LMFDB - Embedded newform 825.6.a.w.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 $ x^13 - 3x^12 - 318x^11 + 776x^10 + 37929x^9 - 75673x^8 - 2114192x^7 + 3527024x^6 + 55736416x^5 - 75128864x^4 - 615221248x^3 + 450708480x^2 + 2256236800x + 1037920000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-8.30687$ 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)

$f(q)$	$=$	\(q+8.30687 q^{2} +9.00000 q^{3} +37.0041 q^{4} +74.7618 q^{6} -54.6353 q^{7} +41.5686 q^{8} +81.0000 q^{9} -121.000 q^{11} +333.037 q^{12} -175.063 q^{13} -453.849 q^{14} -838.827 q^{16} -1248.20 q^{17} +672.857 q^{18} +1248.23 q^{19} -491.718 q^{21} -1005.13 q^{22} +2563.30 q^{23} +374.117 q^{24} -1454.23 q^{26} +729.000 q^{27} -2021.73 q^{28} +3450.78 q^{29} +3171.29 q^{31} -8298.22 q^{32} -1089.00 q^{33} -10368.6 q^{34} +2997.33 q^{36} -15357.2 q^{37} +10368.9 q^{38} -1575.57 q^{39} -14538.0 q^{41} -4084.64 q^{42} -7671.37 q^{43} -4477.50 q^{44} +21293.0 q^{46} -21772.6 q^{47} -7549.44 q^{48} -13822.0 q^{49} -11233.8 q^{51} -6478.06 q^{52} -38910.5 q^{53} +6055.71 q^{54} -2271.11 q^{56} +11234.1 q^{57} +28665.2 q^{58} -4608.63 q^{59} -5627.67 q^{61} +26343.5 q^{62} -4425.46 q^{63} -42089.8 q^{64} -9046.18 q^{66} -35573.8 q^{67} -46188.5 q^{68} +23069.7 q^{69} +33448.0 q^{71} +3367.05 q^{72} +18523.8 q^{73} -127571. q^{74} +46189.6 q^{76} +6610.88 q^{77} -13088.0 q^{78} +15606.1 q^{79} +6561.00 q^{81} -120765. q^{82} +76298.1 q^{83} -18195.6 q^{84} -63725.1 q^{86} +31057.0 q^{87} -5029.80 q^{88} -59295.1 q^{89} +9564.63 q^{91} +94852.8 q^{92} +28541.6 q^{93} -180863. q^{94} -74684.0 q^{96} -77264.9 q^{97} -114817. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) $ 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * 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 $

$2$	8.30687	1.46846	0.734231	−	0.678900i	$-0.237543\pi$
$2$	8.30687	1.46846	0.734231	+	0.678900i	$0.237543\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	37.0041	1.15638
$5$	0	0
$5$	0	0
$6$	74.7618	0.847817
$7$	−54.6353	−0.421433	−0.210717	−	0.977547i	$-0.567580\pi$
$7$	−54.6353	−0.421433	−0.210717	+	0.977547i	$0.567580\pi$
$8$	41.5686	0.229636
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	333.037	0.667636
$13$	−175.063	−0.287300	−0.143650	−	0.989629i	$-0.545884\pi$
$13$	−175.063	−0.287300	−0.143650	+	0.989629i	$0.545884\pi$
$14$	−453.849	−0.618858
$15$	0	0
$16$	−838.827	−0.819167
$17$	−1248.20	−1.04752	−0.523758	−	0.851867i	$-0.675471\pi$
$17$	−1248.20	−1.04752	−0.523758	+	0.851867i	$0.675471\pi$
$18$	672.857	0.489487
$19$	1248.23	0.793251	0.396625	−	0.917981i	$-0.370181\pi$
$19$	1248.23	0.793251	0.396625	+	0.917981i	$0.370181\pi$
$20$	0	0
$21$	−491.718	−0.243314
$22$	−1005.13	−0.442758
$23$	2563.30	1.01037	0.505185	−	0.863011i	$-0.331424\pi$
$23$	2563.30	1.01037	0.505185	+	0.863011i	$0.331424\pi$
$24$	374.117	0.132580
$25$	0	0
$26$	−1454.23	−0.421890
$27$	729.000	0.192450
$28$	−2021.73	−0.487336
$29$	3450.78	0.761943	0.380972	−	0.924587i	$-0.375590\pi$
$29$	3450.78	0.761943	0.380972	+	0.924587i	$0.375590\pi$
$30$	0	0
$31$	3171.29	0.592695	0.296348	−	0.955080i	$-0.404231\pi$
$31$	3171.29	0.592695	0.296348	+	0.955080i	$0.404231\pi$
$32$	−8298.22	−1.43255
$33$	−1089.00	−0.174078
$34$	−10368.6	−1.53824
$35$	0	0
$36$	2997.33	0.385460
$37$	−15357.2	−1.84420	−0.922102	−	0.386947i	$-0.873530\pi$
$37$	−15357.2	−1.84420	−0.922102	+	0.386947i	$0.873530\pi$
$38$	10368.9	1.16486
$39$	−1575.57	−0.165873
$40$	0	0
$41$	−14538.0	−1.35066	−0.675329	−	0.737517i	$-0.735998\pi$
$41$	−14538.0	−1.35066	−0.675329	+	0.737517i	$0.735998\pi$
$42$	−4084.64	−0.357298
$43$	−7671.37	−0.632706	−0.316353	−	0.948642i	$-0.602458\pi$
$43$	−7671.37	−0.632706	−0.316353	+	0.948642i	$0.602458\pi$
$44$	−4477.50	−0.348661
$45$	0	0
$46$	21293.0	1.48369
$47$	−21772.6	−1.43769	−0.718847	−	0.695168i	$-0.755330\pi$
$47$	−21772.6	−1.43769	−0.718847	+	0.695168i	$0.755330\pi$
$48$	−7549.44	−0.472946
$49$	−13822.0	−0.822394
$50$	0	0
$51$	−11233.8	−0.604784
$52$	−6478.06	−0.332228
$53$	−38910.5	−1.90273	−0.951364	−	0.308070i	$-0.900317\pi$
$53$	−38910.5	−1.90273	−0.951364	+	0.308070i	$0.900317\pi$
$54$	6055.71	0.282606
$55$	0	0
$56$	−2271.11	−0.0967762
$57$	11234.1	0.457984
$58$	28665.2	1.11888
$59$	−4608.63	−0.172362	−0.0861810	−	0.996279i	$-0.527466\pi$
$59$	−4608.63	−0.172362	−0.0861810	+	0.996279i	$0.527466\pi$
$60$	0	0
$61$	−5627.67	−0.193644	−0.0968220	−	0.995302i	$-0.530868\pi$
$61$	−5627.67	−0.193644	−0.0968220	+	0.995302i	$0.530868\pi$
$62$	26343.5	0.870350
$63$	−4425.46	−0.140478
$64$	−42089.8	−1.28448
$65$	0	0
$66$	−9046.18	−0.255626
$67$	−35573.8	−0.968150	−0.484075	−	0.875026i	$-0.660844\pi$
$67$	−35573.8	−0.968150	−0.484075	+	0.875026i	$0.660844\pi$
$68$	−46188.5	−1.21133
$69$	23069.7	0.583338
$70$	0	0
$71$	33448.0	0.787453	0.393727	−	0.919228i	$-0.371186\pi$
$71$	33448.0	0.787453	0.393727	+	0.919228i	$0.371186\pi$
$72$	3367.05	0.0765454
$73$	18523.8	0.406839	0.203420	−	0.979092i	$-0.434794\pi$
$73$	18523.8	0.406839	0.203420	+	0.979092i	$0.434794\pi$
$74$	−127571.	−2.70814
$75$	0	0
$76$	46189.6	0.917298
$77$	6610.88	0.127067
$78$	−13088.0	−0.243578
$79$	15606.1	0.281336	0.140668	−	0.990057i	$-0.455075\pi$
