LMFDB - Embedded newform 725.6.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,6,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	725.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:	$116.278269364$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.17343$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.863638 q^{2} -7.07413 q^{3} -31.2541 q^{4} -6.10949 q^{6} +36.7447 q^{7} -54.6287 q^{8} -192.957 q^{9} +O(q^{10})$	\(q+0.863638 q^{2} -7.07413 q^{3} -31.2541 q^{4} -6.10949 q^{6} +36.7447 q^{7} -54.6287 q^{8} -192.957 q^{9} -302.283 q^{11} +221.096 q^{12} +373.472 q^{13} +31.7341 q^{14} +952.953 q^{16} -280.365 q^{17} -166.645 q^{18} +1371.41 q^{19} -259.937 q^{21} -261.063 q^{22} -1861.10 q^{23} +386.450 q^{24} +322.545 q^{26} +3084.01 q^{27} -1148.42 q^{28} -841.000 q^{29} +1472.03 q^{31} +2571.12 q^{32} +2138.39 q^{33} -242.134 q^{34} +6030.69 q^{36} +11730.4 q^{37} +1184.40 q^{38} -2641.99 q^{39} -2177.39 q^{41} -224.491 q^{42} +9679.03 q^{43} +9447.58 q^{44} -1607.32 q^{46} +15909.1 q^{47} -6741.31 q^{48} -15456.8 q^{49} +1983.34 q^{51} -11672.6 q^{52} -24359.3 q^{53} +2663.47 q^{54} -2007.32 q^{56} -9701.53 q^{57} -726.320 q^{58} +36304.7 q^{59} -22316.1 q^{61} +1271.30 q^{62} -7090.14 q^{63} -28274.0 q^{64} +1846.79 q^{66} +54808.6 q^{67} +8762.58 q^{68} +13165.7 q^{69} +27790.4 q^{71} +10541.0 q^{72} -31685.5 q^{73} +10130.8 q^{74} -42862.2 q^{76} -11107.3 q^{77} -2281.73 q^{78} -55328.4 q^{79} +25071.8 q^{81} -1880.47 q^{82} +46888.8 q^{83} +8124.10 q^{84} +8359.18 q^{86} +5949.34 q^{87} +16513.3 q^{88} +2564.30 q^{89} +13723.1 q^{91} +58167.2 q^{92} -10413.3 q^{93} +13739.7 q^{94} -18188.5 q^{96} -34940.3 q^{97} -13349.1 q^{98} +58327.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9}+O(q^{10}) $ 4 * q + 28 * q^3 + 10 * q^4 - 194 * q^6 + 208 * q^7 + 504 * q^8 - 280 * q^9	\( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9} - 124 q^{11} - 20 q^{12} + 460 q^{13} + 768 q^{14} - 414 q^{16} - 184 q^{17} - 3208 q^{18} - 2392 q^{19} + 992 q^{21} - 5538 q^{22} + 1192 q^{23} + 6786 q^{24} + 4724 q^{26} - 2468 q^{27} - 44 q^{28} - 3364 q^{29} - 19212 q^{31} - 6552 q^{32} + 10580 q^{33} - 7612 q^{34} - 7468 q^{36} + 10928 q^{37} + 456 q^{38} - 8732 q^{39} - 1120 q^{41} - 1844 q^{42} + 21420 q^{43} - 1932 q^{44} - 7588 q^{46} - 23772 q^{47} - 33060 q^{48} + 10452 q^{49} + 12744 q^{51} + 29062 q^{52} - 8860 q^{53} + 35410 q^{54} + 34304 q^{56} - 48944 q^{57} - 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 27488 q^{63} - 20734 q^{64} - 47744 q^{66} + 7840 q^{67} - 20724 q^{68} + 58792 q^{69} - 48744 q^{71} - 8088 q^{72} + 74992 q^{73} - 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 2982 q^{78} - 106076 q^{79} - 59692 q^{81} + 234132 q^{82} - 62888 q^{83} - 59832 q^{84} - 216014 q^{86} - 23548 q^{87} + 39426 q^{88} + 107568 q^{89} - 268896 q^{91} + 26268 q^{92} - 221460 q^{93} + 30542 q^{94} - 78606 q^{96} + 49520 q^{97} - 242304 q^{98} + 166720 q^{99}+O(q^{100}) \) 4 * q + 28 * q^3 + 10 * q^4 - 194 * q^6 + 208 * q^7 + 504 * q^8 - 280 * q^9 - 124 * q^11 - 20 * q^12 + 460 * q^13 + 768 * q^14 - 414 * q^16 - 184 * q^17 - 3208 * q^18 - 2392 * q^19 + 992 * q^21 - 5538 * q^22 + 1192 * q^23 + 6786 * q^24 + 4724 * q^26 - 2468 * q^27 - 44 * q^28 - 3364 * q^29 - 19212 * q^31 - 6552 * q^32 + 10580 * q^33 - 7612 * q^34 - 7468 * q^36 + 10928 * q^37 + 456 * q^38 - 8732 * q^39 - 1120 * q^41 - 1844 * q^42 + 21420 * q^43 - 1932 * q^44 - 7588 * q^46 - 23772 * q^47 - 33060 * q^48 + 10452 * q^49 + 12744 * q^51 + 29062 * q^52 - 8860 * q^53 + 35410 * q^54 + 34304 * q^56 - 48944 * q^57 - 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 27488 * q^63 - 20734 * q^64 - 47744 * q^66 + 7840 * q^67 - 20724 * q^68 + 58792 * q^69 - 48744 * q^71 - 8088 * q^72 + 74992 * q^73 - 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 2982 * q^78 - 106076 * q^79 - 59692 * q^81 + 234132 * q^82 - 62888 * q^83 - 59832 * q^84 - 216014 * q^86 - 23548 * q^87 + 39426 * q^88 + 107568 * q^89 - 268896 * q^91 + 26268 * q^92 - 221460 * q^93 + 30542 * q^94 - 78606 * q^96 + 49520 * q^97 - 242304 * q^98 + 166720 * 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 $

$2$	0.863638	0.152671	0.0763356	−	0.997082i	$-0.475678\pi$
$2$	0.863638	0.152671	0.0763356	+	0.997082i	$0.475678\pi$
$3$	−7.07413	−0.453806	−0.226903	−	0.973917i	$-0.572860\pi$
$3$	−7.07413	−0.453806	−0.226903	+	0.973917i	$0.572860\pi$
$4$	−31.2541	−0.976692
$5$	0	0
$5$	0	0
$6$	−6.10949	−0.0692830
$7$	36.7447	0.283433	0.141716	−	0.989907i	$-0.454738\pi$
$7$	36.7447	0.283433	0.141716	+	0.989907i	$0.454738\pi$
$8$	−54.6287	−0.301784
$9$	−192.957	−0.794060
$10$	0	0
$11$	−302.283	−0.753237	−0.376619	−	0.926368i	$-0.622913\pi$
$11$	−302.283	−0.753237	−0.376619	+	0.926368i	$0.622913\pi$
$12$	221.096	0.443228
$13$	373.472	0.612915	0.306457	−	0.951884i	$-0.400856\pi$
$13$	373.472	0.612915	0.306457	+	0.951884i	$0.400856\pi$
$14$	31.7341	0.0432720
$15$	0	0
$16$	952.953	0.930618
$17$	−280.365	−0.235289	−0.117645	−	0.993056i	$-0.537534\pi$
$17$	−280.365	−0.235289	−0.117645	+	0.993056i	$0.537534\pi$
$18$	−166.645	−0.121230
$19$	1371.41	0.871532	0.435766	−	0.900060i	$-0.356478\pi$
$19$	1371.41	0.871532	0.435766	+	0.900060i	$0.356478\pi$
$20$	0	0
$21$	−259.937	−0.128623
$22$	−261.063	−0.114998
$23$	−1861.10	−0.733586	−0.366793	−	0.930303i	$-0.619544\pi$
$23$	−1861.10	−0.733586	−0.366793	+	0.930303i	$0.619544\pi$
$24$	386.450	0.136951
$25$	0	0
$26$	322.545	0.0935744
$27$	3084.01	0.814155
$28$	−1148.42	−0.276826
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	1472.03	0.275114	0.137557	−	0.990494i	$-0.456075\pi$
$31$	1472.03	0.275114	0.137557	+	0.990494i	$0.456075\pi$
$32$	2571.12	0.443862
$33$	2138.39	0.341823
$34$	−242.134	−0.0359219
$35$	0	0
$36$	6030.69	0.775552
$37$	11730.4	1.40867	0.704335	−	0.709868i	$-0.251245\pi$
$37$	11730.4	1.40867	0.704335	+	0.709868i	$0.251245\pi$
