LMFDB - Embedded newform 531.6.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83028 q^{2} +1.99222 q^{4} +62.8474 q^{5} -155.625 q^{7} +174.954 q^{8} +O(q^{10})$	\(q-5.83028 q^{2} +1.99222 q^{4} +62.8474 q^{5} -155.625 q^{7} +174.954 q^{8} -366.418 q^{10} +463.795 q^{11} -160.954 q^{13} +907.339 q^{14} -1083.78 q^{16} -389.283 q^{17} -766.828 q^{19} +125.206 q^{20} -2704.06 q^{22} +16.6555 q^{23} +824.798 q^{25} +938.407 q^{26} -310.039 q^{28} +3705.06 q^{29} -438.095 q^{31} +720.232 q^{32} +2269.63 q^{34} -9780.64 q^{35} +4070.62 q^{37} +4470.82 q^{38} +10995.4 q^{40} -2940.10 q^{41} -7610.41 q^{43} +923.980 q^{44} -97.1064 q^{46} -15633.5 q^{47} +7412.17 q^{49} -4808.81 q^{50} -320.655 q^{52} +703.825 q^{53} +29148.3 q^{55} -27227.2 q^{56} -21601.5 q^{58} -3481.00 q^{59} -39498.4 q^{61} +2554.22 q^{62} +30481.9 q^{64} -10115.5 q^{65} +53596.2 q^{67} -775.537 q^{68} +57023.9 q^{70} +48621.8 q^{71} +44549.1 q^{73} -23732.9 q^{74} -1527.69 q^{76} -72178.1 q^{77} +25736.5 q^{79} -68112.9 q^{80} +17141.6 q^{82} +122226. q^{83} -24465.5 q^{85} +44370.9 q^{86} +81142.7 q^{88} +5907.23 q^{89} +25048.5 q^{91} +33.1814 q^{92} +91147.9 q^{94} -48193.1 q^{95} -52477.9 q^{97} -43215.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 12 q^{2} + 400 q^{4} - 200 q^{5} - 38 q^{7} - 576 q^{8}+O(q^{10}) $ 25 * q - 12 * q^2 + 400 * q^4 - 200 * q^5 - 38 * q^7 - 576 * q^8	$ 25 q - 12 q^{2} + 400 q^{4} - 200 q^{5} - 38 q^{7} - 576 q^{8} + 370 q^{10} - 1210 q^{11} - 136 q^{13} - 1176 q^{14} + 5884 q^{16} - 2312 q^{17} + 1722 q^{19} - 944 q^{20} + 1700 q^{22} - 4476 q^{23} + 17547 q^{25} - 10320 q^{26} - 552 q^{28} - 30036 q^{29} - 822 q^{31} - 44684 q^{32} - 4240 q^{34} - 14700 q^{35} - 1220 q^{37} - 58378 q^{38} - 11890 q^{40} - 43656 q^{41} + 18476 q^{43} - 58080 q^{44} + 25322 q^{46} - 43962 q^{47} + 62015 q^{49} - 167260 q^{50} + 11740 q^{52} - 79780 q^{53} - 42830 q^{55} - 161888 q^{56} + 14274 q^{58} - 87025 q^{59} - 24390 q^{61} - 73224 q^{62} - 13574 q^{64} - 43378 q^{65} - 14168 q^{67} - 202548 q^{68} + 518 q^{70} - 27924 q^{71} + 44646 q^{73} - 186092 q^{74} + 56816 q^{76} - 207738 q^{77} + 21128 q^{79} - 150020 q^{80} + 84138 q^{82} - 84150 q^{83} + 7454 q^{85} - 376480 q^{86} - 54018 q^{88} - 227092 q^{89} + 21768 q^{91} - 470414 q^{92} - 83352 q^{94} - 230580 q^{95} + 177344 q^{97} - 299700 q^{98}+O(q^{100}) $ 25 * q - 12 * q^2 + 400 * q^4 - 200 * q^5 - 38 * q^7 - 576 * q^8 + 370 * q^10 - 1210 * q^11 - 136 * q^13 - 1176 * q^14 + 5884 * q^16 - 2312 * q^17 + 1722 * q^19 - 944 * q^20 + 1700 * q^22 - 4476 * q^23 + 17547 * q^25 - 10320 * q^26 - 552 * q^28 - 30036 * q^29 - 822 * q^31 - 44684 * q^32 - 4240 * q^34 - 14700 * q^35 - 1220 * q^37 - 58378 * q^38 - 11890 * q^40 - 43656 * q^41 + 18476 * q^43 - 58080 * q^44 + 25322 * q^46 - 43962 * q^47 + 62015 * q^49 - 167260 * q^50 + 11740 * q^52 - 79780 * q^53 - 42830 * q^55 - 161888 * q^56 + 14274 * q^58 - 87025 * q^59 - 24390 * q^61 - 73224 * q^62 - 13574 * q^64 - 43378 * q^65 - 14168 * q^67 - 202548 * q^68 + 518 * q^70 - 27924 * q^71 + 44646 * q^73 - 186092 * q^74 + 56816 * q^76 - 207738 * q^77 + 21128 * q^79 - 150020 * q^80 + 84138 * q^82 - 84150 * q^83 + 7454 * q^85 - 376480 * q^86 - 54018 * q^88 - 227092 * q^89 + 21768 * q^91 - 470414 * q^92 - 83352 * q^94 - 230580 * q^95 + 177344 * q^97 - 299700 * q^98

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 $

$2$	−5.83028	−1.03066	−0.515329	−	0.856992i	$-0.672330\pi$
$2$	−5.83028	−1.03066	−0.515329	+	0.856992i	$0.672330\pi$
$3$	0	0
$3$	0	0
$4$	1.99222	0.0622568
$5$	62.8474	1.12425	0.562124	−	0.827053i	$-0.309984\pi$
$5$	62.8474	1.12425	0.562124	+	0.827053i	$0.309984\pi$
$6$	0	0
$7$	−155.625	−1.20042	−0.600212	−	0.799841i	$-0.704917\pi$
$7$	−155.625	−1.20042	−0.600212	+	0.799841i	$0.704917\pi$
$8$	174.954	0.966493
$9$	0	0
$10$	−366.418	−1.15872
$11$	463.795	1.15570	0.577849	−	0.816144i	$-0.303892\pi$
$11$	463.795	1.15570	0.577849	+	0.816144i	$0.303892\pi$
$12$	0	0
$13$	−160.954	−0.264145	−0.132073	−	0.991240i	$-0.542163\pi$
$13$	−160.954	−0.264145	−0.132073	+	0.991240i	$0.542163\pi$
$14$	907.339	1.23723
$15$	0	0
$16$	−1083.78	−1.05838
$17$	−389.283	−0.326696	−0.163348	−	0.986569i	$-0.552229\pi$
$17$	−389.283	−0.326696	−0.163348	+	0.986569i	$0.552229\pi$
$18$	0	0
$19$	−766.828	−0.487319	−0.243660	−	0.969861i	$-0.578348\pi$
$19$	−766.828	−0.487319	−0.243660	+	0.969861i	$0.578348\pi$
$20$	125.206	0.0699921
$21$	0	0
$22$	−2704.06	−1.19113
$23$	16.6555	0.00656506	0.00328253	−	0.999995i	$-0.498955\pi$
$23$	16.6555	0.00656506	0.00328253	+	0.999995i	$0.498955\pi$
$24$	0	0
$25$	824.798	0.263935
$26$	938.407	0.272244
$27$	0	0
$28$	−310.039	−0.0747345
$29$	3705.06	0.818088	0.409044	−	0.912515i	$-0.365862\pi$
$29$	3705.06	0.818088	0.409044	+	0.912515i	$0.365862\pi$
$30$	0	0
$31$	−438.095	−0.0818773	−0.0409387	−	0.999162i	$-0.513035\pi$
$31$	−438.095	−0.0818773	−0.0409387	+	0.999162i	$0.513035\pi$
$32$	720.232	0.124336
$33$	0	0
$34$	2269.63	0.336712
$35$	−9780.64	−1.34957
$36$	0	0
$37$	4070.62	0.488829	0.244414	−	0.969671i	$-0.421404\pi$
$37$	4070.62	0.488829	0.244414	+	0.969671i	$0.421404\pi$
$38$	4470.82	0.502260
$39$	0	0
$40$	10995.4	1.08658
$41$	−2940.10	−0.273151	−0.136576	−	0.990630i	$-0.543610\pi$
$41$	−2940.10	−0.273151	−0.136576	+	0.990630i	$0.543610\pi$
$42$	0	0
$43$	−7610.41	−0.627678	−0.313839	−	0.949476i	$-0.601615\pi$
$43$	−7610.41	−0.627678	−0.313839	+	0.949476i	$0.601615\pi$
$44$	923.980	0.0719500
