LMFDB - Embedded newform 825.6.a.k.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 $ x^5 - x^4 - 143x^3 + 71x^2 + 4216*x - 3740
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.898099$ 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)

$f(q)$	$=$	$q+0.898099 q^{2} +9.00000 q^{3} -31.1934 q^{4} +8.08289 q^{6} -120.732 q^{7} -56.7540 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+0.898099 q^{2} +9.00000 q^{3} -31.1934 q^{4} +8.08289 q^{6} -120.732 q^{7} -56.7540 q^{8} +81.0000 q^{9} -121.000 q^{11} -280.741 q^{12} -667.387 q^{13} -108.430 q^{14} +947.219 q^{16} +74.5586 q^{17} +72.7460 q^{18} -708.549 q^{19} -1086.59 q^{21} -108.670 q^{22} -1238.63 q^{23} -510.786 q^{24} -599.380 q^{26} +729.000 q^{27} +3766.06 q^{28} -3855.89 q^{29} -8236.92 q^{31} +2666.82 q^{32} -1089.00 q^{33} +66.9611 q^{34} -2526.67 q^{36} +5725.33 q^{37} -636.348 q^{38} -6006.49 q^{39} +672.977 q^{41} -975.867 q^{42} -16058.1 q^{43} +3774.40 q^{44} -1112.41 q^{46} +7572.96 q^{47} +8524.97 q^{48} -2230.69 q^{49} +671.028 q^{51} +20818.1 q^{52} +5420.79 q^{53} +654.714 q^{54} +6852.04 q^{56} -6376.94 q^{57} -3462.97 q^{58} -33128.4 q^{59} +18038.6 q^{61} -7397.57 q^{62} -9779.32 q^{63} -27915.9 q^{64} -978.030 q^{66} +60276.1 q^{67} -2325.74 q^{68} -11147.7 q^{69} +12430.5 q^{71} -4597.07 q^{72} +1230.79 q^{73} +5141.92 q^{74} +22102.1 q^{76} +14608.6 q^{77} -5394.42 q^{78} +47015.0 q^{79} +6561.00 q^{81} +604.401 q^{82} +29681.0 q^{83} +33894.5 q^{84} -14421.7 q^{86} -34703.0 q^{87} +6867.23 q^{88} -125061. q^{89} +80575.2 q^{91} +38637.2 q^{92} -74132.3 q^{93} +6801.27 q^{94} +24001.4 q^{96} -74465.3 q^{97} -2003.38 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9} - 605 q^{11} + 1143 q^{12} + 926 q^{13} + 368 q^{14} + 1891 q^{16} + 246 q^{17} + 81 q^{18} + 3420 q^{19} - 1044 q^{21} - 121 q^{22} + 4244 q^{23} + 1377 q^{24} - 8862 q^{26} + 3645 q^{27} + 4904 q^{28} - 2922 q^{29} - 6112 q^{31} + 24757 q^{32} - 5445 q^{33} + 10866 q^{34} + 10287 q^{36} - 6654 q^{37} + 45692 q^{38} + 8334 q^{39} - 14934 q^{41} + 3312 q^{42} - 10804 q^{43} - 15367 q^{44} - 101500 q^{46} + 41460 q^{47} + 17019 q^{48} - 12099 q^{49} + 2214 q^{51} + 97742 q^{52} + 62398 q^{53} + 729 q^{54} - 74368 q^{56} + 30780 q^{57} + 27822 q^{58} + 8524 q^{59} + 59010 q^{61} + 142624 q^{62} - 9396 q^{63} + 13799 q^{64} - 1089 q^{66} + 15772 q^{67} + 83686 q^{68} + 38196 q^{69} + 88124 q^{71} + 12393 q^{72} + 118358 q^{73} + 67194 q^{74} + 100668 q^{76} + 14036 q^{77} - 79758 q^{78} + 57324 q^{79} + 32805 q^{81} - 29102 q^{82} + 7268 q^{83} + 44136 q^{84} - 35288 q^{86} - 26298 q^{87} - 18513 q^{88} + 72978 q^{89} - 1464 q^{91} - 62148 q^{92} - 55008 q^{93} + 344836 q^{94} + 222813 q^{96} + 59174 q^{97} - 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9 - 605 * q^11 + 1143 * q^12 + 926 * q^13 + 368 * q^14 + 1891 * q^16 + 246 * q^17 + 81 * q^18 + 3420 * q^19 - 1044 * q^21 - 121 * q^22 + 4244 * q^23 + 1377 * q^24 - 8862 * q^26 + 3645 * q^27 + 4904 * q^28 - 2922 * q^29 - 6112 * q^31 + 24757 * q^32 - 5445 * q^33 + 10866 * q^34 + 10287 * q^36 - 6654 * q^37 + 45692 * q^38 + 8334 * q^39 - 14934 * q^41 + 3312 * q^42 - 10804 * q^43 - 15367 * q^44 - 101500 * q^46 + 41460 * q^47 + 17019 * q^48 - 12099 * q^49 + 2214 * q^51 + 97742 * q^52 + 62398 * q^53 + 729 * q^54 - 74368 * q^56 + 30780 * q^57 + 27822 * q^58 + 8524 * q^59 + 59010 * q^61 + 142624 * q^62 - 9396 * q^63 + 13799 * q^64 - 1089 * q^66 + 15772 * q^67 + 83686 * q^68 + 38196 * q^69 + 88124 * q^71 + 12393 * q^72 + 118358 * q^73 + 67194 * q^74 + 100668 * q^76 + 14036 * q^77 - 79758 * q^78 + 57324 * q^79 + 32805 * q^81 - 29102 * q^82 + 7268 * q^83 + 44136 * q^84 - 35288 * q^86 - 26298 * q^87 - 18513 * q^88 + 72978 * q^89 - 1464 * q^91 - 62148 * q^92 - 55008 * q^93 + 344836 * q^94 + 222813 * q^96 + 59174 * q^97 - 272767 * q^98 - 49005 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.898099	0.158763	0.0793815	−	0.996844i	$-0.474705\pi$
$2$	0.898099	0.158763	0.0793815	+	0.996844i	$0.474705\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.1934	−0.974794
$5$	0	0
$5$	0	0
$6$	8.08289	0.0916619
$7$	−120.732	−0.931277	−0.465638	−	0.884975i	$-0.654175\pi$
$7$	−120.732	−0.931277	−0.465638	+	0.884975i	$0.654175\pi$
$8$	−56.7540	−0.313524
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−280.741	−0.562798
$13$	−667.387	−1.09527	−0.547633	−	0.836719i	$-0.684471\pi$
$13$	−667.387	−1.09527	−0.547633	+	0.836719i	$0.684471\pi$
$14$	−108.430	−0.147852
$15$	0	0
$16$	947.219	0.925018
$17$	74.5586	0.0625714	0.0312857	−	0.999510i	$-0.490040\pi$
$17$	74.5586	0.0625714	0.0312857	+	0.999510i	$0.490040\pi$
$18$	72.7460	0.0529210
$19$	−708.549	−0.450283	−0.225142	−	0.974326i	$-0.572285\pi$
$19$	−708.549	−0.450283	−0.225142	+	0.974326i	$0.572285\pi$
$20$	0	0
$21$	−1086.59	−0.537673
$22$	−108.670	−0.0478689
$23$	−1238.63	−0.488228	−0.244114	−	0.969747i	$-0.578497\pi$
$23$	−1238.63	−0.488228	−0.244114	+	0.969747i	$0.578497\pi$
$24$	−510.786	−0.181013
$25$	0	0
$26$	−599.380	−0.173888
$27$	729.000	0.192450
$28$	3766.06	0.907803
$29$	−3855.89	−0.851392	−0.425696	−	0.904866i	$-0.639971\pi$
$29$	−3855.89	−0.851392	−0.425696	+	0.904866i	$0.639971\pi$
$30$	0	0
$31$	−8236.92	−1.53943	−0.769716	−	0.638387i	$-0.779602\pi$
$31$	−8236.92	−1.53943	−0.769716	+	0.638387i	$0.779602\pi$
$32$	2666.82	0.460383
$33$	−1089.00	−0.174078
$34$	66.9611	0.00993402
$35$	0	0
$36$	−2526.67	−0.324931
$37$	5725.33	0.687538	0.343769	−	0.939054i	$-0.388296\pi$
$37$	5725.33	0.687538	0.343769	+	0.939054i	$0.388296\pi$
$38$	−636.348	−0.0714884
$39$	−6006.49	−0.632352
$40$	0	0
$41$	672.977	0.0625231	0.0312616	−	0.999511i	$-0.490048\pi$
$41$	672.977	0.0625231	0.0312616	+	0.999511i	$0.490048\pi$
$42$	−975.867	−0.0853626
$43$	−16058.1	−1.32441	−0.662204	−	0.749323i	$-0.730379\pi$
