LMFDB - Embedded newform 1050.6.g.i.799.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,6,Mod(799,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.799"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1050.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32,0,-72,0,0,-162,0,1328] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$168.403010804$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.6.g.i.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -49.0000i q^{7} +64.0000i q^{8} -81.0000 q^{9} +664.000 q^{11} +144.000i q^{12} -318.000i q^{13} -196.000 q^{14} +256.000 q^{16} +1582.00i q^{17} +324.000i q^{18} -236.000 q^{19} -441.000 q^{21} -2656.00i q^{22} -2212.00i q^{23} +576.000 q^{24} -1272.00 q^{26} +729.000i q^{27} +784.000i q^{28} +4954.00 q^{29} -7128.00 q^{31} -1024.00i q^{32} -5976.00i q^{33} +6328.00 q^{34} +1296.00 q^{36} +4358.00i q^{37} +944.000i q^{38} -2862.00 q^{39} +10542.0 q^{41} +1764.00i q^{42} +8452.00i q^{43} -10624.0 q^{44} -8848.00 q^{46} +5352.00i q^{47} -2304.00i q^{48} -2401.00 q^{49} +14238.0 q^{51} +5088.00i q^{52} +33354.0i q^{53} +2916.00 q^{54} +3136.00 q^{56} +2124.00i q^{57} -19816.0i q^{58} +15436.0 q^{59} -36762.0 q^{61} +28512.0i q^{62} +3969.00i q^{63} -4096.00 q^{64} -23904.0 q^{66} +40972.0i q^{67} -25312.0i q^{68} -19908.0 q^{69} -9092.00 q^{71} -5184.00i q^{72} +73454.0i q^{73} +17432.0 q^{74} +3776.00 q^{76} -32536.0i q^{77} +11448.0i q^{78} -89400.0 q^{79} +6561.00 q^{81} -42168.0i q^{82} +6428.00i q^{83} +7056.00 q^{84} +33808.0 q^{86} -44586.0i q^{87} +42496.0i q^{88} +122658. q^{89} -15582.0 q^{91} +35392.0i q^{92} +64152.0i q^{93} +21408.0 q^{94} -9216.00 q^{96} +21370.0i q^{97} +9604.00i q^{98} -53784.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} + 1328 q^{11} - 392 q^{14} + 512 q^{16} - 472 q^{19} - 882 q^{21} + 1152 q^{24} - 2544 q^{26} + 9908 q^{29} - 14256 q^{31} + 12656 q^{34} + 2592 q^{36} - 5724 q^{39}+ \cdots - 107568 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 + 1328 * q^11 - 392 * q^14 + 512 * q^16 - 472 * q^19 - 882 * q^21 + 1152 * q^24 - 2544 * q^26 + 9908 * q^29 - 14256 * q^31 + 12656 * q^34 + 2592 * q^36 - 5724 * q^39 + 21084 * q^41 - 21248 * q^44 - 17696 * q^46 - 4802 * q^49 + 28476 * q^51 + 5832 * q^54 + 6272 * q^56 + 30872 * q^59 - 73524 * q^61 - 8192 * q^64 - 47808 * q^66 - 39816 * q^69 - 18184 * q^71 + 34864 * q^74 + 7552 * q^76 - 178800 * q^79 + 13122 * q^81 + 14112 * q^84 + 67616 * q^86 + 245316 * q^89 - 31164 * q^91 + 42816 * q^94 - 18432 * q^96 - 107568 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	64.0000i	0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	664.000	1.65457	0.827287	−	0.561779i	$-0.189883\pi$
$11$	664.000	1.65457	0.827287	+	0.561779i	$0.189883\pi$
$12$	144.000i	0.288675i
$13$	− 318.000i	− 0.521878i	−0.965355	−	0.260939i	$-0.915968\pi$
$13$	− 318.000i	− 0.521878i	0.965355	−	0.260939i	$-0.0840321\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1582.00i	1.32765i	0.747887	+	0.663826i	$0.231068\pi$
$17$	1582.00i	1.32765i	−0.747887	+	0.663826i	$0.768932\pi$
$18$	324.000i	0.235702i
$19$	−236.000	−0.149978	−0.0749891	−	0.997184i	$-0.523892\pi$
$19$	−236.000	−0.149978	−0.0749891	+	0.997184i	$0.523892\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	− 2656.00i	− 1.16996i
$23$	− 2212.00i	− 0.871898i	−0.899971	−	0.435949i	$-0.856413\pi$
$23$	− 2212.00i	− 0.871898i	0.899971	−	0.435949i	$-0.143587\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−1272.00	−0.369023
$27$	729.000i	0.192450i
$28$	784.000i	0.188982i
$29$	4954.00	1.09386	0.546929	−	0.837179i	$-0.315797\pi$
$29$	4954.00	1.09386	0.546929	+	0.837179i	$0.315797\pi$
$30$	0	0
$31$	−7128.00	−1.33218	−0.666091	−	0.745871i	$-0.732034\pi$
$31$	−7128.00	−1.33218	−0.666091	+	0.745871i	$0.732034\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 5976.00i	− 0.955269i
$34$	6328.00	0.938792
$35$	0	0
$36$	1296.00	0.166667
$37$	4358.00i	0.523339i	0.965158	+	0.261669i	$0.0842730\pi$
$37$	4358.00i	0.523339i	−0.965158	+	0.261669i	$0.915727\pi$
$38$	944.000i	0.106051i
$39$	−2862.00	−0.301306
$40$	0	0
$41$	10542.0	0.979407	0.489704	−	0.871889i	$-0.337105\pi$
$41$	10542.0	0.979407	0.489704	+	0.871889i	$0.337105\pi$
$42$	1764.00i	0.154303i
$43$	8452.00i	0.697089i	0.937292	+	0.348545i	$0.113324\pi$
$43$	8452.00i	0.697089i	−0.937292	+	0.348545i	$0.886676\pi$
$44$	−10624.0	−0.827287
$45$	0	0
$46$	−8848.00	−0.616525
$47$	5352.00i	0.353404i	0.984264	+	0.176702i	$0.0565429\pi$
$47$	5352.00i	0.353404i	−0.984264	+	0.176702i	$0.943457\pi$
$48$	− 2304.00i	− 0.144338i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	14238.0	0.766520
$52$	5088.00i	0.260939i
$53$	33354.0i	1.63102i	0.578746	+	0.815508i	$0.303542\pi$
$53$	33354.0i	1.63102i	−0.578746	+	0.815508i	$0.696458\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	3136.00	0.133631
$57$	2124.00i	0.0865899i
$58$	− 19816.0i	− 0.773475i
$59$	15436.0	0.577304	0.288652	−	0.957434i	$-0.406793\pi$
$59$	15436.0	0.577304	0.288652	+	0.957434i	$0.406793\pi$
$60$	0	0
$61$	−36762.0	−1.26495	−0.632477	−	0.774579i	$-0.717962\pi$
$61$	−36762.0	−1.26495	−0.632477	+	0.774579i	$0.717962\pi$
$62$	28512.0i	0.941995i
$63$	3969.00i	0.125988i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−23904.0	−0.675477
$67$	40972.0i	1.11506i	0.830155	+	0.557532i	$0.188252\pi$
$67$	40972.0i	1.11506i	−0.830155	+	0.557532i	$0.811748\pi$
$68$	− 25312.0i	− 0.663826i
$69$	−19908.0	−0.503390
$70$	0	0
$71$	−9092.00	−0.214049	−0.107025	−	0.994256i	$-0.534132\pi$
$71$	−9092.00	−0.214049	−0.107025	+	0.994256i	$0.534132\pi$
$72$	− 5184.00i	− 0.117851i
$73$	73454.0i	1.61327i	0.591047	+	0.806637i	$0.298715\pi$
$73$	73454.0i	1.61327i	−0.591047	+	0.806637i	$0.701285\pi$
