.23315e6	0.179213
$690$	0	0
$691$	−2.39450e7	−1.90775	−0.953873	−	0.300211i	$-0.902943\pi$
$691$	−2.39450e7	−1.90775	−0.953873	+	0.300211i	$0.902943\pi$
$692$	0	0
$693$	6.17176e6	0.488176
$694$	0	0
$695$	1.69636e7	1.33216
$696$	0	0
$697$	−1.43576e7	−1.11944
$698$	0	0
$699$	−4.35762e6	−0.337331
$700$	0	0
$701$	−6.24393e6	−0.479914	−0.239957	−	0.970784i	$-0.577133\pi$
$701$	−6.24393e6	−0.479914	−0.239957	+	0.970784i	$0.577133\pi$
$702$	0	0
$703$	9.37242e6	0.715259
$704$	0	0
$705$	−5.12938e6	−0.388680
$706$	0	0
$707$	6.81079e6	0.512447
$708$	0	0
$709$	−9.29599e6	−0.694513	−0.347256	−	0.937770i	$-0.612887\pi$
$709$	−9.29599e6	−0.694513	−0.347256	+	0.937770i	$0.612887\pi$
$710$	0	0
$711$	−1.67333e6	−0.124139
$712$	0	0
$713$	−1.20047e7	−0.884355
$714$	0	0
$715$	−1.70209e7	−1.24514
$716$	0	0
$717$	2.41172e7	1.75198
$718$	0	0
$719$	−1.77934e7	−1.28362	−0.641811	−	0.766863i	$-0.721816\pi$
$719$	−1.77934e7	−1.28362	−0.641811	+	0.766863i	$0.721816\pi$
$720$	0	0
$721$	3.53393e6	0.253174
$722$	0	0
$723$	−1.90124e7	−1.35267
$724$	0	0
$725$	1.47539e7	1.04247
$726$	0	0
$727$	1.60920e7	1.12921	0.564604	−	0.825362i	$-0.309029\pi$
$727$	1.60920e7	1.12921	0.564604	+	0.825362i	$0.309029\pi$
$728$	0	0
$729$	5.39607e6	0.376061
$730$	0	0
$731$	−2.43943e6	−0.168848
$732$	0	0
$733$	2.32899e7	1.60106	0.800529	−	0.599295i	$-0.204552\pi$
$733$	2.32899e7	1.60106	0.800529	+	0.599295i	$0.204552\pi$
$734$	0	0
$735$	−9.92950e6	−0.677968
$736$	0	0
$737$	3.96849e7	2.69126
$738$	0	0
$739$	1.68335e7	1.13387	0.566934	−	0.823763i	$-0.308129\pi$
$739$	1.68335e7	1.13387	0.566934	+	0.823763i	$0.308129\pi$
$740$	0	0
$741$	4.67233e6	0.312599
$742$	0	0
$743$	−5.88483e6	−0.391077	−0.195538	−	0.980696i	$-0.562645\pi$
$743$	−5.88483e6	−0.391077	−0.195538	+	0.980696i	$0.562645\pi$
$744$	0	0
$745$	−1.70837e7	−1.12769
$746$	0	0
$747$	−3.72137e6	−0.244006
$748$	0	0
$749$	4.66858e6	0.304074
$750$	0	0
$751$	2.45954e7	1.59131	0.795655	−	0.605750i	$-0.207127\pi$
$751$	2.45954e7	1.59131	0.795655	+	0.605750i	$0.207127\pi$
$752$	0	0
$753$	−3.06400e7	−1.96925
$754$	0	0
$755$	−7.71324e6	−0.492458
$756$	0	0
$757$	1.55533e7	0.986467	0.493233	−	0.869897i	$-0.335815\pi$
$757$	1.55533e7	0.986467	0.493233	+	0.869897i	$0.335815\pi$
$758$	0	0
$759$	3.97309e7	2.50336
$760$	0	0
$761$	−1.20979e7	−0.757264	−0.378632	−	0.925547i	$-0.623605\pi$
$761$	−1.20979e7	−0.757264	−0.378632	+	0.925547i	$0.623605\pi$
$762$	0	0
$763$	−7.32247e6	−0.455351
$764$	0	0
$765$	−8.90502e6	−0.550150
$766$	0	0
$767$	−1.19769e7	−0.735115
$768$	0	0
$769$	6.89705e6	0.420579	0.210289	−	0.977639i	$-0.432559\pi$
$769$	6.89705e6	0.420579	0.210289	+	0.977639i	$0.432559\pi$
$770$	0	0
$771$	−5.72063e6	−0.346584
$772$	0	0
$773$	2.76899e7	1.66676	0.833380	−	0.552701i	$-0.186403\pi$
$773$	2.76899e7	1.66676	0.833380	+	0.552701i	$0.186403\pi$
$774$	0	0
