LMFDB - Embedded newform 1028.6.a.a.1.19

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:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-9.24112 q^{3} +59.2564 q^{5} -65.2931 q^{7} -157.602 q^{9} +O(q^{10})$	$q-9.24112 q^{3} +59.2564 q^{5} -65.2931 q^{7} -157.602 q^{9} -166.876 q^{11} -746.939 q^{13} -547.595 q^{15} +831.242 q^{17} +1047.69 q^{19} +603.381 q^{21} +1799.47 q^{23} +386.317 q^{25} +3702.01 q^{27} +8551.07 q^{29} -1558.64 q^{31} +1542.12 q^{33} -3869.03 q^{35} +792.972 q^{37} +6902.55 q^{39} +2676.33 q^{41} +16136.6 q^{43} -9338.91 q^{45} -29137.2 q^{47} -12543.8 q^{49} -7681.61 q^{51} -16829.7 q^{53} -9888.47 q^{55} -9681.85 q^{57} +45750.4 q^{59} +24718.0 q^{61} +10290.3 q^{63} -44260.9 q^{65} -21258.3 q^{67} -16629.1 q^{69} -4479.06 q^{71} -51199.4 q^{73} -3570.00 q^{75} +10895.9 q^{77} -34798.8 q^{79} +4086.53 q^{81} -76475.1 q^{83} +49256.4 q^{85} -79021.4 q^{87} +89791.4 q^{89} +48769.9 q^{91} +14403.6 q^{93} +62082.5 q^{95} +7163.99 q^{97} +26300.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * 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
$3$	−9.24112	−0.592818	−0.296409	−	0.955061i	$-0.595789\pi$
$3$	−9.24112	−0.592818	−0.296409	+	0.955061i	$0.595789\pi$
$4$	0	0
$5$	59.2564	1.06001	0.530005	−	0.847994i	$-0.322190\pi$
$5$	59.2564	1.06001	0.530005	+	0.847994i	$0.322190\pi$
$6$	0	0
$7$	−65.2931	−0.503642	−0.251821	−	0.967774i	$-0.581029\pi$
$7$	−65.2931	−0.503642	−0.251821	+	0.967774i	$0.581029\pi$
$8$	0	0
$9$	−157.602	−0.648567
$10$	0	0
$11$	−166.876	−0.415827	−0.207913	−	0.978147i	$-0.566667\pi$
$11$	−166.876	−0.415827	−0.207913	+	0.978147i	$0.566667\pi$
$12$	0	0
$13$	−746.939	−1.22582	−0.612910	−	0.790153i	$-0.710001\pi$
$13$	−746.939	−1.22582	−0.612910	+	0.790153i	$0.710001\pi$
$14$	0	0
$15$	−547.595	−0.628393
$16$	0	0
$17$	831.242	0.697598	0.348799	−	0.937198i	$-0.386590\pi$
$17$	831.242	0.697598	0.348799	+	0.937198i	$0.386590\pi$
$18$	0	0
$19$	1047.69	0.665809	0.332905	−	0.942960i	$-0.391971\pi$
$19$	1047.69	0.665809	0.332905	+	0.942960i	$0.391971\pi$
$20$	0	0
$21$	603.381	0.298568
$22$	0	0
$23$	1799.47	0.709292	0.354646	−	0.935001i	$-0.384601\pi$
$23$	1799.47	0.709292	0.354646	+	0.935001i	$0.384601\pi$
$24$	0	0
$25$	386.317	0.123621
$26$	0	0
$27$	3702.01	0.977300
$28$	0	0
$29$	8551.07	1.88810	0.944051	−	0.329800i	$-0.106981\pi$
$29$	8551.07	1.88810	0.944051	+	0.329800i	$0.106981\pi$
$30$	0	0
$31$	−1558.64	−0.291302	−0.145651	−	0.989336i	$-0.546528\pi$
$31$	−1558.64	−0.291302	−0.145651	+	0.989336i	$0.546528\pi$
$32$	0	0
$33$	1542.12	0.246510
$34$	0	0
$35$	−3869.03	−0.533866
$36$	0	0
$37$	792.972	0.0952256	0.0476128	−	0.998866i	$-0.484839\pi$
$37$	792.972	0.0952256	0.0476128	+	0.998866i	$0.484839\pi$
$38$	0	0
$39$	6902.55	0.726688
$40$	0	0
$41$	2676.33	0.248645	0.124323	−	0.992242i	$-0.460324\pi$
$41$	2676.33	0.248645	0.124323	+	0.992242i	$0.460324\pi$
$42$	0	0
$43$	16136.6	1.33088	0.665442	−	0.746450i	$-0.268243\pi$
$43$	16136.6	1.33088	0.665442	+	0.746450i	$0.268243\pi$
$44$	0	0
$45$	−9338.91	−0.687487
$46$	0	0
$47$	−29137.2	−1.92399	−0.961995	−	0.273067i	$-0.911962\pi$
$47$	−29137.2	−1.92399	−0.961995	+	0.273067i	$0.911962\pi$
$48$	0	0
$49$	−12543.8	−0.746345
$50$	0	0
$51$	−7681.61	−0.413549
$52$	0	0
$53$	−16829.7	−0.822974	−0.411487	−	0.911416i	$-0.634991\pi$
$53$	−16829.7	−0.822974	−0.411487	+	0.911416i	$0.634991\pi$
$54$	0	0
$55$	−9888.47	−0.440781
$56$	0	0
$57$	−9681.85	−0.394704
$58$	0	0
$59$	45750.4	1.71106	0.855529	−	0.517754i	$-0.173232\pi$
$59$	45750.4	1.71106	0.855529	+	0.517754i	$0.173232\pi$
$60$	0	0
$61$	24718.0	0.850528	0.425264	−	0.905069i	$-0.360181\pi$
$61$	24718.0	0.850528	0.425264	+	0.905069i	$0.360181\pi$
$62$	0	0
$63$	10290.3	0.326645
$64$	0	0
$65$	−44260.9	−1.29938
$66$	0	0
$67$	−21258.3	−0.578550	−0.289275	−	0.957246i	$-0.593414\pi$
$67$	−21258.3	−0.578550	−0.289275	+	0.957246i	$0.593414\pi$
$68$	0	0
$69$	−16629.1	−0.420481
$70$	0	0
$71$	−4479.06	−0.105449	−0.0527244	−	0.998609i	$-0.516790\pi$
$71$	−4479.06	−0.105449	−0.0527244	+	0.998609i	$0.516790\pi$
$72$	0	0
$73$	−51199.4	−1.12450	−0.562248	−	0.826969i	$-0.690063\pi$
$73$	−51199.4	−1.12450	−0.562248	+	0.826969i	$0.690063\pi$
$74$	0	0
$75$	−3570.00	−0.0732850
$76$	0	0
$77$	10895.9	0.209428
$78$	0	0
$79$	−34798.8	−0.627330	−0.313665	−	0.949534i	$-0.601557\pi$
$79$	−34798.8	−0.627330	−0.313665	+	0.949534i	$0.601557\pi$
$80$	0	0
$81$	4086.53	0.0692057
$82$	0	0
$83$	−76475.1	−1.21850	−0.609249	−	0.792979i	$-0.708529\pi$
$83$	−76475.1	−1.21850	−0.609249	+	0.792979i	$0.708529\pi$
