LMFDB - Embedded newform 650.6.b.k.599.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.b (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:	$104.249482878$
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} + 1094x^{6} + 343849x^{4} + 27023076x^{2} + 498182400 $ x^8 + 1094x^6 + 343849x^4 + 27023076*x^2 + 498182400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.8
Root			$20.3395i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.k.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} +25.3395i q^{3} -16.0000 q^{4} -101.358 q^{6} -154.564i q^{7} -64.0000i q^{8} -399.089 q^{9} -63.7546 q^{11} -405.431i q^{12} -169.000i q^{13} +618.256 q^{14} +256.000 q^{16} -374.136i q^{17} -1596.35i q^{18} +42.3586 q^{19} +3916.57 q^{21} -255.018i q^{22} -1131.66i q^{23} +1621.73 q^{24} +676.000 q^{26} -3955.20i q^{27} +2473.03i q^{28} -2487.62 q^{29} +9742.11 q^{31} +1024.00i q^{32} -1615.51i q^{33} +1496.54 q^{34} +6385.42 q^{36} -5341.14i q^{37} +169.435i q^{38} +4282.37 q^{39} -7131.82 q^{41} +15666.3i q^{42} +6754.09i q^{43} +1020.07 q^{44} +4526.62 q^{46} +14373.8i q^{47} +6486.90i q^{48} -7083.06 q^{49} +9480.40 q^{51} +2704.00i q^{52} +13363.6i q^{53} +15820.8 q^{54} -9892.10 q^{56} +1073.35i q^{57} -9950.47i q^{58} -38502.8 q^{59} +28521.0 q^{61} +38968.4i q^{62} +61684.8i q^{63} -4096.00 q^{64} +6462.03 q^{66} +43953.6i q^{67} +5986.17i q^{68} +28675.6 q^{69} +40991.3 q^{71} +25541.7i q^{72} +68431.1i q^{73} +21364.6 q^{74} -677.738 q^{76} +9854.17i q^{77} +17129.5i q^{78} -20233.0 q^{79} +3244.21 q^{81} -28527.3i q^{82} +63147.4i q^{83} -62665.2 q^{84} -27016.4 q^{86} -63034.9i q^{87} +4080.29i q^{88} +116225. q^{89} -26121.3 q^{91} +18106.5i q^{92} +246860. i q^{93} -57495.1 q^{94} -25947.6 q^{96} -59187.1i q^{97} -28332.2i q^{98} +25443.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} - 160 q^{6} - 444 q^{9} - 588 q^{11} + 1424 q^{14} + 2048 q^{16} - 928 q^{19} - 2016 q^{21} + 2560 q^{24} + 5408 q^{26} + 9088 q^{29} + 11236 q^{31} + 14448 q^{34} + 7104 q^{36} + 6760 q^{39}+ \cdots - 89060 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 - 160 * q^6 - 444 * q^9 - 588 * q^11 + 1424 * q^14 + 2048 * q^16 - 928 * q^19 - 2016 * q^21 + 2560 * q^24 + 5408 * q^26 + 9088 * q^29 + 11236 * q^31 + 14448 * q^34 + 7104 * q^36 + 6760 * q^39 + 26464 * q^41 + 9408 * q^44 + 13344 * q^46 - 2932 * q^49 + 3840 * q^51 + 18784 * q^54 - 22784 * q^56 + 89180 * q^59 + 19336 * q^61 - 32768 * q^64 - 25120 * q^66 + 22636 * q^69 + 2320 * q^71 + 89184 * q^74 + 14848 * q^76 + 211480 * q^79 - 110664 * q^81 + 32256 * q^84 - 165184 * q^86 + 145544 * q^89 - 60164 * q^91 + 70224 * q^94 - 40960 * q^96 - 89060 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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
$3$	25.3395i	1.62553i	0.582593	+	0.812764i	$0.302038\pi$
$3$	25.3395i	1.62553i	−0.582593	+	0.812764i	$0.697962\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−101.358	−1.14942
$7$	− 154.564i	− 1.19224i	−0.802896	−	0.596120i	$-0.796708\pi$
$7$	− 154.564i	− 1.19224i	0.802896	−	0.596120i	$-0.203292\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−399.089	−1.64234
$10$	0	0
$11$	−63.7546	−0.158866	−0.0794328	−	0.996840i	$-0.525311\pi$
$11$	−63.7546	−0.158866	−0.0794328	+	0.996840i	$0.525311\pi$
$12$	− 405.431i	− 0.812764i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	618.256	0.843041
$15$	0	0
$16$	256.000	0.250000
$17$	− 374.136i	− 0.313984i	−0.987600	−	0.156992i	$-0.949820\pi$
$17$	− 374.136i	− 0.313984i	0.987600	−	0.156992i	$-0.0501796\pi$
$18$	− 1596.35i	− 1.16131i
$19$	42.3586	0.0269189	0.0134595	−	0.999909i	$-0.495716\pi$
$19$	42.3586	0.0269189	0.0134595	+	0.999909i	$0.495716\pi$
$20$	0	0
$21$	3916.57	1.93802
$22$	− 255.018i	− 0.112335i
$23$	− 1131.66i	− 0.446062i	−0.974811	−	0.223031i	$-0.928405\pi$
$23$	− 1131.66i	− 0.446062i	0.974811	−	0.223031i	$-0.0715950\pi$
$24$	1621.73	0.574711
$25$	0	0
$26$	676.000	0.196116
$27$	− 3955.20i	− 1.04414i
$28$	2473.03i	0.596120i
$29$	−2487.62	−0.549273	−0.274637	−	0.961548i	$-0.588558\pi$
$29$	−2487.62	−0.549273	−0.274637	+	0.961548i	$0.588558\pi$
$30$	0	0
$31$	9742.11	1.82074	0.910371	−	0.413792i	$-0.135796\pi$
$31$	9742.11	1.82074	0.910371	+	0.413792i	$0.135796\pi$
$32$	1024.00i	0.176777i
$33$	− 1615.51i	− 0.258240i
$34$	1496.54	0.222020
$35$	0	0
$36$	6385.42	0.821170
$37$	− 5341.14i	− 0.641401i	−0.947181	−	0.320700i	$-0.896082\pi$
$37$	− 5341.14i	− 0.641401i	0.947181	−	0.320700i	$-0.103918\pi$
$38$	169.435i	0.0190346i
$39$	4282.37	0.450840
$40$	0	0
$41$	−7131.82	−0.662584	−0.331292	−	0.943528i	$-0.607485\pi$
$41$	−7131.82	−0.662584	−0.331292	+	0.943528i	$0.607485\pi$
$42$	15666.3i	1.37039i
$43$	6754.09i	0.557052i	0.960429	+	0.278526i	$0.0898459\pi$
$43$	6754.09i	0.557052i	−0.960429	+	0.278526i	$0.910154\pi$
$44$	1020.07	0.0794328
$45$	0	0
$46$	4526.62	0.315413
$47$	14373.8i	0.949132i	0.880220	+	0.474566i	$0.157395\pi$
$47$	14373.8i	0.949132i	−0.880220	+	0.474566i	$0.842605\pi$
$48$	6486.90i	0.406382i
$49$	−7083.06	−0.421435
$50$	0	0
$51$	9480.40	0.510389
$52$	2704.00i	0.138675i
$53$	13363.6i	0.653483i	0.945114	+	0.326741i	$0.105951\pi$
$53$	13363.6i	0.653483i	−0.945114	+	0.326741i	$0.894049\pi$
$54$	15820.8	0.738320
$55$	0	0
$56$	−9892.10	−0.421520
$57$	1073.35i	0.0437575i
$58$	− 9950.47i	− 0.388395i
$59$	−38502.8	−1.44000	−0.720000	−	0.693974i	$-0.755858\pi$
$59$	−38502.8	−1.44000	−0.720000	+	0.693974i	$0.755858\pi$
$60$	0	0
$61$	28521.0	0.981388	0.490694	−	0.871332i	$-0.336743\pi$
$61$	28521.0	0.981388	0.490694	+	0.871332i	$0.336743\pi$
$62$	38968.4i	1.28746i
$63$	61684.8i	1.95806i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	6462.03	0.182603
