LMFDB - Embedded newform 650.6.b.f.599.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{145})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 73x^{2} + 1296 $ x^4 + 73*x^2 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-6.52080i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.f.599.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} -15.0416i q^{3} -16.0000 q^{4} -60.1664 q^{6} +52.9584i q^{7} +64.0000i q^{8} +16.7504 q^{9} -259.248 q^{11} +240.666i q^{12} +169.000i q^{13} +211.834 q^{14} +256.000 q^{16} -2273.15i q^{17} -67.0017i q^{18} -730.329 q^{19} +796.579 q^{21} +1036.99i q^{22} -1973.36i q^{23} +962.662 q^{24} +676.000 q^{26} -3907.06i q^{27} -847.334i q^{28} +949.515 q^{29} +225.331 q^{31} -1024.00i q^{32} +3899.50i q^{33} -9092.62 q^{34} -268.007 q^{36} -954.367i q^{37} +2921.32i q^{38} +2542.03 q^{39} -17371.4 q^{41} -3186.32i q^{42} +10088.2i q^{43} +4147.97 q^{44} -7893.46 q^{46} -6069.90i q^{47} -3850.65i q^{48} +14002.4 q^{49} -34191.9 q^{51} -2704.00i q^{52} +16177.7i q^{53} -15628.2 q^{54} -3389.34 q^{56} +10985.3i q^{57} -3798.06i q^{58} -326.933 q^{59} -46814.4 q^{61} -901.324i q^{62} +887.076i q^{63} -4096.00 q^{64} +15598.0 q^{66} +43583.3i q^{67} +36370.5i q^{68} -29682.5 q^{69} +51745.5 q^{71} +1072.03i q^{72} +13295.3i q^{73} -3817.47 q^{74} +11685.3 q^{76} -13729.4i q^{77} -10168.1i q^{78} -76268.5 q^{79} -54698.1 q^{81} +69485.7i q^{82} -30147.4i q^{83} -12745.3 q^{84} +40352.9 q^{86} -14282.2i q^{87} -16591.9i q^{88} +66771.0 q^{89} -8949.97 q^{91} +31573.8i q^{92} -3389.34i q^{93} -24279.6 q^{94} -15402.6 q^{96} +151425. i q^{97} -56009.6i q^{98} -4342.51 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} - 48 q^{6} + 356 q^{9} + 408 q^{11} + 1040 q^{14} + 1024 q^{16} - 224 q^{19} + 200 q^{21} + 768 q^{24} + 2704 q^{26} + 13624 q^{29} - 640 q^{31} - 3232 q^{34} - 5696 q^{36} + 2028 q^{39}+ \cdots + 140712 q^{99}+O(q^{100}) $ 4 * q - 64 * q^4 - 48 * q^6 + 356 * q^9 + 408 * q^11 + 1040 * q^14 + 1024 * q^16 - 224 * q^19 + 200 * q^21 + 768 * q^24 + 2704 * q^26 + 13624 * q^29 - 640 * q^31 - 3232 * q^34 - 5696 * q^36 + 2028 * q^39 - 70160 * q^41 - 6528 * q^44 - 2096 * q^46 + 49748 * q^49 - 102184 * q^51 - 2016 * q^54 - 16640 * q^56 + 43776 * q^59 - 82544 * q^61 - 16384 * q^64 + 64704 * q^66 - 90312 * q^69 - 30960 * q^71 - 106208 * q^74 + 3584 * q^76 - 54416 * q^79 - 97124 * q^81 - 3200 * q^84 - 20272 * q^86 + 136264 * q^89 - 43940 * q^91 - 176304 * q^94 - 12288 * q^96 + 140712 * 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
$3$	− 15.0416i	− 0.964919i	−0.875918	−	0.482459i	$-0.839744\pi$