$43$	−16058.1	−1.32441	−0.662204	+	0.749323i	$0.730379\pi$
$44$	3774.40	0.293912
$45$	0	0
$46$	−1112.41	−0.0775125
$47$	7572.96	0.500059	0.250029	−	0.968238i	$-0.419560\pi$
$47$	7572.96	0.500059	0.250029	+	0.968238i	$0.419560\pi$
$48$	8524.97	0.534060
$49$	−2230.69	−0.132724
$50$	0	0
$51$	671.028	0.0361256
$52$	20818.1	1.06766
$53$	5420.79	0.265078	0.132539	−	0.991178i	$-0.457687\pi$
$53$	5420.79	0.265078	0.132539	+	0.991178i	$0.457687\pi$
$54$	654.714	0.0305540
$55$	0	0
$56$	6852.04	0.291978
$57$	−6376.94	−0.259971
$58$	−3462.97	−0.135170
$59$	−33128.4	−1.23900	−0.619499	−	0.784997i	$-0.712664\pi$
$59$	−33128.4	−1.23900	−0.619499	+	0.784997i	$0.712664\pi$
$60$	0	0
$61$	18038.6	0.620695	0.310348	−	0.950623i	$-0.399555\pi$
$61$	18038.6	0.620695	0.310348	+	0.950623i	$0.399555\pi$
$62$	−7397.57	−0.244405
$63$	−9779.32	−0.310426
$64$	−27915.9	−0.851926
$65$	0	0
$66$	−978.030	−0.0276371
$67$	60276.1	1.64043	0.820216	−	0.572054i	$-0.193853\pi$
$67$	60276.1	1.64043	0.820216	+	0.572054i	$0.193853\pi$
$68$	−2325.74	−0.0609942
$69$	−11147.7	−0.281878
$70$	0	0
$71$	12430.5	0.292646	0.146323	−	0.989237i	$-0.453256\pi$
$71$	12430.5	0.292646	0.146323	+	0.989237i	$0.453256\pi$
$72$	−4597.07	−0.104508
$73$	1230.79	0.0270320	0.0135160	−	0.999909i	$-0.495698\pi$
$73$	1230.79	0.0270320	0.0135160	+	0.999909i	$0.495698\pi$
$74$	5141.92	0.109156
$75$	0	0
$76$	22102.1	0.438934
$77$	14608.6	0.280790
$78$	−5394.42	−0.100394
$79$	47015.0	0.847557	0.423778	−	0.905766i	$-0.360704\pi$
$79$	47015.0	0.847557	0.423778	+	0.905766i	$0.360704\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	604.401	0.00992636
$83$	29681.0	0.472915	0.236457	−	0.971642i	$-0.424014\pi$
$83$	29681.0	0.472915	0.236457	+	0.971642i	$0.424014\pi$
$84$	33894.5	0.524120
$85$	0	0
$86$	−14421.7	−0.210267
$87$	−34703.0	−0.491551
$88$	6867.23	0.0945311
$89$	−125061.	−1.67358	−0.836791	−	0.547522i	$-0.815571\pi$
$89$	−125061.	−1.67358	−0.836791	+	0.547522i	$0.815571\pi$
$90$	0	0
$91$	80575.2	1.02000
$92$	38637.2	0.475922
$93$	−74132.3	−0.888791
$94$	6801.27	0.0793908
$95$	0	0
$96$	24001.4	0.265802
$97$	−74465.3	−0.803572	−0.401786	−	0.915734i	$-0.631610\pi$
$97$	−74465.3	−0.803572	−0.401786	+	0.915734i	$0.631610\pi$
$98$	−2003.38	−0.0210717
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−65393.5	−0.637868	−0.318934	−	0.947777i	$-0.603325\pi$
$101$	−65393.5	−0.637868	−0.318934	+	0.947777i	$0.603325\pi$
$102$	602.650	0.00573541
$103$	35955.1	0.333939	0.166969	−	0.985962i	$-0.446602\pi$
$103$	35955.1	0.333939	0.166969	+	0.985962i	$0.446602\pi$
$104$	37876.9	0.343392
$105$	0	0
$106$	4868.41	0.0420845
$107$	118141.	0.997565	0.498783	−	0.866727i	$-0.333781\pi$
$107$	118141.	0.997565	0.498783	+	0.866727i	$0.333781\pi$
$108$	−22740.0	−0.187599
$109$	216333.	1.74404	0.872019	−	0.489472i	$-0.162811\pi$
$109$	216333.	1.74404	0.872019	+	0.489472i	$0.162811\pi$
$110$	0	0
$111$	51528.0	0.396950
$112$	−114360.	−0.861448
$113$	82987.0	0.611384	0.305692	−	0.952130i	$-0.401112\pi$
$113$	82987.0	0.611384	0.305692	+	0.952130i	$0.401112\pi$
$114$	−5727.13	−0.0412738
$115$	0	0
$116$	120278.	0.829932
$117$	−54058.4	−0.365089
$118$	−29752.6	−0.196707
$119$	−9001.64	−0.0582712
$120$	0	0
$121$	14641.0	0.0909091
$122$	16200.5	0.0985434
$123$	6056.80	0.0360978
$124$	256938.	1.50063
$125$	0	0
$126$	−8782.80	−0.0492841
$127$	112754.	0.620330	0.310165	−	0.950683i	$-0.399616\pi$
$127$	112754.	0.620330	0.310165	+	0.950683i	$0.399616\pi$
$128$	−110410.	−0.595637
$129$	−144523.	−0.764648
$130$	0	0
$131$	177226.	0.902297	0.451149	−	0.892449i	$-0.351014\pi$
$131$	177226.	0.902297	0.451149	+	0.892449i	$0.351014\pi$
$132$	33969.6	0.169690
$133$	85544.8	0.419338
$134$	54133.9	0.260440
$135$	0	0
$136$	−4231.50	−0.0196176
$137$	179083.	0.815181	0.407590	−	0.913165i	$-0.366369\pi$
$137$	179083.	0.815181	0.407590	+	0.913165i	$0.366369\pi$
$138$	−10011.7	−0.0447519
$139$	140181.	0.615392	0.307696	−	0.951485i	$-0.400442\pi$
$139$	140181.	0.615392	0.307696	+	0.951485i	$0.400442\pi$
$140$	0	0
$141$	68156.6	0.288709
$142$	11163.8	0.0464614
$143$	80753.9	0.330235
$144$	76724.7	0.308339
$145$	0	0
$146$	1105.37	0.00429168
$147$	−20076.2	−0.0766283
$148$	−178593.	−0.670208
$149$	−300863.	−1.11020	−0.555102	−	0.831783i	$-0.687321\pi$
$149$	−300863.	−1.11020	−0.555102	+	0.831783i	$0.687321\pi$
$150$	0	0
$151$	225360.	0.804330	0.402165	−	0.915567i	$-0.368258\pi$
$151$	225360.	0.804330	0.402165	+	0.915567i	$0.368258\pi$
$152$	40213.0	0.141175
$153$	6039.25	0.0208571
$154$	13120.0	0.0445791
$155$	0	0
$156$	187363.	0.616413
$157$	344938.	1.11684	0.558421	−	0.829558i	$-0.311407\pi$
$157$	344938.	1.11684	0.558421	+	0.829558i	$0.311407\pi$
$158$	42224.2	0.134561
$159$	48787.1	0.153043
$160$	0	0
$161$	149543.	0.454675
$162$	5892.43	0.0176403
$163$	134781.	0.397337	0.198669	−	0.980067i	$-0.436338\pi$
$163$	134781.	0.397337	0.198669	+	0.980067i	$0.436338\pi$
$164$	−20992.5	−0.0609472
$165$	0	0
$166$	26656.5	0.0750814
$167$	−476574.	−1.32233	−0.661163	−	0.750242i	$-0.729937\pi$
$167$	−476574.	−1.32233	−0.661163	+	0.750242i	$0.729937\pi$
$168$	61668.4	0.168573
$169$	74112.7	0.199607
$170$	0	0
$171$	−57392.5	−0.150094
$172$	500906.	1.29103
$173$	607031.	1.54204	0.771020	−	0.636811i	$-0.219747\pi$
$173$	607031.	1.54204	0.771020	+	0.636811i	$0.219747\pi$
$174$	−31166.7	−0.0780402
$175$	0	0
$176$	−114613.	−0.278903
$177$	−298156.	−0.715336
$178$	−112317.	−0.265703
$179$	−588915.	−1.37379	−0.686895	−	0.726757i	$-0.741027\pi$
$179$	−588915.	−1.37379	−0.686895	+	0.726757i	$0.741027\pi$
$180$	0	0
$181$	−403724.	−0.915984	−0.457992	−	0.888956i	$-0.651431\pi$
$181$	−403724.	−0.915984	−0.457992	+	0.888956i	$0.651431\pi$
$182$	72364.6	0.161938
$183$	162347.	0.358358
$184$	70297.3	0.153071
$185$	0	0
$186$	−66578.1	−0.141107
$187$	−9021.60	−0.0188660
$188$	−236226.	−0.487454
$189$	−88013.9	−0.179224
