LMFDB - Embedded newform 825.6.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$5$
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} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 $ x^5 - 2x^4 - 119x^3 + 206x^2 + 1428x - 1320
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			1.2
Root			$-3.34733$ 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)

$f(q)$	$=$	$q-3.34733 q^{2} -9.00000 q^{3} -20.7954 q^{4} +30.1260 q^{6} -42.9514 q^{7} +176.724 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.34733 q^{2} -9.00000 q^{3} -20.7954 q^{4} +30.1260 q^{6} -42.9514 q^{7} +176.724 q^{8} +81.0000 q^{9} -121.000 q^{11} +187.158 q^{12} -1198.06 q^{13} +143.772 q^{14} +73.9006 q^{16} +203.386 q^{17} -271.134 q^{18} +1574.20 q^{19} +386.562 q^{21} +405.027 q^{22} -1037.51 q^{23} -1590.51 q^{24} +4010.32 q^{26} -729.000 q^{27} +893.191 q^{28} +3263.32 q^{29} +4561.02 q^{31} -5902.52 q^{32} +1089.00 q^{33} -680.799 q^{34} -1684.43 q^{36} -11450.2 q^{37} -5269.38 q^{38} +10782.6 q^{39} -12442.9 q^{41} -1293.95 q^{42} +16091.9 q^{43} +2516.24 q^{44} +3472.87 q^{46} -16446.0 q^{47} -665.105 q^{48} -14962.2 q^{49} -1830.47 q^{51} +24914.2 q^{52} +11571.5 q^{53} +2440.20 q^{54} -7590.52 q^{56} -14167.8 q^{57} -10923.4 q^{58} +28773.2 q^{59} +39996.6 q^{61} -15267.2 q^{62} -3479.06 q^{63} +17392.9 q^{64} -3645.24 q^{66} +30728.9 q^{67} -4229.49 q^{68} +9337.55 q^{69} +19369.8 q^{71} +14314.6 q^{72} +7371.45 q^{73} +38327.7 q^{74} -32736.2 q^{76} +5197.12 q^{77} -36092.9 q^{78} +102237. q^{79} +6561.00 q^{81} +41650.6 q^{82} +36076.2 q^{83} -8038.71 q^{84} -53864.7 q^{86} -29369.9 q^{87} -21383.5 q^{88} +67261.9 q^{89} +51458.5 q^{91} +21575.3 q^{92} -41049.2 q^{93} +55050.2 q^{94} +53122.7 q^{96} +106710. q^{97} +50083.3 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} - 45 q^{3} + 82 q^{4} - 18 q^{6} - 184 q^{7} - 24 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + 2 * q^2 - 45 * q^3 + 82 * q^4 - 18 * q^6 - 184 * q^7 - 24 * q^8 + 405 * q^9	\( 5 q + 2 q^{2} - 45 q^{3} + 82 q^{4} - 18 q^{6} - 184 q^{7} - 24 q^{8} + 405 q^{9} - 605 q^{11} - 738 q^{12} - 1082 q^{13} + 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 162 q^{18} + 1632 q^{19} + 1656 q^{21} - 242 q^{22} - 1212 q^{23} + 216 q^{24} + 5600 q^{26} - 3645 q^{27} - 16508 q^{28} + 82 q^{29} + 12120 q^{31} + 4864 q^{32} + 5445 q^{33} - 4524 q^{34} + 6642 q^{36} + 6530 q^{37} + 15132 q^{38} + 9738 q^{39} + 6782 q^{41} - 3888 q^{42} - 46184 q^{43} - 9922 q^{44} + 12048 q^{46} + 11692 q^{47} - 42930 q^{48} + 34445 q^{49} + 19566 q^{51} - 50020 q^{52} - 10314 q^{53} - 1458 q^{54} + 54928 q^{56} - 14688 q^{57} - 75048 q^{58} + 92892 q^{59} + 106 q^{61} - 97160 q^{62} - 14904 q^{63} + 44550 q^{64} + 2178 q^{66} - 100476 q^{67} - 119928 q^{68} + 10908 q^{69} - 13772 q^{71} - 1944 q^{72} - 94154 q^{73} - 47924 q^{74} - 51524 q^{76} + 22264 q^{77} - 50400 q^{78} + 178744 q^{79} + 32805 q^{81} + 299848 q^{82} + 100116 q^{83} + 148572 q^{84} - 167704 q^{86} - 738 q^{87} + 2904 q^{88} + 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 109080 q^{93} - 310288 q^{94} - 43776 q^{96} - 100682 q^{97} + 16434 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - 45 * q^3 + 82 * q^4 - 18 * q^6 - 184 * q^7 - 24 * q^8 + 405 * q^9 - 605 * q^11 - 738 * q^12 - 1082 * q^13 + 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 162 * q^18 + 1632 * q^19 + 1656 * q^21 - 242 * q^22 - 1212 * q^23 + 216 * q^24 + 5600 * q^26 - 3645 * q^27 - 16508 * q^28 + 82 * q^29 + 12120 * q^31 + 4864 * q^32 + 5445 * q^33 - 4524 * q^34 + 6642 * q^36 + 6530 * q^37 + 15132 * q^38 + 9738 * q^39 + 6782 * q^41 - 3888 * q^42 - 46184 * q^43 - 9922 * q^44 + 12048 * q^46 + 11692 * q^47 - 42930 * q^48 + 34445 * q^49 + 19566 * q^51 - 50020 * q^52 - 10314 * q^53 - 1458 * q^54 + 54928 * q^56 - 14688 * q^57 - 75048 * q^58 + 92892 * q^59 + 106 * q^61 - 97160 * q^62 - 14904 * q^63 + 44550 * q^64 + 2178 * q^66 - 100476 * q^67 - 119928 * q^68 + 10908 * q^69 - 13772 * q^71 - 1944 * q^72 - 94154 * q^73 - 47924 * q^74 - 51524 * q^76 + 22264 * q^77 - 50400 * q^78 + 178744 * q^79 + 32805 * q^81 + 299848 * q^82 + 100116 * q^83 + 148572 * q^84 - 167704 * q^86 - 738 * q^87 + 2904 * q^88 + 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 109080 * q^93 - 310288 * q^94 - 43776 * q^96 - 100682 * q^97 + 16434 * 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 $

$2$	−3.34733	−0.591730	−0.295865	−	0.955230i	$-0.595608\pi$
$2$	−3.34733	−0.591730	−0.295865	+	0.955230i	$0.595608\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−20.7954	−0.649856
$5$	0	0
$5$	0	0
$6$	30.1260	0.341635
$7$	−42.9514	−0.331308	−0.165654	−	0.986184i	$-0.552974\pi$
$7$	−42.9514	−0.331308	−0.165654	+	0.986184i	$0.552974\pi$
$8$	176.724	0.976269
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	187.158	0.375194
$13$	−1198.06	−1.96617	−0.983087	−	0.183139i	$-0.941374\pi$
$13$	−1198.06	−1.96617	−0.983087	+	0.183139i	$0.941374\pi$
$14$	143.772	0.196045
$15$	0	0
$16$	73.9006	0.0721685
$17$	203.386	0.170686	0.0853431	−	0.996352i	$-0.472801\pi$
$17$	203.386	0.170686	0.0853431	+	0.996352i	$0.472801\pi$
$18$	−271.134	−0.197243
$19$	1574.20	1.00041	0.500204	−	0.865908i	$-0.333258\pi$
$19$	1574.20	1.00041	0.500204	+	0.865908i	$0.333258\pi$
$20$	0	0
$21$	386.562	0.191281
$22$	405.027	0.178413
$23$	−1037.51	−0.408950	−0.204475	−	0.978872i	$-0.565549\pi$
$23$	−1037.51	−0.408950	−0.204475	+	0.978872i	$0.565549\pi$
$24$	−1590.51	−0.563649
$25$	0	0
$26$	4010.32	1.16344
$27$	−729.000	−0.192450
$28$	893.191	0.215302
$29$	3263.32	0.720550	0.360275	−	0.932846i	$-0.382683\pi$
$29$	3263.32	0.720550	0.360275	+	0.932846i	$0.382683\pi$
$30$	0	0
$31$	4561.02	0.852428	0.426214	−	0.904622i	$-0.359847\pi$
$31$	4561.02	0.852428	0.426214	+	0.904622i	$0.359847\pi$
$32$	−5902.52	−1.01897
$33$	1089.00	0.174078
$34$	−680.799	−0.101000
$35$	0	0
$36$	−1684.43	−0.216619