$79$	15606.1	0.281336	0.140668	+	0.990057i	$0.455075\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−120765.	−1.98339
$83$	76298.1	1.21568	0.607839	−	0.794060i	$-0.292037\pi$
$83$	76298.1	1.21568	0.607839	+	0.794060i	$0.292037\pi$
$84$	−18195.6	−0.281364
$85$	0	0
$86$	−63725.1	−0.929104
$87$	31057.0	0.439908
$88$	−5029.80	−0.0692379
$89$	−59295.1	−0.793494	−0.396747	−	0.917928i	$-0.629861\pi$
$89$	−59295.1	−0.793494	−0.396747	+	0.917928i	$0.629861\pi$
$90$	0	0
$91$	9564.63	0.121078
$92$	94852.8	1.16837
$93$	28541.6	0.342193
$94$	−180863.	−2.11120
$95$	0	0
$96$	−74684.0	−0.827084
$97$	−77264.9	−0.833783	−0.416891	−	0.908956i	$-0.636880\pi$
$97$	−77264.9	−0.833783	−0.416891	+	0.908956i	$0.636880\pi$
$98$	−114817.	−1.20765
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−143856.	−1.40322	−0.701609	−	0.712562i	$-0.747535\pi$
$101$	−143856.	−1.40322	−0.701609	+	0.712562i	$0.747535\pi$
$102$	−93317.6	−0.888102
$103$	84193.3	0.781960	0.390980	−	0.920399i	$-0.372136\pi$
$103$	84193.3	0.781960	0.390980	+	0.920399i	$0.372136\pi$
$104$	−7277.12	−0.0659746
$105$	0	0
$106$	−323224.	−2.79408
$107$	148939.	1.25762	0.628808	−	0.777560i	$-0.283543\pi$
$107$	148939.	1.25762	0.628808	+	0.777560i	$0.283543\pi$
$108$	26976.0	0.222545
$109$	88621.2	0.714449	0.357225	−	0.934018i	$-0.383723\pi$
$109$	88621.2	0.714449	0.357225	+	0.934018i	$0.383723\pi$
$110$	0	0
$111$	−138215.	−1.06475
$112$	45829.6	0.345224
$113$	−76122.3	−0.560810	−0.280405	−	0.959882i	$-0.590469\pi$
$113$	−76122.3	−0.560810	−0.280405	+	0.959882i	$0.590469\pi$
$114$	93320.0	0.672531
$115$	0	0
$116$	127693.	0.881095
$117$	−14180.1	−0.0957668
$118$	−38283.3	−0.253107
$119$	68195.7	0.441458
$120$	0	0
$121$	14641.0	0.0909091
$122$	−46748.3	−0.284359
$123$	−130842.	−0.779803
$124$	117351.	0.685380
$125$	0	0
$126$	−36761.8	−0.206286
$127$	−133153.	−0.732559	−0.366280	−	0.930505i	$-0.619369\pi$
$127$	−133153.	−0.732559	−0.366280	+	0.930505i	$0.619369\pi$
$128$	−84091.5	−0.453657
$129$	−69042.3	−0.365293
$130$	0	0
$131$	−128189.	−0.652639	−0.326319	−	0.945260i	$-0.605808\pi$
$131$	−128189.	−0.652639	−0.326319	+	0.945260i	$0.605808\pi$
$132$	−40297.5	−0.201300
$133$	−68197.5	−0.334302
$134$	−295507.	−1.42169
$135$	0	0
$136$	−51885.8	−0.240548
$137$	221707.	1.00920	0.504601	−	0.863353i	$-0.331640\pi$
$137$	221707.	1.00920	0.504601	+	0.863353i	$0.331640\pi$
$138$	191637.	0.856609
$139$	105276.	0.462160	0.231080	−	0.972935i	$-0.425774\pi$
$139$	105276.	0.462160	0.231080	+	0.972935i	$0.425774\pi$
$140$	0	0
$141$	−195954.	−0.830053
$142$	277849.	1.15634
$143$	21182.6	0.0866243
$144$	−67945.0	−0.273056
$145$	0	0
$146$	153875.	0.597428
$147$	−124398.	−0.474810
$148$	−568281.	−2.13260
$149$	−408871.	−1.50876	−0.754381	−	0.656437i	$-0.772063\pi$
$149$	−408871.	−1.50876	−0.754381	+	0.656437i	$0.772063\pi$
$150$	0	0
$151$	382934.	1.36673	0.683364	−	0.730078i	$-0.260516\pi$
$151$	382934.	1.36673	0.683364	+	0.730078i	$0.260516\pi$
$152$	51887.1	0.182159
$153$	−101104.	−0.349172
$154$	54915.7	0.186593
$155$	0	0
$156$	−58302.5	−0.191812
$157$	367155.	1.18878	0.594389	−	0.804178i	$-0.297394\pi$
$157$	367155.	1.18878	0.594389	+	0.804178i	$0.297394\pi$
$158$	129637.	0.413131
$159$	−350194.	−1.09854
$160$	0	0
$161$	−140047.	−0.425803
$162$	54501.4	0.163162
$163$	32509.1	0.0958377	0.0479188	−	0.998851i	$-0.484741\pi$
$163$	32509.1	0.0958377	0.0479188	+	0.998851i	$0.484741\pi$
$164$	−537966.	−1.56187
$165$	0	0
$166$	633799.	1.78518
$167$	526634.	1.46123	0.730613	−	0.682791i	$-0.239234\pi$
$167$	526634.	1.46123	0.730613	+	0.682791i	$0.239234\pi$
$168$	−20440.0	−0.0558738
$169$	−340646.	−0.917458
$170$	0	0
$171$	101107.	0.264417
$172$	−283872.	−0.731648
$173$	−162431.	−0.412623	−0.206312	−	0.978486i	$-0.566146\pi$
$173$	−162431.	−0.412623	−0.206312	+	0.978486i	$0.566146\pi$
$174$	257987.	0.645988
$175$	0	0
$176$	101498.	0.246988
$177$	−41477.6	−0.0995132
$178$	−492557.	−1.16522
$179$	632163.	1.47468	0.737338	−	0.675524i	$-0.236082\pi$
$179$	632163.	1.47468	0.737338	+	0.675524i	$0.236082\pi$
$180$	0	0
$181$	−447428.	−1.01514	−0.507571	−	0.861610i	$-0.669456\pi$
$181$	−447428.	−1.01514	−0.507571	+	0.861610i	$0.669456\pi$
$182$	79452.2	0.177798
$183$	−50649.0	−0.111800
$184$	106553.	0.232017
$185$	0	0
$186$	237091.	0.502497
$187$	151032.	0.315838
$188$	−805677.	−1.66252
$189$	−39829.2	−0.0811048
$190$	0	0
$191$	−79686.3	−0.158052	−0.0790261	−	0.996873i	$-0.525181\pi$
$191$	−79686.3	−0.158052	−0.0790261	+	0.996873i	$0.525181\pi$
$192$	−378808.	−0.741594
$193$	−115133.	−0.222488	−0.111244	−	0.993793i	$-0.535484\pi$
$193$	−115133.	−0.222488	−0.111244	+	0.993793i	$0.535484\pi$
$194$	−641830.	−1.22438
$195$	0	0
$196$	−511470.	−0.950999
$197$	359835.	0.660600	0.330300	−	0.943876i	$-0.392850\pi$
$197$	359835.	0.660600	0.330300	+	0.943876i	$0.392850\pi$
$198$	−81415.7	−0.147586
$199$	−281811.	−0.504459	−0.252229	−	0.967667i	$-0.581164\pi$
$199$	−281811.	−0.504459	−0.252229	+	0.967667i	$0.581164\pi$
$200$	0	0
$201$	−320164.	−0.558962
$202$	−1.19500e6	−2.06057
$203$	−188535.	−0.321108
$204$	−415696.	−0.699360
$205$	0	0
$206$	699383.	1.14828
$207$	207628.	0.336790
$208$	146848.	0.235347
$209$	−151036.	−0.239174
$210$	0	0
$211$	791279.	1.22356	0.611778	−	0.791030i	$-0.290455\pi$
$211$	791279.	1.22356	0.611778	+	0.791030i	$0.290455\pi$