$190$	0	0
$191$	−395279.	−0.784007	−0.392004	−	0.919964i	$-0.628218\pi$
$191$	−395279.	−0.784007	−0.392004	+	0.919964i	$0.628218\pi$
$192$	−251243.	−0.491860
$193$	−192334.	−0.371675	−0.185837	−	0.982581i	$-0.559500\pi$
$193$	−192334.	−0.371675	−0.185837	+	0.982581i	$0.559500\pi$
$194$	−66877.3	−0.127578
$195$	0	0
$196$	69582.9	0.129379
$197$	537571.	0.986894	0.493447	−	0.869776i	$-0.335737\pi$
$197$	537571.	0.986894	0.493447	+	0.869776i	$0.335737\pi$
$198$	−8802.27	−0.0159563
$199$	−619032.	−1.10810	−0.554052	−	0.832482i	$-0.686919\pi$
$199$	−619032.	−1.10810	−0.554052	+	0.832482i	$0.686919\pi$
$200$	0	0
$201$	542485.	0.947104
$202$	−58729.8	−0.101270
$203$	465530.	0.792881
$204$	−20931.6	−0.0352150
$205$	0	0
$206$	32291.2	0.0530172
$207$	−100329.	−0.162743
$208$	−632162.	−1.01314
$209$	85734.5	0.135766
$210$	0	0
$211$	937975.	1.45039	0.725196	−	0.688543i	$-0.241749\pi$
$211$	937975.	1.45039	0.725196	+	0.688543i	$0.241749\pi$
$212$	−169093.	−0.258396
$213$	111874.	0.168959
$214$	106102.	0.158376
$215$	0	0
$216$	−41373.6	−0.0603378
$217$	994463.	1.43364
$218$	194288.	0.276889
$219$	11077.1	0.0156069
$220$	0	0
$221$	−49759.5	−0.0685323
$222$	46277.3	0.0630210
$223$	1.05781e6	1.42445	0.712224	−	0.701952i	$-0.247688\pi$
$223$	1.05781e6	1.42445	0.712224	+	0.701952i	$0.247688\pi$
$224$	−321972.	−0.428744
$225$	0	0
$226$	74530.6	0.0970651
$227$	1.38495e6	1.78389	0.891945	−	0.452145i	$-0.149341\pi$
$227$	1.38495e6	1.78389	0.891945	+	0.452145i	$0.149341\pi$
$228$	198919.	0.253419
$229$	−56996.7	−0.0718225	−0.0359113	−	0.999355i	$-0.511433\pi$
$229$	−56996.7	−0.0718225	−0.0359113	+	0.999355i	$0.511433\pi$
$230$	0	0
$231$	131478.	0.162114
$232$	218837.	0.266932
$233$	27988.8	0.0337750	0.0168875	−	0.999857i	$-0.494624\pi$
$233$	27988.8	0.0337750	0.0168875	+	0.999857i	$0.494624\pi$
$234$	−48549.8	−0.0579626
$235$	0	0
$236$	1.03339e6	1.20777
$237$	423135.	0.489337
$238$	−8084.37	−0.00925132
$239$	−1.21406e6	−1.37482	−0.687408	−	0.726272i	$-0.741252\pi$
$239$	−1.21406e6	−1.37482	−0.687408	+	0.726272i	$0.741252\pi$
$240$	0	0
$241$	1.06213e6	1.17797	0.588987	−	0.808142i	$-0.299527\pi$
$241$	1.06213e6	1.17797	0.588987	+	0.808142i	$0.299527\pi$
$242$	13149.1	0.0144330
$243$	59049.0	0.0641500
$244$	−562686.	−0.605050
$245$	0	0
$246$	5439.60	0.00573099
$247$	472877.	0.493180
$248$	467478.	0.482649
$249$	267129.	0.273037
$250$	0	0
$251$	−75353.1	−0.0754948	−0.0377474	−	0.999287i	$-0.512018\pi$
$251$	−75353.1	−0.0754948	−0.0377474	+	0.999287i	$0.512018\pi$
$252$	305050.	0.302601
$253$	149874.	0.147206
$254$	101264.	0.0984855
$255$	0	0
$256$	794151.	0.757361
$257$	−1.14147e6	−1.07803	−0.539016	−	0.842296i	$-0.681204\pi$
$257$	−1.14147e6	−1.07803	−0.539016	+	0.842296i	$0.681204\pi$
$258$	−129796.	−0.121398
$259$	−691233.	−0.640288
$260$	0	0
$261$	−312327.	−0.283797
$262$	159167.	0.143251
$263$	780299.	0.695619	0.347810	−	0.937565i	$-0.386926\pi$
$263$	780299.	0.695619	0.347810	+	0.937565i	$0.386926\pi$
$264$	61805.1	0.0545776
$265$	0	0
$266$	76827.8	0.0665754
$267$	−1.12555e6	−0.966244
$268$	−1.88022e6	−1.59908
$269$	210163.	0.177082	0.0885411	−	0.996073i	$-0.471780\pi$
$269$	210163.	0.177082	0.0885411	+	0.996073i	$0.471780\pi$
$270$	0	0
$271$	−4596.78	−0.00380216	−0.00190108	−	0.999998i	$-0.500605\pi$
$271$	−4596.78	−0.00380216	−0.00190108	+	0.999998i	$0.500605\pi$
$272$	70623.3	0.0578797
$273$	725177.	0.588895
$274$	160835.	0.129421
$275$	0	0
$276$	347734.	0.274774
$277$	689589.	0.539996	0.269998	−	0.962861i	$-0.412977\pi$
$277$	689589.	0.539996	0.269998	+	0.962861i	$0.412977\pi$
$278$	125896.	0.0977016
$279$	−667190.	−0.513144
$280$	0	0
$281$	1.95020e6	1.47338	0.736688	−	0.676233i	$-0.236389\pi$
$281$	1.95020e6	1.47338	0.736688	+	0.676233i	$0.236389\pi$
$282$	61211.4	0.0458363
$283$	−307845.	−0.228489	−0.114245	−	0.993453i	$-0.536445\pi$
$283$	−307845.	−0.228489	−0.114245	+	0.993453i	$0.536445\pi$
$284$	−387750.	−0.285270
$285$	0	0
$286$	72525.0	0.0524291
$287$	−81250.2	−0.0582263
$288$	216013.	0.153461
$289$	−1.41430e6	−0.996085
$290$	0	0
$291$	−670188.	−0.463942
$292$	−38392.7	−0.0263506
$293$	−2.19630e6	−1.49459	−0.747297	−	0.664491i	$-0.768649\pi$
$293$	−2.19630e6	−1.49459	−0.747297	+	0.664491i	$0.768649\pi$
$294$	−18030.5	−0.0121657
$295$	0	0
$296$	−324935.	−0.215560
$297$	−88209.0	−0.0580259
$298$	−270204.	−0.176259
$299$	826647.	0.534739
$300$	0	0
$301$	1.93873e6	1.23339
$302$	202396.	0.127698
$303$	−588541.	−0.368273
$304$	−671151.	−0.416520
$305$	0	0
$306$	5423.85	0.00331134
$307$	905979.	0.548621	0.274310	−	0.961641i	$-0.411550\pi$
$307$	905979.	0.548621	0.274310	+	0.961641i	$0.411550\pi$
$308$	−455693.	−0.273713
$309$	323596.	0.192800
$310$	0	0
$311$	1.05496e6	0.618495	0.309247	−	0.950982i	$-0.399923\pi$
$311$	1.05496e6	0.618495	0.309247	+	0.950982i	$0.399923\pi$
$312$	340892.	0.198258
$313$	−2.98094e6	−1.71985	−0.859927	−	0.510417i	$-0.829491\pi$
$313$	−2.98094e6	−1.71985	−0.859927	+	0.510417i	$0.829491\pi$
$314$	309788.	0.177313
$315$	0	0
$316$	−1.46656e6	−0.826194
$317$	272200.	0.152139	0.0760694	−	0.997103i	$-0.475763\pi$
$317$	272200.	0.152139	0.0760694	+	0.997103i	$0.475763\pi$
$318$	43815.7	0.0242975
$319$	466562.	0.256704
$320$	0	0
$321$	1.06327e6	0.575944
$322$	134304.	0.0721856
$323$	−52828.5	−0.0281749
$324$	−204660.	−0.108310
$325$	0	0
$326$	121047.	0.0630824
$327$	1.94699e6	1.00692
$328$	−38194.1	−0.0196025
$329$	−914301.	−0.465693
$330$	0	0
$331$	−984627.	−0.493971	−0.246986	−	0.969019i	$-0.579440\pi$
$331$	−984627.	−0.493971	−0.246986	+	0.969019i	$0.579440\pi$
$332$	−925851.	−0.460995
$333$	463752.	0.229179
$334$	−428010.	−0.209937
$335$	0	0
$336$	−1.02924e6	−0.497357
$337$	366865.	0.175967	0.0879836	−	0.996122i	$-0.471958\pi$
$337$	366865.	0.175967	0.0879836	+	0.996122i	$0.471958\pi$
$338$	66560.6	0.0316902
$339$	746883.	0.352983
$340$	0	0
