LMFDB - Embedded newform 1075.6.a.f.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.65370 q^{2} +28.3267 q^{3} -10.3431 q^{4} +131.824 q^{6} -208.159 q^{7} -197.052 q^{8} +559.405 q^{9} +O(q^{10})$	\(q+4.65370 q^{2} +28.3267 q^{3} -10.3431 q^{4} +131.824 q^{6} -208.159 q^{7} -197.052 q^{8} +559.405 q^{9} -621.451 q^{11} -292.985 q^{12} -846.155 q^{13} -968.710 q^{14} -586.044 q^{16} +907.124 q^{17} +2603.30 q^{18} +2284.57 q^{19} -5896.47 q^{21} -2892.05 q^{22} +3343.99 q^{23} -5581.84 q^{24} -3937.75 q^{26} +8962.71 q^{27} +2153.00 q^{28} +3724.21 q^{29} +5592.01 q^{31} +3578.39 q^{32} -17603.7 q^{33} +4221.48 q^{34} -5785.95 q^{36} -4210.03 q^{37} +10631.7 q^{38} -23968.8 q^{39} -2090.32 q^{41} -27440.4 q^{42} +1849.00 q^{43} +6427.70 q^{44} +15562.0 q^{46} -4061.55 q^{47} -16600.7 q^{48} +26523.2 q^{49} +25695.9 q^{51} +8751.82 q^{52} +30601.8 q^{53} +41709.8 q^{54} +41018.1 q^{56} +64714.5 q^{57} +17331.4 q^{58} -35306.4 q^{59} +13391.4 q^{61} +26023.6 q^{62} -116445. q^{63} +35406.2 q^{64} -81922.3 q^{66} +53656.3 q^{67} -9382.43 q^{68} +94724.5 q^{69} +64990.8 q^{71} -110232. q^{72} -33693.7 q^{73} -19592.2 q^{74} -23629.5 q^{76} +129361. q^{77} -111544. q^{78} -73996.1 q^{79} +117949. q^{81} -9727.74 q^{82} -66744.2 q^{83} +60987.5 q^{84} +8604.70 q^{86} +105495. q^{87} +122458. q^{88} +71521.6 q^{89} +176135. q^{91} -34587.1 q^{92} +158404. q^{93} -18901.3 q^{94} +101364. q^{96} -36042.7 q^{97} +123431. q^{98} -347642. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 5 q^{2} - 20 q^{3} + 427 q^{4} + 248 q^{6} - 118 q^{7} - 561 q^{8} + 2618 q^{9}+O(q^{10}) $ 22 * q - 5 * q^2 - 20 * q^3 + 427 * q^4 + 248 * q^6 - 118 * q^7 - 561 * q^8 + 2618 * q^9	\( 22 q - 5 q^{2} - 20 q^{3} + 427 q^{4} + 248 q^{6} - 118 q^{7} - 561 q^{8} + 2618 q^{9} + 1206 q^{11} - 2175 q^{12} - 1942 q^{13} + 2531 q^{14} + 8851 q^{16} - 2470 q^{17} - 1279 q^{18} + 3020 q^{19} + 5632 q^{21} + 3227 q^{22} + 1326 q^{23} + 11040 q^{24} - 3415 q^{26} - 5156 q^{27} + 11489 q^{28} + 17906 q^{29} + 7982 q^{31} - 2427 q^{32} - 10100 q^{33} + 25248 q^{34} - 14813 q^{36} - 22640 q^{37} + 13695 q^{38} + 29048 q^{39} + 29112 q^{41} - 9163 q^{42} + 40678 q^{43} + 63924 q^{44} - 14944 q^{46} - 57080 q^{47} - 54894 q^{48} + 165560 q^{49} - 1576 q^{51} - 97639 q^{52} + 8054 q^{53} + 167379 q^{54} + 269326 q^{56} - 125424 q^{57} - 49485 q^{58} + 193484 q^{59} + 107466 q^{61} - 162441 q^{62} - 183778 q^{63} + 412603 q^{64} + 240489 q^{66} - 109764 q^{67} - 144300 q^{68} + 202444 q^{69} + 182964 q^{71} - 341504 q^{72} - 134468 q^{73} + 198067 q^{74} + 247729 q^{76} + 28416 q^{77} + 7286 q^{78} + 11148 q^{79} + 385246 q^{81} - 23657 q^{82} - 33850 q^{83} + 176749 q^{84} - 9245 q^{86} + 298280 q^{87} + 111354 q^{88} + 244912 q^{89} + 158092 q^{91} + 124762 q^{92} - 239860 q^{93} - 192166 q^{94} - 147719 q^{96} - 232826 q^{97} + 482463 q^{98} - 346894 q^{99}+O(q^{100}) \) 22 * q - 5 * q^2 - 20 * q^3 + 427 * q^4 + 248 * q^6 - 118 * q^7 - 561 * q^8 + 2618 * q^9 + 1206 * q^11 - 2175 * q^12 - 1942 * q^13 + 2531 * q^14 + 8851 * q^16 - 2470 * q^17 - 1279 * q^18 + 3020 * q^19 + 5632 * q^21 + 3227 * q^22 + 1326 * q^23 + 11040 * q^24 - 3415 * q^26 - 5156 * q^27 + 11489 * q^28 + 17906 * q^29 + 7982 * q^31 - 2427 * q^32 - 10100 * q^33 + 25248 * q^34 - 14813 * q^36 - 22640 * q^37 + 13695 * q^38 + 29048 * q^39 + 29112 * q^41 - 9163 * q^42 + 40678 * q^43 + 63924 * q^44 - 14944 * q^46 - 57080 * q^47 - 54894 * q^48 + 165560 * q^49 - 1576 * q^51 - 97639 * q^52 + 8054 * q^53 + 167379 * q^54 + 269326 * q^56 - 125424 * q^57 - 49485 * q^58 + 193484 * q^59 + 107466 * q^61 - 162441 * q^62 - 183778 * q^63 + 412603 * q^64 + 240489 * q^66 - 109764 * q^67 - 144300 * q^68 + 202444 * q^69 + 182964 * q^71 - 341504 * q^72 - 134468 * q^73 + 198067 * q^74 + 247729 * q^76 + 28416 * q^77 + 7286 * q^78 + 11148 * q^79 + 385246 * q^81 - 23657 * q^82 - 33850 * q^83 + 176749 * q^84 - 9245 * q^86 + 298280 * q^87 + 111354 * q^88 + 244912 * q^89 + 158092 * q^91 + 124762 * q^92 - 239860 * q^93 - 192166 * q^94 - 147719 * q^96 - 232826 * q^97 + 482463 * q^98 - 346894 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.65370	0.822666	0.411333	−	0.911485i	$-0.365063\pi$
$2$	4.65370	0.822666	0.411333	+	0.911485i	$0.365063\pi$
$3$	28.3267	1.81716	0.908581	−	0.417709i	$-0.137167\pi$
$3$	28.3267	1.81716	0.908581	+	0.417709i	$0.137167\pi$
$4$	−10.3431	−0.323220
$5$	0	0
$5$	0	0
$6$	131.824	1.49492
$7$	−208.159	−1.60565	−0.802823	−	0.596217i	$-0.796670\pi$
$7$	−208.159	−1.60565	−0.802823	+	0.596217i	$0.796670\pi$
$8$	−197.052	−1.08857
$9$	559.405	2.30208
$10$	0	0
$11$	−621.451	−1.54855	−0.774275	−	0.632850i	$-0.781885\pi$
$11$	−621.451	−1.54855	−0.774275	+	0.632850i	$0.781885\pi$
$12$	−292.985	−0.587344
$13$	−846.155	−1.38865	−0.694323	−	0.719664i	$-0.744296\pi$
$13$	−846.155	−1.38865	−0.694323	+	0.719664i	$0.744296\pi$
$14$	−968.710	−1.32091
$15$	0	0
$16$	−586.044	−0.572308
$17$	907.124	0.761280	0.380640	−	0.924723i	$-0.375704\pi$
$17$	907.124	0.761280	0.380640	+	0.924723i	$0.375704\pi$
$18$	2603.30	1.89384
$19$	2284.57	1.45185	0.725924	−	0.687775i	$-0.241412\pi$
$19$	2284.57	1.45185	0.725924	+	0.687775i	$0.241412\pi$
$20$	0	0
$21$	−5896.47	−2.91772
$22$	−2892.05	−1.27394
$23$	3343.99	1.31809	0.659046	−	0.752102i	$-0.270960\pi$
$23$	3343.99	1.31809	0.659046	+	0.752102i	$0.270960\pi$
$24$	−5581.84	−1.97811
$25$	0	0
$26$	−3937.75	−1.14239
$27$	8962.71	2.36608
$28$	2153.00	0.518978
$29$	3724.21	0.822317	0.411158	−	0.911564i	$-0.365124\pi$
$29$	3724.21	0.822317	0.411158	+	0.911564i	$0.365124\pi$
$30$	0	0
$31$	5592.01	1.04511	0.522557	−	0.852604i	$-0.324978\pi$
$31$	5592.01	1.04511	0.522557	+	0.852604i	$0.324978\pi$
$32$	3578.39	0.617750
$33$	−17603.7	−2.81396
$34$	4221.48	0.626279
$35$	0	0
$36$	−5785.95	−0.744078
$37$	−4210.03	−0.505569	−0.252784	−	0.967523i	$-0.581346\pi$