$3$	− 15.0416i	− 0.964919i	0.875918	−	0.482459i	$-0.160256\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−60.1664	−0.682301
$7$	52.9584i	0.408498i	0.978919	+	0.204249i	$0.0654752\pi$
$7$	52.9584i	0.408498i	−0.978919	+	0.204249i	$0.934525\pi$
$8$	64.0000i	0.353553i
$9$	16.7504	0.0689318
$10$	0	0
$11$	−259.248	−0.646001	−0.323001	−	0.946399i	$-0.604692\pi$
$11$	−259.248	−0.646001	−0.323001	+	0.946399i	$0.604692\pi$
$12$	240.666i	0.482459i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	211.834	0.288852
$15$	0	0
$16$	256.000	0.250000
$17$	− 2273.15i	− 1.90768i	−0.300308	−	0.953842i	$-0.597089\pi$
$17$	− 2273.15i	− 1.90768i	0.300308	−	0.953842i	$-0.402911\pi$
$18$	− 67.0017i	− 0.0487422i
$19$	−730.329	−0.464125	−0.232062	−	0.972701i	$-0.574547\pi$
$19$	−730.329	−0.464125	−0.232062	+	0.972701i	$0.574547\pi$
$20$	0	0
$21$	796.579	0.394167
$22$	1036.99i	0.456792i
$23$	− 1973.36i	− 0.777835i	−0.921273	−	0.388918i	$-0.872849\pi$
$23$	− 1973.36i	− 0.777835i	0.921273	−	0.388918i	$-0.127151\pi$
$24$	962.662	0.341150
$25$	0	0
$26$	676.000	0.196116
$27$	− 3907.06i	− 1.03143i
$28$	− 847.334i	− 0.204249i
$29$	949.515	0.209656	0.104828	−	0.994490i	$-0.466571\pi$
$29$	949.515	0.209656	0.104828	+	0.994490i	$0.466571\pi$
$30$	0	0
$31$	225.331	0.0421131	0.0210565	−	0.999778i	$-0.493297\pi$
$31$	225.331	0.0421131	0.0210565	+	0.999778i	$0.493297\pi$
$32$	− 1024.00i	− 0.176777i
$33$	3899.50i	0.623339i
$34$	−9092.62	−1.34894
$35$	0	0
$36$	−268.007	−0.0344659
$37$	− 954.367i	− 0.114607i	−0.998357	−	0.0573035i	$-0.981750\pi$
$37$	− 954.367i	− 0.114607i	0.998357	−	0.0573035i	$-0.0182503\pi$
$38$	2921.32i	0.328186i
$39$	2542.03	0.267620
$40$	0	0
$41$	−17371.4	−1.61390	−0.806948	−	0.590622i	$-0.798882\pi$
$41$	−17371.4	−1.61390	−0.806948	+	0.590622i	$0.798882\pi$
$42$	− 3186.32i	− 0.278718i
$43$	10088.2i	0.832039i	0.909356	+	0.416019i	$0.136575\pi$
$43$	10088.2i	0.832039i	−0.909356	+	0.416019i	$0.863425\pi$
$44$	4147.97	0.323001
$45$	0	0
$46$	−7893.46	−0.550013
$47$	− 6069.90i	− 0.400809i	−0.979713	−	0.200404i	$-0.935774\pi$
$47$	− 6069.90i	− 0.400809i	0.979713	−	0.200404i	$-0.0642256\pi$
$48$	− 3850.65i	− 0.241230i
$49$	14002.4	0.833129
$50$	0	0
$51$	−34191.9	−1.84076
$52$	− 2704.00i	− 0.138675i
$53$	16177.7i	0.791092i	0.918446	+	0.395546i	$0.129445\pi$
$53$	16177.7i	0.791092i	−0.918446	+	0.395546i	$0.870555\pi$
$54$	−15628.2	−0.729333
$55$	0	0
$56$	−3389.34	−0.144426
$57$	10985.3i	0.447843i
$58$	− 3798.06i	− 0.148249i
$59$	−326.933	−0.0122272	−0.00611362	−	0.999981i	$-0.501946\pi$
$59$	−326.933	−0.0122272	−0.00611362	+	0.999981i	$0.501946\pi$
$60$	0	0