$341$	996667.	0.464156
$342$	−51544.2	−0.0238295
$343$	2.29847e6	1.05488
$344$	911359.	0.415234
$345$	0	0
$346$	545174.	0.244819
$347$	558457.	0.248981	0.124490	−	0.992221i	$-0.460270\pi$
$347$	558457.	0.248981	0.124490	+	0.992221i	$0.460270\pi$
$348$	1.08250e6	0.479161
$349$	1.53340e6	0.673895	0.336947	−	0.941524i	$-0.390606\pi$
$349$	1.53340e6	0.673895	0.336947	+	0.941524i	$0.390606\pi$
$350$	0	0
$351$	−486525.	−0.210784
$352$	−322686.	−0.138811
$353$	−2.43850e6	−1.04156	−0.520781	−	0.853690i	$-0.674359\pi$
$353$	−2.43850e6	−1.04156	−0.520781	+	0.853690i	$0.674359\pi$
$354$	−267773.	−0.113569
$355$	0	0
$356$	3.90108e6	1.63140
$357$	−81014.8	−0.0336429
$358$	−528904.	−0.218107
$359$	1.17648e6	0.481778	0.240889	−	0.970553i	$-0.422561\pi$
$359$	1.17648e6	0.481778	0.240889	+	0.970553i	$0.422561\pi$
$360$	0	0
$361$	−1.97406e6	−0.797245
$362$	−362584.	−0.145424
$363$	131769.	0.0524864
$364$	−2.51342e6	−0.994286
$365$	0	0
$366$	145804.	0.0568941
$367$	2.74188e6	1.06263	0.531316	−	0.847174i	$-0.321698\pi$
$367$	2.74188e6	1.06263	0.531316	+	0.847174i	$0.321698\pi$
$368$	−1.17325e6	−0.451620
$369$	54511.2	0.0208410
$370$	0	0
$371$	−654465.	−0.246861
$372$	2.31244e6	0.866389
$373$	−3.46639e6	−1.29005	−0.645023	−	0.764163i	$-0.723152\pi$
$373$	−3.46639e6	−1.29005	−0.645023	+	0.764163i	$0.723152\pi$
$374$	−8102.29	−0.00299522
$375$	0	0
$376$	−429795.	−0.156781
$377$	2.57337e6	0.932500
$378$	−79045.2	−0.0284542
$379$	4.07201e6	1.45617	0.728083	−	0.685489i	$-0.240412\pi$
$379$	4.07201e6	1.45617	0.728083	+	0.685489i	$0.240412\pi$
$380$	0	0
$381$	1.01479e6	0.358148
$382$	−355000.	−0.124471
$383$	−436616.	−0.152091	−0.0760454	−	0.997104i	$-0.524229\pi$
$383$	−436616.	−0.152091	−0.0760454	+	0.997104i	$0.524229\pi$
$384$	−993687.	−0.343891
$385$	0	0
$386$	−172735.	−0.0590082
$387$	−1.30070e6	−0.441470
$388$	2.32283e6	0.783317
$389$	4.27642e6	1.43287	0.716435	−	0.697654i	$-0.245773\pi$
$389$	4.27642e6	1.43287	0.716435	+	0.697654i	$0.245773\pi$
$390$	0	0
$391$	−92350.7	−0.0305491
$392$	126601.	0.0416122
$393$	1.59504e6	0.520942
$394$	482792.	0.156682
$395$	0	0
$396$	305727.	0.0979705
$397$	1.61356e6	0.513817	0.256909	−	0.966436i	$-0.417296\pi$
$397$	1.61356e6	0.513817	0.256909	+	0.966436i	$0.417296\pi$
$398$	−555952.	−0.175926
$399$	769903.	0.242105
$400$	0	0
$401$	4.58182e6	1.42291	0.711455	−	0.702732i	$-0.248037\pi$
$401$	4.58182e6	1.42291	0.711455	+	0.702732i	$0.248037\pi$
$402$	487205.	0.150365
$403$	5.49721e6	1.68609
$404$	2.03985e6	0.621790
$405$	0	0
$406$	418093.	0.125880
$407$	−692766.	−0.207300
$408$	−38083.5	−0.0113263
$409$	2.55861e6	0.756304	0.378152	−	0.925743i	$-0.376560\pi$
$409$	2.55861e6	0.756304	0.378152	+	0.925743i	$0.376560\pi$
$410$	0	0
$411$	1.61175e6	0.470645
$412$	−1.12156e6	−0.325522
$413$	3.99967e6	1.15385
$414$	−90105.6	−0.0258375
$415$	0	0
$416$	−1.77980e6	−0.504242
$417$	1.26163e6	0.355297
$418$	76998.1	0.0215546
$419$	−4.36111e6	−1.21356	−0.606781	−	0.794869i	$-0.707540\pi$
$419$	−4.36111e6	−1.21356	−0.606781	+	0.794869i	$0.707540\pi$
$420$	0	0
$421$	−4.39997e6	−1.20989	−0.604943	−	0.796269i	$-0.706804\pi$
$421$	−4.39997e6	−1.20989	−0.604943	+	0.796269i	$0.706804\pi$
$422$	842395.	0.230269
$423$	613410.	0.166686
$424$	−307651.	−0.0831083
$425$	0	0
$426$	100474.	0.0268245
$427$	−2.17784e6	−0.578039
$428$	−3.68522e6	−0.972421
$429$	726785.	0.190661
$430$	0	0
$431$	5.08667e6	1.31899	0.659493	−	0.751711i	$-0.270771\pi$
$431$	5.08667e6	1.31899	0.659493	+	0.751711i	$0.270771\pi$
$432$	690522.	0.178020
$433$	−1.75095e6	−0.448800	−0.224400	−	0.974497i	$-0.572042\pi$
$433$	−1.75095e6	−0.448800	−0.224400	+	0.974497i	$0.572042\pi$
$434$	893126.	0.227609
$435$	0	0
$436$	−6.74816e6	−1.70008
$437$	877631.	0.219841
$438$	9948.37	0.00247780
$439$	−2.66084e6	−0.658957	−0.329479	−	0.944163i	$-0.606873\pi$
$439$	−2.66084e6	−0.658957	−0.329479	+	0.944163i	$0.606873\pi$
$440$	0	0
$441$	−180686.	−0.0442414
$442$	−44689.0	−0.0108804
$443$	6.98400e6	1.69081	0.845406	−	0.534125i	$-0.179359\pi$
$443$	6.98400e6	1.69081	0.845406	+	0.534125i	$0.179359\pi$
$444$	−1.60733e6	−0.386945
$445$	0	0
$446$	950021.	0.226150
$447$	−2.70776e6	−0.640976
$448$	3.37036e6	0.793379
$449$	7.66547e6	1.79441	0.897207	−	0.441610i	$-0.145593\pi$
$449$	7.66547e6	1.79441	0.897207	+	0.441610i	$0.145593\pi$
$450$	0	0
$451$	−81430.3	−0.0188514
$452$	−2.58865e6	−0.595973
$453$	2.02824e6	0.464380
$454$	1.24382e6	0.283216
$455$	0	0
$456$	361917.	0.0815073
$457$	−3.50188e6	−0.784351	−0.392175	−	0.919890i	$-0.628277\pi$
$457$	−3.50188e6	−0.784351	−0.392175	+	0.919890i	$0.628277\pi$
$458$	−51188.7	−0.0114028
$459$	54353.2	0.0120419
$460$	0	0
$461$	−6.63603e6	−1.45431	−0.727153	−	0.686475i	$-0.759157\pi$
$461$	−6.63603e6	−1.45431	−0.727153	+	0.686475i	$0.759157\pi$
$462$	118080.	0.0257378
$463$	−1.13310e6	−0.245649	−0.122825	−	0.992428i	$-0.539195\pi$
$463$	−1.13310e6	−0.245649	−0.122825	+	0.992428i	$0.539195\pi$
$464$	−3.65237e6	−0.787553
$465$	0	0
$466$	25136.8	0.00536222
$467$	−1.37315e6	−0.291358	−0.145679	−	0.989332i	$-0.546537\pi$
$467$	−1.37315e6	−0.291358	−0.145679	+	0.989332i	$0.546537\pi$
$468$	1.68627e6	0.355886
$469$	−7.27728e6	−1.52770
$470$	0	0
$471$	3.10444e6	0.644809
$472$	1.88017e6	0.388456
$473$	1.94303e6	0.399324
$474$	380017.	0.0776887
$475$	0	0
$476$	280792.	0.0568025
$477$	439084.	0.0883592
$478$	−1.09034e6	−0.218270
$479$	−7.71987e6	−1.53735	−0.768673	−	0.639642i	$-0.779082\pi$
$479$	−7.71987e6	−1.53735	−0.768673	+	0.639642i	$0.779082\pi$
$480$	0	0
$481$	−3.82102e6	−0.753037
$482$	953900.	0.187019
$483$	1.34589e6	0.262507
$484$	−456703.	−0.0886177
$485$	0	0
$486$	53031.9	0.0101847
$487$	−2.76688e6	−0.528650	−0.264325	−	0.964434i	$-0.585149\pi$
$487$	−2.76688e6	−0.528650	−0.264325	+	0.964434i	$0.585149\pi$
$488$	−1.02376e6	−0.194603
$489$	1.21303e6	0.229403
$490$	0	0