$37$	−4210.03	−0.505569	−0.252784	+	0.967523i	$0.581346\pi$
$38$	10631.7	1.19439
$39$	−23968.8	−2.52339
$40$	0	0
$41$	−2090.32	−0.194202	−0.0971010	−	0.995275i	$-0.530957\pi$
$41$	−2090.32	−0.194202	−0.0971010	+	0.995275i	$0.530957\pi$
$42$	−27440.4	−2.40031
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	6427.70	0.500523
$45$	0	0
$46$	15562.0	1.08435
$47$	−4061.55	−0.268193	−0.134096	−	0.990968i	$-0.542813\pi$
$47$	−4061.55	−0.268193	−0.134096	+	0.990968i	$0.542813\pi$
$48$	−16600.7	−1.03998
$49$	26523.2	1.57810
$50$	0	0
$51$	25695.9	1.38337
$52$	8751.82	0.448839
$53$	30601.8	1.49643	0.748217	−	0.663454i	$-0.230910\pi$
$53$	30601.8	1.49643	0.748217	+	0.663454i	$0.230910\pi$
$54$	41709.8	1.94650
$55$	0	0
$56$	41018.1	1.74786
$57$	64714.5	2.63824
$58$	17331.4	0.676492
$59$	−35306.4	−1.32046	−0.660228	−	0.751066i	$-0.729540\pi$
$59$	−35306.4	−1.32046	−0.660228	+	0.751066i	$0.729540\pi$
$60$	0	0
$61$	13391.4	0.460788	0.230394	−	0.973097i	$-0.425999\pi$
$61$	13391.4	0.460788	0.230394	+	0.973097i	$0.425999\pi$
$62$	26023.6	0.859780
$63$	−116445.	−3.69632
$64$	35406.2	1.08051
$65$	0	0
$66$	−81922.3	−2.31495
$67$	53656.3	1.46027	0.730136	−	0.683302i	$-0.239457\pi$
$67$	53656.3	1.46027	0.730136	+	0.683302i	$0.239457\pi$
$68$	−9382.43	−0.246061
$69$	94724.5	2.39519
$70$	0	0
$71$	64990.8	1.53005	0.765026	−	0.643999i	$-0.222726\pi$
$71$	64990.8	1.53005	0.765026	+	0.643999i	$0.222726\pi$
$72$	−110232.	−2.50597
$73$	−33693.7	−0.740017	−0.370009	−	0.929028i	$-0.620645\pi$
$73$	−33693.7	−0.740017	−0.370009	+	0.929028i	$0.620645\pi$
$74$	−19592.2	−0.415914
$75$	0	0
$76$	−23629.5	−0.469267
$77$	129361.	2.48642
$78$	−111544.	−2.07591
$79$	−73996.1	−1.33395	−0.666977	−	0.745078i	$-0.732412\pi$
$79$	−73996.1	−1.33395	−0.666977	+	0.745078i	$0.732412\pi$
$80$	0	0
$81$	117949.	1.99748
$82$	−9727.74	−0.159763
$83$	−66744.2	−1.06345	−0.531727	−	0.846916i	$-0.678457\pi$
$83$	−66744.2	−1.06345	−0.531727	+	0.846916i	$0.678457\pi$
$84$	60987.5	0.943066
$85$	0	0
$86$	8604.70	0.125455
$87$	105495.	1.49428
$88$	122458.	1.68570
$89$	71521.6	0.957110	0.478555	−	0.878057i	$-0.341161\pi$
$89$	71521.6	0.957110	0.478555	+	0.878057i	$0.341161\pi$
$90$	0	0
$91$	176135.	2.22967
$92$	−34587.1	−0.426034
$93$	158404.	1.89914
$94$	−18901.3	−0.220633
$95$	0	0
$96$	101364.	1.12255
$97$	−36042.7	−0.388944	−0.194472	−	0.980908i	$-0.562299\pi$
$97$	−36042.7	−0.388944	−0.194472	+	0.980908i	$0.562299\pi$
$98$	123431.	1.29825
$99$	−347642.	−3.56488
$100$	0	0
$101$	39455.0	0.384856	0.192428	−	0.981311i	$-0.438364\pi$
$101$	39455.0	0.384856	0.192428	+	0.981311i	$0.438364\pi$
$102$	119581.	1.13805
$103$	38771.0	0.360092	0.180046	−	0.983658i	$-0.442375\pi$
$103$	38771.0	0.360092	0.180046	+	0.983658i	$0.442375\pi$
$104$	166736.	1.51164
$105$	0	0
$106$	142412.	1.23107
$107$	53214.8	0.449338	0.224669	−	0.974435i	$-0.427870\pi$
$107$	53214.8	0.449338	0.224669	+	0.974435i	$0.427870\pi$
$108$	−92701.8	−0.764766
$109$	−2537.78	−0.0204591	−0.0102296	−	0.999948i	$-0.503256\pi$
$109$	−2537.78	−0.0204591	−0.0102296	+	0.999948i	$0.503256\pi$
$110$	0	0
$111$	−119256.	−0.918700
$112$	121990.	0.918925
$113$	−100535.	−0.740661	−0.370330	−	0.928900i	$-0.620756\pi$
$113$	−100535.	−0.740661	−0.370330	+	0.928900i	$0.620756\pi$
$114$	301162.	2.17039
$115$	0	0
$116$	−38519.7	−0.265790
$117$	−473343.	−3.19677
$118$	−164306.	−1.08629
$119$	−188826.	−1.22235
$120$	0	0
$121$	225150.	1.39801
$122$	62319.5	0.379075
$123$	−59212.0	−0.352896
$124$	−57838.5	−0.337802
$125$	0	0
$126$	−541901.	−3.04084
$127$	96836.1	0.532755	0.266378	−	0.963869i	$-0.414173\pi$
$127$	96836.1	0.532755	0.266378	+	0.963869i	$0.414173\pi$
$128$	50261.3	0.271149
$129$	52376.2	0.277115
$130$	0	0
$131$	204686.	1.04210	0.521050	−	0.853526i	$-0.325540\pi$
$131$	204686.	1.04210	0.521050	+	0.853526i	$0.325540\pi$
$132$	182076.	0.909531
$133$	−475554.	−2.33116
$134$	249700.	1.20132
$135$	0	0
$136$	−178750.	−0.828705
$137$	−401809.	−1.82902	−0.914511	−	0.404562i	$-0.867424\pi$
$137$	−401809.	−1.82902	−0.914511	+	0.404562i	$0.867424\pi$
$138$	440820.	1.97044
$139$	299963.	1.31683	0.658415	−	0.752655i	$-0.271227\pi$
$139$	299963.	1.31683	0.658415	+	0.752655i	$0.271227\pi$
$140$	0	0
$141$	−115051.	−0.487350
$142$	302448.	1.25872
$143$	525844.	2.15039
$144$	−327835.	−1.31750
$145$	0	0
$146$	−156801.	−0.608787
$147$	751315.	2.86767
$148$	43544.5	0.163410
$149$	48992.9	0.180787	0.0903936	−	0.995906i	$-0.471187\pi$
$149$	48992.9	0.180787	0.0903936	+	0.995906i	$0.471187\pi$
$150$	0	0
$151$	−180144.	−0.642949	−0.321474	−	0.946918i	$-0.604178\pi$
$151$	−180144.	−0.642949	−0.321474	+	0.946918i	$0.604178\pi$
$152$	−450180.	−1.58044
$153$	507449.	1.75252
$154$	602006.	2.04550
$155$	0	0
$156$	247911.	0.815612
$157$	−51406.8	−0.166445	−0.0832226	−	0.996531i	$-0.526521\pi$
$157$	−51406.8	−0.166445	−0.0832226	+	0.996531i	$0.526521\pi$
$158$	−344356.	−1.09740
$159$	866850.	2.71926
$160$	0	0
$161$	−696083.	−2.11639
$162$	548900.	1.64326
$163$	318132.	0.937860	0.468930	−	0.883235i	$-0.344640\pi$
$163$	318132.	0.937860	0.468930	+	0.883235i	$0.344640\pi$
$164$	21620.3	0.0627700
$165$	0	0
$166$	−310608.	−0.874867
$167$	92130.5	0.255630	0.127815	−	0.991798i	$-0.459204\pi$
$167$	92130.5	0.255630	0.127815	+	0.991798i	$0.459204\pi$
$168$	1.16191e6	3.17614
$169$	344685.	0.928337
$170$	0	0
$171$	1.27800e6	3.34226
$172$	−19124.3	−0.0492906
$173$	293581.	0.745784	0.372892	−	0.927875i	$-0.378366\pi$
$173$	293581.	0.745784	0.372892	+	0.927875i	$0.378366\pi$
$174$	490941.	1.22930
$175$	0	0
$176$	364197.	0.886248
$177$	−1.00012e6	−2.39948
$178$	332840.	0.787382
$179$	−261441.	−0.609877	−0.304938	−	0.952372i	$-0.598636\pi$
$179$	−261441.	−0.609877	−0.304938	+	0.952372i	$0.598636\pi$
$180$	0	0
$181$	−211634.	−0.480164	−0.240082	−	0.970753i	$-0.577174\pi$
$181$	−211634.	−0.480164	−0.240082	+	0.970753i	$0.577174\pi$
$182$	819679.	1.83428
