LMFDB - Embedded newform 825.6.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,6,Mod(1,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,27,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.47894$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.47894 q^{2} +9.00000 q^{3} -11.9391 q^{4} +40.3105 q^{6} +168.818 q^{7} -196.801 q^{8} +81.0000 q^{9} +121.000 q^{11} -107.452 q^{12} -290.259 q^{13} +756.126 q^{14} -499.408 q^{16} -623.964 q^{17} +362.794 q^{18} -398.456 q^{19} +1519.36 q^{21} +541.952 q^{22} -3788.06 q^{23} -1771.21 q^{24} -1300.06 q^{26} +729.000 q^{27} -2015.53 q^{28} +4220.43 q^{29} -5594.57 q^{31} +4060.80 q^{32} +1089.00 q^{33} -2794.70 q^{34} -967.065 q^{36} -301.266 q^{37} -1784.66 q^{38} -2612.33 q^{39} -14636.2 q^{41} +6805.14 q^{42} -151.418 q^{43} -1444.63 q^{44} -16966.5 q^{46} +13535.0 q^{47} -4494.67 q^{48} +11692.5 q^{49} -5615.68 q^{51} +3465.43 q^{52} +18116.1 q^{53} +3265.15 q^{54} -33223.5 q^{56} -3586.10 q^{57} +18903.0 q^{58} -50582.4 q^{59} +5984.35 q^{61} -25057.8 q^{62} +13674.3 q^{63} +34169.1 q^{64} +4877.57 q^{66} -22450.7 q^{67} +7449.56 q^{68} -34092.6 q^{69} +10017.6 q^{71} -15940.8 q^{72} +476.148 q^{73} -1349.35 q^{74} +4757.19 q^{76} +20427.0 q^{77} -11700.5 q^{78} -85024.7 q^{79} +6561.00 q^{81} -65554.5 q^{82} +25643.4 q^{83} -18139.8 q^{84} -678.194 q^{86} +37983.8 q^{87} -23812.9 q^{88} +1807.91 q^{89} -49001.0 q^{91} +45226.0 q^{92} -50351.2 q^{93} +60622.4 q^{94} +36547.2 q^{96} -11688.5 q^{97} +52370.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22}+ \cdots + 29403 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 + 363 * q^11 - 378 * q^12 - 290 * q^13 + 916 * q^14 - 590 * q^16 - 434 * q^17 - 162 * q^18 - 2856 * q^19 + 612 * q^21 - 242 * q^22 + 640 * q^23 + 216 * q^24 + 2132 * q^26 + 2187 * q^27 + 580 * q^28 - 4538 * q^29 - 14968 * q^31 + 2496 * q^32 + 3267 * q^33 - 13704 * q^34 - 3402 * q^36 + 6190 * q^37 + 11668 * q^38 - 2610 * q^39 - 8926 * q^41 + 8244 * q^42 + 33592 * q^43 - 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 5310 * q^48 - 14693 * q^49 - 3906 * q^51 - 18780 * q^52 + 22934 * q^53 - 1458 * q^54 - 40012 * q^56 - 25704 * q^57 + 32304 * q^58 - 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 5508 * q^63 + 35474 * q^64 - 2178 * q^66 - 16868 * q^67 + 71288 * q^68 + 5760 * q^69 + 4856 * q^71 + 1944 * q^72 - 1910 * q^73 + 29404 * q^74 + 6116 * q^76 + 8228 * q^77 + 19188 * q^78 - 36844 * q^79 + 19683 * q^81 - 84000 * q^82 + 48796 * q^83 + 5220 * q^84 - 83492 * q^86 - 40842 * q^87 + 2904 * q^88 - 188978 * q^89 - 93208 * q^91 + 6976 * q^92 - 134712 * q^93 + 70472 * q^94 + 22464 * q^96 - 247526 * q^97 + 154654 * q^98 + 29403 * q^99

Coefficient data

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

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

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

$2$	4.47894	0.791773	0.395886	−	0.918300i	$-0.370437\pi$
$2$	4.47894	0.791773	0.395886	+	0.918300i	$0.370437\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−11.9391	−0.373096
$5$	0	0
$5$	0	0
$6$	40.3105	0.457130
$7$	168.818	1.30219	0.651094	−	0.758997i	$-0.274310\pi$
$7$	168.818	1.30219	0.651094	+	0.758997i	$0.274310\pi$
$8$	−196.801	−1.08718
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−107.452	−0.215407
$13$	−290.259	−0.476352	−0.238176	−	0.971222i	$-0.576549\pi$
$13$	−290.259	−0.476352	−0.238176	+	0.971222i	$0.576549\pi$
$14$	756.126	1.03104
$15$	0	0
$16$	−499.408	−0.487703
$17$	−623.964	−0.523645	−0.261823	−	0.965116i	$-0.584324\pi$
$17$	−623.964	−0.523645	−0.261823	+	0.965116i	$0.584324\pi$
$18$	362.794	0.263924
$19$	−398.456	−0.253219	−0.126609	−	0.991953i	$-0.540409\pi$
$19$	−398.456	−0.253219	−0.126609	+	0.991953i	$0.540409\pi$
$20$	0	0
$21$	1519.36	0.751819
$22$	541.952	0.238728
$23$	−3788.06	−1.49313	−0.746565	−	0.665312i	$-0.768298\pi$
$23$	−3788.06	−1.49313	−0.746565	+	0.665312i	$0.768298\pi$
$24$	−1771.21	−0.627684
$25$	0	0
$26$	−1300.06	−0.377162
$27$	729.000	0.192450
$28$	−2015.53	−0.485841
$29$	4220.43	0.931883	0.465941	−	0.884816i	$-0.345716\pi$
$29$	4220.43	0.931883	0.465941	+	0.884816i	$0.345716\pi$
$30$	0	0
$31$	−5594.57	−1.04559	−0.522797	−	0.852457i	$-0.675111\pi$
$31$	−5594.57	−1.04559	−0.522797	+	0.852457i	$0.675111\pi$
$32$	4060.80	0.701030
$33$	1089.00	0.174078
$34$	−2794.70	−0.414608
$35$	0	0
$36$	−967.065	−0.124365
$37$	−301.266	−0.0361781	−0.0180890	−	0.999836i	$-0.505758\pi$
$37$	−301.266	−0.0361781	−0.0180890	+	0.999836i	$0.505758\pi$
$38$	−1784.66	−0.200492
$39$	−2612.33	−0.275022
$40$	0	0
$41$	−14636.2	−1.35978	−0.679889	−	0.733315i	$-0.737972\pi$
$41$	−14636.2	−1.35978	−0.679889	+	0.733315i	$0.737972\pi$
$42$	6805.14	0.595269
$43$	−151.418	−0.0124884	−0.00624421	−	0.999981i	$-0.501988\pi$
$43$	−151.418	−0.0124884	−0.00624421	+	0.999981i	$0.501988\pi$
$44$	−1444.63	−0.112493
$45$	0	0
$46$	−16966.5	−1.18222
$47$	13535.0	0.893743	0.446872	−	0.894598i	$-0.352538\pi$
$47$	13535.0	0.893743	0.446872	+	0.894598i	$0.352538\pi$
$48$	−4494.67	−0.281575
$49$	11692.5	0.695694
$50$	0	0
$51$	−5615.68	−0.302327
$52$	3465.43	0.177725
$53$	18116.1	0.885880	0.442940	−	0.896551i	$-0.353935\pi$
$53$	18116.1	0.885880	0.442940	+	0.896551i	$0.353935\pi$
$54$	3265.15	0.152377
$55$	0	0
$56$	−33223.5	−1.41571
$57$	−3586.10	−0.146196
$58$	18903.0	0.737839
$59$	−50582.4	−1.89177	−0.945887	−	0.324495i	$-0.894806\pi$
$59$	−50582.4	−1.89177	−0.945887	+	0.324495i	$0.894806\pi$
$60$	0	0
$61$	5984.35	0.205917	0.102958	−	0.994686i	$-0.467169\pi$