$61$	−46814.4	−1.61085	−0.805425	−	0.592698i	$-0.798063\pi$
$61$	−46814.4	−1.61085	−0.805425	+	0.592698i	$0.798063\pi$
$62$	− 901.324i	− 0.0297784i
$63$	887.076i	0.0281585i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	15598.0	0.440767
$67$	43583.3i	1.18613i	0.805154	+	0.593066i	$0.202082\pi$
$67$	43583.3i	1.18613i	−0.805154	+	0.593066i	$0.797918\pi$
$68$	36370.5i	0.953842i
$69$	−29682.5	−0.750548
$70$	0	0
$71$	51745.5	1.21822	0.609111	−	0.793085i	$-0.291526\pi$
$71$	51745.5	1.21822	0.609111	+	0.793085i	$0.291526\pi$
$72$	1072.03i	0.0243711i
$73$	13295.3i	0.292006i	0.989284	+	0.146003i	$0.0466410\pi$
$73$	13295.3i	0.292006i	−0.989284	+	0.146003i	$0.953359\pi$
$74$	−3817.47	−0.0810394
$75$	0	0
$76$	11685.3	0.232062
$77$	− 13729.4i	− 0.263890i
$78$	− 10168.1i	− 0.189236i
$79$	−76268.5	−1.37492	−0.687460	−	0.726223i	$-0.741274\pi$
$79$	−76268.5	−1.37492	−0.687460	+	0.726223i	$0.741274\pi$
$80$	0	0
$81$	−54698.1	−0.926317
$82$	69485.7i	1.14120i
$83$	− 30147.4i	− 0.480347i	−0.970730	−	0.240173i	$-0.922796\pi$
$83$	− 30147.4i	− 0.480347i	0.970730	−	0.240173i	$-0.0772043\pi$
$84$	−12745.3	−0.197084
$85$	0	0
$86$	40352.9	0.588340
$87$	− 14282.2i	− 0.202301i
$88$	− 16591.9i	− 0.228396i
$89$	66771.0	0.893537	0.446769	−	0.894650i	$-0.352575\pi$
$89$	66771.0	0.893537	0.446769	+	0.894650i	$0.352575\pi$
$90$	0	0
$91$	−8949.97	−0.113297
$92$	31573.8i	0.388918i
$93$	− 3389.34i	− 0.0406357i
$94$	−24279.6	−0.283415
$95$	0	0
$96$	−15402.6	−0.170575
$97$	151425.i	1.63406i	0.576592	+	0.817032i	$0.304382\pi$
$97$	151425.i	1.63406i	−0.576592	+	0.817032i	$0.695618\pi$
$98$	− 56009.6i	− 0.589112i
$99$	−4342.51	−0.0445300
$100$	0	0
$101$	−145198.	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$101$	−145198.	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$102$	136767.i	1.30161i
$103$	71524.1i	0.664293i	0.943228	+	0.332146i	$0.107773\pi$
$103$	71524.1i	0.664293i	−0.943228	+	0.332146i	$0.892227\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	64710.8	0.559387
$107$	93356.6i	0.788289i	0.919048	+	0.394145i	$0.128959\pi$
$107$	93356.6i	0.788289i	−0.919048	+	0.394145i	$0.871041\pi$
$108$	62513.0i	0.515716i
$109$	−102373.	−0.825315	−0.412658	−	0.910886i	$-0.635399\pi$
$109$	−102373.	−0.825315	−0.412658	+	0.910886i	$0.635399\pi$
$110$	0	0
$111$	−14355.2	−0.110586
$112$	13557.4i	0.102124i
$113$	− 132979.i	− 0.979685i	−0.871811	−	0.489843i	$-0.837054\pi$
$113$	− 132979.i	− 0.979685i	0.871811	−	0.489843i	$-0.162946\pi$
$114$	43941.3	0.316673
$115$	0	0
$116$	−15192.2	−0.104828
$117$	2830.82i	0.0191182i
$118$	1307.73i	0.00864596i
$119$	120383.	0.779285
$120$	0	0
$121$	−93841.6	−0.582682