$74$	17432.0	0.370056
$75$	0	0
$76$	3776.00	0.0749891
$77$	− 32536.0i	− 0.625370i
$78$	11448.0i	0.213056i
$79$	−89400.0	−1.61165	−0.805823	−	0.592156i	$-0.798277\pi$
$79$	−89400.0	−1.61165	−0.805823	+	0.592156i	$0.798277\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 42168.0i	− 0.692546i
$83$	6428.00i	0.102419i	0.998688	+	0.0512095i	$0.0163076\pi$
$83$	6428.00i	0.102419i	−0.998688	+	0.0512095i	$0.983692\pi$
$84$	7056.00	0.109109
$85$	0	0
$86$	33808.0	0.492916
$87$	− 44586.0i	− 0.631539i
$88$	42496.0i	0.584980i
$89$	122658.	1.64142	0.820712	−	0.571342i	$-0.193577\pi$
$89$	122658.	1.64142	0.820712	+	0.571342i	$0.193577\pi$
$90$	0	0
$91$	−15582.0	−0.197251
$92$	35392.0i	0.435949i
$93$	64152.0i	0.769135i
$94$	21408.0	0.249894
$95$	0	0
$96$	−9216.00	−0.102062
$97$	21370.0i	0.230608i	0.993330	+	0.115304i	$0.0367843\pi$
$97$	21370.0i	0.230608i	−0.993330	+	0.115304i	$0.963216\pi$
$98$	9604.00i	0.101015i
$99$	−53784.0	−0.551525
$100$	0	0
$101$	−36814.0	−0.359095	−0.179548	−	0.983749i	$-0.557463\pi$
$101$	−36814.0	−0.359095	−0.179548	+	0.983749i	$0.557463\pi$
$102$	− 56952.0i	− 0.542012i
$103$	− 104528.i	− 0.970822i	−0.874286	−	0.485411i	$-0.838670\pi$
$103$	− 104528.i	− 0.970822i	0.874286	−	0.485411i	$-0.161330\pi$
$104$	20352.0	0.184512
$105$	0	0
$106$	133416.	1.15330
$107$	214440.i	1.81070i	0.424667	+	0.905350i	$0.360391\pi$
$107$	214440.i	1.81070i	−0.424667	+	0.905350i	$0.639609\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	−28798.0	−0.232165	−0.116082	−	0.993240i	$-0.537034\pi$
$109$	−28798.0	−0.232165	−0.116082	+	0.993240i	$0.537034\pi$
$110$	0	0
$111$	39222.0	0.302150
$112$	− 12544.0i	− 0.0944911i
$113$	56014.0i	0.412668i	0.978482	+	0.206334i	$0.0661533\pi$
$113$	56014.0i	0.412668i	−0.978482	+	0.206334i	$0.933847\pi$
$114$	8496.00	0.0612283
$115$	0	0
$116$	−79264.0	−0.546929
$117$	25758.0i	0.173959i
$118$	− 61744.0i	− 0.408216i
$119$	77518.0	0.501805
$120$	0	0
$121$	279845.	1.73762
$122$	147048.i	0.894457i
$123$	− 94878.0i	− 0.565461i
$124$	114048.	0.666091
$125$	0	0
$126$	15876.0	0.0890871
$127$	185400.i	1.02000i	0.860174	+	0.510000i	$0.170355\pi$
$127$	185400.i	1.02000i	−0.860174	+	0.510000i	$0.829645\pi$
$128$	16384.0i	0.0883883i
$129$	76068.0	0.402465
$130$	0	0
$131$	64532.0	0.328547	0.164273	−	0.986415i	$-0.447472\pi$
$131$	64532.0	0.328547	0.164273	+	0.986415i	$0.447472\pi$
$132$	95616.0i	0.477635i
$133$	11564.0i	0.0566864i
$134$	163888.	0.788470
$135$	0	0
$136$	−101248.	−0.469396
$137$	152930.i	0.696131i	0.937470	+	0.348066i	$0.113161\pi$
$137$	152930.i	0.696131i	−0.937470	+	0.348066i	$0.886839\pi$
$138$	79632.0i	0.355951i
$139$	343460.	1.50778	0.753892	−	0.656998i	$-0.228174\pi$
$139$	343460.	1.50778	0.753892	+	0.656998i	$0.228174\pi$
$140$	0	0
$141$	48168.0	0.204038
$142$	36368.0i	0.151356i
$143$	− 211152.i	− 0.863486i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	293816.	1.14076
$147$	21609.0i	0.0824786i
$148$	− 69728.0i	− 0.261669i
$149$	174858.	0.645238	0.322619	−	0.946529i	$-0.395437\pi$
$149$	174858.	0.645238	0.322619	+	0.946529i	$0.395437\pi$
$150$	0	0
$151$	−452552.	−1.61520	−0.807600	−	0.589731i	$-0.799234\pi$
$151$	−452552.	−1.61520	−0.807600	+	0.589731i	$0.799234\pi$
$152$	− 15104.0i	− 0.0530253i
$153$	− 128142.i	− 0.442551i
$154$	−130144.	−0.442204
$155$	0	0
$156$	45792.0	0.150653
$157$	− 499066.i	− 1.61588i	−0.589265	−	0.807940i	$-0.700583\pi$
$157$	− 499066.i	− 1.61588i	0.589265	−	0.807940i	$-0.299417\pi$
$158$	357600.i	1.13961i
$159$	300186.	0.941668
$160$	0	0
$161$	−108388.	−0.329546
$162$	− 26244.0i	− 0.0785674i
$163$	475588.i	1.40204i	0.713139	+	0.701022i	$0.247273\pi$
$163$	475588.i	1.40204i	−0.713139	+	0.701022i	$0.752727\pi$
$164$	−168672.	−0.489704
$165$	0	0
$166$	25712.0	0.0724212
$167$	120224.i	0.333580i	0.985992	+	0.166790i	$0.0533402\pi$
$167$	120224.i	0.333580i	−0.985992	+	0.166790i	$0.946660\pi$
$168$	− 28224.0i	− 0.0771517i
$169$	270169.	0.727644
$170$	0	0
$171$	19116.0	0.0499927
$172$	− 135232.i	− 0.348545i
$173$	− 508874.i	− 1.29269i	−0.763045	−	0.646346i	$-0.776296\pi$
$173$	− 508874.i	− 1.29269i	0.763045	−	0.646346i	$-0.223704\pi$
$174$	−178344.	−0.446566
$175$	0	0
$176$	169984.	0.413644
$177$	− 138924.i	− 0.333307i
$178$	− 490632.i	− 1.16066i
$179$	−487560.	−1.13735	−0.568677	−	0.822561i	$-0.692544\pi$
$179$	−487560.	−1.13735	−0.568677	+	0.822561i	$0.692544\pi$
$180$	0	0
$181$	−544410.	−1.23518	−0.617589	−	0.786501i	$-0.711891\pi$
$181$	−544410.	−1.23518	−0.617589	+	0.786501i	$0.711891\pi$
$182$	62328.0i	0.139478i
$183$	330858.i	0.730321i
$184$	141568.	0.308262
$185$	0	0
$186$	256608.	0.543861
$187$	1.05045e6i	2.19670i
$188$	− 85632.0i	− 0.176702i
$189$	35721.0	0.0727393
$190$	0	0
$191$	376404.	0.746570	0.373285	−	0.927717i	$-0.378231\pi$
$191$	376404.	0.746570	0.373285	+	0.927717i	$0.378231\pi$
$192$	36864.0i	0.0721688i
$193$	− 844946.i	− 1.63281i	−0.577480	−	0.816405i	$-0.695964\pi$
$193$	− 844946.i	− 1.63281i	0.577480	−	0.816405i	$-0.304036\pi$
$194$	85480.0	0.163065
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 492794.i	− 0.904690i	−0.891843	−	0.452345i	$-0.850588\pi$
$197$	− 492794.i	− 0.904690i	0.891843	−	0.452345i	$-0.149412\pi$
$198$	215136.i	0.389987i
$199$	914776.	1.63750	0.818751	−	0.574148i	$-0.194667\pi$
$199$	914776.	1.63750	0.818751	+	0.574148i	$0.194667\pi$
$200$	0	0
$201$	368748.	0.643783
$202$	147256.i	0.253919i
$203$	− 242746.i	− 0.413440i
$204$	−227808.	−0.383260
$205$	0	0
$206$	−418112.	−0.686475
$207$	179172.i	0.290633i
$208$	− 81408.0i	− 0.130469i
$209$	−156704.	−0.248150
$210$	0	0
$211$	311780.	0.482106	0.241053	−	0.970512i	$-0.422507\pi$
$211$	311780.	0.482106	0.241053	+	0.970512i	$0.422507\pi$