$37$	−11450.2	−1.37502	−0.687512	−	0.726173i	$-0.741297\pi$
$37$	−11450.2	−1.37502	−0.687512	+	0.726173i	$0.741297\pi$
$38$	−5269.38	−0.591971
$39$	10782.6	1.13517
$40$	0	0
$41$	−12442.9	−1.15601	−0.578007	−	0.816032i	$-0.696169\pi$
$41$	−12442.9	−1.15601	−0.578007	+	0.816032i	$0.696169\pi$
$42$	−1293.95	−0.113187
$43$	16091.9	1.32720	0.663598	−	0.748089i	$-0.269029\pi$
$43$	16091.9	1.32720	0.663598	+	0.748089i	$0.269029\pi$
$44$	2516.24	0.195939
$45$	0	0
$46$	3472.87	0.241988
$47$	−16446.0	−1.08597	−0.542983	−	0.839744i	$-0.682705\pi$
$47$	−16446.0	−1.08597	−0.542983	+	0.839744i	$0.682705\pi$
$48$	−665.105	−0.0416665
$49$	−14962.2	−0.890235
$50$	0	0
$51$	−1830.47	−0.0985457
$52$	24914.2	1.27773
$53$	11571.5	0.565849	0.282925	−	0.959142i	$-0.408695\pi$
$53$	11571.5	0.565849	0.282925	+	0.959142i	$0.408695\pi$
$54$	2440.20	0.113878
$55$	0	0
$56$	−7590.52	−0.323446
$57$	−14167.8	−0.577586
$58$	−10923.4	−0.426371
$59$	28773.2	1.07611	0.538056	−	0.842909i	$-0.319159\pi$
$59$	28773.2	1.07611	0.538056	+	0.842909i	$0.319159\pi$
$60$	0	0
$61$	39996.6	1.37625	0.688126	−	0.725591i	$-0.258433\pi$
$61$	39996.6	1.37625	0.688126	+	0.725591i	$0.258433\pi$
$62$	−15267.2	−0.504407
$63$	−3479.06	−0.110436
$64$	17392.9	0.530788
$65$	0	0
$66$	−3645.24	−0.103007
$67$	30728.9	0.836296	0.418148	−	0.908379i	$-0.362679\pi$
$67$	30728.9	0.836296	0.418148	+	0.908379i	$0.362679\pi$
$68$	−4229.49	−0.110921
$69$	9337.55	0.236108
$70$	0	0
$71$	19369.8	0.456015	0.228007	−	0.973659i	$-0.426779\pi$
$71$	19369.8	0.456015	0.228007	+	0.973659i	$0.426779\pi$
$72$	14314.6	0.325423
$73$	7371.45	0.161900	0.0809498	−	0.996718i	$-0.474205\pi$
$73$	7371.45	0.161900	0.0809498	+	0.996718i	$0.474205\pi$
$74$	38327.7	0.813642
$75$	0	0
$76$	−32736.2	−0.650121
$77$	5197.12	0.0998931
$78$	−36092.9	−0.671715
$79$	102237.	1.84307	0.921536	−	0.388293i	$-0.126935\pi$
$79$	102237.	1.84307	0.921536	+	0.388293i	$0.126935\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	41650.6	0.684048
$83$	36076.2	0.574811	0.287406	−	0.957809i	$-0.407207\pi$
$83$	36076.2	0.574811	0.287406	+	0.957809i	$0.407207\pi$
$84$	−8038.71	−0.124305
$85$	0	0
$86$	−53864.7	−0.785341
$87$	−29369.9	−0.416010
$88$	−21383.5	−0.294356
$89$	67261.9	0.900107	0.450053	−	0.893002i	$-0.351405\pi$
$89$	67261.9	0.900107	0.450053	+	0.893002i	$0.351405\pi$
$90$	0	0
$91$	51458.5	0.651409
$92$	21575.3	0.265759
$93$	−41049.2	−0.492150
$94$	55050.2	0.642598
$95$	0	0
$96$	53122.7	0.588304
$97$	106710.	1.15153	0.575766	−	0.817614i	$-0.304704\pi$
$97$	106710.	1.15153	0.575766	+	0.817614i	$0.304704\pi$
$98$	50083.3	0.526779
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−87328.7	−0.851831	−0.425915	−	0.904763i	$-0.640048\pi$
$101$	−87328.7	−0.851831	−0.425915	+	0.904763i	$0.640048\pi$
$102$	6127.19	0.0583124
$103$	89983.1	0.835733	0.417867	−	0.908508i	$-0.362778\pi$
$103$	89983.1	0.835733	0.417867	+	0.908508i	$0.362778\pi$
$104$	−211726.	−1.91951
$105$	0	0
$106$	−38733.7	−0.334830
$107$	107015.	0.903618	0.451809	−	0.892115i	$-0.350779\pi$
$107$	107015.	0.903618	0.451809	+	0.892115i	$0.350779\pi$
$108$	15159.8	0.125065
$109$	−5519.13	−0.0444943	−0.0222471	−	0.999753i	$-0.507082\pi$
$109$	−5519.13	−0.0444943	−0.0222471	+	0.999753i	$0.507082\pi$
$110$	0	0
$111$	103052.	0.793870
$112$	−3174.13	−0.0239100
$113$	76126.3	0.560840	0.280420	−	0.959877i	$-0.409526\pi$
$113$	76126.3	0.560840	0.280420	+	0.959877i	$0.409526\pi$
$114$	47424.4	0.341775
$115$	0	0
$116$	−67862.0	−0.468254
$117$	−97043.3	−0.655391
$118$	−96313.2	−0.636768
$119$	−8735.70	−0.0565497
$120$	0	0
$121$	14641.0	0.0909091
$122$	−133882.	−0.814370
$123$	111986.	0.667425
$124$	−94848.2	−0.553956
$125$	0	0
$126$	11645.6	0.0653483
$127$	−251824.	−1.38544	−0.692719	−	0.721207i	$-0.743588\pi$
$127$	−251824.	−1.38544	−0.692719	+	0.721207i	$0.743588\pi$
$128$	130661.	0.704890
$129$	−144827.	−0.766257
$130$	0	0
$131$	−160653.	−0.817921	−0.408961	−	0.912552i	$-0.634109\pi$
$131$	−160653.	−0.817921	−0.408961	+	0.912552i	$0.634109\pi$
$132$	−22646.2	−0.113125
$133$	−67614.2	−0.331443
$134$	−102860.	−0.494861
$135$	0	0
$136$	35943.1	0.166636
$137$	−374788.	−1.70602	−0.853011	−	0.521893i	$-0.825226\pi$
$137$	−374788.	−1.70602	−0.853011	+	0.521893i	$0.825226\pi$
$138$	−31255.8	−0.139712
$139$	−106272.	−0.466531	−0.233266	−	0.972413i	$-0.574941\pi$
$139$	−106272.	−0.466531	−0.233266	+	0.972413i	$0.574941\pi$
$140$	0	0
$141$	148014.	0.626983
$142$	−64837.0	−0.269837
$143$	144966.	0.592824
$144$	5985.95	0.0240562
$145$	0	0
$146$	−24674.7	−0.0958008
$147$	134660.	0.513977
$148$	238112.	0.893567
$149$	−257539.	−0.950335	−0.475168	−	0.879895i	$-0.657613\pi$
$149$	−257539.	−0.950335	−0.475168	+	0.879895i	$0.657613\pi$
$150$	0	0
$151$	−274368.	−0.979243	−0.489622	−	0.871935i	$-0.662865\pi$
$151$	−274368.	−0.979243	−0.489622	+	0.871935i	$0.662865\pi$
$152$	278199.	0.976667
$153$	16474.3	0.0568954
$154$	−17396.5	−0.0591097
$155$	0	0
$156$	−224228.	−0.737698
$157$	−459484.	−1.48772	−0.743860	−	0.668336i	$-0.767007\pi$
$157$	−459484.	−1.48772	−0.743860	+	0.668336i	$0.767007\pi$
$158$	−342222.	−1.09060
$159$	−104144.	−0.326693
$160$	0	0
$161$	44562.3	0.135489
$162$	−21961.8	−0.0657478
$163$	51600.5	0.152119	0.0760597	−	0.997103i	$-0.475766\pi$
$163$	51600.5	0.152119	0.0760597	+	0.997103i	$0.475766\pi$
$164$	258756.	0.751243
$165$	0	0
$166$	−120759.	−0.340133
$167$	522877.	1.45080	0.725401	−	0.688326i	$-0.241654\pi$
$167$	522877.	1.45080	0.725401	+	0.688326i	$0.241654\pi$
$168$	68314.7	0.186741
$169$	1.06407e6	2.86584
$170$	0	0
$171$	127511.	0.333469
$172$	−334636.	−0.862486
$173$	562670.	1.42935	0.714675	−	0.699457i	$-0.246575\pi$
$173$	562670.	1.42935	0.714675	+	0.699457i	$0.246575\pi$
$174$	98310.6	0.246165
$175$	0	0
$176$	−8941.97	−0.0217596
$177$	−258958.	−0.621294
$178$	−225148.	−0.532620
$179$	90980.4	0.212234	0.106117	−	0.994354i	$-0.466158\pi$
$179$	90980.4	0.212234	0.106117	+	0.994354i	$0.466158\pi$