$67$	43953.6i	1.19621i	0.801418	+	0.598104i	$0.204079\pi$
$67$	43953.6i	1.19621i	−0.801418	+	0.598104i	$0.795921\pi$
$68$	5986.17i	0.156992i
$69$	28675.6	0.725085
$70$	0	0
$71$	40991.3	0.965040	0.482520	−	0.875885i	$-0.339722\pi$
$71$	40991.3	0.965040	0.482520	+	0.875885i	$0.339722\pi$
$72$	25541.7i	0.580655i
$73$	68431.1i	1.50296i	0.659758	+	0.751478i	$0.270659\pi$
$73$	68431.1i	1.50296i	−0.659758	+	0.751478i	$0.729341\pi$
$74$	21364.6	0.453539
$75$	0	0
$76$	−677.738	−0.0134595
$77$	9854.17i	0.189406i
$78$	17129.5i	0.318792i
$79$	−20233.0	−0.364747	−0.182374	−	0.983229i	$-0.558378\pi$
$79$	−20233.0	−0.364747	−0.182374	+	0.983229i	$0.558378\pi$
$80$	0	0
$81$	3244.21	0.0549409
$82$	− 28527.3i	− 0.468518i
$83$	63147.4i	1.00614i	0.864244	+	0.503072i	$0.167797\pi$
$83$	63147.4i	1.00614i	−0.864244	+	0.503072i	$0.832203\pi$
$84$	−62665.2	−0.969009
$85$	0	0
$86$	−27016.4	−0.393895
$87$	− 63034.9i	− 0.892859i
$88$	4080.29i	0.0561674i
$89$	116225.	1.55534	0.777668	−	0.628675i	$-0.216402\pi$
$89$	116225.	1.55534	0.777668	+	0.628675i	$0.216402\pi$
$90$	0	0
$91$	−26121.3	−0.330668
$92$	18106.5i	0.223031i
$93$	246860.i	2.95967i
$94$	−57495.1	−0.671138
$95$	0	0
$96$	−25947.6	−0.287355
$97$	− 59187.1i	− 0.638701i	−0.947637	−	0.319351i	$-0.896535\pi$
$97$	− 59187.1i	− 0.638701i	0.947637	−	0.319351i	$-0.103465\pi$
$98$	− 28332.2i	− 0.298000i
$99$	25443.7	0.260911
$100$	0	0
$101$	−10761.5	−0.104971	−0.0524856	−	0.998622i	$-0.516714\pi$
$101$	−10761.5	−0.104971	−0.0524856	+	0.998622i	$0.516714\pi$
$102$	37921.6i	0.360899i
$103$	94434.4i	0.877076i	0.898713	+	0.438538i	$0.144504\pi$
$103$	94434.4i	0.877076i	−0.898713	+	0.438538i	$0.855496\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−53454.4	−0.462082
$107$	− 81403.2i	− 0.687357i	−0.939087	−	0.343678i	$-0.888327\pi$
$107$	− 81403.2i	− 0.687357i	0.939087	−	0.343678i	$-0.111673\pi$
$108$	63283.3i	0.522071i
$109$	145882.	1.17607	0.588037	−	0.808834i	$-0.299901\pi$
$109$	145882.	1.17607	0.588037	+	0.808834i	$0.299901\pi$
$110$	0	0
$111$	135342.	1.04261
$112$	− 39568.4i	− 0.298060i
$113$	− 86838.6i	− 0.639760i	−0.947458	−	0.319880i	$-0.896357\pi$
$113$	− 86838.6i	− 0.639760i	0.947458	−	0.319880i	$-0.103643\pi$
$114$	−4293.38	−0.0309412
$115$	0	0
$116$	39801.9	0.274637
$117$	67446.0i	0.455503i
$118$	− 154011.i	− 1.01823i
$119$	−57827.9	−0.374344
$120$	0	0
$121$	−156986.	−0.974762
$122$	114084.i	0.693946i
$123$	− 180717.i	− 1.07705i
$124$	−155874.	−0.910371
$125$	0	0
$126$	−246739.	−1.38456
$127$	184919.i	1.01736i	0.860957	+	0.508678i	$0.169866\pi$
$127$	184919.i	1.01736i	−0.860957	+	0.508678i	$0.830134\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−171145.	−0.905504
$130$	0	0
$131$	−60089.4	−0.305928	−0.152964	−	0.988232i	$-0.548882\pi$
$131$	−60089.4	−0.305928	−0.152964	+	0.988232i	$0.548882\pi$
$132$	25848.1i	0.129120i
$133$	− 6547.12i	− 0.0320938i
$134$	−175814.	−0.845847
$135$	0	0
$136$	−23944.7	−0.111010
$137$	− 179721.i	− 0.818083i	−0.912516	−	0.409042i	$-0.865863\pi$
$137$	− 179721.i	− 0.818083i	0.912516	−	0.409042i	$-0.134137\pi$
$138$	114702.i	0.512713i
$139$	146077.	0.641276	0.320638	−	0.947202i	$-0.396103\pi$
$139$	146077.	0.641276	0.320638	+	0.947202i	$0.396103\pi$
$140$	0	0
$141$	−364224.	−1.54284
$142$	163965.i	0.682386i
$143$	10774.5i	0.0440614i
$144$	−102167.	−0.410585
$145$	0	0
$146$	−273724.	−1.06275
$147$	− 179481.i	− 0.685055i
$148$	85458.2i	0.320700i
$149$	295863.	1.09175	0.545877	−	0.837865i	$-0.316197\pi$
$149$	295863.	1.09175	0.545877	+	0.837865i	$0.316197\pi$
$150$	0	0
$151$	−53761.2	−0.191879	−0.0959394	−	0.995387i	$-0.530585\pi$
$151$	−53761.2	−0.191879	−0.0959394	+	0.995387i	$0.530585\pi$
$152$	− 2710.95i	− 0.00951728i
$153$	149313.i	0.515668i
$154$	−39416.7	−0.133930
$155$	0	0
$156$	−68517.9	−0.225420
$157$	− 199374.i	− 0.645536i	−0.946478	−	0.322768i	$-0.895387\pi$
$157$	− 199374.i	− 0.645536i	0.946478	−	0.322768i	$-0.104613\pi$
$158$	− 80931.9i	− 0.257915i
$159$	−338627.	−1.06225
$160$	0	0
$161$	−174913.	−0.531812
$162$	12976.8i	0.0388491i
$163$	132038.i	0.389253i	0.980877	+	0.194626i	$0.0623494\pi$
$163$	132038.i	0.389253i	−0.980877	+	0.194626i	$0.937651\pi$
$164$	114109.	0.331292
$165$	0	0
$166$	−252590.	−0.711452
$167$	125924.i	0.349396i	0.984622	+	0.174698i	$0.0558949\pi$
$167$	125924.i	0.349396i	−0.984622	+	0.174698i	$0.944105\pi$
$168$	− 250661.i	− 0.685193i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−16904.9	−0.0442101
$172$	− 108065.i	− 0.278526i
$173$	− 553428.i	− 1.40587i	−0.711252	−	0.702937i	$-0.751872\pi$
$173$	− 553428.i	− 1.40587i	0.711252	−	0.702937i	$-0.248128\pi$
$174$	252140.	0.631347
$175$	0	0
$176$	−16321.2	−0.0397164
$177$	− 975641.i	− 2.34076i
$178$	464900.i	1.09979i
$179$	471297.	1.09941	0.549707	−	0.835357i	$-0.314739\pi$
$179$	471297.	1.09941	0.549707	+	0.835357i	$0.314739\pi$
$180$	0	0
$181$	471171.	1.06901	0.534506	−	0.845165i	$-0.320498\pi$
$181$	471171.	1.06901	0.534506	+	0.845165i	$0.320498\pi$
$182$	− 104485.i	− 0.233817i
$183$	722708.i	1.59527i
$184$	−72426.0	−0.157707
$185$	0	0
$186$	−987439.	−2.09280
$187$	23852.9i	0.0498812i
$188$	− 229981.i	− 0.474566i
$189$	−611332.	−1.24487
$190$	0	0
$191$	319864.	0.634427	0.317214	−	0.948354i	$-0.397253\pi$
$191$	319864.	0.634427	0.317214	+	0.948354i	$0.397253\pi$
$192$	− 103790.i	− 0.203191i
$193$	655179.i	1.26610i	0.774112	+	0.633048i	$0.218197\pi$
$193$	655179.i	1.26610i	−0.774112	+	0.633048i	$0.781803\pi$
$194$	236748.	0.451630
$195$	0	0
$196$	113329.	0.210718
$197$	− 680349.i	− 1.24901i	−0.781021	−	0.624505i	$-0.785301\pi$
$197$	− 680349.i	− 1.24901i	0.781021	−	0.624505i	$-0.214699\pi$
$198$	101775.i	0.184492i