$491$	−2.58549e6	−0.483993	−0.241997	−	0.970277i	$-0.577802\pi$
$491$	−2.58549e6	−0.483993	−0.241997	+	0.970277i	$0.577802\pi$
$492$	−188932.	−0.0351879
$493$	−287490.	−0.0532727
$494$	424690.	0.0782988
$495$	0	0
$496$	−7.80216e6	−1.42400
$497$	−1.50076e6	−0.272534
$498$	239908.	0.0433483
$499$	−580259.	−0.104321	−0.0521604	−	0.998639i	$-0.516611\pi$
$499$	−580259.	−0.104321	−0.0521604	+	0.998639i	$0.516611\pi$
$500$	0	0
$501$	−4.28916e6	−0.763446
$502$	−67674.6	−0.0119858
$503$	8.01677e6	1.41280	0.706398	−	0.707815i	$-0.250319\pi$
$503$	8.01677e6	1.41280	0.706398	+	0.707815i	$0.250319\pi$
$504$	555015.	0.0973260
$505$	0	0
$506$	134602.	0.0233709
$507$	667015.	0.115243
$508$	−3.51718e6	−0.604694
$509$	−375241.	−0.0641971	−0.0320986	−	0.999485i	$-0.510219\pi$
$509$	−375241.	−0.0641971	−0.0320986	+	0.999485i	$0.510219\pi$
$510$	0	0
$511$	−148597.	−0.0251743
$512$	4.24633e6	0.715878
$513$	−516532.	−0.0866571
$514$	−1.02515e6	−0.171152
$515$	0	0
$516$	4.50815e6	0.745374
$517$	−916328.	−0.150773
$518$	−620796.	−0.101654
$519$	5.46328e6	0.890298
$520$	0	0
$521$	−4.40189e6	−0.710469	−0.355235	−	0.934777i	$-0.615599\pi$
$521$	−4.40189e6	−0.710469	−0.355235	+	0.934777i	$0.615599\pi$
$522$	−280501.	−0.0450565
$523$	−3.26635e6	−0.522166	−0.261083	−	0.965316i	$-0.584080\pi$
$523$	−3.26635e6	−0.522166	−0.261083	+	0.965316i	$0.584080\pi$
$524$	−5.52829e6	−0.879554
$525$	0	0
$526$	700786.	0.110439
$527$	−614133.	−0.0963244
$528$	−1.03152e6	−0.161025
$529$	−4.90213e6	−0.761634
$530$	0	0
$531$	−2.68340e6	−0.412999
$532$	−2.66844e6	−0.408769
$533$	−449136.	−0.0684795
$534$	−1.01086e6	−0.153404
$535$	0	0
$536$	−3.42091e6	−0.514315
$537$	−5.30024e6	−0.793158
$538$	188747.	0.0281141
$539$	269914.	0.0400178
$540$	0	0
$541$	8.43137e6	1.23853	0.619263	−	0.785184i	$-0.287432\pi$
$541$	8.43137e6	1.23853	0.619263	+	0.785184i	$0.287432\pi$
$542$	−4128.37	−0.000603643 0
$543$	−3.63351e6	−0.528844
$544$	198835.	0.0288068
$545$	0	0
$546$	651281.	0.0934947
$547$	4.69753e6	0.671276	0.335638	−	0.941991i	$-0.391048\pi$
$547$	4.69753e6	0.671276	0.335638	+	0.941991i	$0.391048\pi$
$548$	−5.58622e6	−0.794633
$549$	1.46113e6	0.206898
$550$	0	0
$551$	2.73209e6	0.383368
$552$	632675.	0.0883758
$553$	−5.67623e6	−0.789310
$554$	619319.	0.0857315
$555$	0	0
$556$	−4.37272e6	−0.599881
$557$	−1.06786e7	−1.45840	−0.729200	−	0.684300i	$-0.760108\pi$
$557$	−1.06786e7	−1.45840	−0.729200	+	0.684300i	$0.760108\pi$
$558$	−599203.	−0.0814683
$559$	1.07169e7	1.45058
$560$	0	0
$561$	−81194.4	−0.0108923
$562$	1.75147e6	0.233917
$563$	−6.73487e6	−0.895485	−0.447742	−	0.894163i	$-0.647772\pi$
$563$	−6.73487e6	−0.895485	−0.447742	+	0.894163i	$0.647772\pi$
$564$	−2.12604e6	−0.281432
$565$	0	0
$566$	−276475.	−0.0362756
$567$	−792125.	−0.103475
$568$	−705480.	−0.0917516
$569$	6.36492e6	0.824161	0.412081	−	0.911147i	$-0.364802\pi$
$569$	6.36492e6	0.824161	0.412081	+	0.911147i	$0.364802\pi$
$570$	0	0
$571$	−3.10897e6	−0.399049	−0.199524	−	0.979893i	$-0.563940\pi$
$571$	−3.10897e6	−0.399049	−0.199524	+	0.979893i	$0.563940\pi$
$572$	−2.51899e6	−0.321911
$573$	−3.55751e6	−0.452647
$574$	−72970.7	−0.00924419
$575$	0	0
$576$	−2.26119e6	−0.283975
$577$	−1.35678e7	−1.69656	−0.848280	−	0.529548i	$-0.822361\pi$
$577$	−1.35678e7	−1.69656	−0.848280	+	0.529548i	$0.822361\pi$
$578$	−1.27018e6	−0.158141
$579$	−1.73101e6	−0.214586
$580$	0	0
$581$	−3.58345e6	−0.440414
$582$	−601895.	−0.0736569
$583$	−655916.	−0.0799239
$584$	−69852.4	−0.00847519
$585$	0	0
$586$	−1.97250e6	−0.237286
$587$	−1.34652e7	−1.61293	−0.806466	−	0.591280i	$-0.798623\pi$
$587$	−1.34652e7	−1.61293	−0.806466	+	0.591280i	$0.798623\pi$
$588$	626247.	0.0746968
$589$	5.83626e6	0.693181
$590$	0	0
$591$	4.83814e6	0.569783
$592$	5.42314e6	0.635985
$593$	−1.49874e7	−1.75021	−0.875105	−	0.483933i	$-0.839208\pi$
$593$	−1.49874e7	−1.75021	−0.875105	+	0.483933i	$0.839208\pi$
$594$	−79220.4	−0.00921237
$595$	0	0
$596$	9.38493e6	1.08222
$597$	−5.57129e6	−0.639764
$598$	742411.	0.0848968
$599$	4.88452e6	0.556231	0.278115	−	0.960548i	$-0.410290\pi$
$599$	4.88452e6	0.556231	0.278115	+	0.960548i	$0.410290\pi$
$600$	0	0
$601$	−801632.	−0.0905293	−0.0452646	−	0.998975i	$-0.514413\pi$
$601$	−801632.	−0.0905293	−0.0452646	+	0.998975i	$0.514413\pi$
$602$	1.74117e6	0.195817
$603$	4.88236e6	0.546811
$604$	−7.02975e6	−0.784056
$605$	0	0
$606$	−528569.	−0.0584682
$607$	−1.38864e7	−1.52974	−0.764872	−	0.644182i	$-0.777198\pi$
$607$	−1.38864e7	−1.52974	−0.764872	+	0.644182i	$0.777198\pi$
$608$	−1.88958e6	−0.207303
$609$	4.18977e6	0.457770
$610$	0	0
$611$	−5.05410e6	−0.547697
$612$	−188385.	−0.0203314
$613$	−1.48685e6	−0.159815	−0.0799073	−	0.996802i	$-0.525462\pi$
$613$	−1.48685e6	−0.159815	−0.0799073	+	0.996802i	$0.525462\pi$
$614$	813659.	0.0871007
$615$	0	0
$616$	−829097.	−0.0880346
$617$	−2.96802e6	−0.313873	−0.156937	−	0.987609i	$-0.550162\pi$
$617$	−2.96802e6	−0.313873	−0.156937	+	0.987609i	$0.550162\pi$
$618$	290621.	0.0306095
$619$	−1.40553e7	−1.47440	−0.737198	−	0.675677i	$-0.763851\pi$
$619$	−1.40553e7	−1.47440	−0.737198	+	0.675677i	$0.763851\pi$
$620$	0	0
$621$	−902962.	−0.0939595
$622$	947461.	0.0981941
$623$	1.50989e7	1.55857
$624$	−5.68945e6	−0.584937
$625$	0	0
$626$	−2.67718e6	−0.273049
$627$	771610.	0.0783843
$628$	−1.07598e7	−1.08869
$629$	426873.	0.0430202
$630$	0	0
$631$	−7.30424e6	−0.730301	−0.365150	−	0.930949i	$-0.618982\pi$
$631$	−7.30424e6	−0.730301	−0.365150	+	0.930949i	$0.618982\pi$
$632$	−2.66829e6	−0.265730
$633$	8.44178e6	0.837384
$634$	244463.	0.0241540
$635$	0	0
$636$	−1.52184e6	−0.149185
$637$	1.48874e6	0.145368
$638$	419019.	0.0407551
$639$	1.00687e6	0.0975487
$640$	0	0
$641$	−7.28278e6	−0.700087	−0.350043	−	0.936733i	$-0.613833\pi$
$641$	−7.28278e6	−0.700087	−0.350043	+	0.936733i	$0.613833\pi$
$642$	954921.	0.0914387
$643$	−1.44225e7	−1.37567	−0.687833	−	0.725869i	$-0.741438\pi$