$61$	5984.35	0.205917	0.102958	+	0.994686i	$0.467169\pi$
$62$	−25057.8	−0.827872
$63$	13674.3	0.434063
$64$	34169.1	1.04276
$65$	0	0
$66$	4877.57	0.137830
$67$	−22450.7	−0.611002	−0.305501	−	0.952192i	$-0.598824\pi$
$67$	−22450.7	−0.611002	−0.305501	+	0.952192i	$0.598824\pi$
$68$	7449.56	0.195370
$69$	−34092.6	−0.862059
$70$	0	0
$71$	10017.6	0.235841	0.117921	−	0.993023i	$-0.462377\pi$
$71$	10017.6	0.235841	0.117921	+	0.993023i	$0.462377\pi$
$72$	−15940.8	−0.362393
$73$	476.148	0.0104577	0.00522883	−	0.999986i	$-0.498336\pi$
$73$	476.148	0.0104577	0.00522883	+	0.999986i	$0.498336\pi$
$74$	−1349.35	−0.0286448
$75$	0	0
$76$	4757.19	0.0944750
$77$	20427.0	0.392624
$78$	−11700.5	−0.217755
$79$	−85024.7	−1.53277	−0.766386	−	0.642380i	$-0.777947\pi$
$79$	−85024.7	−1.53277	−0.766386	+	0.642380i	$0.777947\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−65554.5	−1.07663
$83$	25643.4	0.408583	0.204292	−	0.978910i	$-0.434511\pi$
$83$	25643.4	0.408583	0.204292	+	0.978910i	$0.434511\pi$
$84$	−18139.8	−0.280501
$85$	0	0
$86$	−678.194	−0.00988799
$87$	37983.8	0.538023
$88$	−23812.9	−0.327797
$89$	1807.91	0.0241937	0.0120968	−	0.999927i	$-0.496149\pi$
$89$	1807.91	0.0241937	0.0120968	+	0.999927i	$0.496149\pi$
$90$	0	0
$91$	−49001.0	−0.620300
$92$	45226.0	0.557081
$93$	−50351.2	−0.603674
$94$	60622.4	0.707642
$95$	0	0
$96$	36547.2	0.404740
$97$	−11688.5	−0.126133	−0.0630666	−	0.998009i	$-0.520088\pi$
$97$	−11688.5	−0.126133	−0.0630666	+	0.998009i	$0.520088\pi$
$98$	52370.1	0.550831
$99$	9801.00	0.100504
$100$	0	0
$101$	−135573.	−1.32242	−0.661210	−	0.750200i	$-0.729957\pi$
$101$	−135573.	−1.32242	−0.661210	+	0.750200i	$0.729957\pi$
$102$	−25152.3	−0.239374
$103$	88490.5	0.821871	0.410935	−	0.911664i	$-0.365202\pi$
$103$	88490.5	0.821871	0.410935	+	0.911664i	$0.365202\pi$
$104$	57123.2	0.517880
$105$	0	0
$106$	81140.9	0.701416
$107$	−218742.	−1.84703	−0.923514	−	0.383564i	$-0.874696\pi$
$107$	−218742.	−1.84703	−0.923514	+	0.383564i	$0.874696\pi$
$108$	−8703.59	−0.0718024
$109$	238399.	1.92193	0.960967	−	0.276663i	$-0.0892287\pi$
$109$	238399.	1.92193	0.960967	+	0.276663i	$0.0892287\pi$
$110$	0	0
$111$	−2711.39	−0.0208874
$112$	−84309.1	−0.635081
$113$	−187225.	−1.37933	−0.689664	−	0.724130i	$-0.742242\pi$
$113$	−187225.	−1.37933	−0.689664	+	0.724130i	$0.742242\pi$
$114$	−16061.9	−0.115754
$115$	0	0
$116$	−50388.0	−0.347682
$117$	−23511.0	−0.158784
$118$	−226556.	−1.49786
$119$	−105336.	−0.681885
$120$	0	0
$121$	14641.0	0.0909091
$122$	26803.5	0.163039
$123$	−131725.	−0.785068
$124$	66794.1	0.390107
$125$	0	0
$126$	61246.2	0.343679
$127$	−27727.6	−0.152547	−0.0762733	−	0.997087i	$-0.524302\pi$
$127$	−27727.6	−0.152547	−0.0762733	+	0.997087i	$0.524302\pi$
$128$	23096.0	0.124598
$129$	−1362.77	−0.00721019
$130$	0	0
$131$	−324623.	−1.65273	−0.826363	−	0.563138i	$-0.809594\pi$
$131$	−324623.	−1.65273	−0.826363	+	0.563138i	$0.809594\pi$
$132$	−13001.7	−0.0649477
$133$	−67266.5	−0.329739
$134$	−100555.	−0.483775
$135$	0	0
$136$	122797.	0.569297
$137$	59748.8	0.271974	0.135987	−	0.990711i	$-0.456579\pi$
$137$	59748.8	0.271974	0.135987	+	0.990711i	$0.456579\pi$
$138$	−152699.	−0.682555
$139$	−117646.	−0.516462	−0.258231	−	0.966083i	$-0.583140\pi$
$139$	−117646.	−0.516462	−0.258231	+	0.966083i	$0.583140\pi$
$140$	0	0
$141$	121815.	0.516003
$142$	44868.5	0.186733
$143$	−35121.4	−0.143626
$144$	−40452.0	−0.162568
$145$	0	0
$146$	2132.64	0.00828009
$147$	105233.	0.401659
$148$	3596.84	0.0134979
$149$	52786.3	0.194785	0.0973925	−	0.995246i	$-0.468950\pi$
$149$	52786.3	0.194785	0.0973925	+	0.995246i	$0.468950\pi$
$150$	0	0
$151$	−430119.	−1.53513	−0.767567	−	0.640969i	$-0.778533\pi$
$151$	−430119.	−1.53513	−0.767567	+	0.640969i	$0.778533\pi$
$152$	78416.3	0.275294
$153$	−50541.1	−0.174548
$154$	91491.3	0.310869
$155$	0	0
$156$	31188.9	0.102610
$157$	150989.	0.488873	0.244436	−	0.969665i	$-0.421397\pi$
$157$	150989.	0.488873	0.244436	+	0.969665i	$0.421397\pi$
$158$	−380821.	−1.21361
$159$	163045.	0.511463
$160$	0	0
$161$	−639493.	−1.94434
$162$	29386.3	0.0879747
$163$	−142404.	−0.419812	−0.209906	−	0.977722i	$-0.567316\pi$
$163$	−142404.	−0.419812	−0.209906	+	0.977722i	$0.567316\pi$
$164$	174742.	0.507328
$165$	0	0
$166$	114855.	0.323505
$167$	−55468.4	−0.153906	−0.0769528	−	0.997035i	$-0.524519\pi$
$167$	−55468.4	−0.153906	−0.0769528	+	0.997035i	$0.524519\pi$
$168$	−299011.	−0.817362
$169$	−287042.	−0.773089
$170$	0	0
$171$	−32274.9	−0.0844063
$172$	1807.80	0.00465938
$173$	−133612.	−0.339414	−0.169707	−	0.985495i	$-0.554282\pi$
$173$	−133612.	−0.339414	−0.169707	+	0.985495i	$0.554282\pi$
$174$	170127.	0.425992
$175$	0	0
$176$	−60428.4	−0.147048
$177$	−455242.	−1.09222
$178$	8097.52	0.0191559
$179$	−188254.	−0.439150	−0.219575	−	0.975596i	$-0.570467\pi$
$179$	−188254.	−0.439150	−0.219575	+	0.975596i	$0.570467\pi$
$180$	0	0
$181$	−652252.	−1.47986	−0.739928	−	0.672686i	$-0.765140\pi$
$181$	−652252.	−1.47986	−0.739928	+	0.672686i	$0.765140\pi$
$182$	−219473.	−0.491136
$183$	53859.1	0.118886
$184$	745493.	1.62330
$185$	0	0
$186$	−225520.	−0.477972
$187$	−75499.7	−0.157885
$188$	−161595.	−0.333452
$189$	123068.	0.250606
$190$	0	0
$191$	−705620.	−1.39955	−0.699774	−	0.714365i	$-0.746716\pi$
$191$	−705620.	−1.39955	−0.699774	+	0.714365i	$0.746716\pi$
$192$	307522.	0.602037
$193$	87143.5	0.168400	0.0841999	−	0.996449i	$-0.473167\pi$
$193$	87143.5	0.168400	0.0841999	+	0.996449i	$0.473167\pi$
$194$	−52352.1	−0.0998688
$195$	0	0
$196$	−139598.	−0.259561
$197$	667596.	1.22560	0.612800	−	0.790238i	$-0.290043\pi$