$199$	520113.	0.931033	0.465516	−	0.885039i	$-0.345869\pi$
$199$	520113.	0.931033	0.465516	+	0.885039i	$0.345869\pi$
$200$	0	0
$201$	−1.11376e6	−1.94447
$202$	− 43046.1i	− 0.0742259i
$203$	384496.i	0.654866i
$204$	−151686.	−0.255194
$205$	0	0
$206$	−377738.	−0.620186
$207$	451631.i	0.732585i
$208$	− 43264.0i	− 0.0693375i
$209$	−2700.56	−0.00427649
$210$	0	0
$211$	121818.	0.188367	0.0941836	−	0.995555i	$-0.469976\pi$
$211$	121818.	0.188367	0.0941836	+	0.995555i	$0.469976\pi$
$212$	− 213818.i	− 0.326741i
$213$	1.03870e6i	1.56870i
$214$	325613.	0.486035
$215$	0	0
$216$	−253133.	−0.369160
$217$	− 1.50578e6i	− 2.17076i
$218$	583527.i	0.831609i
$219$	−1.73401e6	−2.44310
$220$	0	0
$221$	−63228.9	−0.0870834
$222$	541366.i	0.737240i
$223$	826276.i	1.11266i	0.830961	+	0.556331i	$0.187791\pi$
$223$	826276.i	1.11266i	−0.830961	+	0.556331i	$0.812209\pi$
$224$	158274.	0.210760
$225$	0	0
$226$	347355.	0.452378
$227$	− 256788.i	− 0.330758i	−0.986230	−	0.165379i	$-0.947115\pi$
$227$	− 256788.i	− 0.330758i	0.986230	−	0.165379i	$-0.0528848\pi$
$228$	− 17173.5i	− 0.0218787i
$229$	−1.19410e6	−1.50470	−0.752351	−	0.658763i	$-0.771080\pi$
$229$	−1.19410e6	−1.50470	−0.752351	+	0.658763i	$0.771080\pi$
$230$	0	0
$231$	−249699.	−0.307884
$232$	159208.i	0.194197i
$233$	1.24372e6i	1.50083i	0.660965	+	0.750416i	$0.270147\pi$
$233$	1.24372e6i	1.50083i	−0.660965	+	0.750416i	$0.729853\pi$
$234$	−269784.	−0.322089
$235$	0	0
$236$	616045.	0.720000
$237$	− 512693.i	− 0.592906i
$238$	− 231312.i	− 0.264701i
$239$	−450986.	−0.510703	−0.255351	−	0.966848i	$-0.582191\pi$
$239$	−450986.	−0.510703	−0.255351	+	0.966848i	$0.582191\pi$
$240$	0	0
$241$	626635.	0.694979	0.347490	−	0.937684i	$-0.387034\pi$
$241$	626635.	0.694979	0.347490	+	0.937684i	$0.387034\pi$
$242$	− 627945.i	− 0.689261i
$243$	− 878908.i	− 0.954834i
$244$	−456337.	−0.490694
$245$	0	0
$246$	722867.	0.761588
$247$	− 7158.61i	− 0.00746597i
$248$	− 623495.i	− 0.643730i
$249$	−1.60012e6	−1.63552
$250$	0	0
$251$	−1.70187e6	−1.70507	−0.852533	−	0.522674i	$-0.824934\pi$
$251$	−1.70187e6	−1.70507	−0.852533	+	0.522674i	$0.824934\pi$
$252$	− 986956.i	− 0.979031i
$253$	72148.2i	0.0708638i
$254$	−739677.	−0.719379
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 64954.5i	− 0.0613446i	−0.999529	−	0.0306723i	$-0.990235\pi$
$257$	− 64954.5i	− 0.0613446i	0.999529	−	0.0306723i	$-0.00976483\pi$
$258$	− 684581.i	− 0.640288i
$259$	−825548.	−0.764703
$260$	0	0
$261$	992780.	0.902094
$262$	− 240358.i	− 0.216324i
$263$	− 263092.i	− 0.234541i	−0.993100	−	0.117270i	$-0.962586\pi$
$263$	− 263092.i	− 0.234541i	0.993100	−	0.117270i	$-0.0374144\pi$
$264$	−103392.	−0.0913017
$265$	0	0
$266$	26188.5	0.0226938
$267$	2.94508e6i	2.52824i
$268$	− 703257.i	− 0.598104i
$269$	1.13941e6	0.960060	0.480030	−	0.877252i	$-0.340626\pi$
$269$	1.13941e6	0.960060	0.480030	+	0.877252i	$0.340626\pi$
$270$	0	0
$271$	2.28682e6	1.89151	0.945754	−	0.324884i	$-0.105325\pi$
$271$	2.28682e6	1.89151	0.945754	+	0.324884i	$0.105325\pi$
$272$	− 95778.7i	− 0.0784959i
$273$	− 661901.i	− 0.537510i
$274$	718884.	0.578472
$275$	0	0
$276$	−458809.	−0.362543
$277$	2.43492e6i	1.90671i	0.301851	+	0.953355i	$0.402395\pi$
$277$	2.43492e6i	1.90671i	−0.301851	+	0.953355i	$0.597605\pi$
$278$	584308.i	0.453451i
$279$	−3.88796e6	−2.99028
$280$	0	0
$281$	−466127.	−0.352159	−0.176079	−	0.984376i	$-0.556342\pi$
$281$	−466127.	−0.352159	−0.176079	+	0.984376i	$0.556342\pi$
$282$	− 1.45690e6i	− 1.09095i
$283$	838973.i	0.622704i	0.950295	+	0.311352i	$0.100782\pi$
$283$	838973.i	0.622704i	−0.950295	+	0.311352i	$0.899218\pi$
$284$	−655860.	−0.482520
$285$	0	0
$286$	−43098.1	−0.0311561
$287$	1.10232e6i	0.789959i
$288$	− 408667.i	− 0.290327i
$289$	1.27988e6	0.901414
$290$	0	0
$291$	1.49977e6	1.03823
$292$	− 1.09490e6i	− 0.751478i
$293$	1.40254e6i	0.954438i	0.878784	+	0.477219i	$0.158355\pi$
$293$	1.40254e6i	0.954438i	−0.878784	+	0.477219i	$0.841645\pi$
$294$	717924.	0.484407
$295$	0	0
$296$	−341833.	−0.226769
$297$	252162.i	0.165878i
$298$	1.18345e6i	0.771986i
$299$	−191250.	−0.123715
$300$	0	0
$301$	1.04394e6	0.664140
$302$	− 215045.i	− 0.135679i
$303$	− 272691.i	− 0.170634i
$304$	10843.8	0.00672974
$305$	0	0
$306$	−597253.	−0.364632
$307$	408221.i	0.247200i	0.992332	+	0.123600i	$0.0394440\pi$
$307$	408221.i	0.247200i	−0.992332	+	0.123600i	$0.960556\pi$
$308$	− 157667.i	− 0.0947029i
$309$	−2.39292e6	−1.42571
$310$	0	0
$311$	2.26426e6	1.32747	0.663735	−	0.747968i	$-0.268970\pi$
$311$	2.26426e6	1.32747	0.663735	+	0.747968i	$0.268970\pi$
$312$	− 274072.i	− 0.159396i
$313$	2.56466e6i	1.47969i	0.672780	+	0.739843i	$0.265100\pi$
$313$	2.56466e6i	1.47969i	−0.672780	+	0.739843i	$0.734900\pi$
$314$	797498.	0.456463
$315$	0	0
$316$	323727.	0.182374
$317$	1.85051e6i	1.03429i	0.855897	+	0.517146i	$0.173006\pi$
$317$	1.85051e6i	1.03429i	−0.855897	+	0.517146i	$0.826994\pi$
$318$	− 1.35451e6i	− 0.751127i
$319$	158597.	0.0872606
$320$	0	0
$321$	2.06272e6	1.11732
$322$	− 699653.i	− 0.376048i
$323$	− 15847.9i	− 0.00845210i
$324$	−51907.3	−0.0274705
$325$	0	0
$326$	−528154.	−0.275243
$327$	3.69656e6i	1.91174i
$328$	456437.i	0.234259i
$329$	2.22167e6	1.13159
$330$	0	0
$331$	−1.94787e6	−0.977215	−0.488607	−	0.872504i	$-0.662495\pi$
$331$	−1.94787e6	−0.977215	−0.488607	+	0.872504i	$0.662495\pi$
$332$	− 1.01036e6i	− 0.503072i
$333$	2.13159e6i	1.05340i
$334$	−503697.	−0.247060
$335$	0	0
$336$	1.00264e6	0.484505
$337$	3.42580e6i	1.64319i	0.570073	+	0.821594i	$0.306915\pi$
$337$	3.42580e6i	1.64319i	−0.570073	+	0.821594i	$0.693085\pi$
$338$	− 114244.i	− 0.0543928i
$339$	2.20044e6	1.03995
$340$	0	0
$341$	−621104.	−0.289253
$342$	− 67619.4i	− 0.0312612i
$343$	− 1.50297e6i	− 0.689788i
