LMFDB - Embedded newform 825.4.c.p.199.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 913x^{4} + 5292x^{2} + 8464 $ x^8 + 54x^6 + 913x^4 + 5292*x^2 + 8464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.5
Root			$1.60719i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.p.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.607192i q^{2} +3.00000i q^{3} +7.63132 q^{4} -1.82158 q^{6} -8.95080i q^{7} +9.49121i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+0.607192i q^{2} +3.00000i q^{3} +7.63132 q^{4} -1.82158 q^{6} -8.95080i q^{7} +9.49121i q^{8} -9.00000 q^{9} -11.0000 q^{11} +22.8940i q^{12} -0.460387i q^{13} +5.43485 q^{14} +55.2876 q^{16} -128.395i q^{17} -5.46473i q^{18} +0.0245858 q^{19} +26.8524 q^{21} -6.67911i q^{22} +171.528i q^{23} -28.4736 q^{24} +0.279543 q^{26} -27.0000i q^{27} -68.3064i q^{28} +226.938 q^{29} +195.637 q^{31} +109.500i q^{32} -33.0000i q^{33} +77.9604 q^{34} -68.6819 q^{36} -338.584i q^{37} +0.0149283i q^{38} +1.38116 q^{39} +136.972 q^{41} +16.3046i q^{42} +336.083i q^{43} -83.9445 q^{44} -104.151 q^{46} +540.292i q^{47} +165.863i q^{48} +262.883 q^{49} +385.185 q^{51} -3.51336i q^{52} -622.387i q^{53} +16.3942 q^{54} +84.9539 q^{56} +0.0737574i q^{57} +137.795i q^{58} +9.86955 q^{59} +902.712 q^{61} +118.789i q^{62} +80.5572i q^{63} +375.813 q^{64} +20.0373 q^{66} -146.979i q^{67} -979.823i q^{68} -514.585 q^{69} -893.798 q^{71} -85.4209i q^{72} -1149.71i q^{73} +205.585 q^{74} +0.187622 q^{76} +98.4588i q^{77} +0.838629i q^{78} +459.528 q^{79} +81.0000 q^{81} +83.1686i q^{82} -125.876i q^{83} +204.919 q^{84} -204.067 q^{86} +680.813i q^{87} -104.403i q^{88} -150.461 q^{89} -4.12083 q^{91} +1308.99i q^{92} +586.912i q^{93} -328.061 q^{94} -328.499 q^{96} +1264.58i q^{97} +159.621i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9}+O(q^{10}) $ 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9	$ 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9} - 88 q^{11} + 104 q^{14} + 132 q^{16} - 272 q^{19} + 204 q^{21} - 288 q^{24} - 640 q^{26} - 104 q^{29} + 984 q^{31} - 488 q^{34} + 468 q^{36} - 12 q^{39} + 536 q^{41} + 572 q^{44} + 736 q^{46} + 992 q^{49} + 444 q^{51} - 216 q^{54} - 1704 q^{56} + 2064 q^{59} + 232 q^{61} + 1836 q^{64} - 264 q^{66} + 384 q^{69} - 1840 q^{71} + 5712 q^{74} + 3144 q^{76} - 2304 q^{79} + 648 q^{81} + 120 q^{84} + 472 q^{86} + 2128 q^{89} + 5560 q^{91} + 2864 q^{94} + 1248 q^{96} + 792 q^{99}+O(q^{100}) $ 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9 - 88 * q^11 + 104 * q^14 + 132 * q^16 - 272 * q^19 + 204 * q^21 - 288 * q^24 - 640 * q^26 - 104 * q^29 + 984 * q^31 - 488 * q^34 + 468 * q^36 - 12 * q^39 + 536 * q^41 + 572 * q^44 + 736 * q^46 + 992 * q^49 + 444 * q^51 - 216 * q^54 - 1704 * q^56 + 2064 * q^59 + 232 * q^61 + 1836 * q^64 - 264 * q^66 + 384 * q^69 - 1840 * q^71 + 5712 * q^74 + 3144 * q^76 - 2304 * q^79 + 648 * q^81 + 120 * q^84 + 472 * q^86 + 2128 * q^89 + 5560 * q^91 + 2864 * q^94 + 1248 * q^96 + 792 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0.607192i	0.214675i	0.994223	+	0.107337i	$0.0342325\pi$
$2$	0.607192i	0.214675i	−0.994223	+	0.107337i	$0.965768\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	7.63132	0.953915
$5$	0	0
$5$	0	0
$6$	−1.82158	−0.123943
$7$	− 8.95080i	− 0.483298i	−0.970364	−	0.241649i	$-0.922312\pi$
$7$	− 8.95080i	− 0.483298i	0.970364	−	0.241649i	$-0.0776882\pi$
$8$	9.49121i	0.419456i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	22.8940i	0.550743i
$13$	− 0.460387i	− 0.00982218i	−0.999988	−	0.00491109i	$-0.998437\pi$
$13$	− 0.460387i	− 0.00982218i	0.999988	−	0.00491109i	$-0.00156325\pi$
$14$	5.43485	0.103752
$15$	0	0
$16$	55.2876	0.863868
$17$	− 128.395i	− 1.83179i	−0.401423	−	0.915893i	$-0.631484\pi$
$17$	− 128.395i	− 1.83179i	0.401423	−	0.915893i	$-0.368516\pi$
$18$	− 5.46473i	− 0.0715582i
$19$	0.0245858	0.000296862 0	0.000148431	−	1.00000i	$-0.499953\pi$
$19$	0.0245858	0.000296862 0	0.000148431		1.00000i	$0.499953\pi$
$20$	0	0
$21$	26.8524	0.279032
$22$	− 6.67911i	− 0.0647269i
$23$	171.528i	1.55505i	0.628852	+	0.777525i	$0.283525\pi$
$23$	171.528i	1.55505i	−0.628852	+	0.777525i	$0.716475\pi$
$24$	−28.4736	−0.242173
$25$	0	0
$26$	0.279543	0.00210857
$27$	− 27.0000i	− 0.192450i
$28$	− 68.3064i	− 0.461025i
$29$	226.938	1.45315	0.726574	−	0.687088i	$-0.241111\pi$
$29$	226.938	1.45315	0.726574	+	0.687088i	$0.241111\pi$
$30$	0	0
$31$	195.637	1.13347	0.566734	−	0.823901i	$-0.308207\pi$
$31$	195.637	1.13347	0.566734	+	0.823901i	$0.308207\pi$
$32$	109.500i	0.604907i
$33$	− 33.0000i	− 0.174078i
$34$	77.9604	0.393238
$35$	0	0
$36$	−68.6819	−0.317972
$37$	− 338.584i	− 1.50440i	−0.658935	−	0.752200i	$-0.728993\pi$
$37$	− 338.584i	− 1.50440i	0.658935	−	0.752200i	$-0.271007\pi$
$38$	0.0149283i	0 6.37287e-5i
$39$	1.38116	0.00567084
$40$	0	0
$41$	136.972	0.521744	0.260872	−	0.965373i	$-0.415990\pi$
$41$	136.972	0.521744	0.260872	+	0.965373i	$0.415990\pi$
$42$	16.3046i	0.0599011i
$43$	336.083i	1.19191i	0.803017	+	0.595956i	$0.203227\pi$
$43$	336.083i	1.19191i	−0.803017	+	0.595956i	$0.796773\pi$
$44$	−83.9445	−0.287616
$45$	0	0
$46$	−104.151	−0.333830
$47$	540.292i	1.67680i	0.545055	+	0.838400i	$0.316509\pi$
$47$	540.292i	1.67680i	−0.545055	+	0.838400i	$0.683491\pi$
$48$	165.863i	0.498754i
$49$	262.883	0.766423
$50$	0	0
$51$	385.185	1.05758
$52$	− 3.51336i	− 0.00936952i
$53$	− 622.387i	− 1.61305i	−0.591203	−	0.806523i	$-0.701347\pi$
$53$	− 622.387i	− 1.61305i	0.591203	−	0.806523i	$-0.298653\pi$
$54$	16.3942	0.0413142
$55$	0	0
$56$	84.9539	0.202722
$57$	0.0737574i	0 0.000171393i
$58$	137.795i	0.311954i
$59$	9.86955	0.0217781	0.0108890	−	0.999941i	$-0.496534\pi$
$59$	9.86955	0.0217781	0.0108890	+	0.999941i	$0.496534\pi$