$212$	− 533664.i	− 0.815508i
$213$	81828.0i	0.123581i
$214$	857760.	1.28036
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	349272.i	0.503517i
$218$	115192.i	0.164165i
$219$	661086.	0.931425
$220$	0	0
$221$	503076.	0.692872
$222$	− 156888.i	− 0.213652i
$223$	1.28776e6i	1.73409i	0.498226	+	0.867047i	$0.333985\pi$
$223$	1.28776e6i	1.73409i	−0.498226	+	0.867047i	$0.666015\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	224056.	0.291800
$227$	1.28905e6i	1.66037i	0.557485	+	0.830187i	$0.311766\pi$
$227$	1.28905e6i	1.66037i	−0.557485	+	0.830187i	$0.688234\pi$
$228$	− 33984.0i	− 0.0432950i
$229$	−678214.	−0.854630	−0.427315	−	0.904103i	$-0.640540\pi$
$229$	−678214.	−0.854630	−0.427315	+	0.904103i	$0.640540\pi$
$230$	0	0
$231$	−292824.	−0.361058
$232$	317056.i	0.386737i
$233$	1.11731e6i	1.34829i	0.738598	+	0.674146i	$0.235488\pi$
$233$	1.11731e6i	1.34829i	−0.738598	+	0.674146i	$0.764512\pi$
$234$	103032.	0.123008
$235$	0	0
$236$	−246976.	−0.288652
$237$	804600.i	0.930485i
$238$	− 310072.i	− 0.354830i
$239$	1.26196e6	1.42906	0.714528	−	0.699606i	$-0.246641\pi$
$239$	1.26196e6	1.42906	0.714528	+	0.699606i	$0.246641\pi$
$240$	0	0
$241$	948218.	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$241$	948218.	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$242$	− 1.11938e6i	− 1.22868i
$243$	− 59049.0i	− 0.0641500i
$244$	588192.	0.632477
$245$	0	0
$246$	−379512.	−0.399841
$247$	75048.0i	0.0782703i
$248$	− 456192.i	− 0.470997i
$249$	57852.0	0.0591317
$250$	0	0
$251$	−486396.	−0.487310	−0.243655	−	0.969862i	$-0.578347\pi$
$251$	−486396.	−0.487310	−0.243655	+	0.969862i	$0.578347\pi$
$252$	− 63504.0i	− 0.0629941i
$253$	− 1.46877e6i	− 1.44262i
$254$	741600.	0.721249
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.03910e6i	− 0.981349i	−0.871343	−	0.490675i	$-0.836750\pi$
$257$	− 1.03910e6i	− 0.981349i	0.871343	−	0.490675i	$-0.163250\pi$
$258$	− 304272.i	− 0.284585i
$259$	213542.	0.197803
$260$	0	0
$261$	−401274.	−0.364619
$262$	− 258128.i	− 0.232317i
$263$	− 1.35104e6i	− 1.20443i	−0.798335	−	0.602213i	$-0.794286\pi$
$263$	− 1.35104e6i	− 1.20443i	0.798335	−	0.602213i	$-0.205714\pi$
$264$	382464.	0.337739
$265$	0	0
$266$	46256.0	0.0400833
$267$	− 1.10392e6i	− 0.947677i
$268$	− 655552.i	− 0.557532i
$269$	1.11811e6	0.942115	0.471057	−	0.882103i	$-0.343872\pi$
$269$	1.11811e6	0.942115	0.471057	+	0.882103i	$0.343872\pi$
$270$	0	0
$271$	−190104.	−0.157242	−0.0786209	−	0.996905i	$-0.525052\pi$
$271$	−190104.	−0.157242	−0.0786209	+	0.996905i	$0.525052\pi$
$272$	404992.i	0.331913i
$273$	140238.i	0.113883i
$274$	611720.	0.492239
$275$	0	0
$276$	318528.	0.251695
$277$	− 200506.i	− 0.157010i	−0.996914	−	0.0785051i	$-0.974985\pi$
$277$	− 200506.i	− 0.157010i	0.996914	−	0.0785051i	$-0.0250147\pi$
$278$	− 1.37384e6i	− 1.06616i
$279$	577368.	0.444061
$280$	0	0
$281$	1.09237e6	0.825285	0.412643	−	0.910893i	$-0.364606\pi$
$281$	1.09237e6	0.825285	0.412643	+	0.910893i	$0.364606\pi$
$282$	− 192672.i	− 0.144277i
$283$	− 1.81258e6i	− 1.34534i	−0.739944	−	0.672669i	$-0.765148\pi$
$283$	− 1.81258e6i	− 1.34534i	0.739944	−	0.672669i	$-0.234852\pi$
$284$	145472.	0.107025
$285$	0	0
$286$	−844608.	−0.610577
$287$	− 516558.i	− 0.370181i
$288$	82944.0i	0.0589256i
$289$	−1.08287e6	−0.762659
$290$	0	0
$291$	192330.	0.133142
$292$	− 1.17526e6i	− 0.806637i
$293$	− 2.10031e6i	− 1.42927i	−0.699499	−	0.714634i	$-0.746593\pi$
$293$	− 2.10031e6i	− 1.42927i	0.699499	−	0.714634i	$-0.253407\pi$
$294$	86436.0	0.0583212
$295$	0	0
$296$	−278912.	−0.185028
$297$	484056.i	0.318423i
$298$	− 699432.i	− 0.456252i
$299$	−703416.	−0.455024
$300$	0	0
$301$	414148.	0.263475
$302$	1.81021e6i	1.14212i
$303$	331326.i	0.207324i
$304$	−60416.0	−0.0374945
$305$	0	0
$306$	−512568.	−0.312931
$307$	− 1.64104e6i	− 0.993743i	−0.867824	−	0.496872i	$-0.834482\pi$
$307$	− 1.64104e6i	− 0.993743i	0.867824	−	0.496872i	$-0.165518\pi$
$308$	520576.i	0.312685i
$309$	−940752.	−0.560504
$310$	0	0
$311$	−945232.	−0.554163	−0.277081	−	0.960846i	$-0.589367\pi$
$311$	−945232.	−0.554163	−0.277081	+	0.960846i	$0.589367\pi$
$312$	− 183168.i	− 0.106528i
$313$	− 415354.i	− 0.239639i	−0.992796	−	0.119820i	$-0.961768\pi$
$313$	− 415354.i	− 0.239639i	0.992796	−	0.119820i	$-0.0382316\pi$
$314$	−1.99626e6	−1.14260
$315$	0	0
$316$	1.43040e6	0.805823
$317$	1.18481e6i	0.662220i	0.943592	+	0.331110i	$0.107423\pi$
$317$	1.18481e6i	0.662220i	−0.943592	+	0.331110i	$0.892577\pi$
$318$	− 1.20074e6i	− 0.665860i
$319$	3.28946e6	1.80987
$320$	0	0
$321$	1.92996e6	1.04541
$322$	433552.i	0.233024i
$323$	− 373352.i	− 0.199119i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.90235e6	0.991395
$327$	259182.i	0.134040i
$328$	674688.i	0.346273i
$329$	262248.	0.133574
$330$	0	0
$331$	1.37155e6	0.688083	0.344042	−	0.938954i	$-0.388204\pi$
$331$	1.37155e6	0.688083	0.344042	+	0.938954i	$0.388204\pi$
$332$	− 102848.i	− 0.0512095i
$333$	− 352998.i	− 0.174446i
$334$	480896.	0.235877
$335$	0	0
$336$	−112896.	−0.0545545
$337$	963522.i	0.462154i	0.972935	+	0.231077i	$0.0742250\pi$
$337$	963522.i	0.462154i	−0.972935	+	0.231077i	$0.925775\pi$
$338$	− 1.08068e6i	− 0.514522i
$339$	504126.	0.238254
$340$	0	0
$341$	−4.73299e6	−2.20419
$342$	− 76464.0i	− 0.0353502i
$343$	117649.i	0.0539949i
$344$	−540928.	−0.246458
$345$	0	0
$346$	−2.03550e6	−0.914071
$347$	2.57731e6i	1.14906i	0.818483	+	0.574531i	$0.194815\pi$
$347$	2.57731e6i	1.14906i	−0.818483	+	0.574531i	$0.805185\pi$
$348$	713376.i	0.315770i
$349$	3.06751e6	1.34810	0.674051	−	0.738684i	$-0.264553\pi$
$349$	3.06751e6	1.34810	0.674051	+	0.738684i	$0.264553\pi$
$350$	0	0
$351$	231822.	0.100435
$352$	− 679936.i	− 0.292490i
$353$	3.10144e6i	1.32473i	0.749182	+	0.662364i	$0.230447\pi$