$344$	432262.	0.196948
$345$	0	0
$346$	2.21371e6	0.994103
$347$	− 3.01951e6i	− 1.34621i	−0.739547	−	0.673105i	$-0.764960\pi$
$347$	− 3.01951e6i	− 1.34621i	0.739547	−	0.673105i	$-0.235040\pi$
$348$	1.00856e6i	0.446430i
$349$	−1.99404e6	−0.876335	−0.438168	−	0.898893i	$-0.644372\pi$
$349$	−1.99404e6	−0.876335	−0.438168	+	0.898893i	$0.644372\pi$
$350$	0	0
$351$	−668429.	−0.289593
$352$	− 65284.7i	− 0.0280837i
$353$	− 383180.i	− 0.163669i	−0.996646	−	0.0818344i	$-0.973922\pi$
$353$	− 383180.i	− 0.163669i	0.996646	−	0.0818344i	$-0.0260779\pi$
$354$	3.90256e6	1.65517
$355$	0	0
$356$	−1.85960e6	−0.777668
$357$	− 1.46533e6i	− 0.608506i
$358$	1.88519e6i	0.777404i
$359$	3.75111e6	1.53612	0.768058	−	0.640381i	$-0.221223\pi$
$359$	3.75111e6	1.53612	0.768058	+	0.640381i	$0.221223\pi$
$360$	0	0
$361$	−2.47430e6	−0.999275
$362$	1.88469e6i	0.755905i
$363$	− 3.97795e6i	− 1.58450i
$364$	417941.	0.165334
$365$	0	0
$366$	−2.89083e6	−1.12803
$367$	391556.i	0.151750i	0.997117	+	0.0758750i	$0.0241750\pi$
$367$	391556.i	0.151750i	−0.997117	+	0.0758750i	$0.975825\pi$
$368$	− 289704.i	− 0.111515i
$369$	2.84623e6	1.08819
$370$	0	0
$371$	2.06553e6	0.779108
$372$	− 3.94976e6i	− 1.47983i
$373$	9831.94i	0.00365904i	0.999998	+	0.00182952i	$0.000582355\pi$
$373$	9831.94i	0.00365904i	−0.999998	+	0.00182952i	$0.999418\pi$
$374$	−95411.4	−0.0352713
$375$	0	0
$376$	919922.	0.335569
$377$	420407.i	0.152341i
$378$	− 2.44533e6i	− 0.880254i
$379$	−1.87968e6	−0.672180	−0.336090	−	0.941830i	$-0.609105\pi$
$379$	−1.87968e6	−0.672180	−0.336090	+	0.941830i	$0.609105\pi$
$380$	0	0
$381$	−4.68576e6	−1.65374
$382$	1.27946e6i	0.448608i
$383$	− 2.14234e6i	− 0.746263i	−0.927778	−	0.373132i	$-0.878284\pi$
$383$	− 2.14234e6i	− 0.746263i	0.927778	−	0.373132i	$-0.121716\pi$
$384$	415162.	0.143678
$385$	0	0
$386$	−2.62072e6	−0.895265
$387$	− 2.69548e6i	− 0.914869i
$388$	946994.i	0.319351i
$389$	−1.86897e6	−0.626220	−0.313110	−	0.949717i	$-0.601371\pi$
$389$	−1.86897e6	−0.626220	−0.313110	+	0.949717i	$0.601371\pi$
$390$	0	0
$391$	−423393.	−0.140056
$392$	453316.i	0.149000i
$393$	− 1.52263e6i	− 0.497295i
$394$	2.72139e6	0.883184
$395$	0	0
$396$	−407100.	−0.130456
$397$	1.49951e6i	0.477499i	0.971081	+	0.238750i	$0.0767375\pi$
$397$	1.49951e6i	0.477499i	−0.971081	+	0.238750i	$0.923263\pi$
$398$	2.08045e6i	0.658339i
$399$	165901.	0.0521694
$400$	0	0
$401$	2.26959e6	0.704833	0.352416	−	0.935843i	$-0.385360\pi$
$401$	2.26959e6	0.704833	0.352416	+	0.935843i	$0.385360\pi$
$402$	− 4.45504e6i	− 1.37495i
$403$	− 1.64642e6i	− 0.504983i
$404$	172184.	0.0524856
$405$	0	0
$406$	−1.53799e6	−0.463060
$407$	340522.i	0.101896i
$408$	− 606746.i	− 0.180450i
$409$	−4.88657e6	−1.44443	−0.722214	−	0.691669i	$-0.756876\pi$
$409$	−4.88657e6	−1.44443	−0.722214	+	0.691669i	$0.756876\pi$
$410$	0	0
$411$	4.55404e6	1.32982
$412$	− 1.51095e6i	− 0.438538i
$413$	5.95115e6i	1.71682i
$414$	−1.80652e6	−0.518016
$415$	0	0
$416$	173056.	0.0490290
$417$	3.70151e6i	1.04241i
$418$	− 10802.2i	− 0.00302394i
$419$	3.07655e6	0.856110	0.428055	−	0.903753i	$-0.359199\pi$
$419$	3.07655e6	0.856110	0.428055	+	0.903753i	$0.359199\pi$
$420$	0	0
$421$	991694.	0.272692	0.136346	−	0.990661i	$-0.456464\pi$
$421$	991694.	0.272692	0.136346	+	0.990661i	$0.456464\pi$
$422$	487272.i	0.133196i
$423$	− 5.73642e6i	− 1.55880i
$424$	855271.	0.231041
$425$	0	0
$426$	−4.15479e6	−1.10924
$427$	− 4.40833e6i	− 1.17005i
$428$	1.30245e6i	0.343678i
$429$	−273021.	−0.0716230
$430$	0	0
$431$	−4.97181e6	−1.28920	−0.644602	−	0.764519i	$-0.722977\pi$
$431$	−4.97181e6	−1.28920	−0.644602	+	0.764519i	$0.722977\pi$
$432$	− 1.01253e6i	− 0.261035i
$433$	929752.i	0.238313i	0.992875	+	0.119156i	$0.0380190\pi$
$433$	929752.i	0.238313i	−0.992875	+	0.119156i	$0.961981\pi$
$434$	6.02312e6	1.53496
$435$	0	0
$436$	−2.33411e6	−0.588037
$437$	− 47935.4i	− 0.0120075i
$438$	− 6.93603e6i	− 1.72753i
$439$	6.24276e6	1.54602	0.773011	−	0.634393i	$-0.218750\pi$
$439$	6.24276e6	1.54602	0.773011	+	0.634393i	$0.218750\pi$
$440$	0	0
$441$	2.82677e6	0.692140
$442$	− 252916.i	− 0.0615772i
$443$	− 6.12191e6i	− 1.48210i	−0.671449	−	0.741051i	$-0.734328\pi$
$443$	− 6.12191e6i	− 1.48210i	0.671449	−	0.741051i	$-0.265672\pi$
$444$	−2.16547e6	−0.521307
$445$	0	0
$446$	−3.30510e6	−0.786771
$447$	7.49701e6i	1.77468i
$448$	633095.i	0.149030i
$449$	−6.81454e6	−1.59522	−0.797611	−	0.603173i	$-0.793903\pi$
$449$	−6.81454e6	−1.59522	−0.797611	+	0.603173i	$0.793903\pi$
$450$	0	0
$451$	454686.	0.105262
$452$	1.38942e6i	0.319880i
$453$	− 1.36228e6i	− 0.311904i
$454$	1.02715e6	0.233881
$455$	0	0
$456$	68694.1	0.0154706
$457$	1.15462e6i	0.258611i	0.991605	+	0.129306i	$0.0412749\pi$
$457$	1.15462e6i	0.258611i	−0.991605	+	0.129306i	$0.958725\pi$
$458$	− 4.77638e6i	− 1.06398i
$459$	−1.47978e6	−0.327843
$460$	0	0
$461$	4.66540e6	1.02244	0.511218	−	0.859451i	$-0.329194\pi$
$461$	4.66540e6	1.02244	0.511218	+	0.859451i	$0.329194\pi$
$462$	− 998798.i	− 0.217707i
$463$	7.33891e6i	1.59103i	0.605932	+	0.795517i	$0.292800\pi$
$463$	7.33891e6i	1.59103i	−0.605932	+	0.795517i	$0.707200\pi$
$464$	−636830.	−0.137318
$465$	0	0
$466$	−4.97487e6	−1.06125
$467$	1.40701e6i	0.298541i	0.988796	+	0.149270i	$0.0476925\pi$
$467$	1.40701e6i	0.298541i	−0.988796	+	0.149270i	$0.952307\pi$
$468$	− 1.07914e6i	− 0.227752i
$469$	6.79364e6	1.42617
$470$	0	0
$471$	5.05204e6	1.04934
$472$	2.46418e6i	0.509117i
$473$	− 430604.i	− 0.0884964i
$474$	2.05077e6	0.419248
$475$	0	0
$476$	925247.	0.187172
$477$	− 5.33327e6i	− 1.07324i
$478$	− 1.80394e6i	− 0.361121i
$479$	8.02191e6	1.59749	0.798746	−	0.601668i	$-0.205497\pi$
$479$	8.02191e6	1.59749	0.798746	+	0.601668i	$0.205497\pi$
$480$	0	0
$481$	−902652.	−0.177893
$482$	2.50654e6i	0.491425i