$643$	−1.44225e7	−1.37567	−0.687833	+	0.725869i	$0.741438\pi$
$644$	−4.66476e6	−0.443215
$645$	0	0
$646$	−47445.2	−0.00447313
$647$	1.62591e6	0.152699	0.0763495	−	0.997081i	$-0.475674\pi$
$647$	1.62591e6	0.152699	0.0763495	+	0.997081i	$0.475674\pi$
$648$	−372363.	−0.0348360
$649$	4.00854e6	0.373572
$650$	0	0
$651$	8.95016e6	0.827711
$652$	−4.20427e6	−0.387322
$653$	1.73130e7	1.58887	0.794435	−	0.607349i	$-0.207767\pi$
$653$	1.73130e7	1.58887	0.794435	+	0.607349i	$0.207767\pi$
$654$	1.74859e6	0.159862
$655$	0	0
$656$	637457.	0.0578350
$657$	99694.3	0.00901066
$658$	−821133.	−0.0739348
$659$	2.47926e6	0.222387	0.111193	−	0.993799i	$-0.464533\pi$
$659$	2.47926e6	0.222387	0.111193	+	0.993799i	$0.464533\pi$
$660$	0	0
$661$	−7.28403e6	−0.648437	−0.324219	−	0.945982i	$-0.605101\pi$
$661$	−7.28403e6	−0.648437	−0.324219	+	0.945982i	$0.605101\pi$
$662$	−884293.	−0.0784244
$663$	−447835.	−0.0395671
$664$	−1.68451e6	−0.148270
$665$	0	0
$666$	416495.	0.0363852
$667$	4.77602e6	0.415673
$668$	1.48660e7	1.28900
$669$	9.52032e6	0.822406
$670$	0	0
$671$	−2.18267e6	−0.187147
$672$	−2.89775e6	−0.247535
$673$	−1.73888e7	−1.47990	−0.739951	−	0.672661i	$-0.765151\pi$
$673$	−1.73888e7	−1.47990	−0.739951	+	0.672661i	$0.765151\pi$
$674$	329481.	0.0279371
$675$	0	0
$676$	−2.31183e6	−0.194576
$677$	1.51982e7	1.27444	0.637221	−	0.770681i	$-0.280084\pi$
$677$	1.51982e7	1.27444	0.637221	+	0.770681i	$0.280084\pi$
$678$	670775.	0.0560406
$679$	8.99038e6	0.748348
$680$	0	0
$681$	1.24645e7	1.02993
$682$	895106.	0.0736908
$683$	2.02075e7	1.65753	0.828765	−	0.559596i	$-0.189044\pi$
$683$	2.02075e7	1.65753	0.828765	+	0.559596i	$0.189044\pi$
$684$	1.79027e6	0.146311
$685$	0	0
$686$	2.06425e6	0.167476
$687$	−512970.	−0.0414668
$688$	−1.52105e7	−1.22510
$689$	−3.61777e6	−0.290330
$690$	0	0
$691$	−1.88904e7	−1.50503	−0.752517	−	0.658573i	$-0.771160\pi$
$691$	−1.88904e7	−1.50503	−0.752517	+	0.658573i	$0.771160\pi$
$692$	−1.89354e7	−1.50317
$693$	1.18330e6	0.0935968
$694$	501550.	0.0395290
$695$	0	0
$696$	1.96953e6	0.154113
$697$	50176.3	0.00391216
$698$	1.37715e6	0.106990
$699$	251900.	0.0195000
$700$	0	0
$701$	−1.63467e7	−1.25642	−0.628210	−	0.778044i	$-0.716212\pi$
$701$	−1.63467e7	−1.25642	−0.628210	+	0.778044i	$0.716212\pi$
$702$	−436948.	−0.0334647
$703$	−4.05668e6	−0.309587
$704$	3.37783e6	0.256865
$705$	0	0
$706$	−2.19001e6	−0.165362
$707$	7.89511e6	0.594032
$708$	9.30050e6	0.697305
$709$	2.28062e7	1.70387	0.851936	−	0.523646i	$-0.175429\pi$
$709$	2.28062e7	1.70387	0.851936	+	0.523646i	$0.175429\pi$
$710$	0	0
$711$	3.80822e6	0.282519
$712$	7.09771e6	0.524709
$713$	1.02025e7	0.751594
$714$	−72759.3	−0.00534125
$715$	0	0
$716$	1.83703e7	1.33916
$717$	−1.09265e7	−0.793750
$718$	1.05659e6	0.0764886
$719$	1.42246e6	0.102617	0.0513083	−	0.998683i	$-0.483661\pi$
$719$	1.42246e6	0.102617	0.0513083	+	0.998683i	$0.483661\pi$
$720$	0	0
$721$	−4.34094e6	−0.310989
$722$	−1.77290e6	−0.126573
$723$	9.55919e6	0.680104
$724$	1.25935e7	0.892896
$725$	0	0
$726$	118342.	0.00833290
$727$	−1.58260e7	−1.11054	−0.555272	−	0.831669i	$-0.687386\pi$
$727$	−1.58260e7	−1.11054	−0.555272	+	0.831669i	$0.687386\pi$
$728$	−4.57297e6	−0.319793
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.19727e6	−0.0828701
$732$	−5.06417e6	−0.349326
$733$	−1.04557e7	−0.718776	−0.359388	−	0.933188i	$-0.617015\pi$
$733$	−1.04557e7	−0.718776	−0.359388	+	0.933188i	$0.617015\pi$
$734$	2.46248e6	0.168707
$735$	0	0
$736$	−3.30321e6	−0.224772
$737$	−7.29341e6	−0.494609
$738$	48956.4	0.00330879
$739$	−5.11969e6	−0.344852	−0.172426	−	0.985023i	$-0.555160\pi$
$739$	−5.11969e6	−0.344852	−0.172426	+	0.985023i	$0.555160\pi$
$740$	0	0
$741$	4.25589e6	0.284738
$742$	−587775.	−0.0391923
$743$	4.44877e6	0.295643	0.147822	−	0.989014i	$-0.452774\pi$
$743$	4.44877e6	0.295643	0.147822	+	0.989014i	$0.452774\pi$
$744$	4.20730e6	0.278658
$745$	0	0
$746$	−3.11316e6	−0.204812
$747$	2.40416e6	0.157638
$748$	281414.	0.0183904
$749$	−1.42634e7	−0.929009
$750$	0	0
$751$	−1.91736e7	−1.24052	−0.620262	−	0.784395i	$-0.712974\pi$
$751$	−1.91736e7	−1.24052	−0.620262	+	0.784395i	$0.712974\pi$
$752$	7.17325e6	0.462563
$753$	−678178.	−0.0435869
$754$	2.31114e6	0.148047
$755$	0	0
$756$	2.74545e6	0.174707
$757$	1.24339e7	0.788621	0.394310	−	0.918977i	$-0.370983\pi$
$757$	1.24339e7	0.788621	0.394310	+	0.918977i	$0.370983\pi$
$758$	3.65707e6	0.231185
$759$	1.34887e6	0.0849896
$760$	0	0
$761$	−1.41772e7	−0.887421	−0.443710	−	0.896170i	$-0.646338\pi$
$761$	−1.41772e7	−0.887421	−0.443710	+	0.896170i	$0.646338\pi$
$762$	911379.	0.0568606
$763$	−2.61184e7	−1.62418
$764$	1.23301e7	0.764246
$765$	0	0
$766$	−392125.	−0.0241464
$767$	2.21095e7	1.35703
$768$	7.14736e6	0.437263
$769$	−4.53694e6	−0.276660	−0.138330	−	0.990386i	$-0.544174\pi$
$769$	−4.53694e6	−0.276660	−0.138330	+	0.990386i	$0.544174\pi$
$770$	0	0
$771$	−1.02732e7	−0.622402
$772$	5.99956e6	0.362306
$773$	2.11780e7	1.27479	0.637393	−	0.770539i	$-0.280013\pi$
$773$	2.11780e7	1.27479	0.637393	+	0.770539i	$0.280013\pi$
$774$	−1.16816e6	−0.0700890
$775$	0	0
$776$	4.22620e6	0.251939
$777$	−6.22110e6	−0.369670
$778$	3.84065e6	0.227487
$779$	−476838.	−0.0281531
$780$	0	0
$781$	−1.50409e6	−0.0882361
$782$	−82940.1	−0.00485007
$783$	−2.81094e6	−0.163850
$784$	−2.11295e6	−0.122772
$785$	0	0
$786$	1.43250e6	0.0827063
$787$	2.73190e7	1.57228	0.786138	−	0.618051i	$-0.212078\pi$
$787$	2.73190e7	1.57228	0.786138	+	0.618051i	$0.212078\pi$
$788$	−1.67687e7	−0.962018
$789$	7.02269e6	0.401616
$790$	0	0
$791$	−1.00192e7	−0.569367
$792$	556246.	0.0315104
$793$	−1.20387e7	−0.679826
$794$	1.44914e6	0.0815752
$795$	0	0
$796$	1.93097e7	1.08017
$797$	8.71204e6	0.485818	0.242909	−	0.970049i	$-0.421898\pi$
$797$	8.71204e6	0.485818	0.242909	+	0.970049i	$0.421898\pi$
$798$	691450.	0.0384373
$799$	564630.	0.0312894
$800$	0	0
$801$	−1.01299e7	−0.557861