$38$	1184.40	0.133058
$39$	−2641.99	−0.278144
$40$	0	0
$41$	−2177.39	−0.202291	−0.101145	−	0.994872i	$-0.532251\pi$
$41$	−2177.39	−0.202291	−0.101145	+	0.994872i	$0.532251\pi$
$42$	−224.491	−0.0196371
$43$	9679.03	0.798290	0.399145	−	0.916888i	$-0.369307\pi$
$43$	9679.03	0.798290	0.399145	+	0.916888i	$0.369307\pi$
$44$	9447.58	0.735680
$45$	0	0
$46$	−1607.32	−0.111997
$47$	15909.1	1.05051	0.525255	−	0.850945i	$-0.323970\pi$
$47$	15909.1	1.05051	0.525255	+	0.850945i	$0.323970\pi$
$48$	−6741.31	−0.422320
$49$	−15456.8	−0.919666
$50$	0	0
$51$	1983.34	0.106776
$52$	−11672.6	−0.598629
$53$	−24359.3	−1.19117	−0.595586	−	0.803291i	$-0.703080\pi$
$53$	−24359.3	−1.19117	−0.595586	+	0.803291i	$0.703080\pi$
$54$	2663.47	0.124298
$55$	0	0
$56$	−2007.32	−0.0855353
$57$	−9701.53	−0.395506
$58$	−726.320	−0.0283503
$59$	36304.7	1.35779	0.678894	−	0.734236i	$-0.262459\pi$
$59$	36304.7	1.35779	0.678894	+	0.734236i	$0.262459\pi$
$60$	0	0
$61$	−22316.1	−0.767880	−0.383940	−	0.923358i	$-0.625433\pi$
$61$	−22316.1	−0.767880	−0.383940	+	0.923358i	$0.625433\pi$
$62$	1271.30	0.0420019
$63$	−7090.14	−0.225063
$64$	−28274.0	−0.862853
$65$	0	0
$66$	1846.79	0.0521865
$67$	54808.6	1.49163	0.745817	−	0.666151i	$-0.232060\pi$
$67$	54808.6	1.49163	0.745817	+	0.666151i	$0.232060\pi$
$68$	8762.58	0.229805
$69$	13165.7	0.332905
$70$	0	0
$71$	27790.4	0.654258	0.327129	−	0.944980i	$-0.393919\pi$
$71$	27790.4	0.654258	0.327129	+	0.944980i	$0.393919\pi$
$72$	10541.0	0.239635
$73$	−31685.5	−0.695910	−0.347955	−	0.937511i	$-0.613124\pi$
$73$	−31685.5	−0.695910	−0.347955	+	0.937511i	$0.613124\pi$
$74$	10130.8	0.215063
$75$	0	0
$76$	−42862.2	−0.851218
$77$	−11107.3	−0.213492
$78$	−2281.73	−0.0424646
$79$	−55328.4	−0.997426	−0.498713	−	0.866767i	$-0.666194\pi$
$79$	−55328.4	−0.997426	−0.498713	+	0.866767i	$0.666194\pi$
$80$	0	0
$81$	25071.8	0.424592
$82$	−1880.47	−0.0308839
$83$	46888.8	0.747092	0.373546	−	0.927612i	$-0.378142\pi$
$83$	46888.8	0.747092	0.373546	+	0.927612i	$0.378142\pi$
$84$	8124.10	0.125625
$85$	0	0
$86$	8359.18	0.121876
$87$	5949.34	0.0842696
$88$	16513.3	0.227315
$89$	2564.30	0.0343157	0.0171579	−	0.999853i	$-0.494538\pi$
$89$	2564.30	0.0343157	0.0171579	+	0.999853i	$0.494538\pi$
$90$	0	0
$91$	13723.1	0.173720
$92$	58167.2	0.716487
$93$	−10413.3	−0.124848
$94$	13739.7	0.160382
$95$	0	0
$96$	−18188.5	−0.201427
$97$	−34940.3	−0.377048	−0.188524	−	0.982069i	$-0.560370\pi$
$97$	−34940.3	−0.377048	−0.188524	+	0.982069i	$0.560370\pi$
$98$	−13349.1	−0.140406
$99$	58327.5	0.598116
$100$	0	0
$101$	170224.	1.66042	0.830210	−	0.557451i	$-0.188221\pi$
$101$	170224.	1.66042	0.830210	+	0.557451i	$0.188221\pi$
$102$	1712.89	0.0163015
$103$	−83962.9	−0.779820	−0.389910	−	0.920853i	$-0.627494\pi$
$103$	−83962.9	−0.779820	−0.389910	+	0.920853i	$0.627494\pi$
$104$	−20402.3	−0.184968
$105$	0	0
$106$	−21037.6	−0.181858
$107$	−16842.4	−0.142215	−0.0711074	−	0.997469i	$-0.522653\pi$
$107$	−16842.4	−0.142215	−0.0711074	+	0.997469i	$0.522653\pi$
$108$	−96388.2	−0.795178
$109$	−3855.63	−0.0310834	−0.0155417	−	0.999879i	$-0.504947\pi$
$109$	−3855.63	−0.0310834	−0.0155417	+	0.999879i	$0.504947\pi$
$110$	0	0
$111$	−82982.5	−0.639262
$112$	35016.0	0.263767
$113$	−107319.	−0.790644	−0.395322	−	0.918543i	$-0.629367\pi$
$113$	−107319.	−0.790644	−0.395322	+	0.918543i	$0.629367\pi$
$114$	−8378.61	−0.0603823
$115$	0	0
$116$	26284.7	0.181367
$117$	−72064.0	−0.486691
$118$	31354.1	0.207295
$119$	−10301.9	−0.0666886
$120$	0	0
$121$	−69676.1	−0.432634
$122$	−19273.0	−0.117233
$123$	15403.1	0.0918006
$124$	−46007.0	−0.268701
$125$	0	0
$126$	−6123.31	−0.0343606
$127$	14823.9	0.0815554	0.0407777	−	0.999168i	$-0.487016\pi$
$127$	14823.9	0.0815554	0.0407777	+	0.999168i	$0.487016\pi$
$128$	−106694.	−0.575595
$129$	−68470.7	−0.362269
$130$	0	0
$131$	−313171.	−1.59442	−0.797211	−	0.603701i	$-0.793692\pi$
$131$	−313171.	−1.59442	−0.797211	+	0.603701i	$0.793692\pi$
$132$	−66833.4	−0.333856
$133$	50392.1	0.247020
$134$	47334.8	0.227729
$135$	0	0
$136$	15316.0	0.0710065
$137$	254312.	1.15762	0.578810	−	0.815463i	$-0.303517\pi$
$137$	254312.	1.15762	0.578810	+	0.815463i	$0.303517\pi$
$138$	11370.4	0.0508250
$139$	−390931.	−1.71618	−0.858091	−	0.513498i	$-0.828349\pi$
$139$	−390931.	−1.71618	−0.858091	+	0.513498i	$0.828349\pi$
$140$	0	0
$141$	−112543.	−0.476727
$142$	24000.9	0.0998863
$143$	−112894.	−0.461670
$144$	−183879.	−0.738967
$145$	0	0
$146$	−27364.8	−0.106245
$147$	109344.	0.417350
$148$	−366624.	−1.37584
$149$	72345.7	0.266961	0.133480	−	0.991051i	$-0.457385\pi$
$149$	72345.7	0.266961	0.133480	+	0.991051i	$0.457385\pi$
$150$	0	0
$151$	−227419.	−0.811679	−0.405840	−	0.913944i	$-0.633021\pi$
$151$	−227419.	−0.811679	−0.405840	+	0.913944i	$0.633021\pi$
$152$	−74918.3	−0.263014
$153$	54098.4	0.186834
$154$	−9592.68	−0.0325940
$155$	0	0
$156$	82573.2	0.271661
$157$	−539281.	−1.74609	−0.873044	−	0.487641i	$-0.837857\pi$
$157$	−539281.	−1.74609	−0.873044	+	0.487641i	$0.837857\pi$
$158$	−47783.7	−0.152278
$159$	172321.	0.540561
$160$	0	0
$161$	−68385.7	−0.207922
$162$	21652.9	0.0648230
$163$	−87996.5	−0.259416	−0.129708	−	0.991552i	$-0.541404\pi$
$163$	−87996.5	−0.259416	−0.129708	+	0.991552i	$0.541404\pi$
$164$	68052.3	0.197576
$165$	0	0
$166$	40495.0	0.114059
$167$	−199897.	−0.554644	−0.277322	−	0.960777i	$-0.589447\pi$
$167$	−199897.	−0.554644	−0.277322	+	0.960777i	$0.589447\pi$
$168$	14200.0	0.0388164
$169$	−231811.	−0.624335
$170$	0	0
$171$	−264623.	−0.692049
$172$	−302510.	−0.779683
$173$	313797.	0.797139	0.398569	−	0.917138i	$-0.369507\pi$
$173$	313797.	0.797139	0.398569	+	0.917138i	$0.369507\pi$
$174$	5138.08	0.0128655
$175$	0	0
$176$	−288061.	−0.700976
$177$	−256824.	−0.616172
$178$	2214.63	0.00523902
$179$	−59814.7	−0.139533	−0.0697663	−	0.997563i	$-0.522225\pi$
$179$	−59814.7	−0.139533	−0.0697663	+	0.997563i	$0.522225\pi$
$180$	0	0