$84$	0	0
$85$	49256.4	0.739461
$86$	0	0
$87$	−79021.4	−1.11930
$88$	0	0
$89$	89791.4	1.20160	0.600800	−	0.799399i	$-0.294849\pi$
$89$	89791.4	1.20160	0.600800	+	0.799399i	$0.294849\pi$
$90$	0	0
$91$	48769.9	0.617375
$92$	0	0
$93$	14403.6	0.172689
$94$	0	0
$95$	62082.5	0.705765
$96$	0	0
$97$	7163.99	0.0773082	0.0386541	−	0.999253i	$-0.487693\pi$
$97$	7163.99	0.0773082	0.0386541	+	0.999253i	$0.487693\pi$
$98$	0	0
$99$	26300.0	0.269691
$100$	0	0
$101$	80667.9	0.786859	0.393430	−	0.919355i	$-0.371288\pi$
$101$	80667.9	0.786859	0.393430	+	0.919355i	$0.371288\pi$
$102$	0	0
$103$	−24108.5	−0.223911	−0.111956	−	0.993713i	$-0.535711\pi$
$103$	−24108.5	−0.223911	−0.111956	+	0.993713i	$0.535711\pi$
$104$	0	0
$105$	35754.2	0.316485
$106$	0	0
$107$	−43662.1	−0.368677	−0.184338	−	0.982863i	$-0.559014\pi$
$107$	−43662.1	−0.368677	−0.184338	+	0.982863i	$0.559014\pi$
$108$	0	0
$109$	87565.1	0.705935	0.352967	−	0.935636i	$-0.385173\pi$
$109$	87565.1	0.705935	0.352967	+	0.935636i	$0.385173\pi$
$110$	0	0
$111$	−7327.95	−0.0564515
$112$	0	0
$113$	−191117.	−1.40800	−0.704001	−	0.710199i	$-0.748605\pi$
$113$	−191117.	−1.40800	−0.704001	+	0.710199i	$0.748605\pi$
$114$	0	0
$115$	106630.	0.751857
$116$	0	0
$117$	117719.	0.795026
$118$	0	0
$119$	−54274.4	−0.351340
$120$	0	0
$121$	−133203.	−0.827088
$122$	0	0
$123$	−24732.3	−0.147401
$124$	0	0
$125$	−162284.	−0.928970
$126$	0	0
$127$	232810.	1.28083	0.640417	−	0.768027i	$-0.278762\pi$
$127$	232810.	1.28083	0.640417	+	0.768027i	$0.278762\pi$
$128$	0	0
$129$	−149120.	−0.788972
$130$	0	0
$131$	−14746.4	−0.0750771	−0.0375385	−	0.999295i	$-0.511952\pi$
$131$	−14746.4	−0.0750771	−0.0375385	+	0.999295i	$0.511952\pi$
$132$	0	0
$133$	−68407.1	−0.335330
$134$	0	0
$135$	219368.	1.03595
$136$	0	0
$137$	−167967.	−0.764581	−0.382290	−	0.924042i	$-0.624865\pi$
$137$	−167967.	−0.764581	−0.382290	+	0.924042i	$0.624865\pi$
$138$	0	0
$139$	39565.2	0.173690	0.0868452	−	0.996222i	$-0.472321\pi$
$139$	39565.2	0.173690	0.0868452	+	0.996222i	$0.472321\pi$
$140$	0	0
$141$	269260.	1.14058
$142$	0	0
$143$	124646.	0.509729
$144$	0	0
$145$	506705.	2.00141
$146$	0	0
$147$	115919.	0.442447
$148$	0	0
$149$	−88058.7	−0.324942	−0.162471	−	0.986713i	$-0.551946\pi$
$149$	−88058.7	−0.324942	−0.162471	+	0.986713i	$0.551946\pi$
$150$	0	0
$151$	−175784.	−0.627388	−0.313694	−	0.949524i	$-0.601567\pi$
$151$	−175784.	−0.627388	−0.313694	+	0.949524i	$0.601567\pi$
$152$	0	0
$153$	−131005.	−0.452439
$154$	0	0
$155$	−92359.6	−0.308783
$156$	0	0
$157$	−13165.6	−0.0426275	−0.0213138	−	0.999773i	$-0.506785\pi$
$157$	−13165.6	−0.0426275	−0.0213138	+	0.999773i	$0.506785\pi$
$158$	0	0
$159$	155525.	0.487874
$160$	0	0
$161$	−117493.	−0.357229
$162$	0	0
$163$	91069.9	0.268476	0.134238	−	0.990949i	$-0.457141\pi$
$163$	91069.9	0.268476	0.134238	+	0.990949i	$0.457141\pi$
$164$	0	0
$165$	91380.5	0.261303
$166$	0	0
$167$	−322019.	−0.893491	−0.446745	−	0.894661i	$-0.647417\pi$
$167$	−322019.	−0.893491	−0.446745	+	0.894661i	$0.647417\pi$
$168$	0	0
$169$	186625.	0.502636
$170$	0	0
$171$	−165118.	−0.431822
$172$	0	0
$173$	−669348.	−1.70034	−0.850172	−	0.526505i	$-0.823502\pi$
$173$	−669348.	−1.70034	−0.850172	+	0.526505i	$0.823502\pi$
$174$	0	0
$175$	−25223.8	−0.0622609
$176$	0	0
$177$	−422785.	−1.01435
$178$	0	0
$179$	217002.	0.506210	0.253105	−	0.967439i	$-0.418548\pi$
$179$	217002.	0.506210	0.253105	+	0.967439i	$0.418548\pi$
$180$	0	0
$181$	−321319.	−0.729021	−0.364510	−	0.931199i	$-0.618764\pi$
$181$	−321319.	−0.729021	−0.364510	+	0.931199i	$0.618764\pi$
$182$	0	0
$183$	−228422.	−0.504208
$184$	0	0
$185$	46988.7	0.100940
$186$	0	0
$187$	−138715.	−0.290080
$188$	0	0
$189$	−241715.	−0.492209
$190$	0	0
$191$	−80502.9	−0.159672	−0.0798359	−	0.996808i	$-0.525440\pi$
$191$	−80502.9	−0.159672	−0.0798359	+	0.996808i	$0.525440\pi$
$192$	0	0
$193$	−395447.	−0.764178	−0.382089	−	0.924125i	$-0.624795\pi$
$193$	−395447.	−0.764178	−0.382089	+	0.924125i	$0.624795\pi$
$194$	0	0
$195$	409020.	0.770297
$196$	0	0
$197$	652703.	1.19826	0.599128	−	0.800653i	$-0.295514\pi$
$197$	652703.	1.19826	0.599128	+	0.800653i	$0.295514\pi$
$198$	0	0
$199$	401162.	0.718103	0.359051	−	0.933318i	$-0.383100\pi$
$199$	401162.	0.718103	0.359051	+	0.933318i	$0.383100\pi$
$200$	0	0
$201$	196450.	0.342975
$202$	0	0
$203$	−558325.	−0.950927
$204$	0	0
$205$	158590.	0.263567
$206$	0	0
$207$	−283600.	−0.460023
$208$	0	0
$209$	−174835.	−0.276861
$210$	0	0
$211$	602604.	0.931807	0.465903	−	0.884836i	$-0.345729\pi$
$211$	602604.	0.931807	0.465903	+	0.884836i	$0.345729\pi$