$197$	667596.	1.22560	0.612800	+	0.790238i	$0.290043\pi$
$198$	43898.1	0.0795761
$199$	−698609.	−1.25055	−0.625276	−	0.780404i	$-0.715013\pi$
$199$	−698609.	−1.25055	−0.625276	+	0.780404i	$0.715013\pi$
$200$	0	0
$201$	−202056.	−0.352762
$202$	−607224.	−1.04706
$203$	712484.	1.21349
$204$	67046.0	0.112797
$205$	0	0
$206$	396344.	0.650735
$207$	−306833.	−0.497710
$208$	144958.	0.232318
$209$	−48213.1	−0.0763484
$210$	0	0
$211$	−558238.	−0.863203	−0.431602	−	0.902064i	$-0.642051\pi$
$211$	−558238.	−0.863203	−0.431602	+	0.902064i	$0.642051\pi$
$212$	−216289.	−0.330518
$213$	90158.8	0.136163
$214$	−979735.	−1.46243
$215$	0	0
$216$	−143468.	−0.209228
$217$	−944465.	−1.36156
$218$	1.06778e6	1.52173
$219$	4285.33	0.00603773
$220$	0	0
$221$	181111.	0.249440
$222$	−12144.2	−0.0165381
$223$	382897.	0.515608	0.257804	−	0.966197i	$-0.417001\pi$
$223$	382897.	0.515608	0.257804	+	0.966197i	$0.417001\pi$
$224$	685536.	0.912873
$225$	0	0
$226$	−838569.	−1.09211
$227$	404735.	0.521322	0.260661	−	0.965430i	$-0.416059\pi$
$227$	404735.	0.521322	0.260661	+	0.965430i	$0.416059\pi$
$228$	42814.7	0.0545452
$229$	664894.	0.837845	0.418922	−	0.908022i	$-0.362408\pi$
$229$	664894.	0.837845	0.418922	+	0.908022i	$0.362408\pi$
$230$	0	0
$231$	183843.	0.226682
$232$	−830582.	−1.01312
$233$	−61531.4	−0.0742518	−0.0371259	−	0.999311i	$-0.511820\pi$
$233$	−61531.4	−0.0742518	−0.0371259	+	0.999311i	$0.511820\pi$
$234$	−105304.	−0.125721
$235$	0	0
$236$	603907.	0.705814
$237$	−765223.	−0.884946
$238$	−471796.	−0.539898
$239$	1.71207e6	1.93877	0.969384	−	0.245549i	$-0.0789682\pi$
$239$	1.71207e6	1.93877	0.969384	+	0.245549i	$0.0789682\pi$
$240$	0	0
$241$	1.31915e6	1.46302	0.731512	−	0.681828i	$-0.238815\pi$
$241$	1.31915e6	1.46302	0.731512	+	0.681828i	$0.238815\pi$
$242$	65576.2	0.0719793
$243$	59049.0	0.0641500
$244$	−71447.6	−0.0768268
$245$	0	0
$246$	−589991.	−0.621595
$247$	115656.	0.120621
$248$	1.10102e6	1.13675
$249$	230791.	0.235896
$250$	0	0
$251$	−237992.	−0.238440	−0.119220	−	0.992868i	$-0.538039\pi$
$251$	−237992.	−0.238440	−0.119220	+	0.992868i	$0.538039\pi$
$252$	−163258.	−0.161947
$253$	−458356.	−0.450196
$254$	−124190.	−0.120782
$255$	0	0
$256$	−989967.	−0.944106
$257$	1.90368e6	1.79789	0.898943	−	0.438066i	$-0.144336\pi$
$257$	1.90368e6	1.79789	0.898943	+	0.438066i	$0.144336\pi$
$258$	−6103.75	−0.00570883
$259$	−50859.1	−0.0471107
$260$	0	0
$261$	341854.	0.310628
$262$	−1.45397e6	−1.30858
$263$	907019.	0.808588	0.404294	−	0.914629i	$-0.367517\pi$
$263$	907019.	0.808588	0.404294	+	0.914629i	$0.367517\pi$
$264$	−214316.	−0.189254
$265$	0	0
$266$	−301283.	−0.261078
$267$	16271.2	0.0139682
$268$	268041.	0.227963
$269$	698485.	0.588541	0.294270	−	0.955722i	$-0.404923\pi$
$269$	698485.	0.588541	0.294270	+	0.955722i	$0.404923\pi$
$270$	0	0
$271$	1.19097e6	0.985098	0.492549	−	0.870285i	$-0.336065\pi$
$271$	1.19097e6	0.985098	0.492549	+	0.870285i	$0.336065\pi$
$272$	311613.	0.255383
$273$	−441009.	−0.358130
$274$	267611.	0.215342
$275$	0	0
$276$	407034.	0.321631
$277$	1.15110e6	0.901392	0.450696	−	0.892677i	$-0.351176\pi$
$277$	1.15110e6	0.901392	0.450696	+	0.892677i	$0.351176\pi$
$278$	−526928.	−0.408921
$279$	−453161.	−0.348531
$280$	0	0
$281$	−1.54201e6	−1.16498	−0.582492	−	0.812836i	$-0.697922\pi$
$281$	−1.54201e6	−1.16498	−0.582492	+	0.812836i	$0.697922\pi$
$282$	545601.	0.408557
$283$	−1.28590e6	−0.954421	−0.477211	−	0.878789i	$-0.658352\pi$
$283$	−1.28590e6	−0.954421	−0.477211	+	0.878789i	$0.658352\pi$
$284$	−119601.	−0.0879915
$285$	0	0
$286$	−157307.	−0.113719
$287$	−2.47085e6	−1.77069
$288$	328925.	0.233677
$289$	−1.03053e6	−0.725795
$290$	0	0
$291$	−105196.	−0.0728230
$292$	−5684.77	−0.00390171
$293$	−1.08802e6	−0.740404	−0.370202	−	0.928951i	$-0.620712\pi$
$293$	−1.08802e6	−0.740404	−0.370202	+	0.928951i	$0.620712\pi$
$294$	471331.	0.318023
$295$	0	0
$296$	59289.3	0.0393321
$297$	88209.0	0.0580259
$298$	236427.	0.154225
$299$	1.09952e6	0.711255
$300$	0	0
$301$	−25562.2	−0.0162623
$302$	−1.92648e6	−1.21548
$303$	−1.22016e6	−0.763500
$304$	198992.	0.123496
$305$	0	0
$306$	−226371.	−0.138203
$307$	580340.	0.351428	0.175714	−	0.984441i	$-0.443777\pi$
$307$	580340.	0.351428	0.175714	+	0.984441i	$0.443777\pi$
$308$	−243879.	−0.146487
$309$	796414.	0.474507
$310$	0	0
$311$	−1.68606e6	−0.988492	−0.494246	−	0.869322i	$-0.664556\pi$
$311$	−1.68606e6	−0.988492	−0.494246	+	0.869322i	$0.664556\pi$
$312$	514109.	0.298998
$313$	1.50328e6	0.867319	0.433659	−	0.901077i	$-0.357222\pi$
$313$	1.50328e6	0.867319	0.433659	+	0.901077i	$0.357222\pi$
$314$	676270.	0.387076
$315$	0	0
$316$	1.01512e6	0.571871
$317$	−1.31016e6	−0.732281	−0.366140	−	0.930560i	$-0.619321\pi$
$317$	−1.31016e6	−0.732281	−0.366140	+	0.930560i	$0.619321\pi$
$318$	730269.	0.404962
$319$	510671.	0.280973
$320$	0	0
$321$	−1.96868e6	−1.06638
$322$	−2.86425e6	−1.53947
$323$	248622.	0.132597
$324$	−78332.3	−0.0414551
$325$	0	0
$326$	−637821.	−0.332395
$327$	2.14559e6	1.10963
$328$	2.88041e6	1.47832
$329$	2.28495e6	1.16382
$330$	0	0
$331$	2.91035e6	1.46007	0.730037	−	0.683408i	$-0.239503\pi$
$331$	2.91035e6	1.46007	0.730037	+	0.683408i	$0.239503\pi$
$332$	−306159.	−0.152441
$333$	−24402.5	−0.0120594
$334$	−248440.	−0.121858
$335$	0	0
$336$	−758782.	−0.366664
$337$	−2.47133e6	−1.18538	−0.592688	−	0.805432i	$-0.701933\pi$
$337$	−2.47133e6	−1.18538	−0.592688	+	0.805432i	$0.701933\pi$
$338$	−1.28565e6	−0.612111
$339$	−1.68502e6	−0.796355
$340$	0	0
$341$	−676944.	−0.315258
$342$	−144557.	−0.0668306
$343$	−863415.	−0.396264
$344$	29799.2	0.0135772