$181$	402231.	0.912597	0.456298	−	0.889827i	$-0.349175\pi$
$181$	402231.	0.912597	0.456298	+	0.889827i	$0.349175\pi$
$182$	11851.8	0.0265220
$183$	157867.	0.348468
$184$	101670.	0.221384
$185$	0	0
$186$	−8993.35	−0.0190607
$187$	84749.6	0.177229
$188$	−497224.	−1.02602
$189$	113321.	0.230758
$190$	0	0
$191$	−903529.	−1.79209	−0.896043	−	0.443967i	$-0.853571\pi$
$191$	−903529.	−1.79209	−0.896043	+	0.443967i	$0.853571\pi$
$192$	200014.	0.391568
$193$	−1.02870e6	−1.98791	−0.993956	−	0.109780i	$-0.964985\pi$
$193$	−1.02870e6	−1.98791	−0.993956	+	0.109780i	$0.964985\pi$
$194$	−30175.7	−0.0575644
$195$	0	0
$196$	483090.	0.898230
$197$	866099.	1.59002	0.795009	−	0.606598i	$-0.207466\pi$
$197$	866099.	1.59002	0.795009	+	0.606598i	$0.207466\pi$
$198$	50373.8	0.0913150
$199$	285836.	0.511663	0.255832	−	0.966721i	$-0.417651\pi$
$199$	285836.	0.511663	0.255832	+	0.966721i	$0.417651\pi$
$200$	0	0
$201$	−387723.	−0.676911
$202$	147012.	0.253498
$203$	−30902.3	−0.0526321
$204$	−61987.6	−0.104287
$205$	0	0
$206$	−72513.6	−0.119056
$207$	359112.	0.582512
$208$	355901.	0.570389
$209$	−414554.	−0.656470
$210$	0	0
$211$	−918930.	−1.42094	−0.710471	−	0.703727i	$-0.751518\pi$
$211$	−918930.	−1.42094	−0.710471	+	0.703727i	$0.751518\pi$
$212$	761328.	1.16341
$213$	−196593.	−0.296906
$214$	−14545.8	−0.0217121
$215$	0	0
$216$	−168476.	−0.245699
$217$	54089.3	0.0779762
$218$	−3329.87	−0.00474554
$219$	224147.	0.315808
$220$	0	0
$221$	−104709.	−0.144212
$222$	−71666.8	−0.0975969
$223$	−486134.	−0.654626	−0.327313	−	0.944916i	$-0.606143\pi$
$223$	−486134.	−0.654626	−0.327313	+	0.944916i	$0.606143\pi$
$224$	94475.2	0.125805
$225$	0	0
$226$	−92684.9	−0.120708
$227$	557639.	0.718271	0.359136	−	0.933285i	$-0.383072\pi$
$227$	557639.	0.718271	0.359136	+	0.933285i	$0.383072\pi$
$228$	303213.	0.386287
$229$	440289.	0.554816	0.277408	−	0.960752i	$-0.410525\pi$
$229$	440289.	0.554816	0.277408	+	0.960752i	$0.410525\pi$
$230$	0	0
$231$	78574.4	0.0968838
$232$	45942.7	0.0560398
$233$	−1.46183e6	−1.76403	−0.882015	−	0.471221i	$-0.843813\pi$
$233$	−1.46183e6	−1.76403	−0.882015	+	0.471221i	$0.843813\pi$
$234$	−62237.2	−0.0743037
$235$	0	0
$236$	−1.13467e6	−1.32614
$237$	391400.	0.452637
$238$	−8897.16	−0.0101814
$239$	1.51292e6	1.71325	0.856627	−	0.515936i	$-0.172556\pi$
$239$	1.51292e6	1.71325	0.856627	+	0.515936i	$0.172556\pi$
$240$	0	0
$241$	−800679.	−0.888006	−0.444003	−	0.896025i	$-0.646442\pi$
$241$	−800679.	−0.888006	−0.444003	+	0.896025i	$0.646442\pi$
$242$	−60175.0	−0.0660507
$243$	−926776.	−1.00684
$244$	697470.	0.749982
$245$	0	0
$246$	13302.7	0.0140153
$247$	512184.	0.534175
$248$	−80415.0	−0.0830249
$249$	−331698.	−0.339035
$250$	0	0
$251$	1.66023e6	1.66335	0.831676	−	0.555261i	$-0.187382\pi$
$251$	1.66023e6	1.66335	0.831676	+	0.555261i	$0.187382\pi$
$252$	221596.	0.219817
$253$	562580.	0.552564
$254$	12802.5	0.0124512
$255$	0	0
$256$	812621.	0.774976
$257$	1.00723e6	0.951250	0.475625	−	0.879648i	$-0.342222\pi$
$257$	1.00723e6	0.951250	0.475625	+	0.879648i	$0.342222\pi$
$258$	−59133.9	−0.0553079
$259$	431031.	0.399263
$260$	0	0
$261$	162277.	0.147453
$262$	−270466.	−0.243422
$263$	−1.01326e6	−0.903298	−0.451649	−	0.892196i	$-0.649164\pi$
$263$	−1.01326e6	−0.903298	−0.451649	+	0.892196i	$0.649164\pi$
$264$	−116817.	−0.103157
$265$	0	0
$266$	43520.5	0.0377129
$267$	−18140.2	−0.0155727
$268$	−1.71300e6	−1.45687
$269$	−1.45518e6	−1.22613	−0.613064	−	0.790034i	$-0.710063\pi$
$269$	−1.45518e6	−1.22613	−0.613064	+	0.790034i	$0.710063\pi$
$270$	0	0
$271$	−1.61146e6	−1.33290	−0.666450	−	0.745550i	$-0.732187\pi$
$271$	−1.61146e6	−1.33290	−0.666450	+	0.745550i	$0.732187\pi$
$272$	−267175.	−0.218964
$273$	−97079.2	−0.0788351
$274$	219634.	0.176735
$275$	0	0
$276$	−411482.	−0.325146
$277$	−2.06358e6	−1.61593	−0.807966	−	0.589230i	$-0.799431\pi$
$277$	−2.06358e6	−1.61593	−0.807966	+	0.589230i	$0.799431\pi$
$278$	−337623.	−0.262011
$279$	−284038.	−0.218457
$280$	0	0
$281$	21664.7	0.0163676	0.00818382	−	0.999967i	$-0.497395\pi$
$281$	21664.7	0.0163676	0.00818382	+	0.999967i	$0.497395\pi$
$282$	−97196.2	−0.0727824
$283$	552726.	0.410245	0.205123	−	0.978736i	$-0.434241\pi$
$283$	552726.	0.410245	0.205123	+	0.978736i	$0.434241\pi$
$284$	−868565.	−0.639008
$285$	0	0
$286$	−97499.8	−0.0704837
$287$	−80007.4	−0.0573358
$288$	−496116.	−0.352453
$289$	−1.34125e6	−0.944639
$290$	0	0
$291$	247172.	0.171107
$292$	990301.	0.679689
$293$	1.53503e6	1.04459	0.522297	−	0.852764i	$-0.325075\pi$
$293$	1.53503e6	1.04459	0.522297	+	0.852764i	$0.325075\pi$
$294$	94433.3	0.0637172
$295$	0	0
$296$	−640817.	−0.425114
$297$	−932244.	−0.613251
$298$	62480.5	0.0407572
$299$	−695071.	−0.449626
$300$	0	0
$301$	355653.	0.226261
$302$	−196408.	−0.123920
$303$	−1.20419e6	−0.753508
$304$	1.30689e6	0.811063
$305$	0	0
$306$	46721.4	0.0285241
$307$	2.69140e6	1.62979	0.814895	−	0.579609i	$-0.196795\pi$
$307$	2.69140e6	1.62979	0.814895	+	0.579609i	$0.196795\pi$
$308$	347149.	0.208516
$309$	593964.	0.353887
$310$	0	0
$311$	−1.88553e6	−1.10543	−0.552717	−	0.833369i	$-0.686409\pi$
$311$	−1.88553e6	−1.10543	−0.552717	+	0.833369i	$0.686409\pi$
$312$	144329.	0.0839394
$313$	2.48142e6	1.43166	0.715828	−	0.698276i	$-0.246049\pi$
$313$	2.48142e6	1.43166	0.715828	+	0.698276i	$0.246049\pi$
$314$	−465744.	−0.266577
$315$	0	0
$316$	1.72924e6	0.974177
$317$	−204476.	−0.114286	−0.0571432	−	0.998366i	$-0.518199\pi$
$317$	−204476.	−0.114286	−0.0571432	+	0.998366i	$0.518199\pi$
$318$	148823.	0.0825280
$319$	254220.	0.139873
$320$	0	0
$321$	119145.	0.0645379
$322$	−59060.5	−0.0317437
$323$	−384496.	−0.205062
$324$	−783596.	−0.414696
$325$	0	0
$326$	−75997.1	−0.0396053
$327$	27275.2	0.0141058
$328$	118948.	0.0610480
$329$	584574.	0.297749
$330$	0	0
$331$	1.58800e6	0.796672	0.398336	−	0.917240i	$-0.369588\pi$
$331$	1.58800e6	0.796672	0.398336	+	0.917240i	$0.369588\pi$
$332$	−1.46547e6	−0.729678
$333$	−2.26346e6	−1.11857
$334$	−172638.	−0.0846782