$122$	187258.i	1.13904i
$123$	261294.i	1.55728i
$124$	−3605.30	−0.0210565
$125$	0	0
$126$	3548.30	0.0199111
$127$	53858.8i	0.296311i	0.988964	+	0.148155i	$0.0473336\pi$
$127$	53858.8i	0.296311i	−0.988964	+	0.148155i	$0.952666\pi$
$128$	16384.0i	0.0883883i
$129$	151743.	0.802850
$130$	0	0
$131$	125815.	0.640552	0.320276	−	0.947324i	$-0.396224\pi$
$131$	125815.	0.640552	0.320276	+	0.947324i	$0.396224\pi$
$132$	− 62392.0i	− 0.311669i
$133$	− 38677.1i	− 0.189594i
$134$	174333.	0.838721
$135$	0	0
$136$	145482.	0.674468
$137$	− 55609.4i	− 0.253132i	−0.991958	−	0.126566i	$-0.959604\pi$
$137$	− 55609.4i	− 0.253132i	0.991958	−	0.126566i	$-0.0403955\pi$
$138$	118730.i	0.530717i
$139$	200026.	0.878113	0.439056	−	0.898460i	$-0.355313\pi$
$139$	200026.	0.878113	0.439056	+	0.898460i	$0.355313\pi$
$140$	0	0
$141$	−91301.0	−0.386748
$142$	− 206982.i	− 0.861413i
$143$	− 43812.9i	− 0.179169i
$144$	4288.11	0.0172330
$145$	0	0
$146$	53181.4	0.206480
$147$	− 210619.i	− 0.803902i
$148$	15269.9i	0.0573035i
$149$	126007.	0.464974	0.232487	−	0.972599i	$-0.425314\pi$
$149$	126007.	0.464974	0.232487	+	0.972599i	$0.425314\pi$
$150$	0	0
$151$	13524.5	0.0482701	0.0241351	−	0.999709i	$-0.492317\pi$
$151$	13524.5	0.0482701	0.0241351	+	0.999709i	$0.492317\pi$
$152$	− 46741.1i	− 0.164093i
$153$	− 38076.3i	− 0.131500i
$154$	−54917.4	−0.186599
$155$	0	0
$156$	−40672.5	−0.133810
$157$	422943.i	1.36941i	0.728821	+	0.684704i	$0.240069\pi$
$157$	422943.i	1.36941i	−0.728821	+	0.684704i	$0.759931\pi$
$158$	305074.i	0.972215i
$159$	243338.	0.763340
$160$	0	0
$161$	104506.	0.317744
$162$	218792.i	0.655005i
$163$	− 668710.i	− 1.97137i	−0.168585	−	0.985687i	$-0.553920\pi$
$163$	− 668710.i	− 1.97137i	0.168585	−	0.985687i	$-0.446080\pi$
$164$	277943.	0.806948
$165$	0	0
$166$	−120590.	−0.339656
$167$	328083.i	0.910316i	0.890411	+	0.455158i	$0.150417\pi$
$167$	328083.i	0.910316i	−0.890411	+	0.455158i	$0.849583\pi$
$168$	50981.0i	0.139359i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−12233.3	−0.0319930
$172$	− 161412.i	− 0.416019i
$173$	− 62008.7i	− 0.157520i	−0.996894	−	0.0787602i	$-0.974904\pi$
$173$	− 62008.7i	− 0.157520i	0.996894	−	0.0787602i	$-0.0250962\pi$
$174$	−57128.9	−0.143048
$175$	0	0
$176$	−66367.4	−0.161500
$177$	4917.59i	0.0117983i
$178$	− 267084.i	− 0.631826i
$179$	−110227.	−0.257131	−0.128566	−	0.991701i	$-0.541037\pi$
$179$	−110227.	−0.257131	−0.128566	+	0.991701i	$0.541037\pi$
$180$	0	0
$181$	−371375.	−0.842589	−0.421295	−	0.906924i	$-0.638424\pi$
$181$	−371375.	−0.842589	−0.421295	+	0.906924i	$0.638424\pi$
$182$	35799.9i	0.0801130i
$183$	704164.i	1.55434i
$184$	126295.	0.275006
$185$	0	0