$180$	0	0
$181$	−364545.	−0.827093	−0.413546	−	0.910483i	$-0.635710\pi$
$181$	−364545.	−0.827093	−0.413546	+	0.910483i	$0.635710\pi$
$182$	−172249.	−0.385458
$183$	−359969.	−0.794580
$184$	−183352.	−0.399246
$185$	0	0
$186$	137405.	0.291220
$187$	−24609.7	−0.0514638
$188$	342001.	0.705721
$189$	31311.6	0.0637603
$190$	0	0
$191$	−252683.	−0.501179	−0.250589	−	0.968093i	$-0.580624\pi$
$191$	−252683.	−0.501179	−0.250589	+	0.968093i	$0.580624\pi$
$192$	−156536.	−0.306451
$193$	−779943.	−1.50720	−0.753598	−	0.657336i	$-0.771683\pi$
$193$	−779943.	−1.50720	−0.753598	+	0.657336i	$0.771683\pi$
$194$	−357194.	−0.681396
$195$	0	0
$196$	311144.	0.578524
$197$	−332964.	−0.611268	−0.305634	−	0.952149i	$-0.598868\pi$
$197$	−332964.	−0.611268	−0.305634	+	0.952149i	$0.598868\pi$
$198$	32807.2	0.0594711
$199$	114756.	0.205419	0.102710	−	0.994711i	$-0.467249\pi$
$199$	114756.	0.205419	0.102710	+	0.994711i	$0.467249\pi$
$200$	0	0
$201$	−276560.	−0.482836
$202$	292318.	0.504054
$203$	−140164.	−0.238724
$204$	38065.4	0.0640405
$205$	0	0
$206$	−301203.	−0.494528
$207$	−84037.9	−0.136317
$208$	−88537.7	−0.141896
$209$	−190479.	−0.301634
$210$	0	0
$211$	−27949.2	−0.0432179	−0.0216090	−	0.999766i	$-0.506879\pi$
$211$	−27949.2	−0.0432179	−0.0216090	+	0.999766i	$0.506879\pi$
$212$	−240634.	−0.367720
$213$	−174328.	−0.263280
$214$	−358214.	−0.534698
$215$	0	0
$216$	−128831.	−0.187883
$217$	−195902.	−0.282416
$218$	18474.3	0.0263286
$219$	−66343.0	−0.0934727
$220$	0	0
$221$	−243669.	−0.335599
$222$	−344949.	−0.469757
$223$	−746134.	−1.00474	−0.502371	−	0.864652i	$-0.667539\pi$
$223$	−746134.	−1.00474	−0.502371	+	0.864652i	$0.667539\pi$
$224$	253521.	0.337594
$225$	0	0
$226$	−254820.	−0.331865
$227$	−1.43433e6	−1.84749	−0.923747	−	0.383004i	$-0.874890\pi$
$227$	−1.43433e6	−1.84749	−0.923747	+	0.383004i	$0.874890\pi$
$228$	294626.	0.375347
$229$	851409.	1.07288	0.536438	−	0.843940i	$-0.319769\pi$
$229$	851409.	1.07288	0.536438	+	0.843940i	$0.319769\pi$
$230$	0	0
$231$	−46774.0	−0.0576733
$232$	576705.	0.703451
$233$	−353222.	−0.426243	−0.213122	−	0.977026i	$-0.568363\pi$
$233$	−353222.	−0.426243	−0.213122	+	0.977026i	$0.568363\pi$
$234$	324836.	0.387815
$235$	0	0
$236$	−598349.	−0.699318
$237$	−920137.	−1.06410
$238$	29241.3	0.0334621
$239$	1.49058e6	1.68795	0.843974	−	0.536384i	$-0.180210\pi$
$239$	1.49058e6	1.68795	0.843974	+	0.536384i	$0.180210\pi$
$240$	0	0
$241$	1.08471e6	1.20301	0.601507	−	0.798868i	$-0.294567\pi$
$241$	1.08471e6	1.20301	0.601507	+	0.798868i	$0.294567\pi$
$242$	−49008.2	−0.0537936
$243$	−59049.0	−0.0641500
$244$	−831744.	−0.894366
$245$	0	0
$246$	−374855.	−0.394935
$247$	−1.88600e6	−1.96698
$248$	806040.	0.832199
$249$	−324685.	−0.331867
$250$	0	0
$251$	−1.12437e6	−1.12648	−0.563242	−	0.826292i	$-0.690446\pi$
$251$	−1.12437e6	−1.12648	−0.563242	+	0.826292i	$0.690446\pi$
$252$	72348.4	0.0717675
$253$	125538.	0.123303
$254$	842937.	0.819805
$255$	0	0
$256$	−993937.	−0.947893
$257$	−801413.	−0.756874	−0.378437	−	0.925627i	$-0.623538\pi$
$257$	−801413.	−0.756874	−0.378437	+	0.925627i	$0.623538\pi$
$258$	484783.	0.453417
$259$	491803.	0.455556
$260$	0	0
$261$	264329.	0.240183
$262$	537760.	0.483988
$263$	339788.	0.302914	0.151457	−	0.988464i	$-0.451604\pi$
$263$	339788.	0.302914	0.151457	+	0.988464i	$0.451604\pi$
$264$	192452.	0.169947
$265$	0	0
$266$	226327.	0.196125
$267$	−605357.	−0.519677
$268$	−639020.	−0.543472
$269$	−1.79086e6	−1.50897	−0.754485	−	0.656318i	$-0.772113\pi$
$269$	−1.79086e6	−1.50897	−0.754485	+	0.656318i	$0.772113\pi$
$270$	0	0
$271$	−990817.	−0.819540	−0.409770	−	0.912189i	$-0.634391\pi$
$271$	−990817.	−0.819540	−0.409770	+	0.912189i	$0.634391\pi$
$272$	15030.3	0.0123182
$273$	−463127.	−0.376091
$274$	1.25454e6	1.00950
$275$	0	0
$276$	−194178.	−0.153436
$277$	1.06956e6	0.837544	0.418772	−	0.908091i	$-0.362461\pi$
$277$	1.06956e6	0.837544	0.418772	+	0.908091i	$0.362461\pi$
$278$	355727.	0.276060
$279$	369443.	0.284143
$280$	0	0
$281$	53772.3	0.0406249	0.0203125	−	0.999794i	$-0.493534\pi$
$281$	53772.3	0.0406249	0.0203125	+	0.999794i	$0.493534\pi$
$282$	−495452.	−0.371004
$283$	−2.33534e6	−1.73334	−0.866671	−	0.498880i	$-0.833745\pi$
$283$	−2.33534e6	−1.73334	−0.866671	+	0.498880i	$0.833745\pi$
$284$	−402802.	−0.296344
$285$	0	0
$286$	−485248.	−0.350791
$287$	534441.	0.382997
$288$	−478104.	−0.339658
$289$	−1.37849e6	−0.970866
$290$	0	0
$291$	−960391.	−0.664838
$292$	−153292.	−0.105211
$293$	127807.	0.0869730	0.0434865	−	0.999054i	$-0.486153\pi$
$293$	127807.	0.0869730	0.0434865	+	0.999054i	$0.486153\pi$
$294$	−450750.	−0.304136
$295$	0	0
$296$	−2.02353e6	−1.34239
$297$	88209.0	0.0580259
$298$	862067.	0.562342
$299$	1.24300e6	0.804068
$300$	0	0
$301$	−691167.	−0.439711
$302$	918399.	0.579447
$303$	785958.	0.491805
$304$	116335.	0.0721980
$305$	0	0
$306$	−55144.7	−0.0336667
$307$	852929.	0.516496	0.258248	−	0.966079i	$-0.416855\pi$
$307$	852929.	0.516496	0.258248	+	0.966079i	$0.416855\pi$
$308$	−108076.	−0.0649161
$309$	−809848.	−0.482511
$310$	0	0
$311$	1.68106e6	0.985560	0.492780	−	0.870154i	$-0.335981\pi$
$311$	1.68106e6	0.985560	0.492780	+	0.870154i	$0.335981\pi$
$312$	1.90554e6	1.10823
$313$	3.10759e6	1.79293	0.896464	−	0.443117i	$-0.146127\pi$
$313$	3.10759e6	1.79293	0.896464	+	0.443117i	$0.146127\pi$
$314$	1.53804e6	0.880328
$315$	0	0
$316$	−2.12607e6	−1.19773
$317$	−815872.	−0.456009	−0.228005	−	0.973660i	$-0.573220\pi$
$317$	−815872.	−0.456009	−0.228005	+	0.973660i	$0.573220\pi$
$318$	348603.	0.193314
$319$	−394861.	−0.217254
$320$	0	0
$321$	−963134.	−0.521704
$322$	−149165.	−0.0801726
$323$	320171.	0.170756
$324$	−136439.	−0.0722062
$325$	0	0
$326$	−172724.	−0.0900136
$327$	49672.1	0.0256888
$328$	−2.19896e6	−1.12858
$329$	706379.	0.359789
$330$	0	0
$331$	−2.62012e6	−1.31447	−0.657235	−	0.753685i	$-0.728274\pi$
$331$	−2.62012e6	−1.31447	−0.657235	+	0.753685i	$0.728274\pi$
$332$	−750218.	−0.373544
$333$	−927469.	−0.458341