$483$	− 4.43221e6i	− 0.864475i
$484$	2.51178e6	0.487381
$485$	0	0
$486$	3.51563e6	0.675169
$487$	5.05570e6i	0.965960i	0.875632	+	0.482980i	$0.160446\pi$
$487$	5.05570e6i	0.965960i	−0.875632	+	0.482980i	$0.839554\pi$
$488$	− 1.82535e6i	− 0.346973i
$489$	−3.34579e6	−0.632741
$490$	0	0
$491$	−7.01316e6	−1.31284	−0.656418	−	0.754398i	$-0.727929\pi$
$491$	−7.01316e6	−1.31284	−0.656418	+	0.754398i	$0.727929\pi$
$492$	2.89147e6i	0.538524i
$493$	930706.i	0.172463i
$494$	28634.4	0.00527924
$495$	0	0
$496$	2.49398e6	0.455186
$497$	− 6.33578e6i	− 1.15056i
$498$	− 6.40049e6i	− 1.15648i
$499$	9.64773e6	1.73450	0.867249	−	0.497875i	$-0.165886\pi$
$499$	9.64773e6	1.73450	0.867249	+	0.497875i	$0.165886\pi$
$500$	0	0
$501$	−3.19085e6	−0.567953
$502$	− 6.80746e6i	− 1.20566i
$503$	− 9.27398e6i	− 1.63435i	−0.576386	−	0.817177i	$-0.695538\pi$
$503$	− 9.27398e6i	− 1.63435i	0.576386	−	0.817177i	$-0.304462\pi$
$504$	3.94783e6	0.692280
$505$	0	0
$506$	−288593.	−0.0501083
$507$	− 723721.i	− 0.125041i
$508$	− 2.95871e6i	− 0.508678i
$509$	1.79545e6	0.307170	0.153585	−	0.988135i	$-0.450918\pi$
$509$	1.79545e6	0.307170	0.153585	+	0.988135i	$0.450918\pi$
$510$	0	0
$511$	1.05770e7	1.79188
$512$	262144.i	0.0441942i
$513$	− 167537.i	− 0.0281072i
$514$	259818.	0.0433772
$515$	0	0
$516$	2.73832e6	0.452752
$517$	− 916395.i	− 0.150784i
$518$	− 3.30219e6i	− 0.540727i
$519$	1.40236e7	2.28529
$520$	0	0
$521$	1.04699e7	1.68986	0.844928	−	0.534880i	$-0.179643\pi$
$521$	1.04699e7	1.68986	0.844928	+	0.534880i	$0.179643\pi$
$522$	3.97112e6i	0.637877i
$523$	− 239742.i	− 0.0383257i	−0.999816	−	0.0191628i	$-0.993900\pi$
$523$	− 239742.i	− 0.0383257i	0.999816	−	0.0191628i	$-0.00610009\pi$
$524$	961431.	0.152964
$525$	0	0
$526$	1.05237e6	0.165845
$527$	− 3.64487e6i	− 0.571683i
$528$	− 413570.i	− 0.0645601i
$529$	5.15570e6	0.801029
$530$	0	0
$531$	1.53660e7	2.36497
$532$	104754.i	0.0160469i
$533$	1.20528e6i	0.183768i
$534$	−1.17803e7	−1.78774
$535$	0	0
$536$	2.81303e6	0.422924
$537$	1.19424e7i	1.78713i
$538$	4.55763e6i	0.678865i
$539$	451578.	0.0669515
$540$	0	0
$541$	−9.72501e6	−1.42855	−0.714277	−	0.699863i	$-0.753244\pi$
$541$	−9.72501e6	−1.42855	−0.714277	+	0.699863i	$0.753244\pi$
$542$	9.14727e6i	1.33750i
$543$	1.19392e7i	1.73771i
$544$	383115.	0.0555050
$545$	0	0
$546$	2.64760e6	0.380077
$547$	− 7.17342e6i	− 1.02508i	−0.858663	−	0.512540i	$-0.828705\pi$
$547$	− 7.17342e6i	− 1.02508i	0.858663	−	0.512540i	$-0.171295\pi$
$548$	2.87554e6i	0.409042i
$549$	−1.13824e7	−1.61177
$550$	0	0
$551$	−105372.	−0.0147859
$552$	− 1.83524e6i	− 0.256356i
$553$	3.12729e6i	0.434866i
$554$	−9.73967e6	−1.34825
$555$	0	0
$556$	−2.33723e6	−0.320638
$557$	− 4.85134e6i	− 0.662558i	−0.943533	−	0.331279i	$-0.892520\pi$
$557$	− 4.85134e6i	− 0.662558i	0.943533	−	0.331279i	$-0.107480\pi$
$558$	− 1.55519e7i	− 2.11445i
$559$	1.14144e6	0.154498
$560$	0	0
$561$	−604419.	−0.0810832
$562$	− 1.86451e6i	− 0.249014i
$563$	2.46602e6i	0.327888i	0.986470	+	0.163944i	$0.0524217\pi$
$563$	2.46602e6i	0.327888i	−0.986470	+	0.163944i	$0.947578\pi$
$564$	5.82759e6	0.771420
$565$	0	0
$566$	−3.35589e6	−0.440318
$567$	− 501438.i	− 0.0655028i
$568$	− 2.62344e6i	− 0.341193i
$569$	1.04367e7	1.35139	0.675697	−	0.737179i	$-0.263843\pi$
$569$	1.04367e7	1.35139	0.675697	+	0.737179i	$0.263843\pi$
$570$	0	0
$571$	4.39937e6	0.564677	0.282338	−	0.959315i	$-0.408890\pi$
$571$	4.39937e6	0.564677	0.282338	+	0.959315i	$0.408890\pi$
$572$	− 172392.i	− 0.0220307i
$573$	8.10518e6i	1.03128i
$574$	−4.40930e6	−0.558585
$575$	0	0
$576$	1.63467e6	0.205293
$577$	− 2.03897e6i	− 0.254959i	−0.991841	−	0.127480i	$-0.959311\pi$
$577$	− 2.03897e6i	− 0.254959i	0.991841	−	0.127480i	$-0.0406887\pi$
$578$	5.11952e6i	0.637396i
$579$	−1.66019e7	−2.05807
$580$	0	0
$581$	9.76032e6	1.19957
$582$	5.99908e6i	0.734137i
$583$	− 851991.i	− 0.103816i
$584$	4.37959e6	0.531376
$585$	0	0
$586$	−5.61018e6	−0.674890
$587$	9.19255e6i	1.10113i	0.834791	+	0.550567i	$0.185589\pi$
$587$	9.19255e6i	1.10113i	−0.834791	+	0.550567i	$0.814411\pi$
$588$	2.87170e6i	0.342527i
$589$	412662.	0.0490125
$590$	0	0
$591$	1.72397e7	2.03030
$592$	− 1.36733e6i	− 0.160350i
$593$	7.34348e6i	0.857561i	0.903409	+	0.428780i	$0.141057\pi$
$593$	7.34348e6i	0.857561i	−0.903409	+	0.428780i	$0.858943\pi$
$594$	−1.00865e6	−0.117294
$595$	0	0
$596$	−4.73380e6	−0.545877
$597$	1.31794e7i	1.51342i
$598$	− 764999.i	− 0.0874799i
$599$	1.04212e7	1.18672	0.593361	−	0.804937i	$-0.297801\pi$
$599$	1.04212e7	1.18672	0.593361	+	0.804937i	$0.297801\pi$
$600$	0	0
$601$	7.92846e6	0.895370	0.447685	−	0.894191i	$-0.352249\pi$
$601$	7.92846e6	0.895370	0.447685	+	0.894191i	$0.352249\pi$
$602$	4.17576e6i	0.469618i
$603$	− 1.75414e7i	− 1.96458i
$604$	860180.	0.0959394
$605$	0	0
$606$	1.09077e6	0.120656
$607$	6.56441e6i	0.723143i	0.932344	+	0.361572i	$0.117760\pi$
$607$	6.56441e6i	0.723143i	−0.932344	+	0.361572i	$0.882240\pi$
$608$	43375.2i	0.00475864i
$609$	−9.74293e6	−1.06450
$610$	0	0
$611$	2.42917e6	0.263242
$612$	− 2.38901e6i	− 0.257834i
$613$	9.93845e6i	1.06824i	0.845410	+	0.534118i	$0.179356\pi$
$613$	9.93845e6i	1.06824i	−0.845410	+	0.534118i	$0.820644\pi$
$614$	−1.63288e6	−0.174797
$615$	0	0
$616$	630667.	0.0669650
$617$	1.49986e7i	1.58613i	0.609138	+	0.793064i	$0.291516\pi$
$617$	1.49986e7i	1.58613i	−0.609138	+	0.793064i	$0.708484\pi$
$618$	− 9.57167e6i	− 1.00813i
$619$	−6.67132e6	−0.699818	−0.349909	−	0.936784i	$-0.613788\pi$
$619$	−6.67132e6	−0.699818	−0.349909	+	0.936784i	$0.613788\pi$
$620$	0	0
$621$	−4.47593e6	−0.465751
$622$	9.05703e6i	0.938663i
$623$	− 1.79642e7i	− 1.85433i
$624$	1.09629e6	0.112710
$625$	0	0
$626$	−1.02587e7	−1.04630
$627$	− 68430.7i	− 0.00695156i
