LMFDB - Embedded newform 825.6.a.u.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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + \cdots - 7036608 $ x^10 - x^9 - 271x^8 + 309x^7 + 24456x^6 - 33410x^5 - 822204x^4 + 1367872x^3 + 7443872x^2 - 12856224x - 7036608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.20891$ 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)

$f(q)$	$=$	$q-9.20891 q^{2} -9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} +195.275 q^{7} -191.582 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.20891 q^{2} -9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} +195.275 q^{7} -191.582 q^{8} +81.0000 q^{9} +121.000 q^{11} -475.236 q^{12} +457.051 q^{13} -1798.27 q^{14} +74.5305 q^{16} -1237.33 q^{17} -745.921 q^{18} +2140.85 q^{19} -1757.47 q^{21} -1114.28 q^{22} -2272.80 q^{23} +1724.23 q^{24} -4208.94 q^{26} -729.000 q^{27} +10311.3 q^{28} +6561.91 q^{29} +6349.26 q^{31} +5444.27 q^{32} -1089.00 q^{33} +11394.5 q^{34} +4277.12 q^{36} +3376.37 q^{37} -19714.9 q^{38} -4113.46 q^{39} +3970.52 q^{41} +16184.4 q^{42} -3901.82 q^{43} +6389.28 q^{44} +20930.0 q^{46} +1267.66 q^{47} -670.775 q^{48} +21325.2 q^{49} +11136.0 q^{51} +24134.1 q^{52} -10563.3 q^{53} +6713.29 q^{54} -37411.0 q^{56} -19267.6 q^{57} -60428.0 q^{58} +43579.8 q^{59} +49086.8 q^{61} -58469.8 q^{62} +15817.2 q^{63} -52520.7 q^{64} +10028.5 q^{66} -14785.3 q^{67} -65336.1 q^{68} +20455.2 q^{69} +8961.61 q^{71} -15518.1 q^{72} +83352.2 q^{73} -31092.6 q^{74} +113045. q^{76} +23628.2 q^{77} +37880.5 q^{78} -17858.5 q^{79} +6561.00 q^{81} -36564.1 q^{82} +66830.4 q^{83} -92801.5 q^{84} +35931.5 q^{86} -59057.2 q^{87} -23181.4 q^{88} -117907. q^{89} +89250.5 q^{91} -120013. q^{92} -57143.4 q^{93} -11673.8 q^{94} -48998.4 q^{96} -156797. q^{97} -196382. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) $ 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * 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 $

$2$	−9.20891	−1.62792	−0.813960	−	0.580921i	$-0.802693\pi$
$2$	−9.20891	−1.62792	−0.813960	+	0.580921i	$0.802693\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	52.8039	1.65012
$5$	0	0
$5$	0	0
$6$	82.8802	0.939880
$7$	195.275	1.50626	0.753131	−	0.657870i	$-0.228542\pi$
$7$	195.275	1.50626	0.753131	+	0.657870i	$0.228542\pi$
$8$	−191.582	−1.05835
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−475.236	−0.952699
$13$	457.051	0.750078	0.375039	−	0.927009i	$-0.377629\pi$
$13$	457.051	0.750078	0.375039	+	0.927009i	$0.377629\pi$
$14$	−1798.27	−2.45208
$15$	0	0
$16$	74.5305	0.0727837
$17$	−1237.33	−1.03840	−0.519200	−	0.854653i	$-0.673770\pi$
$17$	−1237.33	−1.03840	−0.519200	+	0.854653i	$0.673770\pi$
$18$	−745.921	−0.542640
$19$	2140.85	1.36051	0.680255	−	0.732976i	$-0.261869\pi$
$19$	2140.85	1.36051	0.680255	+	0.732976i	$0.261869\pi$
$20$	0	0
$21$	−1757.47	−0.869641
$22$	−1114.28	−0.490836
$23$	−2272.80	−0.895861	−0.447931	−	0.894068i	$-0.647839\pi$
$23$	−2272.80	−0.895861	−0.447931	+	0.894068i	$0.647839\pi$
$24$	1724.23	0.611038
$25$	0	0
$26$	−4208.94	−1.22107
$27$	−729.000	−0.192450
$28$	10311.3	2.48552
$29$	6561.91	1.44889	0.724445	−	0.689333i	$-0.242096\pi$
$29$	6561.91	1.44889	0.724445	+	0.689333i	$0.242096\pi$
$30$	0	0
$31$	6349.26	1.18664	0.593320	−	0.804967i	$-0.297817\pi$
$31$	6349.26	1.18664	0.593320	+	0.804967i	$0.297817\pi$
$32$	5444.27	0.939863
$33$	−1089.00	−0.174078
$34$	11394.5	1.69043
$35$	0	0
$36$	4277.12	0.550041
$37$	3376.37	0.405457	0.202729	−	0.979235i	$-0.435019\pi$
$37$	3376.37	0.405457	0.202729	+	0.979235i	$0.435019\pi$
$38$	−19714.9	−2.21480
$39$	−4113.46	−0.433058
$40$	0	0
$41$	3970.52	0.368882	0.184441	−	0.982844i	$-0.440953\pi$
$41$	3970.52	0.368882	0.184441	+	0.982844i	$0.440953\pi$
$42$	16184.4	1.41571
$43$	−3901.82	−0.321807	−0.160904	−	0.986970i	$-0.551441\pi$
$43$	−3901.82	−0.321807	−0.160904	+	0.986970i	$0.551441\pi$
$44$	6389.28	0.497531
$45$	0	0
$46$	20930.0	1.45839
$47$	1267.66	0.0837065	0.0418532	−	0.999124i	$-0.486674\pi$
$47$	1267.66	0.0837065	0.0418532	+	0.999124i	$0.486674\pi$
$48$	−670.775	−0.0420217
$49$	21325.2	1.26883
$50$	0	0
$51$	11136.0	0.599521
$52$	24134.1	1.23772
$53$	−10563.3	−0.516546	−0.258273	−	0.966072i	$-0.583153\pi$
$53$	−10563.3	−0.516546	−0.258273	+	0.966072i	$0.583153\pi$
$54$	6713.29	0.313293
$55$	0	0
$56$	−37411.0	−1.59415
$57$	−19267.6	−0.785490
$58$	−60428.0	−2.35868
$59$	43579.8	1.62988	0.814938	−	0.579548i	$-0.196771\pi$
$59$	43579.8	1.62988	0.814938	+	0.579548i	$0.196771\pi$
$60$	0	0
$61$	49086.8	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$61$	49086.8	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$62$	−58469.8	−1.93176
$63$	15817.2	0.502088
$64$	−52520.7	−1.60280
$65$	0	0
$66$	10028.5	0.283384
$67$	−14785.3	−0.402386	−0.201193	−	0.979552i	$-0.564482\pi$
$67$	−14785.3	−0.402386	−0.201193	+	0.979552i	$0.564482\pi$
$68$	−65336.1	−1.71349
$69$	20455.2	0.517226
$70$	0	0
$71$	8961.61	0.210979	0.105490	−	0.994420i	$-0.466359\pi$
$71$	8961.61	0.210979	0.105490	+	0.994420i	$0.466359\pi$
$72$	−15518.1	−0.352783
$73$	83352.2	1.83067	0.915335	−	0.402694i	$-0.131926\pi$
$73$	83352.2	1.83067	0.915335	+	0.402694i	$0.131926\pi$
$74$	−31092.6	−0.660052
$75$	0	0
$76$	113045.	2.24501
$77$	23628.2	0.454155