$60$	0	0
$61$	902.712	1.89476	0.947380	−	0.320110i	$-0.103720\pi$
$61$	902.712	1.89476	0.947380	+	0.320110i	$0.103720\pi$
$62$	118.789i	0.243327i
$63$	80.5572i	0.161099i
$64$	375.813	0.734010
$65$	0	0
$66$	20.0373	0.0373701
$67$	− 146.979i	− 0.268005i	−0.990981	−	0.134003i	$-0.957217\pi$
$67$	− 146.979i	− 0.268005i	0.990981	−	0.134003i	$-0.0427831\pi$
$68$	− 979.823i	− 1.74737i
$69$	−514.585	−0.897809
$70$	0	0
$71$	−893.798	−1.49400	−0.747002	−	0.664821i	$-0.768508\pi$
$71$	−893.798	−1.49400	−0.747002	+	0.664821i	$0.768508\pi$
$72$	− 85.4209i	− 0.139819i
$73$	− 1149.71i	− 1.84333i	−0.387988	−	0.921664i	$-0.626830\pi$
$73$	− 1149.71i	− 1.84333i	0.387988	−	0.921664i	$-0.373170\pi$
$74$	205.585	0.322957
$75$	0	0
$76$	0.187622	0.000283181 0
$77$	98.4588i	0.145720i
$78$	0.838629i	0.00121739i
$79$	459.528	0.654443	0.327221	−	0.944948i	$-0.393888\pi$
$79$	459.528	0.654443	0.327221	+	0.944948i	$0.393888\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	83.1686i	0.112005i
$83$	− 125.876i	− 0.166466i	−0.996530	−	0.0832331i	$-0.973475\pi$
$83$	− 125.876i	− 0.166466i	0.996530	−	0.0832331i	$-0.0265246\pi$
$84$	204.919	0.266173
$85$	0	0
$86$	−204.067	−0.255873
$87$	680.813i	0.838975i
$88$	− 104.403i	− 0.126471i
$89$	−150.461	−0.179201	−0.0896003	−	0.995978i	$-0.528559\pi$
$89$	−150.461	−0.179201	−0.0896003	+	0.995978i	$0.528559\pi$
$90$	0	0
$91$	−4.12083	−0.00474704
$92$	1308.99i	1.48339i
$93$	586.912i	0.654408i
$94$	−328.061	−0.359967
$95$	0	0
$96$	−328.499	−0.349243
$97$	1264.58i	1.32370i	0.749635	+	0.661851i	$0.230229\pi$
$97$	1264.58i	1.32370i	−0.749635	+	0.661851i	$0.769771\pi$
$98$	159.621i	0.164532i
$99$	99.0000	0.100504
$100$	0	0
$101$	690.792	0.680558	0.340279	−	0.940324i	$-0.389478\pi$
$101$	690.792	0.680558	0.340279	+	0.940324i	$0.389478\pi$
$102$	233.881i	0.227036i
$103$	− 11.7823i	− 0.0112713i	−0.999984	−	0.00563563i	$-0.998206\pi$
$103$	− 11.7823i	− 0.0112713i	0.999984	−	0.00563563i	$-0.00179389\pi$
$104$	4.36963	0.00411997
$105$	0	0
$106$	377.908	0.346280
$107$	462.884i	0.418212i	0.977893	+	0.209106i	$0.0670554\pi$
$107$	462.884i	0.418212i	−0.977893	+	0.209106i	$0.932945\pi$
$108$	− 206.046i	− 0.183581i
$109$	−330.525	−0.290446	−0.145223	−	0.989399i	$-0.546390\pi$
$109$	−330.525	−0.290446	−0.145223	+	0.989399i	$0.546390\pi$
$110$	0	0
$111$	1015.75	0.868566
$112$	− 494.868i	− 0.417506i
$113$	1142.31i	0.950965i	0.879725	+	0.475483i	$0.157727\pi$
$113$	1142.31i	0.950965i	−0.879725	+	0.475483i	$0.842273\pi$
$114$	−0.0447849	−3.67938e−5 0
$115$	0	0
$116$	1731.83	1.38618
$117$	4.14348i	0.00327406i
$118$	5.99271i	0.00467520i
$119$	−1149.24	−0.885298
$120$	0	0
$121$	121.000	0.0909091
$122$	548.119i	0.406757i
$123$	410.917i	0.301229i
$124$	1492.97	1.08123
$125$	0	0
$126$	−48.9137	−0.0345839
$127$	− 629.589i	− 0.439898i	−0.975511	−	0.219949i	$-0.929411\pi$
$127$	− 629.589i	− 0.439898i	0.975511	−	0.219949i	$-0.0705891\pi$
$128$	1104.19i	0.762480i
$129$	−1008.25	−0.688151
$130$	0	0
$131$	572.566	0.381873	0.190937	−	0.981602i	$-0.438848\pi$
$131$	572.566	0.381873	0.190937	+	0.981602i	$0.438848\pi$
$132$	− 251.833i	− 0.166055i
$133$	− 0.220063i	0 0.000143473i
$134$	89.2446	0.0575340
$135$	0	0
$136$	1218.62	0.768354
$137$	948.680i	0.591615i	0.955248	+	0.295807i	$0.0955887\pi$
$137$	948.680i	0.591615i	−0.955248	+	0.295807i	$0.904411\pi$
$138$	− 312.452i	− 0.192737i
$139$	−2488.86	−1.51872	−0.759362	−	0.650668i	$-0.774489\pi$
$139$	−2488.86	−1.51872	−0.759362	+	0.650668i	$0.774489\pi$
$140$	0	0
$141$	−1620.87	−0.968101
$142$	− 542.707i	− 0.320725i
$143$	5.06425i	0.00296150i
$144$	−497.588	−0.287956
$145$	0	0
$146$	698.093	0.395716
$147$	788.650i	0.442495i
$148$	− 2583.84i	− 1.43507i
$149$	2186.43	1.20214	0.601072	−	0.799195i	$-0.294741\pi$
$149$	2186.43	1.20214	0.601072	+	0.799195i	$0.294741\pi$
$150$	0	0
$151$	3668.21	1.97692	0.988459	−	0.151488i	$-0.0484064\pi$
$151$	3668.21	1.97692	0.988459	+	0.151488i	$0.0484064\pi$
$152$	0.233349i	0 0.000124520i
$153$	1155.55i	0.610595i
$154$	−59.7834	−0.0312824
$155$	0	0
$156$	10.5401	0.00540950
$157$	− 1418.28i	− 0.720960i	−0.932767	−	0.360480i	$-0.882613\pi$
$157$	− 1418.28i	− 0.720960i	0.932767	−	0.360480i	$-0.117387\pi$
$158$	279.022i	0.140492i
$159$	1867.16	0.931293
$160$	0	0
$161$	1535.32	0.751552
$162$	49.1825i	0.0238527i
$163$	− 601.009i	− 0.288802i	−0.989519	−	0.144401i	$-0.953875\pi$
$163$	− 601.009i	− 0.288802i	0.989519	−	0.144401i	$-0.0461255\pi$
$164$	1045.28	0.497699
$165$	0	0
$166$	76.4309	0.0357361
$167$	2221.19i	1.02923i	0.857422	+	0.514614i	$0.172065\pi$
$167$	2221.19i	1.02923i	−0.857422	+	0.514614i	$0.827935\pi$
$168$	254.862i	0.117042i
$169$	2196.79	0.999904
$170$	0	0
$171$	−0.221272	−9.89539e−5 0
$172$	2564.76i	1.13698i
$173$	− 237.280i	− 0.104278i	−0.998640	−	0.0521390i	$-0.983396\pi$
$173$	− 237.280i	− 0.104278i	0.998640	−	0.0521390i	$-0.0166039\pi$
$174$	−413.384	−0.180107
$175$	0	0
$176$	−608.163	−0.260466
$177$	29.6086i	0.0125736i
$178$	− 91.3588i	− 0.0384698i
$179$	−1065.77	−0.445023	−0.222512	−	0.974930i	$-0.571426\pi$
$179$	−1065.77	−0.445023	−0.222512	+	0.974930i	$0.571426\pi$
$180$	0	0
$181$	−1241.00	−0.509627	−0.254813	−	0.966990i	$-0.582014\pi$
$181$	−1241.00	−0.509627	−0.254813	+	0.966990i	$0.582014\pi$
$182$	− 2.50213i	− 0.00101907i
$183$	2708.14i	1.09394i
$184$	−1628.01	−0.652275
$185$	0	0
$186$	−356.368	−0.140485
$187$	1412.34i	0.552304i
$188$	4123.14i	1.59952i
$189$	−241.672	−0.0930107
$190$	0	0
$191$	−1956.80	−0.741305	−0.370653	−	0.928772i	$-0.620866\pi$
$191$	−1956.80	−0.741305	−0.370653	+	0.928772i	$0.620866\pi$
$192$	1127.44i	0.423781i
$193$	2778.02i	1.03610i	0.855352	+	0.518048i	$0.173341\pi$
$193$	2778.02i	1.03610i	−0.855352	+	0.518048i	$0.826659\pi$