$212$	−1.43985e6	−2.20027
$213$	301032.	0.454636
$214$	1.23722e6	1.84676
$215$	0	0
$216$	30303.5	0.0441935
$217$	−173264.	−0.249781
$218$	736165.	1.04914
$219$	166714.	0.234889
$220$	0	0
$221$	218513.	0.300952
$222$	−1.14814e6	−1.56355
$223$	80187.6	0.107980	0.0539902	−	0.998541i	$-0.482806\pi$
$223$	80187.6	0.107980	0.0539902	+	0.998541i	$0.482806\pi$
$224$	453376.	0.603724
$225$	0	0
$226$	−632338.	−0.823528
$227$	−538847.	−0.694066	−0.347033	−	0.937853i	$-0.612811\pi$
$227$	−538847.	−0.694066	−0.347033	+	0.937853i	$0.612811\pi$
$228$	415707.	0.529602
$229$	431400.	0.543616	0.271808	−	0.962352i	$-0.412379\pi$
$229$	431400.	0.543616	0.271808	+	0.962352i	$0.412379\pi$
$230$	0	0
$231$	59497.9	0.0733621
$232$	143444.	0.174970
$233$	−1.56748e6	−1.89152	−0.945760	−	0.324866i	$-0.894681\pi$
$233$	−1.56748e6	−1.89152	−0.945760	+	0.324866i	$0.894681\pi$
$234$	−117792.	−0.140630
$235$	0	0
$236$	−170538.	−0.199316
$237$	140454.	0.162429
$238$	566493.	0.648264
$239$	−402820.	−0.456159	−0.228079	−	0.973643i	$-0.573245\pi$
$239$	−402820.	−0.456159	−0.228079	+	0.973643i	$0.573245\pi$
$240$	0	0
$241$	1.08492e6	1.20325	0.601625	−	0.798779i	$-0.294520\pi$
$241$	1.08492e6	1.20325	0.601625	+	0.798779i	$0.294520\pi$
$242$	121621.	0.133496
$243$	59049.0	0.0641500
$244$	−208247.	−0.223926
$245$	0	0
$246$	−1.08689e6	−1.14511
$247$	−218519.	−0.227901
$248$	131826.	0.136104
$249$	686683.	0.701872
$250$	0	0
$251$	705007.	0.706332	0.353166	−	0.935561i	$-0.385105\pi$
$251$	705007.	0.706332	0.353166	+	0.935561i	$0.385105\pi$
$252$	−163760.	−0.162445
$253$	−310160.	−0.304638
$254$	−1.10609e6	−1.07573
$255$	0	0
$256$	648336.	0.618302
$257$	247273.	0.233530	0.116765	−	0.993160i	$-0.462748\pi$
$257$	247273.	0.233530	0.116765	+	0.993160i	$0.462748\pi$
$258$	−573526.	−0.536418
$259$	839048.	0.777209
$260$	0	0
$261$	279513.	0.253981
$262$	−1.06485e6	−0.958375
$263$	−270166.	−0.240847	−0.120423	−	0.992723i	$-0.538425\pi$
$263$	−270166.	−0.240847	−0.120423	+	0.992723i	$0.538425\pi$
$264$	−45268.2	−0.0399745
$265$	0	0
$266$	−566507.	−0.490910
$267$	−533656.	−0.458124
$268$	−1.31638e6	−1.11955
$269$	838616.	0.706614	0.353307	−	0.935507i	$-0.385057\pi$
$269$	838616.	0.706614	0.353307	+	0.935507i	$0.385057\pi$
$270$	0	0
$271$	1.01547e6	0.839930	0.419965	−	0.907540i	$-0.362042\pi$
$271$	1.01547e6	0.839930	0.419965	+	0.907540i	$0.362042\pi$
$272$	1.04702e6	0.858091
$273$	86081.7	0.0699044
$274$	1.84169e6	1.48197
$275$	0	0
$276$	853675.	0.674559
$277$	−688627.	−0.539243	−0.269622	−	0.962966i	$-0.586899\pi$
$277$	−688627.	−0.539243	−0.269622	+	0.962966i	$0.586899\pi$
$278$	874514.	0.678664
$279$	256874.	0.197565
$280$	0	0
$281$	−14371.5	−0.0108577	−0.00542885	−	0.999985i	$-0.501728\pi$
$281$	−14371.5	−0.0108577	−0.00542885	+	0.999985i	$0.501728\pi$
$282$	−1.62776e6	−1.21890
$283$	−1.23554e6	−0.917044	−0.458522	−	0.888683i	$-0.651621\pi$
$283$	−1.23554e6	−0.917044	−0.458522	+	0.888683i	$0.651621\pi$
$284$	1.23772e6	0.910594
$285$	0	0
$286$	175961.	0.127205
$287$	794289.	0.569212
$288$	−672156.	−0.477517
$289$	138140.	0.0972918
$290$	0	0
$291$	−695384.	−0.481385
$292$	685457.	0.470460
$293$	794280.	0.540511	0.270256	−	0.962789i	$-0.412892\pi$
$293$	794280.	0.540511	0.270256	+	0.962789i	$0.412892\pi$
$294$	−1.03336e6	−0.697239
$295$	0	0
$296$	−638379.	−0.423496
$297$	−88209.0	−0.0580259
$298$	−3.39644e6	−2.21556
$299$	−448740.	−0.290280
$300$	0	0
$301$	419128.	0.266643
$302$	3.18099e6	2.00699
$303$	−1.29471e6	−0.810148
$304$	−1.04705e6	−0.649805
$305$	0	0
$306$	−839858.	−0.512746
$307$	−2.78716e6	−1.68778	−0.843890	−	0.536516i	$-0.819740\pi$
$307$	−2.78716e6	−1.68778	−0.843890	+	0.536516i	$0.819740\pi$
$308$	244630.	0.146937
$309$	757740.	0.451465
$310$	0	0
$311$	−2.91123e6	−1.70677	−0.853386	−	0.521280i	$-0.825455\pi$
$311$	−2.91123e6	−1.70677	−0.853386	+	0.521280i	$0.825455\pi$
$312$	−65494.1	−0.0380904
$313$	−563042.	−0.324848	−0.162424	−	0.986721i	$-0.551931\pi$
$313$	−563042.	−0.324848	−0.162424	+	0.986721i	$0.551931\pi$
$314$	3.04991e6	1.74567
$315$	0	0
$316$	577488.	0.325331
$317$	442425.	0.247282	0.123641	−	0.992327i	$-0.460543\pi$
$317$	442425.	0.247282	0.123641	+	0.992327i	$0.460543\pi$
$318$	−2.90902e6	−1.61316
$319$	−417545.	−0.229735
$320$	0	0
$321$	1.34045e6	0.726085
$322$	−1.16335e6	−0.625276
$323$	−1.55804e6	−0.830944
$324$	242784.	0.128487
$325$	0	0
$326$	270049.	0.140734
$327$	797591.	0.412487
$328$	−604324.	−0.310160
$329$	1.18956e6	0.605892
$330$	0	0
$331$	3.35851e6	1.68491	0.842454	−	0.538768i	$-0.181110\pi$
$331$	3.35851e6	1.68491	0.842454	+	0.538768i	$0.181110\pi$
$332$	2.82335e6	1.40578
$333$	−1.24394e6	−0.614735
$334$	4.37468e6	2.14576
$335$	0	0
$336$	412466.	0.199315
$337$	−460497.	−0.220878	−0.110439	−	0.993883i	$-0.535226\pi$
$337$	−460497.	−0.220878	−0.110439	+	0.993883i	$0.535226\pi$
$338$	−2.82970e6	−1.34725
$339$	−685100.	−0.323784
$340$	0	0
$341$	−383726.	−0.178704
$342$	839880.	0.388286
$343$	1.67342e6	0.768017
$344$	−318888.	−0.145292
$345$	0	0
$346$	−1.34929e6	−0.605922
$347$	−115478.	−0.0514846	−0.0257423	−	0.999669i	$-0.508195\pi$
$347$	−115478.	−0.0514846	−0.0257423	+	0.999669i	$0.508195\pi$
$348$	1.14924e6	0.508700
$349$	−3.65056e6	−1.60434	−0.802168	−	0.597098i	$-0.796320\pi$