$334$	−1.75024e6	−0.858483
$335$	0	0
$336$	28567.2	0.0138045
$337$	296698.	0.142311	0.0711557	−	0.997465i	$-0.477331\pi$
$337$	296698.	0.142311	0.0711557	+	0.997465i	$0.477331\pi$
$338$	−3.56178e6	−1.69580
$339$	−685137.	−0.323801
$340$	0	0
$341$	−551884.	−0.257017
$342$	−426820.	−0.197324
$343$	1.36453e6	0.626250
$344$	2.84381e6	1.29570
$345$	0	0
$346$	−1.88344e6	−0.845789
$347$	769240.	0.342956	0.171478	−	0.985188i	$-0.445146\pi$
$347$	769240.	0.342956	0.171478	+	0.985188i	$0.445146\pi$
$348$	610758.	0.270346
$349$	−1.10268e6	−0.484603	−0.242302	−	0.970201i	$-0.577902\pi$
$349$	−1.10268e6	−0.484603	−0.242302	+	0.970201i	$0.577902\pi$
$350$	0	0
$351$	873389.	0.378390
$352$	714205.	0.307232
$353$	3.19975e6	1.36672	0.683359	−	0.730082i	$-0.260518\pi$
$353$	3.19975e6	1.36672	0.683359	+	0.730082i	$0.260518\pi$
$354$	866819.	0.367638
$355$	0	0
$356$	−1.39874e6	−0.584940
$357$	78621.3	0.0326490
$358$	−304541.	−0.125585
$359$	2.06929e6	0.847392	0.423696	−	0.905805i	$-0.360733\pi$
$359$	2.06929e6	0.847392	0.423696	+	0.905805i	$0.360733\pi$
$360$	0	0
$361$	2020.29	0.000815918 0
$362$	1.22025e6	0.489415
$363$	−131769.	−0.0524864
$364$	−1.07010e6	−0.423322
$365$	0	0
$366$	1.20494e6	0.470177
$367$	1.00025e6	0.387653	0.193827	−	0.981036i	$-0.437910\pi$
$367$	1.00025e6	0.387653	0.193827	+	0.981036i	$0.437910\pi$
$368$	−76672.2	−0.0295134
$369$	−1.00788e6	−0.385338
$370$	0	0
$371$	−497013.	−0.187470
$372$	853634.	0.319826
$373$	−1.88034e6	−0.699784	−0.349892	−	0.936790i	$-0.613782\pi$
$373$	−1.88034e6	−0.699784	−0.349892	+	0.936790i	$0.613782\pi$
$374$	82376.7	0.0304527
$375$	0	0
$376$	−2.90640e6	−1.06019
$377$	−3.90967e6	−1.41673
$378$	−104810.	−0.0377288
$379$	−549797.	−0.196610	−0.0983048	−	0.995156i	$-0.531342\pi$
$379$	−549797.	−0.196610	−0.0983048	+	0.995156i	$0.531342\pi$
$380$	0	0
$381$	2.26641e6	0.799883
$382$	845814.	0.296563
$383$	−106710.	−0.0371714	−0.0185857	−	0.999827i	$-0.505916\pi$
$383$	−106710.	−0.0371714	−0.0185857	+	0.999827i	$0.505916\pi$
$384$	−1.17595e6	−0.406968
$385$	0	0
$386$	2.61073e6	0.891852
$387$	1.30344e6	0.442399
$388$	−2.21908e6	−0.748330
$389$	4.40706e6	1.47664	0.738321	−	0.674449i	$-0.235619\pi$
$389$	4.40706e6	1.47664	0.738321	+	0.674449i	$0.235619\pi$
$390$	0	0
$391$	−211014.	−0.0698022
$392$	−2.64417e6	−0.869109
$393$	1.44588e6	0.472227
$394$	1.11454e6	0.361705
$395$	0	0
$396$	203816.	0.0653130
$397$	−4.08650e6	−1.30129	−0.650646	−	0.759381i	$-0.725502\pi$
$397$	−4.08650e6	−1.30129	−0.650646	+	0.759381i	$0.725502\pi$
$398$	−384125.	−0.121553
$399$	608528.	0.191359
$400$	0	0
$401$	−597759.	−0.185637	−0.0928187	−	0.995683i	$-0.529588\pi$
$401$	−597759.	−0.185637	−0.0928187	+	0.995683i	$0.529588\pi$
$402$	925738.	0.285708
$403$	−5.46440e6	−1.67602
$404$	1.81603e6	0.553567
$405$	0	0
$406$	469175.	0.141260
$407$	1.38548e6	0.414585
$408$	−323488.	−0.0962071
$409$	−4.60230e6	−1.36040	−0.680199	−	0.733027i	$-0.738107\pi$
$409$	−4.60230e6	−1.36040	−0.680199	+	0.733027i	$0.738107\pi$
$410$	0	0
$411$	3.37310e6	0.984972
$412$	−1.87123e6	−0.543106
$413$	−1.23585e6	−0.356525
$414$	281303.	0.0806627
$415$	0	0
$416$	7.07160e6	2.00348
$417$	956446.	0.269352
$418$	637595.	0.178486
$419$	−2.26948e6	−0.631525	−0.315763	−	0.948838i	$-0.602260\pi$
$419$	−2.26948e6	−0.631525	−0.315763	+	0.948838i	$0.602260\pi$
$420$	0	0
$421$	−3.20225e6	−0.880543	−0.440271	−	0.897865i	$-0.645118\pi$
$421$	−3.20225e6	−0.880543	−0.440271	+	0.897865i	$0.645118\pi$
$422$	93555.3	0.0255733
$423$	−1.33213e6	−0.361989
$424$	2.04496e6	0.552421
$425$	0	0
$426$	583533.	0.155791
$427$	−1.71791e6	−0.455964
$428$	−2.22542e6	−0.587221
$429$	−1.30469e6	−0.342267
$430$	0	0
$431$	−5.52813e6	−1.43346	−0.716729	−	0.697352i	$-0.754361\pi$
$431$	−5.52813e6	−1.43346	−0.716729	+	0.697352i	$0.754361\pi$
$432$	−53873.5	−0.0138888
$433$	424723.	0.108864	0.0544322	−	0.998517i	$-0.482665\pi$
$433$	424723.	0.108864	0.0544322	+	0.998517i	$0.482665\pi$
$434$	655749.	0.167114
$435$	0	0
$436$	114772.	0.0289149
$437$	−1.63325e6	−0.409117
$438$	222072.	0.0553106
$439$	378053.	0.0936248	0.0468124	−	0.998904i	$-0.485094\pi$
$439$	378053.	0.0936248	0.0468124	+	0.998904i	$0.485094\pi$
$440$	0	0
$441$	−1.21194e6	−0.296745
$442$	815642.	0.198584
$443$	1.87420e6	0.453739	0.226869	−	0.973925i	$-0.427151\pi$
$443$	1.87420e6	0.453739	0.226869	+	0.973925i	$0.427151\pi$
$444$	−2.14301e6	−0.515901
$445$	0	0
$446$	2.49756e6	0.594536
$447$	2.31785e6	0.548676
$448$	−747048.	−0.175854
$449$	−4.20432e6	−0.984192	−0.492096	−	0.870541i	$-0.663769\pi$
$449$	−4.20432e6	−0.984192	−0.492096	+	0.870541i	$0.663769\pi$
$450$	0	0
$451$	1.50560e6	0.348551
$452$	−1.58308e6	−0.364465
$453$	2.46931e6	0.565366
$454$	4.80116e6	1.09322
$455$	0	0
$456$	−2.50379e6	−0.563879
$457$	1.30433e6	0.292144	0.146072	−	0.989274i	$-0.453337\pi$
$457$	1.30433e6	0.292144	0.146072	+	0.989274i	$0.453337\pi$
$458$	−2.84995e6	−0.634853
$459$	−148268.	−0.0328486
$460$	0	0
$461$	−3.72345e6	−0.816005	−0.408002	−	0.912981i	$-0.633774\pi$
$461$	−3.72345e6	−0.816005	−0.408002	+	0.912981i	$0.633774\pi$
$462$	156568.	0.0341270
$463$	1.32143e6	0.286479	0.143240	−	0.989688i	$-0.454248\pi$
$463$	1.32143e6	0.286479	0.143240	+	0.989688i	$0.454248\pi$
$464$	241161.	0.0520011
$465$	0	0
$466$	1.18235e6	0.252221
$467$	7.53508e6	1.59881	0.799403	−	0.600795i	$-0.205149\pi$
$467$	7.53508e6	1.59881	0.799403	+	0.600795i	$0.205149\pi$
$468$	2.01805e6	0.425910
$469$	−1.31985e6	−0.277072
$470$	0	0
$471$	4.13535e6	0.858935
$472$	5.08490e6	1.05057
$473$	−1.94711e6	−0.400165
$474$	3.08000e6	0.629659
$475$	0	0
$476$	181662.	0.0367492
$477$	937293.	0.188616
$478$	−4.98945e6	−0.998810
$479$	485218.	0.0966269	0.0483134	−	0.998832i	$-0.484615\pi$
$479$	485218.	0.0966269	0.0483134	+	0.998832i	$0.484615\pi$
$480$	0	0
$481$	1.37181e7	2.70353
$482$	−3.63088e6	−0.711859
$483$	−401060.	−0.0782244
$484$	−304465.	−0.0590778
$485$	0	0
$486$	197656.	0.0379595