$212$	0	0
$213$	41391.6	0.0625119
$214$	0	0
$215$	956194.	1.41075
$216$	0	0
$217$	101769.	0.146712
$218$	0	0
$219$	473140.	0.666621
$220$	0	0
$221$	−620887.	−0.855130
$222$	0	0
$223$	455756.	0.613720	0.306860	−	0.951755i	$-0.400722\pi$
$223$	455756.	0.613720	0.306860	+	0.951755i	$0.400722\pi$
$224$	0	0
$225$	−60884.2	−0.0801767
$226$	0	0
$227$	1.25214e6	1.61283	0.806417	−	0.591348i	$-0.201404\pi$
$227$	1.25214e6	1.61283	0.806417	+	0.591348i	$0.201404\pi$
$228$	0	0
$229$	−894688.	−1.12741	−0.563706	−	0.825975i	$-0.690625\pi$
$229$	−894688.	−1.12741	−0.563706	+	0.825975i	$0.690625\pi$
$230$	0	0
$231$	−100690.	−0.124153
$232$	0	0
$233$	−1.45924e6	−1.76091	−0.880457	−	0.474125i	$-0.842764\pi$
$233$	−1.45924e6	−1.76091	−0.880457	+	0.474125i	$0.842764\pi$
$234$	0	0
$235$	−1.72656e6	−2.03945
$236$	0	0
$237$	321580.	0.371893
$238$	0	0
$239$	−1.66469e6	−1.88512	−0.942560	−	0.334036i	$-0.891589\pi$
$239$	−1.66469e6	−1.88512	−0.942560	+	0.334036i	$0.891589\pi$
$240$	0	0
$241$	−787562.	−0.873458	−0.436729	−	0.899593i	$-0.643863\pi$
$241$	−787562.	−0.873458	−0.436729	+	0.899593i	$0.643863\pi$
$242$	0	0
$243$	−937352.	−1.01833
$244$	0	0
$245$	−743301.	−0.791133
$246$	0	0
$247$	−782563.	−0.816163
$248$	0	0
$249$	706715.	0.722348
$250$	0	0
$251$	−743902.	−0.745300	−0.372650	−	0.927972i	$-0.621551\pi$
$251$	−743902.	−0.745300	−0.372650	+	0.927972i	$0.621551\pi$
$252$	0	0
$253$	−300289.	−0.294943
$254$	0	0
$255$	−455184.	−0.438366
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−51775.6	−0.0479596
$260$	0	0
$261$	−1.34766e6	−1.22456
$262$	0	0
$263$	1.71767e6	1.53127	0.765635	−	0.643276i	$-0.222425\pi$
$263$	1.71767e6	1.53127	0.765635	+	0.643276i	$0.222425\pi$
$264$	0	0
$265$	−997266.	−0.872361
$266$	0	0
$267$	−829773.	−0.712330
$268$	0	0
$269$	−1.00441e6	−0.846313	−0.423156	−	0.906057i	$-0.639078\pi$
$269$	−1.00441e6	−0.846313	−0.423156	+	0.906057i	$0.639078\pi$
$270$	0	0
$271$	−1.98828e6	−1.64457	−0.822287	−	0.569072i	$-0.807302\pi$
$271$	−1.98828e6	−1.64457	−0.822287	+	0.569072i	$0.807302\pi$
$272$	0	0
$273$	−450689.	−0.365991
$274$	0	0
$275$	−64467.1	−0.0514051
$276$	0	0
$277$	729024.	0.570877	0.285438	−	0.958397i	$-0.407861\pi$
$277$	729024.	0.570877	0.285438	+	0.958397i	$0.407861\pi$
$278$	0	0
$279$	245645.	0.188929
$280$	0	0
$281$	−2.54408e6	−1.92205	−0.961024	−	0.276464i	$-0.910837\pi$
$281$	−2.54408e6	−1.92205	−0.961024	+	0.276464i	$0.910837\pi$
$282$	0	0
$283$	318367.	0.236299	0.118150	−	0.992996i	$-0.462304\pi$
$283$	318367.	0.236299	0.118150	+	0.992996i	$0.462304\pi$
$284$	0	0
$285$	−573711.	−0.418390
$286$	0	0
$287$	−174746.	−0.125228
$288$	0	0
$289$	−728893.	−0.513357
$290$	0	0
$291$	−66203.3	−0.0458297
$292$	0	0
$293$	−156990.	−0.106832	−0.0534162	−	0.998572i	$-0.517011\pi$
$293$	−156990.	−0.106832	−0.0534162	+	0.998572i	$0.517011\pi$
$294$	0	0
$295$	2.71100e6	1.81374
$296$	0	0
$297$	−617777.	−0.406388
$298$	0	0
$299$	−1.34410e6	−0.869465
$300$	0	0
$301$	−1.05361e6	−0.670289
$302$	0	0
$303$	−745461.	−0.466464
$304$	0	0
$305$	1.46470e6	0.901568
$306$	0	0
$307$	−2.52289e6	−1.52775	−0.763875	−	0.645364i	$-0.776706\pi$
$307$	−2.52289e6	−1.52775	−0.763875	+	0.645364i	$0.776706\pi$
$308$	0	0
$309$	222789.	0.132739
$310$	0	0
$311$	273216.	0.160179	0.0800893	−	0.996788i	$-0.474479\pi$
$311$	273216.	0.160179	0.0800893	+	0.996788i	$0.474479\pi$
$312$	0	0
$313$	853525.	0.492442	0.246221	−	0.969214i	$-0.420811\pi$
$313$	853525.	0.492442	0.246221	+	0.969214i	$0.420811\pi$
$314$	0	0
$315$	609766.	0.346247
$316$	0	0
$317$	−2.97414e6	−1.66231	−0.831157	−	0.556038i	$-0.812321\pi$
$317$	−2.97414e6	−1.66231	−0.831157	+	0.556038i	$0.812321\pi$
$318$	0	0
$319$	−1.42697e6	−0.785123
$320$	0	0
$321$	403487.	0.218558
$322$	0	0
$323$	870887.	0.464467
$324$	0	0
$325$	−288555.	−0.151538
$326$	0	0
$327$	−809199.	−0.418491
$328$	0	0
$329$	1.90246e6	0.969002
$330$	0	0
$331$	1.02834e6	0.515900	0.257950	−	0.966158i	$-0.416953\pi$
$331$	1.02834e6	0.515900	0.257950	+	0.966158i	$0.416953\pi$
$332$	0	0
$333$	−124974.	−0.0617602
$334$	0	0
$335$	−1.25969e6	−0.613269
$336$	0	0
$337$	−1.89694e6	−0.909868	−0.454934	−	0.890525i	$-0.650337\pi$
$337$	−1.89694e6	−0.909868	−0.454934	+	0.890525i	$0.650337\pi$
$338$	0	0
$339$	1.76613e6	0.834688
$340$	0	0
$341$	260101.	0.121131
$342$	0	0
$343$	1.91640e6	0.879532
$344$	0	0
$345$	−985381.	−0.445714
$346$	0	0
$347$	2.19513e6	0.978673	0.489336	−	0.872095i	$-0.337239\pi$
$347$	2.19513e6	0.978673	0.489336	+	0.872095i	$0.337239\pi$
$348$	0	0
$349$	1.28087e6	0.562913	0.281457	−	0.959574i	$-0.409182\pi$