$194$	−767.846	−0.284165
$195$	0	0
$196$	2006.15	0.731102
$197$	− 800.242i	− 0.289416i	−0.989474	−	0.144708i	$-0.953776\pi$
$197$	− 800.242i	− 0.289416i	0.989474	−	0.144708i	$-0.0462242\pi$
$198$	60.1120i	0.0215756i
$199$	2536.78	0.903657	0.451829	−	0.892105i	$-0.350772\pi$
$199$	2536.78	0.903657	0.451829	+	0.892105i	$0.350772\pi$
$200$	0	0
$201$	440.938	0.154733
$202$	419.443i	0.146099i
$203$	− 2031.27i	− 0.702303i
$204$	2939.47	1.00884
$205$	0	0
$206$	7.15409	0.00241966
$207$	− 1543.76i	− 0.518350i
$208$	− 25.4537i	− 0.00848507i
$209$	−0.270444	−8.95071e−5 0
$210$	0	0
$211$	1598.95	0.521688	0.260844	−	0.965381i	$-0.415999\pi$
$211$	1598.95	0.521688	0.260844	+	0.965381i	$0.415999\pi$
$212$	− 4749.63i	− 1.53871i
$213$	− 2681.40i	− 0.862564i
$214$	−281.060	−0.0897797
$215$	0	0
$216$	256.263	0.0807244
$217$	− 1751.11i	− 0.547802i
$218$	− 200.692i	− 0.0623513i
$219$	3449.12	1.06425
$220$	0	0
$221$	−59.1113	−0.0179921
$222$	616.756i	0.186459i
$223$	− 5074.55i	− 1.52384i	−0.647669	−	0.761922i	$-0.724256\pi$
$223$	− 5074.55i	− 1.52384i	0.647669	−	0.761922i	$-0.275744\pi$
$224$	980.111	0.292350
$225$	0	0
$226$	−693.599	−0.204148
$227$	− 1048.19i	− 0.306480i	−0.988189	−	0.153240i	$-0.951029\pi$
$227$	− 1048.19i	− 0.306480i	0.988189	−	0.153240i	$-0.0489708\pi$
$228$	0.562866i	0 0.000163494i
$229$	734.662	0.211999	0.106000	−	0.994366i	$-0.466196\pi$
$229$	734.662	0.211999	0.106000	+	0.994366i	$0.466196\pi$
$230$	0	0
$231$	−295.376	−0.0841313
$232$	2153.91i	0.609532i
$233$	− 2012.78i	− 0.565929i	−0.959130	−	0.282965i	$-0.908682\pi$
$233$	− 2012.78i	− 0.565929i	0.959130	−	0.282965i	$-0.0913179\pi$
$234$	−2.51589	−0.000702858 0
$235$	0	0
$236$	75.3176	0.0207744
$237$	1378.59i	0.377843i
$238$	− 697.808i	− 0.190051i
$239$	−5566.41	−1.50653	−0.753267	−	0.657715i	$-0.771523\pi$
$239$	−5566.41	−1.50653	−0.753267	+	0.657715i	$0.771523\pi$
$240$	0	0
$241$	−3361.25	−0.898412	−0.449206	−	0.893428i	$-0.648293\pi$
$241$	−3361.25	−0.898412	−0.449206	+	0.893428i	$0.648293\pi$
$242$	73.4702i	0.0195159i
$243$	243.000i	0.0641500i
$244$	6888.88	1.80744
$245$	0	0
$246$	−249.506	−0.0646663
$247$	− 0.0113190i	0 2.91583e-6i
$248$	1856.83i	0.475440i
$249$	377.628	0.0961093
$250$	0	0
$251$	−2930.46	−0.736929	−0.368464	−	0.929642i	$-0.620116\pi$
$251$	−2930.46	−0.736929	−0.368464	+	0.929642i	$0.620116\pi$
$252$	614.758i	0.153675i
$253$	− 1886.81i	− 0.468865i
$254$	382.281	0.0944349
$255$	0	0
$256$	2336.05	0.570325
$257$	3412.89i	0.828367i	0.910193	+	0.414184i	$0.135933\pi$
$257$	3412.89i	0.828367i	−0.910193	+	0.414184i	$0.864067\pi$
$258$	− 612.201i	− 0.147729i
$259$	−3030.59	−0.727073
$260$	0	0
$261$	−2042.44	−0.484383
$262$	347.658i	0.0819785i
$263$	3709.06i	0.869622i	0.900522	+	0.434811i	$0.143185\pi$
$263$	3709.06i	0.869622i	−0.900522	+	0.434811i	$0.856815\pi$
$264$	313.210	0.0730179
$265$	0	0
$266$	0.133620	3.07999e−5 0
$267$	− 451.384i	− 0.103462i
$268$	− 1121.64i	− 0.255654i
$269$	5764.39	1.30655	0.653273	−	0.757122i	$-0.273395\pi$
$269$	5764.39	1.30655	0.653273	+	0.757122i	$0.273395\pi$
$270$	0	0
$271$	−1886.33	−0.422827	−0.211414	−	0.977397i	$-0.567807\pi$
$271$	−1886.33	−0.422827	−0.211414	+	0.977397i	$0.567807\pi$
$272$	− 7098.64i	− 1.58242i
$273$	− 12.3625i	− 0.00274070i
$274$	−576.031	−0.127005
$275$	0	0
$276$	−3926.97	−0.856433
$277$	4095.99i	0.888462i	0.895912	+	0.444231i	$0.146523\pi$
$277$	4095.99i	0.888462i	−0.895912	+	0.444231i	$0.853477\pi$
$278$	− 1511.22i	− 0.326032i
$279$	−1760.73	−0.377822
$280$	0	0
$281$	−8788.69	−1.86580	−0.932899	−	0.360138i	$-0.882729\pi$
$281$	−8788.69	−1.86580	−0.932899	+	0.360138i	$0.882729\pi$
$282$	− 984.182i	− 0.207827i
$283$	− 1767.75i	− 0.371314i	−0.982615	−	0.185657i	$-0.940559\pi$
$283$	− 1767.75i	− 0.371314i	0.982615	−	0.185657i	$-0.0594414\pi$
$284$	−6820.86	−1.42515
$285$	0	0
$286$	−3.07497	−0.000635759 0
$287$	− 1226.01i	− 0.252158i
$288$	− 985.498i	− 0.201636i
$289$	−11572.3	−2.35544
$290$	0	0
$291$	−3793.75	−0.764240
$292$	− 8773.78i	− 1.75838i
$293$	3079.85i	0.614085i	0.951696	+	0.307043i	$0.0993394\pi$
$293$	3079.85i	0.614085i	−0.951696	+	0.307043i	$0.900661\pi$
$294$	−478.862	−0.0949924
$295$	0	0
$296$	3213.57	0.631030
$297$	297.000i	0.0580259i
$298$	1327.58i	0.258070i
$299$	78.9695	0.0152740
$300$	0	0
$301$	3008.21	0.576048
$302$	2227.31i	0.424394i
$303$	2072.38i	0.392921i
$304$	1.35929	0.000256449 0
$305$	0	0
$306$	−701.643	−0.131079
$307$	− 2872.16i	− 0.533950i	−0.963703	−	0.266975i	$-0.913976\pi$
$307$	− 2872.16i	− 0.533950i	0.963703	−	0.266975i	$-0.0860242\pi$
$308$	751.370i	0.139004i
$309$	35.3468	0.00650747
$310$	0	0
$311$	−317.978	−0.0579771	−0.0289885	−	0.999580i	$-0.509229\pi$
$311$	−317.978	−0.0579771	−0.0289885	+	0.999580i	$0.509229\pi$
$312$	13.1089i	0.00237867i
$313$	− 8274.81i	− 1.49431i	−0.664648	−	0.747156i	$-0.731419\pi$
$313$	− 8274.81i	− 1.49431i	0.664648	−	0.747156i	$-0.268581\pi$
$314$	861.166	0.154772
$315$	0	0
$316$	3506.81	0.624283
$317$	− 1024.34i	− 0.181492i	−0.995874	−	0.0907459i	$-0.971075\pi$
$317$	− 1024.34i	− 0.181492i	0.995874	−	0.0907459i	$-0.0289251\pi$
$318$	1133.73i	0.199925i
$319$	−2496.32	−0.438141
$320$	0	0
$321$	−1388.65	−0.241455
$322$	932.232i	0.161339i
$323$	− 3.15669i	0 0.000543787i
$324$	618.137	0.105991
$325$	0	0
$326$	364.928	0.0619984
$327$	− 991.575i	− 0.167689i
$328$	1300.03i	0.218849i
$329$	4836.04	0.810394
$330$	0	0
$331$	4575.83	0.759850	0.379925	−	0.925017i	$-0.375950\pi$
$331$	4575.83	0.759850	0.379925	+	0.925017i	$0.375950\pi$
$332$	− 960.600i	− 0.158795i
$333$	3047.25i	0.501467i
$334$	−1348.69	−0.220949
$335$	0	0
$336$	1484.60	0.241047
$337$	917.888i	0.148370i	0.997245	+	0.0741848i	$0.0236355\pi$