$487$	−4.00604e6	−0.765407	−0.382704	−	0.923871i	$-0.625007\pi$
$487$	−4.00604e6	−0.765407	−0.382704	+	0.923871i	$0.625007\pi$
$488$	7.06834e6	1.34359
$489$	−464404.	−0.0878262
$490$	0	0
$491$	8.49092e6	1.58946	0.794732	−	0.606960i	$-0.207611\pi$
$491$	8.49092e6	1.58946	0.794732	+	0.606960i	$0.207611\pi$
$492$	−2.32880e6	−0.433730
$493$	663713.	0.122988
$494$	6.31306e6	1.16392
$495$	0	0
$496$	337062.	0.0615185
$497$	−831959.	−0.151081
$498$	1.08683e6	0.196376
$499$	−2.93701e6	−0.528024	−0.264012	−	0.964519i	$-0.585046\pi$
$499$	−2.93701e6	−0.528024	−0.264012	+	0.964519i	$0.585046\pi$
$500$	0	0
$501$	−4.70589e6	−0.837621
$502$	3.76364e6	0.666574
$503$	110230.	0.0194259	0.00971296	−	0.999953i	$-0.496908\pi$
$503$	110230.	0.0194259	0.00971296	+	0.999953i	$0.496908\pi$
$504$	−614832.	−0.107815
$505$	0	0
$506$	−420217.	−0.0729622
$507$	−9.57660e6	−1.65459
$508$	5.23677e6	0.900336
$509$	−9.46630e6	−1.61952	−0.809759	−	0.586763i	$-0.800402\pi$
$509$	−9.46630e6	−1.61952	−0.809759	+	0.586763i	$0.800402\pi$
$510$	0	0
$511$	−316614.	−0.0536386
$512$	−854119.	−0.143994
$513$	−1.14759e6	−0.192529
$514$	2.68259e6	0.447865
$515$	0	0
$516$	3.01173e6	0.497957
$517$	1.98997e6	0.327431
$518$	−1.64623e6	−0.269566
$519$	−5.06403e6	−0.825235
$520$	0	0
$521$	−8.69981e6	−1.40416	−0.702078	−	0.712100i	$-0.747744\pi$
$521$	−8.69981e6	−1.40416	−0.702078	+	0.712100i	$0.747744\pi$
$522$	−884795.	−0.142124
$523$	−7.54038e6	−1.20542	−0.602711	−	0.797959i	$-0.705913\pi$
$523$	−7.54038e6	−1.20542	−0.602711	+	0.797959i	$0.705913\pi$
$524$	3.34085e6	0.531531
$525$	0	0
$526$	−1.13738e6	−0.179243
$527$	927647.	0.145498
$528$	80477.7	0.0125629
$529$	−5.35993e6	−0.832760
$530$	0	0
$531$	2.33063e6	0.358704
$532$	1.40606e6	0.215390
$533$	1.49074e7	2.27292
$534$	2.02633e6	0.307508
$535$	0	0
$536$	5.43052e6	0.816450
$537$	−818824.	−0.122534
$538$	5.99459e6	0.892902
$539$	1.81042e6	0.268416
$540$	0	0
$541$	406273.	0.0596795	0.0298398	−	0.999555i	$-0.490500\pi$
$541$	406273.	0.0596795	0.0298398	+	0.999555i	$0.490500\pi$
$542$	3.31659e6	0.484946
$543$	3.28090e6	0.477522
$544$	−1.20049e6	−0.173925
$545$	0	0
$546$	1.55024e6	0.222544
$547$	−9.14738e6	−1.30716	−0.653579	−	0.756858i	$-0.726733\pi$
$547$	−9.14738e6	−1.30716	−0.653579	+	0.756858i	$0.726733\pi$
$548$	7.79387e6	1.10867
$549$	3.23972e6	0.458751
$550$	0	0
$551$	5.13713e6	0.720844
$552$	1.65016e6	0.230505
$553$	−4.39124e6	−0.610625
$554$	−3.58019e6	−0.495600
$555$	0	0
$556$	2.20996e6	0.303178
$557$	4.83596e6	0.660457	0.330229	−	0.943901i	$-0.392874\pi$
$557$	4.83596e6	0.660457	0.330229	+	0.943901i	$0.392874\pi$
$558$	−1.23665e6	−0.168136
$559$	−1.92791e7	−2.60950
$560$	0	0
$561$	221487.	0.0297127
$562$	−179993.	−0.0240390
$563$	9.45620e6	1.25732	0.628660	−	0.777681i	$-0.283604\pi$
$563$	9.45620e6	1.25732	0.628660	+	0.777681i	$0.283604\pi$
$564$	−3.07801e6	−0.407448
$565$	0	0
$566$	7.81716e6	1.02567
$567$	−281804.	−0.0368120
$568$	3.42310e6	0.445193
$569$	9.69694e6	1.25561	0.627804	−	0.778372i	$-0.283954\pi$
$569$	9.69694e6	1.25561	0.627804	+	0.778372i	$0.283954\pi$
$570$	0	0
$571$	359699.	0.0461689	0.0230844	−	0.999734i	$-0.492651\pi$
$571$	359699.	0.0461689	0.0230844	+	0.999734i	$0.492651\pi$
$572$	−3.01462e6	−0.385250
$573$	2.27415e6	0.289356
$574$	−1.78895e6	−0.226631
$575$	0	0
$576$	1.40882e6	0.176929
$577$	1.40087e7	1.75170	0.875848	−	0.482588i	$-0.160303\pi$
$577$	1.40087e7	1.75170	0.875848	+	0.482588i	$0.160303\pi$
$578$	4.61426e6	0.574490
$579$	7.01949e6	0.870180
$580$	0	0
$581$	−1.54952e6	−0.190440
$582$	3.21474e6	0.393404
$583$	−1.40015e6	−0.170610
$584$	1.30271e6	0.158057
$585$	0	0
$586$	−427811.	−0.0514645
$587$	−1.26846e7	−1.51943	−0.759714	−	0.650257i	$-0.774661\pi$
$587$	−1.26846e7	−1.51943	−0.759714	+	0.650257i	$0.774661\pi$
$588$	−2.80030e6	−0.334011
$589$	7.17998e6	0.852776
$590$	0	0
$591$	2.99668e6	0.352916
$592$	−846179.	−0.0992334
$593$	1.37638e7	1.60732	0.803661	−	0.595088i	$-0.202883\pi$
$593$	1.37638e7	1.60732	0.803661	+	0.595088i	$0.202883\pi$
$594$	−295265.	−0.0343356
$595$	0	0
$596$	5.35562e6	0.617581
$597$	−1.03280e6	−0.118599
$598$	−4.16072e6	−0.475791
$599$	−1.24063e7	−1.41279	−0.706393	−	0.707820i	$-0.749679\pi$
$599$	−1.24063e7	−1.41279	−0.706393	+	0.707820i	$0.749679\pi$
$600$	0	0
$601$	−9.81045e6	−1.10791	−0.553953	−	0.832548i	$-0.686881\pi$
$601$	−9.81045e6	−1.10791	−0.553953	+	0.832548i	$0.686881\pi$
$602$	2.31356e6	0.260190
$603$	2.48904e6	0.278765
$604$	5.70558e6	0.636367
$605$	0	0
$606$	−2.63086e6	−0.291016
$607$	4.03465e6	0.444461	0.222231	−	0.974994i	$-0.428666\pi$
$607$	4.03465e6	0.444461	0.222231	+	0.974994i	$0.428666\pi$
$608$	−9.29178e6	−1.01939
$609$	1.26148e6	0.137827
$610$	0	0
$611$	1.97034e7	2.13520
$612$	−342588.	−0.0369738
$613$	−8.86622e6	−0.952988	−0.476494	−	0.879178i	$-0.658093\pi$
$613$	−8.86622e6	−0.952988	−0.476494	+	0.879178i	$0.658093\pi$
$614$	−2.85503e6	−0.305626
$615$	0	0
$616$	918453.	0.0975226
$617$	1.00022e7	1.05775	0.528876	−	0.848699i	$-0.322614\pi$
$617$	1.00022e7	1.05775	0.528876	+	0.848699i	$0.322614\pi$
$618$	2.71083e6	0.285516
$619$	−247292.	−0.0259408	−0.0129704	−	0.999916i	$-0.504129\pi$
$619$	−247292.	−0.0259408	−0.0129704	+	0.999916i	$0.504129\pi$
$620$	0	0
$621$	756341.	0.0787025
$622$	−5.62707e6	−0.583185
$623$	−2.88899e6	−0.298213
$624$	796839.	0.0819236
$625$	0	0
$626$	−1.04021e7	−1.06093
$627$	1.71431e6	0.174149
$628$	9.55514e6	0.966803
$629$	−2.32882e6	−0.234697
$630$	0	0
$631$	−4.01296e6	−0.401228	−0.200614	−	0.979670i	$-0.564294\pi$
$631$	−4.01296e6	−0.401228	−0.200614	+	0.979670i	$0.564294\pi$
$632$	1.80678e7	1.79933
$633$	251543.	0.0249519
$634$	2.73099e6	0.269834
$635$	0	0
$636$	2.16571e6	0.212303
$637$	1.79257e7	1.75036
$638$	1.32173e6	0.128556
$639$	1.56895e6	0.152005
$640$	0	0
$641$	1.59047e7	1.52891	0.764454	−	0.644678i	$-0.223009\pi$
$641$	1.59047e7	1.52891	0.764454	+	0.644678i	$0.223009\pi$
$642$	3.22393e6	0.308708