$775$	7.90953e6	0.473039
$776$	0	0
$777$	2.24456e7	1.33376
$778$	0	0
$779$	8.04237e6	0.474833
$780$	0	0
$781$	1.60620e7	0.942261
$782$	0	0
$783$	1.93973e7	1.13068
$784$	0	0
$785$	−2.28116e7	−1.32124
$786$	0	0
$787$	−2.46071e6	−0.141620	−0.0708098	−	0.997490i	$-0.522558\pi$
$787$	−2.46071e6	−0.141620	−0.0708098	+	0.997490i	$0.522558\pi$
$788$	0	0
$789$	−6.26632e6	−0.358360
$790$	0	0
$791$	−2.04689e7	−1.16320
$792$	0	0
$793$	9.11712e6	0.514843
$794$	0	0
$795$	8.57993e6	0.481466
$796$	0	0
$797$	1.86050e7	1.03749	0.518746	−	0.854928i	$-0.326399\pi$
$797$	1.86050e7	1.03749	0.518746	+	0.854928i	$0.326399\pi$
$798$	0	0
$799$	−5.11052e6	−0.283203
$800$	0	0
$801$	−2.88010e6	−0.158608
$802$	0	0
$803$	3.09123e7	1.69177
$804$	0	0
$805$	2.20488e7	1.19921
$806$	0	0
$807$	−1.58776e7	−0.858223
$808$	0	0
$809$	−2.26776e7	−1.21822	−0.609110	−	0.793085i	$-0.708473\pi$
$809$	−2.26776e7	−1.21822	−0.609110	+	0.793085i	$0.708473\pi$
$810$	0	0
$811$	1.84184e7	0.983329	0.491665	−	0.870785i	$-0.336389\pi$
$811$	1.84184e7	0.983329	0.491665	+	0.870785i	$0.336389\pi$
$812$	0	0
$813$	2.73774e7	1.45267
$814$	0	0
$815$	−3.37791e7	−1.78137
$816$	0	0
$817$	1.36644e6	0.0716202
$818$	0	0
$819$	3.10922e6	0.161973
$820$	0	0
$821$	1.39401e7	0.721783	0.360891	−	0.932608i	$-0.382472\pi$
$821$	1.39401e7	0.721783	0.360891	+	0.932608i	$0.382472\pi$
$822$	0	0
$823$	−1.86303e6	−0.0958781	−0.0479391	−	0.998850i	$-0.515265\pi$
$823$	−1.86303e6	−0.0958781	−0.0479391	+	0.998850i	$0.515265\pi$
$824$	0	0
$825$	−2.61775e7	−1.33904
$826$	0	0
$827$	−1.84625e7	−0.938698	−0.469349	−	0.883013i	$-0.655511\pi$
$827$	−1.84625e7	−0.938698	−0.469349	+	0.883013i	$0.655511\pi$
$828$	0	0
$829$	6.01093e6	0.303777	0.151889	−	0.988398i	$-0.451464\pi$
$829$	6.01093e6	0.303777	0.151889	+	0.988398i	$0.451464\pi$
$830$	0	0
$831$	7.26353e6	0.364876
$832$	0	0
$833$	−9.89301e6	−0.493987
$834$	0	0
$835$	2.37597e7	1.17930
$836$	0	0
$837$	1.03989e7	0.513065
$838$	0	0
$839$	2.60373e7	1.27700	0.638500	−	0.769622i	$-0.279555\pi$
$839$	2.60373e7	1.27700	0.638500	+	0.769622i	$0.279555\pi$
$840$	0	0
$841$	2.95191e7	1.43917
$842$	0	0
$843$	−4.02898e7	−1.95266
$844$	0	0
$845$	1.82275e7	0.878184
$846$	0	0
$847$	2.96187e7	1.41859
$848$	0	0
$849$	1.03214e7	0.491441
$850$	0	0
$851$	4.01503e7	1.90049
$852$	0	0
$853$	1.81066e7	0.852050	0.426025	−	0.904711i	$-0.359914\pi$
$853$	1.81066e7	0.852050	0.426025	+	0.904711i	$0.359914\pi$
$854$	0	0
$855$	4.98812e6	0.233358
$856$	0	0
$857$	2.48924e7	1.15775	0.578874	−	0.815417i	$-0.303492\pi$
$857$	2.48924e7	1.15775	0.578874	+	0.815417i	$0.303492\pi$
$858$	0	0
$859$	8.53406e6	0.394614	0.197307	−	0.980342i	$-0.436780\pi$
$859$	8.53406e6	0.394614	0.197307	+	0.980342i	$0.436780\pi$
$860$	0	0
$861$	1.92603e7	0.885434
$862$	0	0
$863$	2.14592e7	0.980815	0.490407	−	0.871493i	$-0.336848\pi$