$628$	3.18999e6i	0.322768i
$629$	−1.99831e6	−0.201389
$630$	0	0
$631$	−1.43883e7	−1.43859	−0.719295	−	0.694705i	$-0.755535\pi$
$631$	−1.43883e7	−1.43859	−0.719295	+	0.694705i	$0.755535\pi$
$632$	1.29491e6i	0.128958i
$633$	3.08680e6i	0.306196i
$634$	−7.40204e6	−0.731355
$635$	0	0
$636$	5.41803e6	0.531127
$637$	1.19704e6i	0.116885i
$638$	634388.i	0.0617026i
$639$	−1.63591e7	−1.58492
$640$	0	0
$641$	−5.54650e6	−0.533180	−0.266590	−	0.963810i	$-0.585897\pi$
$641$	−5.54650e6	−0.533180	−0.266590	+	0.963810i	$0.585897\pi$
$642$	8.25086e6i	0.790063i
$643$	− 1.56374e7i	− 1.49155i	−0.666199	−	0.745774i	$-0.732080\pi$
$643$	− 1.56374e7i	− 1.49155i	0.666199	−	0.745774i	$-0.267920\pi$
$644$	2.79861e6	0.265906
$645$	0	0
$646$	63391.5	0.00597654
$647$	− 9.51432e6i	− 0.893546i	−0.894647	−	0.446773i	$-0.852573\pi$
$647$	− 9.51432e6i	− 0.893546i	0.894647	−	0.446773i	$-0.147427\pi$
$648$	− 207629.i	− 0.0194246i
$649$	2.45473e6	0.228766
$650$	0	0
$651$	3.81557e7	3.52863
$652$	− 2.11262e6i	− 0.194626i
$653$	4.66400e6i	0.428032i	0.976830	+	0.214016i	$0.0686544\pi$
$653$	4.66400e6i	0.428032i	−0.976830	+	0.214016i	$0.931346\pi$
$654$	−1.47863e7	−1.35180
$655$	0	0
$656$	−1.82575e6	−0.165646
$657$	− 2.73101e7i	− 2.46837i
$658$	8.88669e6i	0.800157i
$659$	−3.19305e6	−0.286413	−0.143206	−	0.989693i	$-0.545741\pi$
$659$	−3.19305e6	−0.286413	−0.143206	+	0.989693i	$0.545741\pi$
$660$	0	0
$661$	−2.20448e7	−1.96246	−0.981232	−	0.192829i	$-0.938234\pi$
$661$	−2.20448e7	−1.96246	−0.981232	+	0.192829i	$0.938234\pi$
$662$	− 7.79148e6i	− 0.690995i
$663$	− 1.60219e6i	− 0.141556i
$664$	4.04143e6	0.355726
$665$	0	0
$666$	−8.52635e6	−0.744865
$667$	2.81513e6i	0.245010i
$668$	− 2.01479e6i	− 0.174698i
$669$	−2.09374e7	−1.80866
$670$	0	0
$671$	−1.81835e6	−0.155909
$672$	4.01057e6i	0.342596i
$673$	1.75530e7i	1.49387i	0.664896	+	0.746936i	$0.268476\pi$
$673$	1.75530e7i	1.49387i	−0.664896	+	0.746936i	$0.731524\pi$
$674$	−1.37032e7	−1.16191
$675$	0	0
$676$	456976.	0.0384615
$677$	4.12340e6i	0.345767i	0.984942	+	0.172883i	$0.0553084\pi$
$677$	4.12340e6i	0.345767i	−0.984942	+	0.172883i	$0.944692\pi$
$678$	8.80178e6i	0.735354i
$679$	−9.14820e6	−0.761485
$680$	0	0
$681$	6.50688e6	0.537657
$682$	− 2.48442e6i	− 0.204533i
$683$	1.68043e7i	1.37838i	0.724581	+	0.689190i	$0.242034\pi$
$683$	1.68043e7i	1.37838i	−0.724581	+	0.689190i	$0.757966\pi$
$684$	270478.	0.0221050
$685$	0	0
$686$	6.01189e6	0.487754
$687$	− 3.02577e7i	− 2.44593i
$688$	1.72905e6i	0.139263i
$689$	2.25845e6	0.181244
$690$	0	0
$691$	5.42208e6	0.431987	0.215994	−	0.976395i	$-0.430701\pi$
$691$	5.42208e6	0.431987	0.215994	+	0.976395i	$0.430701\pi$
$692$	8.85486e6i	0.702937i
$693$	− 3.93269e6i	− 0.311069i
$694$	1.20780e7	0.951915
$695$	0	0
$696$	−4.03423e6	−0.315673
$697$	2.66827e6i	0.208040i
$698$	− 7.97616e6i	− 0.619662i
$699$	−3.15152e7	−2.43965
$700$	0	0
$701$	−1.78689e7	−1.37342	−0.686710	−	0.726932i	$-0.740946\pi$
$701$	−1.78689e7	−1.37342	−0.686710	+	0.726932i	$0.740946\pi$
$702$	− 2.67372e6i	− 0.204773i
$703$	− 226243.i	− 0.0172658i
$704$	261139.	0.0198582
$705$	0	0
$706$	1.53272e6	0.115731
$707$	1.66335e6i	0.125151i
$708$	1.56103e7i	1.17038i
$709$	7.86353e6	0.587492	0.293746	−	0.955884i	$-0.405098\pi$
$709$	7.86353e6	0.587492	0.293746	+	0.955884i	$0.405098\pi$
$710$	0	0
$711$	8.07475e6	0.599039
$712$	− 7.43840e6i	− 0.549894i
$713$	− 1.10247e7i	− 0.812163i
$714$	5.86132e6	0.430279
$715$	0	0
$716$	−7.54074e6	−0.549707
$717$	− 1.14277e7i	− 0.830161i
$718$	1.50044e7i	1.08620i
$719$	1.54686e7	1.11591	0.557956	−	0.829870i	$-0.311586\pi$
$719$	1.54686e7	1.11591	0.557956	+	0.829870i	$0.311586\pi$
$720$	0	0
$721$	1.45962e7	1.04568
$722$	− 9.89722e6i	− 0.706594i
$723$	1.58786e7i	1.12971i
$724$	−7.53874e6	−0.534506
$725$	0	0
$726$	1.59118e7	1.12041
$727$	6.05401e6i	0.424822i	0.977180	+	0.212411i	$0.0681316\pi$
$727$	6.05401e6i	0.424822i	−0.977180	+	0.212411i	$0.931868\pi$
$728$	1.67177e6i	0.116909i
$729$	2.30594e7	1.60705
$730$	0	0
$731$	2.52695e6	0.174905
$732$	− 1.15633e7i	− 0.797637i
$733$	− 1.88061e7i	− 1.29282i	−0.762989	−	0.646412i	$-0.776269\pi$
$733$	− 1.88061e7i	− 1.29282i	0.762989	−	0.646412i	$-0.223731\pi$
$734$	−1.56622e6	−0.107303
$735$	0	0
$736$	1.15882e6	0.0788533
$737$	− 2.80224e6i	− 0.190036i
$738$	1.13849e7i	0.769465i
$739$	1.12832e7	0.760011	0.380006	−	0.924984i	$-0.375922\pi$
$739$	1.12832e7	0.760011	0.380006	+	0.924984i	$0.375922\pi$
$740$	0	0
$741$	181395.	0.0121361
$742$	8.26214e6i	0.550913i
$743$	1.08397e6i	0.0720351i	0.999351	+	0.0360175i	$0.0114672\pi$
$743$	1.08397e6i	0.0720351i	−0.999351	+	0.0360175i	$0.988533\pi$
$744$	1.57990e7	1.04640
$745$	0	0
$746$	−39327.8	−0.00258733
$747$	− 2.52014e7i	− 1.65243i
$748$	− 381646.i	− 0.0249406i
$749$	−1.25820e7	−0.819494
$750$	0	0
$751$	−1.03591e7	−0.670230	−0.335115	−	0.942177i	$-0.608775\pi$
$751$	−1.03591e7	−0.670230	−0.335115	+	0.942177i	$0.608775\pi$
$752$	3.67969e6i	0.237283i
$753$	− 4.31244e7i	− 2.77163i
$754$	−1.68163e6	−0.107721
$755$	0	0
$756$	9.78132e6	0.622433
$757$	1.53049e7i	0.970710i	0.874317	+	0.485355i	$0.161310\pi$
$757$	1.53049e7i	0.970710i	−0.874317	+	0.485355i	$0.838690\pi$
$758$	− 7.51871e6i	− 0.475303i
$759$	−1.82820e6	−0.115191
$760$	0	0
$761$	−2.78390e7	−1.74258	−0.871289	−	0.490771i	$-0.836715\pi$
$761$	−2.78390e7	−1.74258	−0.871289	+	0.490771i	$0.836715\pi$
$762$	− 1.87430e7i	− 1.16937i
$763$	− 2.25481e7i	− 1.40216i
$764$	−5.11782e6	−0.317214
$765$	0	0
$766$	8.56937e6	0.527688
$767$	6.50698e6i	0.399384i
$768$	1.66065e6i	0.101595i
$769$	1.88871e7	1.15173	0.575864	−	0.817545i	$-0.304666\pi$
$769$	1.88871e7	1.15173	0.575864	+	0.817545i	$0.304666\pi$
$770$	0	0
$771$	1.64591e6	0.0997173
$772$	− 1.04829e7i	− 0.633048i