$78$	37880.5	0.704983
$79$	−17858.5	−0.321942	−0.160971	−	0.986959i	$-0.551463\pi$
$79$	−17858.5	−0.321942	−0.160971	+	0.986959i	$0.551463\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−36564.1	−0.600511
$83$	66830.4	1.06483	0.532413	−	0.846485i	$-0.321285\pi$
$83$	66830.4	1.06483	0.532413	+	0.846485i	$0.321285\pi$
$84$	−92801.5	−1.43502
$85$	0	0
$86$	35931.5	0.523877
$87$	−59057.2	−0.836517
$88$	−23181.4	−0.319104
$89$	−117907.	−1.57784	−0.788921	−	0.614495i	$-0.789360\pi$
$89$	−117907.	−1.57784	−0.788921	+	0.614495i	$0.789360\pi$
$90$	0	0
$91$	89250.5	1.12981
$92$	−120013.	−1.47828
$93$	−57143.4	−0.685107
$94$	−11673.8	−0.136267
$95$	0	0
$96$	−48998.4	−0.542630
$97$	−156797.	−1.69203	−0.846017	−	0.533155i	$-0.821006\pi$
$97$	−156797.	−1.69203	−0.846017	+	0.533155i	$0.821006\pi$
$98$	−196382.	−2.06555
$99$	9801.00	0.100504
$100$	0	0
$101$	−89018.4	−0.868313	−0.434156	−	0.900838i	$-0.642953\pi$
$101$	−89018.4	−0.868313	−0.434156	+	0.900838i	$0.642953\pi$
$102$	−102550.	−0.975972
$103$	−100268.	−0.931253	−0.465626	−	0.884981i	$-0.654171\pi$
$103$	−100268.	−0.931253	−0.465626	+	0.884981i	$0.654171\pi$
$104$	−87562.6	−0.793844
$105$	0	0
$106$	97276.1	0.840895
$107$	132681.	1.12033	0.560167	−	0.828380i	$-0.310737\pi$
$107$	132681.	1.12033	0.560167	+	0.828380i	$0.310737\pi$
$108$	−38494.1	−0.317566
$109$	88547.8	0.713857	0.356929	−	0.934132i	$-0.383824\pi$
$109$	88547.8	0.713857	0.356929	+	0.934132i	$0.383824\pi$
$110$	0	0
$111$	−30387.3	−0.234091
$112$	14553.9	0.109631
$113$	−92355.2	−0.680401	−0.340201	−	0.940353i	$-0.610495\pi$
$113$	−92355.2	−0.680401	−0.340201	+	0.940353i	$0.610495\pi$
$114$	177434.	1.27872
$115$	0	0
$116$	346495.	2.39085
$117$	37021.1	0.250026
$118$	−401322.	−2.65331
$119$	−241620.	−1.56410
$120$	0	0
$121$	14641.0	0.0909091
$122$	−452036.	−2.74963
$123$	−35734.7	−0.212974
$124$	335266.	1.95810
$125$	0	0
$126$	−145660.	−0.817358
$127$	70624.6	0.388550	0.194275	−	0.980947i	$-0.437765\pi$
$127$	70624.6	0.388550	0.194275	+	0.980947i	$0.437765\pi$
$128$	309442.	1.66938
$129$	35116.4	0.185796
$130$	0	0
$131$	155328.	0.790809	0.395404	−	0.918507i	$-0.370604\pi$
$131$	155328.	0.790809	0.395404	+	0.918507i	$0.370604\pi$
$132$	−57503.5	−0.287250
$133$	418053.	2.04928
$134$	136156.	0.655053
$135$	0	0
$136$	237051.	1.09899
$137$	−70583.9	−0.321295	−0.160647	−	0.987012i	$-0.551358\pi$
$137$	−70583.9	−0.321295	−0.160647	+	0.987012i	$0.551358\pi$
$138$	−188370.	−0.842002
$139$	−400133.	−1.75658	−0.878290	−	0.478129i	$-0.841315\pi$
$139$	−400133.	−1.75658	−0.878290	+	0.478129i	$0.841315\pi$
$140$	0	0
$141$	−11409.0	−0.0483279
$142$	−82526.6	−0.343458
$143$	55303.2	0.226157
$144$	6036.97	0.0242612
$145$	0	0
$146$	−767583.	−2.98018
$147$	−191927.	−0.732558
$148$	178285.	0.669055
$149$	27434.0	0.101233	0.0506166	−	0.998718i	$-0.483881\pi$
$149$	27434.0	0.101233	0.0506166	+	0.998718i	$0.483881\pi$
$150$	0	0
$151$	111764.	0.398894	0.199447	−	0.979909i	$-0.436085\pi$
$151$	111764.	0.398894	0.199447	+	0.979909i	$0.436085\pi$
$152$	−410147.	−1.43989
$153$	−100224.	−0.346133
$154$	−217590.	−0.739329
$155$	0	0
$156$	−217207.	−0.714598
$157$	−231260.	−0.748775	−0.374388	−	0.927272i	$-0.622147\pi$
$157$	−231260.	−0.748775	−0.374388	+	0.927272i	$0.622147\pi$
$158$	164458.	0.524096
$159$	95069.4	0.298228
$160$	0	0
$161$	−443819.	−1.34940
$162$	−60419.6	−0.180880
$163$	219113.	0.645949	0.322974	−	0.946408i	$-0.395317\pi$
$163$	219113.	0.645949	0.322974	+	0.946408i	$0.395317\pi$
$164$	209659.	0.608701
$165$	0	0
$166$	−615435.	−1.73345
$167$	−138345.	−0.383859	−0.191929	−	0.981409i	$-0.561474\pi$
$167$	−138345.	−0.383859	−0.191929	+	0.981409i	$0.561474\pi$
$168$	336699.	0.920384
$169$	−162397.	−0.437383
$170$	0	0
$171$	173409.	0.453503
$172$	−206031.	−0.531022
$173$	314475.	0.798860	0.399430	−	0.916764i	$-0.369208\pi$
$173$	314475.	0.798860	0.399430	+	0.916764i	$0.369208\pi$
$174$	543852.	1.36178
$175$	0	0
$176$	9018.19	0.0219451
$177$	−392218.	−0.941010
$178$	1.08579e6	2.56860
$179$	−725112.	−1.69150	−0.845751	−	0.533578i	$-0.820847\pi$
$179$	−725112.	−1.69150	−0.845751	+	0.533578i	$0.820847\pi$
$180$	0	0
$181$	843502.	1.91377	0.956885	−	0.290466i	$-0.0938102\pi$
$181$	843502.	1.91377	0.956885	+	0.290466i	$0.0938102\pi$
$182$	−821899.	−1.83925
$183$	−441782.	−0.975169
$184$	435426.	0.948134
$185$	0	0
$186$	526228.	1.11530
$187$	−149717.	−0.313089
$188$	66937.6	0.138126
$189$	−142355.	−0.289880
$190$	0	0
$191$	−659387.	−1.30785	−0.653923	−	0.756561i	$-0.726878\pi$
$191$	−659387.	−1.30785	−0.653923	+	0.756561i	$0.726878\pi$
$192$	472686.	0.925380
$193$	350322.	0.676978	0.338489	−	0.940970i	$-0.390084\pi$
$193$	350322.	0.676978	0.338489	+	0.940970i	$0.390084\pi$
$194$	1.44393e6	2.75450
$195$	0	0
$196$	1.12605e6	2.09372
$197$	−504269.	−0.925757	−0.462878	−	0.886422i	$-0.653183\pi$
$197$	−504269.	−0.925757	−0.462878	+	0.886422i	$0.653183\pi$
$198$	−90256.5	−0.163612
$199$	−939943.	−1.68255	−0.841276	−	0.540605i	$-0.818195\pi$
$199$	−939943.	−1.68255	−0.841276	+	0.540605i	$0.818195\pi$
$200$	0	0
$201$	133068.	0.232318
$202$	819762.	1.41354
$203$	1.28137e6	2.18241
$204$	588025.	0.989283
$205$	0	0
$206$	923355.	1.51601
$207$	−184096.	−0.298620
$208$	34064.2	0.0545934
$209$	259042.	0.410209
$210$	0	0