$335$	0	0
$336$	−247708.	−0.119699
$337$	813616.	0.390252	0.195126	−	0.980778i	$-0.437488\pi$
$337$	813616.	0.390252	0.195126	+	0.980778i	$0.437488\pi$
$338$	−200201.	−0.0953180
$339$	759189.	0.358799
$340$	0	0
$341$	−444969.	−0.207226
$342$	−228538.	−0.105656
$343$	−1.18553e6	−0.544096
$344$	−528753.	−0.240911
$345$	0	0
$346$	271007.	0.121700
$347$	−516038.	−0.230069	−0.115034	−	0.993362i	$-0.536698\pi$
$347$	−516038.	−0.230069	−0.115034	+	0.993362i	$0.536698\pi$
$348$	−185942.	−0.0823054
$349$	−2.65164e6	−1.16533	−0.582667	−	0.812711i	$-0.697991\pi$
$349$	−2.65164e6	−1.16533	−0.582667	+	0.812711i	$0.697991\pi$
$350$	0	0
$351$	1.15179e6	0.499007
$352$	−777207.	−0.334333
$353$	−391323.	−0.167147	−0.0835734	−	0.996502i	$-0.526633\pi$
$353$	−391323.	−0.167147	−0.0835734	+	0.996502i	$0.526633\pi$
$354$	−221803.	−0.0940717
$355$	0	0
$356$	−80144.9	−0.0335159
$357$	72877.3	0.0302637
$358$	−51658.3	−0.0213026
$359$	3.03988e6	1.24486	0.622430	−	0.782675i	$-0.286146\pi$
$359$	3.03988e6	1.24486	0.622430	+	0.782675i	$0.286146\pi$
$360$	0	0
$361$	−595334.	−0.240432
$362$	347382.	0.139327
$363$	492898.	0.196332
$364$	−428905.	−0.169671
$365$	0	0
$366$	136340.	0.0532010
$367$	−1.02897e6	−0.398785	−0.199392	−	0.979920i	$-0.563897\pi$
$367$	−1.02897e6	−0.398785	−0.199392	+	0.979920i	$0.563897\pi$
$368$	−1.77354e6	−0.682688
$369$	420141.	0.160631
$370$	0	0
$371$	−895074.	−0.337617
$372$	325460.	0.121938
$373$	4.42358e6	1.64627	0.823136	−	0.567844i	$-0.192222\pi$
$373$	4.42358e6	1.64627	0.823136	+	0.567844i	$0.192222\pi$
$374$	73193.0	0.0270577
$375$	0	0
$376$	−869091.	−0.317027
$377$	−314090.	−0.113815
$378$	97868.5	0.0352301
$379$	−4.23499e6	−1.51445	−0.757223	−	0.653156i	$-0.773445\pi$
$379$	−4.23499e6	−1.51445	−0.757223	+	0.653156i	$0.773445\pi$
$380$	0	0
$381$	−104866.	−0.0370103
$382$	−780323.	−0.273600
$383$	894484.	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$383$	894484.	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$384$	754770.	0.261208
$385$	0	0
$386$	−888428.	−0.303497
$387$	−1.86763e6	−0.633891
$388$	1.09203e6	0.368260
$389$	3.60641e6	1.20837	0.604186	−	0.796843i	$-0.293498\pi$
$389$	3.60641e6	1.20837	0.604186	+	0.796843i	$0.293498\pi$
$390$	0	0
$391$	521789.	0.172605
$392$	844386.	0.277540
$393$	2.21541e6	0.723557
$394$	747996.	0.242750
$395$	0	0
$396$	−1.82297e6	−0.584175
$397$	1.96195e6	0.624757	0.312379	−	0.949958i	$-0.398874\pi$
$397$	1.96195e6	0.624757	0.312379	+	0.949958i	$0.398874\pi$
$398$	246859.	0.0781162
$399$	−356480.	−0.112099
$400$	0	0
$401$	−721116.	−0.223947	−0.111973	−	0.993711i	$-0.535717\pi$
$401$	−721116.	−0.223947	−0.111973	+	0.993711i	$0.535717\pi$
$402$	−334853.	−0.103345
$403$	549762.	0.168621
$404$	−5.32021e6	−1.62172
$405$	0	0
$406$	−26688.4	−0.00803540
$407$	−3.54590e6	−1.06106
$408$	−108347.	−0.0322231
$409$	1.33383e6	0.394269	0.197135	−	0.980376i	$-0.436836\pi$
$409$	1.33383e6	0.394269	0.197135	+	0.980376i	$0.436836\pi$
$410$	0	0
$411$	−1.79904e6	−0.525334
$412$	2.62419e6	0.761644
$413$	1.33400e6	0.384842
$414$	310143.	0.0889327
$415$	0	0
$416$	960244.	0.272050
$417$	2.76550e6	0.778813
$418$	−358024.	−0.100224
$419$	−4.51294e6	−1.25581	−0.627905	−	0.778290i	$-0.716087\pi$
$419$	−4.51294e6	−1.25581	−0.627905	+	0.778290i	$0.716087\pi$
$420$	0	0
$421$	982181.	0.270076	0.135038	−	0.990840i	$-0.456884\pi$
$421$	982181.	0.270076	0.135038	+	0.990840i	$0.456884\pi$
$422$	−793623.	−0.216937
$423$	−3.06976e6	−0.834168
$424$	1.33071e6	0.359476
$425$	0	0
$426$	−169785.	−0.0453290
$427$	−819998.	−0.217642
$428$	526395.	0.138900
$429$	798629.	0.209508
$430$	0	0
$431$	5.31784e6	1.37893	0.689465	−	0.724319i	$-0.257846\pi$
$431$	5.31784e6	1.37893	0.689465	+	0.724319i	$0.257846\pi$
$432$	2.93892e6	0.757667
$433$	−679278.	−0.174112	−0.0870558	−	0.996203i	$-0.527746\pi$
$433$	−679278.	−0.174112	−0.0870558	+	0.996203i	$0.527746\pi$
$434$	46713.6	0.0119047
$435$	0	0
$436$	120504.	0.0303589
$437$	−2.55234e6	−0.639344
$438$	193582.	0.0482147
$439$	3.53619e6	0.875739	0.437870	−	0.899039i	$-0.355733\pi$
$439$	3.53619e6	0.875739	0.437870	+	0.899039i	$0.355733\pi$
$440$	0	0
$441$	2.98250e6	0.730270
$442$	−90430.5	−0.0220170
$443$	1.35298e6	0.327553	0.163776	−	0.986497i	$-0.447632\pi$
$443$	1.35298e6	0.327553	0.163776	+	0.986497i	$0.447632\pi$
$444$	2.59355e6	0.624362
$445$	0	0
$446$	−419844.	−0.0999425
$447$	−511783.	−0.121148
$448$	−1.03892e6	−0.244561
$449$	−2.04554e6	−0.478843	−0.239421	−	0.970916i	$-0.576958\pi$
$449$	−2.04554e6	−0.478843	−0.239421	+	0.970916i	$0.576958\pi$
$450$	0	0
$451$	658186.	0.152373
$452$	3.35416e6	0.772215
$453$	1.60879e6	0.368345
$454$	481598.	0.109659
$455$	0	0
$456$	529982.	0.119357
$457$	−8.48379e6	−1.90020	−0.950100	−	0.311944i	$-0.899020\pi$
$457$	−8.48379e6	−1.90020	−0.950100	+	0.311944i	$0.899020\pi$
$458$	380250.	0.0847044
$459$	−864651.	−0.191562
$460$	0	0
$461$	−2.06795e6	−0.453198	−0.226599	−	0.973988i	$-0.572761\pi$
$461$	−2.06795e6	−0.453198	−0.226599	+	0.973988i	$0.572761\pi$
$462$	67859.9	0.0147914
$463$	368290.	0.0798430	0.0399215	−	0.999203i	$-0.487289\pi$
$463$	368290.	0.0798430	0.0399215	+	0.999203i	$0.487289\pi$
$464$	−801433.	−0.172811
$465$	0	0
$466$	−1.26249e6	−0.269317
$467$	2.78365e6	0.590640	0.295320	−	0.955398i	$-0.404574\pi$
$467$	2.78365e6	0.590640	0.295320	+	0.955398i	$0.404574\pi$
$468$	2.25230e6	0.475347
$469$	2.01393e6	0.422777
$470$	0	0
$471$	3.81494e6	0.792385
$472$	−1.98328e6	−0.409759
$473$	−2.92580e6	−0.601302
$474$	338028.	0.0691047
$475$	0	0
$476$	321978.	0.0651342
$477$	4.70028e6	0.945863
$478$	1.30662e6	0.261564
$479$	1.49820e6	0.298352	0.149176	−	0.988811i	$-0.452338\pi$
$479$	1.49820e6	0.298352	0.149176	+	0.988811i	$0.452338\pi$
$480$	0	0
$481$	4.38099e6	0.863394
$482$	−691497.	−0.135573
$483$	483770.	0.0943563
$484$	2.17767e6	0.422550
$485$	0	0
$486$	−800399.	−0.153715
$487$	−7.95675e6	−1.52024	−0.760122	−	0.649781i	$-0.774861\pi$