$45$	0	0
$46$	−97.1064	−0.00676633
$47$	−15633.5	−1.03231	−0.516157	−	0.856494i	$-0.672638\pi$
$47$	−15633.5	−1.03231	−0.516157	+	0.856494i	$0.672638\pi$
$48$	0	0
$49$	7412.17	0.441017
$50$	−4808.81	−0.272027
$51$	0	0
$52$	−320.655	−0.0164448
$53$	703.825	0.0344172	0.0172086	−	0.999852i	$-0.494522\pi$
$53$	703.825	0.0344172	0.0172086	+	0.999852i	$0.494522\pi$
$54$	0	0
$55$	29148.3	1.29929
$56$	−27227.2	−1.16020
$57$	0	0
$58$	−21601.5	−0.843169
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−39498.4	−1.35911	−0.679555	−	0.733625i	$-0.737827\pi$
$61$	−39498.4	−1.35911	−0.679555	+	0.733625i	$0.737827\pi$
$62$	2554.22	0.0843876
$63$	0	0
$64$	30481.9	0.930233
$65$	−10115.5	−0.296965
$66$	0	0
$67$	53596.2	1.45864	0.729319	−	0.684174i	$-0.239837\pi$
$67$	53596.2	1.45864	0.729319	+	0.684174i	$0.239837\pi$
$68$	−775.537	−0.0203390
$69$	0	0
$70$	57023.9	1.39095
$71$	48621.8	1.14468	0.572341	−	0.820016i	$-0.306035\pi$
$71$	48621.8	1.14468	0.572341	+	0.820016i	$0.306035\pi$
$72$	0	0
$73$	44549.1	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$73$	44549.1	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$74$	−23732.9	−0.503815
$75$	0	0
$76$	−1527.69	−0.0303389
$77$	−72178.1	−1.38733
$78$	0	0
$79$	25736.5	0.463961	0.231980	−	0.972720i	$-0.425479\pi$
$79$	25736.5	0.463961	0.231980	+	0.972720i	$0.425479\pi$
$80$	−68112.9	−1.18988
$81$	0	0
$82$	17141.6	0.281525
$83$	122226.	1.94746	0.973730	−	0.227708i	$-0.0731231\pi$
$83$	122226.	1.94746	0.973730	+	0.227708i	$0.0731231\pi$
$84$	0	0
$85$	−24465.5	−0.367287
$86$	44370.9	0.646922
$87$	0	0
$88$	81142.7	1.11697
$89$	5907.23	0.0790512	0.0395256	−	0.999219i	$-0.487415\pi$
$89$	5907.23	0.0790512	0.0395256	+	0.999219i	$0.487415\pi$
$90$	0	0
$91$	25048.5	0.317086
$92$	33.1814	0.000408720 0
$93$	0	0
$94$	91147.9	1.06396
$95$	−48193.1	−0.547868
$96$	0	0
$97$	−52477.9	−0.566300	−0.283150	−	0.959076i	$-0.591379\pi$
$97$	−52477.9	−0.566300	−0.283150	+	0.959076i	$0.591379\pi$
$98$	−43215.1	−0.454538
$99$	0	0
$100$	1643.18	0.0164318
$101$	−89590.7	−0.873896	−0.436948	−	0.899487i	$-0.643941\pi$
$101$	−89590.7	−0.873896	−0.436948	+	0.899487i	$0.643941\pi$
$102$	0	0
$103$	43300.5	0.402160	0.201080	−	0.979575i	$-0.435555\pi$
$103$	43300.5	0.402160	0.201080	+	0.979575i	$0.435555\pi$
$104$	−28159.5	−0.255295
$105$	0	0
$106$	−4103.50	−0.0354723
$107$	−77792.2	−0.656866	−0.328433	−	0.944527i	$-0.606521\pi$
$107$	−77792.2	−0.656866	−0.328433	+	0.944527i	$0.606521\pi$
$108$	0	0
$109$	−110991.	−0.894788	−0.447394	−	0.894337i	$-0.647648\pi$
$109$	−110991.	−0.894788	−0.447394	+	0.894337i	$0.647648\pi$
$110$	−169943.	−1.33913
$111$	0	0
$112$	168664.	1.27051
$113$	95926.3	0.706710	0.353355	−	0.935489i	$-0.385041\pi$
$113$	95926.3	0.706710	0.353355	+	0.935489i	$0.385041\pi$
$114$	0	0
$115$	1046.76	0.00738076
$116$	7381.28	0.0509315
$117$	0	0
$118$	20295.2	0.134180
$119$	60582.3	0.392173
$120$	0	0
$121$	54054.6	0.335637
$122$	230287.	1.40078
$123$	0	0
$124$	−872.780	−0.00509742
$125$	−144562.	−0.827520
$126$	0	0
$127$	−312964.	−1.72181	−0.860904	−	0.508767i	$-0.830101\pi$
$127$	−312964.	−1.72181	−0.860904	+	0.508767i	$0.830101\pi$
$128$	−200765.	−1.08309
$129$	0	0
$130$	58976.4	0.306070
$131$	−162787.	−0.828782	−0.414391	−	0.910099i	$-0.636005\pi$
$131$	−162787.	−0.828782	−0.414391	+	0.910099i	$0.636005\pi$
$132$	0	0
$133$	119338.	0.584990
$134$	−312481.	−1.50336
$135$	0	0
$136$	−68106.7	−0.315749
$137$	−76854.8	−0.349840	−0.174920	−	0.984583i	$-0.555967\pi$
$137$	−76854.8	−0.349840	−0.174920	+	0.984583i	$0.555967\pi$
$138$	0	0
$139$	−300022.	−1.31709	−0.658546	−	0.752540i	$-0.728828\pi$
$139$	−300022.	−1.31709	−0.658546	+	0.752540i	$0.728828\pi$
$140$	−19485.2	−0.0840202
$141$	0	0
$142$	−283479.	−1.17978
$143$	−74649.6	−0.305272
$144$	0	0
$145$	232853.	0.919735
$146$	−259734.	−1.00843
$147$	0	0
$148$	8109.57	0.0304329
$149$	100575.	0.371128	0.185564	−	0.982632i	$-0.440589\pi$
$149$	100575.	0.371128	0.185564	+	0.982632i	$0.440589\pi$
$150$	0	0
$151$	−121727.	−0.434456	−0.217228	−	0.976121i	$-0.569701\pi$
$151$	−121727.	−0.434456	−0.217228	+	0.976121i	$0.569701\pi$
$152$	−134159.	−0.470991
$153$	0	0
$154$	420819.	1.42986
$155$	−27533.1	−0.0920505
$156$	0	0
$157$	305840.	0.990249	0.495125	−	0.868822i	$-0.335122\pi$
$157$	305840.	0.990249	0.495125	+	0.868822i	$0.335122\pi$
$158$	−150051.	−0.478185
$159$	0	0
$160$	45264.7	0.139785
$161$	−2592.02	−0.00788085
$162$	0	0
$163$	−225162.	−0.663784	−0.331892	−	0.943317i	$-0.607687\pi$
$163$	−225162.	−0.663784	−0.331892	+	0.943317i	$0.607687\pi$
$164$	−5857.32	−0.0170055
$165$	0	0
$166$	−712612.	−2.00717
$167$	179691.	0.498581	0.249291	−	0.968429i	$-0.419803\pi$
$167$	179691.	0.498581	0.249291	+	0.968429i	$0.419803\pi$
$168$	0	0
$169$	−345387.	−0.930227
$170$	142641.	0.378548
$171$	0	0
$172$	−15161.6	−0.0390772
$173$	−691603.	−1.75688	−0.878439	−	0.477855i	$-0.841414\pi$
$173$	−691603.	−1.75688	−0.878439	+	0.477855i	$0.841414\pi$
$174$	0	0
$175$	−128359.	−0.316834
$176$	−502652.	−1.22317
$177$	0	0
$178$	−34440.8	−0.0814748
$179$	−172853.	−0.403221	−0.201611	−	0.979466i	$-0.564618\pi$
$179$	−172853.	−0.403221	−0.201611	+	0.979466i	$0.564618\pi$
$180$	0	0
$181$	−198427.	−0.450200	−0.225100	−	0.974336i	$-0.572271\pi$
$181$	−198427.	−0.450200	−0.225100	+	0.974336i	$0.572271\pi$
$182$	−146040.	−0.326808
$183$	0	0
$184$	2913.95	0.00634508
$185$	255828.	0.549565
$186$	0	0
$187$	−180548.	−0.377562
$188$	−31145.4	−0.0642686
$189$	0	0