$337$	917.888i	0.148370i	−0.997245	+	0.0741848i	$0.976365\pi$
$338$	1333.87i	0.214654i
$339$	−3426.92	−0.549040
$340$	0	0
$341$	−2152.01	−0.341753
$342$	− 0.134355i	0 2.12429e-5i
$343$	− 5423.14i	− 0.853708i
$344$	−3189.84	−0.499955
$345$	0	0
$346$	144.075	0.0223859
$347$	− 5402.26i	− 0.835760i	−0.908502	−	0.417880i	$-0.862773\pi$
$347$	− 5402.26i	− 0.835760i	0.908502	−	0.417880i	$-0.137227\pi$
$348$	5195.50i	0.800311i
$349$	−5115.85	−0.784656	−0.392328	−	0.919825i	$-0.628330\pi$
$349$	−5115.85	−0.784656	−0.392328	+	0.919825i	$0.628330\pi$
$350$	0	0
$351$	−12.4304	−0.00189028
$352$	− 1204.50i	− 0.182386i
$353$	− 3102.03i	− 0.467718i	−0.972270	−	0.233859i	$-0.924865\pi$
$353$	− 3102.03i	− 0.467718i	0.972270	−	0.233859i	$-0.0751355\pi$
$354$	−17.9781	−0.00269923
$355$	0	0
$356$	−1148.22	−0.170942
$357$	− 3447.71i	− 0.511127i
$358$	− 647.125i	− 0.0955352i
$359$	−3496.32	−0.514008	−0.257004	−	0.966410i	$-0.582735\pi$
$359$	−3496.32	−0.514008	−0.257004	+	0.966410i	$0.582735\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	− 753.522i	− 0.109404i
$363$	363.000i	0.0524864i
$364$	−31.4474	−0.00452827
$365$	0	0
$366$	−1644.36	−0.234841
$367$	− 7644.54i	− 1.08731i	−0.839309	−	0.543654i	$-0.817040\pi$
$367$	− 7644.54i	− 1.08731i	0.839309	−	0.543654i	$-0.182960\pi$
$368$	9483.39i	1.34336i
$369$	−1232.75	−0.173915
$370$	0	0
$371$	−5570.86	−0.779582
$372$	4478.91i	0.624249i
$373$	− 5189.90i	− 0.720437i	−0.932868	−	0.360218i	$-0.882702\pi$
$373$	− 5189.90i	− 0.720437i	0.932868	−	0.360218i	$-0.117298\pi$
$374$	−857.564	−0.118566
$375$	0	0
$376$	−5128.02	−0.703344
$377$	− 104.479i	− 0.0142731i
$378$	− 146.741i	− 0.0199670i
$379$	−8573.91	−1.16204	−0.581019	−	0.813890i	$-0.697346\pi$
$379$	−8573.91	−1.16204	−0.581019	+	0.813890i	$0.697346\pi$
$380$	0	0
$381$	1888.77	0.253975
$382$	− 1188.15i	− 0.159140i
$383$	− 11740.2i	− 1.56631i	−0.621824	−	0.783157i	$-0.713608\pi$
$383$	− 11740.2i	− 1.56631i	0.621824	−	0.783157i	$-0.286392\pi$
$384$	−3312.57	−0.440218
$385$	0	0
$386$	−1686.79	−0.222424
$387$	− 3024.75i	− 0.397304i
$388$	9650.45i	1.26270i
$389$	3763.68	0.490556	0.245278	−	0.969453i	$-0.421121\pi$
$389$	3763.68	0.490556	0.245278	+	0.969453i	$0.421121\pi$
$390$	0	0
$391$	22023.4	2.84852
$392$	2495.08i	0.321481i
$393$	1717.70i	0.220474i
$394$	485.900	0.0621302
$395$	0	0
$396$	755.500	0.0958720
$397$	− 1151.35i	− 0.145554i	−0.997348	−	0.0727768i	$-0.976814\pi$
$397$	− 1151.35i	− 0.145554i	0.997348	−	0.0727768i	$-0.0231861\pi$
$398$	1540.31i	0.193992i
$399$	0.660188	8.28339e−5 0
$400$	0	0
$401$	−10051.0	−1.25168	−0.625842	−	0.779950i	$-0.715245\pi$
$401$	−10051.0	−1.25168	−0.625842	+	0.779950i	$0.715245\pi$
$402$	267.734i	0.0332173i
$403$	− 90.0688i	− 0.0111331i
$404$	5271.66	0.649195
$405$	0	0
$406$	1233.37	0.150767
$407$	3724.42i	0.453594i
$408$	3655.87i	0.443609i
$409$	6263.39	0.757225	0.378612	−	0.925555i	$-0.376401\pi$
$409$	6263.39	0.757225	0.378612	+	0.925555i	$0.376401\pi$
$410$	0	0
$411$	−2846.04	−0.341569
$412$	− 89.9142i	− 0.0107518i
$413$	− 88.3403i	− 0.0105253i
$414$	937.356	0.111277
$415$	0	0
$416$	50.4123	0.00594150
$417$	− 7466.59i	− 0.876836i
$418$	− 0.164211i	0 1.92149e-5i
$419$	−8625.67	−1.00571	−0.502854	−	0.864372i	$-0.667717\pi$
$419$	−8625.67	−1.00571	−0.502854	+	0.864372i	$0.667717\pi$
$420$	0	0
$421$	−9095.24	−1.05291	−0.526455	−	0.850203i	$-0.676479\pi$
$421$	−9095.24	−1.05291	−0.526455	+	0.850203i	$0.676479\pi$
$422$	970.868i	0.111993i
$423$	− 4862.62i	− 0.558934i
$424$	5907.21	0.676602
$425$	0	0
$426$	1628.12	0.185171
$427$	− 8079.99i	− 0.915733i
$428$	3532.42i	0.398939i
$429$	−15.1928	−0.00170982
$430$	0	0
$431$	−4008.72	−0.448013	−0.224006	−	0.974588i	$-0.571914\pi$
$431$	−4008.72	−0.448013	−0.224006	+	0.974588i	$0.571914\pi$
$432$	− 1492.76i	− 0.166251i
$433$	2715.38i	0.301369i	0.988582	+	0.150684i	$0.0481477\pi$
$433$	2715.38i	0.301369i	−0.988582	+	0.150684i	$0.951852\pi$
$434$	1063.26	0.117599
$435$	0	0
$436$	−2522.34	−0.277060
$437$	4.21717i	0 0.000461635i
$438$	2094.28i	0.228467i
$439$	11087.5	1.20541	0.602707	−	0.797962i	$-0.294089\pi$
$439$	11087.5	1.20541	0.602707	+	0.797962i	$0.294089\pi$
$440$	0	0
$441$	−2365.95	−0.255474
$442$	− 35.8919i	− 0.00386245i
$443$	7132.98i	0.765007i	0.923954	+	0.382503i	$0.124938\pi$
$443$	7132.98i	0.765007i	−0.923954	+	0.382503i	$0.875062\pi$
$444$	7751.52	0.828538
$445$	0	0
$446$	3081.23	0.327131
$447$	6559.29i	0.694058i
$448$	− 3363.83i	− 0.354745i
$449$	−6291.28	−0.661255	−0.330628	−	0.943761i	$-0.607260\pi$
$449$	−6291.28	−0.661255	−0.330628	+	0.943761i	$0.607260\pi$
$450$	0	0
$451$	−1506.70	−0.157312
$452$	8717.30i	0.907140i
$453$	11004.6i	1.14137i
$454$	636.455	0.0657936
$455$	0	0
$456$	−0.700047	−7.18919e−5 0
$457$	− 7291.98i	− 0.746400i	−0.927751	−	0.373200i	$-0.878261\pi$
$457$	− 7291.98i	− 0.746400i	0.927751	−	0.373200i	$-0.121739\pi$
$458$	446.081i	0.0455109i
$459$	−3466.66	−0.352527
$460$	0	0
$461$	−16805.4	−1.69784	−0.848922	−	0.528519i	$-0.822748\pi$
$461$	−16805.4	−1.69784	−0.848922	+	0.528519i	$0.822748\pi$
$462$	− 179.350i	− 0.0180609i
$463$	7478.48i	0.750658i	0.926892	+	0.375329i	$0.122470\pi$
$463$	7478.48i	0.750658i	−0.926892	+	0.375329i	$0.877530\pi$
$464$	12546.8	1.25533
$465$	0	0
$466$	1222.14	0.121491
$467$	7526.83i	0.745824i	0.927867	+	0.372912i	$0.121641\pi$
$467$	7526.83i	0.745824i	−0.927867	+	0.372912i	$0.878359\pi$
$468$	31.6202i	0.00312317i
$469$	−1315.58	−0.129526
$470$	0	0
$471$	4254.83	0.416247
$472$	93.6739i	0.00913494i
$473$	− 3696.92i	− 0.359375i
$474$	−837.066	−0.0811133
$475$	0	0
$476$	−8770.20	−0.844499
$477$	5601.48i	0.537682i
$478$	− 3379.88i	− 0.323415i
$479$	−1258.80	−0.120075	−0.0600376	−	0.998196i	$-0.519122\pi$
$479$	−1258.80	−0.120075	−0.0600376	+	0.998196i	$0.519122\pi$