$487$	−7.95675e6	−1.52024	−0.760122	+	0.649781i	$0.774861\pi$
$488$	1.21910e6	0.231734
$489$	622498.	0.117724
$490$	0	0
$491$	−1.87342e6	−0.350697	−0.175348	−	0.984506i	$-0.556105\pi$
$491$	−1.87342e6	−0.350697	−0.175348	+	0.984506i	$0.556105\pi$
$492$	−481411.	−0.0896609
$493$	235787.	0.0436921
$494$	442341.	0.0815530
$495$	0	0
$496$	1.40277e6	0.256026
$497$	1.02115e6	0.185438
$498$	−286467.	−0.0517608
$499$	−352286.	−0.0633350	−0.0316675	−	0.999498i	$-0.510082\pi$
$499$	−352286.	−0.0633350	−0.0316675	+	0.999498i	$0.510082\pi$
$500$	0	0
$501$	1.41410e6	0.251701
$502$	1.43384e6	0.253946
$503$	3.57266e6	0.629611	0.314805	−	0.949156i	$-0.398061\pi$
$503$	3.57266e6	0.629611	0.314805	+	0.949156i	$0.398061\pi$
$504$	387325.	0.0679202
$505$	0	0
$506$	485865.	0.0843606
$507$	1.63986e6	0.283327
$508$	−463308.	−0.0796545
$509$	−1.05719e7	−1.80867	−0.904337	−	0.426819i	$-0.859634\pi$
$509$	−1.05719e7	−1.80867	−0.904337	+	0.426819i	$0.859634\pi$
$510$	0	0
$511$	−1.16427e6	−0.197243
$512$	4.11603e6	0.693911
$513$	4.22945e6	0.709562
$514$	869880.	0.145228
$515$	0	0
$516$	2.13999e6	0.353825
$517$	−4.80903e6	−0.791282
$518$	372255.	0.0609559
$519$	−2.21984e6	−0.361746
$520$	0	0
$521$	7.42701e6	1.19872	0.599362	−	0.800478i	$-0.295421\pi$
$521$	7.42701e6	1.19872	0.599362	+	0.800478i	$0.295421\pi$
$522$	140148.	0.0225119
$523$	1.03292e7	1.65125	0.825623	−	0.564222i	$-0.190824\pi$
$523$	1.03292e7	1.65125	0.825623	+	0.564222i	$0.190824\pi$
$524$	9.78788e6	1.55726
$525$	0	0
$526$	−875089.	−0.137908
$527$	−412706.	−0.0647313
$528$	2.03778e6	0.318107
$529$	−2.97263e6	−0.461851
$530$	0	0
$531$	−7.00523e6	−1.07817
$532$	−1.57496e6	−0.241263
$533$	−813193.	−0.123987
$534$	−15666.5	−0.00237750
$535$	0	0
$536$	−2.99412e6	−0.450151
$537$	423137.	0.0633207
$538$	−1.25675e6	−0.187194
$539$	4.67233e6	0.692726
$540$	0	0
$541$	4.60833e6	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$541$	4.60833e6	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$542$	−1.39172e6	−0.203495
$543$	−2.84543e6	−0.414142
$544$	−720854.	−0.104436
$545$	0	0
$546$	−83841.3	−0.0120358
$547$	1.79146e6	0.255999	0.127999	−	0.991774i	$-0.459144\pi$
$547$	1.79146e6	0.255999	0.127999	+	0.991774i	$0.459144\pi$
$548$	−7.94831e6	−1.13064
$549$	4.30604e6	0.609743
$550$	0	0
$551$	−1.15336e6	−0.161839
$552$	−719224.	−0.100465
$553$	−2.03303e6	−0.282703
$554$	−1.78219e6	−0.246706
$555$	0	0
$556$	1.22182e7	1.67618
$557$	−1.07840e7	−1.47280	−0.736399	−	0.676548i	$-0.763475\pi$
$557$	−1.07840e7	−1.47280	−0.736399	+	0.676548i	$0.763475\pi$
$558$	−245306.	−0.0333521
$559$	3.61485e6	0.489284
$560$	0	0
$561$	−599530.	−0.0804273
$562$	18710.4	0.00249887
$563$	1.40598e7	1.86942	0.934710	−	0.355411i	$-0.115659\pi$
$563$	1.40598e7	1.86942	0.934710	+	0.355411i	$0.115659\pi$
$564$	3.51743e6	0.465615
$565$	0	0
$566$	477355.	0.0626326
$567$	921255.	0.120343
$568$	−1.51815e6	−0.197444
$569$	4.09890e6	0.530746	0.265373	−	0.964146i	$-0.414505\pi$
$569$	4.09890e6	0.530746	0.265373	+	0.964146i	$0.414505\pi$
$570$	0	0
$571$	−2.46148e6	−0.315941	−0.157971	−	0.987444i	$-0.550495\pi$
$571$	−2.46148e6	−0.315941	−0.157971	+	0.987444i	$0.550495\pi$
$572$	3.52841e6	0.450909
$573$	6.39168e6	0.813259
$574$	−69097.5	−0.00875351
$575$	0	0
$576$	5.45565e6	0.685157
$577$	−7.09785e6	−0.887539	−0.443770	−	0.896141i	$-0.646359\pi$
$577$	−7.09785e6	−0.887539	−0.443770	+	0.896141i	$0.646359\pi$
$578$	−1.15836e6	−0.144219
$579$	7.27718e6	0.902125
$580$	0	0
$581$	1.72292e6	0.211750
$582$	213467.	0.0261230
$583$	7.36339e6	0.897235
$584$	1.73094e6	0.210014
$585$	0	0
$586$	1.32571e6	0.159479
$587$	1.43549e7	1.71951	0.859753	−	0.510710i	$-0.170617\pi$
$587$	1.43549e7	1.71951	0.859753	+	0.510710i	$0.170617\pi$
$588$	−3.41744e6	−0.407622
$589$	2.01876e6	0.239770
$590$	0	0
$591$	−6.12690e6	−0.721559
$592$	1.11785e7	1.31093
$593$	−1.01492e7	−1.18521	−0.592606	−	0.805493i	$-0.701901\pi$
$593$	−1.01492e7	−1.18521	−0.592606	+	0.805493i	$0.701901\pi$
$594$	−805122.	−0.0936258
$595$	0	0
$596$	−2.26110e6	−0.260738
$597$	−2.02204e6	−0.232196
$598$	−600290.	−0.0686449
$599$	−8.76777e6	−0.998440	−0.499220	−	0.866475i	$-0.666380\pi$
$599$	−8.76777e6	−0.998440	−0.499220	+	0.866475i	$0.666380\pi$
$600$	0	0
$601$	1.49619e7	1.68966	0.844831	−	0.535034i	$-0.179701\pi$
$601$	1.49619e7	1.68966	0.844831	+	0.535034i	$0.179701\pi$
$602$	307156.	0.0345436
$603$	−1.05757e7	−1.18445
$604$	7.10779e6	0.792760
$605$	0	0
$606$	−1.03998e6	−0.115039
$607$	−7.03290e6	−0.774752	−0.387376	−	0.921922i	$-0.626618\pi$
$607$	−7.03290e6	−0.774752	−0.387376	+	0.921922i	$0.626618\pi$
$608$	3.52606e6	0.386840
$609$	218607.	0.0238847
$610$	0	0
$611$	5.94159e6	0.643873
$612$	−1.69080e6	−0.182479
$613$	806845.	0.0867239	0.0433620	−	0.999059i	$-0.486193\pi$
$613$	806845.	0.0867239	0.0433620	+	0.999059i	$0.486193\pi$
$614$	2.32439e6	0.248822
$615$	0	0
$616$	606777.	0.0644284
$617$	−7.36483e6	−0.778843	−0.389421	−	0.921060i	$-0.627325\pi$
$617$	−7.36483e6	−0.778843	−0.389421	+	0.921060i	$0.627325\pi$
$618$	512970.	0.0540283
$619$	−1.30528e7	−1.36923	−0.684617	−	0.728903i	$-0.740031\pi$
$619$	−1.30528e7	−1.36923	−0.684617	+	0.728903i	$0.740031\pi$
$620$	0	0
$621$	−5.73967e6	−0.597253
$622$	−1.62842e6	−0.168768
$623$	94224.4	0.00972620
$624$	−2.51769e6	−0.258846
$625$	0	0
$626$	2.14305e6	0.218573
$627$	2.93261e6	0.297910
$628$	1.68548e7	1.70539
$629$	−3.28880e6	−0.331445
$630$	0	0
$631$	2.89105e6	0.289056	0.144528	−	0.989501i	$-0.453834\pi$
$631$	2.89105e6	0.289056	0.144528	+	0.989501i	$0.453834\pi$
$632$	3.02252e6	0.301007
$633$	6.50063e6	0.644831
$634$	−176594.	−0.0174482
$635$	0	0
$636$	−5.38573e6	−0.527961
$637$	−5.77270e6	−0.563677
$638$	219554.	0.0213545
$639$	−5.36234e6	−0.519520
$640$	0	0
$641$	−1.31397e7	−1.26311	−0.631555	−	0.775331i	$-0.717583\pi$
$641$	−1.31397e7	−1.26311	−0.631555	+	0.775331i	$0.717583\pi$
$642$	102899.	0.00985307