$190$	280980.	0.564665
$191$	−347371.	−0.688986	−0.344493	−	0.938789i	$-0.611949\pi$
$191$	−347371.	−0.688986	−0.344493	+	0.938789i	$0.611949\pi$
$192$	0	0
$193$	−366073.	−0.707414	−0.353707	−	0.935356i	$-0.615079\pi$
$193$	−366073.	−0.707414	−0.353707	+	0.935356i	$0.615079\pi$
$194$	305961.	0.583662
$195$	0	0
$196$	14766.7	0.0274563
$197$	−307108.	−0.563801	−0.281901	−	0.959444i	$-0.590965\pi$
$197$	−307108.	−0.563801	−0.281901	+	0.959444i	$0.590965\pi$
$198$	0	0
$199$	−980675.	−1.75547	−0.877733	−	0.479149i	$-0.840945\pi$
$199$	−980675.	−1.75547	−0.877733	+	0.479149i	$0.840945\pi$
$200$	144302.	0.255092
$201$	0	0
$202$	522339.	0.900688
$203$	−576600.	−0.982052
$204$	0	0
$205$	−184778.	−0.307090
$206$	−252454.	−0.414490
$207$	0	0
$208$	174439.	0.279566
$209$	−355651.	−0.563194
$210$	0	0
$211$	390489.	0.603814	0.301907	−	0.953337i	$-0.402377\pi$
$211$	390489.	0.603814	0.301907	+	0.953337i	$0.402377\pi$
$212$	1402.17	0.00214270
$213$	0	0
$214$	453551.	0.677005
$215$	−478295.	−0.705667
$216$	0	0
$217$	68178.5	0.0982875
$218$	647107.	0.922221
$219$	0	0
$220$	58069.8	0.0808897
$221$	62656.7	0.0862952
$222$	0	0
$223$	635250.	0.855425	0.427713	−	0.903915i	$-0.359319\pi$
$223$	635250.	0.855425	0.427713	+	0.903915i	$0.359319\pi$
$224$	−112086.	−0.149256
$225$	0	0
$226$	−559277.	−0.728377
$227$	−122167.	−0.157359	−0.0786793	−	0.996900i	$-0.525070\pi$
$227$	−122167.	−0.157359	−0.0786793	+	0.996900i	$0.525070\pi$
$228$	0	0
$229$	320208.	0.403499	0.201750	−	0.979437i	$-0.435337\pi$
$229$	320208.	0.403499	0.201750	+	0.979437i	$0.435337\pi$
$230$	−6102.89	−0.00760704
$231$	0	0
$232$	648214.	0.790676
$233$	390727.	0.471503	0.235751	−	0.971813i	$-0.424245\pi$
$233$	390727.	0.471503	0.235751	+	0.971813i	$0.424245\pi$
$234$	0	0
$235$	−982527.	−1.16058
$236$	−6934.91	−0.00810514
$237$	0	0
$238$	−353212.	−0.404197
$239$	1.04343e6	1.18160	0.590799	−	0.806819i	$-0.298813\pi$
$239$	1.04343e6	1.18160	0.590799	+	0.806819i	$0.298813\pi$
$240$	0	0
$241$	−1.11425e6	−1.23578	−0.617891	−	0.786264i	$-0.712013\pi$
$241$	−1.11425e6	−1.23578	−0.617891	+	0.786264i	$0.712013\pi$
$242$	−315154.	−0.345927
$243$	0	0
$244$	−78689.3	−0.0846138
$245$	465836.	0.495813
$246$	0	0
$247$	123424.	0.128723
$248$	−76646.4	−0.0791339
$249$	0	0
$250$	842836.	0.852890
$251$	−125909.	−0.126145	−0.0630727	−	0.998009i	$-0.520090\pi$
$251$	−125909.	−0.126145	−0.0630727	+	0.998009i	$0.520090\pi$
$252$	0	0
$253$	7724.74	0.00758722
$254$	1.82467e6	1.77460
$255$	0	0
$256$	195100.	0.186061
$257$	−1.45678e6	−1.37581	−0.687907	−	0.725799i	$-0.741470\pi$
$257$	−1.45678e6	−1.37581	−0.687907	+	0.725799i	$0.741470\pi$
$258$	0	0
$259$	−633491.	−0.586801
$260$	−20152.3	−0.0184881
$261$	0	0
$262$	949092.	0.854191
$263$	−549123.	−0.489531	−0.244765	−	0.969582i	$-0.578711\pi$
$263$	−549123.	−0.489531	−0.244765	+	0.969582i	$0.578711\pi$
$264$	0	0
$265$	44233.6	0.0386935
$266$	−695772.	−0.602925
$267$	0	0
$268$	106775.	0.0908101
$269$	−1.08480e6	−0.914048	−0.457024	−	0.889454i	$-0.651085\pi$
$269$	−1.08480e6	−0.914048	−0.457024	+	0.889454i	$0.651085\pi$
$270$	0	0
$271$	−419141.	−0.346687	−0.173343	−	0.984861i	$-0.555457\pi$
$271$	−419141.	−0.346687	−0.173343	+	0.984861i	$0.555457\pi$
$272$	421898.	0.345769
$273$	0	0
$274$	448086.	0.360566
$275$	382537.	0.305029
$276$	0	0
$277$	−201151.	−0.157516	−0.0787578	−	0.996894i	$-0.525095\pi$
$277$	−201151.	−0.157516	−0.0787578	+	0.996894i	$0.525095\pi$
$278$	1.74921e6	1.35747
$279$	0	0
$280$	−1.71116e6	−1.30435
$281$	−1.34807e6	−1.01846	−0.509232	−	0.860629i	$-0.670071\pi$
$281$	−1.34807e6	−1.01846	−0.509232	+	0.860629i	$0.670071\pi$
$282$	0	0
$283$	156785.	0.116369	0.0581845	−	0.998306i	$-0.481469\pi$
$283$	156785.	0.116369	0.0581845	+	0.998306i	$0.481469\pi$
$284$	96865.1	0.0712642
$285$	0	0
$286$	435228.	0.314631
$287$	457554.	0.327897
$288$	0	0
$289$	−1.26832e6	−0.893270
$290$	−1.35760e6	−0.947932
$291$	0	0
$292$	88751.4	0.0609142
$293$	−1.94271e6	−1.32203	−0.661013	−	0.750375i	$-0.729873\pi$
$293$	−1.94271e6	−1.32203	−0.661013	+	0.750375i	$0.729873\pi$
$294$	0	0
$295$	−218772.	−0.146365
$296$	712171.	0.472449
$297$	0	0
$298$	−586379.	−0.382506
$299$	−2680.77	−0.00173413
$300$	0	0
$301$	1.18437e6	0.753480
$302$	709705.	0.447775
$303$	0	0
$304$	831074.	0.515770
$305$	−2.48237e6	−1.52798
$306$	0	0
$307$	−199642.	−0.120894	−0.0604471	−	0.998171i	$-0.519253\pi$
$307$	−199642.	−0.120894	−0.0604471	+	0.998171i	$0.519253\pi$
$308$	−143794.	−0.0863705
$309$	0	0
$310$	160526.	0.0948726
$311$	750035.	0.439725	0.219862	−	0.975531i	$-0.429439\pi$
$311$	750035.	0.439725	0.219862	+	0.975531i	$0.429439\pi$
$312$	0	0
$313$	882012.	0.508878	0.254439	−	0.967089i	$-0.418109\pi$
$313$	882012.	0.508878	0.254439	+	0.967089i	$0.418109\pi$
$314$	−1.78313e6	−1.02061
$315$	0	0
$316$	51272.6	0.0288847
$317$	751345.	0.419944	0.209972	−	0.977707i	$-0.432663\pi$
$317$	751345.	0.419944	0.209972	+	0.977707i	$0.432663\pi$
$318$	0	0
$319$	1.71839e6	0.945462
$320$	1.91571e6	1.04581
$321$	0	0
$322$	15112.2	0.00812247
$323$	298513.	0.159205
$324$	0	0
$325$	−132754.	−0.0697173
$326$	1.31276e6	0.684135
$327$	0	0
$328$	−514383.	−0.263999
$329$	2.43297e6	1.23922
$330$	0	0
$331$	−2.56565e6	−1.28715	−0.643573	−	0.765385i	$-0.722549\pi$
$331$	−2.56565e6	−1.28715	−0.643573	+	0.765385i	$0.722549\pi$
$332$	243501.	0.121243
$333$	0	0
$334$	−1.04765e6	−0.513867
$335$	3.36839e6	1.63987
$336$	0	0
$337$	−147287.	−0.0706465	−0.0353233	−	0.999376i	$-0.511246\pi$