$345$	0	0
$346$	−598440.	−0.268739
$347$	−1.75988e6	−0.784619	−0.392309	−	0.919833i	$-0.628324\pi$
$347$	−1.75988e6	−0.784619	−0.392309	+	0.919833i	$0.628324\pi$
$348$	−453492.	−0.200734
$349$	−386142.	−0.169700	−0.0848502	−	0.996394i	$-0.527041\pi$
$349$	−386142.	−0.169700	−0.0848502	+	0.996394i	$0.527041\pi$
$350$	0	0
$351$	−211599.	−0.0916740
$352$	491357.	0.211368
$353$	2.44184e6	1.04299	0.521495	−	0.853254i	$-0.325375\pi$
$353$	2.44184e6	1.04299	0.521495	+	0.853254i	$0.325375\pi$
$354$	−2.03900e6	−0.864787
$355$	0	0
$356$	−21584.8	−0.00902657
$357$	−948028.	−0.393686
$358$	−843180.	−0.347707
$359$	3.77524e6	1.54600	0.772998	−	0.634408i	$-0.218756\pi$
$359$	3.77524e6	1.54600	0.772998	+	0.634408i	$0.218756\pi$
$360$	0	0
$361$	−2.31733e6	−0.935880
$362$	−2.92140e6	−1.17171
$363$	131769.	0.0524864
$364$	585027.	0.231431
$365$	0	0
$366$	241232.	0.0941309
$367$	3.98351e6	1.54383	0.771917	−	0.635723i	$-0.219298\pi$
$367$	3.98351e6	1.54383	0.771917	+	0.635723i	$0.219298\pi$
$368$	1.89179e6	0.728204
$369$	−1.18553e6	−0.453259
$370$	0	0
$371$	3.05832e6	1.15358
$372$	601147.	0.225228
$373$	−3.13497e6	−1.16670	−0.583352	−	0.812219i	$-0.698259\pi$
$373$	−3.13497e6	−1.16670	−0.583352	+	0.812219i	$0.698259\pi$
$374$	−338159.	−0.125009
$375$	0	0
$376$	−2.66369e6	−0.971660
$377$	−1.22502e6	−0.443904
$378$	551216.	0.198423
$379$	−2.50995e6	−0.897567	−0.448784	−	0.893640i	$-0.648143\pi$
$379$	−2.50995e6	−0.897567	−0.448784	+	0.893640i	$0.648143\pi$
$380$	0	0
$381$	−249548.	−0.0880728
$382$	−3.16043e6	−1.10812
$383$	4.20287e6	1.46403	0.732014	−	0.681290i	$-0.238581\pi$
$383$	4.20287e6	1.46403	0.732014	+	0.681290i	$0.238581\pi$
$384$	207864.	0.0719368
$385$	0	0
$386$	390311.	0.133334
$387$	−12264.9	−0.00416281
$388$	139550.	0.0470598
$389$	−1.91171e6	−0.640543	−0.320271	−	0.947326i	$-0.603774\pi$
$389$	−1.91171e6	−0.640543	−0.320271	+	0.947326i	$0.603774\pi$
$390$	0	0
$391$	2.36362e6	0.781871
$392$	−2.30110e6	−0.756344
$393$	−2.92161e6	−0.954202
$394$	2.99013e6	0.970396
$395$	0	0
$396$	−117015.	−0.0374976
$397$	−3.62426e6	−1.15410	−0.577049	−	0.816709i	$-0.695796\pi$
$397$	−3.62426e6	−1.15410	−0.577049	+	0.816709i	$0.695796\pi$
$398$	−3.12903e6	−0.990152
$399$	−605399.	−0.190375
$400$	0	0
$401$	−2.22052e6	−0.689593	−0.344797	−	0.938677i	$-0.612052\pi$
$401$	−2.22052e6	−0.689593	−0.344797	+	0.938677i	$0.612052\pi$
$402$	−904998.	−0.279307
$403$	1.62388e6	0.498070
$404$	1.61862e6	0.493390
$405$	0	0
$406$	3.19117e6	0.960805
$407$	−36453.2	−0.0109081
$408$	1.10517e6	0.328684
$409$	20564.7	0.00607876	0.00303938	−	0.999995i	$-0.499033\pi$
$409$	20564.7	0.00607876	0.00303938	+	0.999995i	$0.499033\pi$
$410$	0	0
$411$	537739.	0.157024
$412$	−1.05649e6	−0.306637
$413$	−8.53922e6	−2.46345
$414$	−1.37429e6	−0.394073
$415$	0	0
$416$	−1.17869e6	−0.333937
$417$	−1.05881e6	−0.298180
$418$	−215944.	−0.0604505
$419$	−2.30684e6	−0.641923	−0.320962	−	0.947092i	$-0.604006\pi$
$419$	−2.30684e6	−0.641923	−0.320962	+	0.947092i	$0.604006\pi$
$420$	0	0
$421$	5.00382e6	1.37593	0.687965	−	0.725744i	$-0.258504\pi$
$421$	5.00382e6	1.37593	0.687965	+	0.725744i	$0.258504\pi$
$422$	−2.50032e6	−0.683461
$423$	1.09633e6	0.297914
$424$	−3.56526e6	−0.963111
$425$	0	0
$426$	403816.	0.107810
$427$	1.01027e6	0.268143
$428$	2.61158e6	0.689119
$429$	−316092.	−0.0829222
$430$	0	0
$431$	−2.70202e6	−0.700641	−0.350320	−	0.936630i	$-0.613927\pi$
$431$	−2.70202e6	−0.700641	−0.350320	+	0.936630i	$0.613927\pi$
$432$	−364068.	−0.0938585
$433$	3.92321e6	1.00559	0.502796	−	0.864405i	$-0.332305\pi$
$433$	3.92321e6	1.00559	0.502796	+	0.864405i	$0.332305\pi$
$434$	−4.23020e6	−1.07805
$435$	0	0
$436$	−2.84627e6	−0.717066
$437$	1.50938e6	0.378089
$438$	19193.7	0.00478051
$439$	−53261.3	−0.0131902	−0.00659508	−	0.999978i	$-0.502099\pi$
$439$	−53261.3	−0.0131902	−0.00659508	+	0.999978i	$0.502099\pi$
$440$	0	0
$441$	947095.	0.231898
$442$	811188.	0.197499
$443$	−3.30457e6	−0.800029	−0.400015	−	0.916509i	$-0.630995\pi$
$443$	−3.30457e6	−0.800029	−0.400015	+	0.916509i	$0.630995\pi$
$444$	32371.5	0.00779302
$445$	0	0
$446$	1.71497e6	0.408244
$447$	475077.	0.112459
$448$	5.76837e6	1.35787
$449$	3.04596e6	0.713031	0.356516	−	0.934289i	$-0.383965\pi$
$449$	3.04596e6	0.713031	0.356516	+	0.934289i	$0.383965\pi$
$450$	0	0
$451$	−1.77098e6	−0.409988
$452$	2.23529e6	0.514622
$453$	−3.87107e6	−0.886309
$454$	1.81279e6	0.412769
$455$	0	0
$456$	705747.	0.158941
$457$	2.67154e6	0.598371	0.299186	−	0.954195i	$-0.403285\pi$
$457$	2.67154e6	0.598371	0.299186	+	0.954195i	$0.403285\pi$
$458$	2.97802e6	0.663382
$459$	−454870.	−0.100776
$460$	0	0
$461$	−3.30382e6	−0.724042	−0.362021	−	0.932170i	$-0.617913\pi$
$461$	−3.30382e6	−0.724042	−0.362021	+	0.932170i	$0.617913\pi$
$462$	823421.	0.179480
$463$	7.66989e6	1.66279	0.831394	−	0.555684i	$-0.187543\pi$
$463$	7.66989e6	1.66279	0.831394	+	0.555684i	$0.187543\pi$
$464$	−2.10771e6	−0.454482
$465$	0	0
$466$	−275596.	−0.0587905
$467$	7.91117e6	1.67860	0.839302	−	0.543665i	$-0.182964\pi$
$467$	7.91117e6	1.67860	0.839302	+	0.543665i	$0.182964\pi$
$468$	280700.	0.0592417
$469$	−3.79008e6	−0.795640
$470$	0	0
$471$	1.35890e6	0.282251
$472$	9.95465e6	2.05670
$473$	−18321.6	−0.00376540
$474$	−3.42739e6	−0.700676
$475$	0	0
$476$	1.25762e6	0.254409
$477$	1.46740e6	0.295293
$478$	7.66825e6	1.53506
$479$	5.44099e6	1.08352	0.541762	−	0.840532i	$-0.317757\pi$
$479$	5.44099e6	1.08352	0.541762	+	0.840532i	$0.317757\pi$
$480$	0	0
$481$	87445.3	0.0172335
$482$	5.90840e6	1.15838
$483$	−5.75544e6	−1.12256
$484$	−174800.	−0.0339178