$643$	−1.59597e7	−1.52229	−0.761144	−	0.648583i	$-0.775362\pi$
$643$	−1.59597e7	−1.52229	−0.761144	+	0.648583i	$0.775362\pi$
$644$	−926690.	−0.0880480
$645$	0	0
$646$	−1.07172e6	−0.101041
$647$	1.02052e7	0.958432	0.479216	−	0.877697i	$-0.340921\pi$
$647$	1.02052e7	0.958432	0.479216	+	0.877697i	$0.340921\pi$
$648$	1.15948e6	0.108474
$649$	−3.48155e6	−0.324460
$650$	0	0
$651$	1.76312e6	0.163053
$652$	−1.07305e6	−0.0988557
$653$	7.35098e6	0.674625	0.337312	−	0.941393i	$-0.390482\pi$
$653$	7.35098e6	0.674625	0.337312	+	0.941393i	$0.390482\pi$
$654$	−166269.	−0.0152008
$655$	0	0
$656$	−919540.	−0.0834278
$657$	597087.	0.0539665
$658$	−2.36448e6	−0.212898
$659$	6.56355e6	0.588743	0.294371	−	0.955691i	$-0.404890\pi$
$659$	6.56355e6	0.588743	0.294371	+	0.955691i	$0.404890\pi$
$660$	0	0
$661$	1.71194e7	1.52400	0.762000	−	0.647576i	$-0.224217\pi$
$661$	1.71194e7	1.52400	0.762000	+	0.647576i	$0.224217\pi$
$662$	8.77040e6	0.777812
$663$	2.19302e6	0.193758
$664$	6.37551e6	0.561170
$665$	0	0
$666$	3.10454e6	0.271214
$667$	−3.38571e6	−0.294669
$668$	−1.08734e7	−0.942812
$669$	6.71521e6	0.580089
$670$	0	0
$671$	−4.83959e6	−0.414956
$672$	−2.28169e6	−0.194910
$673$	3.23866e6	0.275631	0.137815	−	0.990458i	$-0.455992\pi$
$673$	3.23866e6	0.275631	0.137815	+	0.990458i	$0.455992\pi$
$674$	−993145.	−0.0842099
$675$	0	0
$676$	−2.21277e7	−1.86238
$677$	1.78321e7	1.49531	0.747656	−	0.664086i	$-0.231179\pi$
$677$	1.78321e7	1.49531	0.747656	+	0.664086i	$0.231179\pi$
$678$	2.29338e6	0.191603
$679$	−4.58335e6	−0.381512
$680$	0	0
$681$	1.29089e7	1.06665
$682$	1.84734e6	0.152084
$683$	1.21915e7	1.00001	0.500006	−	0.866022i	$-0.333331\pi$
$683$	1.21915e7	1.00001	0.500006	+	0.866022i	$0.333331\pi$
$684$	−2.65163e6	−0.216707
$685$	0	0
$686$	−4.56753e6	−0.370571
$687$	−7.66269e6	−0.619426
$688$	1.18920e6	0.0957818
$689$	−1.38634e7	−1.11256
$690$	0	0
$691$	−7.48762e6	−0.596553	−0.298276	−	0.954480i	$-0.596412\pi$
$691$	−7.48762e6	−0.596553	−0.298276	+	0.954480i	$0.596412\pi$
$692$	−1.17009e7	−0.928871
$693$	420966.	0.0332977
$694$	−2.57490e6	−0.202937
$695$	0	0
$696$	−5.19034e6	−0.406138
$697$	−2.53072e6	−0.197316
$698$	3.69104e6	0.286754
$699$	3.17900e6	0.246092
$700$	0	0
$701$	−6.40179e6	−0.492047	−0.246023	−	0.969264i	$-0.579124\pi$
$701$	−6.40179e6	−0.492047	−0.246023	+	0.969264i	$0.579124\pi$
$702$	−2.92352e6	−0.223905
$703$	−1.80250e7	−1.37558
$704$	−2.10454e6	−0.160039
$705$	0	0
$706$	−1.07106e7	−0.808728
$707$	3.75089e6	0.282218
$708$	5.38514e6	0.403751
$709$	−411411.	−0.0307369	−0.0153685	−	0.999882i	$-0.504892\pi$
$709$	−411411.	−0.0307369	−0.0153685	+	0.999882i	$0.504892\pi$
$710$	0	0
$711$	8.28123e6	0.614357
$712$	1.18868e7	0.878746
$713$	−4.73208e6	−0.348601
$714$	−263171.	−0.0193194
$715$	0	0
$716$	−1.89197e6	−0.137922
$717$	−1.34152e7	−0.974538
$718$	−6.92658e6	−0.501427
$719$	−2.22633e7	−1.60608	−0.803039	−	0.595927i	$-0.796785\pi$
$719$	−2.22633e7	−1.60608	−0.803039	+	0.595927i	$0.796785\pi$
$720$	0	0
$721$	−3.86490e6	−0.276885
$722$	−6762.59	−0.000482803 0
$723$	−9.76238e6	−0.694560
$724$	7.58084e6	0.537491
$725$	0	0
$726$	441074.	0.0310578
$727$	−1.25469e7	−0.880439	−0.440219	−	0.897890i	$-0.645099\pi$
$727$	−1.25469e7	−0.880439	−0.440219	+	0.897890i	$0.645099\pi$
$728$	9.09393e6	0.635951
$729$	531441.	0.0370370
$730$	0	0
$731$	3.27286e6	0.226534
$732$	7.48570e6	0.516362
$733$	−1.43081e6	−0.0983608	−0.0491804	−	0.998790i	$-0.515661\pi$
$733$	−1.43081e6	−0.0983608	−0.0491804	+	0.998790i	$0.515661\pi$
$734$	−3.34816e6	−0.229386
$735$	0	0
$736$	6.12390e6	0.416710
$737$	−3.71820e6	−0.252153
$738$	3.37370e6	0.228016
$739$	−687576.	−0.0463137	−0.0231568	−	0.999732i	$-0.507372\pi$
$739$	−687576.	−0.0463137	−0.0231568	+	0.999732i	$0.507372\pi$
$740$	0	0
$741$	1.69740e7	1.13563
$742$	1.66366e6	0.110932
$743$	−2.22502e7	−1.47864	−0.739320	−	0.673354i	$-0.764853\pi$
$743$	−2.22502e7	−1.47864	−0.739320	+	0.673354i	$0.764853\pi$
$744$	−7.25436e6	−0.480470
$745$	0	0
$746$	6.29411e6	0.414083
$747$	2.92217e6	0.191604
$748$	511768.	0.0334441
$749$	−4.59644e6	−0.299376
$750$	0	0
$751$	−8.53330e6	−0.552099	−0.276050	−	0.961143i	$-0.589025\pi$
$751$	−8.53330e6	−0.552099	−0.276050	+	0.961143i	$0.589025\pi$
$752$	−1.21537e6	−0.0783725
$753$	1.01193e7	0.650376
$754$	1.30869e7	0.838320
$755$	0	0
$756$	−651136.	−0.0414350
$757$	8.67518e6	0.550223	0.275111	−	0.961412i	$-0.411285\pi$
$757$	8.67518e6	0.550223	0.275111	+	0.961412i	$0.411285\pi$
$758$	1.84035e6	0.116340
$759$	−1.12984e6	−0.0711891
$760$	0	0
$761$	2.25693e7	1.41272	0.706359	−	0.707853i	$-0.250336\pi$
$761$	2.25693e7	1.41272	0.706359	+	0.707853i	$0.250336\pi$
$762$	−7.58643e6	−0.473315
$763$	237054.	0.0147413
$764$	5.25464e6	0.325694
$765$	0	0
$766$	357194.	0.0219954
$767$	−3.44721e7	−2.11582
$768$	8.94544e6	0.547266
$769$	−8.97286e6	−0.547161	−0.273580	−	0.961849i	$-0.588208\pi$
$769$	−8.97286e6	−0.547161	−0.273580	+	0.961849i	$0.588208\pi$
$770$	0	0
$771$	7.21272e6	0.436982
$772$	1.62192e7	0.979460
$773$	−1.40901e6	−0.0848133	−0.0424067	−	0.999100i	$-0.513503\pi$
$773$	−1.40901e6	−0.0848133	−0.0424067	+	0.999100i	$0.513503\pi$
$774$	−4.36304e6	−0.261780
$775$	0	0
$776$	1.88582e7	1.12421
$777$	−4.42623e6	−0.263016
$778$	−1.47519e7	−0.873773
$779$	−1.95877e7	−1.15649
$780$	0	0
$781$	−2.34374e6	−0.137494
$782$	706333.	0.0413040
$783$	−2.37896e6	−0.138670
$784$	−1.10571e6	−0.0642469
$785$	0	0
$786$	−4.83984e6	−0.279431
$787$	2.56288e6	0.147500	0.0737500	−	0.997277i	$-0.476503\pi$
$787$	2.56288e6	0.147500	0.0737500	+	0.997277i	$0.476503\pi$
$788$	6.92411e6	0.397236
$789$	−3.05809e6	−0.174887
$790$	0	0
$791$	−3.26973e6	−0.185811
$792$	−1.73207e6	−0.0981187
$793$	−4.79185e7	−2.70595
$794$	1.36789e7	0.770014
$795$	0	0
$796$	−2.38639e6	−0.133493
$797$	1.67033e7	0.931444	0.465722	−	0.884931i	$-0.345795\pi$
$797$	1.67033e7	0.931444	0.465722	+	0.884931i	$0.345795\pi$
$798$	−2.03694e6	−0.113233
$799$	−3.34489e6	−0.185359
$800$	0	0
$801$	5.44821e6	0.300036