$349$	1.28087e6	0.562913	0.281457	+	0.959574i	$0.409182\pi$
$350$	0	0
$351$	−2.76517e6	−1.19799
$352$	0	0
$353$	−4.24805e6	−1.81448	−0.907241	−	0.420611i	$-0.861816\pi$
$353$	−4.24805e6	−1.81448	−0.907241	+	0.420611i	$0.861816\pi$
$354$	0	0
$355$	−265413.	−0.111777
$356$	0	0
$357$	501556.	0.208281
$358$	0	0
$359$	1.83351e6	0.750839	0.375420	−	0.926855i	$-0.377499\pi$
$359$	1.83351e6	0.750839	0.375420	+	0.926855i	$0.377499\pi$
$360$	0	0
$361$	−1.37844e6	−0.556698
$362$	0	0
$363$	1.23095e6	0.490313
$364$	0	0
$365$	−3.03389e6	−1.19198
$366$	0	0
$367$	3.03425e6	1.17594	0.587971	−	0.808882i	$-0.299927\pi$
$367$	3.03425e6	1.17594	0.587971	+	0.808882i	$0.299927\pi$
$368$	0	0
$369$	−421795.	−0.161263
$370$	0	0
$371$	1.09886e6	0.414484
$372$	0	0
$373$	−556599.	−0.207143	−0.103571	−	0.994622i	$-0.533027\pi$
$373$	−556599.	−0.207143	−0.103571	+	0.994622i	$0.533027\pi$
$374$	0	0
$375$	1.49969e6	0.550710
$376$	0	0
$377$	−6.38713e6	−2.31447
$378$	0	0
$379$	1.44637e6	0.517229	0.258614	−	0.965981i	$-0.416734\pi$
$379$	1.44637e6	0.517229	0.258614	+	0.965981i	$0.416734\pi$
$380$	0	0
$381$	−2.15143e6	−0.759301
$382$	0	0
$383$	2.34864e6	0.818125	0.409063	−	0.912506i	$-0.365856\pi$
$383$	2.34864e6	0.818125	0.409063	+	0.912506i	$0.365856\pi$
$384$	0	0
$385$	645649.	0.221996
$386$	0	0
$387$	−2.54315e6	−0.863167
$388$	0	0
$389$	1.14081e6	0.382241	0.191121	−	0.981567i	$-0.438788\pi$
$389$	1.14081e6	0.382241	0.191121	+	0.981567i	$0.438788\pi$
$390$	0	0
$391$	1.49580e6	0.494801
$392$	0	0
$393$	136273.	0.0445070
$394$	0	0
$395$	−2.06205e6	−0.664976
$396$	0	0
$397$	4.47920e6	1.42634	0.713172	−	0.700989i	$-0.247258\pi$
$397$	4.47920e6	1.42634	0.713172	+	0.700989i	$0.247258\pi$
$398$	0	0
$399$	632158.	0.198789
$400$	0	0
$401$	4.25030e6	1.31995	0.659977	−	0.751286i	$-0.270566\pi$
$401$	4.25030e6	1.31995	0.659977	+	0.751286i	$0.270566\pi$
$402$	0	0
$403$	1.16421e6	0.357084
$404$	0	0
$405$	242153.	0.0733588
$406$	0	0
$407$	−132328.	−0.0395974
$408$	0	0
$409$	2.43070e6	0.718495	0.359247	−	0.933242i	$-0.383033\pi$
$409$	2.43070e6	0.718495	0.359247	+	0.933242i	$0.383033\pi$
$410$	0	0
$411$	1.55221e6	0.453257
$412$	0	0
$413$	−2.98718e6	−0.861761
$414$	0	0
$415$	−4.53164e6	−1.29162
$416$	0	0
$417$	−365626.	−0.102967
$418$	0	0
$419$	3.08140e6	0.857459	0.428730	−	0.903433i	$-0.358961\pi$
$419$	3.08140e6	0.857459	0.428730	+	0.903433i	$0.358961\pi$
$420$	0	0
$421$	1.08132e6	0.297338	0.148669	−	0.988887i	$-0.452501\pi$
$421$	1.08132e6	0.297338	0.148669	+	0.988887i	$0.452501\pi$
$422$	0	0
$423$	4.59207e6	1.24784
$424$	0	0
$425$	321123.	0.0862381
$426$	0	0
$427$	−1.61391e6	−0.428362
$428$	0	0
$429$	−1.15187e6	−0.302177
$430$	0	0
$431$	−4.66136e6	−1.20870	−0.604351	−	0.796718i	$-0.706567\pi$
$431$	−4.66136e6	−1.20870	−0.604351	+	0.796718i	$0.706567\pi$
$432$	0	0
$433$	551696.	0.141410	0.0707051	−	0.997497i	$-0.477475\pi$
$433$	551696.	0.141410	0.0707051	+	0.997497i	$0.477475\pi$
$434$	0	0
$435$	−4.68252e6	−1.18647
$436$	0	0
$437$	1.88529e6	0.472253
$438$	0	0
$439$	−143480.	−0.0355329	−0.0177664	−	0.999842i	$-0.505656\pi$
$439$	−143480.	−0.0355329	−0.0177664	+	0.999842i	$0.505656\pi$
$440$	0	0
$441$	1.97693e6	0.484054
$442$	0	0
$443$	−4.86089e6	−1.17681	−0.588405	−	0.808566i	$-0.700244\pi$
$443$	−4.86089e6	−1.17681	−0.588405	+	0.808566i	$0.700244\pi$
$444$	0	0
$445$	5.32071e6	1.27371
$446$	0	0
$447$	813760.	0.192632
$448$	0	0
$449$	−5.03991e6	−1.17980	−0.589898	−	0.807478i	$-0.700832\pi$
$449$	−5.03991e6	−1.17980	−0.589898	+	0.807478i	$0.700832\pi$
$450$	0	0
$451$	−446616.	−0.103393
$452$	0	0
$453$	1.62444e6	0.371927
$454$	0	0
$455$	2.88993e6	0.654423
$456$	0	0
$457$	−3.03826e6	−0.680509	−0.340255	−	0.940333i	$-0.610513\pi$
$457$	−3.03826e6	−0.680509	−0.340255	+	0.940333i	$0.610513\pi$
$458$	0	0
$459$	3.07727e6	0.681763
$460$	0	0
$461$	−3.21786e6	−0.705203	−0.352602	−	0.935773i	$-0.614703\pi$
$461$	−3.21786e6	−0.705203	−0.352602	+	0.935773i	$0.614703\pi$
$462$	0	0
$463$	−1.02701e6	−0.222649	−0.111325	−	0.993784i	$-0.535509\pi$
$463$	−1.02701e6	−0.222649	−0.111325	+	0.993784i	$0.535509\pi$
$464$	0	0
$465$	853506.	0.183052
$466$	0	0
$467$	−2.63845e6	−0.559831	−0.279916	−	0.960025i	$-0.590306\pi$
$467$	−2.63845e6	−0.559831	−0.279916	+	0.960025i	$0.590306\pi$
$468$	0	0
$469$	1.38802e6	0.291382
$470$	0	0
$471$	121664.	0.0252704
$472$	0	0
$473$	−2.69281e6	−0.553417
$474$	0	0
$475$	404741.	0.0823083
$476$	0	0
$477$	2.65239e6	0.533754
$478$	0	0
$479$	−85397.7	−0.0170062	−0.00850310	−	0.999964i	$-0.502707\pi$