$485$	0	0
$486$	264477.	0.0507922
$487$	8.50740e6	1.62545	0.812727	−	0.582645i	$-0.197982\pi$
$487$	8.50740e6	1.62545	0.812727	+	0.582645i	$0.197982\pi$
$488$	−1.17772e6	−0.223869
$489$	−1.28164e6	−0.242378
$490$	0	0
$491$	−334168.	−0.0625549	−0.0312775	−	0.999511i	$-0.509958\pi$
$491$	−334168.	−0.0625549	−0.0312775	+	0.999511i	$0.509958\pi$
$492$	1.57268e6	0.292906
$493$	−2.63339e6	−0.487976
$494$	518014.	0.0955046
$495$	0	0
$496$	2.79397e6	0.509939
$497$	1.69116e6	0.307110
$498$	1.03370e6	0.186776
$499$	−3.50055e6	−0.629339	−0.314669	−	0.949201i	$-0.601894\pi$
$499$	−3.50055e6	−0.629339	−0.314669	+	0.949201i	$0.601894\pi$
$500$	0	0
$501$	−499216.	−0.0888574
$502$	−1.06595e6	−0.188790
$503$	−6.56301e6	−1.15660	−0.578300	−	0.815824i	$-0.696284\pi$
$503$	−6.56301e6	−1.15660	−0.578300	+	0.815824i	$0.696284\pi$
$504$	−2.69110e6	−0.471904
$505$	0	0
$506$	−2.05295e6	−0.356453
$507$	−2.58338e6	−0.446343
$508$	331042.	0.0569146
$509$	109860.	0.0187952	0.00939760	−	0.999956i	$-0.497009\pi$
$509$	109860.	0.0187952	0.00939760	+	0.999956i	$0.497009\pi$
$510$	0	0
$511$	80382.3	0.0136178
$512$	−5.17308e6	−0.872115
$513$	−290474.	−0.0487320
$514$	8.52649e6	1.42352
$515$	0	0
$516$	16270.2	0.00269010
$517$	1.63773e6	0.269474
$518$	−227795.	−0.0373009
$519$	−1.20251e6	−0.195961
$520$	0	0
$521$	−914334.	−0.147574	−0.0737872	−	0.997274i	$-0.523509\pi$
$521$	−914334.	−0.147574	−0.0737872	+	0.997274i	$0.523509\pi$
$522$	1.53115e6	0.245946
$523$	−4.00814e6	−0.640750	−0.320375	−	0.947291i	$-0.603809\pi$
$523$	−4.00814e6	−0.640750	−0.320375	+	0.947291i	$0.603809\pi$
$524$	3.87570e6	0.616626
$525$	0	0
$526$	4.06249e6	0.640218
$527$	3.49081e6	0.547520
$528$	−543855.	−0.0848982
$529$	7.91308e6	1.22944
$530$	0	0
$531$	−4.09717e6	−0.630592
$532$	803100.	0.123024
$533$	4.24829e6	0.647732
$534$	72877.7	0.0110597
$535$	0	0
$536$	4.41831e6	0.664269
$537$	−1.69429e6	−0.253543
$538$	3.12847e6	0.465990
$539$	1.41480e6	0.209760
$540$	0	0
$541$	−1.87833e6	−0.275918	−0.137959	−	0.990438i	$-0.544054\pi$
$541$	−1.87833e6	−0.275918	−0.137959	+	0.990438i	$0.544054\pi$
$542$	5.33431e6	0.779974
$543$	−5.87027e6	−0.854395
$544$	−2.53379e6	−0.367091
$545$	0	0
$546$	−1.97525e6	−0.283558
$547$	−87797.6	−0.0125463	−0.00627313	−	0.999980i	$-0.501997\pi$
$547$	−87797.6	−0.0125463	−0.00627313	+	0.999980i	$0.501997\pi$
$548$	−713346.	−0.101473
$549$	484732.	0.0686390
$550$	0	0
$551$	−1.68165e6	−0.235970
$552$	6.70944e6	0.937213
$553$	−1.43537e7	−1.99596
$554$	5.15571e6	0.713698
$555$	0	0
$556$	1.40458e6	0.192690
$557$	3.46548e6	0.473288	0.236644	−	0.971596i	$-0.423952\pi$
$557$	3.46548e6	0.473288	0.236644	+	0.971596i	$0.423952\pi$
$558$	−2.02968e6	−0.275957
$559$	43950.6	0.00594888
$560$	0	0
$561$	−679497.	−0.0911550
$562$	−6.90655e6	−0.922403
$563$	2.85737e6	0.379923	0.189961	−	0.981792i	$-0.439164\pi$
$563$	2.85737e6	0.379923	0.189961	+	0.981792i	$0.439164\pi$
$564$	−1.45436e6	−0.192519
$565$	0	0
$566$	−5.75946e6	−0.755685
$567$	1.10762e6	0.144688
$568$	−1.97148e6	−0.256402
$569$	−5.48030e6	−0.709617	−0.354808	−	0.934939i	$-0.615454\pi$
$569$	−5.48030e6	−0.709617	−0.354808	+	0.934939i	$0.615454\pi$
$570$	0	0
$571$	8.81180e6	1.13103	0.565515	−	0.824738i	$-0.308677\pi$
$571$	8.81180e6	1.13103	0.565515	+	0.824738i	$0.308677\pi$
$572$	419317.	0.0535861
$573$	−6.35058e6	−0.808029
$574$	−1.10668e7	−1.40198
$575$	0	0
$576$	2.76770e6	0.347586
$577$	−502598.	−0.0628465	−0.0314233	−	0.999506i	$-0.510004\pi$
$577$	−502598.	−0.0628465	−0.0314233	+	0.999506i	$0.510004\pi$
$578$	−4.61566e6	−0.574665
$579$	784291.	0.0972257
$580$	0	0
$581$	4.32907e6	0.532053
$582$	−471169.	−0.0576593
$583$	2.19205e6	0.267103
$584$	−93706.2	−0.0113694
$585$	0	0
$586$	−4.87319e6	−0.586232
$587$	7.45878e6	0.893455	0.446728	−	0.894670i	$-0.352589\pi$
$587$	7.45878e6	0.893455	0.446728	+	0.894670i	$0.352589\pi$
$588$	−1.25638e6	−0.149857
$589$	2.22919e6	0.264764
$590$	0	0
$591$	6.00837e6	0.707600
$592$	150455.	0.0176442
$593$	−6.69352e6	−0.781660	−0.390830	−	0.920463i	$-0.627812\pi$
$593$	−6.69352e6	−0.781660	−0.390830	+	0.920463i	$0.627812\pi$
$594$	395083.	0.0459433
$595$	0	0
$596$	−630220.	−0.0726736
$597$	−6.28748e6	−0.722006
$598$	4.92469e6	0.563153
$599$	−1.64186e7	−1.86968	−0.934842	−	0.355065i	$-0.884459\pi$
$599$	−1.64186e7	−1.86968	−0.934842	+	0.355065i	$0.884459\pi$
$600$	0	0
$601$	1.42657e7	1.61105	0.805523	−	0.592564i	$-0.201884\pi$
$601$	1.42657e7	1.61105	0.805523	+	0.592564i	$0.201884\pi$
$602$	−114491.	−0.0128760
$603$	−1.81851e6	−0.203667
$604$	5.13522e6	0.572752
$605$	0	0
$606$	−5.46501e6	−0.604518
$607$	−2.38734e6	−0.262992	−0.131496	−	0.991317i	$-0.541978\pi$
$607$	−2.38734e6	−0.262992	−0.131496	+	0.991317i	$0.541978\pi$
$608$	−1.61805e6	−0.177514
$609$	6.41235e6	0.700607
$610$	0	0
$611$	−3.92865e6	−0.425736
$612$	603414.	0.0651234
$613$	1.36125e6	0.146314	0.0731570	−	0.997320i	$-0.476693\pi$
$613$	1.36125e6	0.146314	0.0731570	+	0.997320i	$0.476693\pi$
$614$	2.59931e6	0.278251
$615$	0	0
$616$	−4.02004e6	−0.426853
$617$	1.33759e6	0.141453	0.0707264	−	0.997496i	$-0.477468\pi$
$617$	1.33759e6	0.141453	0.0707264	+	0.997496i	$0.477468\pi$
$618$	3.56709e6	0.375702
$619$	1.70842e7	1.79212	0.896062	−	0.443930i	$-0.146416\pi$
$619$	1.70842e7	1.79212	0.896062	+	0.443930i	$0.146416\pi$
$620$	0	0
$621$	−2.76150e6	−0.287353
$622$	−7.55179e6	−0.782661
$623$	305208.	0.0315047
$624$	1.30462e6	0.134129
$625$	0	0
$626$	6.73310e6	0.686719
$627$	−433918.	−0.0440797
$628$	−1.80267e6	−0.182397
$629$	187979.	0.0189445
$630$	0	0
$631$	1.70136e7	1.70107	0.850534	−	0.525920i	$-0.176279\pi$