$643$	1.17266e7	1.11852	0.559259	−	0.828993i	$-0.311086\pi$
$643$	1.17266e7	1.11852	0.559259	+	0.828993i	$0.311086\pi$
$644$	2.13734e6	0.203076
$645$	0	0
$646$	−332065.	−0.0313071
$647$	−1.10662e7	−1.03929	−0.519646	−	0.854382i	$-0.673936\pi$
$647$	−1.10662e7	−1.03929	−0.519646	+	0.854382i	$0.673936\pi$
$648$	−1.36964e6	−0.128135
$649$	−1.09743e7	−1.02274
$650$	0	0
$651$	−382635.	−0.0353860
$652$	2.75025e6	0.253369
$653$	3.44597e6	0.316249	0.158124	−	0.987419i	$-0.449455\pi$
$653$	3.44597e6	0.316249	0.158124	+	0.987419i	$0.449455\pi$
$654$	23555.9	0.00215355
$655$	0	0
$656$	−2.07495e6	−0.188255
$657$	6.11392e6	0.552594
$658$	504860.	0.0454576
$659$	−9.12181e6	−0.818215	−0.409108	−	0.912486i	$-0.634160\pi$
$659$	−9.12181e6	−0.818215	−0.409108	+	0.912486i	$0.634160\pi$
$660$	0	0
$661$	1.68113e7	1.49657	0.748287	−	0.663375i	$-0.230877\pi$
$661$	1.68113e7	1.49657	0.748287	+	0.663375i	$0.230877\pi$
$662$	1.37145e6	0.121629
$663$	740723.	0.0654443
$664$	−2.56147e6	−0.225460
$665$	0	0
$666$	−1.95481e6	−0.170773
$667$	1.56519e6	0.136224
$668$	6.24760e6	0.541716
$669$	3.43897e6	0.297073
$670$	0	0
$671$	6.74577e6	0.578396
$672$	−668330.	−0.0570910
$673$	−8.53511e6	−0.726393	−0.363196	−	0.931713i	$-0.618315\pi$
$673$	−8.53511e6	−0.726393	−0.363196	+	0.931713i	$0.618315\pi$
$674$	702670.	0.0595802
$675$	0	0
$676$	7.24506e6	0.609783
$677$	912690.	0.0765335	0.0382667	−	0.999268i	$-0.487816\pi$
$677$	912690.	0.0765335	0.0382667	+	0.999268i	$0.487816\pi$
$678$	655665.	0.0547782
$679$	−1.28387e6	−0.106868
$680$	0	0
$681$	−3.94481e6	−0.325955
$682$	−384292.	−0.0316374
$683$	4.20567e6	0.344972	0.172486	−	0.985012i	$-0.444820\pi$
$683$	4.20567e6	0.344972	0.172486	+	0.985012i	$0.444820\pi$
$684$	8.27055e6	0.675918
$685$	0	0
$686$	−1.02386e6	−0.0830677
$687$	−3.11466e6	−0.251779
$688$	9.22366e6	0.742903
$689$	−9.09751e6	−0.730087
$690$	0	0
$691$	6.23030e6	0.496379	0.248190	−	0.968711i	$-0.420164\pi$
$691$	6.23030e6	0.496379	0.248190	+	0.968711i	$0.420164\pi$
$692$	−9.80746e6	−0.778559
$693$	2.14323e6	0.169525
$694$	−445670.	−0.0351249
$695$	0	0
$696$	−325005.	−0.0254312
$697$	610464.	0.0475968
$698$	−2.29005e6	−0.177913
$699$	1.03412e7	0.800527
$700$	0	0
$701$	1.69453e7	1.30243	0.651215	−	0.758893i	$-0.274259\pi$
$701$	1.69453e7	1.30243	0.651215	+	0.758893i	$0.274259\pi$
$702$	994733.	0.0761840
$703$	1.60872e7	1.22770
$704$	8.54673e6	0.649933
$705$	0	0
$706$	−337961.	−0.0255185
$707$	6.25484e6	0.470617
$708$	8.02680e6	0.601810
$709$	−1.87326e7	−1.39953	−0.699766	−	0.714372i	$-0.746712\pi$
$709$	−1.87326e7	−1.39953	−0.699766	+	0.714372i	$0.746712\pi$
$710$	0	0
$711$	1.06760e7	0.792016
$712$	−140084.	−0.0103559
$713$	−2.73960e6	−0.201820
$714$	62939.6	0.00462039
$715$	0	0
$716$	1.86946e6	0.136280
$717$	−1.07026e7	−0.777485
$718$	2.62536e6	0.190054
$719$	2.15611e6	0.155542	0.0777711	−	0.996971i	$-0.475220\pi$
$719$	2.15611e6	0.155542	0.0777711	+	0.996971i	$0.475220\pi$
$720$	0	0
$721$	−3.08519e6	−0.221026
$722$	−514154.	−0.0367071
$723$	5.66410e6	0.402982
$724$	−1.25714e7	−0.891326
$725$	0	0
$726$	425686.	0.0299742
$727$	−4.87466e6	−0.342065	−0.171032	−	0.985265i	$-0.554710\pi$
$727$	−4.87466e6	−0.342065	−0.171032	+	0.985265i	$0.554710\pi$
$728$	−749677.	−0.0524259
$729$	463697.	0.0323159
$730$	0	0
$731$	−2.71367e6	−0.187829
$732$	−4.93399e6	−0.340346
$733$	−5.15920e6	−0.354669	−0.177334	−	0.984151i	$-0.556747\pi$
$733$	−5.15920e6	−0.354669	−0.177334	+	0.984151i	$0.556747\pi$
$734$	−888660.	−0.0608829
$735$	0	0
$736$	−4.78513e6	−0.325611
$737$	−1.65677e7	−1.12355
$738$	362850.	0.0245237
$739$	−4.60821e6	−0.310400	−0.155200	−	0.987883i	$-0.549602\pi$
$739$	−4.60821e6	−0.310400	−0.155200	+	0.987883i	$0.549602\pi$
$740$	0	0
$741$	−3.62325e6	−0.242411
$742$	−773020.	−0.0515444
$743$	−2.50473e7	−1.66452	−0.832261	−	0.554384i	$-0.812954\pi$
$743$	−2.50473e7	−1.66452	−0.832261	+	0.554384i	$0.812954\pi$
$744$	568866.	0.0376771
$745$	0	0
$746$	3.82037e6	0.251338
$747$	−9.04751e6	−0.593236
$748$	−2.64878e6	−0.173098
$749$	−618870.	−0.0403083
$750$	0	0
$751$	−1.87279e7	−1.21168	−0.605841	−	0.795586i	$-0.707163\pi$
$751$	−1.87279e7	−1.21168	−0.605841	+	0.795586i	$0.707163\pi$
$752$	1.51606e7	0.977623
$753$	−1.17447e7	−0.754839
$754$	−271260.	−0.0173763
$755$	0	0
$756$	−3.54176e6	−0.225379
$757$	5.50814e6	0.349354	0.174677	−	0.984626i	$-0.444112\pi$
$757$	5.50814e6	0.349354	0.174677	+	0.984626i	$0.444112\pi$
$758$	−3.65750e6	−0.231212
$759$	−3.97976e6	−0.250757
$760$	0	0
$761$	−2.67826e7	−1.67645	−0.838226	−	0.545323i	$-0.816407\pi$
$761$	−2.67826e7	−1.67645	−0.838226	+	0.545323i	$0.816407\pi$
$762$	−90566.4	−0.00565040
$763$	−141674.	−0.00881005
$764$	2.82390e7	1.75032
$765$	0	0
$766$	772511.	0.0475699
$767$	1.35588e7	0.832209
$768$	−5.74859e6	−0.351689
$769$	1.76374e7	1.07552	0.537761	−	0.843097i	$-0.319270\pi$
$769$	1.76374e7	1.07552	0.537761	+	0.843097i	$0.319270\pi$
$770$	0	0
$771$	−7.12526e6	−0.431683
$772$	3.21512e7	1.94158
$773$	6.59079e6	0.396724	0.198362	−	0.980129i	$-0.436438\pi$
$773$	6.59079e6	0.396724	0.198362	+	0.980129i	$0.436438\pi$
$774$	−1.61296e6	−0.0967768
$775$	0	0
$776$	1.90874e6	0.113787
$777$	−3.04917e6	−0.181188
$778$	3.11463e6	0.184484
$779$	−2.98609e6	−0.176303
$780$	0	0
$781$	−8.40056e6	−0.492811
$782$	450637.	0.0263518
$783$	−2.59366e6	−0.151185
$784$	−1.47296e7	−0.855858
$785$	0	0
$786$	1.91331e6	0.110466
$787$	−1.83730e7	−1.05741	−0.528704	−	0.848806i	$-0.677322\pi$
$787$	−1.83730e7	−1.05741	−0.528704	+	0.848806i	$0.677322\pi$
$788$	−2.70692e7	−1.55296
$789$	7.16792e6	0.409922
$790$	0	0
$791$	−3.94341e6	−0.224094
$792$	−3.18635e6	−0.180502
$793$	−8.33444e6	−0.470645
$794$	1.69441e6	0.0953824
$795$	0	0
$796$	−8.93356e6	−0.499737
$797$	3.06769e7	1.71067	0.855334	−	0.518078i	$-0.173352\pi$
$797$	3.06769e7	1.71067	0.855334	+	0.518078i	$0.173352\pi$
$798$	−307870.	−0.0171143
$799$	−4.46035e6	−0.247174
$800$	0	0