$211$	666610.	1.03078	0.515390	−	0.856956i	$-0.327647\pi$
$211$	666610.	1.03078	0.515390	+	0.856956i	$0.327647\pi$
$212$	−557782.	−0.852364
$213$	−80654.5	−0.121809
$214$	−1.22184e6	−1.82381
$215$	0	0
$216$	139663.	0.203679
$217$	1.23985e6	1.78739
$218$	−815428.	−1.16210
$219$	−750170.	−1.05694
$220$	0	0
$221$	−565525.	−0.778881
$222$	279834.	0.381081
$223$	459829.	0.619205	0.309602	−	0.950866i	$-0.399804\pi$
$223$	459829.	0.619205	0.309602	+	0.950866i	$0.399804\pi$
$224$	1.06313e6	1.41568
$225$	0	0
$226$	850490.	1.10764
$227$	−1.11522e6	−1.43647	−0.718236	−	0.695799i	$-0.755050\pi$
$227$	−1.11522e6	−1.43647	−0.718236	+	0.695799i	$0.755050\pi$
$228$	−1.01741e6	−1.29616
$229$	1.19306e6	1.50340	0.751700	−	0.659505i	$-0.229234\pi$
$229$	1.19306e6	1.50340	0.751700	+	0.659505i	$0.229234\pi$
$230$	0	0
$231$	−212654.	−0.262207
$232$	−1.25714e6	−1.53343
$233$	−675874.	−0.815598	−0.407799	−	0.913072i	$-0.633704\pi$
$233$	−675874.	−0.815598	−0.407799	+	0.913072i	$0.633704\pi$
$234$	−340924.	−0.407022
$235$	0	0
$236$	2.30118e6	2.68950
$237$	160727.	0.185873
$238$	2.22506e6	2.54624
$239$	436348.	0.494127	0.247063	−	0.968999i	$-0.420534\pi$
$239$	436348.	0.494127	0.247063	+	0.968999i	$0.420534\pi$
$240$	0	0
$241$	597010.	0.662123	0.331062	−	0.943609i	$-0.392593\pi$
$241$	597010.	0.662123	0.331062	+	0.943609i	$0.392593\pi$
$242$	−134828.	−0.147993
$243$	−59049.0	−0.0641500
$244$	2.59198e6	2.78713
$245$	0	0
$246$	329077.	0.346705
$247$	978476.	1.02049
$248$	−1.21640e6	−1.25588
$249$	−601474.	−0.614778
$250$	0	0
$251$	−75812.0	−0.0759545	−0.0379773	−	0.999279i	$-0.512091\pi$
$251$	−75812.0	−0.0759545	−0.0379773	+	0.999279i	$0.512091\pi$
$252$	835213.	0.828507
$253$	−275008.	−0.270112
$254$	−650376.	−0.632528
$255$	0	0
$256$	−1.16896e6	−1.11480
$257$	1.20303e6	1.13617	0.568087	−	0.822969i	$-0.307684\pi$
$257$	1.20303e6	1.13617	0.568087	+	0.822969i	$0.307684\pi$
$258$	−323383.	−0.302460
$259$	659319.	0.610725
$260$	0	0
$261$	531515.	0.482963
$262$	−1.43040e6	−1.28737
$263$	−332575.	−0.296483	−0.148242	−	0.988951i	$-0.547361\pi$
$263$	−332575.	−0.296483	−0.148242	+	0.988951i	$0.547361\pi$
$264$	208632.	0.184235
$265$	0	0
$266$	−3.84981e6	−3.33607
$267$	1.06116e6	0.910968
$268$	−780722.	−0.663987
$269$	1.60565e6	1.35291	0.676455	−	0.736484i	$-0.263515\pi$
$269$	1.60565e6	1.35291	0.676455	+	0.736484i	$0.263515\pi$
$270$	0	0
$271$	−973760.	−0.805432	−0.402716	−	0.915325i	$-0.631934\pi$
$271$	−973760.	−0.805432	−0.402716	+	0.915325i	$0.631934\pi$
$272$	−92219.2	−0.0755786
$273$	−803254.	−0.652299
$274$	650000.	0.523043
$275$	0	0
$276$	1.08011e6	0.853486
$277$	430301.	0.336956	0.168478	−	0.985705i	$-0.446115\pi$
$277$	430301.	0.336956	0.168478	+	0.985705i	$0.446115\pi$
$278$	3.68479e6	2.85957
$279$	514290.	0.395547
$280$	0	0
$281$	−566489.	−0.427982	−0.213991	−	0.976836i	$-0.568646\pi$
$281$	−566489.	−0.427982	−0.213991	+	0.976836i	$0.568646\pi$
$282$	105064.	0.0786740
$283$	962280.	0.714225	0.357113	−	0.934061i	$-0.383761\pi$
$283$	962280.	0.714225	0.357113	+	0.934061i	$0.383761\pi$
$284$	473208.	0.348142
$285$	0	0
$286$	−509282.	−0.368165
$287$	775342.	0.555633
$288$	440986.	0.313288
$289$	111140.	0.0782753
$290$	0	0
$291$	1.41118e6	0.976897
$292$	4.40133e6	3.02083
$293$	762793.	0.519084	0.259542	−	0.965732i	$-0.416428\pi$
$293$	762793.	0.519084	0.259542	+	0.965732i	$0.416428\pi$
$294$	1.76744e6	1.19255
$295$	0	0
$296$	−646850.	−0.429115
$297$	−88209.0	−0.0580259
$298$	−252637.	−0.164800
$299$	−1.03878e6	−0.671966
$300$	0	0
$301$	−761926.	−0.484726
$302$	−1.02922e6	−0.649368
$303$	801165.	0.501321
$304$	159558.	0.0990229
$305$	0	0
$306$	922954.	0.563478
$307$	−3.02419e6	−1.83132	−0.915658	−	0.401958i	$-0.868330\pi$
$307$	−3.02419e6	−1.83132	−0.915658	+	0.401958i	$0.868330\pi$
$308$	1.24766e6	0.749412
$309$	902408.	0.537659
$310$	0	0
$311$	2.33903e6	1.37131	0.685654	−	0.727928i	$-0.259516\pi$
$311$	2.33903e6	1.37131	0.685654	+	0.727928i	$0.259516\pi$
$312$	788063.	0.458326
$313$	−1.42601e6	−0.822740	−0.411370	−	0.911469i	$-0.634950\pi$
$313$	−1.42601e6	−0.822740	−0.411370	+	0.911469i	$0.634950\pi$
$314$	2.12965e6	1.21895
$315$	0	0
$316$	−943001.	−0.531244
$317$	1.60632e6	0.897809	0.448904	−	0.893580i	$-0.351814\pi$
$317$	1.60632e6	0.897809	0.448904	+	0.893580i	$0.351814\pi$
$318$	−875485.	−0.485491
$319$	793991.	0.436857
$320$	0	0
$321$	−1.19412e6	−0.646825
$322$	4.08709e6	2.19672
$323$	−2.64894e6	−1.41275
$324$	346447.	0.183347
$325$	0	0
$326$	−2.01779e6	−1.05155
$327$	−796930.	−0.412146
$328$	−760678.	−0.390406
$329$	247542.	0.126084
$330$	0	0
$331$	3.11538e6	1.56294	0.781469	−	0.623945i	$-0.214471\pi$
$331$	3.11538e6	1.56294	0.781469	+	0.623945i	$0.214471\pi$
$332$	3.52891e6	1.75710
$333$	273486.	0.135152
$334$	1.27400e6	0.624892
$335$	0	0
$336$	−130985.	−0.0632957
$337$	1.61111e6	0.772768	0.386384	−	0.922338i	$-0.373724\pi$
$337$	1.61111e6	0.772768	0.386384	+	0.922338i	$0.373724\pi$
$338$	1.49550e6	0.712025
$339$	831197.	0.392830
$340$	0	0
$341$	768261.	0.357786
$342$	−1.59690e6	−0.738267
$343$	882288.	0.404925
$344$	747517.	0.340584
$345$	0	0
$346$	−2.89597e6	−1.30048
$347$	−847409.	−0.377806	−0.188903	−	0.981996i	$-0.560493\pi$
$347$	−847409.	−0.377806	−0.188903	+	0.981996i	$0.560493\pi$
$348$	−3.11845e6	−1.38036