$186$	−13557.4	−0.0287338
$187$	589310.i	1.23237i
$188$	97118.5i	0.200404i
$189$	206912.	0.421338
$190$	0	0
$191$	−973641.	−1.93115	−0.965573	−	0.260131i	$-0.916234\pi$
$191$	−973641.	−1.93115	−0.965573	+	0.260131i	$0.916234\pi$
$192$	61610.4i	0.120615i
$193$	17937.3i	0.0346628i	0.999850	+	0.0173314i	$0.00551704\pi$
$193$	17937.3i	0.0346628i	−0.999850	+	0.0173314i	$0.994483\pi$
$194$	605701.	1.15546
$195$	0	0
$196$	−224039.	−0.416565
$197$	396284.i	0.727514i	0.931494	+	0.363757i	$0.118506\pi$
$197$	396284.i	0.727514i	−0.931494	+	0.363757i	$0.881494\pi$
$198$	17370.1i	0.0314875i
$199$	374690.	0.670718	0.335359	−	0.942090i	$-0.391142\pi$
$199$	374690.	0.670718	0.335359	+	0.942090i	$0.391142\pi$
$200$	0	0
$201$	655562.	1.14452
$202$	580794.i	1.00148i
$203$	50284.8i	0.0856439i
$204$	547070.	0.920380
$205$	0	0
$206$	286096.	0.469726
$207$	− 33054.7i	− 0.0536176i
$208$	43264.0i	0.0693375i
$209$	189336.	0.299825
$210$	0	0
$211$	666091.	1.02998	0.514988	−	0.857197i	$-0.327796\pi$
$211$	666091.	1.02998	0.514988	+	0.857197i	$0.327796\pi$
$212$	− 258843.i	− 0.395546i
$213$	− 778334.i	− 1.17549i
$214$	373426.	0.557405
$215$	0	0
$216$	250052.	0.364666
$217$	11933.2i	0.0172031i
$218$	409493.i	0.583586i
$219$	199983.	0.281762
$220$	0	0
$221$	384163.	0.529097
$222$	57420.8i	0.0781964i
$223$	840415.i	1.13170i	0.824508	+	0.565851i	$0.191452\pi$
$223$	840415.i	1.13170i	−0.824508	+	0.565851i	$0.808548\pi$
$224$	54229.4	0.0722129
$225$	0	0
$226$	−531915.	−0.692742
$227$	− 880953.i	− 1.13472i	−0.823470	−	0.567359i	$-0.807965\pi$
$227$	− 880953.i	− 1.13472i	0.823470	−	0.567359i	$-0.192035\pi$
$228$	− 175765.i	− 0.223921i
$229$	622076.	0.783889	0.391945	−	0.919989i	$-0.371802\pi$
$229$	622076.	0.783889	0.391945	+	0.919989i	$0.371802\pi$
$230$	0	0
$231$	−206511.	−0.254633
$232$	60768.9i	0.0741245i
$233$	1.09326e6i	1.31927i	0.751588	+	0.659633i	$0.229288\pi$
$233$	1.09326e6i	1.31927i	−0.751588	+	0.659633i	$0.770712\pi$
$234$	11323.3	0.0135186
$235$	0	0
$236$	5230.92	0.00611362
$237$	1.14720e6i	1.32669i
$238$	− 481531.i	− 0.551038i
$239$	−153202.	−0.173488	−0.0867442	−	0.996231i	$-0.527646\pi$
$239$	−153202.	−0.173488	−0.0867442	+	0.996231i	$0.527646\pi$
$240$	0	0
$241$	1.24027e6	1.37554	0.687771	−	0.725927i	$-0.258589\pi$
$241$	1.24027e6	1.37554	0.687771	+	0.725927i	$0.258589\pi$
$242$	375366.i	0.412019i
$243$	− 126670.i	− 0.137612i
$244$	749031.	0.805425
$245$	0	0
$246$	1.04518e6	1.10116
$247$	− 123426.i	− 0.128725i
$248$	14421.2i	0.0148892i
$249$	−453465.	−0.463495
$250$	0	0
$251$	−1.52653e6	−1.52940	−0.764702	−	0.644384i	$-0.777114\pi$
$251$	−1.52653e6	−1.52940	−0.764702	+	0.644384i	$0.777114\pi$