$631$	1.70136e7	1.70107	0.850534	+	0.525920i	$0.176279\pi$
$632$	1.67329e7	1.66640
$633$	−5.02414e6	−0.498371
$634$	−5.86815e6	−0.579800
$635$	0	0
$636$	−1.94661e6	−0.190825
$637$	−3.39387e6	−0.331395
$638$	2.28727e6	0.222467
$639$	811429.	0.0786138
$640$	0	0
$641$	1.04366e7	1.00326	0.501632	−	0.865081i	$-0.332733\pi$
$641$	1.04366e7	1.00326	0.501632	+	0.865081i	$0.332733\pi$
$642$	−8.81761e6	−0.844332
$643$	−1.32098e7	−1.25999	−0.629996	−	0.776598i	$-0.716944\pi$
$643$	−1.32098e7	−1.25999	−0.629996	+	0.776598i	$0.716944\pi$
$644$	7.63496e6	0.725424
$645$	0	0
$646$	1.11356e6	0.104987
$647$	−3.14204e6	−0.295087	−0.147544	−	0.989056i	$-0.547137\pi$
$647$	−3.14204e6	−0.295087	−0.147544	+	0.989056i	$0.547137\pi$
$648$	−1.29121e6	−0.120798
$649$	−6.12047e6	−0.570391
$650$	0	0
$651$	−8.50019e6	−0.786097
$652$	1.70018e6	0.156630
$653$	−9.82598e6	−0.901764	−0.450882	−	0.892583i	$-0.648891\pi$
$653$	−9.82598e6	−0.901764	−0.450882	+	0.892583i	$0.648891\pi$
$654$	9.60998e6	0.878574
$655$	0	0
$656$	7.30942e6	0.663167
$657$	38568.0	0.00348589
$658$	1.02341e7	0.921482
$659$	−1.78524e7	−1.60134	−0.800670	−	0.599105i	$-0.795523\pi$
$659$	−1.78524e7	−1.60134	−0.800670	+	0.599105i	$0.795523\pi$
$660$	0	0
$661$	1.02807e7	0.915207	0.457603	−	0.889156i	$-0.348708\pi$
$661$	1.02807e7	0.915207	0.457603	+	0.889156i	$0.348708\pi$
$662$	1.30353e7	1.15605
$663$	1.63000e6	0.144014
$664$	−5.04664e6	−0.444204
$665$	0	0
$666$	−109298.	−0.00954827
$667$	−1.59872e7	−1.39142
$668$	662242.	0.0574216
$669$	3.44607e6	0.297686
$670$	0	0
$671$	724106.	0.0620863
$672$	6.16983e6	0.527047
$673$	2.22873e7	1.89679	0.948394	−	0.317094i	$-0.102707\pi$
$673$	2.22873e7	1.89679	0.948394	+	0.317094i	$0.102707\pi$
$674$	−1.10690e7	−0.938549
$675$	0	0
$676$	3.42702e6	0.288436
$677$	1.56465e7	1.31204	0.656018	−	0.754745i	$-0.272239\pi$
$677$	1.56465e7	1.31204	0.656018	+	0.754745i	$0.272239\pi$
$678$	−7.54712e6	−0.630532
$679$	−1.97323e6	−0.164249
$680$	0	0
$681$	3.64262e6	0.300986
$682$	−3.03199e6	−0.249613
$683$	1.82208e7	1.49457	0.747283	−	0.664506i	$-0.231358\pi$
$683$	1.82208e7	1.49457	0.747283	+	0.664506i	$0.231358\pi$
$684$	385333.	0.0314917
$685$	0	0
$686$	−3.86719e6	−0.313751
$687$	5.98404e6	0.483730
$688$	75619.5	0.00609064
$689$	−5.25837e6	−0.421991
$690$	0	0
$691$	1.07957e7	0.860116	0.430058	−	0.902801i	$-0.358493\pi$
$691$	1.07957e7	0.860116	0.430058	+	0.902801i	$0.358493\pi$
$692$	1.59520e6	0.126634
$693$	1.65459e6	0.130875
$694$	−7.88239e6	−0.621240
$695$	0	0
$696$	−7.47524e6	−0.584927
$697$	9.13244e6	0.712041
$698$	−1.72951e6	−0.134364
$699$	−553783.	−0.0428693
$700$	0	0
$701$	−7.59781e6	−0.583973	−0.291987	−	0.956422i	$-0.594316\pi$
$701$	−7.59781e6	−0.583973	−0.291987	+	0.956422i	$0.594316\pi$
$702$	−947740.	−0.0725849
$703$	120041.	0.00916098
$704$	4.13447e6	0.314404
$705$	0	0
$706$	1.09369e7	0.825811
$707$	−2.28872e7	−1.72204
$708$	5.43516e6	0.407502
$709$	1.26638e7	0.946122	0.473061	−	0.881030i	$-0.343149\pi$
$709$	1.26638e7	0.946122	0.473061	+	0.881030i	$0.343149\pi$
$710$	0	0
$711$	−6.88700e6	−0.510924
$712$	−355798.	−0.0263029
$713$	2.11926e7	1.56121
$714$	−4.24616e6	−0.311710
$715$	0	0
$716$	2.24758e6	0.163845
$717$	1.54086e7	1.11935
$718$	1.69091e7	1.22408
$719$	−8.16704e6	−0.589172	−0.294586	−	0.955625i	$-0.595182\pi$
$719$	−8.16704e6	−0.589172	−0.294586	+	0.955625i	$0.595182\pi$
$720$	0	0
$721$	1.49388e7	1.07023
$722$	−1.03792e7	−0.741004
$723$	1.18723e7	0.844677
$724$	7.78729e6	0.552128
$725$	0	0
$726$	590186.	0.0415573
$727$	−1.99934e6	−0.140298	−0.0701488	−	0.997537i	$-0.522347\pi$
$727$	−1.99934e6	−0.140298	−0.0701488	+	0.997537i	$0.522347\pi$
$728$	9.64343e6	0.674377
$729$	531441.	0.0370370
$730$	0	0
$731$	94479.6	0.00653950
$732$	−643028.	−0.0443560
$733$	−2.26768e7	−1.55892	−0.779458	−	0.626455i	$-0.784505\pi$
$733$	−2.26768e7	−1.55892	−0.779458	+	0.626455i	$0.784505\pi$
$734$	1.78419e7	1.22237
$735$	0	0
$736$	−1.53826e7	−1.04673
$737$	−2.71653e6	−0.184224
$738$	−5.30992e6	−0.358878
$739$	−1.49990e7	−1.01030	−0.505151	−	0.863031i	$-0.668563\pi$
$739$	−1.49990e7	−1.01030	−0.505151	+	0.863031i	$0.668563\pi$
$740$	0	0
$741$	1.04090e6	0.0696407
$742$	1.36981e7	0.913375
$743$	8.72196e6	0.579618	0.289809	−	0.957084i	$-0.406408\pi$
$743$	8.72196e6	0.579618	0.289809	+	0.957084i	$0.406408\pi$
$744$	9.90914e6	0.656302
$745$	0	0
$746$	−1.40413e7	−0.923765
$747$	2.07712e6	0.136194
$748$	901396.	0.0589063
$749$	−3.69277e7	−2.40518
$750$	0	0
$751$	3.03235e7	1.96191	0.980957	−	0.194227i	$-0.0622198\pi$
$751$	3.03235e7	1.96191	0.980957	+	0.194227i	$0.0622198\pi$
$752$	−6.75947e6	−0.435881
$753$	−2.14193e6	−0.137663
$754$	−5.48679e6	−0.351471
$755$	0	0
$756$	−1.46932e6	−0.0935002
$757$	−999401.	−0.0633870	−0.0316935	−	0.999498i	$-0.510090\pi$
$757$	−999401.	−0.0633870	−0.0316935	+	0.999498i	$0.510090\pi$
$758$	−1.12419e7	−0.710669
$759$	−4.12520e6	−0.259921
$760$	0	0
$761$	1.23318e7	0.771904	0.385952	−	0.922519i	$-0.373873\pi$
$761$	1.23318e7	0.771904	0.385952	+	0.922519i	$0.373873\pi$
$762$	−1.11771e6	−0.0697336
$763$	4.02461e7	2.50272
$764$	8.42445e6	0.522166
$765$	0	0
$766$	1.88244e7	1.15918
$767$	1.46820e7	0.901150
$768$	−8.90970e6	−0.545080
$769$	−3.88179e6	−0.236710	−0.118355	−	0.992971i	$-0.537762\pi$
$769$	−3.88179e6	−0.236710	−0.118355	+	0.992971i	$0.537762\pi$
$770$	0	0
$771$	1.71332e7	1.03801
$772$	−1.04041e6	−0.0628293
$773$	−2.35138e7	−1.41539	−0.707693	−	0.706520i	$-0.750264\pi$
$773$	−2.35138e7	−1.41539	−0.707693	+	0.706520i	$0.750264\pi$
$774$	−54933.7	−0.00329600