$802$	2.00090e6	0.109847
$803$	−891945.	−0.0488145
$804$	5.75118e6	0.313774
$805$	0	0
$806$	1.82911e7	0.991752
$807$	1.61177e7	0.871204
$808$	−1.54330e7	−0.831616
$809$	3.17360e6	0.170483	0.0852414	−	0.996360i	$-0.472834\pi$
$809$	3.17360e6	0.170483	0.0852414	+	0.996360i	$0.472834\pi$
$810$	0	0
$811$	2.08296e7	1.11206	0.556031	−	0.831161i	$-0.312323\pi$
$811$	2.08296e7	1.11206	0.556031	+	0.831161i	$0.312323\pi$
$812$	2.91476e6	0.155136
$813$	8.91735e6	0.473162
$814$	−4.63765e6	−0.245322
$815$	0	0
$816$	−135273.	−0.00711190
$817$	2.53319e7	1.32774
$818$	1.54054e7	0.804988
$819$	4.16814e6	0.217136
$820$	0	0
$821$	−3.57622e7	−1.85168	−0.925841	−	0.377913i	$-0.876642\pi$
$821$	−3.57622e7	−1.85168	−0.925841	+	0.377913i	$0.876642\pi$
$822$	−1.12909e7	−0.582838
$823$	−2.20862e7	−1.13664	−0.568319	−	0.822808i	$-0.692406\pi$
$823$	−2.20862e7	−1.13664	−0.568319	+	0.822808i	$0.692406\pi$
$824$	1.59021e7	0.815901
$825$	0	0
$826$	4.13679e6	0.210966
$827$	−1.95550e7	−0.994244	−0.497122	−	0.867681i	$-0.665610\pi$
$827$	−1.95550e7	−0.994244	−0.497122	+	0.867681i	$0.665610\pi$
$828$	1.74760e6	0.0885863
$829$	−2.29453e6	−0.115960	−0.0579800	−	0.998318i	$-0.518466\pi$
$829$	−2.29453e6	−0.115960	−0.0579800	+	0.998318i	$0.518466\pi$
$830$	0	0
$831$	−9.62608e6	−0.483556
$832$	−2.08378e7	−1.04362
$833$	−3.04310e6	−0.151951
$834$	−3.20154e6	−0.159384
$835$	0	0
$836$	3.96108e6	0.196019
$837$	−3.32498e6	−0.164050
$838$	7.59668e6	0.373692
$839$	2.17508e7	1.06677	0.533383	−	0.845874i	$-0.320920\pi$
$839$	2.17508e7	1.06677	0.533383	+	0.845874i	$0.320920\pi$
$840$	0	0
$841$	−9.86191e6	−0.480807
$842$	1.07190e7	0.521043
$843$	−483950.	−0.0234548
$844$	581215.	0.0280854
$845$	0	0
$846$	4.45907e6	0.214199
$847$	−628851.	−0.0301189
$848$	855142.	0.0408365
$849$	2.10181e7	1.00075
$850$	0	0
$851$	1.18797e7	0.562316
$852$	3.62522e6	0.171094
$853$	−2.12849e7	−1.00161	−0.500807	−	0.865559i	$-0.666963\pi$
$853$	−2.12849e7	−1.00161	−0.500807	+	0.865559i	$0.666963\pi$
$854$	5.75040e6	0.269807
$855$	0	0
$856$	1.89121e7	0.882174
$857$	−2.29708e7	−1.06838	−0.534188	−	0.845365i	$-0.679383\pi$
$857$	−2.29708e7	−1.06838	−0.534188	+	0.845365i	$0.679383\pi$
$858$	4.36724e6	0.202530
$859$	3.02509e7	1.39880	0.699401	−	0.714730i	$-0.253450\pi$
$859$	3.02509e7	1.39880	0.699401	+	0.714730i	$0.253450\pi$
$860$	0	0
$861$	−4.80997e6	−0.221123
$862$	1.85045e7	0.848219
$863$	−2.58158e7	−1.17994	−0.589968	−	0.807427i	$-0.700860\pi$
$863$	−2.58158e7	−1.17994	−0.589968	+	0.807427i	$0.700860\pi$
$864$	4.30294e6	0.196101
$865$	0	0
$866$	−1.42169e6	−0.0644183
$867$	1.24064e7	0.560530
$868$	4.07386e6	0.183530
$869$	−1.23707e7	−0.555707
$870$	0	0
$871$	−3.68152e7	−1.64430
$872$	−975360.	−0.0434384
$873$	8.64352e6	0.383844
$874$	5.46701e6	0.242087
$875$	0	0
$876$	1.37963e6	0.0607438
$877$	2.52567e7	1.10886	0.554431	−	0.832229i	$-0.312936\pi$
$877$	2.52567e7	1.10886	0.554431	+	0.832229i	$0.312936\pi$
$878$	−1.26547e6	−0.0554006
$879$	−1.15026e6	−0.0502139
$880$	0	0
$881$	−1.80305e7	−0.782650	−0.391325	−	0.920253i	$-0.627983\pi$
$881$	−1.80305e7	−0.782650	−0.391325	+	0.920253i	$0.627983\pi$
$882$	4.05675e6	0.175593
$883$	3.27242e7	1.41243	0.706215	−	0.707998i	$-0.250401\pi$
$883$	3.27242e7	1.41243	0.706215	+	0.707998i	$0.250401\pi$
$884$	5.06720e6	0.218091
$885$	0	0
$886$	−6.27356e6	−0.268491
$887$	−3.14768e7	−1.34333	−0.671663	−	0.740857i	$-0.734420\pi$
$887$	−3.14768e7	−1.34333	−0.671663	+	0.740857i	$0.734420\pi$
$888$	1.82117e7	0.775031
$889$	1.08162e7	0.459007
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.55162e7	0.652938
$893$	−2.58894e7	−1.08641
$894$	−7.75860e6	−0.324668
$895$	0	0
$896$	−5.61207e6	−0.233536
$897$	−1.11870e7	−0.464229
$898$	1.40732e7	0.582376
$899$	1.48841e7	0.614217
$900$	0	0
$901$	2.35348e6	0.0965826
$902$	−5.03972e6	−0.206248
$903$	6.22051e6	0.253867
$904$	1.34533e7	0.547530
$905$	0	0
$906$	−8.26559e6	−0.334544
$907$	2.18917e7	0.883613	0.441807	−	0.897110i	$-0.354338\pi$
$907$	2.18917e7	0.883613	0.441807	+	0.897110i	$0.354338\pi$
$908$	2.98273e7	1.20060
$909$	−7.07362e6	−0.283944
$910$	0	0
$911$	1.12623e7	0.449605	0.224802	−	0.974404i	$-0.427826\pi$
$911$	1.12623e7	0.449605	0.224802	+	0.974404i	$0.427826\pi$
$912$	−1.04701e6	−0.0416835
$913$	−4.36522e6	−0.173312
$914$	−4.36602e6	−0.172870
$915$	0	0
$916$	−1.77054e7	−0.697215
$917$	6.90028e6	0.270984
$918$	496303.	0.0194375
$919$	1.75699e7	0.686247	0.343123	−	0.939290i	$-0.388515\pi$
$919$	1.75699e7	0.686247	0.343123	+	0.939290i	$0.388515\pi$
$920$	0	0
$921$	−7.67636e6	−0.298199
$922$	1.24636e7	0.482854
$923$	−2.32063e7	−0.896604
$924$	972685.	0.0374794
$925$	0	0
$926$	−4.42328e6	−0.169518
$927$	7.28863e6	0.278578
$928$	−1.92618e7	−0.734221
$929$	−4.50980e7	−1.71442	−0.857212	−	0.514964i	$-0.827805\pi$
$929$	−4.50980e7	−1.71442	−0.857212	+	0.514964i	$0.827805\pi$
$930$	0	0
$931$	−2.35535e7	−0.890598
$932$	7.34538e6	0.276997
$933$	−1.51296e7	−0.569013
$934$	−2.52224e7	−0.946061
$935$	0	0
$936$	−1.71498e7	−0.639838
$937$	3.23937e7	1.20535	0.602674	−	0.797988i	$-0.294102\pi$
$937$	3.23937e7	1.20535	0.602674	+	0.797988i	$0.294102\pi$
$938$	4.41797e6	0.163952
$939$	−2.79683e7	−1.03515
$940$	0	0
$941$	−3.56704e7	−1.31321	−0.656605	−	0.754235i	$-0.728008\pi$
$941$	−3.56704e7	−1.31321	−0.656605	+	0.754235i	$0.728008\pi$
$942$	−1.38424e7	−0.508257
$943$	1.29096e7	0.472753
$944$	2.12635e6	0.0776614
$945$	0	0
$946$	6.51763e6	0.236789
$947$	−1.95070e7	−0.706831	−0.353415	−	0.935467i	$-0.614980\pi$
$947$	−1.95070e7	−0.706831	−0.353415	+	0.935467i	$0.614980\pi$
$948$	1.91346e7	0.691510
$949$	−8.83147e6	−0.318323
$950$	0	0
$951$	7.34285e6	0.263277
$952$	−1.54380e6	−0.0552077
$953$	−5.09834e7	−1.81843	−0.909216	−	0.416325i	$-0.863318\pi$
$953$	−5.09834e7	−1.81843	−0.909216	+	0.416325i	$0.863318\pi$
$954$	−3.13743e6	−0.111610
$955$	0	0
$956$	−3.09971e7	−1.09692
$957$	3.55375e6	0.125432
$958$	−1.62418e6	−0.0571770
$959$	1.60977e7	0.565219