$349$	−3.65056e6	−1.60434	−0.802168	+	0.597098i	$0.796320\pi$
$350$	0	0
$351$	−127621.	−0.0552910
$352$	1.00408e6	0.431930
$353$	2.90387e6	1.24034	0.620170	−	0.784467i	$-0.287064\pi$
$353$	2.90387e6	1.24034	0.620170	+	0.784467i	$0.287064\pi$
$354$	−344549.	−0.146131
$355$	0	0
$356$	−2.19416e6	−0.917580
$357$	613761.	0.254876
$358$	5.25130e6	2.16551
$359$	4.14381e6	1.69693	0.848464	−	0.529254i	$-0.177528\pi$
$359$	4.14381e6	1.69693	0.848464	+	0.529254i	$0.177528\pi$
$360$	0	0
$361$	−918022.	−0.370753
$362$	−3.71673e6	−1.49070
$363$	131769.	0.0524864
$364$	353931.	0.140012
$365$	0	0
$366$	−420735.	−0.164175
$367$	−527417.	−0.204404	−0.102202	−	0.994764i	$-0.532589\pi$
$367$	−527417.	−0.204404	−0.102202	+	0.994764i	$0.532589\pi$
$368$	−2.15017e6	−0.827662
$369$	−1.17758e6	−0.450219
$370$	0	0
$371$	2.12589e6	0.801872
$372$	1.05616e6	0.395704
$373$	−2.76612e6	−1.02944	−0.514718	−	0.857360i	$-0.672103\pi$
$373$	−2.76612e6	−1.02944	−0.514718	+	0.857360i	$0.672103\pi$
$374$	1.25460e6	0.463796
$375$	0	0
$376$	−905057.	−0.330146
$377$	−604105.	−0.218907
$378$	−330856.	−0.119099
$379$	2.05750e6	0.735769	0.367884	−	0.929872i	$-0.380082\pi$
$379$	2.05750e6	0.735769	0.367884	+	0.929872i	$0.380082\pi$
$380$	0	0
$381$	−1.19838e6	−0.422943
$382$	−661944.	−0.232093
$383$	4.97186e6	1.73190	0.865949	−	0.500133i	$-0.166715\pi$
$383$	4.97186e6	1.73190	0.865949	+	0.500133i	$0.166715\pi$
$384$	−756824.	−0.261919
$385$	0	0
$386$	−956396.	−0.326715
$387$	−621381.	−0.210902
$388$	−2.85912e6	−0.964169
$389$	4.75117e6	1.59194	0.795970	−	0.605336i	$-0.206961\pi$
$389$	4.75117e6	1.59194	0.795970	+	0.605336i	$0.206961\pi$
$390$	0	0
$391$	−3.19951e6	−1.05838
$392$	−574560.	−0.188851
$393$	−1.15370e6	−0.376801
$394$	2.98911e6	0.970065
$395$	0	0
$396$	−362677.	−0.116220
$397$	−405.982	−0.000129280 0	−6.46398e−5	−	1.00000i	$-0.500021\pi$
$397$	−405.982	−0.000129280 0	−6.46398e−5		1.00000i	$0.500021\pi$
$398$	−2.34097e6	−0.740778
$399$	−613777.	−0.193009
$400$	0	0
$401$	21666.1	0.00672851	0.00336426	−	0.999994i	$-0.498929\pi$
$401$	21666.1	0.00672851	0.00336426	+	0.999994i	$0.498929\pi$
$402$	−2.65956e6	−0.820814
$403$	−555175.	−0.170282
$404$	−5.32327e6	−1.62265
$405$	0	0
$406$	−1.56613e6	−0.471535
$407$	1.85823e6	0.556049
$408$	−466972.	−0.138880
$409$	142989.	0.0422664	0.0211332	−	0.999777i	$-0.493273\pi$
$409$	142989.	0.0422664	0.0211332	+	0.999777i	$0.493273\pi$
$410$	0	0
$411$	1.99536e6	0.582663
$412$	3.11550e6	0.904242
$413$	251794.	0.0726390
$414$	1.72474e6	0.494563
$415$	0	0
$416$	1.45271e6	0.411573
$417$	947484.	0.266828
$418$	−1.25463e6	−0.351218
$419$	−454799.	−0.126557	−0.0632783	−	0.997996i	$-0.520156\pi$
$419$	−454799.	−0.126557	−0.0632783	+	0.997996i	$0.520156\pi$
$420$	0	0
$421$	3.37870e6	0.929062	0.464531	−	0.885557i	$-0.346223\pi$
$421$	3.37870e6	0.929062	0.464531	+	0.885557i	$0.346223\pi$
$422$	6.57305e6	1.79674
$423$	−1.76358e6	−0.479231
$424$	−1.61745e6	−0.436935
$425$	0	0
$426$	2.50064e6	0.667616
$427$	307470.	0.0816080
$428$	5.51135e6	1.45428
$429$	190644.	0.0500126
$430$	0	0
$431$	−5.68143e6	−1.47321	−0.736604	−	0.676324i	$-0.763572\pi$
$431$	−5.68143e6	−1.47321	−0.736604	+	0.676324i	$0.763572\pi$
$432$	−611505.	−0.157649
$433$	1.86171e6	0.477191	0.238596	−	0.971119i	$-0.423313\pi$
$433$	1.86171e6	0.477191	0.238596	+	0.971119i	$0.423313\pi$
$434$	−1.43928e6	−0.366794
$435$	0	0
$436$	3.27935e6	0.826174
$437$	3.19959e6	0.801477
$438$	1.38487e6	0.344925
$439$	−801463.	−0.198482	−0.0992412	−	0.995063i	$-0.531642\pi$
$439$	−801463.	−0.198482	−0.0992412	+	0.995063i	$0.531642\pi$
$440$	0	0
$441$	−1.11958e6	−0.274131
$442$	1.81516e6	0.441937
$443$	2.24766e6	0.544154	0.272077	−	0.962275i	$-0.412289\pi$
$443$	2.24766e6	0.544154	0.272077	+	0.962275i	$0.412289\pi$
$444$	−5.11453e6	−1.23126
$445$	0	0
$446$	666108.	0.158565
$447$	−3.67984e6	−0.871084
$448$	2.29959e6	0.541322
$449$	−4.10754e6	−0.961537	−0.480769	−	0.876848i	$-0.659642\pi$
$449$	−4.10754e6	−0.961537	−0.480769	+	0.876848i	$0.659642\pi$
$450$	0	0
$451$	1.75910e6	0.407239
$452$	−2.81684e6	−0.648509
$453$	3.44641e6	0.789081
$454$	−4.47613e6	−1.01921
$455$	0	0
$456$	466984.	0.105170
$457$	−7.14569e6	−1.60049	−0.800246	−	0.599672i	$-0.795298\pi$
$457$	−7.14569e6	−1.60049	−0.800246	+	0.599672i	$0.795298\pi$
$458$	3.58359e6	0.798278
$459$	−909936.	−0.201595
$460$	0	0
$461$	4.42326e6	0.969372	0.484686	−	0.874688i	$-0.338934\pi$
$461$	4.42326e6	0.969372	0.484686	+	0.874688i	$0.338934\pi$
$462$	494241.	0.107729
$463$	−2.72107e6	−0.589912	−0.294956	−	0.955511i	$-0.595305\pi$
$463$	−2.72107e6	−0.589912	−0.294956	+	0.955511i	$0.595305\pi$
$464$	−2.89461e6	−0.624159
$465$	0	0
$466$	−1.30208e7	−2.77762
$467$	−2.07874e6	−0.441070	−0.220535	−	0.975379i	$-0.570780\pi$
$467$	−2.07874e6	−0.441070	−0.220535	+	0.975379i	$0.570780\pi$
$468$	−524723.	−0.110743
$469$	1.94358e6	0.408010
$470$	0	0
$471$	3.30440e6	0.686341
$472$	−191574.	−0.0395805
$473$	928236.	0.190768
$474$	1.16674e6	0.238521
$475$	0	0
$476$	2.52352e6	0.510493
$477$	−3.15175e6	−0.634243
$478$	−3.34617e6	−0.669851
$479$	−7.38214e6	−1.47009	−0.735044	−	0.678019i	$-0.762839\pi$
$479$	−7.38214e6	−1.47009	−0.735044	+	0.678019i	$0.762839\pi$
$480$	0	0
$481$	2.68849e6	0.529841
$482$	9.01230e6	1.76693
$483$	−1.26042e6	−0.245838
$484$	541777.	0.105125