$863$	2.14592e7	0.980815	0.490407	+	0.871493i	$0.336848\pi$
$864$	0	0
$865$	−1.42833e7	−0.649066
$866$	0	0
$867$	−5.88400e6	−0.265843
$868$	0	0
$869$	−1.22433e7	−0.549981
$870$	0	0
$871$	1.99925e7	0.892939
$872$	0	0
$873$	2.10612e6	0.0935292
$874$	0	0
$875$	7.23698e6	0.319549
$876$	0	0
$877$	3.45569e7	1.51717	0.758587	−	0.651572i	$-0.225890\pi$
$877$	3.45569e7	1.51717	0.758587	+	0.651572i	$0.225890\pi$
$878$	0	0
$879$	−4.12495e7	−1.80072
$880$	0	0
$881$	780223.	0.0338672	0.0169336	−	0.999857i	$-0.494610\pi$
$881$	780223.	0.0338672	0.0169336	+	0.999857i	$0.494610\pi$
$882$	0	0
$883$	1.51541e7	0.654077	0.327039	−	0.945011i	$-0.393949\pi$
$883$	1.51541e7	0.654077	0.327039	+	0.945011i	$0.393949\pi$
$884$	0	0
$885$	−4.60161e7	−1.97493
$886$	0	0
$887$	2.42062e7	1.03304	0.516520	−	0.856275i	$-0.327227\pi$
$887$	2.42062e7	1.03304	0.516520	+	0.856275i	$0.327227\pi$
$888$	0	0
$889$	7.19240e6	0.305225
$890$	0	0
$891$	−4.99607e7	−2.10831
$892$	0	0
$893$	2.86265e6	0.120127
$894$	0	0
$895$	5.13320e6	0.214205
$896$	0	0
$897$	2.00157e7	0.830594
$898$	0	0
$899$	2.68211e7	1.10682
$900$	0	0
$901$	8.54840e6	0.350811
$902$	0	0
$903$	3.27243e6	0.133552
$904$	0	0
$905$	−8.01696e6	−0.325378
$906$	0	0
$907$	3.38285e7	1.36541	0.682706	−	0.730693i	$-0.260803\pi$
$907$	3.38285e7	1.36541	0.682706	+	0.730693i	$0.260803\pi$
$908$	0	0
$909$	6.60064e6	0.264958
$910$	0	0
$911$	3.51743e7	1.40420	0.702101	−	0.712077i	$-0.252245\pi$
$911$	3.51743e7	1.40420	0.702101	+	0.712077i	$0.252245\pi$
$912$	0	0
$913$	−2.72281e7	−1.08104
$914$	0	0
$915$	3.50287e7	1.38315
$916$	0	0
$917$	5.56612e6	0.218590
$918$	0	0
$919$	−7.15839e6	−0.279593	−0.139797	−	0.990180i	$-0.544645\pi$
$919$	−7.15839e6	−0.279593	−0.139797	+	0.990180i	$0.544645\pi$
$920$	0	0
$921$	3.94021e7	1.53063
$922$	0	0
$923$	8.09172e6	0.312634
$924$	0	0
$925$	−2.64539e7	−1.01656
$926$	0	0
$927$	3.42489e6	0.130902
$928$	0	0
$929$	−4.60865e7	−1.75200	−0.876000	−	0.482311i	$-0.839798\pi$
$929$	−4.60865e7	−1.75200	−0.876000	+	0.482311i	$0.839798\pi$
$930$	0	0
$931$	5.54154e6	0.209535
$932$	0	0
$933$	−4.79484e7	−1.80331
$934$	0	0
$935$	−6.51553e7	−2.43737
$936$	0	0
$937$	2.18537e7	0.813160	0.406580	−	0.913615i	$-0.366721\pi$
$937$	2.18537e7	0.813160	0.406580	+	0.913615i	$0.366721\pi$
$938$	0	0
$939$	−1.77698e7	−0.657686
$940$	0	0
$941$	−1.41230e7	−0.519941	−0.259970	−	0.965617i	$-0.583713\pi$
$941$	−1.41230e7	−0.519941	−0.259970	+	0.965617i	$0.583713\pi$
$942$	0	0
$943$	3.44525e7	1.26166
$944$	0	0
$945$	−1.90994e7	−0.695731
$946$	0	0
$947$	1.31237e6	0.0475535	0.0237768	−	0.999717i	$-0.492431\pi$
$947$	1.31237e6	0.0475535	0.0237768	+	0.999717i	$0.492431\pi$
$948$	0	0
$949$	1.55730e7	0.561316
$950$	0	0
$951$	−5.30836e6	−0.190331
$952$	0	0
$953$	2.64202e7	0.942333	0.471167	−	0.882044i	$-0.343833\pi$
$953$	2.64202e7	0.942333	0.471167	+	0.882044i	$0.343833\pi$