$479$	−85397.7	−0.0170062	−0.00850310	+	0.999964i	$0.502707\pi$
$480$	0	0
$481$	−592302.	−0.116730
$482$	0	0
$483$	1.08577e6	0.211772
$484$	0	0
$485$	424512.	0.0819475
$486$	0	0
$487$	−6.57806e6	−1.25683	−0.628414	−	0.777879i	$-0.716295\pi$
$487$	−6.57806e6	−1.25683	−0.628414	+	0.777879i	$0.716295\pi$
$488$	0	0
$489$	−841588.	−0.159158
$490$	0	0
$491$	−5.69650e6	−1.06636	−0.533180	−	0.846002i	$-0.679003\pi$
$491$	−5.69650e6	−1.06636	−0.533180	+	0.846002i	$0.679003\pi$
$492$	0	0
$493$	7.10801e6	1.31714
$494$	0	0
$495$	1.55844e6	0.285876
$496$	0	0
$497$	292452.	0.0531084
$498$	0	0
$499$	−2.50395e6	−0.450167	−0.225083	−	0.974339i	$-0.572265\pi$
$499$	−2.50395e6	−0.450167	−0.225083	+	0.974339i	$0.572265\pi$
$500$	0	0
$501$	2.97581e6	0.529678
$502$	0	0
$503$	7.13115e6	1.25672	0.628361	−	0.777922i	$-0.283726\pi$
$503$	7.13115e6	1.25672	0.628361	+	0.777922i	$0.283726\pi$
$504$	0	0
$505$	4.78008e6	0.834079
$506$	0	0
$507$	−1.72462e6	−0.297971
$508$	0	0
$509$	−4.78458e6	−0.818558	−0.409279	−	0.912409i	$-0.634220\pi$
$509$	−4.78458e6	−0.818558	−0.409279	+	0.912409i	$0.634220\pi$
$510$	0	0
$511$	3.34296e6	0.566343
$512$	0	0
$513$	3.87857e6	0.650696
$514$	0	0
$515$	−1.42858e6	−0.237348
$516$	0	0
$517$	4.86230e6	0.800047
$518$	0	0
$519$	6.18553e6	1.00799
$520$	0	0
$521$	−6.45156e6	−1.04129	−0.520644	−	0.853774i	$-0.674308\pi$
$521$	−6.45156e6	−1.04129	−0.520644	+	0.853774i	$0.674308\pi$
$522$	0	0
$523$	−2.55623e6	−0.408645	−0.204322	−	0.978904i	$-0.565499\pi$
$523$	−2.55623e6	−0.408645	−0.204322	+	0.978904i	$0.565499\pi$
$524$	0	0
$525$	233096.	0.0369094
$526$	0	0
$527$	−1.29561e6	−0.203212
$528$	0	0
$529$	−3.19825e6	−0.496904
$530$	0	0
$531$	−7.21034e6	−1.10974
$532$	0	0
$533$	−1.99906e6	−0.304795
$534$	0	0
$535$	−2.58726e6	−0.390801
$536$	0	0
$537$	−2.00534e6	−0.300090
$538$	0	0
$539$	2.09326e6	0.310350
$540$	0	0
$541$	−3.47789e6	−0.510885	−0.255442	−	0.966824i	$-0.582221\pi$
$541$	−3.47789e6	−0.510885	−0.255442	+	0.966824i	$0.582221\pi$
$542$	0	0
$543$	2.96935e6	0.432177
$544$	0	0
$545$	5.18879e6	0.748298
$546$	0	0
$547$	−4.50428e6	−0.643661	−0.321831	−	0.946797i	$-0.604298\pi$
$547$	−4.50428e6	−0.643661	−0.321831	+	0.946797i	$0.604298\pi$
$548$	0	0
$549$	−3.89560e6	−0.551624
$550$	0	0
$551$	8.95889e6	1.25712
$552$	0	0
$553$	2.27212e6	0.315950
$554$	0	0
$555$	−434228.	−0.0598391
$556$	0	0
$557$	−3.03718e6	−0.414794	−0.207397	−	0.978257i	$-0.566499\pi$
$557$	−3.03718e6	−0.414794	−0.207397	+	0.978257i	$0.566499\pi$
$558$	0	0
$559$	−1.20530e7	−1.63142
$560$	0	0
$561$	1.28188e6	0.171965
$562$	0	0
$563$	8.64541e6	1.14951	0.574757	−	0.818324i	$-0.305096\pi$
$563$	8.64541e6	1.14951	0.574757	+	0.818324i	$0.305096\pi$
$564$	0	0
$565$	−1.13249e7	−1.49250
$566$	0	0
$567$	−266822.	−0.0348549
$568$	0	0
$569$	6.50893e6	0.842809	0.421404	−	0.906873i	$-0.361537\pi$
$569$	6.50893e6	0.842809	0.421404	+	0.906873i	$0.361537\pi$
$570$	0	0
$571$	4.53579e6	0.582187	0.291093	−	0.956695i	$-0.405981\pi$
$571$	4.53579e6	0.582187	0.291093	+	0.956695i	$0.405981\pi$
$572$	0	0
$573$	743937.	0.0946563
$574$	0	0
$575$	695166.	0.0876837
$576$	0	0
$577$	5.67052e6	0.709060	0.354530	−	0.935045i	$-0.384641\pi$
$577$	5.67052e6	0.709060	0.354530	+	0.935045i	$0.384641\pi$
$578$	0	0
$579$	3.65437e6	0.453019
$580$	0	0
$581$	4.99329e6	0.613687
$582$	0	0
$583$	2.80847e6	0.342215
$584$	0	0
$585$	6.97559e6	0.842736
$586$	0	0
$587$	−1.11759e7	−1.33871	−0.669354	−	0.742943i	$-0.733429\pi$
$587$	−1.11759e7	−1.33871	−0.669354	+	0.742943i	$0.733429\pi$
$588$	0	0
$589$	−1.63298e6	−0.193951
$590$	0	0
$591$	−6.03170e6	−0.710348
$592$	0	0
$593$	−946380.	−0.110517	−0.0552585	−	0.998472i	$-0.517598\pi$
$593$	−946380.	−0.110517	−0.0552585	+	0.998472i	$0.517598\pi$
$594$	0	0
$595$	−3.21610e6	−0.372424
$596$	0	0
$597$	−3.70718e6	−0.425704
$598$	0	0
$599$	−5.84615e6	−0.665738	−0.332869	−	0.942973i	$-0.608017\pi$
$599$	−5.84615e6	−0.665738	−0.332869	+	0.942973i	$0.608017\pi$
$600$	0	0
$601$	3.98045e6	0.449517	0.224758	−	0.974415i	$-0.427841\pi$
$601$	3.98045e6	0.449517	0.224758	+	0.974415i	$0.427841\pi$
$602$	0	0
$603$	3.35034e6	0.375228
$604$	0	0
$605$	−7.89315e6	−0.876722
$606$	0	0
$607$	−1.08032e7	−1.19009	−0.595045	−	0.803692i	$-0.702866\pi$
$607$	−1.08032e7	−1.19009	−0.595045	+	0.803692i	$0.702866\pi$
$608$	0	0
$609$	5.15955e6	0.563727
$610$	0	0
$611$	2.17637e7	2.35847
$612$	0	0
$613$	−7.86974e6	−0.845881	−0.422941	−	0.906157i	$-0.639002\pi$
$613$	−7.86974e6	−0.845881	−0.422941	+	0.906157i	$0.639002\pi$
$614$	0	0
$615$	−1.46555e6	−0.156247
$616$	0	0