$183$	379334.	0.837326
$184$	−658941.	−1.43483
$185$	0	0
$186$	737163.	1.56236
$187$	−563733.	−1.17888
$188$	42008.8	0.0866854
$189$	−1.86567e6	−3.79909
$190$	0	0
$191$	523175.	1.03768	0.518840	−	0.854871i	$-0.326364\pi$
$191$	523175.	1.03768	0.518840	+	0.854871i	$0.326364\pi$
$192$	1.00294e6	1.96346
$193$	−540663.	−1.04480	−0.522401	−	0.852700i	$-0.674963\pi$
$193$	−540663.	−1.04480	−0.522401	+	0.852700i	$0.674963\pi$
$194$	−167732.	−0.319971
$195$	0	0
$196$	−274330.	−0.510075
$197$	991111.	1.81952	0.909759	−	0.415136i	$-0.136266\pi$
$197$	991111.	1.81952	0.909759	+	0.415136i	$0.136266\pi$
$198$	−1.61782e6	−2.93271
$199$	346596.	0.620428	0.310214	−	0.950667i	$-0.399599\pi$
$199$	346596.	0.620428	0.310214	+	0.950667i	$0.399599\pi$
$200$	0	0
$201$	1.51991e6	2.65355
$202$	183612.	0.316608
$203$	−775228.	−1.32035
$204$	−265774.	−0.447133
$205$	0	0
$206$	180429.	0.296236
$207$	1.87065e6	3.03435
$208$	495884.	0.794733
$209$	−1.41975e6	−2.24826
$210$	0	0
$211$	−30207.8	−0.0467103	−0.0233552	−	0.999727i	$-0.507435\pi$
$211$	−30207.8	−0.0467103	−0.0233552	+	0.999727i	$0.507435\pi$
$212$	−316516.	−0.483678
$213$	1.84098e6	2.78035
$214$	247646.	0.369655
$215$	0	0
$216$	−1.76612e6	−2.57564
$217$	−1.16403e6	−1.67808
$218$	−11810.1	−0.0168310
$219$	−954433.	−1.34473
$220$	0	0
$221$	−767567.	−1.05715
$222$	−554983.	−0.755784
$223$	−623016.	−0.838952	−0.419476	−	0.907766i	$-0.637786\pi$
$223$	−623016.	−0.838952	−0.419476	+	0.907766i	$0.637786\pi$
$224$	−744874.	−0.991888
$225$	0	0
$226$	−467858.	−0.609317
$227$	−1.52909e6	−1.96955	−0.984776	−	0.173825i	$-0.944387\pi$
$227$	−1.52909e6	−1.96955	−0.984776	+	0.173825i	$0.944387\pi$
$228$	−669346.	−0.852734
$229$	443829.	0.559277	0.279638	−	0.960105i	$-0.409785\pi$
$229$	443829.	0.559277	0.279638	+	0.960105i	$0.409785\pi$
$230$	0	0
$231$	3.66436e6	4.51823
$232$	−733863.	−0.895148
$233$	−977455.	−1.17952	−0.589762	−	0.807577i	$-0.700778\pi$
$233$	−977455.	−1.17952	−0.589762	+	0.807577i	$0.700778\pi$
$234$	−2.20280e6	−2.62987
$235$	0	0
$236$	365176.	0.426798
$237$	−2.09607e6	−2.42401
$238$	−878740.	−1.00558
$239$	−394340.	−0.446556	−0.223278	−	0.974755i	$-0.571676\pi$
$239$	−394340.	−0.446556	−0.223278	+	0.974755i	$0.571676\pi$
$240$	0	0
$241$	607979.	0.674288	0.337144	−	0.941453i	$-0.390539\pi$
$241$	607979.	0.674288	0.337144	+	0.941453i	$0.390539\pi$
$242$	1.04778e6	1.15009
$243$	1.16318e6	1.26366
$244$	−138508.	−0.148936
$245$	0	0
$246$	−275555.	−0.290316
$247$	−1.93310e6	−2.01610
$248$	−1.10192e6	−1.13768
$249$	−1.89065e6	−1.93247
$250$	0	0
$251$	802918.	0.804427	0.402214	−	0.915546i	$-0.368241\pi$
$251$	802918.	0.804427	0.402214	+	0.915546i	$0.368241\pi$
$252$	1.20440e6	1.19473
$253$	−2.07813e6	−2.04113
$254$	450646.	0.438280
$255$	0	0
$256$	−899096.	−0.857445
$257$	364918.	0.344637	0.172319	−	0.985041i	$-0.444874\pi$
$257$	364918.	0.344637	0.172319	+	0.985041i	$0.444874\pi$
$258$	243743.	0.227973
$259$	876355.	0.811765
$260$	0	0
$261$	2.08334e6	1.89304
$262$	952547.	0.857301
$263$	1.08117e6	0.963836	0.481918	−	0.876216i	$-0.339940\pi$
$263$	1.08117e6	0.963836	0.481918	+	0.876216i	$0.339940\pi$
$264$	3.46884e6	3.06319
$265$	0	0
$266$	−2.21309e6	−1.91776
$267$	2.02597e6	1.73922
$268$	−554970.	−0.471990
$269$	1.58654e6	1.33681	0.668405	−	0.743797i	$-0.266977\pi$
$269$	1.58654e6	1.33681	0.668405	+	0.743797i	$0.266977\pi$
$270$	0	0
$271$	1.47887e6	1.22322	0.611612	−	0.791158i	$-0.290522\pi$
$271$	1.47887e6	1.22322	0.611612	+	0.791158i	$0.290522\pi$
$272$	−531614.	−0.435687
$273$	4.98932e6	4.05168
$274$	−1.86990e6	−1.50467
$275$	0	0
$276$	−979740.	−0.774173
$277$	2.04574e6	1.60195	0.800977	−	0.598695i	$-0.204314\pi$
$277$	2.04574e6	1.60195	0.800977	+	0.598695i	$0.204314\pi$
$278$	1.39594e6	1.08331
$279$	3.12820e6	2.40593
$280$	0	0
$281$	1.81785e6	1.37339	0.686694	−	0.726947i	$-0.259061\pi$
$281$	1.81785e6	1.37339	0.686694	+	0.726947i	$0.259061\pi$
$282$	−535411.	−0.400926
$283$	−2.13614e6	−1.58549	−0.792746	−	0.609553i	$-0.791349\pi$
$283$	−2.13614e6	−1.58549	−0.792746	+	0.609553i	$0.791349\pi$
$284$	−672204.	−0.494544
$285$	0	0
$286$	2.44712e6	1.76905
$287$	435119.	0.311820
$288$	2.00177e6	1.42211
$289$	−596984.	−0.420454
$290$	0	0
$291$	−1.02097e6	−0.706775
$292$	348496.	0.239189
$293$	−871941.	−0.593360	−0.296680	−	0.954977i	$-0.595879\pi$
$293$	−871941.	−0.593360	−0.296680	+	0.954977i	$0.595879\pi$
$294$	3.49640e6	2.35913
$295$	0	0
$296$	829594.	0.550346
$297$	−5.56988e6	−3.66400
$298$	227999.	0.148728
$299$	−2.82954e6	−1.83036
$300$	0	0
$301$	−384886.	−0.244859
$302$	−838334.	−0.528932
$303$	1.11763e6	0.699345
$304$	−1.33886e6	−0.830905
$305$	0	0
$306$	2.36152e6	1.44174
$307$	−45665.0	−0.0276527	−0.0138263	−	0.999904i	$-0.504401\pi$
$307$	−45665.0	−0.0276527	−0.0138263	+	0.999904i	$0.504401\pi$
$308$	−1.33798e6	−0.803663
$309$	1.09826e6	0.654346
$310$	0	0
$311$	−55396.0	−0.0324771	−0.0162385	−	0.999868i	$-0.505169\pi$
$311$	−55396.0	−0.0324771	−0.0162385	+	0.999868i	$0.505169\pi$
$312$	4.72310e6	2.74689
$313$	−838174.	−0.483586	−0.241793	−	0.970328i	$-0.577735\pi$
$313$	−838174.	−0.483586	−0.241793	+	0.970328i	$0.577735\pi$
$314$	−239232.	−0.136929
$315$	0	0
$316$	765345.	0.431161
$317$	−2.45573e6	−1.37256	−0.686281	−	0.727337i	$-0.740758\pi$
$317$	−2.45573e6	−1.37256	−0.686281	+	0.727337i	$0.740758\pi$
$318$	4.03406e6	2.23705
$319$	−2.31441e6	−1.27340
$320$	0	0
$321$	1.50740e6	0.816520
$322$	−3.23936e6	−1.74108
$323$	2.07239e6	1.10526
$324$	−1.21995e6	−0.645626
$325$	0	0
$326$	1.48049e6	0.771546
$327$	−71886.9	−0.0371775
$328$	411902.	0.211402
$329$	845449.	0.430623
$330$	0	0
$331$	−3.39433e6	−1.70288	−0.851440	−	0.524453i	$-0.824270\pi$
$331$	−3.39433e6	−1.70288	−0.851440	+	0.524453i	$0.824270\pi$
$332$	690339.	0.343730
$333$	−2.35511e6	−1.16386
$334$	428748.	0.210298
$335$	0	0
$336$	3.45559e6	1.66984
$337$	3.35177e6	1.60768	0.803839	−	0.594847i	$-0.202787\pi$