$480$	0	0
$481$	−155.879	−0.0147765
$482$	− 2040.93i	− 0.192866i
$483$	4605.95i	0.433909i
$484$	923.389	0.0867195
$485$	0	0
$486$	−147.548	−0.0137714
$487$	− 5127.97i	− 0.477147i	−0.971124	−	0.238574i	$-0.923320\pi$
$487$	− 5127.97i	− 0.477147i	0.971124	−	0.238574i	$-0.0766798\pi$
$488$	8567.83i	0.794769i
$489$	1803.03	0.166740
$490$	0	0
$491$	11932.3	1.09673	0.548366	−	0.836238i	$-0.315250\pi$
$491$	11932.3	1.09673	0.548366	+	0.836238i	$0.315250\pi$
$492$	3135.84i	0.287347i
$493$	− 29137.7i	− 2.66185i
$494$	0.00687279	6.25955e−7 0
$495$	0	0
$496$	10816.3	0.979166
$497$	8000.21i	0.722049i
$498$	229.293i	0.0206322i
$499$	−14716.3	−1.32022	−0.660111	−	0.751168i	$-0.729491\pi$
$499$	−14716.3	−1.32022	−0.660111	+	0.751168i	$0.729491\pi$
$500$	0	0
$501$	−6663.57	−0.594225
$502$	− 1779.35i	− 0.158200i
$503$	− 5598.95i	− 0.496312i	−0.968720	−	0.248156i	$-0.920175\pi$
$503$	− 5598.95i	− 0.496312i	0.968720	−	0.248156i	$-0.0798246\pi$
$504$	−764.585	−0.0675741
$505$	0	0
$506$	1145.66	0.100654
$507$	6590.36i	0.577295i
$508$	− 4804.60i	− 0.419625i
$509$	16720.0	1.45600	0.727998	−	0.685579i	$-0.240451\pi$
$509$	16720.0	1.45600	0.727998	+	0.685579i	$0.240451\pi$
$510$	0	0
$511$	−10290.8	−0.890877
$512$	10251.9i	0.884915i
$513$	− 0.663817i	0 5.71310e-5i
$514$	−2072.28	−0.177830
$515$	0	0
$516$	−7694.27	−0.656437
$517$	− 5943.21i	− 0.505574i
$518$	− 1840.15i	− 0.156084i
$519$	711.841	0.0602050
$520$	0	0
$521$	3498.81	0.294214	0.147107	−	0.989121i	$-0.453004\pi$
$521$	3498.81	0.294214	0.147107	+	0.989121i	$0.453004\pi$
$522$	− 1240.15i	− 0.103985i
$523$	− 5681.71i	− 0.475036i	−0.971383	−	0.237518i	$-0.923666\pi$
$523$	− 5681.71i	− 0.475036i	0.971383	−	0.237518i	$-0.0763339\pi$
$524$	4369.44	0.364274
$525$	0	0
$526$	−2252.11	−0.186686
$527$	− 25118.8i	− 2.07627i
$528$	− 1824.49i	− 0.150380i
$529$	−17255.0	−1.41818
$530$	0	0
$531$	−88.8259	−0.00725935
$532$	− 1.67937i	0 0.000136861i
$533$	− 63.0603i	− 0.00512466i
$534$	274.076	0.0222106
$535$	0	0
$536$	1395.01	0.112417
$537$	− 3197.30i	− 0.256934i
$538$	3500.09i	0.280482i
$539$	−2891.72	−0.231085
$540$	0	0
$541$	11641.7	0.925169	0.462585	−	0.886575i	$-0.346922\pi$
$541$	11641.7	0.925169	0.462585	+	0.886575i	$0.346922\pi$
$542$	− 1145.36i	− 0.0907704i
$543$	− 3722.99i	− 0.294233i
$544$	14059.2	1.10806
$545$	0	0
$546$	7.50640	0.000588360 0
$547$	16460.2i	1.28663i	0.765602	+	0.643315i	$0.222441\pi$
$547$	16460.2i	1.28663i	−0.765602	+	0.643315i	$0.777559\pi$
$548$	7239.68i	0.564350i
$549$	−8124.41	−0.631587
$550$	0	0
$551$	5.57945	0.000431384 0
$552$	− 4884.04i	− 0.376591i
$553$	− 4113.15i	− 0.316291i
$554$	−2487.05	−0.190730
$555$	0	0
$556$	−18993.3	−1.44873
$557$	11769.7i	0.895331i	0.894201	+	0.447666i	$0.147745\pi$
$557$	11769.7i	0.895331i	−0.894201	+	0.447666i	$0.852255\pi$
$558$	− 1069.10i	− 0.0811089i
$559$	154.728	0.0117072
$560$	0	0
$561$	−4237.03	−0.318873
$562$	− 5336.42i	− 0.400540i
$563$	21123.9i	1.58129i	0.612276	+	0.790644i	$0.290254\pi$
$563$	21123.9i	1.58129i	−0.612276	+	0.790644i	$0.709746\pi$
$564$	−12369.4	−0.923486
$565$	0	0
$566$	1073.36	0.0797118
$567$	− 725.015i	− 0.0536997i
$568$	− 8483.23i	− 0.626670i
$569$	21890.4	1.61282	0.806410	−	0.591357i	$-0.201408\pi$
$569$	21890.4	1.61282	0.806410	+	0.591357i	$0.201408\pi$
$570$	0	0
$571$	6630.99	0.485986	0.242993	−	0.970028i	$-0.421871\pi$
$571$	6630.99	0.485986	0.242993	+	0.970028i	$0.421871\pi$
$572$	38.6469i	0.00282502i
$573$	− 5870.41i	− 0.427993i
$574$	744.425	0.0541319
$575$	0	0
$576$	−3382.32	−0.244670
$577$	− 13361.3i	− 0.964015i	−0.876167	−	0.482007i	$-0.839908\pi$
$577$	− 13361.3i	− 0.964015i	0.876167	−	0.482007i	$-0.160092\pi$
$578$	− 7026.58i	− 0.505653i
$579$	−8334.07	−0.598190
$580$	0	0
$581$	−1126.69	−0.0804527
$582$	− 2303.54i	− 0.164063i
$583$	6846.26i	0.486352i
$584$	10912.1	0.773196
$585$	0	0
$586$	−1870.06	−0.131829
$587$	9451.34i	0.664563i	0.943180	+	0.332282i	$0.107818\pi$
$587$	9451.34i	0.664563i	−0.943180	+	0.332282i	$0.892182\pi$
$588$	6018.44i	0.422102i
$589$	4.80990	0.000336483 0
$590$	0	0
$591$	2400.73	0.167094
$592$	− 18719.5i	− 1.29960i
$593$	712.344i	0.0493296i	0.999696	+	0.0246648i	$0.00785185\pi$
$593$	712.344i	0.0493296i	−0.999696	+	0.0246648i	$0.992148\pi$
$594$	−180.336	−0.0124567
$595$	0	0
$596$	16685.3	1.14674
$597$	7610.35i	0.521727i
$598$	47.9496i	0.00327894i
$599$	−23951.9	−1.63380	−0.816901	−	0.576777i	$-0.804310\pi$
$599$	−23951.9	−1.63380	−0.816901	+	0.576777i	$0.804310\pi$
$600$	0	0
$601$	18577.3	1.26087	0.630434	−	0.776243i	$-0.282877\pi$
$601$	18577.3	1.26087	0.630434	+	0.776243i	$0.282877\pi$
$602$	1826.56i	0.123663i
$603$	1322.81i	0.0893351i
$604$	27993.3	1.88581
$605$	0	0
$606$	−1258.33	−0.0843501
$607$	6230.72i	0.416634i	0.978061	+	0.208317i	$0.0667986\pi$
$607$	6230.72i	0.416634i	−0.978061	+	0.208317i	$0.933201\pi$
$608$	2.69214i	0 0.000179574i
$609$	6093.82	0.405475
$610$	0	0
$611$	248.743	0.0164698
$612$	8818.40i	0.582456i
$613$	− 17744.8i	− 1.16918i	−0.811330	−	0.584588i	$-0.801256\pi$
$613$	− 17744.8i	− 1.16918i	0.811330	−	0.584588i	$-0.198744\pi$
$614$	1743.95	0.114626
$615$	0	0
$616$	−934.493	−0.0611230
$617$	− 15305.4i	− 0.998656i	−0.866413	−	0.499328i	$-0.833580\pi$
$617$	− 15305.4i	− 0.998656i	0.866413	−	0.499328i	$-0.166420\pi$
$618$	21.4623i	0.00139699i
$619$	−17372.3	−1.12803	−0.564015	−	0.825764i	$-0.690744\pi$
$619$	−17372.3	−1.12803	−0.564015	+	0.825764i	$0.690744\pi$
$620$	0	0
$621$	4631.27	0.299270
$622$	− 193.074i	− 0.0124462i
$623$	1346.75i	0.0866073i
$624$	76.3610	0.00489886
$625$	0	0
$626$	5024.40	0.320791
$627$	− 0.811331i	0 5.16770e-5i
$628$	− 10823.3i	− 0.687735i
$629$	−43472.4	−2.75574
$630$	0	0
$631$	23699.3	1.49517	0.747586	−	0.664165i	$-0.231213\pi$