$485$	0	0
$486$	490512.	0.0942018
$487$	−4.25158e6	−0.812321	−0.406161	−	0.913802i	$-0.633133\pi$
$487$	−4.25158e6	−0.812321	−0.406161	+	0.913802i	$0.633133\pi$
$488$	−233934.	−0.0444677
$489$	292582.	0.0553319
$490$	0	0
$491$	5.75275e6	1.07689	0.538446	−	0.842660i	$-0.319012\pi$
$491$	5.75275e6	1.07689	0.538446	+	0.842660i	$0.319012\pi$
$492$	−4.84170e6	−0.901747
$493$	−4.30726e6	−0.798148
$494$	−1.81521e6	−0.334664
$495$	0	0
$496$	−2.66016e6	−0.485516
$497$	−1.82745e6	−0.331859
$498$	5.70419e6	1.03067
$499$	−7.53164e6	−1.35406	−0.677031	−	0.735955i	$-0.736734\pi$
$499$	−7.53164e6	−1.35406	−0.677031	+	0.735955i	$0.736734\pi$
$500$	0	0
$501$	4.73971e6	0.843640
$502$	5.85640e6	1.03722
$503$	6.85977e6	1.20890	0.604449	−	0.796644i	$-0.293393\pi$
$503$	6.85977e6	1.20890	0.604449	+	0.796644i	$0.293393\pi$
$504$	−183960.	−0.0322587
$505$	0	0
$506$	−2.57646e6	−0.447349
$507$	−3.06581e6	−0.529695
$508$	−4.92722e6	−0.847116
$509$	−3.43084e6	−0.586956	−0.293478	−	0.955966i	$-0.594813\pi$
$509$	−3.43084e6	−0.586956	−0.293478	+	0.955966i	$0.594813\pi$
$510$	0	0
$511$	−1.01205e6	−0.171455
$512$	8.07658e6	1.36161
$513$	909959.	0.152661
$514$	2.05406e6	0.342930
$515$	0	0
$516$	−2.55485e6	−0.422417
$517$	2.63449e6	0.433481
$518$	6.96987e6	1.14130
$519$	−1.46188e6	−0.238228
$520$	0	0
$521$	7.81233e6	1.26092	0.630458	−	0.776223i	$-0.282867\pi$
$521$	7.81233e6	1.26092	0.630458	+	0.776223i	$0.282867\pi$
$522$	2.32188e6	0.372961
$523$	9.26985e6	1.48190	0.740949	−	0.671561i	$-0.234376\pi$
$523$	9.26985e6	1.48190	0.740949	+	0.671561i	$0.234376\pi$
$524$	−4.74352e6	−0.754698
$525$	0	0
$526$	−2.24423e6	−0.353674
$527$	−3.95839e6	−0.620858
$528$	913483.	0.142599
$529$	134185.	0.0208480
$530$	0	0
$531$	−373299.	−0.0574540
$532$	−2.52359e6	−0.386580
$533$	2.54507e6	0.388045
$534$	−4.43301e6	−0.672737
$535$	0	0
$536$	−1.47875e6	−0.222322
$537$	5.68947e6	0.851405
$538$	6.96627e6	1.03764
$539$	1.67246e6	0.247961
$540$	0	0
$541$	−8.86597e6	−1.30237	−0.651183	−	0.758921i	$-0.725727\pi$
$541$	−8.86597e6	−1.30237	−0.651183	+	0.758921i	$0.725727\pi$
$542$	8.43537e6	1.23341
$543$	−4.02685e6	−0.586092
$544$	1.03578e7	1.50062
$545$	0	0
$546$	715070.	0.102652
$547$	−9.18782e6	−1.31294	−0.656469	−	0.754353i	$-0.727951\pi$
$547$	−9.18782e6	−1.31294	−0.656469	+	0.754353i	$0.727951\pi$
$548$	8.20407e6	1.16702
$549$	−455841.	−0.0645480
$550$	0	0
$551$	4.30737e6	0.604412
$552$	958976.	0.133955
$553$	−852642.	−0.118564
$554$	−5.72034e6	−0.791858
$555$	0	0
$556$	3.89565e6	0.534432
$557$	5.08584e6	0.694584	0.347292	−	0.937757i	$-0.387101\pi$
$557$	5.08584e6	0.694584	0.347292	+	0.937757i	$0.387101\pi$
$558$	2.13382e6	0.290117
$559$	1.34297e6	0.181777
$560$	0	0
$561$	1.35929e6	0.182349
$562$	−119383.	−0.0159441
$563$	−9.69174e6	−1.28864	−0.644319	−	0.764757i	$-0.722859\pi$
$563$	−9.69174e6	−1.28864	−0.644319	+	0.764757i	$0.722859\pi$
$564$	−7.25110e6	−0.959855
$565$	0	0
$566$	−1.02635e7	−1.34664
$567$	−358462.	−0.0468259
$568$	1.39039e6	0.180828
$569$	1.19915e6	0.155271	0.0776357	−	0.996982i	$-0.475263\pi$
$569$	1.19915e6	0.155271	0.0776357	+	0.996982i	$0.475263\pi$
$570$	0	0
$571$	2.76815e6	0.355303	0.177652	−	0.984093i	$-0.443150\pi$
$571$	2.76815e6	0.355303	0.177652	+	0.984093i	$0.443150\pi$
$572$	783845.	0.100171
$573$	−717177.	−0.0912515
$574$	6.59806e6	0.835865
$575$	0	0
$576$	−3.40927e6	−0.428160
$577$	1.52867e7	1.91150	0.955748	−	0.294186i	$-0.0950486\pi$
$577$	1.52867e7	1.91150	0.955748	+	0.294186i	$0.0950486\pi$
$578$	1.14751e6	0.142869
$579$	−1.03620e6	−0.128454
$580$	0	0
$581$	−4.16857e6	−0.512327
$582$	−5.77647e6	−0.706895
$583$	4.70817e6	0.573694
$584$	770008.	0.0934250
$585$	0	0
$586$	6.59798e6	0.793720
$587$	−1.19628e7	−1.43298	−0.716489	−	0.697599i	$-0.754252\pi$
$587$	−1.19628e7	−1.43298	−0.716489	+	0.697599i	$0.754252\pi$
$588$	−4.60323e6	−0.549060
$589$	3.95849e6	0.470156
$590$	0	0
$591$	3.23852e6	0.381397
$592$	1.28821e7	1.51071
$593$	−5.73684e6	−0.669940	−0.334970	−	0.942229i	$-0.608726\pi$
$593$	−5.73684e6	−0.669940	−0.334970	+	0.942229i	$0.608726\pi$
$594$	−732741.	−0.0852088
$595$	0	0
$596$	−1.51299e7	−1.74470
$597$	−2.53630e6	−0.291249
$598$	−3.72763e6	−0.426265
$599$	3.70905e6	0.422373	0.211186	−	0.977446i	$-0.432267\pi$
$599$	3.70905e6	0.422373	0.211186	+	0.977446i	$0.432267\pi$
$600$	0	0
$601$	−128805.	−0.0145461	−0.00727306	−	0.999974i	$-0.502315\pi$
$601$	−128805.	−0.0145461	−0.00727306	+	0.999974i	$0.502315\pi$
$602$	3.48164e6	0.391555
$603$	−2.88147e6	−0.322717
$604$	1.41702e7	1.58046
$605$	0	0
$606$	−1.07550e7	−1.18967
$607$	−1.63065e7	−1.79635	−0.898173	−	0.439642i	$-0.855105\pi$
$607$	−1.63065e7	−1.79635	−0.898173	+	0.439642i	$0.855105\pi$
$608$	−1.03581e7	−1.13637
$609$	−1.69681e6	−0.185392
$610$	0	0
$611$	3.81159e6	0.413050
$612$	−3.74126e6	−0.403775
$613$	1.05148e6	0.113019	0.0565094	−	0.998402i	$-0.482003\pi$
$613$	1.05148e6	0.113019	0.0565094	+	0.998402i	$0.482003\pi$
$614$	−2.31526e7	−2.47844
$615$	0	0
$616$	274805.	0.0291791
$617$	−1.47566e7	−1.56053	−0.780265	−	0.625449i	$-0.784916\pi$
$617$	−1.47566e7	−1.56053	−0.780265	+	0.625449i	$0.784916\pi$
$618$	6.29445e6	0.662959
$619$	−3.16617e6	−0.332130	−0.166065	−	0.986115i	$-0.553106\pi$
$619$	−3.16617e6	−0.332130	−0.166065	+	0.986115i	$0.553106\pi$