$337$	3.35177e6	1.60768	0.803839	+	0.594847i	$0.202787\pi$
$338$	1.60406e6	0.763711
$339$	−2.84782e6	−1.34590
$340$	0	0
$341$	−3.47516e6	−1.61841
$342$	5.94744e6	2.74957
$343$	−2.02251e6	−0.928228
$344$	−364349.	−0.166005
$345$	0	0
$346$	1.36624e6	0.613531
$347$	2.74481e6	1.22374	0.611869	−	0.790959i	$-0.290418\pi$
$347$	2.74481e6	1.22374	0.611869	+	0.790959i	$0.290418\pi$
$348$	−1.09114e6	−0.482983
$349$	925319.	0.406657	0.203328	−	0.979111i	$-0.434824\pi$
$349$	925319.	0.406657	0.203328	+	0.979111i	$0.434824\pi$
$350$	0	0
$351$	−7.58384e6	−3.28565
$352$	−2.22379e6	−0.956616
$353$	−1.16092e6	−0.495867	−0.247933	−	0.968777i	$-0.579751\pi$
$353$	−1.16092e6	−0.495867	−0.247933	+	0.968777i	$0.579751\pi$
$354$	−4.65424e6	−1.97397
$355$	0	0
$356$	−739751.	−0.309358
$357$	−5.34882e6	−2.22120
$358$	−1.21667e6	−0.501725
$359$	1.08910e6	0.445995	0.222998	−	0.974819i	$-0.428416\pi$
$359$	1.08910e6	0.445995	0.222998	+	0.974819i	$0.428416\pi$
$360$	0	0
$361$	2.74318e6	1.10786
$362$	−984884.	−0.395015
$363$	6.37777e6	2.54040
$364$	−1.82177e6	−0.720676
$365$	0	0
$366$	1.76531e6	0.688840
$367$	3.10109e6	1.20185	0.600924	−	0.799306i	$-0.294799\pi$
$367$	3.10109e6	1.20185	0.600924	+	0.799306i	$0.294799\pi$
$368$	−1.95973e6	−0.754355
$369$	−1.16934e6	−0.447068
$370$	0	0
$371$	−6.37005e6	−2.40275
$372$	−1.63838e6	−0.613841
$373$	−2.95925e6	−1.10131	−0.550654	−	0.834734i	$-0.685622\pi$
$373$	−2.95925e6	−1.10131	−0.550654	+	0.834734i	$0.685622\pi$
$374$	−2.62344e6	−0.969824
$375$	0	0
$376$	800337.	0.291946
$377$	−3.15126e6	−1.14191
$378$	−8.68227e6	−3.12539
$379$	4.46871e6	1.59803	0.799014	−	0.601312i	$-0.205355\pi$
$379$	4.46871e6	1.59803	0.799014	+	0.601312i	$0.205355\pi$
$380$	0	0
$381$	2.74305e6	0.968103
$382$	2.43470e6	0.853665
$383$	1.56288e6	0.544414	0.272207	−	0.962239i	$-0.412246\pi$
$383$	1.56288e6	0.544414	0.272207	+	0.962239i	$0.412246\pi$
$384$	1.42374e6	0.492722
$385$	0	0
$386$	−2.51609e6	−0.859523
$387$	1.03434e6	0.351063
$388$	372791.	0.125715
$389$	−746865.	−0.250247	−0.125123	−	0.992141i	$-0.539933\pi$
$389$	−746865.	−0.250247	−0.125123	+	0.992141i	$0.539933\pi$
$390$	0	0
$391$	3.03342e6	1.00344
$392$	−5.22644e6	−1.71787
$393$	5.79809e6	1.89367
$394$	4.61233e6	1.49686
$395$	0	0
$396$	3.59568e6	1.15224
$397$	2.44018e6	0.777044	0.388522	−	0.921439i	$-0.372986\pi$
$397$	2.44018e6	0.777044	0.388522	+	0.921439i	$0.372986\pi$
$398$	1.61296e6	0.510405
$399$	−1.34709e7	−4.23609
$400$	0	0
$401$	3.52360e6	1.09427	0.547137	−	0.837043i	$-0.315718\pi$
$401$	3.52360e6	1.09427	0.547137	+	0.837043i	$0.315718\pi$
$402$	7.07320e6	2.18299
$403$	−4.73171e6	−1.45129
$404$	−408085.	−0.124393
$405$	0	0
$406$	−3.60768e6	−1.08621
$407$	2.61632e6	0.782899
$408$	−5.06342e6	−1.50589
$409$	4.58574e6	1.35551	0.677753	−	0.735290i	$-0.262954\pi$
$409$	4.58574e6	1.35551	0.677753	+	0.735290i	$0.262954\pi$
$410$	0	0
$411$	−1.13820e7	−3.32363
$412$	−401010.	−0.116389
$413$	7.34935e6	2.12018
$414$	8.70543e6	2.49626
$415$	0	0
$416$	−3.02787e6	−0.857836
$417$	8.49696e6	2.39289
$418$	−6.60709e6	−1.84957
$419$	−2.99636e6	−0.833793	−0.416896	−	0.908954i	$-0.636882\pi$
$419$	−2.99636e6	−0.833793	−0.416896	+	0.908954i	$0.636882\pi$
$420$	0	0
$421$	3.05478e6	0.839992	0.419996	−	0.907526i	$-0.362031\pi$
$421$	3.05478e6	0.839992	0.419996	+	0.907526i	$0.362031\pi$
$422$	−140578.	−0.0384270
$423$	−2.27205e6	−0.617401
$424$	−6.03015e6	−1.62897
$425$	0	0
$426$	8.56737e6	2.28730
$427$	−2.78754e6	−0.739863
$428$	−550404.	−0.145235
$429$	1.48954e7	3.90760
$430$	0	0
$431$	2.99693e6	0.777111	0.388555	−	0.921425i	$-0.372974\pi$
$431$	2.99693e6	0.777111	0.388555	+	0.921425i	$0.372974\pi$
$432$	−5.25254e6	−1.35413
$433$	70122.4	0.0179737	0.00898685	−	0.999960i	$-0.497139\pi$
$433$	70122.4	0.0179737	0.00898685	+	0.999960i	$0.497139\pi$
$434$	−5.41704e6	−1.38050
$435$	0	0
$436$	26248.3	0.00661281
$437$	7.63960e6	1.91367
$438$	−4.44165e6	−1.10626
$439$	1.21942e6	0.301990	0.150995	−	0.988535i	$-0.451752\pi$
$439$	1.21942e6	0.301990	0.150995	+	0.988535i	$0.451752\pi$
$440$	0	0
$441$	1.48372e7	3.63291
$442$	−3.57203e6	−0.869679
$443$	3.69639e6	0.894889	0.447444	−	0.894312i	$-0.352334\pi$
$443$	3.69639e6	0.894889	0.447444	+	0.894312i	$0.352334\pi$
$444$	1.23347e6	0.296943
$445$	0	0
$446$	−2.89933e6	−0.690178
$447$	1.38781e6	0.328520
$448$	−7.37011e6	−1.73492
$449$	1.50803e6	0.353016	0.176508	−	0.984299i	$-0.443520\pi$
$449$	1.50803e6	0.353016	0.176508	+	0.984299i	$0.443520\pi$
$450$	0	0
$451$	1.29903e6	0.300731
$452$	1.03983e6	0.239397
$453$	−5.10288e6	−1.16834
$454$	−7.11592e6	−1.62028
$455$	0	0
$456$	−1.27521e7	−2.87191
$457$	−3.91836e6	−0.877634	−0.438817	−	0.898576i	$-0.644602\pi$
$457$	−3.91836e6	−0.877634	−0.438817	+	0.898576i	$0.644602\pi$
$458$	2.06545e6	0.460098
$459$	8.13029e6	1.80125
$460$	0	0
$461$	−3.68012e6	−0.806510	−0.403255	−	0.915088i	$-0.632121\pi$
$461$	−3.68012e6	−0.806510	−0.403255	+	0.915088i	$0.632121\pi$
$462$	1.70529e7	3.71700
$463$	−1.76652e6	−0.382972	−0.191486	−	0.981495i	$-0.561331\pi$
$463$	−1.76652e6	−0.382972	−0.191486	+	0.981495i	$0.561331\pi$
$464$	−2.18255e6	−0.470619
$465$	0	0
$466$	−4.54878e6	−0.970355
$467$	940748.	0.199609	0.0998047	−	0.995007i	$-0.468178\pi$
$467$	940748.	0.199609	0.0998047	+	0.995007i	$0.468178\pi$
$468$	4.89581e6	1.03326
$469$	−1.11690e7	−2.34468
$470$	0	0
$471$	−1.45619e6	−0.302458
$472$	6.95720e6	1.43741
$473$	−1.14906e6	−0.236152
$474$	−9.75448e6	−1.99415
$475$	0	0
$476$	1.95304e6	0.395087
$477$	1.71188e7	3.44491
$478$	−1.83514e6	−0.367367
$479$	3.20439e6	0.638126	0.319063	−	0.947734i	$-0.396632\pi$
$479$	3.20439e6	0.638126	0.319063	+	0.947734i	$0.396632\pi$
$480$	0	0
$481$	3.56233e6	0.702056
$482$	2.82935e6	0.554714
$483$	−1.97178e7	−3.84583
$484$	−2.32874e6	−0.451864
$485$	0	0
$486$	5.41308e6	1.03957
$487$	6.83661e6	1.30623	0.653113	−	0.757261i	$-0.273463\pi$
$487$	6.83661e6	1.30623	0.653113	+	0.757261i	$0.273463\pi$