$631$	23699.3	1.49517	0.747586	+	0.664165i	$0.231213\pi$
$632$	4361.48i	0.274510i
$633$	4796.85i	0.301197i
$634$	621.973	0.0389617
$635$	0	0
$636$	14248.9	0.888374
$637$	− 121.028i	− 0.00752795i
$638$	− 1515.74i	− 0.0940577i
$639$	8044.19	0.498002
$640$	0	0
$641$	−15423.9	−0.950400	−0.475200	−	0.879878i	$-0.657624\pi$
$641$	−15423.9	−0.950400	−0.475200	+	0.879878i	$0.657624\pi$
$642$	− 843.179i	− 0.0518343i
$643$	13152.3i	0.806652i	0.915056	+	0.403326i	$0.132146\pi$
$643$	13152.3i	0.806652i	−0.915056	+	0.403326i	$0.867854\pi$
$644$	11716.5	0.716917
$645$	0	0
$646$	1.91672	0.000116737 0
$647$	5601.87i	0.340390i	0.985410	+	0.170195i	$0.0544397\pi$
$647$	5601.87i	0.340390i	−0.985410	+	0.170195i	$0.945560\pi$
$648$	768.788i	0.0466062i
$649$	−108.565	−0.00656633
$650$	0	0
$651$	5253.33	0.316274
$652$	− 4586.49i	− 0.275492i
$653$	− 27505.9i	− 1.64837i	−0.566319	−	0.824186i	$-0.691633\pi$
$653$	− 27505.9i	− 1.64837i	0.566319	−	0.824186i	$-0.308367\pi$
$654$	602.076	0.0359986
$655$	0	0
$656$	7572.87	0.450718
$657$	10347.4i	0.614443i
$658$	2936.41i	0.173971i
$659$	−16490.6	−0.974785	−0.487392	−	0.873183i	$-0.662052\pi$
$659$	−16490.6	−0.974785	−0.487392	+	0.873183i	$0.662052\pi$
$660$	0	0
$661$	12184.4	0.716973	0.358487	−	0.933535i	$-0.383293\pi$
$661$	12184.4	0.716973	0.358487	+	0.933535i	$0.383293\pi$
$662$	2778.41i	0.163121i
$663$	− 177.334i	− 0.0103878i
$664$	1194.72	0.0698253
$665$	0	0
$666$	−1850.27	−0.107652
$667$	38926.3i	2.25972i
$668$	16950.6i	0.981795i
$669$	15223.7	0.879791
$670$	0	0
$671$	−9929.83	−0.571292
$672$	2940.33i	0.168788i
$673$	− 13598.5i	− 0.778876i	−0.921053	−	0.389438i	$-0.872669\pi$
$673$	− 13598.5i	− 0.778876i	0.921053	−	0.389438i	$-0.127331\pi$
$674$	−557.334	−0.0318512
$675$	0	0
$676$	16764.4	0.953823
$677$	− 23318.1i	− 1.32377i	−0.749607	−	0.661883i	$-0.769758\pi$
$677$	− 23318.1i	− 1.32377i	0.749607	−	0.661883i	$-0.230242\pi$
$678$	− 2080.80i	− 0.117865i
$679$	11319.0	0.639742
$680$	0	0
$681$	3144.58	0.176947
$682$	− 1306.68i	− 0.0733658i
$683$	11493.5i	0.643906i	0.946756	+	0.321953i	$0.104339\pi$
$683$	11493.5i	0.643906i	−0.946756	+	0.321953i	$0.895661\pi$
$684$	−1.68860	−9.43935e−5 0
$685$	0	0
$686$	3292.89	0.183270
$687$	2203.99i	0.122398i
$688$	18581.2i	1.02965i
$689$	−286.539	−0.0158436
$690$	0	0
$691$	−20253.1	−1.11500	−0.557501	−	0.830176i	$-0.688240\pi$
$691$	−20253.1	−1.11500	−0.557501	+	0.830176i	$0.688240\pi$
$692$	− 1810.76i	− 0.0994724i
$693$	− 886.129i	− 0.0485733i
$694$	3280.21	0.179416
$695$	0	0
$696$	−6461.74	−0.351913
$697$	− 17586.6i	− 0.955723i
$698$	− 3106.30i	− 0.168446i
$699$	6038.34	0.326739
$700$	0	0
$701$	26895.3	1.44910	0.724551	−	0.689221i	$-0.242047\pi$
$701$	26895.3	1.44910	0.724551	+	0.689221i	$0.242047\pi$
$702$	− 7.54766i	0 0.000405795i
$703$	− 8.32435i	0 0.000446599i
$704$	−4133.94	−0.221312
$705$	0	0
$706$	1883.53	0.100407
$707$	− 6183.14i	− 0.328912i
$708$	225.953i	0.0119941i
$709$	18567.3	0.983512	0.491756	−	0.870733i	$-0.336355\pi$
$709$	18567.3	0.983512	0.491756	+	0.870733i	$0.336355\pi$
$710$	0	0
$711$	−4135.76	−0.218148
$712$	− 1428.06i	− 0.0751668i
$713$	33557.4i	1.76260i
$714$	2093.42	0.109726
$715$	0	0
$716$	−8133.21	−0.424514
$717$	− 16699.2i	− 0.869797i
$718$	− 2122.94i	− 0.110344i
$719$	19223.5	0.997104	0.498552	−	0.866860i	$-0.333865\pi$
$719$	19223.5	0.997104	0.498552	+	0.866860i	$0.333865\pi$
$720$	0	0
$721$	−105.461	−0.00544738
$722$	− 4164.73i	− 0.214675i
$723$	− 10083.8i	− 0.518698i
$724$	−9470.43	−0.486140
$725$	0	0
$726$	−220.411	−0.0112675
$727$	33855.9i	1.72716i	0.504210	+	0.863581i	$0.331784\pi$
$727$	33855.9i	1.72716i	−0.504210	+	0.863581i	$0.668216\pi$
$728$	− 39.1117i	− 0.00199117i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	43151.4	2.18333
$732$	20666.6i	1.04353i
$733$	26532.1i	1.33695i	0.743733	+	0.668477i	$0.233053\pi$
$733$	26532.1i	1.33695i	−0.743733	+	0.668477i	$0.766947\pi$
$734$	4641.71	0.233418
$735$	0	0
$736$	−18782.3	−0.940661
$737$	1616.77i	0.0808067i
$738$	− 748.517i	− 0.0373351i
$739$	15358.9	0.764528	0.382264	−	0.924053i	$-0.375144\pi$
$739$	15358.9	0.764528	0.382264	+	0.924053i	$0.375144\pi$
$740$	0	0
$741$	0.0339569	1.68345e−6 0
$742$	− 3382.58i	− 0.167356i
$743$	− 30995.4i	− 1.53043i	−0.643774	−	0.765216i	$-0.722632\pi$
$743$	− 30995.4i	− 1.53043i	0.643774	−	0.765216i	$-0.277368\pi$
$744$	−5570.50	−0.274495
$745$	0	0
$746$	3151.27	0.154660
$747$	1132.88i	0.0554887i
$748$	10778.0i	0.526851i
$749$	4143.19	0.202121
$750$	0	0
$751$	−32222.4	−1.56566	−0.782831	−	0.622234i	$-0.786225\pi$
$751$	−32222.4	−1.56566	−0.782831	+	0.622234i	$0.786225\pi$
$752$	29871.4i	1.44853i
$753$	− 8791.38i	− 0.425466i
$754$	63.4389	0.00306407
$755$	0	0
$756$	−1844.27	−0.0887243
$757$	36596.5i	1.75710i	0.477651	+	0.878549i	$0.341488\pi$
$757$	36596.5i	1.75710i	−0.477651	+	0.878549i	$0.658512\pi$
$758$	− 5206.01i	− 0.249460i
$759$	5660.44	0.270700
$760$	0	0
$761$	24718.0	1.17743	0.588716	−	0.808340i	$-0.299634\pi$
$761$	24718.0	1.17743	0.588716	+	0.808340i	$0.299634\pi$
$762$	1146.84i	0.0545220i
$763$	2958.46i	0.140372i
$764$	−14933.0	−0.707142
$765$	0	0
$766$	7128.58	0.336248
$767$	− 4.54381i	0 0.000213908i
$768$	7008.15i	0.329277i
$769$	−36376.7	−1.70582	−0.852910	−	0.522058i	$-0.825164\pi$
$769$	−36376.7	−1.70582	−0.852910	+	0.522058i	$0.825164\pi$
$770$	0	0
$771$	−10238.7	−0.478258
$772$	21200.0i	0.988347i
$773$	7525.40i	0.350155i	0.984555	+	0.175078i	$0.0560176\pi$
$773$	7525.40i	0.350155i	−0.984555	+	0.175078i	$0.943982\pi$
$774$	1836.60	0.0852911
$775$	0	0
$776$	−12002.4	−0.555235
$777$	− 9091.78i	− 0.419776i
$778$	2285.28i	0.105310i
$779$	3.36758	0.000154886 0
$780$	0	0
$781$	9831.78	0.450459
$782$	13372.4i	0.611505i