$337$	−147287.	−0.0706465	−0.0353233	+	0.999376i	$0.511246\pi$
$338$	2.01370e6	0.958747
$339$	0	0
$340$	−48740.5	−0.0228661
$341$	−203186.	−0.0946254
$342$	0	0
$343$	1.46207e6	0.671016
$344$	−1.33147e6	−0.606647
$345$	0	0
$346$	4.03224e6	1.81074
$347$	−224073.	−0.0999002	−0.0499501	−	0.998752i	$-0.515906\pi$
$347$	−224073.	−0.0999002	−0.0499501	+	0.998752i	$0.515906\pi$
$348$	0	0
$349$	423627.	0.186174	0.0930871	−	0.995658i	$-0.470326\pi$
$349$	423627.	0.186174	0.0930871	+	0.995658i	$0.470326\pi$
$350$	748371.	0.326548
$351$	0	0
$352$	334040.	0.143695
$353$	2.05559e6	0.878010	0.439005	−	0.898485i	$-0.355331\pi$
$353$	2.05559e6	0.878010	0.439005	+	0.898485i	$0.355331\pi$
$354$	0	0
$355$	3.05575e6	1.28691
$356$	11768.5	0.00492148
$357$	0	0
$358$	1.00778e6	0.415583
$359$	−47825.0	−0.0195848	−0.00979240	−	0.999952i	$-0.503117\pi$
$359$	−47825.0	−0.0195848	−0.00979240	+	0.999952i	$0.503117\pi$
$360$	0	0
$361$	−1.88807e6	−0.762520
$362$	1.15689e6	0.464002
$363$	0	0
$364$	49902.0	0.0197408
$365$	2.79979e6	1.10000
$366$	0	0
$367$	753834.	0.292153	0.146077	−	0.989273i	$-0.453335\pi$
$367$	753834.	0.292153	0.146077	+	0.989273i	$0.453335\pi$
$368$	−18051.0	−0.00694833
$369$	0	0
$370$	−1.49155e6	−0.566414
$371$	−109533.	−0.0413152
$372$	0	0
$373$	3.05748e6	1.13787	0.568934	−	0.822384i	$-0.307356\pi$
$373$	3.05748e6	1.13787	0.568934	+	0.822384i	$0.307356\pi$
$374$	1.05264e6	0.389137
$375$	0	0
$376$	−2.73515e6	−0.997725
$377$	−596343.	−0.216094
$378$	0	0
$379$	3.45858e6	1.23680	0.618401	−	0.785863i	$-0.287781\pi$
$379$	3.45858e6	1.23680	0.618401	+	0.785863i	$0.287781\pi$
$380$	−96011.2	−0.0341085
$381$	0	0
$382$	2.02527e6	0.710109
$383$	−1.69986e6	−0.592129	−0.296065	−	0.955168i	$-0.595674\pi$
$383$	−1.69986e6	−0.592129	−0.296065	+	0.955168i	$0.595674\pi$
$384$	0	0
$385$	−4.53621e6	−1.55970
$386$	2.13431e6	0.729103
$387$	0	0
$388$	−104547.	−0.0352560
$389$	2.23513e6	0.748907	0.374454	−	0.927246i	$-0.377830\pi$
$389$	2.23513e6	0.748907	0.374454	+	0.927246i	$0.377830\pi$
$390$	0	0
$391$	−6483.72	−0.00214478
$392$	1.29679e6	0.426240
$393$	0	0
$394$	1.79053e6	0.581086
$395$	1.61747e6	0.521607
$396$	0	0
$397$	1.97443e6	0.628731	0.314366	−	0.949302i	$-0.398208\pi$
$397$	1.97443e6	0.628731	0.314366	+	0.949302i	$0.398208\pi$
$398$	5.71762e6	1.80929
$399$	0	0
$400$	−893901.	−0.279344
$401$	−712941.	−0.221408	−0.110704	−	0.993853i	$-0.535311\pi$
$401$	−712941.	−0.221408	−0.110704	+	0.993853i	$0.535311\pi$
$402$	0	0
$403$	70513.0	0.0216275
$404$	−178484.	−0.0544059
$405$	0	0
$406$	3.36174e6	1.01216
$407$	1.88793e6	0.564938
$408$	0	0
$409$	4.83618e6	1.42953	0.714767	−	0.699363i	$-0.246533\pi$
$409$	4.83618e6	1.42953	0.714767	+	0.699363i	$0.246533\pi$
$410$	1.07731e6	0.316505
$411$	0	0
$412$	86263.9	0.0250372
$413$	541731.	0.156282
$414$	0	0
$415$	7.68159e6	2.18943
$416$	−115924.	−0.0328428
$417$	0	0
$418$	2.07354e6	0.580460
$419$	568148.	0.158098	0.0790490	−	0.996871i	$-0.474812\pi$
$419$	568148.	0.158098	0.0790490	+	0.996871i	$0.474812\pi$
$420$	0	0
$421$	−4.84229e6	−1.33151	−0.665756	−	0.746169i	$-0.731891\pi$
$421$	−4.84229e6	−1.33151	−0.665756	+	0.746169i	$0.731891\pi$
$422$	−2.27666e6	−0.622326
$423$	0	0
$424$	123137.	0.0332639
$425$	−321080.	−0.0862266
$426$	0	0
$427$	6.14694e6	1.63151
$428$	−154979.	−0.0408944
$429$	0	0
$430$	2.78860e6	0.727301
$431$	−3.21792e6	−0.834415	−0.417208	−	0.908811i	$-0.636991\pi$
$431$	−3.21792e6	−0.834415	−0.417208	+	0.908811i	$0.636991\pi$
$432$	0	0
$433$	−6.18279e6	−1.58477	−0.792383	−	0.610024i	$-0.791160\pi$
$433$	−6.18279e6	−1.58477	−0.792383	+	0.610024i	$0.791160\pi$
$434$	−397500.	−0.101301
$435$	0	0
$436$	−221118.	−0.0557066
$437$	−12771.9	−0.00319928
$438$	0	0
$439$	−518115.	−0.128311	−0.0641557	−	0.997940i	$-0.520435\pi$
$439$	−518115.	−0.128311	−0.0641557	+	0.997940i	$0.520435\pi$
$440$	5.09961e6	1.25576
$441$	0	0
$442$	−365306.	−0.0889409
$443$	492473.	0.119227	0.0596133	−	0.998222i	$-0.481013\pi$
$443$	492473.	0.119227	0.0596133	+	0.998222i	$0.481013\pi$
$444$	0	0
$445$	371254.	0.0888732
$446$	−3.70369e6	−0.881651
$447$	0	0
$448$	−4.74374e6	−1.11667
$449$	−743793.	−0.174115	−0.0870575	−	0.996203i	$-0.527746\pi$
$449$	−743793.	−0.174115	−0.0870575	+	0.996203i	$0.527746\pi$
$450$	0	0
$451$	−1.36360e6	−0.315680
$452$	191106.	0.0439975
$453$	0	0
$454$	712270.	0.162183
$455$	1.57423e6	0.356484
$456$	0	0
$457$	1.90413e6	0.426487	0.213243	−	0.976999i	$-0.431597\pi$
$457$	1.90413e6	0.426487	0.213243	+	0.976999i	$0.431597\pi$
$458$	−1.86690e6	−0.415870
$459$	0	0
$460$	2085.37	0.000459502 0
$461$	−3.48173e6	−0.763031	−0.381516	−	0.924362i	$-0.624598\pi$
$461$	−3.48173e6	−0.763031	−0.381516	+	0.924362i	$0.624598\pi$
$462$	0	0
$463$	−3.41826e6	−0.741058	−0.370529	−	0.928821i	$-0.620824\pi$
$463$	−3.41826e6	−0.741058	−0.370529	+	0.928821i	$0.620824\pi$
$464$	−4.01548e6	−0.865849
$465$	0	0
$466$	−2.27805e6	−0.485958
$467$	7.08422e6	1.50314	0.751571	−	0.659652i	$-0.229297\pi$
$467$	7.08422e6	1.50314	0.751571	+	0.659652i	$0.229297\pi$
$468$	0	0
$469$	−8.34092e6	−1.75098
$470$	5.72841e6	1.19616
$471$	0	0
$472$	−609015.	−0.125827
$473$	−3.52967e6	−0.725406
$474$	0	0
$475$	−632478.	−0.128621
$476$	120693.	0.0244155
$477$	0	0
$478$	−6.08351e6	−1.21782
$479$	3.93012e6	0.782649	0.391325	−	0.920253i	$-0.372017\pi$
$479$	3.93012e6	0.782649	0.391325	+	0.920253i	$0.372017\pi$
$480$	0	0
$481$	−655182.	−0.129122
$482$	6.49642e6	1.27367
$483$	0	0
$484$	107689.	0.0208957
$485$	−3.29810e6	−0.636662
$486$	0	0