$349$	2.75989e6	1.21291	0.606455	−	0.795118i	$-0.292591\pi$
$349$	2.75989e6	1.21291	0.606455	+	0.795118i	$0.292591\pi$
$350$	0	0
$351$	−333190.	−0.144353
$352$	658756.	0.283379
$353$	4.12207e6	1.76067	0.880336	−	0.474351i	$-0.157317\pi$
$353$	4.12207e6	1.76067	0.880336	+	0.474351i	$0.157317\pi$
$354$	3.61190e6	1.53189
$355$	0	0
$356$	−6.22594e6	−2.60363
$357$	2.17458e6	0.903036
$358$	6.67749e6	2.75363
$359$	585248.	0.239664	0.119832	−	0.992794i	$-0.461764\pi$
$359$	585248.	0.239664	0.119832	+	0.992794i	$0.461764\pi$
$360$	0	0
$361$	2.10712e6	0.850985
$362$	−7.76773e6	−3.11547
$363$	−131769.	−0.0524864
$364$	4.71278e6	1.86433
$365$	0	0
$366$	4.06832e6	1.58750
$367$	−3.47515e6	−1.34681	−0.673407	−	0.739272i	$-0.735170\pi$
$367$	−3.47515e6	−1.34681	−0.673407	+	0.739272i	$0.735170\pi$
$368$	−169393.	−0.0652041
$369$	321612.	0.122961
$370$	0	0
$371$	−2.06274e6	−0.778053
$372$	−3.01740e6	−1.13051
$373$	−2.08904e6	−0.777453	−0.388726	−	0.921353i	$-0.627085\pi$
$373$	−2.08904e6	−0.777453	−0.388726	+	0.921353i	$0.627085\pi$
$374$	1.37873e6	0.509685
$375$	0	0
$376$	−242861.	−0.0885906
$377$	2.99913e6	1.08678
$378$	1.31094e6	0.471902
$379$	−1.41198e6	−0.504931	−0.252466	−	0.967606i	$-0.581241\pi$
$379$	−1.41198e6	−0.504931	−0.252466	+	0.967606i	$0.581241\pi$
$380$	0	0
$381$	−635622.	−0.224329
$382$	6.07223e6	2.12907
$383$	2.02171e6	0.704241	0.352121	−	0.935955i	$-0.385461\pi$
$383$	2.02171e6	0.704241	0.352121	+	0.935955i	$0.385461\pi$
$384$	−2.78498e6	−0.963814
$385$	0	0
$386$	−3.22608e6	−1.10207
$387$	−316047.	−0.107269
$388$	−8.27952e6	−2.79207
$389$	−2.12481e6	−0.711943	−0.355971	−	0.934497i	$-0.615850\pi$
$389$	−2.12481e6	−0.711943	−0.355971	+	0.934497i	$0.615850\pi$
$390$	0	0
$391$	2.81221e6	0.930263
$392$	−4.08551e6	−1.34286
$393$	−1.39795e6	−0.456574
$394$	4.64377e6	1.50706
$395$	0	0
$396$	517531.	0.165844
$397$	−2.11404e6	−0.673188	−0.336594	−	0.941650i	$-0.609275\pi$
$397$	−2.11404e6	−0.673188	−0.336594	+	0.941650i	$0.609275\pi$
$398$	8.65584e6	2.73906
$399$	−3.76248e6	−1.18315
$400$	0	0
$401$	1.00967e6	0.313558	0.156779	−	0.987634i	$-0.449889\pi$
$401$	1.00967e6	0.313558	0.156779	+	0.987634i	$0.449889\pi$
$402$	−1.22541e6	−0.378195
$403$	2.90194e6	0.890073
$404$	−4.70052e6	−1.43282
$405$	0	0
$406$	−1.18001e7	−3.55279
$407$	408540.	0.122250
$408$	−2.13345e6	−0.634502
$409$	−190771.	−0.0563904	−0.0281952	−	0.999602i	$-0.508976\pi$
$409$	−190771.	−0.0563904	−0.0281952	+	0.999602i	$0.508976\pi$
$410$	0	0
$411$	635255.	0.185500
$412$	−5.29453e6	−1.53668
$413$	8.51002e6	2.45502
$414$	1.69533e6	0.486130
$415$	0	0
$416$	2.48831e6	0.704970
$417$	3.60120e6	1.01416
$418$	−2.38550e6	−0.667787
$419$	−4.82015e6	−1.34130	−0.670650	−	0.741774i	$-0.733985\pi$
$419$	−4.82015e6	−1.34130	−0.670650	+	0.741774i	$0.733985\pi$
$420$	0	0
$421$	−1.47210e6	−0.404791	−0.202395	−	0.979304i	$-0.564873\pi$
$421$	−1.47210e6	−0.404791	−0.202395	+	0.979304i	$0.564873\pi$
$422$	−6.13875e6	−1.67803
$423$	102681.	0.0279022
$424$	2.02373e6	0.546685
$425$	0	0
$426$	742739.	0.198295
$427$	9.58542e6	2.54414
$428$	7.00605e6	1.84869
$429$	−497729.	−0.130572
$430$	0	0
$431$	−3.75922e6	−0.974774	−0.487387	−	0.873186i	$-0.662050\pi$
$431$	−3.75922e6	−0.974774	−0.487387	+	0.873186i	$0.662050\pi$
$432$	−54332.8	−0.0140072
$433$	3.03901e6	0.778955	0.389478	−	0.921036i	$-0.372656\pi$
$433$	3.03901e6	0.778955	0.389478	+	0.921036i	$0.372656\pi$
$434$	−1.14177e7	−2.90973
$435$	0	0
$436$	4.67567e6	1.17795
$437$	−4.86571e6	−1.21883
$438$	6.90824e6	1.72061
$439$	−395912.	−0.0980477	−0.0490239	−	0.998798i	$-0.515611\pi$
$439$	−395912.	−0.0980477	−0.0490239	+	0.998798i	$0.515611\pi$
$440$	0	0
$441$	1.72734e6	0.422943
$442$	5.20787e6	1.26796
$443$	−6.25166e6	−1.51351	−0.756757	−	0.653697i	$-0.773217\pi$
$443$	−6.25166e6	−1.51351	−0.756757	+	0.653697i	$0.773217\pi$
$444$	−1.60457e6	−0.386279
$445$	0	0
$446$	−4.23452e6	−1.00802
$447$	−246906.	−0.0584470
$448$	−1.02560e7	−2.41425
$449$	−3.94443e6	−0.923355	−0.461678	−	0.887048i	$-0.652752\pi$
$449$	−3.94443e6	−0.923355	−0.461678	+	0.887048i	$0.652752\pi$
$450$	0	0
$451$	480433.	0.111222
$452$	−4.87672e6	−1.12275
$453$	−1.00587e6	−0.230302
$454$	1.02700e7	2.33846
$455$	0	0
$456$	3.69132e6	0.831323
$457$	−6.70635e6	−1.50209	−0.751045	−	0.660251i	$-0.770450\pi$
$457$	−6.70635e6	−1.50209	−0.751045	+	0.660251i	$0.770450\pi$
$458$	−1.09868e7	−2.44741
$459$	902017.	0.199840
$460$	0	0
$461$	−4.66837e6	−1.02309	−0.511544	−	0.859257i	$-0.670926\pi$
$461$	−4.66837e6	−1.02309	−0.511544	+	0.859257i	$0.670926\pi$
$462$	1.95831e6	0.426852
$463$	−5.02305e6	−1.08897	−0.544484	−	0.838771i	$-0.683275\pi$
$463$	−5.02305e6	−1.08897	−0.544484	+	0.838771i	$0.683275\pi$
$464$	489063.	0.105456
$465$	0	0
$466$	6.22406e6	1.32773
$467$	−1.73795e6	−0.368762	−0.184381	−	0.982855i	$-0.559028\pi$
$467$	−1.73795e6	−0.368762	−0.184381	+	0.982855i	$0.559028\pi$
$468$	1.95486e6	0.412574
$469$	−2.88719e6	−0.606099
$470$	0	0
$471$	2.08134e6	0.432306
$472$	−8.34908e6	−1.72498
$473$	−472120.	−0.0970285
$474$	−1.48012e6	−0.302587
$475$	0	0
$476$	−1.27585e7	−2.58096
$477$	−855625.	−0.172182
$478$	−4.01829e6	−0.804398
$479$	−761980.	−0.151742	−0.0758708	−	0.997118i	$-0.524174\pi$
$479$	−761980.	−0.151742	−0.0758708	+	0.997118i	$0.524174\pi$
$480$	0	0