$783$	− 6127.32i	− 0.279658i
$784$	14534.2	0.662089
$785$	0	0
$786$	−1042.97	−0.0473303
$787$	41044.5i	1.85906i	0.368747	+	0.929530i	$0.379787\pi$
$787$	41044.5i	1.85906i	−0.368747	+	0.929530i	$0.620213\pi$
$788$	− 6106.90i	− 0.276078i
$789$	−11127.2	−0.502077
$790$	0	0
$791$	10224.5	0.459599
$792$	939.630i	0.0421569i
$793$	− 415.597i	− 0.0186107i
$794$	699.093	0.0312467
$795$	0	0
$796$	19359.0	0.862012
$797$	10836.9i	0.481635i	0.970570	+	0.240817i	$0.0774155\pi$
$797$	10836.9i	0.481635i	−0.970570	+	0.240817i	$0.922584\pi$
$798$	0.400861i	0 1.77823e-5i
$799$	69370.7	3.07154
$800$	0	0
$801$	1354.15	0.0597335
$802$	− 6102.91i	− 0.268705i
$803$	12646.8i	0.555785i
$804$	3364.93	0.147602
$805$	0	0
$806$	54.6890	0.00239000
$807$	17293.2i	0.754335i
$808$	6556.45i	0.285464i
$809$	16093.5	0.699402	0.349701	−	0.936861i	$-0.386283\pi$
$809$	16093.5	0.699402	0.349701	+	0.936861i	$0.386283\pi$
$810$	0	0
$811$	7289.26	0.315611	0.157805	−	0.987470i	$-0.449558\pi$
$811$	7289.26	0.315611	0.157805	+	0.987470i	$0.449558\pi$
$812$	− 15501.3i	− 0.669937i
$813$	− 5658.98i	− 0.244120i
$814$	−2261.44	−0.0973751
$815$	0	0
$816$	21295.9	0.913611
$817$	8.26288i	0 0.000353833i
$818$	3803.08i	0.162557i
$819$	37.0875	0.00158235
$820$	0	0
$821$	−33153.4	−1.40933	−0.704666	−	0.709539i	$-0.748903\pi$
$821$	−33153.4	−1.40933	−0.704666	+	0.709539i	$0.748903\pi$
$822$	− 1728.09i	− 0.0733262i
$823$	− 2781.57i	− 0.117812i	−0.998264	−	0.0589060i	$-0.981239\pi$
$823$	− 2781.57i	− 0.117812i	0.998264	−	0.0589060i	$-0.0187612\pi$
$824$	111.828	0.00472780
$825$	0	0
$826$	53.6395	0.00225951
$827$	4498.61i	0.189156i	0.995517	+	0.0945780i	$0.0301502\pi$
$827$	4498.61i	0.189156i	−0.995517	+	0.0945780i	$0.969850\pi$
$828$	− 11780.9i	− 0.494462i
$829$	−15630.3	−0.654843	−0.327421	−	0.944878i	$-0.606180\pi$
$829$	−15630.3	−0.654843	−0.327421	+	0.944878i	$0.606180\pi$
$830$	0	0
$831$	−12288.0	−0.512954
$832$	− 173.019i	− 0.00720958i
$833$	− 33752.9i	− 1.40392i
$834$	4533.66	0.188235
$835$	0	0
$836$	−2.06384	−8.53822e−5 0
$837$	− 5282.20i	− 0.218136i
$838$	− 5237.43i	− 0.215900i
$839$	6182.34	0.254396	0.127198	−	0.991877i	$-0.459402\pi$
$839$	6182.34	0.254396	0.127198	+	0.991877i	$0.459402\pi$
$840$	0	0
$841$	27111.8	1.11164
$842$	− 5522.56i	− 0.226033i
$843$	− 26366.1i	− 1.07722i
$844$	12202.1	0.497646
$845$	0	0
$846$	2952.55	0.119989
$847$	− 1083.05i	− 0.0439362i
$848$	− 34410.3i	− 1.39346i
$849$	5303.26	0.214378
$850$	0	0
$851$	58076.7	2.33942
$852$	− 20462.6i	− 0.822813i
$853$	31979.3i	1.28365i	0.766852	+	0.641824i	$0.221822\pi$
$853$	31979.3i	1.28365i	−0.766852	+	0.641824i	$0.778178\pi$
$854$	4906.11	0.196585
$855$	0	0
$856$	−4393.33	−0.175422
$857$	8439.63i	0.336397i	0.985753	+	0.168199i	$0.0537950\pi$
$857$	8439.63i	0.336397i	−0.985753	+	0.168199i	$0.946205\pi$
$858$	− 9.22492i	0 0.000367056i
$859$	−47640.1	−1.89227	−0.946134	−	0.323774i	$-0.895048\pi$
$859$	−47640.1	−1.89227	−0.946134	+	0.323774i	$0.895048\pi$
$860$	0	0
$861$	3678.04	0.145583
$862$	− 2434.06i	− 0.0961770i
$863$	27963.3i	1.10299i	0.834177	+	0.551496i	$0.185943\pi$
$863$	27963.3i	1.10299i	−0.834177	+	0.551496i	$0.814057\pi$
$864$	2956.50	0.116414
$865$	0	0
$866$	−1648.75	−0.0646962
$867$	− 34716.8i	− 1.35991i
$868$	− 13363.3i	− 0.522557i
$869$	−5054.81	−0.197322
$870$	0	0
$871$	−67.6673	−0.00263240
$872$	− 3137.08i	− 0.121829i
$873$	− 11381.3i	− 0.441234i
$874$	−2.56063	−9.91013e−5 0
$875$	0	0
$876$	26321.3	1.01520
$877$	30584.0i	1.17759i	0.808282	+	0.588796i	$0.200398\pi$
$877$	30584.0i	1.17759i	−0.808282	+	0.588796i	$0.799602\pi$
$878$	6732.23i	0.258772i
$879$	−9239.56	−0.354542
$880$	0	0
$881$	14444.3	0.552372	0.276186	−	0.961104i	$-0.410929\pi$
$881$	14444.3	0.552372	0.276186	+	0.961104i	$0.410929\pi$
$882$	− 1436.58i	− 0.0548439i
$883$	− 19921.4i	− 0.759241i	−0.925142	−	0.379620i	$-0.876055\pi$
$883$	− 19921.4i	− 0.759241i	0.925142	−	0.379620i	$-0.123945\pi$
$884$	−451.097	−0.0171630
$885$	0	0
$886$	−4331.09	−0.164228
$887$	13172.5i	0.498635i	0.968422	+	0.249317i	$0.0802063\pi$
$887$	13172.5i	0.498635i	−0.968422	+	0.249317i	$0.919794\pi$
$888$	9640.71i	0.364325i
$889$	−5635.33	−0.212602
$890$	0	0
$891$	−891.000	−0.0335013
$892$	− 38725.5i	− 1.45362i
$893$	13.2835i	0 0.000497778i
$894$	−3982.75	−0.148997
$895$	0	0
$896$	9883.38	0.368505
$897$	236.908i	0.00881844i
$898$	− 3820.01i	− 0.141955i
$899$	44397.5	1.64710
$900$	0	0
$901$	−79911.4	−2.95475
$902$	− 914.854i	− 0.0337709i
$903$	9024.64i	0.332582i
$904$	−10841.9	−0.398888
$905$	0	0
$906$	−6681.92	−0.245024
$907$	− 16309.7i	− 0.597084i	−0.954396	−	0.298542i	$-0.903500\pi$
$907$	− 16309.7i	− 0.597084i	0.954396	−	0.298542i	$-0.0965003\pi$
$908$	− 7999.10i	− 0.292356i
$909$	−6217.13	−0.226853
$910$	0	0
$911$	−14994.4	−0.545320	−0.272660	−	0.962110i	$-0.587903\pi$
$911$	−14994.4	−0.545320	−0.272660	+	0.962110i	$0.587903\pi$
$912$	4.07787i	0 0.000148061i
$913$	1384.64i	0.0501914i
$914$	4427.63	0.160233
$915$	0	0
$916$	5606.44	0.202229
$917$	− 5124.93i	− 0.184558i
$918$	− 2104.93i	− 0.0756787i
$919$	−40263.5	−1.44524	−0.722618	−	0.691248i	$-0.757061\pi$
$919$	−40263.5	−1.44524	−0.722618	+	0.691248i	$0.757061\pi$
$920$	0	0
$921$	8616.48	0.308276
$922$	− 10204.1i	− 0.364484i
$923$	411.493i	0.0146744i
$924$	−2254.11	−0.0802541
$925$	0	0
$926$	−4540.87	−0.161147
$927$	106.040i	0.00375709i
$928$	24849.6i	0.879019i
$929$	9469.52	0.334429	0.167215	−	0.985921i	$-0.446523\pi$
$929$	9469.52	0.334429	0.167215	+	0.985921i	$0.446523\pi$
$930$	0	0
$931$	6.46319	0.000227522 0
$932$	− 15360.2i	− 0.539848i
$933$	− 953.933i	− 0.0334731i
$934$	−4570.23	−0.160110
$935$	0	0
$936$	−39.3266	−0.00137332
$937$	− 6356.51i	− 0.221620i	−0.993842	−	0.110810i	$-0.964655\pi$