$617$	−4.25477e6	−0.449948	−0.224974	−	0.974365i	$-0.572230\pi$
$617$	−4.25477e6	−0.449948	−0.224974	+	0.974365i	$0.572230\pi$
$618$	0	0
$619$	1.35714e7	1.42363	0.711815	−	0.702367i	$-0.247873\pi$
$619$	1.35714e7	1.42363	0.711815	+	0.702367i	$0.247873\pi$
$620$	0	0
$621$	6.66166e6	0.693191
$622$	0	0
$623$	−5.86276e6	−0.605176
$624$	0	0
$625$	−1.08236e7	−1.10834
$626$	0	0
$627$	1.61567e6	0.164128
$628$	0	0
$629$	659152.	0.0664292
$630$	0	0
$631$	785045.	0.0784912	0.0392456	−	0.999230i	$-0.487505\pi$
$631$	785045.	0.0784912	0.0392456	+	0.999230i	$0.487505\pi$
$632$	0	0
$633$	−5.56873e6	−0.552392
$634$	0	0
$635$	1.37955e7	1.35770
$636$	0	0
$637$	9.36947e6	0.914885
$638$	0	0
$639$	705908.	0.0683906
$640$	0	0
$641$	−7.33126e6	−0.704747	−0.352374	−	0.935859i	$-0.614625\pi$
$641$	−7.33126e6	−0.704747	−0.352374	+	0.935859i	$0.614625\pi$
$642$	0	0
$643$	7.36241e6	0.702252	0.351126	−	0.936328i	$-0.385799\pi$
$643$	7.36241e6	0.702252	0.351126	+	0.936328i	$0.385799\pi$
$644$	0	0
$645$	−8.83630e6	−0.836318
$646$	0	0
$647$	−7.03128e6	−0.660349	−0.330175	−	0.943920i	$-0.607108\pi$
$647$	−7.03128e6	−0.660349	−0.330175	+	0.943920i	$0.607108\pi$
$648$	0	0
$649$	−7.63465e6	−0.711504
$650$	0	0
$651$	−940456.	−0.0869733
$652$	0	0
$653$	−1.05259e7	−0.965994	−0.482997	−	0.875622i	$-0.660452\pi$
$653$	−1.05259e7	−0.965994	−0.482997	+	0.875622i	$0.660452\pi$
$654$	0	0
$655$	−873817.	−0.0795824
$656$	0	0
$657$	8.06911e6	0.729310
$658$	0	0
$659$	−1.68740e7	−1.51358	−0.756790	−	0.653658i	$-0.773234\pi$
$659$	−1.68740e7	−1.51358	−0.756790	+	0.653658i	$0.773234\pi$
$660$	0	0
$661$	−1.75057e7	−1.55839	−0.779197	−	0.626780i	$-0.784373\pi$
$661$	−1.75057e7	−1.55839	−0.779197	+	0.626780i	$0.784373\pi$
$662$	0	0
$663$	5.73769e6	0.506937
$664$	0	0
$665$	−4.05355e6	−0.355453
$666$	0	0
$667$	1.53874e7	1.33922
$668$	0	0
$669$	−4.21170e6	−0.363824
$670$	0	0
$671$	−4.12484e6	−0.353672
$672$	0	0
$673$	7.37204e6	0.627408	0.313704	−	0.949521i	$-0.398430\pi$
$673$	7.37204e6	0.627408	0.313704	+	0.949521i	$0.398430\pi$
$674$	0	0
$675$	1.43015e6	0.120815
$676$	0	0
$677$	2.93161e6	0.245830	0.122915	−	0.992417i	$-0.460776\pi$
$677$	2.93161e6	0.245830	0.122915	+	0.992417i	$0.460776\pi$
$678$	0	0
$679$	−467759.	−0.0389357
$680$	0	0
$681$	−1.15712e7	−0.956117
$682$	0	0
$683$	3.94617e6	0.323686	0.161843	−	0.986817i	$-0.448256\pi$
$683$	3.94617e6	0.323686	0.161843	+	0.986817i	$0.448256\pi$
$684$	0	0
$685$	−9.95313e6	−0.810463
$686$	0	0
$687$	8.26792e6	0.668350
$688$	0	0
$689$	1.25708e7	1.00882
$690$	0	0
$691$	7.90403e6	0.629729	0.314865	−	0.949137i	$-0.398041\pi$
$691$	7.90403e6	0.629729	0.314865	+	0.949137i	$0.398041\pi$
$692$	0	0
$693$	−1.71721e6	−0.135828
$694$	0	0
$695$	2.34449e6	0.184114
$696$	0	0
$697$	2.22468e6	0.173455
$698$	0	0
$699$	1.34851e7	1.04390
$700$	0	0
$701$	2.20897e7	1.69783	0.848915	−	0.528529i	$-0.177256\pi$
$701$	2.20897e7	1.69783	0.848915	+	0.528529i	$0.177256\pi$
$702$	0	0
$703$	830791.	0.0634021
$704$	0	0
$705$	1.59554e7	1.20902
$706$	0	0
$707$	−5.26705e6	−0.396295
$708$	0	0
$709$	1.78932e7	1.33682	0.668410	−	0.743793i	$-0.266975\pi$
$709$	1.78932e7	1.33682	0.668410	+	0.743793i	$0.266975\pi$
$710$	0	0
$711$	5.48435e6	0.406866
$712$	0	0
$713$	−2.80474e6	−0.206618
$714$	0	0
$715$	7.38609e6	0.540318
$716$	0	0
$717$	1.53836e7	1.11753
$718$	0	0
$719$	−5.89022e6	−0.424922	−0.212461	−	0.977170i	$-0.568148\pi$
$719$	−5.89022e6	−0.424922	−0.212461	+	0.977170i	$0.568148\pi$
$720$	0	0
$721$	1.57411e6	0.112771
$722$	0	0
$723$	7.27795e6	0.517802
$724$	0	0
$725$	3.30342e6	0.233410
$726$	0	0
$727$	−1.14381e7	−0.802632	−0.401316	−	0.915940i	$-0.631447\pi$
$727$	−1.14381e7	−0.802632	−0.401316	+	0.915940i	$0.631447\pi$
$728$	0	0
$729$	7.66915e6	0.534477
$730$	0	0
$731$	1.34134e7	0.928422
$732$	0	0
$733$	−3.54064e6	−0.243401	−0.121700	−	0.992567i	$-0.538835\pi$
$733$	−3.54064e6	−0.243401	−0.121700	+	0.992567i	$0.538835\pi$
$734$	0	0
$735$	6.86893e6	0.468998
$736$	0	0
$737$	3.54750e6	0.240577
$738$	0	0
$739$	1.49710e7	1.00841	0.504206	−	0.863583i	$-0.331785\pi$
$739$	1.49710e7	1.00841	0.504206	+	0.863583i	$0.331785\pi$
$740$	0	0
$741$	7.23175e6	0.483836
$742$	0	0
$743$	−1.85546e7	−1.23305	−0.616523	−	0.787337i	$-0.711459\pi$
$743$	−1.85546e7	−1.23305	−0.616523	+	0.787337i	$0.711459\pi$
$744$	0	0
$745$	−5.21804e6	−0.344442
$746$	0	0
$747$	1.20526e7	0.790277
$748$	0	0
$749$	2.85083e6	0.185681
$750$	0	0
$751$	−3.87586e6	−0.250766	−0.125383	−	0.992108i	$-0.540016\pi$
$751$	−3.87586e6	−0.250766	−0.125383	+	0.992108i	$0.540016\pi$
$752$	0	0
$753$	6.87448e6	0.441827
$754$	0	0