$488$	−2.63880e6	−0.501599
$489$	9.01164e6	1.70424
$490$	0	0
$491$	6.26039e6	1.17192	0.585960	−	0.810340i	$-0.300718\pi$
$491$	6.26039e6	1.17192	0.585960	+	0.810340i	$0.300718\pi$
$492$	612433.	0.114063
$493$	3.37832e6	0.626013
$494$	−8.99608e6	−1.65858
$495$	0	0
$496$	−3.27716e6	−0.598128
$497$	−1.35284e7	−2.45672
$498$	−8.79851e6	−1.58978
$499$	−3.02293e6	−0.543472	−0.271736	−	0.962372i	$-0.587598\pi$
$499$	−3.02293e6	−0.543472	−0.271736	+	0.962372i	$0.587598\pi$
$500$	0	0
$501$	2.60976e6	0.464521
$502$	3.73654e6	0.661775
$503$	−1.92092e6	−0.338524	−0.169262	−	0.985571i	$-0.554138\pi$
$503$	−1.92092e6	−0.338524	−0.169262	+	0.985571i	$0.554138\pi$
$504$	2.29457e7	4.02370
$505$	0	0
$506$	−9.67099e6	−1.67917
$507$	9.76380e6	1.68694
$508$	−1.00158e6	−0.172197
$509$	869353.	0.148731	0.0743655	−	0.997231i	$-0.476307\pi$
$509$	869353.	0.148731	0.0743655	+	0.997231i	$0.476307\pi$
$510$	0	0
$511$	7.01365e6	1.18821
$512$	−5.79249e6	−0.976540
$513$	2.04760e7	3.43519
$514$	1.69822e6	0.283521
$515$	0	0
$516$	−541729.	−0.0895691
$517$	2.52406e6	0.415310
$518$	4.07829e6	0.667812
$519$	8.31620e6	1.35521
$520$	0	0
$521$	3.08384e6	0.497734	0.248867	−	0.968538i	$-0.419942\pi$
$521$	3.08384e6	0.497734	0.248867	+	0.968538i	$0.419942\pi$
$522$	9.69524e6	1.55734
$523$	4.27186e6	0.682908	0.341454	−	0.939898i	$-0.389081\pi$
$523$	4.27186e6	0.682908	0.341454	+	0.939898i	$0.389081\pi$
$524$	−2.11708e6	−0.336828
$525$	0	0
$526$	5.03142e6	0.792915
$527$	5.07265e6	0.795624
$528$	1.03165e7	1.61046
$529$	4.74596e6	0.737369
$530$	0	0
$531$	−1.97506e7	−3.03979
$532$	4.91868e6	0.753477
$533$	1.76874e6	0.269678
$534$	9.42828e6	1.43080
$535$	0	0
$536$	−1.05731e7	−1.58961
$537$	−7.40579e6	−1.10824
$538$	7.38328e6	1.09975
$539$	−1.64828e7	−2.44377
$540$	0	0
$541$	289066.	0.0424623	0.0212312	−	0.999775i	$-0.493241\pi$
$541$	289066.	0.0424623	0.0212312	+	0.999775i	$0.493241\pi$
$542$	6.88220e6	1.00630
$543$	−5.99492e6	−0.872536
$544$	3.24604e6	0.470280
$545$	0	0
$546$	2.32188e7	3.33318
$547$	−4.63415e6	−0.662219	−0.331109	−	0.943592i	$-0.607423\pi$
$547$	−4.63415e6	−0.662219	−0.331109	+	0.943592i	$0.607423\pi$
$548$	4.15594e6	0.591177
$549$	7.49120e6	1.06077
$550$	0	0
$551$	8.50823e6	1.19388
$552$	−1.86656e7	−2.60733
$553$	1.54030e7	2.14186
$554$	9.52024e6	1.31787
$555$	0	0
$556$	−3.10253e6	−0.425627
$557$	−1.24274e7	−1.69723	−0.848616	−	0.529010i	$-0.822563\pi$
$557$	−1.24274e7	−1.69723	−0.848616	+	0.529010i	$0.822563\pi$
$558$	1.45577e7	1.97928
$559$	−1.56454e6	−0.211766
$560$	0	0
$561$	−1.59687e7	−2.14221
$562$	8.45975e6	1.12984
$563$	6.12085e6	0.813843	0.406922	−	0.913463i	$-0.366602\pi$
$563$	6.12085e6	0.813843	0.406922	+	0.913463i	$0.366602\pi$
$564$	1.18997e6	0.157521
$565$	0	0
$566$	−9.94097e6	−1.30433
$567$	−2.45522e7	−3.20725
$568$	−1.28066e7	−1.66557
$569$	−4.24956e6	−0.550255	−0.275127	−	0.961408i	$-0.588720\pi$
$569$	−4.24956e6	−0.550255	−0.275127	+	0.961408i	$0.588720\pi$
$570$	0	0
$571$	−9.74012e6	−1.25018	−0.625092	−	0.780551i	$-0.714939\pi$
$571$	−9.74012e6	−1.25018	−0.625092	+	0.780551i	$0.714939\pi$
$572$	−5.43883e6	−0.695049
$573$	1.48199e7	1.88563
$574$	2.02492e6	0.256524
$575$	0	0
$576$	1.98064e7	2.48742
$577$	1.40655e7	1.75879	0.879397	−	0.476089i	$-0.157946\pi$
$577$	1.40655e7	1.75879	0.879397	+	0.476089i	$0.157946\pi$
$578$	−2.77819e6	−0.345893
$579$	−1.53152e7	−1.89857
$580$	0	0
$581$	1.38934e7	1.70753
$582$	−4.75130e6	−0.581440
$583$	−1.90175e7	−2.31730
$584$	6.63941e6	0.805560
$585$	0	0
$586$	−4.05776e6	−0.488137
$587$	375468.	0.0449757	0.0224878	−	0.999747i	$-0.492841\pi$
$587$	375468.	0.0449757	0.0224878	+	0.999747i	$0.492841\pi$
$588$	−7.77089e6	−0.926888
$589$	1.27754e7	1.51735
$590$	0	0
$591$	2.80749e7	3.30636
$592$	2.46726e6	0.289341
$593$	−7.07258e6	−0.825926	−0.412963	−	0.910748i	$-0.635506\pi$
$593$	−7.07258e6	−0.825926	−0.412963	+	0.910748i	$0.635506\pi$
$594$	−2.59206e7	−3.01425
$595$	0	0
$596$	−506736.	−0.0584341
$597$	9.81794e6	1.12742
$598$	−1.31678e7	−1.50578
$599$	1.54285e7	1.75694	0.878470	−	0.477797i	$-0.158565\pi$
$599$	1.54285e7	1.75694	0.878470	+	0.477797i	$0.158565\pi$
$600$	0	0
$601$	1.65606e7	1.87021	0.935105	−	0.354370i	$-0.115305\pi$
$601$	1.65606e7	1.87021	0.935105	+	0.354370i	$0.115305\pi$
$602$	−1.79114e6	−0.201437
$603$	3.00156e7	3.36166
$604$	1.86323e6	0.207814
$605$	0	0
$606$	5.20112e6	0.575328
$607$	1.09158e7	1.20250	0.601249	−	0.799061i	$-0.294670\pi$
$607$	1.09158e7	1.20250	0.601249	+	0.799061i	$0.294670\pi$
$608$	8.17509e6	0.896879
$609$	−2.19597e7	−2.39929
$610$	0	0
$611$	3.43670e6	0.372425
$612$	−5.24857e6	−0.566451
$613$	8.35290e6	0.897814	0.448907	−	0.893579i	$-0.351813\pi$
$613$	8.35290e6	0.897814	0.448907	+	0.893579i	$0.351813\pi$
$614$	−212511.	−0.0227489
$615$	0	0
$616$	−2.54908e7	−2.70664
$617$	−1.04273e7	−1.10270	−0.551352	−	0.834273i	$-0.685888\pi$
$617$	−1.04273e7	−1.10270	−0.551352	+	0.834273i	$0.685888\pi$
$618$	5.11096e6	0.538308
$619$	489795.	0.0513792	0.0256896	−	0.999670i	$-0.491822\pi$
$619$	489795.	0.0513792	0.0256896	+	0.999670i	$0.491822\pi$
$620$	0	0
$621$	2.99713e7	3.11872
$622$	−257796.	−0.0267178
$623$	−1.48879e7	−1.53678
$624$	1.40468e7	1.44416
$625$	0	0
$626$	−3.90061e6	−0.397830
$627$	−4.02169e7	−4.08545
$628$	531703.	0.0537985
$629$	−3.81901e6	−0.384879
$630$	0	0
$631$	4.51997e6	0.451920	0.225960	−	0.974137i	$-0.427448\pi$
$631$	4.51997e6	0.451920	0.225960	+	0.974137i	$0.427448\pi$
$632$	1.45811e7	1.45210
$633$	−855688.	−0.0848802
$634$	−1.14282e7	−1.12916
$635$	0	0
$636$	−8.96588e6	−0.878921
$637$	−2.24427e7	−2.19142
$638$	−1.07706e7	−1.04758
$639$	3.63562e7	3.52230
$640$	0	0
$641$	1.56230e7	1.50183	0.750913	−	0.660401i	$-0.229614\pi$
$641$	1.56230e7	1.50183	0.750913	+	0.660401i	$0.229614\pi$
$642$	7.01500e6	0.671723
$643$	1.35160e7	1.28921	0.644603	−	0.764518i	$-0.277023\pi$
$643$	1.35160e7	1.28921	0.644603	+	0.764518i	$0.277023\pi$