$773$	1.66647e7i	1.00311i	0.865125	+	0.501556i	$0.167239\pi$
$773$	1.66647e7i	1.00311i	−0.865125	+	0.501556i	$0.832761\pi$
$774$	1.07819e7	0.646910
$775$	0	0
$776$	−3.78798e6	−0.225815
$777$	− 2.09190e7i	− 1.24305i
$778$	− 7.47586e6i	− 0.442805i
$779$	−302094.	−0.0178361
$780$	0	0
$781$	−2.61338e6	−0.153312
$782$	− 1.69357e6i	− 0.0990345i
$783$	9.83903e6i	0.573519i
$784$	−1.81326e6	−0.105359
$785$	0	0
$786$	6.09054e6	0.351641
$787$	− 1.30313e7i	− 0.749984i	−0.927028	−	0.374992i	$-0.877645\pi$
$787$	− 1.30313e7i	− 0.749984i	0.927028	−	0.374992i	$-0.122355\pi$
$788$	1.08856e7i	0.624505i
$789$	6.66661e6	0.381252
$790$	0	0
$791$	−1.34221e7	−0.762747
$792$	− 1.62840e6i	− 0.0922460i
$793$	− 4.82005e6i	− 0.272188i
$794$	−5.99803e6	−0.337643
$795$	0	0
$796$	−8.32180e6	−0.465516
$797$	2.29513e6i	0.127986i	0.997950	+	0.0639929i	$0.0203835\pi$
$797$	2.29513e6i	0.127986i	−0.997950	+	0.0639929i	$0.979617\pi$
$798$	663603.i	0.0368893i
$799$	5.37775e6	0.298012
$800$	0	0
$801$	−4.63841e7	−2.55439
$802$	9.07835e6i	0.498392i
$803$	− 4.36280e6i	− 0.238768i
$804$	1.78202e7	0.972235
$805$	0	0
$806$	6.58566e6	0.357077
$807$	2.88720e7i	1.56060i
$808$	688738.i	0.0371129i
$809$	8.57148e6	0.460452	0.230226	−	0.973137i	$-0.426053\pi$
$809$	8.57148e6	0.460452	0.230226	+	0.973137i	$0.426053\pi$
$810$	0	0
$811$	−3.45545e6	−0.184481	−0.0922407	−	0.995737i	$-0.529403\pi$
$811$	−3.45545e6	−0.184481	−0.0922407	+	0.995737i	$0.529403\pi$
$812$	− 6.15194e6i	− 0.327433i
$813$	5.79467e7i	3.07470i
$814$	−1.36209e6	−0.0720517
$815$	0	0
$816$	2.42698e6	0.127597
$817$	286094.i	0.0149953i
$818$	− 1.95463e7i	− 1.02137i
$819$	1.04247e7	0.543069
$820$	0	0
$821$	−2.35363e7	−1.21866	−0.609328	−	0.792918i	$-0.708561\pi$
$821$	−2.35363e7	−1.21866	−0.609328	+	0.792918i	$0.708561\pi$
$822$	1.82161e7i	0.940323i
$823$	1.60053e7i	0.823693i	0.911253	+	0.411846i	$0.135116\pi$
$823$	1.60053e7i	0.823693i	−0.911253	+	0.411846i	$0.864884\pi$
$824$	6.04380e6	0.310093
$825$	0	0
$826$	−2.38046e7	−1.21398
$827$	− 6.11399e6i	− 0.310857i	−0.987847	−	0.155429i	$-0.950324\pi$
$827$	− 6.11399e6i	− 0.310857i	0.987847	−	0.155429i	$-0.0496758\pi$
$828$	− 7.22610e6i	− 0.366292i
$829$	−2.09598e7	−1.05925	−0.529627	−	0.848231i	$-0.677668\pi$
$829$	−2.09598e7	−1.05925	−0.529627	+	0.848231i	$0.677668\pi$
$830$	0	0
$831$	−6.16995e7	−3.09941
$832$	692224.i	0.0346688i
$833$	2.65003e6i	0.132324i
$834$	−1.48061e7	−0.737097
$835$	0	0
$836$	43208.9	0.00213825
$837$	− 3.85320e7i	− 1.90111i
$838$	1.23062e7i	0.605361i
$839$	2.39998e7	1.17707	0.588536	−	0.808471i	$-0.299704\pi$
$839$	2.39998e7	1.17707	0.588536	+	0.808471i	$0.299704\pi$
$840$	0	0
$841$	−1.43229e7	−0.698299
$842$	3.96678e6i	0.192822i
$843$	− 1.18114e7i	− 0.572444i
$844$	−1.94909e6	−0.0941836
$845$	0	0
$846$	2.29457e7	1.10224
$847$	2.42645e7i	1.16215i
$848$	3.42108e6i	0.163371i
$849$	−2.12591e7	−1.01222
$850$	0	0
$851$	−6.04433e6	−0.286104
$852$	− 1.66191e7i	− 0.784350i
$853$	8.22205e6i	0.386908i	0.981109	+	0.193454i	$0.0619690\pi$
$853$	8.22205e6i	0.386908i	−0.981109	+	0.193454i	$0.938031\pi$
$854$	1.76333e7	0.827350
$855$	0	0
$856$	−5.20981e6	−0.243017
$857$	1.68142e7i	0.782032i	0.920384	+	0.391016i	$0.127876\pi$
$857$	1.68142e7i	0.782032i	−0.920384	+	0.391016i	$0.872124\pi$
$858$	− 1.09208e6i	− 0.0506451i
$859$	3.26713e6	0.151072	0.0755360	−	0.997143i	$-0.475933\pi$
$859$	3.26713e6	0.151072	0.0755360	+	0.997143i	$0.475933\pi$
$860$	0	0
$861$	−2.79323e7	−1.28410
$862$	− 1.98872e7i	− 0.911604i
$863$	− 3.42932e7i	− 1.56741i	−0.621136	−	0.783703i	$-0.713329\pi$
$863$	− 3.42932e7i	− 1.56741i	0.621136	−	0.783703i	$-0.286671\pi$
$864$	4.05013e6	0.184580
$865$	0	0
$866$	−3.71901e6	−0.168513
$867$	3.24315e7i	1.46527i
$868$	2.40925e7i	1.08538i
$869$	1.28994e6	0.0579457
$870$	0	0
$871$	7.42815e6	0.331769
$872$	− 9.33642e6i	− 0.415805i
$873$	2.36209e7i	1.04896i
$874$	191742.	0.00849059
$875$	0	0
$876$	2.77441e7	1.22155
$877$	3.93564e6i	0.172789i	0.996261	+	0.0863946i	$0.0275346\pi$
$877$	3.93564e6i	0.172789i	−0.996261	+	0.0863946i	$0.972465\pi$
$878$	2.49711e7i	1.09320i
$879$	−3.55397e7	−1.55147
$880$	0	0
$881$	−4.25403e7	−1.84655	−0.923274	−	0.384141i	$-0.874498\pi$
$881$	−4.25403e7	−1.84655	−0.923274	+	0.384141i	$0.874498\pi$
$882$	1.13071e7i	0.489417i
$883$	1.48855e7i	0.642482i	0.946998	+	0.321241i	$0.104100\pi$
$883$	1.48855e7i	0.642482i	−0.946998	+	0.321241i	$0.895900\pi$
$884$	1.01166e6	0.0435417
$885$	0	0
$886$	2.44877e7	1.04800
$887$	3.89095e7i	1.66053i	0.557370	+	0.830264i	$0.311811\pi$
$887$	3.89095e7i	1.66053i	−0.557370	+	0.830264i	$0.688189\pi$
$888$	− 8.66186e6i	− 0.368620i
$889$	2.85819e7	1.21293
$890$	0	0
$891$	−206833.	−0.00872822
$892$	− 1.32204e7i	− 0.556331i
$893$	608854.i	0.0255496i
$894$	−2.99880e7	−1.25489
$895$	0	0
$896$	−2.53238e6	−0.105380
$897$	− 4.84617e6i	− 0.201102i
$898$	− 2.72582e7i	− 1.12799i
$899$	−2.42346e7	−1.00009
$900$	0	0
$901$	4.99980e6	0.205183
$902$	1.81875e6i	0.0744313i
$903$	2.64529e7i	1.07958i
$904$	−5.55767e6	−0.226189
$905$	0	0
$906$	5.44912e6	0.220550
$907$	− 4.26154e6i	− 0.172008i	−0.996295	−	0.0860039i	$-0.972590\pi$
$907$	− 4.26154e6i	− 0.172008i	0.996295	−	0.0860039i	$-0.0274098\pi$
$908$	4.10861e6i	0.165379i
$909$	4.29480e6	0.172399
$910$	0	0
$911$	−2.52991e7	−1.00997	−0.504985	−	0.863128i	$-0.668502\pi$
$911$	−2.52991e7	−1.00997	−0.504985	+	0.863128i	$0.668502\pi$
$912$	274776.i	0.0109394i
$913$	− 4.02594e6i	− 0.159842i
$914$	−4.61847e6	−0.182866
$915$	0	0
$916$	1.91055e7	0.752351
$917$	9.28767e6i	0.364740i
$918$	− 5.91913e6i	− 0.231820i
$919$	2.20112e7	0.859717	0.429858	−	0.902896i	$-0.358563\pi$
$919$	2.20112e7	0.859717	0.429858	+	0.902896i	$0.358563\pi$
$920$	0	0