$755$	−1.04163e7	−0.665038
$756$	0	0
$757$	1.49345e7	0.947223	0.473612	−	0.880734i	$-0.342950\pi$
$757$	1.49345e7	0.947223	0.473612	+	0.880734i	$0.342950\pi$
$758$	0	0
$759$	2.77500e6	0.174847
$760$	0	0
$761$	1.52234e7	0.952903	0.476452	−	0.879201i	$-0.341923\pi$
$761$	1.52234e7	0.952903	0.476452	+	0.879201i	$0.341923\pi$
$762$	0	0
$763$	−5.71739e6	−0.355538
$764$	0	0
$765$	−7.76289e6	−0.479590
$766$	0	0
$767$	−3.41728e7	−2.09745
$768$	0	0
$769$	492277.	0.0300188	0.0150094	−	0.999887i	$-0.495222\pi$
$769$	492277.	0.0300188	0.0150094	+	0.999887i	$0.495222\pi$
$770$	0	0
$771$	−610367.	−0.0369790
$772$	0	0
$773$	−1.57704e7	−0.949277	−0.474639	−	0.880181i	$-0.657421\pi$
$773$	−1.57704e7	−0.949277	−0.474639	+	0.880181i	$0.657421\pi$
$774$	0	0
$775$	−602131.	−0.0360111
$776$	0	0
$777$	478464.	0.0284313
$778$	0	0
$779$	2.80397e6	0.165550
$780$	0	0
$781$	747449.	0.0438484
$782$	0	0
$783$	3.16561e7	1.84524
$784$	0	0
$785$	−780143.	−0.0451856
$786$	0	0
$787$	3.06825e7	1.76585	0.882924	−	0.469515i	$-0.155571\pi$
$787$	3.06825e7	1.76585	0.882924	+	0.469515i	$0.155571\pi$
$788$	0	0
$789$	−1.58732e7	−0.907764
$790$	0	0
$791$	1.24786e7	0.709128
$792$	0	0
$793$	−1.84628e7	−1.04259
$794$	0	0
$795$	9.21585e6	0.517151
$796$	0	0
$797$	−1.83757e7	−1.02471	−0.512353	−	0.858775i	$-0.671226\pi$
$797$	−1.83757e7	−1.02471	−0.512353	+	0.858775i	$0.671226\pi$
$798$	0	0
$799$	−2.42201e7	−1.34217
$800$	0	0
$801$	−1.41513e7	−0.779318
$802$	0	0
$803$	8.54395e6	0.467595
$804$	0	0
$805$	−6.96221e6	−0.378667
$806$	0	0
$807$	9.28188e6	0.501709
$808$	0	0
$809$	2.59874e7	1.39602	0.698009	−	0.716089i	$-0.254070\pi$
$809$	2.59874e7	1.39602	0.698009	+	0.716089i	$0.254070\pi$
$810$	0	0
$811$	1.59506e7	0.851578	0.425789	−	0.904823i	$-0.359997\pi$
$811$	1.59506e7	0.851578	0.425789	+	0.904823i	$0.359997\pi$
$812$	0	0
$813$	1.83739e7	0.974934
$814$	0	0
$815$	5.39647e6	0.284588
$816$	0	0
$817$	1.69062e7	0.886115
$818$	0	0
$819$	−7.68623e6	−0.400409
$820$	0	0
$821$	3.08798e7	1.59889	0.799443	−	0.600742i	$-0.205128\pi$
$821$	3.08798e7	1.59889	0.799443	+	0.600742i	$0.205128\pi$
$822$	0	0
$823$	−1.48817e7	−0.765868	−0.382934	−	0.923776i	$-0.625086\pi$
$823$	−1.48817e7	−0.765868	−0.382934	+	0.923776i	$0.625086\pi$
$824$	0	0
$825$	595748.	0.0304739
$826$	0	0
$827$	1.55948e7	0.792894	0.396447	−	0.918058i	$-0.370243\pi$
$827$	1.55948e7	0.792894	0.396447	+	0.918058i	$0.370243\pi$
$828$	0	0
$829$	3.21558e6	0.162507	0.0812536	−	0.996693i	$-0.474108\pi$
$829$	3.21558e6	0.162507	0.0812536	+	0.996693i	$0.474108\pi$
$830$	0	0
$831$	−6.73700e6	−0.338426
$832$	0	0
$833$	−1.04270e7	−0.520649
$834$	0	0
$835$	−1.90817e7	−0.947109
$836$	0	0
$837$	−5.77012e6	−0.284689
$838$	0	0
$839$	−2.71327e7	−1.33072	−0.665362	−	0.746521i	$-0.731723\pi$
$839$	−2.71327e7	−1.33072	−0.665362	+	0.746521i	$0.731723\pi$
$840$	0	0
$841$	5.26096e7	2.56493
$842$	0	0
$843$	2.35101e7	1.13943
$844$	0	0
$845$	1.10587e7	0.532799
$846$	0	0
$847$	8.69725e6	0.416556
$848$	0	0
$849$	−2.94207e6	−0.140082
$850$	0	0
$851$	1.42693e6	0.0675428
$852$	0	0
$853$	2.01554e7	0.948457	0.474229	−	0.880402i	$-0.342727\pi$
$853$	2.01554e7	0.948457	0.474229	+	0.880402i	$0.342727\pi$
$854$	0	0
$855$	−9.78430e6	−0.457736
$856$	0	0
$857$	−3.96753e7	−1.84531	−0.922653	−	0.385631i	$-0.873984\pi$
$857$	−3.96753e7	−1.84531	−0.922653	+	0.385631i	$0.873984\pi$
$858$	0	0
$859$	2.98669e7	1.38104	0.690521	−	0.723312i	$-0.257381\pi$
$859$	2.98669e7	1.38104	0.690521	+	0.723312i	$0.257381\pi$
$860$	0	0
$861$	1.61485e6	0.0742376
$862$	0	0
$863$	3.57592e7	1.63441	0.817205	−	0.576347i	$-0.195522\pi$
$863$	3.57592e7	1.63441	0.817205	+	0.576347i	$0.195522\pi$
$864$	0	0
$865$	−3.96631e7	−1.80238
$866$	0	0
$867$	6.73579e6	0.304327
$868$	0	0
$869$	5.80708e6	0.260861
$870$	0	0
$871$	1.58786e7	0.709198
$872$	0	0
$873$	−1.12906e6	−0.0501395
$874$	0	0
$875$	1.05960e7	0.467868
$876$	0	0
$877$	3.24836e7	1.42615	0.713075	−	0.701088i	$-0.247302\pi$
$877$	3.24836e7	1.42615	0.713075	+	0.701088i	$0.247302\pi$
$878$	0	0
$879$	1.45076e6	0.0633322
$880$	0	0
$881$	−1.93672e7	−0.840675	−0.420338	−	0.907368i	$-0.638088\pi$
$881$	−1.93672e7	−0.840675	−0.420338	+	0.907368i	$0.638088\pi$
$882$	0	0
$883$	1.91923e7	0.828370	0.414185	−	0.910193i	$-0.364067\pi$
$883$	1.91923e7	0.828370	0.414185	+	0.910193i	$0.364067\pi$
$884$	0	0
$885$	−2.50527e7	−1.07522
$886$	0	0
$887$	1.50822e7	0.643657	0.321829	−	0.946798i	$-0.395702\pi$
$887$	1.50822e7	0.643657	0.321829	+	0.946798i	$0.395702\pi$
$888$	0	0
$889$	−1.52009e7	−0.645082
$890$	0	0
$891$	−681944.	−0.0287776