$487$	−6.02332e6	−1.15084	−0.575418	−	0.817859i	$-0.695161\pi$
$487$	−6.02332e6	−1.15084	−0.575418	+	0.817859i	$0.695161\pi$
$488$	−6.91039e6	−1.31357
$489$	0	0
$490$	−2.71596e6	−0.511014
$491$	−4.15285e6	−0.777396	−0.388698	−	0.921365i	$-0.627075\pi$
$491$	−4.15285e6	−0.777396	−0.388698	+	0.921365i	$0.627075\pi$
$492$	0	0
$493$	−1.44232e6	−0.267266
$494$	−719596.	−0.132670
$495$	0	0
$496$	474799.	0.0866574
$497$	−7.56677e6	−1.37410
$498$	0	0
$499$	−4.87198e6	−0.875899	−0.437949	−	0.899000i	$-0.644295\pi$
$499$	−4.87198e6	−0.875899	−0.437949	+	0.899000i	$0.644295\pi$
$500$	−287998.	−0.0515187
$501$	0	0
$502$	734084.	0.130013
$503$	437433.	0.0770888	0.0385444	−	0.999257i	$-0.487728\pi$
$503$	437433.	0.0770888	0.0385444	+	0.999257i	$0.487728\pi$
$504$	0	0
$505$	−5.63055e6	−0.982476
$506$	−45037.5	−0.00781983
$507$	0	0
$508$	−623492.	−0.107194
$509$	−1.36574e6	−0.233654	−0.116827	−	0.993152i	$-0.537272\pi$
$509$	−1.36574e6	−0.233654	−0.116827	+	0.993152i	$0.537272\pi$
$510$	0	0
$511$	−6.93295e6	−1.17454
$512$	5.28701e6	0.891323
$513$	0	0
$514$	8.49342e6	1.41799
$515$	2.72132e6	0.452128
$516$	0	0
$517$	−7.25075e6	−1.19304
$518$	3.69343e6	0.604792
$519$	0	0
$520$	−1.76975e6	−0.287015
$521$	1.08212e7	1.74655	0.873274	−	0.487230i	$-0.161993\pi$
$521$	1.08212e7	1.74655	0.873274	+	0.487230i	$0.161993\pi$
$522$	0	0
$523$	−9.64293e6	−1.54154	−0.770770	−	0.637114i	$-0.780128\pi$
$523$	−9.64293e6	−1.54154	−0.770770	+	0.637114i	$0.780128\pi$
$524$	−324306.	−0.0515973
$525$	0	0
$526$	3.20154e6	0.504539
$527$	170543.	0.0267490
$528$	0	0
$529$	−6.43607e6	−0.999957
$530$	−257894.	−0.0398797
$531$	0	0
$532$	237746.	0.0364196
$533$	473221.	0.0721516
$534$	0	0
$535$	−4.88904e6	−0.738481
$536$	9.37687e6	1.40976
$537$	0	0
$538$	6.32469e6	0.942071
$539$	3.43773e6	0.509682
$540$	0	0
$541$	−8.05285e6	−1.18292	−0.591462	−	0.806333i	$-0.701449\pi$
$541$	−8.05285e6	−1.18292	−0.591462	+	0.806333i	$0.701449\pi$
$542$	2.44371e6	0.357315
$543$	0	0
$544$	−280375.	−0.0406201
$545$	−6.97548e6	−1.00596
$546$	0	0
$547$	3.80692e6	0.544008	0.272004	−	0.962296i	$-0.412314\pi$
$547$	3.80692e6	0.544008	0.272004	+	0.962296i	$0.412314\pi$
$548$	−153112.	−0.0217799
$549$	0	0
$550$	−2.23030e6	−0.314381
$551$	−2.84114e6	−0.398670
$552$	0	0
$553$	−4.00524e6	−0.556949
$554$	1.17277e6	0.162345
$555$	0	0
$556$	−597709.	−0.0819979
$557$	−4.56487e6	−0.623434	−0.311717	−	0.950175i	$-0.600904\pi$
$557$	−4.56487e6	−0.623434	−0.311717	+	0.950175i	$0.600904\pi$
$558$	0	0
$559$	1.22493e6	0.165798
$560$	1.06001e7	1.42836
$561$	0	0
$562$	7.85962e6	1.04969
$563$	7.33113e6	0.974765	0.487383	−	0.873188i	$-0.337952\pi$
$563$	7.33113e6	0.974765	0.487383	+	0.873188i	$0.337952\pi$
$564$	0	0
$565$	6.02872e6	0.794518
$566$	−914099.	−0.119937
$567$	0	0
$568$	8.50657e6	1.10633
$569$	−3.71618e6	−0.481189	−0.240595	−	0.970626i	$-0.577342\pi$
$569$	−3.71618e6	−0.481189	−0.240595	+	0.970626i	$0.577342\pi$
$570$	0	0
$571$	1.12698e7	1.44653	0.723265	−	0.690570i	$-0.242640\pi$
$571$	1.12698e7	1.44653	0.723265	+	0.690570i	$0.242640\pi$
$572$	−148718.	−0.0190053
$573$	0	0
$574$	−2.66767e6	−0.337950
$575$	13737.4	0.00173275
$576$	0	0
$577$	3.31156e6	0.414089	0.207044	−	0.978332i	$-0.433616\pi$
$577$	3.31156e6	0.414089	0.207044	+	0.978332i	$0.433616\pi$
$578$	7.39464e6	0.920656
$579$	0	0
$580$	463895.	0.0572597
$581$	−1.90214e7	−2.33778
$582$	0	0
$583$	326430.	0.0397758
$584$	7.79403e6	0.945650
$585$	0	0
$586$	1.13266e7	1.36256
$587$	−8.18669e6	−0.980648	−0.490324	−	0.871540i	$-0.663122\pi$
$587$	−8.18669e6	−0.980648	−0.490324	+	0.871540i	$0.663122\pi$
$588$	0	0
$589$	335943.	0.0399004
$590$	1.27550e6	0.150852
$591$	0	0
$592$	−4.41167e6	−0.517367
$593$	−2.97789e6	−0.347754	−0.173877	−	0.984767i	$-0.555630\pi$
$593$	−2.97789e6	−0.347754	−0.173877	+	0.984767i	$0.555630\pi$
$594$	0	0
$595$	3.80744e6	0.440900
$596$	200367.	0.0231052
$597$	0	0
$598$	15629.7	0.00178730
$599$	−1.51735e7	−1.72790	−0.863951	−	0.503575i	$-0.832018\pi$
$599$	−1.51735e7	−1.72790	−0.863951	+	0.503575i	$0.832018\pi$
$600$	0	0
$601$	244714.	0.0276358	0.0138179	−	0.999905i	$-0.495601\pi$
$601$	244714.	0.0276358	0.0138179	+	0.999905i	$0.495601\pi$
$602$	−6.90522e6	−0.776581
$603$	0	0
$604$	−242507.	−0.0270478
$605$	3.39719e6	0.377339
$606$	0	0
$607$	−813197.	−0.0895827	−0.0447914	−	0.998996i	$-0.514262\pi$
$607$	−813197.	−0.0895827	−0.0447914	+	0.998996i	$0.514262\pi$
$608$	−552294.	−0.0605915
$609$	0	0
$610$	1.44729e7	1.57482
$611$	2.51628e6	0.272681
$612$	0	0
$613$	−2.18564e6	−0.234924	−0.117462	−	0.993077i	$-0.537476\pi$
$613$	−2.18564e6	−0.234924	−0.117462	+	0.993077i	$0.537476\pi$
$614$	1.16397e6	0.124601
$615$	0	0
$616$	−1.26278e7	−1.34084
$617$	−2.54414e6	−0.269046	−0.134523	−	0.990910i	$-0.542950\pi$
$617$	−2.54414e6	−0.269046	−0.134523	+	0.990910i	$0.542950\pi$
$618$	0	0
$619$	1.32006e7	1.38474	0.692368	−	0.721545i	$-0.256568\pi$
$619$	1.32006e7	1.38474	0.692368	+	0.721545i	$0.256568\pi$
$620$	−54852.0	−0.00573077
$621$	0	0
$622$	−4.37292e6	−0.453206
$623$	−919313.	−0.0948950
$624$	0	0
$625$	−1.16628e7	−1.19427
$626$	−5.14238e6	−0.524479
$627$	0	0
$628$	609299.	0.0616497
$629$	−1.58463e6	−0.159698
$630$	0	0
$631$	1.36163e7	1.36140	0.680700	−	0.732563i	$-0.261676\pi$
$631$	1.36163e7	1.36140	0.680700	+	0.732563i	$0.261676\pi$
$632$	4.50270e6	0.448415
$633$	0	0
$634$	−4.38056e6	−0.432819
$635$	−1.96690e7	−1.93574
$636$	0	0
$637$	−1.19302e6	−0.116493
$638$	−1.00187e7	−0.974449
$639$	0	0
$640$	−1.26176e7	−1.21766