$620$	0	0
$621$	1.86865e6	0.194446
$622$	−2.41832e7	−2.50633
$623$	3.23961e6	0.334405
$624$	1.32163e6	0.135878
$625$	0	0
$626$	−4.67712e6	−0.477027
$627$	−1.35932e6	−0.138087
$628$	1.35863e7	1.37468
$629$	1.91689e7	1.93184
$630$	0	0
$631$	1.74306e7	1.74276	0.871382	−	0.490605i	$-0.163224\pi$
$631$	1.74306e7	1.74276	0.871382	+	0.490605i	$0.163224\pi$
$632$	648721.	0.0646049
$633$	7.12151e6	0.706420
$634$	3.67517e6	0.363123
$635$	0	0
$636$	−1.29586e7	−1.27033
$637$	2.41972e6	0.236274
$638$	−3.46849e6	−0.337356
$639$	2.70929e6	0.262484
$640$	0	0
$641$	1.80105e6	0.173134	0.0865669	−	0.996246i	$-0.472410\pi$
$641$	1.80105e6	0.173134	0.0865669	+	0.996246i	$0.472410\pi$
$642$	1.11349e7	1.06623
$643$	4.35963e6	0.415836	0.207918	−	0.978146i	$-0.433331\pi$
$643$	4.35963e6	0.415836	0.207918	+	0.978146i	$0.433331\pi$
$644$	−5.18232e6	−0.492390
$645$	0	0
$646$	−1.29424e7	−1.22021
$647$	1.51638e7	1.42412	0.712061	−	0.702117i	$-0.247762\pi$
$647$	1.51638e7	1.42412	0.712061	+	0.702117i	$0.247762\pi$
$648$	272731.	0.0255151
$649$	557644.	0.0519691
$650$	0	0
$651$	−1.55938e6	−0.144211
$652$	1.20297e6	0.110825
$653$	−9.42837e6	−0.865274	−0.432637	−	0.901568i	$-0.642417\pi$
$653$	−9.42837e6	−0.865274	−0.432637	+	0.901568i	$0.642417\pi$
$654$	6.62548e6	0.605722
$655$	0	0
$656$	1.21949e7	1.10641
$657$	1.50043e6	0.135613
$658$	9.88149e6	0.889728
$659$	−1.78022e7	−1.59683	−0.798415	−	0.602107i	$-0.794328\pi$
$659$	−1.78022e7	−1.59683	−0.798415	+	0.602107i	$0.794328\pi$
$660$	0	0
$661$	−1.63561e6	−0.145605	−0.0728027	−	0.997346i	$-0.523194\pi$
$661$	−1.63561e6	−0.145605	−0.0728027	+	0.997346i	$0.523194\pi$
$662$	2.78987e7	2.47422
$663$	1.96662e6	0.173755
$664$	3.17160e6	0.279164
$665$	0	0
$666$	−1.03332e7	−0.902714
$667$	8.84541e6	0.769845
$668$	1.94876e7	1.68973
$669$	721688.	0.0623425
$670$	0	0
$671$	680948.	0.0583859
$672$	4.08039e6	0.348560
$673$	−9.45334e6	−0.804540	−0.402270	−	0.915521i	$-0.631779\pi$
$673$	−9.45334e6	−0.804540	−0.402270	+	0.915521i	$0.631779\pi$
$674$	−3.82529e6	−0.324350
$675$	0	0
$676$	−1.26053e7	−1.06093
$677$	−1.41485e7	−1.18642	−0.593212	−	0.805047i	$-0.702140\pi$
$677$	−1.41485e7	−1.18642	−0.593212	+	0.805047i	$0.702140\pi$
$678$	−5.69104e6	−0.475464
$679$	4.22139e6	0.351384
$680$	0	0
$681$	−4.84962e6	−0.400719
$682$	−3.18756e6	−0.262420
$683$	−1.45731e6	−0.119536	−0.0597681	−	0.998212i	$-0.519036\pi$
$683$	−1.45731e6	−0.119536	−0.0597681	+	0.998212i	$0.519036\pi$
$684$	3.74136e6	0.305766
$685$	0	0
$686$	1.39009e7	1.12780
$687$	3.88260e6	0.313857
$688$	6.43495e6	0.518292
$689$	6.81179e6	0.546655
$690$	0	0
$691$	1.64861e7	1.31348	0.656740	−	0.754117i	$-0.271935\pi$
$691$	1.64861e7	1.31348	0.656740	+	0.754117i	$0.271935\pi$
$692$	−6.01062e6	−0.477149
$693$	535481.	0.0423556
$694$	−959264.	−0.0756031
$695$	0	0
$696$	1.29100e6	0.101019
$697$	1.81463e7	1.41484
$698$	−3.03247e7	−2.35591
$699$	−1.41073e7	−1.09207
$700$	0	0
$701$	−1.82652e7	−1.40388	−0.701939	−	0.712237i	$-0.747682\pi$
$701$	−1.82652e7	−1.40388	−0.701939	+	0.712237i	$0.747682\pi$
$702$	−1.06013e6	−0.0811927
$703$	−1.91694e7	−1.46292
$704$	5.09287e6	0.387285
$705$	0	0
$706$	2.41221e7	1.82139
$707$	7.85963e6	0.591363
$708$	−1.53484e6	−0.115075
$709$	2.53273e7	1.89223	0.946115	−	0.323830i	$-0.104970\pi$
$709$	2.53273e7	1.89223	0.946115	+	0.323830i	$0.104970\pi$
$710$	0	0
$711$	1.26409e6	0.0937787
$712$	−2.46481e6	−0.182215
$713$	8.12897e6	0.598841
$714$	5.09844e6	0.374276
$715$	0	0
$716$	2.33926e7	1.70528
$717$	−3.62538e6	−0.263363
$718$	3.44221e7	2.49187
$719$	−1.50245e7	−1.08387	−0.541937	−	0.840419i	$-0.682309\pi$
$719$	−1.50245e7	−1.08387	−0.541937	+	0.840419i	$0.682309\pi$
$720$	0	0
$721$	−4.59993e6	−0.329544
$722$	−7.62589e6	−0.544437
$723$	9.76429e6	0.694697
$724$	−1.65567e7	−1.17389
$725$	0	0
$726$	1.09459e6	0.0770742
$727$	−1.17483e7	−0.824401	−0.412200	−	0.911093i	$-0.635240\pi$
$727$	−1.17483e7	−0.824401	−0.412200	+	0.911093i	$0.635240\pi$
$728$	397588.	0.0278039
$729$	531441.	0.0370370
$730$	0	0
$731$	9.57539e6	0.662770
$732$	−1.87422e6	−0.129284
$733$	9.57778e6	0.658423	0.329211	−	0.944256i	$-0.393217\pi$
$733$	9.57778e6	0.658423	0.329211	+	0.944256i	$0.393217\pi$
$734$	−4.38119e6	−0.300159
$735$	0	0
$736$	−2.12709e7	−1.44741
$737$	4.30442e6	0.291908
$738$	−9.78200e6	−0.661130
$739$	−1.95084e7	−1.31405	−0.657023	−	0.753871i	$-0.728184\pi$
$739$	−1.95084e7	−1.31405	−0.657023	+	0.753871i	$0.728184\pi$
$740$	0	0
$741$	−1.96667e6	−0.131579
$742$	1.76595e7	1.17752
$743$	1.16476e6	0.0774039	0.0387019	−	0.999251i	$-0.487678\pi$
$743$	1.16476e6	0.0774039	0.0387019	+	0.999251i	$0.487678\pi$
$744$	1.18643e6	0.0785798
$745$	0	0
$746$	−2.29778e7	−1.51169
$747$	6.18015e6	0.405226
$748$	5.58880e6	0.365229
$749$	−8.13732e6	−0.530001
$750$	0	0
$751$	2.66259e7	1.72268	0.861340	−	0.508029i	$-0.169626\pi$
$751$	2.66259e7	1.72268	0.861340	+	0.508029i	$0.169626\pi$
$752$	1.82635e7	1.17771
$753$	6.34506e6	0.407801
$754$	−5.01822e6	−0.321456
$755$	0	0
$756$	−1.47384e6	−0.0937879
$757$	2.27792e7	1.44477	0.722385	−	0.691491i	$-0.243046\pi$
$757$	2.27792e7	1.44477	0.722385	+	0.691491i	$0.243046\pi$
$758$	1.70914e7	1.08045
$759$	−2.79144e6	−0.175883
$760$	0	0
$761$	−2.05310e7	−1.28513	−0.642567	−	0.766229i	$-0.722131\pi$