$644$	7.19962e6	0.684061
$645$	0	0
$646$	9.64429e6	0.909262
$647$	−6.46419e6	−0.607091	−0.303545	−	0.952817i	$-0.598170\pi$
$647$	−6.46419e6	−0.607091	−0.303545	+	0.952817i	$0.598170\pi$
$648$	−2.32421e7	−2.17439
$649$	2.19412e7	2.04479
$650$	0	0
$651$	−3.29731e7	−3.04935
$652$	−3.29045e6	−0.303136
$653$	−5.89885e6	−0.541358	−0.270679	−	0.962670i	$-0.587248\pi$
$653$	−5.89885e6	−0.541358	−0.270679	+	0.962670i	$0.587248\pi$
$654$	−334540.	−0.0305847
$655$	0	0
$656$	1.22502e6	0.111143
$657$	−1.88484e7	−1.70358
$658$	3.93447e6	0.354259
$659$	−1.57029e7	−1.40853	−0.704267	−	0.709935i	$-0.748724\pi$
$659$	−1.57029e7	−1.40853	−0.704267	+	0.709935i	$0.748724\pi$
$660$	0	0
$661$	2.11094e6	0.187920	0.0939598	−	0.995576i	$-0.470047\pi$
$661$	2.11094e6	0.187920	0.0939598	+	0.995576i	$0.470047\pi$
$662$	−1.57962e7	−1.40090
$663$	−2.17427e7	−1.92101
$664$	1.31521e7	1.15764
$665$	0	0
$666$	−1.09600e7	−0.957467
$667$	1.24537e7	1.08389
$668$	−952910.	−0.0826249
$669$	−1.76480e7	−1.52451
$670$	0	0
$671$	−8.32209e6	−0.713553
$672$	−2.10999e7	−1.80242
$673$	−1.60692e7	−1.36759	−0.683795	−	0.729674i	$-0.739672\pi$
$673$	−1.60692e7	−1.36759	−0.683795	+	0.729674i	$0.739672\pi$
$674$	1.55981e7	1.32258
$675$	0	0
$676$	−3.56509e6	−0.300057
$677$	7.12008e6	0.597054	0.298527	−	0.954401i	$-0.403505\pi$
$677$	7.12008e6	0.597054	0.298527	+	0.954401i	$0.403505\pi$
$678$	−1.32529e7	−1.10723
$679$	7.50260e6	0.624507
$680$	0	0
$681$	−4.33141e7	−3.57900
$682$	−1.61724e7	−1.33141
$683$	−9.76118e6	−0.800665	−0.400332	−	0.916370i	$-0.631105\pi$
$683$	−9.76118e6	−0.800665	−0.400332	+	0.916370i	$0.631105\pi$
$684$	−1.32184e7	−1.08029
$685$	0	0
$686$	−9.41215e6	−0.763622
$687$	1.25722e7	1.01630
$688$	−1.08359e6	−0.0872762
$689$	−2.58939e7	−2.07802
$690$	0	0
$691$	5.50494e6	0.438589	0.219294	−	0.975659i	$-0.429625\pi$
$691$	5.50494e6	0.438589	0.219294	+	0.975659i	$0.429625\pi$
$692$	−3.03653e6	−0.241053
$693$	7.23649e7	5.72394
$694$	1.27735e7	1.00673
$695$	0	0
$696$	−2.07879e7	−1.62663
$697$	−1.89618e6	−0.147842
$698$	4.30616e6	0.334543
$699$	−2.76881e7	−2.14339
$700$	0	0
$701$	−7.24878e6	−0.557147	−0.278574	−	0.960415i	$-0.589862\pi$
$701$	−7.24878e6	−0.557147	−0.278574	+	0.960415i	$0.589862\pi$
$702$	−3.52929e7	−2.70299
$703$	−9.61811e6	−0.734009
$704$	−2.20032e7	−1.67322
$705$	0	0
$706$	−5.40257e6	−0.407933
$707$	−8.21291e6	−0.617943
$708$	1.03442e7	0.775561
$709$	9.64950e6	0.720924	0.360462	−	0.932774i	$-0.382619\pi$
$709$	9.64950e6	0.720924	0.360462	+	0.932774i	$0.382619\pi$
$710$	0	0
$711$	−4.13937e7	−3.07087
$712$	−1.40935e7	−1.04188
$713$	1.86997e7	1.37756
$714$	−2.48918e7	−1.82731
$715$	0	0
$716$	2.70410e6	0.197125
$717$	−1.11704e7	−0.811465
$718$	5.06833e6	0.366905
$719$	−2.25875e6	−0.162947	−0.0814735	−	0.996676i	$-0.525963\pi$
$719$	−2.25875e6	−0.162947	−0.0814735	+	0.996676i	$0.525963\pi$
$720$	0	0
$721$	−8.07053e6	−0.578181
$722$	1.27659e7	0.911401
$723$	1.72221e7	1.22529
$724$	2.18895e6	0.155199
$725$	0	0
$726$	2.96803e7	2.08990
$727$	−1.17565e6	−0.0824976	−0.0412488	−	0.999149i	$-0.513134\pi$
$727$	−1.17565e6	−0.0824976	−0.0412488	+	0.999149i	$0.513134\pi$
$728$	−3.47077e7	−2.42715
$729$	4.28736e6	0.298794
$730$	0	0
$731$	1.67727e6	0.116094
$732$	−3.92348e6	−0.270641
$733$	1.44387e7	0.992585	0.496292	−	0.868155i	$-0.334694\pi$
$733$	1.44387e7	0.992585	0.496292	+	0.868155i	$0.334694\pi$
$734$	1.44316e7	0.988720
$735$	0	0
$736$	1.19661e7	0.814252
$737$	−3.33448e7	−2.26130
$738$	−5.44174e6	−0.367788
$739$	996412.	0.0671163	0.0335581	−	0.999437i	$-0.489316\pi$
$739$	996412.	0.0671163	0.0335581	+	0.999437i	$0.489316\pi$
$740$	0	0
$741$	−5.47585e7	−3.66358
$742$	−2.96443e7	−1.97666
$743$	−1.80651e7	−1.20052	−0.600260	−	0.799805i	$-0.704936\pi$
$743$	−1.80651e7	−1.20052	−0.600260	+	0.799805i	$0.704936\pi$
$744$	−3.12137e7	−2.06735
$745$	0	0
$746$	−1.37714e7	−0.906009
$747$	−3.73370e7	−2.44815
$748$	5.83072e6	0.381038
$749$	−1.10771e7	−0.721478
$750$	0	0
$751$	4.09336e6	0.264838	0.132419	−	0.991194i	$-0.457726\pi$
$751$	4.09336e6	0.264838	0.132419	+	0.991194i	$0.457726\pi$
$752$	2.38025e6	0.153489
$753$	2.27440e7	1.46177
$754$	−1.46650e7	−0.939408
$755$	0	0
$756$	1.92967e7	1.22794
$757$	3.12098e7	1.97948	0.989741	−	0.142870i	$-0.0456330\pi$
$757$	3.12098e7	1.97948	0.989741	+	0.142870i	$0.0456330\pi$
$758$	2.07961e7	1.31464
$759$	−5.88666e7	−3.70907
$760$	0	0
$761$	2.93149e6	0.183496	0.0917481	−	0.995782i	$-0.470755\pi$
$761$	2.93149e6	0.183496	0.0917481	+	0.995782i	$0.470755\pi$
$762$	1.27653e7	0.796425
$763$	528261.	0.0328501
$764$	−5.41123e6	−0.335400
$765$	0	0
$766$	7.27320e6	0.447871
$767$	2.98747e7	1.83364
$768$	−2.54685e7	−1.55812
$769$	2.03099e7	1.23849	0.619245	−	0.785198i	$-0.287439\pi$
$769$	2.03099e7	1.23849	0.619245	+	0.785198i	$0.287439\pi$
$770$	0	0
$771$	1.03369e7	0.626261
$772$	5.59211e6	0.337701
$773$	−2.59383e7	−1.56132	−0.780660	−	0.624956i	$-0.785117\pi$
$773$	−2.59383e7	−1.56132	−0.780660	+	0.624956i	$0.785117\pi$
$774$	4.81351e6	0.288808
$775$	0	0
$776$	7.10228e6	0.423393
$777$	2.48243e7	1.47511
$778$	−3.47569e6	−0.205869
$779$	−4.77549e6	−0.281952
$780$	0	0
$781$	−4.03886e7	−2.36936
$782$	1.41166e7	0.825494
$783$	3.33790e7	1.94567
$784$	−1.55437e7	−0.903161
$785$	0	0
$786$	2.69826e7	1.55785
$787$	−1.59170e7	−0.916061	−0.458030	−	0.888937i	$-0.651445\pi$
$787$	−1.59170e7	−0.916061	−0.458030	+	0.888937i	$0.651445\pi$
$788$	−1.02511e7	−0.588105
$789$	3.06259e7	1.75144
$790$	0	0
$791$	2.09272e7	1.18924
$792$	6.85036e7	3.88062
$793$	−1.13312e7	−0.639871
$794$	1.13559e7	0.639248
$795$	0	0
$796$	−3.58486e6	−0.200535
$797$	1.09940e7	0.613070	0.306535	−	0.951859i	$-0.400830\pi$
$797$	1.09940e7	0.613070	0.306535	+	0.951859i	$0.400830\pi$
$798$	−6.26896e7	−3.48488
$799$	−3.68433e6	−0.204170
$800$	0	0
$801$	4.00095e7	2.20334
$802$	1.63978e7	0.900222
$803$	2.09390e7	1.14595
$804$	−1.57205e7	−0.857681