$641$	−1.89460e7	−1.82126	−0.910632	−	0.413218i	$-0.864405\pi$
$641$	−1.89460e7	−1.82126	−0.910632	+	0.413218i	$0.864405\pi$
$642$	0	0
$643$	9.68356e6	0.923651	0.461825	−	0.886971i	$-0.347195\pi$
$643$	9.68356e6	0.923651	0.461825	+	0.886971i	$0.347195\pi$
$644$	−5163.86	−0.000490637 0
$645$	0	0
$646$	−1.74042e6	−0.164086
$647$	1.12969e6	0.106096	0.0530482	−	0.998592i	$-0.483106\pi$
$647$	1.12969e6	0.106096	0.0530482	+	0.998592i	$0.483106\pi$
$648$	0	0
$649$	−1.61447e6	−0.150459
$650$	773996.	0.0718547
$651$	0	0
$652$	−448572.	−0.0413251
$653$	−1.14766e7	−1.05325	−0.526624	−	0.850098i	$-0.676542\pi$
$653$	−1.14766e7	−1.05325	−0.526624	+	0.850098i	$0.676542\pi$
$654$	0	0
$655$	−1.02307e7	−0.931757
$656$	3.18643e6	0.289098
$657$	0	0
$658$	−1.41849e7	−1.27721
$659$	−1.98748e7	−1.78274	−0.891372	−	0.453272i	$-0.850257\pi$
$659$	−1.98748e7	−1.78274	−0.891372	+	0.453272i	$0.850257\pi$
$660$	0	0
$661$	1.44698e7	1.28813	0.644066	−	0.764970i	$-0.277246\pi$
$661$	1.44698e7	1.28813	0.644066	+	0.764970i	$0.277246\pi$
$662$	1.49585e7	1.32661
$663$	0	0
$664$	2.13839e7	1.88221
$665$	7.50006e6	0.657674
$666$	0	0
$667$	61709.7	0.00537080
$668$	357984.	0.0310401
$669$	0	0
$670$	−1.96386e7	−1.69015
$671$	−1.83191e7	−1.57072
$672$	0	0
$673$	228842.	0.0194759	0.00973797	−	0.999953i	$-0.496900\pi$
$673$	228842.	0.0194759	0.00973797	+	0.999953i	$0.496900\pi$
$674$	858727.	0.0728124
$675$	0	0
$676$	−688086.	−0.0579130
$677$	2.30695e6	0.193449	0.0967246	−	0.995311i	$-0.469163\pi$
$677$	2.30695e6	0.193449	0.0967246	+	0.995311i	$0.469163\pi$
$678$	0	0
$679$	8.16687e6	0.679800
$680$	−4.28033e6	−0.354981
$681$	0	0
$682$	1.18463e6	0.0975265
$683$	−1.63779e6	−0.134341	−0.0671703	−	0.997742i	$-0.521397\pi$
$683$	−1.63779e6	−0.134341	−0.0671703	+	0.997742i	$0.521397\pi$
$684$	0	0
$685$	−4.83013e6	−0.393307
$686$	−8.52429e6	−0.691589
$687$	0	0
$688$	8.24803e6	0.664323
$689$	−113283.	−0.00909113
$690$	0	0
$691$	2.85561e6	0.227512	0.113756	−	0.993509i	$-0.463712\pi$
$691$	2.85561e6	0.227512	0.113756	+	0.993509i	$0.463712\pi$
$692$	−1.37782e6	−0.109378
$693$	0	0
$694$	1.30641e6	0.102963
$695$	−1.88556e7	−1.48074
$696$	0	0
$697$	1.14453e6	0.0892373
$698$	−2.46986e6	−0.191882
$699$	0	0
$700$	−255720.	−0.0197251
$701$	−2.04262e7	−1.56997	−0.784986	−	0.619513i	$-0.787330\pi$
$701$	−2.04262e7	−1.56997	−0.784986	+	0.619513i	$0.787330\pi$
$702$	0	0
$703$	−3.12147e6	−0.238216
$704$	1.41373e7	1.07507
$705$	0	0
$706$	−1.19847e7	−0.904929
$707$	1.39426e7	1.04905
$708$	0	0
$709$	9.69596e6	0.724394	0.362197	−	0.932101i	$-0.382027\pi$
$709$	9.69596e6	0.724394	0.362197	+	0.932101i	$0.382027\pi$
$710$	−1.78159e7	−1.32636
$711$	0	0
$712$	1.03349e6	0.0764025
$713$	−7296.69	−0.000537530 0
$714$	0	0
$715$	−4.69153e6	−0.343202
$716$	−344360.	−0.0251033
$717$	0	0
$718$	278834.	0.0201852
$719$	3.89912e6	0.281284	0.140642	−	0.990061i	$-0.455083\pi$
$719$	3.89912e6	0.281284	0.140642	+	0.990061i	$0.455083\pi$
$720$	0	0
$721$	−6.73864e6	−0.482763
$722$	1.10080e7	0.785897
$723$	0	0
$724$	−395311.	−0.0280280
$725$	3.05592e6	0.215922
$726$	0	0
$727$	2.55226e7	1.79097	0.895486	−	0.445090i	$-0.146828\pi$
$727$	2.55226e7	1.79097	0.895486	+	0.445090i	$0.146828\pi$
$728$	4.38233e6	0.306462
$729$	0	0
$730$	−1.63236e7	−1.13373
$731$	2.96261e6	0.205060
$732$	0	0
$733$	1.20778e7	0.830283	0.415142	−	0.909757i	$-0.363732\pi$
$733$	1.20778e7	0.830283	0.415142	+	0.909757i	$0.363732\pi$
$734$	−4.39507e6	−0.301110
$735$	0	0
$736$	11995.8	0.000816275 0
$737$	2.48577e7	1.68574
$738$	0	0
$739$	1.08285e7	0.729386	0.364693	−	0.931128i	$-0.381174\pi$
$739$	1.08285e7	0.729386	0.364693	+	0.931128i	$0.381174\pi$
$740$	509665.	0.0342141
$741$	0	0
$742$	638608.	0.0425818
$743$	1.08360e7	0.720104	0.360052	−	0.932932i	$-0.382759\pi$
$743$	1.08360e7	0.720104	0.360052	+	0.932932i	$0.382759\pi$
$744$	0	0
$745$	6.32086e6	0.417240
$746$	−1.78260e7	−1.17275
$747$	0	0
$748$	−359690.	−0.0235058
$749$	1.21064e7	0.788518
$750$	0	0
$751$	1.77442e7	1.14804	0.574019	−	0.818842i	$-0.305384\pi$
$751$	1.77442e7	1.14804	0.574019	+	0.818842i	$0.305384\pi$
$752$	1.69433e7	1.09258
$753$	0	0
$754$	3.47685e6	0.222719
$755$	−7.65025e6	−0.488436
$756$	0	0
$757$	−1.28953e7	−0.817887	−0.408943	−	0.912560i	$-0.634103\pi$
$757$	−1.28953e7	−0.817887	−0.408943	+	0.912560i	$0.634103\pi$
$758$	−2.01645e7	−1.27472
$759$	0	0
$760$	−8.43158e6	−0.529511
$761$	1.84325e6	0.115378	0.0576891	−	0.998335i	$-0.481627\pi$
$761$	1.84325e6	0.115378	0.0576891	+	0.998335i	$0.481627\pi$
$762$	0	0
$763$	1.72729e7	1.07412
$764$	−692039.	−0.0428941
$765$	0	0
$766$	9.91068e6	0.610283
$767$	560280.	0.0343888
$768$	0	0
$769$	2.15859e7	1.31630	0.658148	−	0.752888i	$-0.271340\pi$
$769$	2.15859e7	1.31630	0.658148	+	0.752888i	$0.271340\pi$
$770$	2.64474e7	1.60752
$771$	0	0
$772$	−729296.	−0.0440414
$773$	−6.72583e6	−0.404853	−0.202426	−	0.979297i	$-0.564883\pi$
$773$	−6.72583e6	−0.404853	−0.202426	+	0.979297i	$0.564883\pi$
$774$	0	0
$775$	−361340.	−0.0216103
$776$	−9.18121e6	−0.547325
$777$	0	0
$778$	−1.30314e7	−0.771867
$779$	2.25455e6	0.133112
$780$	0	0
$781$	2.25505e7	1.32291
$782$	37801.9	0.00221053
$783$	0	0
$784$	−8.03318e6	−0.466764
$785$	1.92212e7	1.11329
$786$	0	0
$787$	2.77363e7	1.59629	0.798144	−	0.602467i	$-0.205815\pi$
$787$	2.77363e7	1.59629	0.798144	+	0.602467i	$0.205815\pi$
$788$	−611826.	−0.0351004
$789$	0	0
$790$	−9.43031e6	−0.537599
$791$	−1.49285e7	−0.848352
$792$	0	0
$793$	6.35741e6	0.359002
$794$	−1.15115e7	−0.648007