$481$	1.54317e6	0.304125
$482$	−5.49781e6	−1.07788
$483$	3.99438e6	0.779078
$484$	773103.	0.150011
$485$	0	0
$486$	543777.	0.104431
$487$	−840046.	−0.160502	−0.0802510	−	0.996775i	$-0.525572\pi$
$487$	−840046.	−0.160502	−0.0802510	+	0.996775i	$0.525572\pi$
$488$	−9.40413e6	−1.78760
$489$	−1.97201e6	−0.372939
$490$	0	0
$491$	1.05708e7	1.97882	0.989409	−	0.145156i	$-0.0463684\pi$
$491$	1.05708e7	1.97882	0.989409	+	0.145156i	$0.0463684\pi$
$492$	−1.88693e6	−0.351434
$493$	−8.11928e6	−1.50453
$494$	−9.01069e6	−1.66127
$495$	0	0
$496$	473214.	0.0863681
$497$	1.74997e6	0.317790
$498$	5.53891e6	1.00081
$499$	2.87438e6	0.516765	0.258382	−	0.966043i	$-0.416811\pi$
$499$	2.87438e6	0.516765	0.258382	+	0.966043i	$0.416811\pi$
$500$	0	0
$501$	1.24510e6	0.221621
$502$	698145.	0.123648
$503$	−1.40949e6	−0.248395	−0.124197	−	0.992258i	$-0.539636\pi$
$503$	−1.40949e6	−0.248395	−0.124197	+	0.992258i	$0.539636\pi$
$504$	−3.03029e6	−0.531384
$505$	0	0
$506$	2.53253e6	0.439721
$507$	1.46158e6	0.252523
$508$	3.72926e6	0.641156
$509$	−7.60203e6	−1.30057	−0.650287	−	0.759688i	$-0.725352\pi$
$509$	−7.60203e6	−1.30057	−0.650287	+	0.759688i	$0.725352\pi$
$510$	0	0
$511$	1.62766e7	2.75747
$512$	862682.	0.145437
$513$	−1.56068e6	−0.261830
$514$	−1.10786e7	−1.84960
$515$	0	0
$516$	1.85428e6	0.306586
$517$	153387.	0.0252384
$518$	−6.07160e6	−0.994212
$519$	−2.83027e6	−0.461222
$520$	0	0
$521$	−2.35016e6	−0.379318	−0.189659	−	0.981850i	$-0.560738\pi$
$521$	−2.35016e6	−0.379318	−0.189659	+	0.981850i	$0.560738\pi$
$522$	−4.89467e6	−0.786225
$523$	8.59353e6	1.37378	0.686891	−	0.726761i	$-0.258975\pi$
$523$	8.59353e6	1.37378	0.686891	+	0.726761i	$0.258975\pi$
$524$	8.20193e6	1.30493
$525$	0	0
$526$	3.06265e6	0.482651
$527$	−7.85616e6	−1.23221
$528$	−81163.7	−0.0126700
$529$	−1.27074e6	−0.197432
$530$	0	0
$531$	3.52996e6	0.543292
$532$	2.20748e7	3.38157
$533$	1.81473e6	0.276690
$534$	−9.77213e6	−1.48298
$535$	0	0
$536$	2.83259e6	0.425865
$537$	6.52601e6	0.976589
$538$	−1.47862e7	−2.20243
$539$	2.58035e6	0.382566
$540$	0	0
$541$	−3.76377e6	−0.552878	−0.276439	−	0.961031i	$-0.589154\pi$
$541$	−3.76377e6	−0.552878	−0.276439	+	0.961031i	$0.589154\pi$
$542$	8.96726e6	1.31118
$543$	−7.59152e6	−1.10492
$544$	−6.73638e6	−0.975954
$545$	0	0
$546$	7.39709e6	1.06189
$547$	8.53624e6	1.21983	0.609914	−	0.792468i	$-0.291204\pi$
$547$	8.53624e6	1.21983	0.609914	+	0.792468i	$0.291204\pi$
$548$	−3.72711e6	−0.530176
$549$	3.97603e6	0.563014
$550$	0	0
$551$	1.40480e7	1.97123
$552$	−3.91883e6	−0.547405
$553$	−3.48732e6	−0.484930
$554$	−3.96260e6	−0.548537
$555$	0	0
$556$	−2.11286e7	−2.89857
$557$	−5.80583e6	−0.792915	−0.396457	−	0.918053i	$-0.629761\pi$
$557$	−5.80583e6	−0.792915	−0.396457	+	0.918053i	$0.629761\pi$
$558$	−4.73605e6	−0.643919
$559$	−1.78333e6	−0.241380
$560$	0	0
$561$	1.34746e6	0.180762
$562$	5.21674e6	0.696720
$563$	7.65501e6	1.01783	0.508914	−	0.860817i	$-0.330047\pi$
$563$	7.65501e6	1.01783	0.508914	+	0.860817i	$0.330047\pi$
$564$	−602438.	−0.0797471
$565$	0	0
$566$	−8.86154e6	−1.16270
$567$	1.28120e6	0.167363
$568$	−1.71688e6	−0.223290
$569$	1.92933e6	0.249819	0.124909	−	0.992168i	$-0.460136\pi$
$569$	1.92933e6	0.249819	0.124909	+	0.992168i	$0.460136\pi$
$570$	0	0
$571$	−1.05928e7	−1.35962	−0.679812	−	0.733386i	$-0.737939\pi$
$571$	−1.05928e7	−1.35962	−0.679812	+	0.733386i	$0.737939\pi$
$572$	2.92023e6	0.373187
$573$	5.93448e6	0.755086
$574$	−7.14005e6	−0.904527
$575$	0	0
$576$	−4.25418e6	−0.534268
$577$	3.01519e6	0.377030	0.188515	−	0.982070i	$-0.439633\pi$
$577$	3.01519e6	0.377030	0.188515	+	0.982070i	$0.439633\pi$
$578$	−1.02348e6	−0.127426
$579$	−3.15290e6	−0.390853
$580$	0	0
$581$	1.30503e7	1.60391
$582$	−1.29954e7	−1.59031
$583$	−1.27816e6	−0.155744
$584$	−1.59687e7	−1.93749
$585$	0	0
$586$	−7.02449e6	−0.845027
$587$	−2.91411e6	−0.349069	−0.174534	−	0.984651i	$-0.555842\pi$
$587$	−2.91411e6	−0.349069	−0.174534	+	0.984651i	$0.555842\pi$
$588$	−1.01345e7	−1.20881
$589$	1.35928e7	1.61444
$590$	0	0
$591$	4.53842e6	0.534486
$592$	251642.	0.0295107
$593$	1.55954e7	1.82121	0.910607	−	0.413274i	$-0.135615\pi$
$593$	1.55954e7	1.82121	0.910607	+	0.413274i	$0.135615\pi$
$594$	812308.	0.0944615
$595$	0	0
$596$	1.44862e6	0.167047
$597$	8.45948e6	0.971422
$598$	9.56606e6	1.09391
$599$	−3.70805e6	−0.422259	−0.211130	−	0.977458i	$-0.567714\pi$
$599$	−3.70805e6	−0.422259	−0.211130	+	0.977458i	$0.567714\pi$
$600$	0	0
$601$	7.22245e6	0.815640	0.407820	−	0.913062i	$-0.366289\pi$
$601$	7.22245e6	0.815640	0.407820	+	0.913062i	$0.366289\pi$
$602$	7.01651e6	0.789096
$603$	−1.19761e6	−0.134129
$604$	5.90156e6	0.658225
$605$	0	0
$606$	−7.37785e6	−0.816110
$607$	1.06431e7	1.17246	0.586228	−	0.810146i	$-0.300612\pi$
$607$	1.06431e7	1.17246	0.586228	+	0.810146i	$0.300612\pi$
$608$	1.16553e7	1.27869
$609$	−1.15324e7	−1.26001
$610$	0	0
$611$	579386.	0.0627863
$612$	−5.29223e6	−0.571163
$613$	7.11493e6	0.764750	0.382375	−	0.924007i	$-0.375106\pi$
$613$	7.11493e6	0.764750	0.382375	+	0.924007i	$0.375106\pi$
$614$	2.78495e7	2.98124
$615$	0	0
$616$	−4.52673e6	−0.480655
$617$	1.83690e6	0.194255	0.0971277	−	0.995272i	$-0.469034\pi$
$617$	1.83690e6	0.194255	0.0971277	+	0.995272i	$0.469034\pi$
$618$	−8.31019e6	−0.875266
$619$	1.10798e7	1.16226	0.581132	−	0.813810i	$-0.302610\pi$