$353$	3.10144e6i	1.32473i	−0.749182	+	0.662364i	$0.769553\pi$
$354$	−555696.	−0.235683
$355$	0	0
$356$	−1.96253e6	−0.820712
$357$	− 697662.i	− 0.289717i
$358$	1.95024e6i	0.804230i
$359$	327508.	0.134118	0.0670588	−	0.997749i	$-0.478638\pi$
$359$	327508.	0.134118	0.0670588	+	0.997749i	$0.478638\pi$
$360$	0	0
$361$	−2.42040e6	−0.977507
$362$	2.17764e6i	0.873403i
$363$	− 2.51860e6i	− 1.00321i
$364$	249312.	0.0986256
$365$	0	0
$366$	1.32343e6	0.516415
$367$	− 2.86739e6i	− 1.11128i	−0.831424	−	0.555638i	$-0.812474\pi$
$367$	− 2.86739e6i	− 1.11128i	0.831424	−	0.555638i	$-0.187526\pi$
$368$	− 566272.i	− 0.217974i
$369$	−853902.	−0.326469
$370$	0	0
$371$	1.63435e6	0.616466
$372$	− 1.02643e6i	− 0.384568i
$373$	− 3.58029e6i	− 1.33244i	−0.745757	−	0.666218i	$-0.767912\pi$
$373$	− 3.58029e6i	− 1.33244i	0.745757	−	0.666218i	$-0.232088\pi$
$374$	4.20179e6	1.55330
$375$	0	0
$376$	−342528.	−0.124947
$377$	− 1.57537e6i	− 0.570860i
$378$	− 142884.i	− 0.0514344i
$379$	−1.64235e6	−0.587310	−0.293655	−	0.955912i	$-0.594872\pi$
$379$	−1.64235e6	−0.587310	−0.293655	+	0.955912i	$0.594872\pi$
$380$	0	0
$381$	1.66860e6	0.588898
$382$	− 1.50562e6i	− 0.527905i
$383$	2.05698e6i	0.716527i	0.933621	+	0.358263i	$0.116631\pi$
$383$	2.05698e6i	0.716527i	−0.933621	+	0.358263i	$0.883369\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−3.37978e6	−1.15457
$387$	− 684612.i	− 0.232363i
$388$	− 341920.i	− 0.115304i
$389$	−616142.	−0.206446	−0.103223	−	0.994658i	$-0.532916\pi$
$389$	−616142.	−0.206446	−0.103223	+	0.994658i	$0.532916\pi$
$390$	0	0
$391$	3.49938e6	1.15758
$392$	− 153664.i	− 0.0505076i
$393$	− 580788.i	− 0.189686i
$394$	−1.97118e6	−0.639713
$395$	0	0
$396$	860544.	0.275762
$397$	2.19212e6i	0.698052i	0.937113	+	0.349026i	$0.113487\pi$
$397$	2.19212e6i	0.698052i	−0.937113	+	0.349026i	$0.886513\pi$
$398$	− 3.65910e6i	− 1.15789i
$399$	104076.	0.0327279
$400$	0	0
$401$	3.28454e6	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$401$	3.28454e6	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$402$	− 1.47499e6i	− 0.455223i
$403$	2.26670e6i	0.695236i
$404$	589024.	0.179548
$405$	0	0
$406$	−970984.	−0.292346
$407$	2.89371e6i	0.865903i
$408$	911232.i	0.271006i
$409$	3.61219e6	1.06773	0.533866	−	0.845569i	$-0.320739\pi$
$409$	3.61219e6	1.06773	0.533866	+	0.845569i	$0.320739\pi$
$410$	0	0
$411$	1.37637e6	0.401912
$412$	1.67245e6i	0.485411i
$413$	− 756364.i	− 0.218200i
$414$	716688.	0.205508
$415$	0	0
$416$	−325632.	−0.0922558
$417$	− 3.09114e6i	− 0.870520i
$418$	626816.i	0.175469i
$419$	−5.41489e6	−1.50680	−0.753398	−	0.657564i	$-0.771587\pi$
$419$	−5.41489e6	−1.50680	−0.753398	+	0.657564i	$0.771587\pi$
$420$	0	0
$421$	3.60629e6	0.991644	0.495822	−	0.868424i	$-0.334867\pi$
$421$	3.60629e6	0.991644	0.495822	+	0.868424i	$0.334867\pi$
$422$	− 1.24712e6i	− 0.340900i
$423$	− 433512.i	− 0.117801i
$424$	−2.13466e6	−0.576651
$425$	0	0
$426$	327312.	0.0873852
$427$	1.80134e6i	0.478107i
$428$	− 3.43104e6i	− 0.905350i
$429$	−1.90037e6	−0.498534
$430$	0	0
$431$	−2.78214e6	−0.721416	−0.360708	−	0.932679i	$-0.617465\pi$
$431$	−2.78214e6	−0.721416	−0.360708	+	0.932679i	$0.617465\pi$
$432$	186624.i	0.0481125i
$433$	− 6.27619e6i	− 1.60871i	−0.594152	−	0.804353i	$-0.702512\pi$
$433$	− 6.27619e6i	− 1.60871i	0.594152	−	0.804353i	$-0.297488\pi$
$434$	1.39709e6	0.356041
$435$	0	0
$436$	460768.	0.116082
$437$	522032.i	0.130766i
$438$	− 2.64434e6i	− 0.658617i
$439$	−641592.	−0.158890	−0.0794452	−	0.996839i	$-0.525315\pi$
$439$	−641592.	−0.158890	−0.0794452	+	0.996839i	$0.525315\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	− 2.01230e6i	− 0.489934i
$443$	− 6.05546e6i	− 1.46601i	−0.680222	−	0.733006i	$-0.738117\pi$
$443$	− 6.05546e6i	− 1.46601i	0.680222	−	0.733006i	$-0.261883\pi$
$444$	−627552.	−0.151075
$445$	0	0
$446$	5.15104e6	1.22619
$447$	− 1.57372e6i	− 0.372528i
$448$	200704.i	0.0472456i
$449$	5.16681e6	1.20950	0.604752	−	0.796414i	$-0.293272\pi$
$449$	5.16681e6	1.20950	0.604752	+	0.796414i	$0.293272\pi$
$450$	0	0
$451$	6.99989e6	1.62050
$452$	− 896224.i	− 0.206334i
$453$	4.07297e6i	0.932536i
$454$	5.15621e6	1.17406
$455$	0	0
$456$	−135936.	−0.0306142
$457$	− 227798.i	− 0.0510222i	−0.999675	−	0.0255111i	$-0.991879\pi$
$457$	− 227798.i	− 0.0510222i	0.999675	−	0.0255111i	$-0.00812132\pi$
$458$	2.71286e6i	0.604315i
$459$	−1.15328e6	−0.255507
$460$	0	0
$461$	585146.	0.128237	0.0641183	−	0.997942i	$-0.479577\pi$
$461$	585146.	0.128237	0.0641183	+	0.997942i	$0.479577\pi$
$462$	1.17130e6i	0.255306i
$463$	3.41454e6i	0.740251i	0.928982	+	0.370126i	$0.120685\pi$
$463$	3.41454e6i	0.740251i	−0.928982	+	0.370126i	$0.879315\pi$
$464$	1.26822e6	0.273465
$465$	0	0
$466$	4.46924e6	0.953386
$467$	716300.i	0.151986i	0.997108	+	0.0759929i	$0.0242126\pi$
$467$	716300.i	0.151986i	−0.997108	+	0.0759929i	$0.975787\pi$
$468$	− 412128.i	− 0.0869796i
$469$	2.00763e6	0.421455
$470$	0	0
$471$	−4.49159e6	−0.932928
$472$	987904.i	0.204108i
$473$	5.61213e6i	1.15339i
$474$	3.21840e6	0.657952
$475$	0	0
$476$	−1.24029e6	−0.250903
$477$	− 2.70167e6i	− 0.543672i
$478$	− 5.04782e6i	− 1.01050i
$479$	−5.24092e6	−1.04368	−0.521842	−	0.853042i	$-0.674755\pi$
$479$	−5.24092e6	−1.04368	−0.521842	+	0.853042i	$0.674755\pi$
$480$	0	0
$481$	1.38584e6	0.273119
$482$	− 3.79287e6i	− 0.743619i
$483$	975492.i	0.190264i
$484$	−4.47752e6	−0.868809
$485$	0	0
$486$	−236196.	−0.0453609
$487$	1.11702e6i	0.213421i	0.994290	+	0.106710i	$0.0340318\pi$
$487$	1.11702e6i	0.213421i	−0.994290	+	0.106710i	$0.965968\pi$
$488$	− 2.35277e6i	− 0.447229i
$489$	4.28029e6	0.809471
$490$	0	0
$491$	1.34458e6	0.251699	0.125850	−	0.992049i	$-0.459834\pi$
$491$	1.34458e6	0.251699	0.125850	+	0.992049i	$0.459834\pi$
$492$	1.51805e6i	0.282731i
$493$	7.83723e6i	1.45226i