$795$	0	0
$796$	−1.95372e6	−0.109290
$797$	−1.62250e7	−0.904770	−0.452385	−	0.891823i	$-0.649427\pi$
$797$	−1.62250e7	−0.904770	−0.452385	+	0.891823i	$0.649427\pi$
$798$	0	0
$799$	6.08587e6	0.337253
$800$	594046.	0.0328167
$801$	0	0
$802$	4.15665e6	0.228196
$803$	2.06616e7	1.13077
$804$	0	0
$805$	−162902.	−0.00886004
$806$	−411111.	−0.0222906
$807$	0	0
$808$	−1.56742e7	−0.844614
$809$	−1.97769e6	−0.106240	−0.0531200	−	0.998588i	$-0.516917\pi$
$809$	−1.97769e6	−0.106240	−0.0531200	+	0.998588i	$0.516917\pi$
$810$	0	0
$811$	−1.88822e7	−1.00809	−0.504046	−	0.863677i	$-0.668156\pi$
$811$	−1.88822e7	−1.00809	−0.504046	+	0.863677i	$0.668156\pi$
$812$	−1.14871e6	−0.0611394
$813$	0	0
$814$	−1.10072e7	−0.582258
$815$	−1.41509e7	−0.746258
$816$	0	0
$817$	5.83588e6	0.305880
$818$	−2.81963e7	−1.47336
$819$	0	0
$820$	−368118.	−0.0191184
$821$	7.00063e6	0.362476	0.181238	−	0.983439i	$-0.441990\pi$
$821$	7.00063e6	0.362476	0.181238	+	0.983439i	$0.441990\pi$
$822$	0	0
$823$	−2.95427e7	−1.52037	−0.760186	−	0.649705i	$-0.774892\pi$
$823$	−2.95427e7	−1.52037	−0.760186	+	0.649705i	$0.774892\pi$
$824$	7.57558e6	0.388685
$825$	0	0
$826$	−3.15845e6	−0.161073
$827$	−2.31544e7	−1.17726	−0.588628	−	0.808404i	$-0.700332\pi$
$827$	−2.31544e7	−1.17726	−0.588628	+	0.808404i	$0.700332\pi$
$828$	0	0
$829$	−4.64372e6	−0.234682	−0.117341	−	0.993092i	$-0.537437\pi$
$829$	−4.64372e6	−0.234682	−0.117341	+	0.993092i	$0.537437\pi$
$830$	−4.47858e7	−2.25655
$831$	0	0
$832$	−4.90617e6	−0.245717
$833$	−2.88544e6	−0.144078
$834$	0	0
$835$	1.12931e7	0.560529
$836$	−708533.	−0.0350626
$837$	0	0
$838$	−3.31247e6	−0.162945
$839$	1.94063e7	0.951783	0.475892	−	0.879504i	$-0.342125\pi$
$839$	1.94063e7	0.951783	0.475892	+	0.879504i	$0.342125\pi$
$840$	0	0
$841$	−6.78369e6	−0.330732
$842$	2.82319e7	1.37233
$843$	0	0
$844$	777940.	0.0375915
$845$	−2.17067e7	−1.04581
$846$	0	0
$847$	−8.41226e6	−0.402906
$848$	−762793.	−0.0364265
$849$	0	0
$850$	1.87199e6	0.0888701
$851$	67798.3	0.00320919
$852$	0	0
$853$	−3.30655e6	−0.155598	−0.0777988	−	0.996969i	$-0.524789\pi$
$853$	−3.30655e6	−0.155598	−0.0777988	+	0.996969i	$0.524789\pi$
$854$	−3.58384e7	−1.68153
$855$	0	0
$856$	−1.36101e7	−0.634856
$857$	1.78379e7	0.829646	0.414823	−	0.909902i	$-0.363844\pi$
$857$	1.78379e7	0.829646	0.414823	+	0.909902i	$0.363844\pi$
$858$	0	0
$859$	2.88455e7	1.33381	0.666907	−	0.745141i	$-0.267618\pi$
$859$	2.88455e7	1.33381	0.666907	+	0.745141i	$0.267618\pi$
$860$	−952867.	−0.0439325
$861$	0	0
$862$	1.87614e7	0.859997
$863$	−1.11663e7	−0.510369	−0.255184	−	0.966892i	$-0.582136\pi$
$863$	−1.11663e7	−0.510369	−0.255184	+	0.966892i	$0.582136\pi$
$864$	0	0
$865$	−4.34654e7	−1.97517
$866$	3.60474e7	1.63335
$867$	0	0
$868$	135826.	0.00611906
$869$	1.19364e7	0.536198
$870$	0	0
$871$	−8.62652e6	−0.385292
$872$	−1.94183e7	−0.864806
$873$	0	0
$874$	74463.9	0.00329737
$875$	2.24974e7	0.993374
$876$	0	0
$877$	6.06570e6	0.266307	0.133153	−	0.991095i	$-0.457490\pi$
$877$	6.06570e6	0.266307	0.133153	+	0.991095i	$0.457490\pi$
$878$	3.02076e6	0.132245
$879$	0	0
$880$	−3.15904e7	−1.37515
$881$	−2.61985e7	−1.13720	−0.568601	−	0.822614i	$-0.692515\pi$
$881$	−2.61985e7	−1.13720	−0.568601	+	0.822614i	$0.692515\pi$
$882$	0	0
$883$	−1.07559e7	−0.464244	−0.232122	−	0.972687i	$-0.574567\pi$
$883$	−1.07559e7	−0.464244	−0.232122	+	0.972687i	$0.574567\pi$
$884$	124826.	0.00537246
$885$	0	0
$886$	−2.87126e6	−0.122882
$887$	−1.01637e7	−0.433754	−0.216877	−	0.976199i	$-0.569587\pi$
$887$	−1.01637e7	−0.433754	−0.216877	+	0.976199i	$0.569587\pi$
$888$	0	0
$889$	4.87050e7	2.06690
$890$	−2.16452e6	−0.0915980
$891$	0	0
$892$	1.26556e6	0.0532560
$893$	1.19882e7	0.503067
$894$	0	0
$895$	−1.08633e7	−0.453321
$896$	3.12441e7	1.30016
$897$	0	0
$898$	4.33653e6	0.179453
$899$	−1.62317e6	−0.0669829
$900$	0	0
$901$	−273987.	−0.0112439
$902$	7.95020e6	0.325358
$903$	0	0
$904$	1.67827e7	0.683031
$905$	−1.24707e7	−0.506137
$906$	0	0
$907$	1.40328e7	0.566404	0.283202	−	0.959060i	$-0.408603\pi$
$907$	1.40328e7	0.566404	0.283202	+	0.959060i	$0.408603\pi$
$908$	−243384.	−0.00979664
$909$	0	0
$910$	−9.17821e6	−0.367413
$911$	1.38761e6	0.0553952	0.0276976	−	0.999616i	$-0.491182\pi$
$911$	1.38761e6	0.0553952	0.0276976	+	0.999616i	$0.491182\pi$
$912$	0	0
$913$	5.66878e7	2.25067
$914$	−1.11016e7	−0.439562
$915$	0	0
$916$	637923.	0.0251206
$917$	2.53337e7	0.994890
$918$	0	0
$919$	−3.05384e7	−1.19277	−0.596387	−	0.802697i	$-0.703397\pi$
$919$	−3.05384e7	−1.19277	−0.596387	+	0.802697i	$0.703397\pi$
$920$	183134.	0.00713345
$921$	0	0
$922$	2.02995e7	0.786425
$923$	−7.82586e6	−0.302362
$924$	0	0
$925$	3.35744e6	0.129019
$926$	1.99294e7	0.763778
$927$	0	0
$928$	2.66850e6	0.101718
$929$	3.43014e7	1.30399	0.651993	−	0.758225i	$-0.273933\pi$
$929$	3.43014e7	1.30399	0.651993	+	0.758225i	$0.273933\pi$
$930$	0	0
$931$	−5.68386e6	−0.214916
$932$	778414.	0.0293542
$933$	0	0
$934$	−4.13030e7	−1.54923
$935$	−1.13470e7	−0.424473
$936$	0	0
$937$	−4.23607e7	−1.57621	−0.788105	−	0.615541i	$-0.788938\pi$
$937$	−4.23607e7	−1.57621	−0.788105	+	0.615541i	$0.788938\pi$
$938$	4.86299e7	1.80467
$939$	0	0
$940$	−1.95741e6	−0.0722539
$941$	2.11497e7	0.778629	0.389315	−	0.921105i	$-0.372712\pi$
$941$	2.11497e7	0.778629	0.389315	+	0.921105i	$0.372712\pi$
$942$	0	0
$943$	−48968.9	−0.00179325
$944$	3.77265e6	0.137789
$945$	0	0
$946$	2.05790e7	0.747646
$947$	8.73518e6	0.316517	0.158258	−	0.987398i	$-0.449412\pi$
$947$	8.73518e6	0.316517	0.158258	+	0.987398i	$0.449412\pi$