$801$	−494798.	−0.0272488
$802$	−622784.	−0.0341902
$803$	9.57797e6	0.524185
$804$	1.21180e7	0.661134
$805$	0	0
$806$	474796.	0.0257436
$807$	1.02941e7	0.556423
$808$	−9.29912e6	−0.501088
$809$	−2.81720e6	−0.151338	−0.0756688	−	0.997133i	$-0.524109\pi$
$809$	−2.81720e6	−0.151338	−0.0756688	+	0.997133i	$0.524109\pi$
$810$	0	0
$811$	−1.57819e7	−0.842574	−0.421287	−	0.906927i	$-0.638421\pi$
$811$	−1.57819e7	−0.842574	−0.421287	+	0.906927i	$0.638421\pi$
$812$	965825.	0.0514053
$813$	1.13997e7	0.604877
$814$	−3.06238e6	−0.161994
$815$	0	0
$816$	1.89003e6	0.0993673
$817$	1.32739e7	0.695735
$818$	1.15195e6	0.0601935
$819$	−2.64797e6	−0.137944
$820$	0	0
$821$	−2.40563e6	−0.124558	−0.0622789	−	0.998059i	$-0.519837\pi$
$821$	−2.40563e6	−0.124558	−0.0622789	+	0.998059i	$0.519837\pi$
$822$	−1.55372e6	−0.0802034
$823$	−4.47397e6	−0.230247	−0.115123	−	0.993351i	$-0.536726\pi$
$823$	−4.47397e6	−0.230247	−0.115123	+	0.993351i	$0.536726\pi$
$824$	4.58678e6	0.235337
$825$	0	0
$826$	1.15210e6	0.0587542
$827$	−2.36054e7	−1.20018	−0.600091	−	0.799932i	$-0.704869\pi$
$827$	−2.36054e7	−1.20018	−0.600091	+	0.799932i	$0.704869\pi$
$828$	−1.12237e7	−0.568934
$829$	−7.52668e6	−0.380380	−0.190190	−	0.981747i	$-0.560910\pi$
$829$	−7.52668e6	−0.380380	−0.190190	+	0.981747i	$0.560910\pi$
$830$	0	0
$831$	1.45981e7	0.733319
$832$	−1.05595e7	−0.528855
$833$	4.33356e6	0.216388
$834$	2.38839e6	0.118902
$835$	0	0
$836$	1.29565e7	0.641169
$837$	4.53976e6	0.223985
$838$	−3.89754e6	−0.191726
$839$	−7.28878e6	−0.357478	−0.178739	−	0.983896i	$-0.557202\pi$
$839$	−7.28878e6	−0.357478	−0.178739	+	0.983896i	$0.557202\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	848249.	0.0412328
$843$	−153259.	−0.00742773
$844$	2.87204e7	1.38782
$845$	0	0
$846$	−2.65116e6	−0.127353
$847$	−2.56023e6	−0.122623
$848$	−2.32132e7	−1.10853
$849$	−3.91005e6	−0.186172
$850$	0	0
$851$	−2.18315e7	−1.03338
$852$	6.14434e6	0.289985
$853$	−2.18767e7	−1.02946	−0.514729	−	0.857353i	$-0.672107\pi$
$853$	−2.18767e7	−1.02946	−0.514729	+	0.857353i	$0.672107\pi$
$854$	−708182.	−0.0332277
$855$	0	0
$856$	920079.	0.0429181
$857$	1.62738e7	0.756897	0.378448	−	0.925622i	$-0.376458\pi$
$857$	1.62738e7	0.756897	0.378448	+	0.925622i	$0.376458\pi$
$858$	689726.	0.0319859
$859$	1.50220e7	0.694615	0.347308	−	0.937751i	$-0.387096\pi$
$859$	1.50220e7	0.694615	0.347308	+	0.937751i	$0.387096\pi$
$860$	0	0
$861$	565983.	0.0260193
$862$	4.59269e6	0.210523
$863$	−2.24278e7	−1.02508	−0.512542	−	0.858662i	$-0.671296\pi$
$863$	−2.24278e7	−1.02508	−0.512542	+	0.858662i	$0.671296\pi$
$864$	7.92938e6	0.361372
$865$	0	0
$866$	−586650.	−0.0265818
$867$	9.48819e6	0.428682
$868$	−1.69051e6	−0.0761587
$869$	1.67248e7	0.751298
$870$	0	0
$871$	2.04695e7	0.914244
$872$	210628.	0.00938047
$873$	6.74196e6	0.299399
$874$	−2.20430e6	−0.0976093
$875$	0	0
$876$	−7.00552e6	−0.308447
$877$	−1.25450e7	−0.550771	−0.275385	−	0.961334i	$-0.588805\pi$
$877$	−1.25450e7	−0.550771	−0.275385	+	0.961334i	$0.588805\pi$
$878$	3.05399e6	0.133700
$879$	−1.08590e7	−0.474043
$880$	0	0
$881$	7.60341e6	0.330042	0.165021	−	0.986290i	$-0.447231\pi$
$881$	7.60341e6	0.330042	0.165021	+	0.986290i	$0.447231\pi$
$882$	2.57580e6	0.111491
$883$	−2.15220e7	−0.928923	−0.464462	−	0.885593i	$-0.653752\pi$
$883$	−2.15220e7	−0.928923	−0.464462	+	0.885593i	$0.653752\pi$
$884$	3.27258e6	0.140851
$885$	0	0
$886$	1.16848e6	0.0500079
$887$	333279.	0.0142232	0.00711162	−	0.999975i	$-0.497736\pi$
$887$	333279.	0.0142232	0.00711162	+	0.999975i	$0.497736\pi$
$888$	4.53322e6	0.192919
$889$	544700.	0.0231155
$890$	0	0
$891$	−7.57876e6	−0.319819
$892$	1.51937e7	0.639368
$893$	2.18178e7	0.915552
$894$	−441995.	−0.0184958
$895$	0	0
$896$	−3.92046e6	−0.163142
$897$	4.91702e6	0.204043
$898$	−1.76661e6	−0.0731055
$899$	−1.23798e6	−0.0510873
$900$	0	0
$901$	6.82950e6	0.280270
$902$	568435.	0.0232629
$903$	−2.51594e6	−0.102679
$904$	5.86270e6	0.238603
$905$	0	0
$906$	1.38941e6	0.0562356
$907$	−1.65687e7	−0.668759	−0.334380	−	0.942439i	$-0.608527\pi$
$907$	−1.65687e7	−0.668759	−0.334380	+	0.942439i	$0.608527\pi$
$908$	−1.74285e7	−0.701529
$909$	−3.28459e7	−1.31847
$910$	0	0
$911$	1.30648e7	0.521561	0.260781	−	0.965398i	$-0.416020\pi$
$911$	1.30648e7	0.521561	0.260781	+	0.965398i	$0.416020\pi$
$912$	−9.24510e6	−0.368065
$913$	−1.41737e7	−0.562737
$914$	−7.32693e6	−0.290106
$915$	0	0
$916$	−1.37608e7	−0.541884
$917$	−1.15074e7	−0.451911
$918$	−746746.	−0.0292460
$919$	−7.30805e6	−0.285439	−0.142719	−	0.989763i	$-0.545585\pi$
$919$	−7.30805e6	−0.285439	−0.142719	+	0.989763i	$0.545585\pi$
$920$	0	0
$921$	−1.90393e7	−0.739608
$922$	−1.78596e6	−0.0691902
$923$	1.03789e7	0.401004
$924$	−2.45578e6	−0.0946256
$925$	0	0
$926$	318069.	0.0121897
$927$	1.62012e7	0.619224
$928$	−2.16232e6	−0.0824231
$929$	2.69866e7	1.02591	0.512955	−	0.858416i	$-0.328551\pi$
$929$	2.69866e7	1.02591	0.512955	+	0.858416i	$0.328551\pi$
$930$	0	0
$931$	−2.11976e7	−0.801518
$932$	4.56881e7	1.72291
$933$	1.33385e7	0.501652
$934$	2.40407e6	0.0901737
$935$	0	0
$936$	3.93676e6	0.146876
$937$	2.15536e7	0.801993	0.400997	−	0.916080i	$-0.368664\pi$
$937$	2.15536e7	0.801993	0.400997	+	0.916080i	$0.368664\pi$
$938$	1.73930e6	0.0645459
$939$	−1.75539e7	−0.649694
$940$	0	0
$941$	−1.22668e7	−0.451602	−0.225801	−	0.974173i	$-0.572500\pi$
$941$	−1.22668e7	−0.451602	−0.225801	+	0.974173i	$0.572500\pi$
$942$	3.29473e6	0.120974
$943$	4.05234e6	0.148398
$944$	3.45966e7	1.26358
$945$	0	0
$946$	−2.52684e6	−0.0918014
$947$	−5.13280e7	−1.85986	−0.929928	−	0.367741i	$-0.880131\pi$
$947$	−5.13280e7	−1.85986	−0.929928	+	0.367741i	$0.880131\pi$
$948$	−1.22329e7	−0.442087
$949$	−1.18336e7	−0.426533
$950$	0	0
$951$	1.44649e6	0.0518638
$952$	562782.	0.0201255
$953$	3.30301e7	1.17809	0.589044	−	0.808101i	$-0.299504\pi$
$953$	3.30301e7	1.17809	0.589044	+	0.808101i	$0.299504\pi$
$954$	4.05934e6	0.144406
$955$	0	0
$956$	−4.72851e7	−1.67332