$494$	300192.	0.0553454
$495$	0	0
$496$	−1.82477e6	−0.333045
$497$	445508.i	0.0809030i
$498$	− 231408.i	− 0.0418124i
$499$	6.54648e6	1.17695	0.588473	−	0.808517i	$-0.299729\pi$
$499$	6.54648e6	1.17695	0.588473	+	0.808517i	$0.299729\pi$
$500$	0	0
$501$	1.08202e6	0.192592
$502$	1.94558e6i	0.344580i
$503$	8.22050e6i	1.44870i	0.689432	+	0.724350i	$0.257860\pi$
$503$	8.22050e6i	1.44870i	−0.689432	+	0.724350i	$0.742140\pi$
$504$	−254016.	−0.0445435
$505$	0	0
$506$	−5.87507e6	−1.02009
$507$	− 2.43152e6i	− 0.420105i
$508$	− 2.96640e6i	− 0.510000i
$509$	5.11045e6	0.874308	0.437154	−	0.899387i	$-0.355987\pi$
$509$	5.11045e6	0.874308	0.437154	+	0.899387i	$0.355987\pi$
$510$	0	0
$511$	3.59925e6	0.609760
$512$	− 262144.i	− 0.0441942i
$513$	− 172044.i	− 0.0288633i
$514$	−4.15639e6	−0.693919
$515$	0	0
$516$	−1.21709e6	−0.201232
$517$	3.55373e6i	0.584733i
$518$	− 854168.i	− 0.139868i
$519$	−4.57987e6	−0.746336
$520$	0	0
$521$	9.69999e6	1.56559	0.782793	−	0.622282i	$-0.213794\pi$
$521$	9.69999e6	1.56559	0.782793	+	0.622282i	$0.213794\pi$
$522$	1.60510e6i	0.257825i
$523$	3.17295e6i	0.507234i	0.967305	+	0.253617i	$0.0816204\pi$
$523$	3.17295e6i	0.507234i	−0.967305	+	0.253617i	$0.918380\pi$
$524$	−1.03251e6	−0.164273
$525$	0	0
$526$	−5.40418e6	−0.851658
$527$	− 1.12765e7i	− 1.76867i
$528$	− 1.52986e6i	− 0.238817i
$529$	1.54340e6	0.239794
$530$	0	0
$531$	−1.25032e6	−0.192435
$532$	− 185024.i	− 0.0283432i
$533$	− 3.35236e6i	− 0.511131i
$534$	−4.41569e6	−0.670109
$535$	0	0
$536$	−2.62221e6	−0.394235
$537$	4.38804e6i	0.656651i
$538$	− 4.47244e6i	− 0.666176i
$539$	−1.59426e6	−0.236368
$540$	0	0
$541$	−6.62575e6	−0.973289	−0.486644	−	0.873600i	$-0.661779\pi$
$541$	−6.62575e6	−0.973289	−0.486644	+	0.873600i	$0.661779\pi$
$542$	760416.i	0.111187i
$543$	4.89969e6i	0.713131i
$544$	1.61997e6	0.234698
$545$	0	0
$546$	560952.	0.0805275
$547$	3.84707e6i	0.549745i	0.961481	+	0.274873i	$0.0886357\pi$
$547$	3.84707e6i	0.549745i	−0.961481	+	0.274873i	$0.911364\pi$
$548$	− 2.44688e6i	− 0.348066i
$549$	2.97772e6	0.421651
$550$	0	0
$551$	−1.16914e6	−0.164055
$552$	− 1.27411e6i	− 0.177975i
$553$	4.38060e6i	0.609145i
$554$	−802024.	−0.111023
$555$	0	0
$556$	−5.49536e6	−0.753892
$557$	5.00176e6i	0.683101i	0.939863	+	0.341550i	$0.110952\pi$
$557$	5.00176e6i	0.683101i	−0.939863	+	0.341550i	$0.889048\pi$
$558$	− 2.30947e6i	− 0.313998i
$559$	2.68774e6	0.363795
$560$	0	0
$561$	9.45403e6	1.26826
$562$	− 4.36948e6i	− 0.583565i
$563$	− 2.27772e6i	− 0.302852i	−0.988469	−	0.151426i	$-0.951614\pi$
$563$	− 2.27772e6i	− 0.302852i	0.988469	−	0.151426i	$-0.0483865\pi$
$564$	−770688.	−0.102019
$565$	0	0
$566$	−7.25032e6	−0.951297
$567$	− 321489.i	− 0.0419961i
$568$	− 581888.i	− 0.0756778i
$569$	−8.86979e6	−1.14850	−0.574252	−	0.818678i	$-0.694707\pi$
$569$	−8.86979e6	−1.14850	−0.574252	+	0.818678i	$0.694707\pi$
$570$	0	0
$571$	1.40102e7	1.79826	0.899132	−	0.437678i	$-0.144199\pi$
$571$	1.40102e7	1.79826	0.899132	+	0.437678i	$0.144199\pi$
$572$	3.37843e6i	0.431743i
$573$	− 3.38764e6i	− 0.431033i
$574$	−2.06623e6	−0.261758
$575$	0	0
$576$	331776.	0.0416667
$577$	8.75327e6i	1.09454i	0.836957	+	0.547269i	$0.184332\pi$
$577$	8.75327e6i	1.09454i	−0.836957	+	0.547269i	$0.815668\pi$
$578$	4.33147e6i	0.539281i
$579$	−7.60451e6	−0.942703
$580$	0	0
$581$	314972.	0.0387108
$582$	− 769320.i	− 0.0941455i
$583$	2.21471e7i	2.69864i
$584$	−4.70106e6	−0.570379
$585$	0	0
$586$	−8.40122e6	−1.01064
$587$	− 1.06117e7i	− 1.27113i	−0.772048	−	0.635564i	$-0.780768\pi$
$587$	− 1.06117e7i	− 1.27113i	0.772048	−	0.635564i	$-0.219232\pi$
$588$	− 345744.i	− 0.0412393i
$589$	1.68221e6	0.199798
$590$	0	0
$591$	−4.43515e6	−0.522323
$592$	1.11565e6i	0.130835i
$593$	− 1.88552e6i	− 0.220188i	−0.993921	−	0.110094i	$-0.964885\pi$
$593$	− 1.88552e6i	− 0.220188i	0.993921	−	0.110094i	$-0.0351152\pi$
$594$	1.93622e6	0.225159
$595$	0	0
$596$	−2.79773e6	−0.322619
$597$	− 8.23298e6i	− 0.945413i
$598$	2.81366e6i	0.321751i
$599$	−1.27256e7	−1.44915	−0.724573	−	0.689198i	$-0.757963\pi$
$599$	−1.27256e7	−1.44915	−0.724573	+	0.689198i	$0.757963\pi$
$600$	0	0
$601$	7.18846e6	0.811801	0.405900	−	0.913917i	$-0.366958\pi$
$601$	7.18846e6	0.811801	0.405900	+	0.913917i	$0.366958\pi$
$602$	− 1.65659e6i	− 0.186305i
$603$	− 3.31873e6i	− 0.371688i
$604$	7.24083e6	0.807600
$605$	0	0
$606$	1.32530e6	0.146600
$607$	1.08494e7i	1.19519i	0.801800	+	0.597593i	$0.203876\pi$
$607$	1.08494e7i	1.19519i	−0.801800	+	0.597593i	$0.796124\pi$
$608$	241664.i	0.0265126i
$609$	−2.18471e6	−0.238699
$610$	0	0
$611$	1.70194e6	0.184434
$612$	2.05027e6i	0.221275i
$613$	4.90511e6i	0.527227i	0.964628	+	0.263614i	$0.0849144\pi$
$613$	4.90511e6i	0.527227i	−0.964628	+	0.263614i	$0.915086\pi$
$614$	−6.56418e6	−0.702683
$615$	0	0
$616$	2.08230e6	0.221102
$617$	2.58445e6i	0.273310i	0.990619	+	0.136655i	$0.0436351\pi$
$617$	2.58445e6i	0.273310i	−0.990619	+	0.136655i	$0.956365\pi$
$618$	3.76301e6i	0.396336i
$619$	4.99336e6	0.523801	0.261901	−	0.965095i	$-0.415651\pi$
$619$	4.99336e6	0.523801	0.261901	+	0.965095i	$0.415651\pi$
$620$	0	0
$621$	1.61255e6	0.167797
$622$	3.78093e6i	0.391852i
$623$	− 6.01024e6i	− 0.620400i
$624$	−732672.	−0.0753266
$625$	0	0
$626$	−1.66142e6	−0.169450
$627$	1.41034e6i	0.143269i
$628$	7.98506e6i	0.807940i
$629$	−6.89436e6	−0.694812
$630$	0	0
$631$	−1.18219e7	−1.18199	−0.590997	−	0.806674i	$-0.701265\pi$
$631$	−1.18219e7	−1.18199	−0.590997	+	0.806674i	$0.701265\pi$
$632$	− 5.72160e6i	− 0.569803i
$633$	− 2.80602e6i	− 0.278344i
$634$	4.73926e6	0.468260
$635$	0	0
$636$	−4.80298e6	−0.470834
$637$	763518.i	0.0745540i
$638$	− 1.31578e7i	− 1.27977i
$639$	736452.	0.0713497
$640$	0	0
$641$	−5.47007e6	−0.525833	−0.262916	−	0.964819i	$-0.584684\pi$