$948$	0	0
$949$	−7.17034e6	−0.258449
$950$	3.68752e6	0.132564
$951$	0	0
$952$	1.05991e7	0.379033
$953$	−2.53043e7	−0.902531	−0.451265	−	0.892390i	$-0.649027\pi$
$953$	−2.53043e7	−0.902531	−0.451265	+	0.892390i	$0.649027\pi$
$954$	0	0
$955$	−2.18314e7	−0.774592
$956$	2.07874e6	0.0735624
$957$	0	0
$958$	−2.29137e7	−0.806644
$959$	1.19605e7	0.419957
$960$	0	0
$961$	−2.84372e7	−0.993296
$962$	3.81990e6	0.133080
$963$	0	0
$964$	−2.21984e6	−0.0769358
$965$	−2.30067e7	−0.795310
$966$	0	0
$967$	6.72304e6	0.231206	0.115603	−	0.993295i	$-0.463120\pi$
$967$	6.72304e6	0.231206	0.115603	+	0.993295i	$0.463120\pi$
$968$	9.45707e6	0.324391
$969$	0	0
$970$	1.92288e7	0.656181
$971$	1.62157e7	0.551935	0.275968	−	0.961167i	$-0.411002\pi$
$971$	1.62157e7	0.551935	0.275968	+	0.961167i	$0.411002\pi$
$972$	0	0
$973$	4.66910e7	1.58107
$974$	3.51177e7	1.18612
$975$	0	0
$976$	4.28076e7	1.43846
$977$	−4.34464e7	−1.45619	−0.728094	−	0.685477i	$-0.759594\pi$
$977$	−4.34464e7	−1.45619	−0.728094	+	0.685477i	$0.759594\pi$
$978$	0	0
$979$	2.73974e6	0.0913593
$980$	928046.	0.0308677
$981$	0	0
$982$	2.42123e7	0.801230
$983$	1.02054e7	0.336857	0.168428	−	0.985714i	$-0.446131\pi$
$983$	1.02054e7	0.336857	0.168428	+	0.985714i	$0.446131\pi$
$984$	0	0
$985$	−1.93010e7	−0.633853
$986$	8.40912e6	0.275460
$987$	0	0
$988$	245887.	0.00801389
$989$	−126755.	−0.00412075
$990$	0	0
$991$	−4.23429e7	−1.36961	−0.684804	−	0.728727i	$-0.740112\pi$
$991$	−4.23429e7	−1.36961	−0.684804	+	0.728727i	$0.740112\pi$
$992$	−315530.	−0.0101803
$993$	0	0
$994$	4.41164e7	1.41623
$995$	−6.16329e7	−1.97358
$996$	0	0
$997$	−4.62288e7	−1.47290	−0.736452	−	0.676490i	$-0.763500\pi$
$997$	−4.62288e7	−1.47290	−0.736452	+	0.676490i	$0.763500\pi$
$998$	2.84050e7	0.902752
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.g.1.8	✓	25
3.2	odd	2		531.6.a.h.1.18	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.6.a.g.1.8	✓	25	1.1	even	1	trivial
531.6.a.h.1.18	yes	25	3.2	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-5.83028 q^{2} +1.99222 q^{4} +62.8474 q^{5} -155.625 q^{7} +174.954 q^{8} +O(q^{10})\)	\(q-5.83028 q^{2} +1.99222 q^{4} +62.8474 q^{5} -155.625 q^{7} +174.954 q^{8} -366.418 q^{10} +463.795 q^{11} -160.954 q^{13} +907.339 q^{14} -1083.78 q^{16} -389.283 q^{17} -766.828 q^{19} +125.206 q^{20} -2704.06 q^{22} +16.6555 q^{23} +824.798 q^{25} +938.407 q^{26} -310.039 q^{28} +3705.06 q^{29} -438.095 q^{31} +720.232 q^{32} +2269.63 q^{34} -9780.64 q^{35} +4070.62 q^{37} +4470.82 q^{38} +10995.4 q^{40} -2940.10 q^{41} -7610.41 q^{43} +923.980 q^{44} -97.1064 q^{46} -15633.5 q^{47} +7412.17 q^{49} -4808.81 q^{50} -320.655 q^{52} +703.825 q^{53} +29148.3 q^{55} -27227.2 q^{56} -21601.5 q^{58} -3481.00 q^{59} -39498.4 q^{61} +2554.22 q^{62} +30481.9 q^{64} -10115.5 q^{65} +53596.2 q^{67} -775.537 q^{68} +57023.9 q^{70} +48621.8 q^{71} +44549.1 q^{73} -23732.9 q^{74} -1527.69 q^{76} -72178.1 q^{77} +25736.5 q^{79} -68112.9 q^{80} +17141.6 q^{82} +122226. q^{83} -24465.5 q^{85} +44370.9 q^{86} +81142.7 q^{88} +5907.23 q^{89} +25048.5 q^{91} +33.1814 q^{92} +91147.9 q^{94} -48193.1 q^{95} -52477.9 q^{97} -43215.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 12 q^{2} + 400 q^{4} - 200 q^{5} - 38 q^{7} - 576 q^{8}+O(q^{10}) \) 25 * q - 12 * q^2 + 400 * q^4 - 200 * q^5 - 38 * q^7 - 576 * q^8	\( 25 q - 12 q^{2} + 400 q^{4} - 200 q^{5} - 38 q^{7} - 576 q^{8} + 370 q^{10} - 1210 q^{11} - 136 q^{13} - 1176 q^{14} + 5884 q^{16} - 2312 q^{17} + 1722 q^{19} - 944 q^{20} + 1700 q^{22} - 4476 q^{23} + 17547 q^{25} - 10320 q^{26} - 552 q^{28} - 30036 q^{29} - 822 q^{31} - 44684 q^{32} - 4240 q^{34} - 14700 q^{35} - 1220 q^{37} - 58378 q^{38} - 11890 q^{40} - 43656 q^{41} + 18476 q^{43} - 58080 q^{44} + 25322 q^{46} - 43962 q^{47} + 62015 q^{49} - 167260 q^{50} + 11740 q^{52} - 79780 q^{53} - 42830 q^{55} - 161888 q^{56} + 14274 q^{58} - 87025 q^{59} - 24390 q^{61} - 73224 q^{62} - 13574 q^{64} - 43378 q^{65} - 14168 q^{67} - 202548 q^{68} + 518 q^{70} - 27924 q^{71} + 44646 q^{73} - 186092 q^{74} + 56816 q^{76} - 207738 q^{77} + 21128 q^{79} - 150020 q^{80} + 84138 q^{82} - 84150 q^{83} + 7454 q^{85} - 376480 q^{86} - 54018 q^{88} - 227092 q^{89} + 21768 q^{91} - 470414 q^{92} - 83352 q^{94} - 230580 q^{95} + 177344 q^{97} - 299700 q^{98}+O(q^{100}) \) 25 * q - 12 * q^2 + 400 * q^4 - 200 * q^5 - 38 * q^7 - 576 * q^8 + 370 * q^10 - 1210 * q^11 - 136 * q^13 - 1176 * q^14 + 5884 * q^16 - 2312 * q^17 + 1722 * q^19 - 944 * q^20 + 1700 * q^22 - 4476 * q^23 + 17547 * q^25 - 10320 * q^26 - 552 * q^28 - 30036 * q^29 - 822 * q^31 - 44684 * q^32 - 4240 * q^34 - 14700 * q^35 - 1220 * q^37 - 58378 * q^38 - 11890 * q^40 - 43656 * q^41 + 18476 * q^43 - 58080 * q^44 + 25322 * q^46 - 43962 * q^47 + 62015 * q^49 - 167260 * q^50 + 11740 * q^52 - 79780 * q^53 - 42830 * q^55 - 161888 * q^56 + 14274 * q^58 - 87025 * q^59 - 24390 * q^61 - 73224 * q^62 - 13574 * q^64 - 43378 * q^65 - 14168 * q^67 - 202548 * q^68 + 518 * q^70 - 27924 * q^71 + 44646 * q^73 - 186092 * q^74 + 56816 * q^76 - 207738 * q^77 + 21128 * q^79 - 150020 * q^80 + 84138 * q^82 - 84150 * q^83 + 7454 * q^85 - 376480 * q^86 - 54018 * q^88 - 227092 * q^89 + 21768 * q^91 - 470414 * q^92 - 83352 * q^94 - 230580 * q^95 + 177344 * q^97 - 299700 * q^98

Introduction

L-functions

Modular forms

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Database

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