$802$	4.11493e6	0.225905
$803$	−148926.	−0.00815045
$804$	−1.69220e7	−0.923231
$805$	0	0
$806$	4.93704e6	0.267688
$807$	1.89146e6	0.102238
$808$	3.71134e6	0.199987
$809$	−8.30436e6	−0.446103	−0.223051	−	0.974807i	$-0.571602\pi$
$809$	−8.30436e6	−0.446103	−0.223051	+	0.974807i	$0.571602\pi$
$810$	0	0
$811$	−2.16579e7	−1.15629	−0.578143	−	0.815936i	$-0.696222\pi$
$811$	−2.16579e7	−1.15629	−0.578143	+	0.815936i	$0.696222\pi$
$812$	−1.45215e7	−0.772896
$813$	−41371.0	−0.00219518
$814$	−622172.	−0.0329116
$815$	0	0
$816$	635610.	0.0334168
$817$	1.13779e7	0.596359
$818$	2.29789e6	0.120073
$819$	6.52660e6	0.339998
$820$	0	0
$821$	−4.66524e6	−0.241555	−0.120777	−	0.992680i	$-0.538539\pi$
$821$	−4.66524e6	−0.241555	−0.120777	+	0.992680i	$0.538539\pi$
$822$	1.44751e6	0.0747210
$823$	3.86223e7	1.98765	0.993823	−	0.110977i	$-0.0353978\pi$
$823$	3.86223e7	1.98765	0.993823	+	0.110977i	$0.0353978\pi$
$824$	−2.04059e6	−0.104698
$825$	0	0
$826$	3.59210e6	0.183189
$827$	−1.64856e7	−0.838187	−0.419094	−	0.907943i	$-0.637652\pi$
$827$	−1.64856e7	−0.838187	−0.419094	+	0.907943i	$0.637652\pi$
$828$	3.12961e6	0.158641
$829$	1.17447e7	0.593546	0.296773	−	0.954948i	$-0.404089\pi$
$829$	1.17447e7	0.593546	0.296773	+	0.954948i	$0.404089\pi$
$830$	0	0
$831$	6.20630e6	0.311767
$832$	1.86307e7	0.933086
$833$	−166317.	−0.00830473
$834$	1.13307e6	0.0564080
$835$	0	0
$836$	−2.67435e6	−0.132344
$837$	−6.00471e6	−0.296264
$838$	−3.91671e6	−0.192669
$839$	2.08153e7	1.02089	0.510444	−	0.859911i	$-0.329481\pi$
$839$	2.08153e7	1.02089	0.510444	+	0.859911i	$0.329481\pi$
$840$	0	0
$841$	−5.64328e6	−0.275132
$842$	−3.95161e6	−0.192085
$843$	1.75518e7	0.850653
$844$	−2.92587e7	−1.41383
$845$	0	0
$846$	550903.	0.0264636
$847$	−1.76764e6	−0.0846615
$848$	5.13467e6	0.245202
$849$	−2.77060e6	−0.131918
$850$	0	0
$851$	−7.09158e6	−0.335675
$852$	−3.48975e6	−0.164700
$853$	−4.06291e6	−0.191190	−0.0955950	−	0.995420i	$-0.530475\pi$
$853$	−4.06291e6	−0.191190	−0.0955950	+	0.995420i	$0.530475\pi$
$854$	−1.95592e6	−0.0917712
$855$	0	0
$856$	−6.70497e6	−0.312761
$857$	−66823.3	−0.00310796	−0.00155398	−	0.999999i	$-0.500495\pi$
$857$	−66823.3	−0.00310796	−0.00155398	+	0.999999i	$0.500495\pi$
$858$	652725.	0.0302700
$859$	9.60662e6	0.444209	0.222105	−	0.975023i	$-0.428707\pi$
$859$	9.60662e6	0.444209	0.222105	+	0.975023i	$0.428707\pi$
$860$	0	0
$861$	−731251.	−0.0336170
$862$	4.56833e6	0.209406
$863$	3.62555e6	0.165709	0.0828547	−	0.996562i	$-0.473596\pi$
$863$	3.62555e6	0.165709	0.0828547	+	0.996562i	$0.473596\pi$
$864$	1.94411e6	0.0886008
$865$	0	0
$866$	−1.57252e6	−0.0712529
$867$	−1.27287e7	−0.575090
$868$	−3.10207e7	−1.39750
$869$	−5.68882e6	−0.255548
$870$	0	0
$871$	−4.02275e7	−1.79671
$872$	−1.22777e7	−0.546798
$873$	−6.03169e6	−0.267857
$874$	788200.	0.0349026
$875$	0	0
$876$	−345534.	−0.0152135
$877$	6.29358e6	0.276311	0.138156	−	0.990411i	$-0.455883\pi$
$877$	6.29358e6	0.276311	0.138156	+	0.990411i	$0.455883\pi$
$878$	−2.38970e6	−0.104618
$879$	−1.97667e7	−0.862904
$880$	0	0
$881$	2.35769e7	1.02340	0.511701	−	0.859163i	$-0.329015\pi$
$881$	2.35769e7	1.02340	0.511701	+	0.859163i	$0.329015\pi$
$882$	−162274.	−0.00702389
$883$	−1.08516e7	−0.468372	−0.234186	−	0.972192i	$-0.575242\pi$
$883$	−1.08516e7	−0.468372	−0.234186	+	0.972192i	$0.575242\pi$
$884$	1.55217e6	0.0668049
$885$	0	0
$886$	6.27233e6	0.268438
$887$	2.44367e7	1.04288	0.521439	−	0.853289i	$-0.325396\pi$
$887$	2.44367e7	1.04288	0.521439	+	0.853289i	$0.325396\pi$
$888$	−2.92442e6	−0.124454
$889$	−1.36131e7	−0.577699
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−3.29968e7	−1.38854
$893$	−5.36581e6	−0.225168
$894$	−2.43184e6	−0.101763
$895$	0	0
$896$	1.33300e7	0.554703
$897$	7.43982e6	0.308732
$898$	6.88435e6	0.284887
$899$	3.17606e7	1.31066
$900$	0	0
$901$	404167.	0.0165863
$902$	−73132.5	−0.00299291
$903$	1.74486e7	0.712098
$904$	−4.70984e6	−0.191684
$905$	0	0
$906$	1.82156e6	0.0737264
$907$	4.71119e7	1.90157	0.950785	−	0.309850i	$-0.100279\pi$
$907$	4.71119e7	1.90157	0.950785	+	0.309850i	$0.100279\pi$
$908$	−4.32012e7	−1.73893
$909$	−5.29687e6	−0.212623
$910$	0	0
$911$	4.74577e7	1.89457	0.947286	−	0.320390i	$-0.103814\pi$
$911$	4.74577e7	1.89457	0.947286	+	0.320390i	$0.103814\pi$
$912$	−6.04036e6	−0.240478
$913$	−3.59140e6	−0.142589
$914$	−3.14503e6	−0.124526
$915$	0	0
$916$	1.77792e6	0.0700122
$917$	−2.13969e7	−0.840288
$918$	48814.6	0.00191180
$919$	1.88222e7	0.735159	0.367580	−	0.929992i	$-0.380187\pi$
$919$	1.88222e7	0.735159	0.367580	+	0.929992i	$0.380187\pi$
$920$	0	0
$921$	8.15381e6	0.316746
$922$	−5.95982e6	−0.230890
$923$	−8.29595e6	−0.320525
$924$	−4.10123e6	−0.158028
$925$	0	0
$926$	−1.01764e6	−0.0390001
$927$	2.91236e6	0.111313
$928$	−1.02830e7	−0.391966
$929$	1.80487e7	0.686132	0.343066	−	0.939311i	$-0.388535\pi$
$929$	1.80487e7	0.686132	0.343066	+	0.939311i	$0.388535\pi$
$930$	0	0
$931$	1.58056e6	0.0597634
$932$	−873067.	−0.0329237
$933$	9.49466e6	0.357088
$934$	−1.23323e6	−0.0462569
$935$	0	0
$936$	3.06803e6	0.114464
$937$	5.09381e7	1.89537	0.947685	−	0.319208i	$-0.103417\pi$
$937$	5.09381e7	1.89537	0.947685	+	0.319208i	$0.103417\pi$
$938$	−6.53572e6	−0.242542
$939$	−2.68284e7	−0.992958
$940$	0	0
$941$	4.08876e7	1.50528	0.752641	−	0.658431i	$-0.228779\pi$
$941$	4.08876e7	1.50528	0.752641	+	0.658431i	$0.228779\pi$
$942$	2.78810e6	0.102372
$943$	−833571.	−0.0305255
$944$	−3.13799e7	−1.14610
$945$	0	0
$946$	1.74503e6	0.0633979
$947$	1.33924e6	0.0485271	0.0242636	−	0.999706i	$-0.492276\pi$
$947$	1.33924e6	0.0485271	0.0242636	+	0.999706i	$0.492276\pi$
$948$	−1.31990e7	−0.477003
$949$	−821416.	−0.0296072
$950$	0	0
$951$	2.44980e6	0.0878374
$952$	510879.	0.0182695
$953$	−1.17556e6	−0.0419288	−0.0209644	−	0.999780i	$-0.506674\pi$
$953$	−1.17556e6	−0.0419288	−0.0209644	+	0.999780i	$0.506674\pi$
$954$	394341.	0.0140282