$252$	− 14193.2i	− 0.0140793i
$253$	511590.i	0.502483i
$254$	215435.	0.209523
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.27524e6i	1.20436i	0.798359	+	0.602182i	$0.205702\pi$
$257$	1.27524e6i	1.20436i	−0.798359	+	0.602182i	$0.794298\pi$
$258$	− 606972.i	− 0.567701i
$259$	50541.8	0.0468167
$260$	0	0
$261$	15904.8	0.0144520
$262$	− 503260.i	− 0.452938i
$263$	− 942046.i	− 0.839813i	−0.907567	−	0.419907i	$-0.862063\pi$
$263$	− 942046.i	− 0.839813i	0.907567	−	0.419907i	$-0.137937\pi$
$264$	−249568.	−0.220384
$265$	0	0
$266$	−154708.	−0.134063
$267$	− 1.00434e6i	− 0.862191i
$268$	− 697332.i	− 0.593066i
$269$	−806087.	−0.679205	−0.339603	−	0.940569i	$-0.610293\pi$
$269$	−806087.	−0.679205	−0.339603	+	0.940569i	$0.610293\pi$
$270$	0	0
$271$	600088.	0.496354	0.248177	−	0.968715i	$-0.420169\pi$
$271$	600088.	0.496354	0.248177	+	0.968715i	$0.420169\pi$
$272$	− 581927.i	− 0.476921i
$273$	134622.i	0.109322i
$274$	−222438.	−0.178991
$275$	0	0
$276$	474921.	0.375274
$277$	1.62719e6i	1.27421i	0.770779	+	0.637103i	$0.219867\pi$
$277$	1.62719e6i	1.27421i	−0.770779	+	0.637103i	$0.780133\pi$
$278$	− 800105.i	− 0.620919i
$279$	3774.39	0.00290293
$280$	0	0
$281$	−228378.	−0.172539	−0.0862697	−	0.996272i	$-0.527495\pi$
$281$	−228378.	−0.172539	−0.0862697	+	0.996272i	$0.527495\pi$
$282$	365204.i	0.273472i
$283$	1.27404e6i	0.945619i	0.881165	+	0.472809i	$0.156760\pi$
$283$	1.27404e6i	0.945619i	−0.881165	+	0.472809i	$0.843240\pi$
$284$	−827928.	−0.609111
$285$	0	0
$286$	−175252.	−0.126691
$287$	− 919963.i	− 0.659273i
$288$	− 17152.4i	− 0.0121855i
$289$	−3.74737e6	−2.63926
$290$	0	0
$291$	2.27768e6	1.57674
$292$	− 212725.i	− 0.146003i
$293$	2.84127e6i	1.93350i	0.255725	+	0.966749i	$0.417686\pi$
$293$	2.84127e6i	1.93350i	−0.255725	+	0.966749i	$0.582314\pi$
$294$	−842474.	−0.568445
$295$	0	0
$296$	61079.5	0.0405197
$297$	1.01290e6i	0.666307i
$298$	− 504028.i	− 0.328786i
$299$	333499.	0.215733
$300$	0	0
$301$	−534256.	−0.339886
$302$	− 54097.9i	− 0.0341321i
$303$	2.18402e6i	1.36662i
$304$	−186964.	−0.116031
$305$	0	0
$306$	−152305.	−0.0929847
$307$	− 452866.i	− 0.274235i	−0.990555	−	0.137118i	$-0.956216\pi$
$307$	− 452866.i	− 0.274235i	0.990555	−	0.137118i	$-0.0437839\pi$
$308$	219670.i	0.131945i
$309$	1.07584e6	0.640988
$310$	0	0
$311$	−1.43388e6	−0.840645	−0.420323	−	0.907375i	$-0.638083\pi$
$311$	−1.43388e6	−0.840645	−0.420323	+	0.907375i	$0.638083\pi$
$312$	162690.i	0.0946181i
$313$	165281.i	0.0953588i	0.998863	+	0.0476794i	$0.0151826\pi$
$313$	165281.i	0.0953588i	−0.998863	+	0.0476794i	$0.984817\pi$
$314$	1.69177e6	0.968318
$315$	0	0