$761$	−2.05310e7	−1.28513	−0.642567	+	0.766229i	$0.722131\pi$
$762$	−9.95479e6	−0.621076
$763$	−4.84185e6	−0.301093
$764$	−2.94872e6	−0.182768
$765$	0	0
$766$	4.13006e7	2.54322
$767$	806801.	0.0495197
$768$	5.83503e6	0.356977
$769$	2.16009e7	1.31721	0.658606	−	0.752488i	$-0.271146\pi$
$769$	2.16009e7	1.31721	0.658606	+	0.752488i	$0.271146\pi$
$770$	0	0
$771$	2.22545e6	0.134829
$772$	−4.26040e6	−0.257281
$773$	−2.83324e7	−1.70543	−0.852717	−	0.522373i	$-0.825047\pi$
$773$	−2.83324e7	−1.70543	−0.852717	+	0.522373i	$0.825047\pi$
$774$	−5.16173e6	−0.309701
$775$	0	0
$776$	−3.21179e6	−0.191467
$777$	7.55144e6	0.448722
$778$	3.94674e7	2.33770
$779$	−1.81468e7	−1.07141
$780$	0	0
$781$	−4.04721e6	−0.237426
$782$	−2.65779e7	−1.55419
$783$	2.51562e6	0.146636
$784$	1.15942e7	0.673678
$785$	0	0
$786$	−9.58365e6	−0.553318
$787$	1.78605e7	1.02791	0.513956	−	0.857816i	$-0.328179\pi$
$787$	1.78605e7	1.02791	0.513956	+	0.857816i	$0.328179\pi$
$788$	1.33154e7	0.763903
$789$	−2.43149e6	−0.139053
$790$	0	0
$791$	4.15897e6	0.236344
$792$	−407414.	−0.0230793
$793$	985198.	0.0556340
$794$	−3372.44	−0.000189842 0
$795$	0	0
$796$	−1.04282e7	−0.583345
$797$	−2.12249e7	−1.18359	−0.591793	−	0.806090i	$-0.701580\pi$
$797$	−2.12249e7	−1.18359	−0.591793	+	0.806090i	$0.701580\pi$
$798$	−5.09857e6	−0.283427
$799$	2.71766e7	1.50601
$800$	0	0
$801$	−4.80290e6	−0.264498
$802$	179977.	0.00988056
$803$	−2.24138e6	−0.122667
$804$	−1.18474e7	−0.646371
$805$	0	0
$806$	−4.61177e6	−0.250052
$807$	7.54754e6	0.407964
$808$	−5.97990e6	−0.322230
$809$	2.98029e7	1.60098	0.800492	−	0.599343i	$-0.204572\pi$
$809$	2.98029e7	1.60098	0.800492	+	0.599343i	$0.204572\pi$
$810$	0	0
$811$	2.84229e6	0.151745	0.0758727	−	0.997118i	$-0.475826\pi$
$811$	2.84229e6	0.151745	0.0758727	+	0.997118i	$0.475826\pi$
$812$	−6.97656e6	−0.371323
$813$	9.13922e6	0.484934
$814$	1.54361e7	0.816536
$815$	0	0
$816$	9.42320e6	0.495419
$817$	−9.57563e6	−0.501894
$818$	1.18779e6	0.0620665
$819$	774735.	0.0403593
$820$	0	0
$821$	3.04342e7	1.57581	0.787906	−	0.615795i	$-0.211165\pi$
$821$	3.04342e7	1.57581	0.787906	+	0.615795i	$0.211165\pi$
$822$	1.65752e7	0.855618
$823$	8.37626e6	0.431073	0.215536	−	0.976496i	$-0.430850\pi$
$823$	8.37626e6	0.431073	0.215536	+	0.976496i	$0.430850\pi$
$824$	3.49980e6	0.179566
$825$	0	0
$826$	2.09162e6	0.106668
$827$	−1.58227e7	−0.804482	−0.402241	−	0.915534i	$-0.631769\pi$
$827$	−1.58227e7	−0.804482	−0.402241	+	0.915534i	$0.631769\pi$
$828$	7.68308e6	0.389457
$829$	1.41878e7	0.717016	0.358508	−	0.933527i	$-0.383286\pi$
$829$	1.41878e7	0.717016	0.358508	+	0.933527i	$0.383286\pi$
$830$	0	0
$831$	−6.19765e6	−0.311332
$832$	7.36837e6	0.369031
$833$	1.72526e7	0.861472
$834$	7.87063e6	0.391827
$835$	0	0
$836$	−5.58895e6	−0.276576
$837$	2.31187e6	0.114064
$838$	−3.77796e6	−0.185843
$839$	3.30757e7	1.62220	0.811099	−	0.584908i	$-0.198869\pi$
$839$	3.30757e7	1.62220	0.811099	+	0.584908i	$0.198869\pi$
$840$	0	0
$841$	−8.60325e6	−0.419442
$842$	2.80665e7	1.36429
$843$	−129344.	−0.00626869
$844$	2.92806e7	1.41489
$845$	0	0
$846$	−1.46499e7	−0.703733
$847$	−799916.	−0.0383121
$848$	3.26391e7	1.55865
$849$	−1.11198e7	−0.529456
$850$	0	0
$851$	−3.93653e7	−1.86333
$852$	1.11394e7	0.525732
$853$	−1.36981e7	−0.644596	−0.322298	−	0.946638i	$-0.604455\pi$
$853$	−1.36981e7	−0.644596	−0.322298	+	0.946638i	$0.604455\pi$
$854$	2.55411e6	0.119838
$855$	0	0
$856$	6.19117e6	0.288794
$857$	−3.02670e7	−1.40773	−0.703863	−	0.710336i	$-0.748543\pi$
$857$	−3.02670e7	−1.40773	−0.703863	+	0.710336i	$0.748543\pi$
$858$	1.58365e6	0.0734416
$859$	3.84178e7	1.77643	0.888217	−	0.459424i	$-0.151944\pi$
$859$	3.84178e7	1.77643	0.888217	+	0.459424i	$0.151944\pi$
$860$	0	0
$861$	7.14860e6	0.328635
$862$	−4.71949e7	−2.16335
$863$	4.21231e7	1.92528	0.962638	−	0.270793i	$-0.0872858\pi$
$863$	4.21231e7	1.92528	0.962638	+	0.270793i	$0.0872858\pi$
$864$	−6.04940e6	−0.275695
$865$	0	0
$866$	1.54650e7	0.700737
$867$	1.24326e6	0.0561714
$868$	−6.41149e6	−0.288842
$869$	−1.88833e6	−0.0848260
$870$	0	0
$871$	6.22765e6	0.278150
$872$	3.68386e6	0.164063
$873$	−6.25846e6	−0.277928
$874$	2.65786e7	1.17694
$875$	0	0
$876$	6.16911e6	0.271620
$877$	2.52548e7	1.10878	0.554391	−	0.832257i	$-0.312951\pi$
$877$	2.52548e7	1.10878	0.554391	+	0.832257i	$0.312951\pi$
$878$	−6.65765e6	−0.291464
$879$	7.14852e6	0.312064
$880$	0	0
$881$	−1.22164e7	−0.530279	−0.265139	−	0.964210i	$-0.585418\pi$
$881$	−1.22164e7	−0.530279	−0.265139	+	0.964210i	$0.585418\pi$
$882$	−9.30021e6	−0.402551
$883$	1.53042e7	0.660554	0.330277	−	0.943884i	$-0.392858\pi$
$883$	1.53042e7	0.660554	0.330277	+	0.943884i	$0.392858\pi$
$884$	8.08590e6	0.348015
$885$	0	0
$886$	1.86711e7	0.799070
$887$	−4.48803e7	−1.91534	−0.957672	−	0.287861i	$-0.907056\pi$
$887$	−4.48803e7	−1.91534	−0.957672	+	0.287861i	$0.907056\pi$
$888$	−5.74541e6	−0.244505
$889$	7.27488e6	0.308725
$890$	0	0
$891$	−793881.	−0.0335013
$892$	2.96727e6	0.124866
$893$	−2.71773e7	−1.14045
$894$	−3.05680e7	−1.27915
$895$	0	0
$896$	4.59437e6	0.191186
$897$	−4.03866e6	−0.167593
$898$	−3.41208e7	−1.41198
$899$	1.09434e7	0.451600
$900$	0	0
$901$	4.85679e7	1.99314
$902$	1.46126e7	0.598014
$903$	3.77215e6	0.153946
$904$	−3.16429e6	−0.128782
$905$	0	0
$906$	2.86289e7	1.15873
$907$	−898531.	−0.0362673	−0.0181336	−	0.999836i	$-0.505772\pi$