$805$	0	0
$806$	−2.20200e7	−1.19393
$807$	4.49415e7	2.42920
$808$	−7.77468e6	−0.418942
$809$	−2.94002e6	−0.157935	−0.0789677	−	0.996877i	$-0.525162\pi$
$809$	−2.94002e6	−0.157935	−0.0789677	+	0.996877i	$0.525162\pi$
$810$	0	0
$811$	−8.30904e6	−0.443607	−0.221804	−	0.975091i	$-0.571194\pi$
$811$	−8.30904e6	−0.443607	−0.221804	+	0.975091i	$0.571194\pi$
$812$	8.01822e6	0.426764
$813$	4.18915e7	2.22279
$814$	1.21756e7	0.644064
$815$	0	0
$816$	−1.50589e7	−0.791713
$817$	4.22418e6	0.221405
$818$	2.13407e7	1.11513
$819$	9.85306e7	5.13288
$820$	0	0
$821$	−2.59182e7	−1.34198	−0.670992	−	0.741465i	$-0.734132\pi$
$821$	−2.59182e7	−1.34198	−0.670992	+	0.741465i	$0.734132\pi$
$822$	−5.29682e7	−2.73424
$823$	−3.07730e7	−1.58369	−0.791844	−	0.610724i	$-0.790879\pi$
$823$	−3.07730e7	−1.58369	−0.791844	+	0.610724i	$0.790879\pi$
$824$	−7.63990e6	−0.391985
$825$	0	0
$826$	3.42017e7	1.74420
$827$	−2.28020e7	−1.15934	−0.579668	−	0.814853i	$-0.696818\pi$
$827$	−2.28020e7	−1.15934	−0.579668	+	0.814853i	$0.696818\pi$
$828$	−1.93482e7	−0.980764
$829$	7.96856e6	0.402711	0.201356	−	0.979518i	$-0.435465\pi$
$829$	7.96856e6	0.402711	0.201356	+	0.979518i	$0.435465\pi$
$830$	0	0
$831$	5.79490e7	2.91101
$832$	−2.99591e7	−1.50045
$833$	2.40598e7	1.20138
$834$	3.95423e7	1.96855
$835$	0	0
$836$	1.46845e7	0.726683
$837$	5.01196e7	2.47283
$838$	−1.39441e7	−0.685933
$839$	−3.73243e7	−1.83057	−0.915285	−	0.402806i	$-0.868035\pi$
$839$	−3.73243e7	−1.83057	−0.915285	+	0.402806i	$0.868035\pi$
$840$	0	0
$841$	−6.64141e6	−0.323795
$842$	1.42161e7	0.691033
$843$	5.14939e7	2.49567
$844$	312441.	0.0150977
$845$	0	0
$846$	−1.05734e7	−0.507915
$847$	−4.68670e7	−2.24470
$848$	−1.79340e7	−0.856422
$849$	−6.05099e7	−2.88109
$850$	0	0
$851$	−1.40783e7	−0.666387
$852$	−1.90413e7	−0.898667
$853$	−1.14613e7	−0.539338	−0.269669	−	0.962953i	$-0.586914\pi$
$853$	−1.14613e7	−0.539338	−0.269669	+	0.962953i	$0.586914\pi$
$854$	−1.29724e7	−0.608660
$855$	0	0
$856$	−1.04861e7	−0.489135
$857$	−1.20339e7	−0.559701	−0.279850	−	0.960044i	$-0.590285\pi$
$857$	−1.20339e7	−0.559701	−0.279850	+	0.960044i	$0.590285\pi$
$858$	6.93189e7	3.21465
$859$	−2.20607e6	−0.102009	−0.0510044	−	0.998698i	$-0.516242\pi$
$859$	−2.20607e6	−0.102009	−0.0510044	+	0.998698i	$0.516242\pi$
$860$	0	0
$861$	1.23255e7	0.566627
$862$	1.39468e7	0.639303
$863$	1.73527e7	0.793124	0.396562	−	0.918008i	$-0.370203\pi$
$863$	1.73527e7	0.793124	0.396562	+	0.918008i	$0.370203\pi$
$864$	3.20721e7	1.46165
$865$	0	0
$866$	326329.	0.0147863
$867$	−1.69106e7	−0.764032
$868$	1.20396e7	0.542391
$869$	4.59849e7	2.06569
$870$	0	0
$871$	−4.54015e7	−2.02780
$872$	500074.	0.0222712
$873$	−2.01624e7	−0.895380
$874$	3.55524e7	1.57431
$875$	0	0
$876$	9.87175e6	0.434644
$877$	−4.58144e6	−0.201142	−0.100571	−	0.994930i	$-0.532067\pi$
$877$	−4.58144e6	−0.201142	−0.100571	+	0.994930i	$0.532067\pi$
$878$	5.67482e6	0.248437
$879$	−2.46993e7	−1.07823
$880$	0	0
$881$	1.60364e7	0.696094	0.348047	−	0.937477i	$-0.386845\pi$
$881$	1.60364e7	0.696094	0.348047	+	0.937477i	$0.386845\pi$
$882$	6.90478e7	2.98867
$883$	−6.65028e6	−0.287037	−0.143519	−	0.989648i	$-0.545842\pi$
$883$	−6.65028e6	−0.287037	−0.143519	+	0.989648i	$0.545842\pi$
$884$	7.93898e6	0.341692
$885$	0	0
$886$	1.72019e7	0.736195
$887$	−2.42167e7	−1.03349	−0.516744	−	0.856140i	$-0.672856\pi$
$887$	−2.42167e7	−1.03349	−0.516744	+	0.856140i	$0.672856\pi$
$888$	2.34997e7	1.00007
$889$	−2.01573e7	−0.855417
$890$	0	0
$891$	−7.32996e7	−3.09319
$892$	6.44389e6	0.271166
$893$	−9.27891e6	−0.389375
$894$	6.45846e6	0.270262
$895$	0	0
$896$	−1.04623e7	−0.435370
$897$	−8.01516e7	−3.32607
$898$	7.01793e6	0.290414
$899$	2.08258e7	0.859415
$900$	0	0
$901$	2.77596e7	1.13920
$902$	6.04531e6	0.247402
$903$	−1.09026e7	−0.444948
$904$	1.98105e7	0.806260
$905$	0	0
$906$	−2.37473e7	−0.961155
$907$	2.41584e7	0.975103	0.487551	−	0.873094i	$-0.337890\pi$
$907$	2.41584e7	0.975103	0.487551	+	0.873094i	$0.337890\pi$
$908$	1.58154e7	0.636600
$909$	2.20713e7	0.885968
$910$	0	0
$911$	1.20076e7	0.479360	0.239680	−	0.970852i	$-0.422957\pi$
$911$	1.20076e7	0.479360	0.239680	+	0.970852i	$0.422957\pi$
$912$	−3.79255e7	−1.50989
$913$	4.14783e7	1.64681
$914$	−1.82349e7	−0.722000
$915$	0	0
$916$	−4.59054e6	−0.180770
$917$	−4.26072e7	−1.67325
$918$	3.78359e7	1.48183
$919$	−2.29070e7	−0.894705	−0.447352	−	0.894358i	$-0.647633\pi$
$919$	−2.29070e7	−0.894705	−0.447352	+	0.894358i	$0.647633\pi$
$920$	0	0
$921$	−1.29354e6	−0.0502494
$922$	−1.71262e7	−0.663489
$923$	−5.49923e7	−2.12470
$924$	−3.79007e7	−1.46039
$925$	0	0
$926$	−8.22087e6	−0.315058
$927$	2.16887e7	0.828960
$928$	1.33267e7	0.507986
$929$	−1.57239e7	−0.597752	−0.298876	−	0.954292i	$-0.596612\pi$
$929$	−1.57239e7	−0.597752	−0.298876	+	0.954292i	$0.596612\pi$
$930$	0	0
$931$	6.05941e7	2.29116
$932$	1.01099e7	0.381246
$933$	−1.56919e6	−0.0590161
$934$	4.37796e6	0.164212
$935$	0	0
$936$	9.32731e7	3.47990
$937$	−4.74231e6	−0.176458	−0.0882289	−	0.996100i	$-0.528121\pi$
$937$	−4.74231e6	−0.176458	−0.0882289	+	0.996100i	$0.528121\pi$
$938$	−5.19774e7	−1.92889
$939$	−2.37427e7	−0.878753
$940$	0	0
$941$	−2.47934e7	−0.912773	−0.456386	−	0.889782i	$-0.650856\pi$
$941$	−2.47934e7	−0.912773	−0.456386	+	0.889782i	$0.650856\pi$
$942$	−6.77666e6	−0.248822
$943$	−6.99003e6	−0.255976
$944$	2.06911e7	0.755707
$945$	0	0
$946$	−5.34740e6	−0.194274
$947$	4.70056e7	1.70324	0.851618	−	0.524163i	$-0.175622\pi$
$947$	4.70056e7	1.70324	0.851618	+	0.524163i	$0.175622\pi$
$948$	2.16797e7	0.783490
$949$	2.85101e7	1.02762
$950$	0	0
$951$	−6.95627e7	−2.49417
$952$	3.72085e7	1.33061
$953$	−4.53325e7	−1.61688	−0.808439	−	0.588579i	$-0.799687\pi$
$953$	−4.53325e7	−1.61688	−0.808439	+	0.588579i	$0.799687\pi$
$954$	7.96658e7	2.83401
$955$	0	0
$956$	4.07868e6	0.144336
$957$	−6.55598e7	−2.31397
$958$	1.49123e7	0.524964
$959$	8.36403e7	2.93676
$960$	0	0
$961$	2.64145e6	0.0922642