$316$	1.22030e6	0.687460
$317$	− 1.53580e6i	− 0.858395i	−0.903211	−	0.429198i	$-0.858796\pi$
$317$	− 1.53580e6i	− 0.858395i	0.903211	−	0.429198i	$-0.141204\pi$
$318$	− 973354.i	− 0.539763i
$319$	−246160.	−0.135438
$320$	0	0
$321$	1.40423e6	0.760635
$322$	− 418025.i	− 0.224679i
$323$	1.66015e6i	0.885404i
$324$	875169.	0.463158
$325$	0	0
$326$	−2.67484e6	−1.39397
$327$	1.53986e6i	0.796362i
$328$	− 1.11177e6i	− 0.570599i
$329$	321452.	0.163730
$330$	0	0
$331$	−1.57759e6	−0.791449	−0.395725	−	0.918369i	$-0.629507\pi$
$331$	−1.57759e6	−0.791449	−0.395725	+	0.918369i	$0.629507\pi$
$332$	482358.i	0.240173i
$333$	− 15986.1i	− 0.00790007i
$334$	1.31233e6	0.643691
$335$	0	0
$336$	203924.	0.0985418
$337$	− 303917.i	− 0.145774i	−0.997340	−	0.0728869i	$-0.976779\pi$
$337$	− 303917.i	− 0.145774i	0.997340	−	0.0728869i	$-0.0232212\pi$
$338$	114244.i	0.0543928i
$339$	−2.00021e6	−0.945316
$340$	0	0
$341$	−58416.6	−0.0272051
$342$	48933.3i	0.0226224i
$343$	1.63162e6i	0.748829i
$344$	−645646.	−0.294170
$345$	0	0
$346$	−248035.	−0.111384
$347$	− 3.29844e6i	− 1.47057i	−0.677760	−	0.735283i	$-0.737049\pi$
$347$	− 3.29844e6i	− 1.47057i	0.677760	−	0.735283i	$-0.262951\pi$
$348$	228515.i	0.101150i
$349$	356002.	0.156455	0.0782273	−	0.996936i	$-0.475074\pi$
$349$	356002.	0.156455	0.0782273	+	0.996936i	$0.475074\pi$
$350$	0	0
$351$	660293.	0.286068
$352$	265470.i	0.114198i
$353$	− 3.72901e6i	− 1.59279i	−0.604780	−	0.796393i	$-0.706739\pi$
$353$	− 3.72901e6i	− 1.59279i	0.604780	−	0.796393i	$-0.293261\pi$
$354$	19670.3	0.00834265
$355$	0	0
$356$	−1.06834e6	−0.446769
$357$	− 1.81075e6i	− 0.751947i
$358$	440907.i	0.181819i
$359$	2.27503e6	0.931647	0.465824	−	0.884878i	$-0.345758\pi$
$359$	2.27503e6	0.931647	0.465824	+	0.884878i	$0.345758\pi$
$360$	0	0
$361$	−1.94272e6	−0.784588
$362$	1.48550e6i	0.595801i
$363$	1.41153e6i	0.562241i
$364$	143200.	0.0566485
$365$	0	0
$366$	2.81665e6	1.09908
$367$	1.21410e6i	0.470534i	0.971931	+	0.235267i	$0.0755964\pi$
$367$	1.21410e6i	0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$368$	− 505181.i	− 0.194459i
$369$	−290979.	−0.111249
$370$	0	0
$371$	−856745.	−0.323160
$372$	54229.4i	0.0203178i
$373$	1.56733e6i	0.583294i	0.956526	+	0.291647i	$0.0942033\pi$
$373$	1.56733e6i	0.583294i	−0.956526	+	0.291647i	$0.905797\pi$
$374$	2.35724e6	0.871415
$375$	0	0
$376$	388474.	0.141707
$377$	160468.i	0.0581480i
$378$	− 827647.i	− 0.297931i
$379$	−4.40321e6	−1.57461	−0.787303	−	0.616567i	$-0.788523\pi$
$379$	−4.40321e6	−1.57461	−0.787303	+	0.616567i	$0.788523\pi$
$380$	0	0
$381$	810122.	0.285916
$382$	3.89456e6i	1.36553i
$383$	3.45596e6i	1.20385i	0.798554	+	0.601924i	$0.205599\pi$