$907$	−898531.	−0.0362673	−0.0181336	+	0.999836i	$0.505772\pi$
$908$	−1.99396e7	−0.802604
$909$	−1.16524e7	−0.467739
$910$	0	0
$911$	1.67133e7	0.667217	0.333609	−	0.942712i	$-0.391734\pi$
$911$	1.67133e7	0.667217	0.333609	+	0.942712i	$0.391734\pi$
$912$	−9.42344e6	−0.375165
$913$	−9.23207e6	−0.366541
$914$	−5.93583e7	−2.35026
$915$	0	0
$916$	1.59636e7	0.628625
$917$	7.00366e6	0.275044
$918$	−7.55872e6	−0.296034
$919$	1.70008e7	0.664019	0.332009	−	0.943276i	$-0.392273\pi$
$919$	1.70008e7	0.664019	0.332009	+	0.943276i	$0.392273\pi$
$920$	0	0
$921$	−2.50844e7	−0.974440
$922$	3.67435e7	1.42349
$923$	−5.85552e6	−0.226236
$924$	2.20167e6	0.0848343
$925$	0	0
$926$	−2.26036e7	−0.866263
$927$	6.81966e6	0.260653
$928$	−2.86354e7	−1.09152
$929$	4.86778e7	1.85051	0.925255	−	0.379345i	$-0.123851\pi$
$929$	4.86778e7	1.85051	0.925255	+	0.379345i	$0.123851\pi$
$930$	0	0
$931$	−1.72530e7	−0.652365
$932$	−5.80030e7	−2.18731
$933$	−2.62011e7	−0.985405
$934$	−1.72678e7	−0.647694
$935$	0	0
$936$	−589447.	−0.0219915
$937$	2.95283e6	0.109873	0.0549363	−	0.998490i	$-0.482504\pi$
$937$	2.95283e6	0.109873	0.0549363	+	0.998490i	$0.482504\pi$
$938$	1.61451e7	0.599148
$939$	−5.06738e6	−0.187551
$940$	0	0
$941$	−2.28430e7	−0.840969	−0.420484	−	0.907300i	$-0.638140\pi$
$941$	−2.28430e7	−0.840969	−0.420484	+	0.907300i	$0.638140\pi$
$942$	2.74492e7	1.00787
$943$	−3.72653e7	−1.36466
$944$	3.86584e6	0.141193
$945$	0	0
$946$	7.71074e6	0.280135
$947$	−2.08580e7	−0.755783	−0.377892	−	0.925850i	$-0.623351\pi$
$947$	−2.08580e7	−0.755783	−0.377892	+	0.925850i	$0.623351\pi$
$948$	5.19739e6	0.187830
$949$	−3.24283e6	−0.116885
$950$	0	0
$951$	3.98183e6	0.142768
$952$	2.83480e6	0.101375
$953$	−4.73251e7	−1.68795	−0.843974	−	0.536384i	$-0.819790\pi$
$953$	−4.73251e7	−1.68795	−0.843974	+	0.536384i	$0.819790\pi$
$954$	−2.61812e7	−0.931361
$955$	0	0
$956$	−1.49060e7	−0.527492
$957$	−3.75790e6	−0.132637
$958$	−6.13225e7	−2.15877
$959$	−1.21130e7	−0.425311
$960$	0	0
$961$	−1.85721e7	−0.648713
$962$	2.23329e7	0.778051
$963$	1.20640e7	0.419206
$964$	4.01466e7	1.39141
$965$	0	0
$966$	−1.04702e7	−0.361003
$967$	2.72981e6	0.0938785	0.0469393	−	0.998898i	$-0.485053\pi$
$967$	2.72981e6	0.0938785	0.0469393	+	0.998898i	$0.485053\pi$
$968$	608605.	0.0208760
$969$	−1.40223e7	−0.479746
$970$	0	0
$971$	−1.09369e6	−0.0372260	−0.0186130	−	0.999827i	$-0.505925\pi$
$971$	−1.09369e6	−0.0372260	−0.0186130	+	0.999827i	$0.505925\pi$
$972$	2.18506e6	0.0741817
$973$	−5.75179e6	−0.194770
$974$	−3.53173e7	−1.19286
$975$	0	0
$976$	4.72064e6	0.158627
$977$	−5.19943e7	−1.74269	−0.871343	−	0.490675i	$-0.836750\pi$
$977$	−5.19943e7	−1.74269	−0.871343	+	0.490675i	$0.836750\pi$
$978$	2.43044e6	0.0812528
$979$	7.17471e6	0.239247
$980$	0	0
$981$	7.17832e6	0.238150
$982$	4.77874e7	1.58137
$983$	−3.13653e7	−1.03530	−0.517649	−	0.855593i	$-0.673193\pi$
$983$	−3.13653e7	−1.03530	−0.517649	+	0.855593i	$0.673193\pi$
$984$	−5.43892e6	−0.179071
$985$	0	0
$986$	−3.57798e7	−1.17205
$987$	1.07060e7	0.349812
$988$	−8.08610e6	−0.263540
$989$	−1.96641e7	−0.639267
$990$	0	0
$991$	−2.56995e7	−0.831266	−0.415633	−	0.909532i	$-0.636440\pi$
$991$	−2.56995e7	−0.831266	−0.415633	+	0.909532i	$0.636440\pi$
$992$	−2.63160e7	−0.849066
$993$	3.02266e7	0.972782
$994$	−1.51804e7	−0.487322
$995$	0	0
$996$	2.54101e7	0.811630
$997$	−3.33461e7	−1.06245	−0.531224	−	0.847231i	$-0.678268\pi$
$997$	−3.33461e7	−1.06245	−0.531224	+	0.847231i	$0.678268\pi$
$998$	−6.25644e7	−1.98839
$999$	−1.11954e7	−0.354917

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.w.1.12		13
5.2	odd	4		165.6.c.a.34.23	yes	26
5.3	odd	4		165.6.c.a.34.4	✓	26
5.4	even	2		825.6.a.x.1.2		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.a.34.4	✓	26	5.3	odd	4
165.6.c.a.34.23	yes	26	5.2	odd	4
825.6.a.w.1.12		13	1.1	even	1	trivial
825.6.a.x.1.2		13	5.4	even	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+8.30687 q^{2} +9.00000 q^{3} +37.0041 q^{4} +74.7618 q^{6} -54.6353 q^{7} +41.5686 q^{8} +81.0000 q^{9} -121.000 q^{11} +333.037 q^{12} -175.063 q^{13} -453.849 q^{14} -838.827 q^{16} -1248.20 q^{17} +672.857 q^{18} +1248.23 q^{19} -491.718 q^{21} -1005.13 q^{22} +2563.30 q^{23} +374.117 q^{24} -1454.23 q^{26} +729.000 q^{27} -2021.73 q^{28} +3450.78 q^{29} +3171.29 q^{31} -8298.22 q^{32} -1089.00 q^{33} -10368.6 q^{34} +2997.33 q^{36} -15357.2 q^{37} +10368.9 q^{38} -1575.57 q^{39} -14538.0 q^{41} -4084.64 q^{42} -7671.37 q^{43} -4477.50 q^{44} +21293.0 q^{46} -21772.6 q^{47} -7549.44 q^{48} -13822.0 q^{49} -11233.8 q^{51} -6478.06 q^{52} -38910.5 q^{53} +6055.71 q^{54} -2271.11 q^{56} +11234.1 q^{57} +28665.2 q^{58} -4608.63 q^{59} -5627.67 q^{61} +26343.5 q^{62} -4425.46 q^{63} -42089.8 q^{64} -9046.18 q^{66} -35573.8 q^{67} -46188.5 q^{68} +23069.7 q^{69} +33448.0 q^{71} +3367.05 q^{72} +18523.8 q^{73} -127571. q^{74} +46189.6 q^{76} +6610.88 q^{77} -13088.0 q^{78} +15606.1 q^{79} +6561.00 q^{81} -120765. q^{82} +76298.1 q^{83} -18195.6 q^{84} -63725.1 q^{86} +31057.0 q^{87} -5029.80 q^{88} -59295.1 q^{89} +9564.63 q^{91} +94852.8 q^{92} +28541.6 q^{93} -180863. q^{94} -74684.0 q^{96} -77264.9 q^{97} -114817. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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