$962$	1.65780e7	0.577558
$963$	2.97686e7	1.03441
$964$	−6.28835e6	−0.217944
$965$	0	0
$966$	−9.17606e7	−3.16383
$967$	−1.62002e7	−0.557128	−0.278564	−	0.960418i	$-0.589858\pi$
$967$	−1.62002e7	−0.557128	−0.278564	+	0.960418i	$0.589858\pi$
$968$	−4.43663e7	−1.52182
$969$	5.87041e7	2.00844
$970$	0	0
$971$	−4.31039e7	−1.46713	−0.733565	−	0.679619i	$-0.762145\pi$
$971$	−4.31039e7	−1.46713	−0.733565	+	0.679619i	$0.762145\pi$
$972$	−1.20308e7	−0.408440
$973$	−6.24399e7	−2.11437
$974$	3.18155e7	1.07459
$975$	0	0
$976$	−7.84794e6	−0.263713
$977$	−1.67475e6	−0.0561325	−0.0280662	−	0.999606i	$-0.508935\pi$
$977$	−1.67475e6	−0.0561325	−0.0280662	+	0.999606i	$0.508935\pi$
$978$	4.19375e7	1.40202
$979$	−4.44471e7	−1.48213
$980$	0	0
$981$	−1.41964e6	−0.0470985
$982$	2.91340e7	0.964098
$983$	−3.55110e7	−1.17214	−0.586069	−	0.810261i	$-0.699325\pi$
$983$	−3.55110e7	−1.17214	−0.586069	+	0.810261i	$0.699325\pi$
$984$	1.16678e7	0.384152
$985$	0	0
$986$	1.57217e7	0.515000
$987$	2.39488e7	0.782512
$988$	1.99942e7	0.651645
$989$	6.18305e6	0.201007
$990$	0	0
$991$	−1.14454e6	−0.0370208	−0.0185104	−	0.999829i	$-0.505892\pi$
$991$	−1.14454e6	−0.0370208	−0.0185104	+	0.999829i	$0.505892\pi$
$992$	2.00104e7	0.645619
$993$	−9.61503e7	−3.09441
$994$	−6.29573e7	−2.02106
$995$	0	0
$996$	1.95551e7	0.624613
$997$	−4.91861e7	−1.56713	−0.783564	−	0.621311i	$-0.786600\pi$
$997$	−4.91861e7	−1.56713	−0.783564	+	0.621311i	$0.786600\pi$
$998$	−1.40678e7	−0.447096
$999$	−3.77332e7	−1.19622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.f.1.15		22
5.4	even	2		215.6.a.d.1.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.a.d.1.8	✓	22	5.4	even	2
1075.6.a.f.1.15		22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+4.65370 q^{2} +28.3267 q^{3} -10.3431 q^{4} +131.824 q^{6} -208.159 q^{7} -197.052 q^{8} +559.405 q^{9} +O(q^{10})\)	\(q+4.65370 q^{2} +28.3267 q^{3} -10.3431 q^{4} +131.824 q^{6} -208.159 q^{7} -197.052 q^{8} +559.405 q^{9} -621.451 q^{11} -292.985 q^{12} -846.155 q^{13} -968.710 q^{14} -586.044 q^{16} +907.124 q^{17} +2603.30 q^{18} +2284.57 q^{19} -5896.47 q^{21} -2892.05 q^{22} +3343.99 q^{23} -5581.84 q^{24} -3937.75 q^{26} +8962.71 q^{27} +2153.00 q^{28} +3724.21 q^{29} +5592.01 q^{31} +3578.39 q^{32} -17603.7 q^{33} +4221.48 q^{34} -5785.95 q^{36} -4210.03 q^{37} +10631.7 q^{38} -23968.8 q^{39} -2090.32 q^{41} -27440.4 q^{42} +1849.00 q^{43} +6427.70 q^{44} +15562.0 q^{46} -4061.55 q^{47} -16600.7 q^{48} +26523.2 q^{49} +25695.9 q^{51} +8751.82 q^{52} +30601.8 q^{53} +41709.8 q^{54} +41018.1 q^{56} +64714.5 q^{57} +17331.4 q^{58} -35306.4 q^{59} +13391.4 q^{61} +26023.6 q^{62} -116445. q^{63} +35406.2 q^{64} -81922.3 q^{66} +53656.3 q^{67} -9382.43 q^{68} +94724.5 q^{69} +64990.8 q^{71} -110232. q^{72} -33693.7 q^{73} -19592.2 q^{74} -23629.5 q^{76} +129361. q^{77} -111544. q^{78} -73996.1 q^{79} +117949. q^{81} -9727.74 q^{82} -66744.2 q^{83} +60987.5 q^{84} +8604.70 q^{86} +105495. q^{87} +122458. q^{88} +71521.6 q^{89} +176135. q^{91} -34587.1 q^{92} +158404. q^{93} -18901.3 q^{94} +101364. q^{96} -36042.7 q^{97} +123431. q^{98} -347642. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 5 q^{2} - 20 q^{3} + 427 q^{4} + 248 q^{6} - 118 q^{7} - 561 q^{8} + 2618 q^{9}+O(q^{10}) \) 22 * q - 5 * q^2 - 20 * q^3 + 427 * q^4 + 248 * q^6 - 118 * q^7 - 561 * q^8 + 2618 * q^9	\( 22 q - 5 q^{2} - 20 q^{3} + 427 q^{4} + 248 q^{6} - 118 q^{7} - 561 q^{8} + 2618 q^{9} + 1206 q^{11} - 2175 q^{12} - 1942 q^{13} + 2531 q^{14} + 8851 q^{16} - 2470 q^{17} - 1279 q^{18} + 3020 q^{19} + 5632 q^{21} + 3227 q^{22} + 1326 q^{23} + 11040 q^{24} - 3415 q^{26} - 5156 q^{27} + 11489 q^{28} + 17906 q^{29} + 7982 q^{31} - 2427 q^{32} - 10100 q^{33} + 25248 q^{34} - 14813 q^{36} - 22640 q^{37} + 13695 q^{38} + 29048 q^{39} + 29112 q^{41} - 9163 q^{42} + 40678 q^{43} + 63924 q^{44} - 14944 q^{46} - 57080 q^{47} - 54894 q^{48} + 165560 q^{49} - 1576 q^{51} - 97639 q^{52} + 8054 q^{53} + 167379 q^{54} + 269326 q^{56} - 125424 q^{57} - 49485 q^{58} + 193484 q^{59} + 107466 q^{61} - 162441 q^{62} - 183778 q^{63} + 412603 q^{64} + 240489 q^{66} - 109764 q^{67} - 144300 q^{68} + 202444 q^{69} + 182964 q^{71} - 341504 q^{72} - 134468 q^{73} + 198067 q^{74} + 247729 q^{76} + 28416 q^{77} + 7286 q^{78} + 11148 q^{79} + 385246 q^{81} - 23657 q^{82} - 33850 q^{83} + 176749 q^{84} - 9245 q^{86} + 298280 q^{87} + 111354 q^{88} + 244912 q^{89} + 158092 q^{91} + 124762 q^{92} - 239860 q^{93} - 192166 q^{94} - 147719 q^{96} - 232826 q^{97} + 482463 q^{98} - 346894 q^{99}+O(q^{100}) \) 22 * q - 5 * q^2 - 20 * q^3 + 427 * q^4 + 248 * q^6 - 118 * q^7 - 561 * q^8 + 2618 * q^9 + 1206 * q^11 - 2175 * q^12 - 1942 * q^13 + 2531 * q^14 + 8851 * q^16 - 2470 * q^17 - 1279 * q^18 + 3020 * q^19 + 5632 * q^21 + 3227 * q^22 + 1326 * q^23 + 11040 * q^24 - 3415 * q^26 - 5156 * q^27 + 11489 * q^28 + 17906 * q^29 + 7982 * q^31 - 2427 * q^32 - 10100 * q^33 + 25248 * q^34 - 14813 * q^36 - 22640 * q^37 + 13695 * q^38 + 29048 * q^39 + 29112 * q^41 - 9163 * q^42 + 40678 * q^43 + 63924 * q^44 - 14944 * q^46 - 57080 * q^47 - 54894 * q^48 + 165560 * q^49 - 1576 * q^51 - 97639 * q^52 + 8054 * q^53 + 167379 * q^54 + 269326 * q^56 - 125424 * q^57 - 49485 * q^58 + 193484 * q^59 + 107466 * q^61 - 162441 * q^62 - 183778 * q^63 + 412603 * q^64 + 240489 * q^66 - 109764 * q^67 - 144300 * q^68 + 202444 * q^69 + 182964 * q^71 - 341504 * q^72 - 134468 * q^73 + 198067 * q^74 + 247729 * q^76 + 28416 * q^77 + 7286 * q^78 + 11148 * q^79 + 385246 * q^81 - 23657 * q^82 - 33850 * q^83 + 176749 * q^84 - 9245 * q^86 + 298280 * q^87 + 111354 * q^88 + 244912 * q^89 + 158092 * q^91 + 124762 * q^92 - 239860 * q^93 - 192166 * q^94 - 147719 * q^96 - 232826 * q^97 + 482463 * q^98 - 346894 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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