$619$	1.10798e7	1.16226	0.581132	+	0.813810i	$0.302610\pi$
$620$	0	0
$621$	1.65687e6	0.172409
$622$	−2.15399e7	−2.23238
$623$	−2.30242e7	−2.37665
$624$	−306578.	−0.0315195
$625$	0	0
$626$	1.31320e7	1.33935
$627$	−2.33138e6	−0.236834
$628$	−1.22114e7	−1.23557
$629$	−4.17769e6	−0.421027
$630$	0	0
$631$	1.34216e7	1.34193	0.670966	−	0.741488i	$-0.265880\pi$
$631$	1.34216e7	1.34193	0.670966	+	0.741488i	$0.265880\pi$
$632$	3.42137e6	0.340727
$633$	−5.99949e6	−0.595121
$634$	−1.47924e7	−1.46156
$635$	0	0
$636$	5.02004e6	0.492112
$637$	9.74670e6	0.951720
$638$	−7.31179e6	−0.711168
$639$	725890.	0.0703265
$640$	0	0
$641$	8.26487e6	0.794495	0.397247	−	0.917712i	$-0.369965\pi$
$641$	8.26487e6	0.794495	0.397247	+	0.917712i	$0.369965\pi$
$642$	1.09966e7	1.05298
$643$	9.43394e6	0.899841	0.449920	−	0.893069i	$-0.351452\pi$
$643$	9.43394e6	0.899841	0.449920	+	0.893069i	$0.351452\pi$
$644$	−2.34354e7	−2.22668
$645$	0	0
$646$	2.43939e7	2.29985
$647$	9.83647e6	0.923802	0.461901	−	0.886932i	$-0.347168\pi$
$647$	9.83647e6	0.923802	0.461901	+	0.886932i	$0.347168\pi$
$648$	−1.25697e6	−0.117594
$649$	5.27315e6	0.491426
$650$	0	0
$651$	−1.11587e7	−1.03195
$652$	1.15700e7	1.06590
$653$	2.07669e7	1.90585	0.952924	−	0.303208i	$-0.0980577\pi$
$653$	2.07669e7	1.90585	0.952924	+	0.303208i	$0.0980577\pi$
$654$	7.33885e6	0.670940
$655$	0	0
$656$	295925.	0.0268486
$657$	6.75153e6	0.610223
$658$	−2.27959e6	−0.205255
$659$	−1.74284e7	−1.56331	−0.781654	−	0.623713i	$-0.785624\pi$
$659$	−1.74284e7	−1.56331	−0.781654	+	0.623713i	$0.785624\pi$
$660$	0	0
$661$	−1.91510e7	−1.70486	−0.852428	−	0.522844i	$-0.824871\pi$
$661$	−1.91510e7	−1.70486	−0.852428	+	0.522844i	$0.824871\pi$
$662$	−2.86893e7	−2.54434
$663$	5.08972e6	0.449687
$664$	−1.28035e7	−1.12696
$665$	0	0
$666$	−2.51850e6	−0.220017
$667$	−1.49139e7	−1.29800
$668$	−7.30515e6	−0.633415
$669$	−4.13846e6	−0.357498
$670$	0	0
$671$	5.93951e6	0.509265
$672$	−9.56815e6	−0.817343
$673$	342635.	0.0291605	0.0145802	−	0.999894i	$-0.495359\pi$
$673$	342635.	0.0291605	0.0145802	+	0.999894i	$0.495359\pi$
$674$	−1.48365e7	−1.25801
$675$	0	0
$676$	−8.57522e6	−0.721737
$677$	4.92746e6	0.413191	0.206596	−	0.978426i	$-0.433762\pi$
$677$	4.92746e6	0.413191	0.206596	+	0.978426i	$0.433762\pi$
$678$	−7.65441e6	−0.639496
$679$	−3.06185e7	−2.54865
$680$	0	0
$681$	1.00370e7	0.829348
$682$	−7.07484e6	−0.582446
$683$	1.73541e7	1.42348	0.711740	−	0.702443i	$-0.247907\pi$
$683$	1.73541e7	1.42348	0.711740	+	0.702443i	$0.247907\pi$
$684$	9.15666e6	0.748336
$685$	0	0
$686$	−8.12491e6	−0.659186
$687$	−1.07376e7	−0.867988
$688$	−290805.	−0.0234223
$689$	−4.82795e6	−0.387449
$690$	0	0
$691$	1.50935e7	1.20253	0.601264	−	0.799051i	$-0.294664\pi$
$691$	1.50935e7	1.20253	0.601264	+	0.799051i	$0.294664\pi$
$692$	1.66055e7	1.31822
$693$	1.91389e6	0.151385
$694$	7.80371e6	0.615039
$695$	0	0
$696$	1.13143e7	0.885326
$697$	−4.91286e6	−0.383047
$698$	−2.54156e7	−1.97452
$699$	6.08287e6	0.470886
$700$	0	0
$701$	1.70526e7	1.31068	0.655338	−	0.755336i	$-0.272526\pi$
$701$	1.70526e7	1.31068	0.655338	+	0.755336i	$0.272526\pi$
$702$	3.06832e6	0.234994
$703$	7.22828e6	0.551628
$704$	−6.35501e6	−0.483264
$705$	0	0
$706$	−3.79597e7	−2.86623
$707$	−1.73830e7	−1.30791
$708$	−2.07106e7	−1.55278
$709$	4.47464e6	0.334305	0.167152	−	0.985931i	$-0.446543\pi$
$709$	4.47464e6	0.334305	0.167152	+	0.985931i	$0.446543\pi$
$710$	0	0
$711$	−1.44654e6	−0.107314
$712$	2.25888e7	1.66991
$713$	−1.44306e7	−1.06307
$714$	−2.00255e7	−1.47007
$715$	0	0
$716$	−3.82888e7	−2.79119
$717$	−3.92713e6	−0.285284
$718$	−5.38949e6	−0.390155
$719$	2.05300e7	1.48104	0.740519	−	0.672036i	$-0.234580\pi$
$719$	2.05300e7	1.48104	0.740519	+	0.672036i	$0.234580\pi$
$720$	0	0
$721$	−1.95797e7	−1.40271
$722$	−1.94043e7	−1.38534
$723$	−5.37309e6	−0.382277
$724$	4.45403e7	3.15796
$725$	0	0
$726$	1.21345e6	0.0854436
$727$	1.49603e7	1.04979	0.524897	−	0.851166i	$-0.324104\pi$
$727$	1.49603e7	1.04979	0.524897	+	0.851166i	$0.324104\pi$
$728$	−1.70987e7	−1.19574
$729$	531441.	0.0370370
$730$	0	0
$731$	4.82785e6	0.334165
$732$	−2.33278e7	−1.60915
$733$	2.26287e7	1.55561	0.777804	−	0.628507i	$-0.216334\pi$
$733$	2.26287e7	1.55561	0.777804	+	0.628507i	$0.216334\pi$
$734$	3.20023e7	2.19251
$735$	0	0
$736$	−1.23737e7	−0.841987
$737$	−1.78902e6	−0.121324
$738$	−2.96169e6	−0.200170
$739$	−1.39790e7	−0.941597	−0.470798	−	0.882241i	$-0.656034\pi$
$739$	−1.39790e7	−0.941597	−0.470798	+	0.882241i	$0.656034\pi$
$740$	0	0
$741$	−8.80628e6	−0.589179
$742$	1.89956e7	1.26661
$743$	−5.66041e6	−0.376163	−0.188081	−	0.982153i	$-0.560227\pi$
$743$	−5.66041e6	−0.376163	−0.188081	+	0.982153i	$0.560227\pi$
$744$	1.09476e7	0.725082
$745$	0	0
$746$	1.92377e7	1.26563
$747$	5.41326e6	0.354942
$748$	−7.90567e6	−0.516636
$749$	2.59091e7	1.68752
$750$	0	0
$751$	−1.77912e7	−1.15108	−0.575540	−	0.817774i	$-0.695208\pi$
$751$	−1.77912e7	−1.15108	−0.575540	+	0.817774i	$0.695208\pi$
$752$	94479.5	0.00609247
$753$	682308.	0.0438524
$754$	−2.76187e7	−1.76919
$755$	0	0
$756$	−7.51692e6	−0.478338
$757$	2.55861e7	1.62280	0.811398	−	0.584494i	$-0.198707\pi$
$757$	2.55861e7	1.62280	0.811398	+	0.584494i	$0.198707\pi$
$758$	1.30028e7	0.821987
$759$	2.47507e6	0.155949
$760$	0	0