$641$	−5.47007e6	−0.525833	−0.262916	+	0.964819i	$0.584684\pi$
$642$	− 7.71984e6i	− 0.739215i
$643$	− 9.64934e6i	− 0.920386i	−0.887819	−	0.460193i	$-0.847780\pi$
$643$	− 9.64934e6i	− 0.920386i	0.887819	−	0.460193i	$-0.152220\pi$
$644$	1.73421e6	0.164773
$645$	0	0
$646$	−1.49341e6	−0.140798
$647$	292368.i	0.0274580i	0.999906	+	0.0137290i	$0.00437022\pi$
$647$	292368.i	0.0274580i	−0.999906	+	0.0137290i	$0.995630\pi$
$648$	419904.i	0.0392837i
$649$	1.02495e7	0.955193
$650$	0	0
$651$	3.14345e6	0.290706
$652$	− 7.60941e6i	− 0.701022i
$653$	− 6.94081e6i	− 0.636982i	−0.947926	−	0.318491i	$-0.896824\pi$
$653$	− 6.94081e6i	− 0.636982i	0.947926	−	0.318491i	$-0.103176\pi$
$654$	1.03673e6	0.0947808
$655$	0	0
$656$	2.69875e6	0.244852
$657$	− 5.94977e6i	− 0.537758i
$658$	− 1.04899e6i	− 0.0944512i
$659$	1.32912e7	1.19221	0.596104	−	0.802908i	$-0.296715\pi$
$659$	1.32912e7	1.19221	0.596104	+	0.802908i	$0.296715\pi$
$660$	0	0
$661$	2.05219e6	0.182690	0.0913448	−	0.995819i	$-0.470883\pi$
$661$	2.05219e6	0.182690	0.0913448	+	0.995819i	$0.470883\pi$
$662$	− 5.48619e6i	− 0.486548i
$663$	− 4.52768e6i	− 0.400030i
$664$	−411392.	−0.0362106
$665$	0	0
$666$	−1.41199e6	−0.123352
$667$	− 1.09582e7i	− 0.953732i
$668$	− 1.92358e6i	− 0.166790i
$669$	1.15898e7	1.00118
$670$	0	0
$671$	−2.44100e7	−2.09296
$672$	451584.i	0.0385758i
$673$	1.57039e7i	1.33650i	0.743935	+	0.668252i	$0.232957\pi$
$673$	1.57039e7i	1.33650i	−0.743935	+	0.668252i	$0.767043\pi$
$674$	3.85409e6	0.326792
$675$	0	0
$676$	−4.32270e6	−0.363822
$677$	− 969534.i	− 0.0813002i	−0.999173	−	0.0406501i	$-0.987057\pi$
$677$	− 969534.i	− 0.0813002i	0.999173	−	0.0406501i	$-0.0129429\pi$
$678$	− 2.01650e6i	− 0.168471i
$679$	1.04713e6	0.0871618
$680$	0	0
$681$	1.16015e7	0.958617
$682$	1.89320e7i	1.55860i
$683$	1.49908e7i	1.22962i	0.788673	+	0.614812i	$0.210768\pi$
$683$	1.49908e7i	1.22962i	−0.788673	+	0.614812i	$0.789232\pi$
$684$	−305856.	−0.0249964
$685$	0	0
$686$	470596.	0.0381802
$687$	6.10393e6i	0.493421i
$688$	2.16371e6i	0.174272i
$689$	1.06066e7	0.851191
$690$	0	0
$691$	−7.16038e6	−0.570481	−0.285240	−	0.958456i	$-0.592073\pi$
$691$	−7.16038e6	−0.570481	−0.285240	+	0.958456i	$0.592073\pi$
$692$	8.14198e6i	0.646346i
$693$	2.63542e6i	0.208457i
$694$	1.03092e7	0.812509
$695$	0	0
$696$	2.85350e6	0.223283
$697$	1.66774e7i	1.30031i
$698$	− 1.22701e7i	− 0.953253i
$699$	1.00558e7	0.778437
$700$	0	0
$701$	−91834.0	−0.00705844	−0.00352922	−	0.999994i	$-0.501123\pi$
$701$	−91834.0	−0.00705844	−0.00352922	+	0.999994i	$0.501123\pi$
$702$	− 927288.i	− 0.0710186i
$703$	− 1.02849e6i	− 0.0784894i
$704$	−2.71974e6	−0.206822
$705$	0	0
$706$	1.24058e7	0.936725
$707$	1.80389e6i	0.135725i
$708$	2.22278e6i	0.166653i
$709$	−2.20981e7	−1.65097	−0.825487	−	0.564422i	$-0.809099\pi$
$709$	−2.20981e7	−1.65097	−0.825487	+	0.564422i	$0.809099\pi$
$710$	0	0
$711$	7.24140e6	0.537216
$712$	7.85011e6i	0.580331i
$713$	1.57671e7i	1.16153i
$714$	−2.79065e6	−0.204861
$715$	0	0
$716$	7.80096e6	0.568677
$717$	− 1.13576e7i	− 0.825066i
$718$	− 1.31003e6i	− 0.0948355i
$719$	−1.58388e7	−1.14262	−0.571308	−	0.820736i	$-0.693564\pi$
$719$	−1.58388e7	−1.14262	−0.571308	+	0.820736i	$0.693564\pi$
$720$	0	0
$721$	−5.12187e6	−0.366936
$722$	9.68161e6i	0.691202i
$723$	− 8.53396e6i	− 0.607163i
$724$	8.71056e6	0.617589
$725$	0	0
$726$	−1.00744e7	−0.709379
$727$	6.31418e6i	0.443078i	0.975151	+	0.221539i	$0.0711081\pi$
$727$	6.31418e6i	0.443078i	−0.975151	+	0.221539i	$0.928892\pi$
$728$	− 997248.i	− 0.0697388i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−1.33711e7	−0.925492
$732$	− 5.29373e6i	− 0.365161i
$733$	− 6.93003e6i	− 0.476404i	−0.971216	−	0.238202i	$-0.923442\pi$
$733$	− 6.93003e6i	− 0.476404i	0.971216	−	0.238202i	$-0.0765580\pi$
$734$	−1.14696e7	−0.785791
$735$	0	0
$736$	−2.26509e6	−0.154131
$737$	2.72054e7i	1.84496i
$738$	3.41561e6i	0.230849i
$739$	−1.42331e7	−0.958714	−0.479357	−	0.877620i	$-0.659130\pi$
$739$	−1.42331e7	−0.958714	−0.479357	+	0.877620i	$0.659130\pi$
$740$	0	0
$741$	675432.	0.0451894
$742$	− 6.53738e6i	− 0.435907i
$743$	5.94460e6i	0.395048i	0.980298	+	0.197524i	$0.0632901\pi$
$743$	5.94460e6i	0.395048i	−0.980298	+	0.197524i	$0.936710\pi$
$744$	−4.10573e6	−0.271930
$745$	0	0
$746$	−1.43212e7	−0.942175
$747$	− 520668.i	− 0.0341397i
$748$	− 1.68072e7i	− 1.09835i
$749$	1.05076e7	0.684380
$750$	0	0
$751$	−682752.	−0.0441736	−0.0220868	−	0.999756i	$-0.507031\pi$
$751$	−682752.	−0.0441736	−0.0220868	+	0.999756i	$0.507031\pi$
$752$	1.37011e6i	0.0883510i
$753$	4.37756e6i	0.281349i
$754$	−6.30149e6	−0.403659
$755$	0	0
$756$	−571536.	−0.0363696
$757$	1.46333e7i	0.928116i	0.885805	+	0.464058i	$0.153607\pi$
$757$	1.46333e7i	0.928116i	−0.885805	+	0.464058i	$0.846393\pi$
$758$	6.56939e6i	0.415291i
$759$	−1.32189e7	−0.832897
$760$	0	0
$761$	−1.16367e7	−0.728399	−0.364200	−	0.931321i	$-0.618657\pi$
$761$	−1.16367e7	−0.728399	−0.364200	+	0.931321i	$0.618657\pi$
$762$	− 6.67440e6i	− 0.416414i
$763$	1.41110e6i	0.0877500i
$764$	−6.02246e6	−0.373285
$765$	0	0
$766$	8.22790e6	0.506661
$767$	− 4.90865e6i	− 0.301282i
$768$	− 589824.i	− 0.0360844i
$769$	−1.91472e7	−1.16759	−0.583793	−	0.811902i	$-0.698432\pi$
$769$	−1.91472e7	−1.16759	−0.583793	+	0.811902i	$0.698432\pi$
$770$	0	0
$771$	−9.35188e6	−0.566582
$772$	1.35191e7i	0.816405i
$773$	5.39261e6i	0.324601i	0.986741	+	0.162301i	$0.0518914\pi$
$773$	5.39261e6i	0.324601i	−0.986741	+	0.162301i	$0.948109\pi$
$774$	−2.73845e6	−0.164305
$775$	0	0
$776$	−1.36768e6	−0.0815324
$777$	− 1.92188e6i	− 0.114202i
$778$	2.46457e6i	0.145979i
$779$	−2.48791e6	−0.146890
$780$	0	0
$781$	−6.03709e6	−0.354160
$782$	− 1.39975e7i	− 0.818530i
$783$	3.61147e6i	0.210513i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	−2.32315e6	−0.134129