$892$	0	0
$893$	−3.05268e7	−1.28101
$894$	0	0
$895$	1.28587e7	0.536588
$896$	0	0
$897$	1.24209e7	0.515434
$898$	0	0
$899$	−1.33281e7	−0.550007
$900$	0	0
$901$	−1.39895e7	−0.574106
$902$	0	0
$903$	9.73649e6	0.397359
$904$	0	0
$905$	−1.90402e7	−0.772770
$906$	0	0
$907$	−1.85028e7	−0.746824	−0.373412	−	0.927666i	$-0.621812\pi$
$907$	−1.85028e7	−0.746824	−0.373412	+	0.927666i	$0.621812\pi$
$908$	0	0
$909$	−1.27134e7	−0.510331
$910$	0	0
$911$	5.95419e6	0.237699	0.118849	−	0.992912i	$-0.462079\pi$
$911$	5.95419e6	0.237699	0.118849	+	0.992912i	$0.462079\pi$
$912$	0	0
$913$	1.27619e7	0.506684
$914$	0	0
$915$	−1.35355e7	−0.534466
$916$	0	0
$917$	962836.	0.0378120
$918$	0	0
$919$	−1.65423e7	−0.646111	−0.323056	−	0.946380i	$-0.604710\pi$
$919$	−1.65423e7	−0.646111	−0.323056	+	0.946380i	$0.604710\pi$
$920$	0	0
$921$	2.33143e7	0.905678
$922$	0	0
$923$	3.34559e6	0.129261
$924$	0	0
$925$	306339.	0.0117719
$926$	0	0
$927$	3.79953e6	0.145222
$928$	0	0
$929$	1.73964e7	0.661333	0.330666	−	0.943748i	$-0.392727\pi$
$929$	1.73964e7	0.661333	0.330666	+	0.943748i	$0.392727\pi$
$930$	0	0
$931$	−1.31421e7	−0.496923
$932$	0	0
$933$	−2.52482e6	−0.0949568
$934$	0	0
$935$	−8.21972e6	−0.307488
$936$	0	0
$937$	−1.63563e6	−0.0608606	−0.0304303	−	0.999537i	$-0.509688\pi$
$937$	−1.63563e6	−0.0608606	−0.0304303	+	0.999537i	$0.509688\pi$
$938$	0	0
$939$	−7.88752e6	−0.291929
$940$	0	0
$941$	310890.	0.0114454	0.00572271	−	0.999984i	$-0.498178\pi$
$941$	310890.	0.0114454	0.00572271	+	0.999984i	$0.498178\pi$
$942$	0	0
$943$	4.81598e6	0.176362
$944$	0	0
$945$	−1.43232e7	−0.521747
$946$	0	0
$947$	2.88996e7	1.04717	0.523585	−	0.851973i	$-0.324594\pi$
$947$	2.88996e7	1.04717	0.523585	+	0.851973i	$0.324594\pi$
$948$	0	0
$949$	3.82428e7	1.37843
$950$	0	0
$951$	2.74844e7	0.985450
$952$	0	0
$953$	3.28773e7	1.17264	0.586319	−	0.810080i	$-0.300577\pi$
$953$	3.28773e7	1.17264	0.586319	+	0.810080i	$0.300577\pi$
$954$	0	0
$955$	−4.77031e6	−0.169254
$956$	0	0
$957$	1.31868e7	0.465435
$958$	0	0
$959$	1.09671e7	0.385075
$960$	0	0
$961$	−2.61998e7	−0.915143
$962$	0	0
$963$	6.88123e6	0.239111
$964$	0	0
$965$	−2.34327e7	−0.810037
$966$	0	0
$967$	−589588.	−0.0202760	−0.0101380	−	0.999949i	$-0.503227\pi$
$967$	−589588.	−0.0202760	−0.0101380	+	0.999949i	$0.503227\pi$
$968$	0	0
$969$	−8.04797e6	−0.275345
$970$	0	0
$971$	−2.79183e7	−0.950255	−0.475128	−	0.879917i	$-0.657598\pi$
$971$	−2.79183e7	−0.950255	−0.475128	+	0.879917i	$0.657598\pi$
$972$	0	0
$973$	−2.58333e6	−0.0874778
$974$	0	0
$975$	2.66657e6	0.0898342
$976$	0	0
$977$	−4.32932e7	−1.45105	−0.725527	−	0.688194i	$-0.758404\pi$
$977$	−4.32932e7	−1.45105	−0.725527	+	0.688194i	$0.758404\pi$
$978$	0	0
$979$	−1.49840e7	−0.499657
$980$	0	0
$981$	−1.38004e7	−0.457846
$982$	0	0
$983$	−4.19469e7	−1.38457	−0.692287	−	0.721623i	$-0.743397\pi$
$983$	−4.19469e7	−1.38457	−0.692287	+	0.721623i	$0.743397\pi$
$984$	0	0
$985$	3.86768e7	1.27016
$986$	0	0
$987$	−1.75808e7	−0.574442
$988$	0	0
$989$	2.90373e7	0.943985
$990$	0	0
$991$	−2.12252e7	−0.686543	−0.343271	−	0.939236i	$-0.611535\pi$
$991$	−2.12252e7	−0.686543	−0.343271	+	0.939236i	$0.611535\pi$
$992$	0	0
$993$	−9.50298e6	−0.305835
$994$	0	0
$995$	2.37714e7	0.761196
$996$	0	0
$997$	−4.54516e7	−1.44814	−0.724071	−	0.689726i	$-0.757731\pi$
$997$	−4.54516e7	−1.44814	−0.724071	+	0.689726i	$0.757731\pi$
$998$	0	0
$999$	2.93559e6	0.0930640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.19	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.19	✓	49	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)

\(f(q)\)	\(=\)	\(q-9.24112 q^{3} +59.2564 q^{5} -65.2931 q^{7} -157.602 q^{9} +O(q^{10})\)	\(q-9.24112 q^{3} +59.2564 q^{5} -65.2931 q^{7} -157.602 q^{9} -166.876 q^{11} -746.939 q^{13} -547.595 q^{15} +831.242 q^{17} +1047.69 q^{19} +603.381 q^{21} +1799.47 q^{23} +386.317 q^{25} +3702.01 q^{27} +8551.07 q^{29} -1558.64 q^{31} +1542.12 q^{33} -3869.03 q^{35} +792.972 q^{37} +6902.55 q^{39} +2676.33 q^{41} +16136.6 q^{43} -9338.91 q^{45} -29137.2 q^{47} -12543.8 q^{49} -7681.61 q^{51} -16829.7 q^{53} -9888.47 q^{55} -9681.85 q^{57} +45750.4 q^{59} +24718.0 q^{61} +10290.3 q^{63} -44260.9 q^{65} -21258.3 q^{67} -16629.1 q^{69} -4479.06 q^{71} -51199.4 q^{73} -3570.00 q^{75} +10895.9 q^{77} -34798.8 q^{79} +4086.53 q^{81} -76475.1 q^{83} +49256.4 q^{85} -79021.4 q^{87} +89791.4 q^{89} +48769.9 q^{91} +14403.6 q^{93} +62082.5 q^{95} +7163.99 q^{97} +26300.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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