$937$	− 6356.51i	− 0.221620i	0.993842	−	0.110810i	$-0.0353445\pi$
$938$	− 798.810i	− 0.0278061i
$939$	24824.4	0.862742
$940$	0	0
$941$	−5764.45	−0.199698	−0.0998489	−	0.995003i	$-0.531836\pi$
$941$	−5764.45	−0.199698	−0.0998489	+	0.995003i	$0.531836\pi$
$942$	2583.50i	0.0893577i
$943$	23494.7i	0.811338i
$944$	545.663	0.0188134
$945$	0	0
$946$	2244.74	0.0771487
$947$	2284.74i	0.0783993i	0.999231	+	0.0391997i	$0.0124808\pi$
$947$	2284.74i	0.0783993i	−0.999231	+	0.0391997i	$0.987519\pi$
$948$	10520.4i	0.360430i
$949$	−529.310	−0.0181055
$950$	0	0
$951$	3073.03	0.104784
$952$	− 10907.7i	− 0.371344i
$953$	− 17209.0i	− 0.584947i	−0.956274	−	0.292474i	$-0.905522\pi$
$953$	− 17209.0i	− 0.584947i	0.956274	−	0.292474i	$-0.0944783\pi$
$954$	−3401.18	−0.115427
$955$	0	0
$956$	−42479.1	−1.43710
$957$	− 7488.95i	− 0.252961i
$958$	− 764.333i	− 0.0257771i
$959$	8491.45	0.285926
$960$	0	0
$961$	8482.92	0.284748
$962$	− 94.6487i	− 0.00317214i
$963$	− 4165.96i	− 0.139404i
$964$	−25650.8	−0.857008
$965$	0	0
$966$	−2796.70	−0.0931493
$967$	7966.25i	0.264920i	0.991188	+	0.132460i	$0.0422876\pi$
$967$	7966.25i	0.264920i	−0.991188	+	0.132460i	$0.957712\pi$
$968$	1148.44i	0.0381324i
$969$	9.47008	0.000313955 0
$970$	0	0
$971$	−25547.5	−0.844343	−0.422172	−	0.906516i	$-0.638732\pi$
$971$	−25547.5	−0.844343	−0.422172	+	0.906516i	$0.638732\pi$
$972$	1854.41i	0.0611937i
$973$	22277.3i	0.733996i
$974$	3113.66	0.102431
$975$	0	0
$976$	49908.7	1.63682
$977$	− 12899.8i	− 0.422417i	−0.977441	−	0.211209i	$-0.932260\pi$
$977$	− 12899.8i	− 0.422417i	0.977441	−	0.211209i	$-0.0677399\pi$
$978$	1094.78i	0.0357948i
$979$	1655.07	0.0540310
$980$	0	0
$981$	2974.73	0.0968152
$982$	7245.18i	0.235441i
$983$	− 7399.28i	− 0.240082i	−0.992769	−	0.120041i	$-0.961697\pi$
$983$	− 7399.28i	− 0.240082i	0.992769	−	0.120041i	$-0.0383026\pi$
$984$	−3900.10	−0.126352
$985$	0	0
$986$	17692.2	0.571433
$987$	14508.1i	0.467881i
$988$	− 0.0863787i	0 2.78145e-6i
$989$	−57647.8	−1.85348
$990$	0	0
$991$	−38466.4	−1.23302	−0.616511	−	0.787346i	$-0.711454\pi$
$991$	−38466.4	−1.23302	−0.616511	+	0.787346i	$0.711454\pi$
$992$	21422.2i	0.685642i
$993$	13727.5i	0.438700i
$994$	−4857.66	−0.155006
$995$	0	0
$996$	2881.80	0.0916801
$997$	62092.5i	1.97241i	0.165538	+	0.986203i	$0.447064\pi$
$997$	62092.5i	1.97241i	−0.165538	+	0.986203i	$0.552936\pi$
$998$	− 8935.60i	− 0.283418i
$999$	−9141.76	−0.289522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.p.199.5		8
5.2	odd	4		165.4.a.h.1.2	✓	4
5.3	odd	4		825.4.a.t.1.3		4
5.4	even	2	inner	825.4.c.p.199.4		8
15.2	even	4		495.4.a.m.1.3		4
15.8	even	4		2475.4.a.be.1.2		4
55.32	even	4		1815.4.a.t.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.h.1.2	✓	4	5.2	odd	4
495.4.a.m.1.3		4	15.2	even	4
825.4.a.t.1.3		4	5.3	odd	4
825.4.c.p.199.4		8	5.4	even	2	inner
825.4.c.p.199.5		8	1.1	even	1	trivial
1815.4.a.t.1.3		4	55.32	even	4
2475.4.a.be.1.2		4	15.8	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.607192i q^{2} +3.00000i q^{3} +7.63132 q^{4} -1.82158 q^{6} -8.95080i q^{7} +9.49121i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+0.607192i q^{2} +3.00000i q^{3} +7.63132 q^{4} -1.82158 q^{6} -8.95080i q^{7} +9.49121i q^{8} -9.00000 q^{9} -11.0000 q^{11} +22.8940i q^{12} -0.460387i q^{13} +5.43485 q^{14} +55.2876 q^{16} -128.395i q^{17} -5.46473i q^{18} +0.0245858 q^{19} +26.8524 q^{21} -6.67911i q^{22} +171.528i q^{23} -28.4736 q^{24} +0.279543 q^{26} -27.0000i q^{27} -68.3064i q^{28} +226.938 q^{29} +195.637 q^{31} +109.500i q^{32} -33.0000i q^{33} +77.9604 q^{34} -68.6819 q^{36} -338.584i q^{37} +0.0149283i q^{38} +1.38116 q^{39} +136.972 q^{41} +16.3046i q^{42} +336.083i q^{43} -83.9445 q^{44} -104.151 q^{46} +540.292i q^{47} +165.863i q^{48} +262.883 q^{49} +385.185 q^{51} -3.51336i q^{52} -622.387i q^{53} +16.3942 q^{54} +84.9539 q^{56} +0.0737574i q^{57} +137.795i q^{58} +9.86955 q^{59} +902.712 q^{61} +118.789i q^{62} +80.5572i q^{63} +375.813 q^{64} +20.0373 q^{66} -146.979i q^{67} -979.823i q^{68} -514.585 q^{69} -893.798 q^{71} -85.4209i q^{72} -1149.71i q^{73} +205.585 q^{74} +0.187622 q^{76} +98.4588i q^{77} +0.838629i q^{78} +459.528 q^{79} +81.0000 q^{81} +83.1686i q^{82} -125.876i q^{83} +204.919 q^{84} -204.067 q^{86} +680.813i q^{87} -104.403i q^{88} -150.461 q^{89} -4.12083 q^{91} +1308.99i q^{92} +586.912i q^{93} -328.061 q^{94} -328.499 q^{96} +1264.58i q^{97} +159.621i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9}+O(q^{10}) \) 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9	\( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9} - 88 q^{11} + 104 q^{14} + 132 q^{16} - 272 q^{19} + 204 q^{21} - 288 q^{24} - 640 q^{26} - 104 q^{29} + 984 q^{31} - 488 q^{34} + 468 q^{36} - 12 q^{39} + 536 q^{41} + 572 q^{44} + 736 q^{46} + 992 q^{49} + 444 q^{51} - 216 q^{54} - 1704 q^{56} + 2064 q^{59} + 232 q^{61} + 1836 q^{64} - 264 q^{66} + 384 q^{69} - 1840 q^{71} + 5712 q^{74} + 3144 q^{76} - 2304 q^{79} + 648 q^{81} + 120 q^{84} + 472 q^{86} + 2128 q^{89} + 5560 q^{91} + 2864 q^{94} + 1248 q^{96} + 792 q^{99}+O(q^{100}) \) 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9 - 88 * q^11 + 104 * q^14 + 132 * q^16 - 272 * q^19 + 204 * q^21 - 288 * q^24 - 640 * q^26 - 104 * q^29 + 984 * q^31 - 488 * q^34 + 468 * q^36 - 12 * q^39 + 536 * q^41 + 572 * q^44 + 736 * q^46 + 992 * q^49 + 444 * q^51 - 216 * q^54 - 1704 * q^56 + 2064 * q^59 + 232 * q^61 + 1836 * q^64 - 264 * q^66 + 384 * q^69 - 1840 * q^71 + 5712 * q^74 + 3144 * q^76 - 2304 * q^79 + 648 * q^81 + 120 * q^84 + 472 * q^86 + 2128 * q^89 + 5560 * q^91 + 2864 * q^94 + 1248 * q^96 + 792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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