$775$	0	0
$776$	2.30030e6	0.137129
$777$	−457732.	−0.0271994
$778$	−8.56244e6	−0.507164
$779$	5.83186e6	0.344321
$780$	0	0
$781$	1.21214e6	0.0711088
$782$	1.05865e7	0.619064
$783$	3.07669e6	0.179341
$784$	−5.83934e6	−0.339292
$785$	0	0
$786$	−1.30857e7	−0.755511
$787$	−2.16876e6	−0.124817	−0.0624086	−	0.998051i	$-0.519878\pi$
$787$	−2.16876e6	−0.124817	−0.0624086	+	0.998051i	$0.519878\pi$
$788$	−7.97049e6	−0.457266
$789$	8.16317e6	0.466838
$790$	0	0
$791$	−3.16069e7	−1.79614
$792$	−1.92884e6	−0.109266
$793$	−1.73701e6	−0.0980890
$794$	−1.62328e7	−0.913784
$795$	0	0
$796$	8.34075e6	0.466576
$797$	1.85869e7	1.03648	0.518241	−	0.855235i	$-0.326587\pi$
$797$	1.85869e7	1.03648	0.518241	+	0.855235i	$0.326587\pi$
$798$	−2.71155e6	−0.150733
$799$	−8.44534e6	−0.468005
$800$	0	0
$801$	146441.	0.00806456
$802$	−9.94556e6	−0.546001
$803$	57613.9	0.00315310
$804$	2.41237e6	0.131614
$805$	0	0
$806$	7.27326e6	0.394359
$807$	6.28637e6	0.339794
$808$	2.66808e7	1.43771
$809$	2.99966e7	1.61139	0.805694	−	0.592332i	$-0.201793\pi$
$809$	2.99966e7	1.61139	0.805694	+	0.592332i	$0.201793\pi$
$810$	0	0
$811$	−1.54479e6	−0.0824740	−0.0412370	−	0.999149i	$-0.513130\pi$
$811$	−1.54479e6	−0.0824740	−0.0412370	+	0.999149i	$0.513130\pi$
$812$	−8.50640e6	−0.452747
$813$	1.07188e7	0.568747
$814$	−163272.	−0.00863674
$815$	0	0
$816$	2.80451e6	0.147446
$817$	60333.5	0.00316230
$818$	92108.3	0.00481299
$819$	−3.96908e6	−0.206767
$820$	0	0
$821$	−2.60445e7	−1.34852	−0.674261	−	0.738493i	$-0.735538\pi$
$821$	−2.60445e7	−1.34852	−0.674261	+	0.738493i	$0.735538\pi$
$822$	2.40850e6	0.124328
$823$	9.56891e6	0.492451	0.246226	−	0.969213i	$-0.420810\pi$
$823$	9.56891e6	0.492451	0.246226	+	0.969213i	$0.420810\pi$
$824$	−1.74150e7	−0.893521
$825$	0	0
$826$	−3.82467e7	−1.95049
$827$	−1.07164e7	−0.544860	−0.272430	−	0.962176i	$-0.587827\pi$
$827$	−1.07164e7	−0.544860	−0.272430	+	0.962176i	$0.587827\pi$
$828$	3.66330e6	0.185694
$829$	−1.64337e7	−0.830519	−0.415259	−	0.909703i	$-0.636309\pi$
$829$	−1.64337e7	−0.830519	−0.415259	+	0.909703i	$0.636309\pi$
$830$	0	0
$831$	1.03599e7	0.520419
$832$	−9.91791e6	−0.496720
$833$	−7.29572e6	−0.364297
$834$	−4.74235e6	−0.236091
$835$	0	0
$836$	575620.	0.0284853
$837$	−4.07845e6	−0.201225
$838$	−1.03322e7	−0.508257
$839$	9.64356e6	0.472969	0.236484	−	0.971635i	$-0.424005\pi$
$839$	9.64356e6	0.472969	0.236484	+	0.971635i	$0.424005\pi$
$840$	0	0
$841$	−2.69916e6	−0.131595
$842$	2.24118e7	1.08942
$843$	−1.38781e7	−0.672604
$844$	6.66485e6	0.322058
$845$	0	0
$846$	4.91041e6	0.235881
$847$	2.47166e6	0.118381
$848$	−9.04732e6	−0.432046
$849$	−1.15731e7	−0.551035
$850$	0	0
$851$	1.14121e6	0.0540186
$852$	−1.07641e6	−0.0508019
$853$	2.52742e7	1.18934	0.594669	−	0.803971i	$-0.297283\pi$
$853$	2.52742e7	1.18934	0.594669	+	0.803971i	$0.297283\pi$
$854$	4.52492e6	0.212308
$855$	0	0
$856$	4.30486e7	2.00805
$857$	262739.	0.0122200	0.00611002	−	0.999981i	$-0.498055\pi$
$857$	262739.	0.0122200	0.00611002	+	0.999981i	$0.498055\pi$
$858$	−1.41576e6	−0.0656555
$859$	−9.43849e6	−0.436435	−0.218218	−	0.975900i	$-0.570024\pi$
$859$	−9.43849e6	−0.436435	−0.218218	+	0.975900i	$0.570024\pi$
$860$	0	0
$861$	−2.22376e7	−1.02231
$862$	−1.21022e7	−0.554748
$863$	−3.78002e7	−1.72769	−0.863847	−	0.503754i	$-0.831952\pi$
$863$	−3.78002e7	−1.72769	−0.863847	+	0.503754i	$0.831952\pi$
$864$	2.96032e6	0.134913
$865$	0	0
$866$	1.75718e7	0.796200
$867$	−9.27473e6	−0.419038
$868$	1.12760e7	0.507993
$869$	−1.02880e7	−0.462148
$870$	0	0
$871$	6.51653e6	0.291052
$872$	−4.69171e7	−2.08949
$873$	−946768.	−0.0420444
$874$	6.76041e6	0.299360
$875$	0	0
$876$	−51162.9	−0.00225266
$877$	−2.42348e7	−1.06400	−0.531999	−	0.846745i	$-0.678559\pi$
$877$	−2.42348e7	−1.06400	−0.531999	+	0.846745i	$0.678559\pi$
$878$	−238554.	−0.0104436
$879$	−9.79220e6	−0.427472
$880$	0	0
$881$	1.97060e7	0.855377	0.427689	−	0.903926i	$-0.359328\pi$
$881$	1.97060e7	0.855377	0.427689	+	0.903926i	$0.359328\pi$
$882$	4.24198e6	0.183610
$883$	2.09867e7	0.905822	0.452911	−	0.891556i	$-0.350386\pi$
$883$	2.09867e7	0.905822	0.452911	+	0.891556i	$0.350386\pi$
$884$	−2.16230e6	−0.0930649
$885$	0	0
$886$	−1.48010e7	−0.633441
$887$	3.47934e7	1.48487	0.742434	−	0.669919i	$-0.233671\pi$
$887$	3.47934e7	1.48487	0.742434	+	0.669919i	$0.233671\pi$
$888$	533604.	0.0227084
$889$	−4.68091e6	−0.198644
$890$	0	0
$891$	793881.	0.0335013
$892$	−4.57143e6	−0.192371
$893$	−5.39309e6	−0.226313
$894$	2.12784e6	0.0890421
$895$	0	0
$896$	3.89902e6	0.162250
$897$	9.89569e6	0.410644
$898$	1.36427e7	0.564559
$899$	−2.36115e7	−0.974370
$900$	0	0
$901$	−1.13038e7	−0.463887
$902$	−7.93210e6	−0.324617
$903$	−230059.	−0.00938903
$904$	3.68460e7	1.49958
$905$	0	0
$906$	−1.73383e7	−0.701756
$907$	4.14396e7	1.67262	0.836310	−	0.548257i	$-0.184709\pi$
$907$	4.14396e7	1.67262	0.836310	+	0.548257i	$0.184709\pi$
$908$	−4.83217e6	−0.194503
$909$	−1.09814e7	−0.440807
$910$	0	0
$911$	−4.55462e7	−1.81826	−0.909131	−	0.416511i	$-0.863253\pi$
$911$	−4.55462e7	−1.81826	−0.909131	+	0.416511i	$0.863253\pi$
$912$	1.79093e6	0.0713002
$913$	3.10285e6	0.123193
$914$	1.19657e7	0.473774
$915$	0	0
$916$	−7.93822e6	−0.312597
$917$	−5.48022e7	−2.15216
$918$	−2.03734e6	−0.0797914
$919$	8.41097e6	0.328517	0.164258	−	0.986417i	$-0.447477\pi$
$919$	8.41097e6	0.328517	0.164258	+	0.986417i	$0.447477\pi$
$920$	0	0
$921$	5.22306e6	0.202897
$922$	−1.47976e7	−0.573277
$923$	−2.90772e6	−0.112343
$924$	−2.19491e6	−0.0845741
$925$	0	0
$926$	3.43530e7	1.31655
$927$	7.16773e6	0.273957
$928$	1.71383e7	0.653278