$921$	−1.03441e7	−0.401831
$922$	1.86616e7i	0.722972i
$923$	− 6.92752e6i	− 0.267654i
$924$	3.99519e6	0.153942
$925$	0	0
$926$	−2.93557e7	−1.12503
$927$	− 3.76877e7i	− 1.44046i
$928$	− 2.54732e6i	− 0.0970987i
$929$	1.80352e6	0.0685615	0.0342808	−	0.999412i	$-0.489086\pi$
$929$	1.80352e6	0.0685615	0.0342808	+	0.999412i	$0.489086\pi$
$930$	0	0
$931$	−300029.	−0.0113446
$932$	− 1.98995e7i	− 0.750416i
$933$	5.73751e7i	2.15784i
$934$	−5.62803e6	−0.211100
$935$	0	0
$936$	4.31654e6	0.161045
$937$	− 3.06903e7i	− 1.14196i	−0.820963	−	0.570981i	$-0.806563\pi$
$937$	− 3.06903e7i	− 1.14196i	0.820963	−	0.570981i	$-0.193437\pi$
$938$	2.71746e7i	1.00845i
$939$	−6.49872e7	−2.40527
$940$	0	0
$941$	4.83823e7	1.78120	0.890600	−	0.454788i	$-0.150285\pi$
$941$	4.83823e7	1.78120	0.890600	+	0.454788i	$0.150285\pi$
$942$	2.02082e7i	0.741993i
$943$	8.07077e6i	0.295553i
$944$	−9.85672e6	−0.360000
$945$	0	0
$946$	1.72242e6	0.0625764
$947$	− 3.69874e7i	− 1.34023i	−0.742259	−	0.670114i	$-0.766245\pi$
$947$	− 3.69874e7i	− 1.34023i	0.742259	−	0.670114i	$-0.233755\pi$
$948$	8.20308e6i	0.296453i
$949$	1.15649e7	0.416845
$950$	0	0
$951$	−4.68910e7	−1.68127
$952$	3.70099e6i	0.132350i
$953$	− 2.30839e7i	− 0.823335i	−0.911334	−	0.411668i	$-0.864946\pi$
$953$	− 2.30839e7i	− 0.823335i	0.911334	−	0.411668i	$-0.135054\pi$
$954$	2.13331e7	0.758896
$955$	0	0
$956$	7.21577e6	0.255351
$957$	4.01876e6i	0.141845i
$958$	3.20876e7i	1.12960i
$959$	−2.77784e7	−0.975351
$960$	0	0
$961$	6.62795e7	2.31510
$962$	− 3.61061e6i	− 0.125789i
$963$	3.24871e7i	1.12887i
$964$	−1.00262e7	−0.347490
$965$	0	0
$966$	1.77288e7	0.611276
$967$	− 2.85570e7i	− 0.982079i	−0.871137	−	0.491040i	$-0.836617\pi$
$967$	− 2.85570e7i	− 0.982079i	0.871137	−	0.491040i	$-0.163383\pi$
$968$	1.00471e7i	0.344630i
$969$	401577.	0.0137391
$970$	0	0
$971$	−1.99121e7	−0.677748	−0.338874	−	0.940832i	$-0.610046\pi$
$971$	−1.99121e7	−0.677748	−0.338874	+	0.940832i	$0.610046\pi$
$972$	1.40625e7i	0.477417i
$973$	− 2.25783e7i	− 0.764555i
$974$	−2.02228e7	−0.683037
$975$	0	0
$976$	7.30138e6	0.245347
$977$	− 9.96441e6i	− 0.333976i	−0.985959	−	0.166988i	$-0.946596\pi$
$977$	− 9.96441e6i	− 0.333976i	0.985959	−	0.166988i	$-0.0534041\pi$
$978$	− 1.33831e7i	− 0.447415i
$979$	−7.40987e6	−0.247089
$980$	0	0
$981$	−5.82197e7	−1.93151
$982$	− 2.80527e7i	− 0.928315i
$983$	3.94695e7i	1.30280i	0.758734	+	0.651400i	$0.225818\pi$
$983$	3.94695e7i	1.30280i	−0.758734	+	0.651400i	$0.774182\pi$
$984$	−1.15659e7	−0.380794
$985$	0	0
$986$	−3.72283e6	−0.121950
$987$	5.62960e7i	1.83943i
$988$	114538.i	0.00373299i
$989$	7.64331e6	0.248480
$990$	0	0
$991$	4.25424e7	1.37606	0.688031	−	0.725682i	$-0.258475\pi$
$991$	4.25424e7	1.37606	0.688031	+	0.725682i	$0.258475\pi$
$992$	9.97592e6i	0.321865i
$993$	− 4.93580e7i	− 1.58849i
$994$	2.53431e7	0.813568
$995$	0	0
$996$	2.56019e7	0.817758
$997$	− 1.32492e6i	− 0.0422136i	−0.999777	−	0.0211068i	$-0.993281\pi$
$997$	− 1.32492e6i	− 0.0422136i	0.999777	−	0.0211068i	$-0.00671900\pi$
$998$	3.85909e7i	1.22647i
$999$	−2.11253e7	−0.669713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.k.599.8		8
5.2	odd	4		650.6.a.l.1.4	✓	4
5.3	odd	4		650.6.a.m.1.1	yes	4
5.4	even	2	inner	650.6.b.k.599.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.l.1.4	✓	4	5.2	odd	4
650.6.a.m.1.1	yes	4	5.3	odd	4
650.6.b.k.599.1		8	5.4	even	2	inner
650.6.b.k.599.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +25.3395i q^{3} -16.0000 q^{4} -101.358 q^{6} -154.564i q^{7} -64.0000i q^{8} -399.089 q^{9} -63.7546 q^{11} -405.431i q^{12} -169.000i q^{13} +618.256 q^{14} +256.000 q^{16} -374.136i q^{17} -1596.35i q^{18} +42.3586 q^{19} +3916.57 q^{21} -255.018i q^{22} -1131.66i q^{23} +1621.73 q^{24} +676.000 q^{26} -3955.20i q^{27} +2473.03i q^{28} -2487.62 q^{29} +9742.11 q^{31} +1024.00i q^{32} -1615.51i q^{33} +1496.54 q^{34} +6385.42 q^{36} -5341.14i q^{37} +169.435i q^{38} +4282.37 q^{39} -7131.82 q^{41} +15666.3i q^{42} +6754.09i q^{43} +1020.07 q^{44} +4526.62 q^{46} +14373.8i q^{47} +6486.90i q^{48} -7083.06 q^{49} +9480.40 q^{51} +2704.00i q^{52} +13363.6i q^{53} +15820.8 q^{54} -9892.10 q^{56} +1073.35i q^{57} -9950.47i q^{58} -38502.8 q^{59} +28521.0 q^{61} +38968.4i q^{62} +61684.8i q^{63} -4096.00 q^{64} +6462.03 q^{66} +43953.6i q^{67} +5986.17i q^{68} +28675.6 q^{69} +40991.3 q^{71} +25541.7i q^{72} +68431.1i q^{73} +21364.6 q^{74} -677.738 q^{76} +9854.17i q^{77} +17129.5i q^{78} -20233.0 q^{79} +3244.21 q^{81} -28527.3i q^{82} +63147.4i q^{83} -62665.2 q^{84} -27016.4 q^{86} -63034.9i q^{87} +4080.29i q^{88} +116225. q^{89} -26121.3 q^{91} +18106.5i q^{92} +246860. i q^{93} -57495.1 q^{94} -25947.6 q^{96} -59187.1i q^{97} -28332.2i q^{98} +25443.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} - 160 q^{6} - 444 q^{9} - 588 q^{11} + 1424 q^{14} + 2048 q^{16} - 928 q^{19} - 2016 q^{21} + 2560 q^{24} + 5408 q^{26} + 9088 q^{29} + 11236 q^{31} + 14448 q^{34} + 7104 q^{36} + 6760 q^{39}+ \cdots - 89060 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 - 160 * q^6 - 444 * q^9 - 588 * q^11 + 1424 * q^14 + 2048 * q^16 - 928 * q^19 - 2016 * q^21 + 2560 * q^24 + 5408 * q^26 + 9088 * q^29 + 11236 * q^31 + 14448 * q^34 + 7104 * q^36 + 6760 * q^39 + 26464 * q^41 + 9408 * q^44 + 13344 * q^46 - 2932 * q^49 + 3840 * q^51 + 18784 * q^54 - 22784 * q^56 + 89180 * q^59 + 19336 * q^61 - 32768 * q^64 - 25120 * q^66 + 22636 * q^69 + 2320 * q^71 + 89184 * q^74 + 14848 * q^76 + 211480 * q^79 - 110664 * q^81 + 32256 * q^84 - 165184 * q^86 + 145544 * q^89 - 60164 * q^91 + 70224 * q^94 - 40960 * q^96 - 89060 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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