$761$	−8.89006e6	−0.556472	−0.278236	−	0.960513i	$-0.589750\pi$
$761$	−8.89006e6	−0.556472	−0.278236	+	0.960513i	$0.589750\pi$
$762$	5.85338e6	0.365190
$763$	1.72911e7	1.07526
$764$	−3.48182e7	−2.15811
$765$	0	0
$766$	−1.86177e7	−1.14645
$767$	1.99182e7	1.22253
$768$	1.05206e7	0.643633
$769$	−2.70789e7	−1.65126	−0.825629	−	0.564213i	$-0.809180\pi$
$769$	−2.70789e7	−1.65126	−0.825629	+	0.564213i	$0.809180\pi$
$770$	0	0
$771$	−1.08273e7	−0.655970
$772$	1.84984e7	1.11710
$773$	−3.17581e7	−1.91164	−0.955820	−	0.293953i	$-0.905029\pi$
$773$	−3.17581e7	−1.91164	−0.955820	+	0.293953i	$0.905029\pi$
$774$	2.91045e6	0.174626
$775$	0	0
$776$	3.00395e7	1.79076
$777$	−5.93387e6	−0.352602
$778$	1.95671e7	1.15899
$779$	8.50027e6	0.501868
$780$	0	0
$781$	1.08435e6	0.0636127
$782$	−2.58974e7	−1.51439
$783$	−4.78363e6	−0.278839
$784$	1.58938e6	0.0923500
$785$	0	0
$786$	1.28736e7	0.743265
$787$	4.19625e6	0.241504	0.120752	−	0.992683i	$-0.461469\pi$
$787$	4.19625e6	0.241504	0.120752	+	0.992683i	$0.461469\pi$
$788$	−2.66274e7	−1.52761
$789$	2.99318e6	0.171175
$790$	0	0
$791$	−1.80346e7	−1.02486
$792$	−1.87769e6	−0.106368
$793$	2.24352e7	1.26691
$794$	1.94680e7	1.09590
$795$	0	0
$796$	−4.96327e7	−2.77642
$797$	−320849.	−0.0178918	−0.00894591	−	0.999960i	$-0.502848\pi$
$797$	−320849.	−0.0178918	−0.00894591	+	0.999960i	$0.502848\pi$
$798$	3.46483e7	1.92608
$799$	−1.56852e6	−0.0869208
$800$	0	0
$801$	−9.55044e6	−0.525947
$802$	−9.29794e6	−0.510447
$803$	1.00856e7	0.551968
$804$	7.02650e6	0.383353
$805$	0	0
$806$	−2.67237e7	−1.44897
$807$	−1.44508e7	−0.781103
$808$	1.70543e7	0.918978
$809$	2.32915e7	1.25120	0.625600	−	0.780144i	$-0.284854\pi$
$809$	2.32915e7	1.25120	0.625600	+	0.780144i	$0.284854\pi$
$810$	0	0
$811$	1.65090e7	0.881389	0.440695	−	0.897657i	$-0.354732\pi$
$811$	1.65090e7	0.881389	0.440695	+	0.897657i	$0.354732\pi$
$812$	6.76616e7	3.60124
$813$	8.76384e6	0.465016
$814$	−3.76221e6	−0.199013
$815$	0	0
$816$	829973.	0.0436353
$817$	−8.35319e6	−0.437822
$818$	1.75680e6	0.0917990
$819$	7.22929e6	0.376605
$820$	0	0
$821$	−3.37457e7	−1.74727	−0.873637	−	0.486578i	$-0.838245\pi$
$821$	−3.37457e7	−1.74727	−0.873637	+	0.486578i	$0.838245\pi$
$822$	−5.85000e6	−0.301979
$823$	−1.19897e7	−0.617036	−0.308518	−	0.951219i	$-0.599833\pi$
$823$	−1.19897e7	−0.617036	−0.308518	+	0.951219i	$0.599833\pi$
$824$	1.92094e7	0.985590
$825$	0	0
$826$	−7.83680e7	−3.99658
$827$	2.29575e7	1.16724	0.583620	−	0.812027i	$-0.301636\pi$
$827$	2.29575e7	1.16724	0.583620	+	0.812027i	$0.301636\pi$
$828$	−9.72102e6	−0.492761
$829$	2.33662e7	1.18087	0.590435	−	0.807085i	$-0.298956\pi$
$829$	2.33662e7	1.18087	0.590435	+	0.807085i	$0.298956\pi$
$830$	0	0
$831$	−3.87271e6	−0.194542
$832$	−2.40046e7	−1.20223
$833$	−2.63864e7	−1.31755
$834$	−3.31631e7	−1.65097
$835$	0	0
$836$	1.36785e7	0.676895
$837$	−4.62861e6	−0.228369
$838$	4.43883e7	2.18353
$839$	1.72966e7	0.848315	0.424157	−	0.905588i	$-0.360570\pi$
$839$	1.72966e7	0.848315	0.424157	+	0.905588i	$0.360570\pi$
$840$	0	0
$841$	2.25475e7	1.09928
$842$	1.35564e7	0.658967
$843$	5.09840e6	0.247096
$844$	3.51997e7	1.70091
$845$	0	0
$846$	−945576.	−0.0454225
$847$	2.85902e6	0.136933
$848$	−787286.	−0.0375961
$849$	−8.66052e6	−0.412358
$850$	0	0
$851$	−7.67379e6	−0.363234
$852$	−4.25887e6	−0.201000
$853$	−9.73772e6	−0.458231	−0.229116	−	0.973399i	$-0.573583\pi$
$853$	−9.73772e6	−0.458231	−0.229116	+	0.973399i	$0.573583\pi$
$854$	−8.82712e7	−4.14166
$855$	0	0
$856$	−2.54191e7	−1.18570
$857$	1.49824e7	0.696832	0.348416	−	0.937340i	$-0.386720\pi$
$857$	1.49824e7	0.696832	0.348416	+	0.937340i	$0.386720\pi$
$858$	4.58353e6	0.212560
$859$	−2.41887e7	−1.11848	−0.559242	−	0.829004i	$-0.688908\pi$
$859$	−2.41887e7	−1.11848	−0.559242	+	0.829004i	$0.688908\pi$
$860$	0	0
$861$	−6.97808e6	−0.320795
$862$	3.46183e7	1.58685
$863$	−1.77029e7	−0.809126	−0.404563	−	0.914510i	$-0.632576\pi$
$863$	−1.77029e7	−0.809126	−0.404563	+	0.914510i	$0.632576\pi$
$864$	−3.96887e6	−0.180877
$865$	0	0
$866$	−2.79860e7	−1.26808
$867$	−1.00026e6	−0.0451923
$868$	6.54690e7	2.94942
$869$	−2.16088e6	−0.0970692
$870$	0	0
$871$	−6.75764e6	−0.301821
$872$	−1.69641e7	−0.755510
$873$	−1.27006e7	−0.564012
$874$	4.48078e7	1.98415
$875$	0	0
$876$	−3.96119e7	−1.74408
$877$	3.95853e7	1.73794	0.868970	−	0.494864i	$-0.164782\pi$
$877$	3.95853e7	1.73794	0.868970	+	0.494864i	$0.164782\pi$
$878$	3.64592e6	0.159614
$879$	−6.86514e6	−0.299693
$880$	0	0
$881$	−1.06184e7	−0.460914	−0.230457	−	0.973082i	$-0.574022\pi$
$881$	−1.06184e7	−0.460914	−0.230457	+	0.973082i	$0.574022\pi$
$882$	−1.59069e7	−0.688517
$883$	−798967.	−0.0344848	−0.0172424	−	0.999851i	$-0.505489\pi$
$883$	−798967.	−0.0344848	−0.0172424	+	0.999851i	$0.505489\pi$
$884$	−2.98619e7	−1.28525
$885$	0	0
$886$	5.75710e7	2.46388
$887$	2.53565e7	1.08213	0.541065	−	0.840981i	$-0.318021\pi$
$887$	2.53565e7	1.08213	0.541065	+	0.840981i	$0.318021\pi$
$888$	5.82165e6	0.247750
$889$	1.37912e7	0.585259
$890$	0	0
$891$	793881.	0.0335013
$892$	2.42808e7	1.02176
$893$	2.71387e6	0.113883
$894$	2.27373e6	0.0951470
$895$	0	0
$896$	6.04261e7	2.51452
$897$	9.34905e6	0.387960
$898$	3.63239e7	1.50315
$899$	4.16633e7	1.71931
$900$	0	0
$901$	1.30703e7	0.536381
$902$	−4.42426e6	−0.181061
$903$	6.85734e6	0.279857
$904$	1.76936e7	0.720102
$905$	0	0