$960$	0	0
$961$	−7.82624e6	−0.273366
$962$	−4.59191e7	−1.59976
$963$	8.66821e6	0.301206
$964$	−2.25569e7	−0.781785
$965$	0	0
$966$	1.34248e6	0.0462877
$967$	−1.54639e7	−0.531806	−0.265903	−	0.964000i	$-0.585670\pi$
$967$	−1.54639e7	−0.531806	−0.265903	+	0.964000i	$0.585670\pi$
$968$	2.58741e6	0.0887517
$969$	−2.88154e6	−0.0985859
$970$	0	0
$971$	4.40584e7	1.49962	0.749810	−	0.661654i	$-0.230145\pi$
$971$	4.40584e7	1.49962	0.749810	+	0.661654i	$0.230145\pi$
$972$	1.22795e6	0.0416883
$973$	4.56452e6	0.154566
$974$	1.34095e7	0.452914
$975$	0	0
$976$	2.95577e6	0.0993221
$977$	2.93714e6	0.0984439	0.0492219	−	0.998788i	$-0.484326\pi$
$977$	2.93714e6	0.0984439	0.0492219	+	0.998788i	$0.484326\pi$
$978$	1.55451e6	0.0519694
$979$	−8.13869e6	−0.271392
$980$	0	0
$981$	−447049.	−0.0148314
$982$	−2.84219e7	−0.940533
$983$	−1.09485e7	−0.361384	−0.180692	−	0.983540i	$-0.557834\pi$
$983$	−1.09485e7	−0.361384	−0.180692	+	0.983540i	$0.557834\pi$
$984$	1.97906e7	0.651586
$985$	0	0
$986$	−2.22166e6	−0.0727757
$987$	−6.35741e6	−0.207724
$988$	3.92201e7	1.27825
$989$	−1.66954e7	−0.542757
$990$	0	0
$991$	−3.00090e7	−0.970662	−0.485331	−	0.874330i	$-0.661301\pi$
$991$	−3.00090e7	−0.970662	−0.485331	+	0.874330i	$0.661301\pi$
$992$	−2.69215e7	−0.868601
$993$	2.35811e7	0.758910
$994$	2.78484e6	0.0893993
$995$	0	0
$996$	6.75196e6	0.215666
$997$	−4.86836e7	−1.55112	−0.775558	−	0.631276i	$-0.782531\pi$
$997$	−4.86836e7	−1.55112	−0.775558	+	0.631276i	$0.782531\pi$
$998$	9.83113e6	0.312448
$999$	8.34722e6	0.264623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.l.1.2		5
5.4	even	2		165.6.a.f.1.4	✓	5
15.14	odd	2		495.6.a.j.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.f.1.4	✓	5	5.4	even	2
495.6.a.j.1.2		5	15.14	odd	2
825.6.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-3.34733 q^{2} -9.00000 q^{3} -20.7954 q^{4} +30.1260 q^{6} -42.9514 q^{7} +176.724 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.34733 q^{2} -9.00000 q^{3} -20.7954 q^{4} +30.1260 q^{6} -42.9514 q^{7} +176.724 q^{8} +81.0000 q^{9} -121.000 q^{11} +187.158 q^{12} -1198.06 q^{13} +143.772 q^{14} +73.9006 q^{16} +203.386 q^{17} -271.134 q^{18} +1574.20 q^{19} +386.562 q^{21} +405.027 q^{22} -1037.51 q^{23} -1590.51 q^{24} +4010.32 q^{26} -729.000 q^{27} +893.191 q^{28} +3263.32 q^{29} +4561.02 q^{31} -5902.52 q^{32} +1089.00 q^{33} -680.799 q^{34} -1684.43 q^{36} -11450.2 q^{37} -5269.38 q^{38} +10782.6 q^{39} -12442.9 q^{41} -1293.95 q^{42} +16091.9 q^{43} +2516.24 q^{44} +3472.87 q^{46} -16446.0 q^{47} -665.105 q^{48} -14962.2 q^{49} -1830.47 q^{51} +24914.2 q^{52} +11571.5 q^{53} +2440.20 q^{54} -7590.52 q^{56} -14167.8 q^{57} -10923.4 q^{58} +28773.2 q^{59} +39996.6 q^{61} -15267.2 q^{62} -3479.06 q^{63} +17392.9 q^{64} -3645.24 q^{66} +30728.9 q^{67} -4229.49 q^{68} +9337.55 q^{69} +19369.8 q^{71} +14314.6 q^{72} +7371.45 q^{73} +38327.7 q^{74} -32736.2 q^{76} +5197.12 q^{77} -36092.9 q^{78} +102237. q^{79} +6561.00 q^{81} +41650.6 q^{82} +36076.2 q^{83} -8038.71 q^{84} -53864.7 q^{86} -29369.9 q^{87} -21383.5 q^{88} +67261.9 q^{89} +51458.5 q^{91} +21575.3 q^{92} -41049.2 q^{93} +55050.2 q^{94} +53122.7 q^{96} +106710. q^{97} +50083.3 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} - 45 q^{3} + 82 q^{4} - 18 q^{6} - 184 q^{7} - 24 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + 2 * q^2 - 45 * q^3 + 82 * q^4 - 18 * q^6 - 184 * q^7 - 24 * q^8 + 405 * q^9	\( 5 q + 2 q^{2} - 45 q^{3} + 82 q^{4} - 18 q^{6} - 184 q^{7} - 24 q^{8} + 405 q^{9} - 605 q^{11} - 738 q^{12} - 1082 q^{13} + 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 162 q^{18} + 1632 q^{19} + 1656 q^{21} - 242 q^{22} - 1212 q^{23} + 216 q^{24} + 5600 q^{26} - 3645 q^{27} - 16508 q^{28} + 82 q^{29} + 12120 q^{31} + 4864 q^{32} + 5445 q^{33} - 4524 q^{34} + 6642 q^{36} + 6530 q^{37} + 15132 q^{38} + 9738 q^{39} + 6782 q^{41} - 3888 q^{42} - 46184 q^{43} - 9922 q^{44} + 12048 q^{46} + 11692 q^{47} - 42930 q^{48} + 34445 q^{49} + 19566 q^{51} - 50020 q^{52} - 10314 q^{53} - 1458 q^{54} + 54928 q^{56} - 14688 q^{57} - 75048 q^{58} + 92892 q^{59} + 106 q^{61} - 97160 q^{62} - 14904 q^{63} + 44550 q^{64} + 2178 q^{66} - 100476 q^{67} - 119928 q^{68} + 10908 q^{69} - 13772 q^{71} - 1944 q^{72} - 94154 q^{73} - 47924 q^{74} - 51524 q^{76} + 22264 q^{77} - 50400 q^{78} + 178744 q^{79} + 32805 q^{81} + 299848 q^{82} + 100116 q^{83} + 148572 q^{84} - 167704 q^{86} - 738 q^{87} + 2904 q^{88} + 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 109080 q^{93} - 310288 q^{94} - 43776 q^{96} - 100682 q^{97} + 16434 q^{98} - 49005 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - 45 * q^3 + 82 * q^4 - 18 * q^6 - 184 * q^7 - 24 * q^8 + 405 * q^9 - 605 * q^11 - 738 * q^12 - 1082 * q^13 + 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 162 * q^18 + 1632 * q^19 + 1656 * q^21 - 242 * q^22 - 1212 * q^23 + 216 * q^24 + 5600 * q^26 - 3645 * q^27 - 16508 * q^28 + 82 * q^29 + 12120 * q^31 + 4864 * q^32 + 5445 * q^33 - 4524 * q^34 + 6642 * q^36 + 6530 * q^37 + 15132 * q^38 + 9738 * q^39 + 6782 * q^41 - 3888 * q^42 - 46184 * q^43 - 9922 * q^44 + 12048 * q^46 + 11692 * q^47 - 42930 * q^48 + 34445 * q^49 + 19566 * q^51 - 50020 * q^52 - 10314 * q^53 - 1458 * q^54 + 54928 * q^56 - 14688 * q^57 - 75048 * q^58 + 92892 * q^59 + 106 * q^61 - 97160 * q^62 - 14904 * q^63 + 44550 * q^64 + 2178 * q^66 - 100476 * q^67 - 119928 * q^68 + 10908 * q^69 - 13772 * q^71 - 1944 * q^72 - 94154 * q^73 - 47924 * q^74 - 51524 * q^76 + 22264 * q^77 - 50400 * q^78 + 178744 * q^79 + 32805 * q^81 + 299848 * q^82 + 100116 * q^83 + 148572 * q^84 - 167704 * q^86 - 738 * q^87 + 2904 * q^88 + 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 109080 * q^93 - 310288 * q^94 - 43776 * q^96 - 100682 * q^97 + 16434 * q^98 - 49005 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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