$787$	3.04348e6i	0.175159i	0.996158	+	0.0875796i	$0.0279132\pi$
$787$	3.04348e6i	0.175159i	−0.996158	+	0.0875796i	$0.972087\pi$
$788$	7.88470e6i	0.452345i
$789$	−1.21594e7	−0.695376
$790$	0	0
$791$	2.74469e6	0.155974
$792$	− 3.44218e6i	− 0.194993i
$793$	1.16903e7i	0.660151i
$794$	8.76847e6	0.493597
$795$	0	0
$796$	−1.46364e7	−0.818751
$797$	2.29652e7i	1.28063i	0.768111	+	0.640316i	$0.221197\pi$
$797$	2.29652e7i	1.28063i	−0.768111	+	0.640316i	$0.778803\pi$
$798$	− 416304.i	− 0.0231421i
$799$	−8.46686e6	−0.469197
$800$	0	0
$801$	−9.93530e6	−0.547141
$802$	− 1.31382e7i	− 0.721271i
$803$	4.87735e7i	2.66928i
$804$	−5.89997e6	−0.321892
$805$	0	0
$806$	9.06682e6	0.491606
$807$	− 1.00630e7i	− 0.543930i
$808$	− 2.35610e6i	− 0.126959i
$809$	−1.90787e7	−1.02489	−0.512445	−	0.858720i	$-0.671260\pi$
$809$	−1.90787e7	−1.02489	−0.512445	+	0.858720i	$0.671260\pi$
$810$	0	0
$811$	1.09414e7	0.584147	0.292074	−	0.956396i	$-0.405655\pi$
$811$	1.09414e7	0.584147	0.292074	+	0.956396i	$0.405655\pi$
$812$	3.88394e6i	0.206720i
$813$	1.71094e6i	0.0907836i
$814$	1.15748e7	0.612286
$815$	0	0
$816$	3.64493e6	0.191630
$817$	− 1.99467e6i	− 0.104548i
$818$	− 1.44488e7i	− 0.755001i
$819$	1.26214e6	0.0657504
$820$	0	0
$821$	2.12594e7	1.10076	0.550380	−	0.834914i	$-0.314483\pi$
$821$	2.12594e7	1.10076	0.550380	+	0.834914i	$0.314483\pi$
$822$	− 5.50548e6i	− 0.284194i
$823$	1.42256e7i	0.732103i	0.930595	+	0.366052i	$0.119291\pi$
$823$	1.42256e7i	0.732103i	−0.930595	+	0.366052i	$0.880709\pi$
$824$	6.68979e6	0.343237
$825$	0	0
$826$	−3.02546e6	−0.154291
$827$	2.76103e6i	0.140381i	0.997534	+	0.0701904i	$0.0223607\pi$
$827$	2.76103e6i	0.140381i	−0.997534	+	0.0701904i	$0.977639\pi$
$828$	− 2.86675e6i	− 0.145316i
$829$	3.82147e7	1.93127	0.965637	−	0.259895i	$-0.0836880\pi$
$829$	3.82147e7	1.93127	0.965637	+	0.259895i	$0.0836880\pi$
$830$	0	0
$831$	−1.80455e6	−0.0906499
$832$	1.30253e6i	0.0652347i
$833$	− 3.79838e6i	− 0.189665i
$834$	−1.23646e7	−0.615550
$835$	0	0
$836$	2.50726e6	0.124075
$837$	− 5.19631e6i	− 0.256378i
$838$	2.16596e7i	1.06547i
$839$	−1.06044e7	−0.520094	−0.260047	−	0.965596i	$-0.583738\pi$
$839$	−1.06044e7	−0.520094	−0.260047	+	0.965596i	$0.583738\pi$
$840$	0	0
$841$	4.03097e6	0.196526
$842$	− 1.44252e7i	− 0.701198i
$843$	− 9.83133e6i	− 0.476479i
$844$	−4.98848e6	−0.241053
$845$	0	0
$846$	−1.73405e6	−0.0832981
$847$	− 1.37124e7i	− 0.656758i
$848$	8.53862e6i	0.407754i
$849$	−1.63132e7	−0.776731
$850$	0	0
$851$	9.63990e6	0.456298
$852$	− 1.30925e6i	− 0.0617907i
$853$	4.07009e7i	1.91527i	0.287977	+	0.957637i	$0.407017\pi$
$853$	4.07009e7i	1.91527i	−0.287977	+	0.957637i	$0.592983\pi$
$854$	7.20535e6	0.338073
$855$	0	0
$856$	−1.37242e7	−0.640179
$857$	− 3.10120e7i	− 1.44237i	−0.692741	−	0.721187i	$-0.743597\pi$
$857$	− 3.10120e7i	− 1.44237i	0.692741	−	0.721187i	$-0.256403\pi$
$858$	7.60147e6i	0.352517i
$859$	−1.09104e7	−0.504495	−0.252247	−	0.967663i	$-0.581170\pi$
$859$	−1.09104e7	−0.504495	−0.252247	+	0.967663i	$0.581170\pi$
$860$	0	0
$861$	−4.64902e6	−0.213724
$862$	1.11286e7i	0.510118i
$863$	− 1.04089e7i	− 0.475751i	−0.971296	−	0.237875i	$-0.923549\pi$
$863$	− 1.04089e7i	− 0.475751i	0.971296	−	0.237875i	$-0.0764510\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−2.51048e7	−1.13753
$867$	9.74580e6i	0.440321i
$868$	− 5.58835e6i	− 0.251759i
$869$	−5.93616e7	−2.66659
$870$	0	0
$871$	1.30291e7	0.581928
$872$	− 1.84307e6i	− 0.0820826i
$873$	− 1.73097e6i	− 0.0768695i
$874$	2.08813e6	0.0924652
$875$	0	0
$876$	−1.05774e7	−0.465712
$877$	1.64064e7i	0.720299i	0.932895	+	0.360150i	$0.117274\pi$
$877$	1.64064e7i	0.720299i	−0.932895	+	0.360150i	$0.882726\pi$
$878$	2.56637e6i	0.112352i
$879$	−1.89028e7	−0.825188
$880$	0	0
$881$	1.48577e7	0.644927	0.322464	−	0.946582i	$-0.395489\pi$
$881$	1.48577e7	0.644927	0.322464	+	0.946582i	$0.395489\pi$
$882$	− 777924.i	− 0.0336718i
$883$	2.72018e7i	1.17407i	0.809560	+	0.587037i	$0.199706\pi$
$883$	2.72018e7i	1.17407i	−0.809560	+	0.587037i	$0.800294\pi$
$884$	−8.04922e6	−0.346436
$885$	0	0
$886$	−2.42218e7	−1.03663
$887$	2.71242e7i	1.15757i	0.815480	+	0.578785i	$0.196473\pi$
$887$	2.71242e7i	1.15757i	−0.815480	+	0.578785i	$0.803527\pi$
$888$	2.51021e6i	0.106826i
$889$	9.08460e6	0.385524
$890$	0	0
$891$	4.35650e6	0.183842
$892$	− 2.06042e7i	− 0.867047i
$893$	− 1.26307e6i	− 0.0530029i
$894$	−6.29489e6	−0.263417
$895$	0	0
$896$	802816.	0.0334077
$897$	6.33074e6i	0.262708i
$898$	− 2.06673e7i	− 0.855248i
$899$	−3.53121e7	−1.45722
$900$	0	0
$901$	−5.27660e7	−2.16542
$902$	− 2.79996e7i	− 1.14587i
$903$	− 3.72733e6i	− 0.152117i
$904$	−3.58490e6	−0.145900
$905$	0	0
$906$	1.62919e7	0.659402
$907$	− 8.42269e6i	− 0.339964i	−0.985447	−	0.169982i	$-0.945629\pi$
$907$	− 8.42269e6i	− 0.339964i	0.985447	−	0.169982i	$-0.0543709\pi$
$908$	− 2.06248e7i	− 0.830187i
$909$	2.98193e6	0.119698
$910$	0	0
$911$	3.08637e7	1.23212	0.616060	−	0.787700i	$-0.288728\pi$
$911$	3.08637e7	1.23212	0.616060	+	0.787700i	$0.288728\pi$
$912$	543744.i	0.0216475i
$913$	4.26819e6i	0.169460i
$914$	−911192.	−0.0360782
$915$	0	0
$916$	1.08514e7	0.427315
$917$	− 3.16207e6i	− 0.124179i
$918$	4.61311e6i	0.180671i
$919$	−4.93895e6	−0.192906	−0.0964531	−	0.995338i	$-0.530750\pi$
$919$	−4.93895e6	−0.192906	−0.0964531	+	0.995338i	$0.530750\pi$
$920$	0	0
$921$	−1.47694e7	−0.573738
$922$	− 2.34058e6i	− 0.0906770i
$923$	2.89126e6i	0.111707i
$924$	4.68518e6	0.180529
$925$	0	0
$926$	1.36581e7	0.523437
$927$	8.46677e6i	0.323607i
$928$	− 5.07290e6i	− 0.193369i
$929$	−5.62575e6	−0.213866	−0.106933	−	0.994266i	$-0.534103\pi$
$929$	−5.62575e6	−0.213866	−0.106933	+	0.994266i	$0.534103\pi$
$930$	0	0
$931$	566636.	0.0214255
$932$	− 1.78770e7i	− 0.674146i
$933$	8.50709e6i	0.319946i