$954$	0	0
$955$	4.48264e7	1.59047
$956$	0	0
$957$	−8.87675e7	−3.13310
$958$	0	0
$959$	−2.28407e6	−0.0801979
$960$	0	0
$961$	−1.42504e7	−0.497760
$962$	0	0
$963$	4.52453e6	0.157220
$964$	0	0
$965$	5.23579e7	1.80994
$966$	0	0
$967$	−1.50682e7	−0.518199	−0.259099	−	0.965851i	$-0.583426\pi$
$967$	−1.50682e7	−0.518199	−0.259099	+	0.965851i	$0.583426\pi$
$968$	0	0
$969$	1.78855e7	0.611914
$970$	0	0
$971$	5.19475e7	1.76814	0.884070	−	0.467354i	$-0.154793\pi$
$971$	5.19475e7	1.76814	0.884070	+	0.467354i	$0.154793\pi$
$972$	0	0
$973$	−2.26725e7	−0.767747
$974$	0	0
$975$	−1.31877e7	−0.444282
$976$	0	0
$977$	−3.80206e7	−1.27433	−0.637166	−	0.770727i	$-0.719893\pi$
$977$	−3.80206e7	−1.27433	−0.637166	+	0.770727i	$0.719893\pi$
$978$	0	0
$979$	−2.10728e7	−0.702693
$980$	0	0
$981$	−7.09654e6	−0.235437
$982$	0	0
$983$	−4.43240e7	−1.46304	−0.731518	−	0.681822i	$-0.761188\pi$
$983$	−4.43240e7	−1.46304	−0.731518	+	0.681822i	$0.761188\pi$
$984$	0	0
$985$	−1.97633e7	−0.649038
$986$	0	0
$987$	6.85563e6	0.224003
$988$	0	0
$989$	5.85366e6	0.190299
$990$	0	0
$991$	−1.06776e7	−0.345373	−0.172687	−	0.984977i	$-0.555245\pi$
$991$	−1.06776e7	−0.345373	−0.172687	+	0.984977i	$0.555245\pi$
$992$	0	0
$993$	5.03457e7	1.62028
$994$	0	0
$995$	−2.26826e7	−0.726333
$996$	0	0
$997$	−2.28612e7	−0.728386	−0.364193	−	0.931324i	$-0.618655\pi$
$997$	−2.28612e7	−0.728386	−0.364193	+	0.931324i	$0.618655\pi$
$998$	0	0
$999$	−3.47796e7	−1.10258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.3		10
4.3	odd	2		43.6.a.b.1.10	✓	10
12.11	even	2		387.6.a.e.1.1		10
20.19	odd	2		1075.6.a.b.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.10	✓	10	4.3	odd	2
387.6.a.e.1.1		10	12.11	even	2
688.6.a.h.1.3		10	1.1	even	1	trivial
1075.6.a.b.1.1		10	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-18.3440 q^{3} -72.1865 q^{5} +96.4803 q^{7} +93.5034 q^{9} +O(q^{10})\)	\(q-18.3440 q^{3} -72.1865 q^{5} +96.4803 q^{7} +93.5034 q^{9} +684.136 q^{11} +344.655 q^{13} +1324.19 q^{15} +1319.32 q^{17} -739.016 q^{19} -1769.84 q^{21} -3165.85 q^{23} +2085.89 q^{25} +2742.37 q^{27} +7073.21 q^{29} +3791.93 q^{31} -12549.8 q^{33} -6964.58 q^{35} -12682.3 q^{37} -6322.37 q^{39} -10882.5 q^{41} -1849.00 q^{43} -6749.68 q^{45} -3873.59 q^{47} -7498.54 q^{49} -24201.7 q^{51} +6479.38 q^{53} -49385.4 q^{55} +13556.5 q^{57} -34750.3 q^{59} +26452.9 q^{61} +9021.24 q^{63} -24879.4 q^{65} +58007.3 q^{67} +58074.5 q^{69} +23477.7 q^{71} +45184.4 q^{73} -38263.6 q^{75} +66005.7 q^{77} -17895.9 q^{79} -73027.4 q^{81} -39799.2 q^{83} -95237.3 q^{85} -129751. q^{87} -30802.0 q^{89} +33252.5 q^{91} -69559.2 q^{93} +53346.9 q^{95} +22524.5 q^{97} +63969.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * q^99

Introduction

L-functions

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