$955$	0	0
$956$	3.78706e7	1.34016
$957$	4.19906e6	0.148208
$958$	−6.93321e6	−0.244074
$959$	−2.16212e7	−0.759159
$960$	0	0
$961$	3.92177e7	1.36985
$962$	−3.43165e6	−0.119554
$963$	9.56942e6	0.332522
$964$	−3.31315e7	−1.14828
$965$	0	0
$966$	1.20874e6	0.0416764
$967$	1.01207e7	0.348053	0.174026	−	0.984741i	$-0.444322\pi$
$967$	1.01207e7	0.348053	0.174026	+	0.984741i	$0.444322\pi$
$968$	−830935.	−0.0285022
$969$	−475456.	−0.0162668
$970$	0	0
$971$	−2.71903e7	−0.925476	−0.462738	−	0.886495i	$-0.653133\pi$
$971$	−2.71903e7	−0.925476	−0.462738	+	0.886495i	$0.653133\pi$
$972$	−1.84194e6	−0.0625331
$973$	−1.69244e7	−0.573101
$974$	−2.48493e6	−0.0839300
$975$	0	0
$976$	1.70865e7	0.574154
$977$	−2.41360e7	−0.808962	−0.404481	−	0.914546i	$-0.632548\pi$
$977$	−2.41360e7	−0.808962	−0.404481	+	0.914546i	$0.632548\pi$
$978$	1.08942e6	0.0364207
$979$	1.51324e7	0.504604
$980$	0	0
$981$	1.75229e7	0.581346
$982$	−2.32203e6	−0.0768402
$983$	2.27328e7	0.750359	0.375180	−	0.926952i	$-0.377581\pi$
$983$	2.27328e7	0.750359	0.375180	+	0.926952i	$0.377581\pi$
$984$	−343747.	−0.0113175
$985$	0	0
$986$	−258194.	−0.00845774
$987$	−8.22871e6	−0.268868
$988$	−1.47506e7	−0.480749
$989$	1.98900e7	0.646613
$990$	0	0
$991$	−1.42957e6	−0.0462403	−0.0231202	−	0.999733i	$-0.507360\pi$
$991$	−1.42957e6	−0.0462403	−0.0231202	+	0.999733i	$0.507360\pi$
$992$	−2.19664e7	−0.708728
$993$	−8.86164e6	−0.285194
$994$	−1.34783e6	−0.0432684
$995$	0	0
$996$	−8.33266e6	−0.266155
$997$	3.51095e6	0.111863	0.0559316	−	0.998435i	$-0.482187\pi$
$997$	3.51095e6	0.111863	0.0559316	+	0.998435i	$0.482187\pi$
$998$	−521130.	−0.0165623
$999$	4.17377e6	0.132317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.k.1.3		5
5.4	even	2		165.6.a.g.1.3	✓	5
15.14	odd	2		495.6.a.i.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.g.1.3	✓	5	5.4	even	2
495.6.a.i.1.3		5	15.14	odd	2
825.6.a.k.1.3		5	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+0.898099 q^{2} +9.00000 q^{3} -31.1934 q^{4} +8.08289 q^{6} -120.732 q^{7} -56.7540 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+0.898099 q^{2} +9.00000 q^{3} -31.1934 q^{4} +8.08289 q^{6} -120.732 q^{7} -56.7540 q^{8} +81.0000 q^{9} -121.000 q^{11} -280.741 q^{12} -667.387 q^{13} -108.430 q^{14} +947.219 q^{16} +74.5586 q^{17} +72.7460 q^{18} -708.549 q^{19} -1086.59 q^{21} -108.670 q^{22} -1238.63 q^{23} -510.786 q^{24} -599.380 q^{26} +729.000 q^{27} +3766.06 q^{28} -3855.89 q^{29} -8236.92 q^{31} +2666.82 q^{32} -1089.00 q^{33} +66.9611 q^{34} -2526.67 q^{36} +5725.33 q^{37} -636.348 q^{38} -6006.49 q^{39} +672.977 q^{41} -975.867 q^{42} -16058.1 q^{43} +3774.40 q^{44} -1112.41 q^{46} +7572.96 q^{47} +8524.97 q^{48} -2230.69 q^{49} +671.028 q^{51} +20818.1 q^{52} +5420.79 q^{53} +654.714 q^{54} +6852.04 q^{56} -6376.94 q^{57} -3462.97 q^{58} -33128.4 q^{59} +18038.6 q^{61} -7397.57 q^{62} -9779.32 q^{63} -27915.9 q^{64} -978.030 q^{66} +60276.1 q^{67} -2325.74 q^{68} -11147.7 q^{69} +12430.5 q^{71} -4597.07 q^{72} +1230.79 q^{73} +5141.92 q^{74} +22102.1 q^{76} +14608.6 q^{77} -5394.42 q^{78} +47015.0 q^{79} +6561.00 q^{81} +604.401 q^{82} +29681.0 q^{83} +33894.5 q^{84} -14421.7 q^{86} -34703.0 q^{87} +6867.23 q^{88} -125061. q^{89} +80575.2 q^{91} +38637.2 q^{92} -74132.3 q^{93} +6801.27 q^{94} +24001.4 q^{96} -74465.3 q^{97} -2003.38 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9	\( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9} - 605 q^{11} + 1143 q^{12} + 926 q^{13} + 368 q^{14} + 1891 q^{16} + 246 q^{17} + 81 q^{18} + 3420 q^{19} - 1044 q^{21} - 121 q^{22} + 4244 q^{23} + 1377 q^{24} - 8862 q^{26} + 3645 q^{27} + 4904 q^{28} - 2922 q^{29} - 6112 q^{31} + 24757 q^{32} - 5445 q^{33} + 10866 q^{34} + 10287 q^{36} - 6654 q^{37} + 45692 q^{38} + 8334 q^{39} - 14934 q^{41} + 3312 q^{42} - 10804 q^{43} - 15367 q^{44} - 101500 q^{46} + 41460 q^{47} + 17019 q^{48} - 12099 q^{49} + 2214 q^{51} + 97742 q^{52} + 62398 q^{53} + 729 q^{54} - 74368 q^{56} + 30780 q^{57} + 27822 q^{58} + 8524 q^{59} + 59010 q^{61} + 142624 q^{62} - 9396 q^{63} + 13799 q^{64} - 1089 q^{66} + 15772 q^{67} + 83686 q^{68} + 38196 q^{69} + 88124 q^{71} + 12393 q^{72} + 118358 q^{73} + 67194 q^{74} + 100668 q^{76} + 14036 q^{77} - 79758 q^{78} + 57324 q^{79} + 32805 q^{81} - 29102 q^{82} + 7268 q^{83} + 44136 q^{84} - 35288 q^{86} - 26298 q^{87} - 18513 q^{88} + 72978 q^{89} - 1464 q^{91} - 62148 q^{92} - 55008 q^{93} + 344836 q^{94} + 222813 q^{96} + 59174 q^{97} - 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + q^2 + 45 * q^3 + 127 * q^4 + 9 * q^6 - 116 * q^7 + 153 * q^8 + 405 * q^9 - 605 * q^11 + 1143 * q^12 + 926 * q^13 + 368 * q^14 + 1891 * q^16 + 246 * q^17 + 81 * q^18 + 3420 * q^19 - 1044 * q^21 - 121 * q^22 + 4244 * q^23 + 1377 * q^24 - 8862 * q^26 + 3645 * q^27 + 4904 * q^28 - 2922 * q^29 - 6112 * q^31 + 24757 * q^32 - 5445 * q^33 + 10866 * q^34 + 10287 * q^36 - 6654 * q^37 + 45692 * q^38 + 8334 * q^39 - 14934 * q^41 + 3312 * q^42 - 10804 * q^43 - 15367 * q^44 - 101500 * q^46 + 41460 * q^47 + 17019 * q^48 - 12099 * q^49 + 2214 * q^51 + 97742 * q^52 + 62398 * q^53 + 729 * q^54 - 74368 * q^56 + 30780 * q^57 + 27822 * q^58 + 8524 * q^59 + 59010 * q^61 + 142624 * q^62 - 9396 * q^63 + 13799 * q^64 - 1089 * q^66 + 15772 * q^67 + 83686 * q^68 + 38196 * q^69 + 88124 * q^71 + 12393 * q^72 + 118358 * q^73 + 67194 * q^74 + 100668 * q^76 + 14036 * q^77 - 79758 * q^78 + 57324 * q^79 + 32805 * q^81 - 29102 * q^82 + 7268 * q^83 + 44136 * q^84 - 35288 * q^86 - 26298 * q^87 - 18513 * q^88 + 72978 * q^89 - 1464 * q^91 - 62148 * q^92 - 55008 * q^93 + 344836 * q^94 + 222813 * q^96 + 59174 * q^97 - 272767 * q^98 - 49005 * q^99

Introduction

L-functions

Modular forms

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Database

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