$929$	1.89074e6	0.0718773	0.0359386	−	0.999354i	$-0.488558\pi$
$929$	1.89074e6	0.0718773	0.0359386	+	0.999354i	$0.488558\pi$
$930$	0	0
$931$	−4.65895e6	−0.176163
$932$	734628.	0.0277031
$933$	−1.51746e7	−0.570706
$934$	3.54337e7	1.32907
$935$	0	0
$936$	4.62698e6	0.172627
$937$	−3.58861e7	−1.33530	−0.667648	−	0.744477i	$-0.732699\pi$
$937$	−3.58861e7	−1.33530	−0.667648	+	0.744477i	$0.732699\pi$
$938$	−1.69756e7	−0.629966
$939$	1.35295e7	0.500747
$940$	0	0
$941$	3.25537e7	1.19847	0.599235	−	0.800573i	$-0.295472\pi$
$941$	3.25537e7	1.19847	0.599235	+	0.800573i	$0.295472\pi$
$942$	6.08643e6	0.223478
$943$	5.54427e7	2.03032
$944$	2.52612e7	0.922624
$945$	0	0
$946$	−82061.5	−0.00298134
$947$	−6.44146e6	−0.233405	−0.116702	−	0.993167i	$-0.537232\pi$
$947$	−6.44146e6	−0.233405	−0.116702	+	0.993167i	$0.537232\pi$
$948$	9.13605e6	0.330170
$949$	−138206.	−0.00498153
$950$	0	0
$951$	−1.17915e7	−0.422782
$952$	2.07303e7	0.741332
$953$	−2.40023e7	−0.856093	−0.428046	−	0.903757i	$-0.640798\pi$
$953$	−2.40023e7	−0.856093	−0.428046	+	0.903757i	$0.640798\pi$
$954$	6.57242e6	0.233805
$955$	0	0
$956$	−2.04405e7	−0.723347
$957$	4.59604e6	0.162220
$958$	2.43699e7	0.857905
$959$	1.00867e7	0.354162
$960$	0	0
$961$	2.67012e6	0.0932656
$962$	391662.	0.0136450
$963$	−1.77181e7	−0.615676
$964$	−1.57494e7	−0.545849
$965$	0	0
$966$	−2.57783e7	−0.888815
$967$	−4.13239e7	−1.42113	−0.710567	−	0.703630i	$-0.751561\pi$
$967$	−4.13239e7	−1.42113	−0.710567	+	0.703630i	$0.751561\pi$
$968$	−2.88136e6	−0.0988345
$969$	2.23760e6	0.0765549
$970$	0	0
$971$	8.46004e6	0.287955	0.143977	−	0.989581i	$-0.454011\pi$
$971$	8.46004e6	0.287955	0.143977	+	0.989581i	$0.454011\pi$
$972$	−704991.	−0.0239341
$973$	−1.98607e7	−0.672531
$974$	3.81042e7	1.28699
$975$	0	0
$976$	−2.98863e6	−0.100426
$977$	−1.33050e7	−0.445942	−0.222971	−	0.974825i	$-0.571575\pi$
$977$	−1.33050e7	−0.445942	−0.222971	+	0.974825i	$0.571575\pi$
$978$	−5.74039e6	−0.191909
$979$	218757.	0.00729467
$980$	0	0
$981$	1.93103e7	0.640645
$982$	−1.49672e6	−0.0495293
$983$	−2.90239e7	−0.958014	−0.479007	−	0.877811i	$-0.659003\pi$
$983$	−2.90239e7	−0.958014	−0.479007	+	0.877811i	$0.659003\pi$
$984$	2.59237e7	0.853510
$985$	0	0
$986$	−1.17948e7	−0.386366
$987$	2.05645e7	0.671933
$988$	−1.38082e6	−0.0450033
$989$	573582.	0.0186468
$990$	0	0
$991$	−4.88551e7	−1.58025	−0.790125	−	0.612946i	$-0.789984\pi$
$991$	−4.88551e7	−1.58025	−0.790125	+	0.612946i	$0.789984\pi$
$992$	−2.27184e7	−0.732992
$993$	2.61931e7	0.842974
$994$	7.57460e6	0.243161
$995$	0	0
$996$	−2.75543e6	−0.0880118
$997$	2.19218e7	0.698453	0.349227	−	0.937038i	$-0.386444\pi$
$997$	2.19218e7	0.698453	0.349227	+	0.937038i	$0.386444\pi$
$998$	−1.56787e7	−0.498293
$999$	−219623.	−0.00696248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.g.1.3		3
5.4	even	2		165.6.a.c.1.1	✓	3
15.14	odd	2		495.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.1	✓	3	5.4	even	2
495.6.a.b.1.3		3	15.14	odd	2
825.6.a.g.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+4.47894 q^{2} +9.00000 q^{3} -11.9391 q^{4} +40.3105 q^{6} +168.818 q^{7} -196.801 q^{8} +81.0000 q^{9} +121.000 q^{11} -107.452 q^{12} -290.259 q^{13} +756.126 q^{14} -499.408 q^{16} -623.964 q^{17} +362.794 q^{18} -398.456 q^{19} +1519.36 q^{21} +541.952 q^{22} -3788.06 q^{23} -1771.21 q^{24} -1300.06 q^{26} +729.000 q^{27} -2015.53 q^{28} +4220.43 q^{29} -5594.57 q^{31} +4060.80 q^{32} +1089.00 q^{33} -2794.70 q^{34} -967.065 q^{36} -301.266 q^{37} -1784.66 q^{38} -2612.33 q^{39} -14636.2 q^{41} +6805.14 q^{42} -151.418 q^{43} -1444.63 q^{44} -16966.5 q^{46} +13535.0 q^{47} -4494.67 q^{48} +11692.5 q^{49} -5615.68 q^{51} +3465.43 q^{52} +18116.1 q^{53} +3265.15 q^{54} -33223.5 q^{56} -3586.10 q^{57} +18903.0 q^{58} -50582.4 q^{59} +5984.35 q^{61} -25057.8 q^{62} +13674.3 q^{63} +34169.1 q^{64} +4877.57 q^{66} -22450.7 q^{67} +7449.56 q^{68} -34092.6 q^{69} +10017.6 q^{71} -15940.8 q^{72} +476.148 q^{73} -1349.35 q^{74} +4757.19 q^{76} +20427.0 q^{77} -11700.5 q^{78} -85024.7 q^{79} +6561.00 q^{81} -65554.5 q^{82} +25643.4 q^{83} -18139.8 q^{84} -678.194 q^{86} +37983.8 q^{87} -23812.9 q^{88} +1807.91 q^{89} -49001.0 q^{91} +45226.0 q^{92} -50351.2 q^{93} +60622.4 q^{94} +36547.2 q^{96} -11688.5 q^{97} +52370.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22}+ \cdots + 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 + 363 * q^11 - 378 * q^12 - 290 * q^13 + 916 * q^14 - 590 * q^16 - 434 * q^17 - 162 * q^18 - 2856 * q^19 + 612 * q^21 - 242 * q^22 + 640 * q^23 + 216 * q^24 + 2132 * q^26 + 2187 * q^27 + 580 * q^28 - 4538 * q^29 - 14968 * q^31 + 2496 * q^32 + 3267 * q^33 - 13704 * q^34 - 3402 * q^36 + 6190 * q^37 + 11668 * q^38 - 2610 * q^39 - 8926 * q^41 + 8244 * q^42 + 33592 * q^43 - 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 5310 * q^48 - 14693 * q^49 - 3906 * q^51 - 18780 * q^52 + 22934 * q^53 - 1458 * q^54 - 40012 * q^56 - 25704 * q^57 + 32304 * q^58 - 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 5508 * q^63 + 35474 * q^64 - 2178 * q^66 - 16868 * q^67 + 71288 * q^68 + 5760 * q^69 + 4856 * q^71 + 1944 * q^72 - 1910 * q^73 + 29404 * q^74 + 6116 * q^76 + 8228 * q^77 + 19188 * q^78 - 36844 * q^79 + 19683 * q^81 - 84000 * q^82 + 48796 * q^83 + 5220 * q^84 - 83492 * q^86 - 40842 * q^87 + 2904 * q^88 - 188978 * q^89 - 93208 * q^91 + 6976 * q^92 - 134712 * q^93 + 70472 * q^94 + 22464 * q^96 - 247526 * q^97 + 154654 * q^98 + 29403 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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