$906$	9.26298e6	0.374913
$907$	−2.39197e6	−0.0965468	−0.0482734	−	0.998834i	$-0.515372\pi$
$907$	−2.39197e6	−0.0965468	−0.0482734	+	0.998834i	$0.515372\pi$
$908$	−5.88882e7	−2.37036
$909$	−7.21049e6	−0.289438
$910$	0	0
$911$	2.84437e7	1.13551	0.567754	−	0.823198i	$-0.307813\pi$
$911$	2.84437e7	1.13551	0.567754	+	0.823198i	$0.307813\pi$
$912$	−1.43603e6	−0.0571709
$913$	8.08648e6	0.321057
$914$	6.17582e7	2.44528
$915$	0	0
$916$	6.29984e7	2.48079
$917$	3.03316e7	1.19117
$918$	−8.30659e6	−0.325324
$919$	−1.05298e7	−0.411273	−0.205636	−	0.978628i	$-0.565926\pi$
$919$	−1.05298e7	−0.411273	−0.205636	+	0.978628i	$0.565926\pi$
$920$	0	0
$921$	2.72177e7	1.05731
$922$	4.29906e7	1.66550
$923$	4.09591e6	0.158251
$924$	−1.12290e7	−0.432673
$925$	0	0
$926$	4.62568e7	1.77275
$927$	−8.12168e6	−0.310418
$928$	3.57248e7	1.36176
$929$	1.19595e7	0.454647	0.227323	−	0.973819i	$-0.427003\pi$
$929$	1.19595e7	0.454647	0.227323	+	0.973819i	$0.427003\pi$
$930$	0	0
$931$	4.56540e7	1.72625
$932$	−3.56888e7	−1.34584
$933$	−2.10513e7	−0.791725
$934$	1.60047e7	0.600315
$935$	0	0
$936$	−7.09257e6	−0.264615
$937$	2.02229e7	0.752480	0.376240	−	0.926522i	$-0.377217\pi$
$937$	2.02229e7	0.752480	0.376240	+	0.926522i	$0.377217\pi$
$938$	2.65879e7	0.986681
$939$	1.28341e7	0.475009
$940$	0	0
$941$	2.13506e7	0.786026	0.393013	−	0.919533i	$-0.371433\pi$
$941$	2.13506e7	0.786026	0.393013	+	0.919533i	$0.371433\pi$
$942$	−1.91669e7	−0.703759
$943$	−9.02418e6	−0.330467
$944$	3.24802e6	0.118628
$945$	0	0
$946$	4.34771e6	0.157955
$947$	−3.13109e6	−0.113454	−0.0567271	−	0.998390i	$-0.518066\pi$
$947$	−3.13109e6	−0.113454	−0.0567271	+	0.998390i	$0.518066\pi$
$948$	8.48701e6	0.306714
$949$	3.80962e7	1.37314
$950$	0	0
$951$	−1.44569e7	−0.518350
$952$	4.62900e7	1.65537
$953$	−4.65632e7	−1.66077	−0.830387	−	0.557188i	$-0.811880\pi$
$953$	−4.65632e7	−1.66077	−0.830387	+	0.557188i	$0.811880\pi$
$954$	7.87937e6	0.280298
$955$	0	0
$956$	2.30409e7	0.815370
$957$	−7.14592e6	−0.252219
$958$	7.01700e6	0.247023
$959$	−1.37832e7	−0.483955
$960$	0	0
$961$	1.16840e7	0.408116
$962$	−1.42109e7	−0.495090
$963$	1.07471e7	0.373445
$964$	3.15245e7	1.09258
$965$	0	0
$966$	−3.67838e7	−1.26828
$967$	−3.64320e7	−1.25290	−0.626450	−	0.779462i	$-0.715493\pi$
$967$	−3.64320e7	−1.25290	−0.626450	+	0.779462i	$0.715493\pi$
$968$	−2.80495e6	−0.0962135
$969$	2.38405e7	0.815653
$970$	0	0
$971$	2.45693e7	0.836267	0.418134	−	0.908386i	$-0.362684\pi$
$971$	2.45693e7	0.836267	0.418134	+	0.908386i	$0.362684\pi$
$972$	−3.11802e6	−0.105855
$973$	−7.81359e7	−2.64587
$974$	7.73591e6	0.261285
$975$	0	0
$976$	3.65847e6	0.122935
$977$	4.28557e7	1.43639	0.718194	−	0.695842i	$-0.244969\pi$
$977$	4.28557e7	1.43639	0.718194	+	0.695842i	$0.244969\pi$
$978$	1.81601e7	0.607115
$979$	−1.42667e7	−0.475737
$980$	0	0
$981$	7.17237e6	0.237952
$982$	−9.73459e7	−3.22136
$983$	3.25942e7	1.07586	0.537931	−	0.842989i	$-0.319206\pi$
$983$	3.25942e7	1.07586	0.537931	+	0.842989i	$0.319206\pi$
$984$	6.84611e6	0.225401
$985$	0	0
$986$	7.47697e7	2.44925
$987$	−2.22788e6	−0.0727946
$988$	5.16674e7	1.68393
$989$	8.86804e6	0.288295
$990$	0	0
$991$	−1.81427e7	−0.586836	−0.293418	−	0.955984i	$-0.594793\pi$
$991$	−1.81427e7	−0.586836	−0.293418	+	0.955984i	$0.594793\pi$
$992$	3.45671e7	1.11528
$993$	−2.80384e7	−0.902362
$994$	−1.61154e7	−0.517337
$995$	0	0
$996$	−3.17602e7	−1.01446
$997$	9.28807e6	0.295929	0.147965	−	0.988993i	$-0.452728\pi$
$997$	9.28807e6	0.295929	0.147965	+	0.988993i	$0.452728\pi$
$998$	−2.64699e7	−0.841252
$999$	−2.46137e6	−0.0780303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.u.1.2	yes	10
5.4	even	2		825.6.a.t.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.9	✓	10	5.4	even	2
825.6.a.u.1.2	yes	10	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)

\(f(q)\)	\(=\)	\(q-9.20891 q^{2} -9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} +195.275 q^{7} -191.582 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.20891 q^{2} -9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} +195.275 q^{7} -191.582 q^{8} +81.0000 q^{9} +121.000 q^{11} -475.236 q^{12} +457.051 q^{13} -1798.27 q^{14} +74.5305 q^{16} -1237.33 q^{17} -745.921 q^{18} +2140.85 q^{19} -1757.47 q^{21} -1114.28 q^{22} -2272.80 q^{23} +1724.23 q^{24} -4208.94 q^{26} -729.000 q^{27} +10311.3 q^{28} +6561.91 q^{29} +6349.26 q^{31} +5444.27 q^{32} -1089.00 q^{33} +11394.5 q^{34} +4277.12 q^{36} +3376.37 q^{37} -19714.9 q^{38} -4113.46 q^{39} +3970.52 q^{41} +16184.4 q^{42} -3901.82 q^{43} +6389.28 q^{44} +20930.0 q^{46} +1267.66 q^{47} -670.775 q^{48} +21325.2 q^{49} +11136.0 q^{51} +24134.1 q^{52} -10563.3 q^{53} +6713.29 q^{54} -37411.0 q^{56} -19267.6 q^{57} -60428.0 q^{58} +43579.8 q^{59} +49086.8 q^{61} -58469.8 q^{62} +15817.2 q^{63} -52520.7 q^{64} +10028.5 q^{66} -14785.3 q^{67} -65336.1 q^{68} +20455.2 q^{69} +8961.61 q^{71} -15518.1 q^{72} +83352.2 q^{73} -31092.6 q^{74} +113045. q^{76} +23628.2 q^{77} +37880.5 q^{78} -17858.5 q^{79} +6561.00 q^{81} -36564.1 q^{82} +66830.4 q^{83} -92801.5 q^{84} +35931.5 q^{86} -59057.2 q^{87} -23181.4 q^{88} -117907. q^{89} +89250.5 q^{91} -120013. q^{92} -57143.4 q^{93} -11673.8 q^{94} -48998.4 q^{96} -156797. q^{97} -196382. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more