$957$	−1.79838e6	−0.0634750
$958$	1.29390e6	0.0455498
$959$	9.34463e6	0.328107
$960$	0	0
$961$	−2.64623e7	−0.924312
$962$	3.78359e6	0.131815
$963$	3.24986e6	0.112927
$964$	2.50245e7	0.867307
$965$	0	0
$966$	417802.	0.0144055
$967$	7.77281e6	0.267308	0.133654	−	0.991028i	$-0.457329\pi$
$967$	7.77281e6	0.267308	0.133654	+	0.991028i	$0.457329\pi$
$968$	3.80632e6	0.130562
$969$	2.71997e6	0.0930583
$970$	0	0
$971$	2.31111e7	0.786635	0.393317	−	0.919403i	$-0.371327\pi$
$971$	2.31111e7	0.786635	0.393317	+	0.919403i	$0.371327\pi$
$972$	2.89656e7	0.983369
$973$	−1.43647e7	−0.486422
$974$	−6.87175e6	−0.232097
$975$	0	0
$976$	−2.12662e7	−0.714603
$977$	−1.04792e7	−0.351231	−0.175616	−	0.984459i	$-0.556192\pi$
$977$	−1.04792e7	−0.351231	−0.175616	+	0.984459i	$0.556192\pi$
$978$	537613.	0.0179731
$979$	−775143.	−0.0258479
$980$	0	0
$981$	743969.	0.0246821
$982$	−1.61796e6	−0.0535413
$983$	−3.94189e7	−1.30113	−0.650565	−	0.759450i	$-0.725468\pi$
$983$	−3.94189e7	−1.30113	−0.650565	+	0.759450i	$0.725468\pi$
$984$	−841452.	−0.0277039
$985$	0	0
$986$	203635.	0.00667052
$987$	−4.13535e6	−0.135120
$988$	−1.60079e7	−0.521724
$989$	−1.80137e7	−0.585615
$990$	0	0
$991$	−4.53294e7	−1.46621	−0.733104	−	0.680117i	$-0.761929\pi$
$991$	−4.53294e7	−1.46621	−0.733104	+	0.680117i	$0.761929\pi$
$992$	3.78477e6	0.122113
$993$	−1.12337e7	−0.361534
$994$	881905.	0.0283110
$995$	0	0
$996$	1.03669e7	0.331132
$997$	−4.74723e7	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$997$	−4.74723e7	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$998$	−304247.	−0.00966942
$999$	3.61768e7	1.14687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.6.a.a.1.3		4
5.4	even	2		29.6.a.a.1.2	✓	4
15.14	odd	2		261.6.a.a.1.3		4
20.19	odd	2		464.6.a.i.1.1		4
145.144	even	2		841.6.a.a.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.2	✓	4	5.4	even	2
261.6.a.a.1.3		4	15.14	odd	2
464.6.a.i.1.1		4	20.19	odd	2
725.6.a.a.1.3		4	1.1	even	1	trivial
841.6.a.a.1.3		4	145.144	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q+0.863638 q^{2} -7.07413 q^{3} -31.2541 q^{4} -6.10949 q^{6} +36.7447 q^{7} -54.6287 q^{8} -192.957 q^{9} +O(q^{10})\)	\(q+0.863638 q^{2} -7.07413 q^{3} -31.2541 q^{4} -6.10949 q^{6} +36.7447 q^{7} -54.6287 q^{8} -192.957 q^{9} -302.283 q^{11} +221.096 q^{12} +373.472 q^{13} +31.7341 q^{14} +952.953 q^{16} -280.365 q^{17} -166.645 q^{18} +1371.41 q^{19} -259.937 q^{21} -261.063 q^{22} -1861.10 q^{23} +386.450 q^{24} +322.545 q^{26} +3084.01 q^{27} -1148.42 q^{28} -841.000 q^{29} +1472.03 q^{31} +2571.12 q^{32} +2138.39 q^{33} -242.134 q^{34} +6030.69 q^{36} +11730.4 q^{37} +1184.40 q^{38} -2641.99 q^{39} -2177.39 q^{41} -224.491 q^{42} +9679.03 q^{43} +9447.58 q^{44} -1607.32 q^{46} +15909.1 q^{47} -6741.31 q^{48} -15456.8 q^{49} +1983.34 q^{51} -11672.6 q^{52} -24359.3 q^{53} +2663.47 q^{54} -2007.32 q^{56} -9701.53 q^{57} -726.320 q^{58} +36304.7 q^{59} -22316.1 q^{61} +1271.30 q^{62} -7090.14 q^{63} -28274.0 q^{64} +1846.79 q^{66} +54808.6 q^{67} +8762.58 q^{68} +13165.7 q^{69} +27790.4 q^{71} +10541.0 q^{72} -31685.5 q^{73} +10130.8 q^{74} -42862.2 q^{76} -11107.3 q^{77} -2281.73 q^{78} -55328.4 q^{79} +25071.8 q^{81} -1880.47 q^{82} +46888.8 q^{83} +8124.10 q^{84} +8359.18 q^{86} +5949.34 q^{87} +16513.3 q^{88} +2564.30 q^{89} +13723.1 q^{91} +58167.2 q^{92} -10413.3 q^{93} +13739.7 q^{94} -18188.5 q^{96} -34940.3 q^{97} -13349.1 q^{98} +58327.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9}+O(q^{10}) \) 4 * q + 28 * q^3 + 10 * q^4 - 194 * q^6 + 208 * q^7 + 504 * q^8 - 280 * q^9	\( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9} - 124 q^{11} - 20 q^{12} + 460 q^{13} + 768 q^{14} - 414 q^{16} - 184 q^{17} - 3208 q^{18} - 2392 q^{19} + 992 q^{21} - 5538 q^{22} + 1192 q^{23} + 6786 q^{24} + 4724 q^{26} - 2468 q^{27} - 44 q^{28} - 3364 q^{29} - 19212 q^{31} - 6552 q^{32} + 10580 q^{33} - 7612 q^{34} - 7468 q^{36} + 10928 q^{37} + 456 q^{38} - 8732 q^{39} - 1120 q^{41} - 1844 q^{42} + 21420 q^{43} - 1932 q^{44} - 7588 q^{46} - 23772 q^{47} - 33060 q^{48} + 10452 q^{49} + 12744 q^{51} + 29062 q^{52} - 8860 q^{53} + 35410 q^{54} + 34304 q^{56} - 48944 q^{57} - 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 27488 q^{63} - 20734 q^{64} - 47744 q^{66} + 7840 q^{67} - 20724 q^{68} + 58792 q^{69} - 48744 q^{71} - 8088 q^{72} + 74992 q^{73} - 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 2982 q^{78} - 106076 q^{79} - 59692 q^{81} + 234132 q^{82} - 62888 q^{83} - 59832 q^{84} - 216014 q^{86} - 23548 q^{87} + 39426 q^{88} + 107568 q^{89} - 268896 q^{91} + 26268 q^{92} - 221460 q^{93} + 30542 q^{94} - 78606 q^{96} + 49520 q^{97} - 242304 q^{98} + 166720 q^{99}+O(q^{100}) \) 4 * q + 28 * q^3 + 10 * q^4 - 194 * q^6 + 208 * q^7 + 504 * q^8 - 280 * q^9 - 124 * q^11 - 20 * q^12 + 460 * q^13 + 768 * q^14 - 414 * q^16 - 184 * q^17 - 3208 * q^18 - 2392 * q^19 + 992 * q^21 - 5538 * q^22 + 1192 * q^23 + 6786 * q^24 + 4724 * q^26 - 2468 * q^27 - 44 * q^28 - 3364 * q^29 - 19212 * q^31 - 6552 * q^32 + 10580 * q^33 - 7612 * q^34 - 7468 * q^36 + 10928 * q^37 + 456 * q^38 - 8732 * q^39 - 1120 * q^41 - 1844 * q^42 + 21420 * q^43 - 1932 * q^44 - 7588 * q^46 - 23772 * q^47 - 33060 * q^48 + 10452 * q^49 + 12744 * q^51 + 29062 * q^52 - 8860 * q^53 + 35410 * q^54 + 34304 * q^56 - 48944 * q^57 - 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 27488 * q^63 - 20734 * q^64 - 47744 * q^66 + 7840 * q^67 - 20724 * q^68 + 58792 * q^69 - 48744 * q^71 - 8088 * q^72 + 74992 * q^73 - 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 2982 * q^78 - 106076 * q^79 - 59692 * q^81 + 234132 * q^82 - 62888 * q^83 - 59832 * q^84 - 216014 * q^86 - 23548 * q^87 + 39426 * q^88 + 107568 * q^89 - 268896 * q^91 + 26268 * q^92 - 221460 * q^93 + 30542 * q^94 - 78606 * q^96 + 49520 * q^97 - 242304 * q^98 + 166720 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more