$934$	2.86520e6	0.107470
$935$	0	0
$936$	−1.64851e6	−0.0615039
$937$	2.60073e7i	0.967714i	0.875147	+	0.483857i	$0.160764\pi$
$937$	2.60073e7i	0.967714i	−0.875147	+	0.483857i	$0.839236\pi$
$938$	− 8.03051e6i	− 0.298014i
$939$	−3.73819e6	−0.138356
$940$	0	0
$941$	3.02160e6	0.111241	0.0556203	−	0.998452i	$-0.482286\pi$
$941$	3.02160e6	0.111241	0.0556203	+	0.998452i	$0.482286\pi$
$942$	1.79664e7i	0.659680i
$943$	− 2.33189e7i	− 0.853943i
$944$	3.95162e6	0.144326
$945$	0	0
$946$	2.24485e7	0.815567
$947$	− 3.48282e7i	− 1.26199i	−0.775787	−	0.630995i	$-0.782647\pi$
$947$	− 3.48282e7i	− 1.26199i	0.775787	−	0.630995i	$-0.217353\pi$
$948$	− 1.28736e7i	− 0.465242i
$949$	2.33584e7	0.841932
$950$	0	0
$951$	1.06633e7	0.382333
$952$	4.96115e6i	0.177415i
$953$	9.39009e6i	0.334917i	0.985879	+	0.167459i	$0.0535560\pi$
$953$	9.39009e6i	0.334917i	−0.985879	+	0.167459i	$0.946444\pi$
$954$	−1.08067e7	−0.384434
$955$	0	0
$956$	−2.01913e7	−0.714528
$957$	− 2.96051e7i	− 1.04493i
$958$	2.09637e7i	0.737996i
$959$	7.49357e6	0.263113
$960$	0	0
$961$	2.21792e7	0.774708
$962$	− 5.54338e6i	− 0.193124i
$963$	− 1.73696e7i	− 0.603566i
$964$	−1.51715e7	−0.525818
$965$	0	0
$966$	3.90197e6	0.134537
$967$	1.44768e7i	0.497860i	0.968521	+	0.248930i	$0.0800789\pi$
$967$	1.44768e7i	0.497860i	−0.968521	+	0.248930i	$0.919921\pi$
$968$	1.79101e7i	0.614340i
$969$	−3.36017e6	−0.114961
$970$	0	0
$971$	9.24976e6	0.314834	0.157417	−	0.987532i	$-0.449683\pi$
$971$	9.24976e6	0.314834	0.157417	+	0.987532i	$0.449683\pi$
$972$	944784.i	0.0320750i
$973$	− 1.68295e7i	− 0.569889i
$974$	4.46806e6	0.150911
$975$	0	0
$976$	−9.41107e6	−0.316238
$977$	− 4.97780e7i	− 1.66840i	−0.551459	−	0.834202i	$-0.685929\pi$
$977$	− 4.97780e7i	− 1.66840i	0.551459	−	0.834202i	$-0.314071\pi$
$978$	− 1.71212e7i	− 0.572382i
$979$	8.14449e7	2.71586
$980$	0	0
$981$	2.33264e6	0.0773882
$982$	− 5.37830e6i	− 0.177978i
$983$	8.95601e6i	0.295618i	0.989016	+	0.147809i	$0.0472221\pi$
$983$	8.95601e6i	0.295618i	−0.989016	+	0.147809i	$0.952778\pi$
$984$	6.07219e6	0.199921
$985$	0	0
$986$	3.13489e7	1.02690
$987$	− 2.36023e6i	− 0.0771191i
$988$	− 1.20077e6i	− 0.0391351i
$989$	1.86958e7	0.607790
$990$	0	0
$991$	2.62400e7	0.848751	0.424376	−	0.905486i	$-0.360494\pi$
$991$	2.62400e7	0.848751	0.424376	+	0.905486i	$0.360494\pi$
$992$	7.29907e6i	0.235499i
$993$	− 1.23439e7i	− 0.397265i
$994$	1.78203e6	0.0572070
$995$	0	0
$996$	−925632.	−0.0295658
$997$	2.80506e7i	0.893727i	0.894602	+	0.446863i	$0.147459\pi$
$997$	2.80506e7i	0.893727i	−0.894602	+	0.446863i	$0.852541\pi$
$998$	− 2.61859e7i	− 0.832226i
$999$	−3.17698e6	−0.100717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.6.g.i.799.1		2
5.2	odd	4		1050.6.a.k.1.1		1
5.3	odd	4		42.6.a.d.1.1	✓	1
5.4	even	2	inner	1050.6.g.i.799.2		2
15.8	even	4		126.6.a.i.1.1		1
20.3	even	4		336.6.a.h.1.1		1
35.3	even	12		294.6.e.p.79.1		2
35.13	even	4		294.6.a.b.1.1		1
35.18	odd	12		294.6.e.i.79.1		2
35.23	odd	12		294.6.e.i.67.1		2
35.33	even	12		294.6.e.p.67.1		2
60.23	odd	4		1008.6.a.j.1.1		1
105.83	odd	4		882.6.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.d.1.1	✓	1	5.3	odd	4
126.6.a.i.1.1		1	15.8	even	4
294.6.a.b.1.1		1	35.13	even	4
294.6.e.i.67.1		2	35.23	odd	12
294.6.e.i.79.1		2	35.18	odd	12
294.6.e.p.67.1		2	35.33	even	12
294.6.e.p.79.1		2	35.3	even	12
336.6.a.h.1.1		1	20.3	even	4
882.6.a.s.1.1		1	105.83	odd	4
1008.6.a.j.1.1		1	60.23	odd	4
1050.6.a.k.1.1		1	5.2	odd	4
1050.6.g.i.799.1		2	1.1	even	1	trivial
1050.6.g.i.799.2		2	5.4	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} -49.0000i q^{7} +64.0000i q^{8} -81.0000 q^{9} +664.000 q^{11} +144.000i q^{12} -318.000i q^{13} -196.000 q^{14} +256.000 q^{16} +1582.00i q^{17} +324.000i q^{18} -236.000 q^{19} -441.000 q^{21} -2656.00i q^{22} -2212.00i q^{23} +576.000 q^{24} -1272.00 q^{26} +729.000i q^{27} +784.000i q^{28} +4954.00 q^{29} -7128.00 q^{31} -1024.00i q^{32} -5976.00i q^{33} +6328.00 q^{34} +1296.00 q^{36} +4358.00i q^{37} +944.000i q^{38} -2862.00 q^{39} +10542.0 q^{41} +1764.00i q^{42} +8452.00i q^{43} -10624.0 q^{44} -8848.00 q^{46} +5352.00i q^{47} -2304.00i q^{48} -2401.00 q^{49} +14238.0 q^{51} +5088.00i q^{52} +33354.0i q^{53} +2916.00 q^{54} +3136.00 q^{56} +2124.00i q^{57} -19816.0i q^{58} +15436.0 q^{59} -36762.0 q^{61} +28512.0i q^{62} +3969.00i q^{63} -4096.00 q^{64} -23904.0 q^{66} +40972.0i q^{67} -25312.0i q^{68} -19908.0 q^{69} -9092.00 q^{71} -5184.00i q^{72} +73454.0i q^{73} +17432.0 q^{74} +3776.00 q^{76} -32536.0i q^{77} +11448.0i q^{78} -89400.0 q^{79} +6561.00 q^{81} -42168.0i q^{82} +6428.00i q^{83} +7056.00 q^{84} +33808.0 q^{86} -44586.0i q^{87} +42496.0i q^{88} +122658. q^{89} -15582.0 q^{91} +35392.0i q^{92} +64152.0i q^{93} +21408.0 q^{94} -9216.00 q^{96} +21370.0i q^{97} +9604.00i q^{98} -53784.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} + 1328 q^{11} - 392 q^{14} + 512 q^{16} - 472 q^{19} - 882 q^{21} + 1152 q^{24} - 2544 q^{26} + 9908 q^{29} - 14256 q^{31} + 12656 q^{34} + 2592 q^{36} - 5724 q^{39}+ \cdots - 107568 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 + 1328 * q^11 - 392 * q^14 + 512 * q^16 - 472 * q^19 - 882 * q^21 + 1152 * q^24 - 2544 * q^26 + 9908 * q^29 - 14256 * q^31 + 12656 * q^34 + 2592 * q^36 - 5724 * q^39 + 21084 * q^41 - 21248 * q^44 - 17696 * q^46 - 4802 * q^49 + 28476 * q^51 + 5832 * q^54 + 6272 * q^56 + 30872 * q^59 - 73524 * q^61 - 8192 * q^64 - 47808 * q^66 - 39816 * q^69 - 18184 * q^71 + 34864 * q^74 + 7552 * q^76 - 178800 * q^79 + 13122 * q^81 + 14112 * q^84 + 67616 * q^86 + 245316 * q^89 - 31164 * q^91 + 42816 * q^94 - 18432 * q^96 - 107568 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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