$383$	3.45596e6i	1.20385i	−0.798554	+	0.601924i	$0.794401\pi$
$384$	246441.	0.0852876
$385$	0	0
$386$	71749.3	0.0245103
$387$	168982.i	0.0573539i
$388$	− 2.42280e6i	− 0.817032i
$389$	−556620.	−0.186503	−0.0932513	−	0.995643i	$-0.529726\pi$
$389$	−556620.	−0.186503	−0.0932513	+	0.995643i	$0.529726\pi$
$390$	0	0
$391$	−4.48576e6	−1.48386
$392$	896154.i	0.294556i
$393$	− 1.89246e6i	− 0.618080i
$394$	1.58514e6	0.514430
$395$	0	0
$396$	69480.2	0.0222650
$397$	− 225504.i	− 0.0718089i	−0.999355	−	0.0359045i	$-0.988569\pi$
$397$	− 225504.i	− 0.0718089i	0.999355	−	0.0359045i	$-0.0114312\pi$
$398$	− 1.49876e6i	− 0.474269i
$399$	−581765.	−0.182943
$400$	0	0
$401$	898293.	0.278970	0.139485	−	0.990224i	$-0.455455\pi$
$401$	898293.	0.278970	0.139485	+	0.990224i	$0.455455\pi$
$402$	− 2.62225e6i	− 0.809298i
$403$	38080.9i	0.0116801i
$404$	2.32317e6	0.708155
$405$	0	0
$406$	201139.	0.0605594
$407$	247418.i	0.0740363i
$408$	− 2.18828e6i	− 0.650807i
$409$	1.77186e6	0.523746	0.261873	−	0.965102i	$-0.415660\pi$
$409$	1.77186e6	0.523746	0.261873	+	0.965102i	$0.415660\pi$
$410$	0	0
$411$	−836454.	−0.244252
$412$	− 1.14439e6i	− 0.332146i
$413$	− 17313.8i	− 0.00499480i
$414$	−132219.	−0.0379134
$415$	0	0
$416$	173056.	0.0490290
$417$	− 3.00871e6i	− 0.847307i
$418$	− 757345.i	− 0.212008i
$419$	−617010.	−0.171695	−0.0858474	−	0.996308i	$-0.527360\pi$
$419$	−617010.	−0.171695	−0.0858474	+	0.996308i	$0.527360\pi$
$420$	0	0
$421$	−6.27988e6	−1.72682	−0.863409	−	0.504505i	$-0.831675\pi$
$421$	−6.27988e6	−1.72682	−0.863409	+	0.504505i	$0.831675\pi$
$422$	− 2.66436e6i	− 0.728303i
$423$	− 101674.i	− 0.0276285i
$424$	−1.03537e6	−0.279693
$425$	0	0
$426$	−3.11334e6	−0.831194
$427$	− 2.47922e6i	− 0.658029i
$428$	− 1.49371e6i	− 0.394145i
$429$	−659016.	−0.172883
$430$	0	0
$431$	2.03052e6	0.526519	0.263259	−	0.964725i	$-0.415202\pi$
$431$	2.03052e6	0.526519	0.263259	+	0.964725i	$0.415202\pi$
$432$	− 1.00021e6i	− 0.257858i
$433$	1.29410e6i	0.331702i	0.986151	+	0.165851i	$0.0530371\pi$
$433$	1.29410e6i	0.331702i	−0.986151	+	0.165851i	$0.946963\pi$
$434$	47732.7	0.0121644
$435$	0	0
$436$	1.63797e6	0.412658
$437$	1.44121e6i	0.361013i
$438$	− 799933.i	− 0.199236i
$439$	2.53819e6	0.628584	0.314292	−	0.949326i	$-0.398233\pi$
$439$	2.53819e6	0.628584	0.314292	+	0.949326i	$0.398233\pi$
$440$	0	0
$441$	234546.	0.0574291
$442$	− 1.53665e6i	− 0.374128i
$443$	− 6.27678e6i	− 1.51959i	−0.650160	−	0.759797i	$-0.725298\pi$
$443$	− 6.27678e6i	− 1.51959i	0.650160	−	0.759797i	$-0.274702\pi$
$444$	229683.	0.0552932
$445$	0	0
$446$	3.36166e6	0.800234
$447$	− 1.89535e6i	− 0.448662i
$448$	− 216918.i	− 0.0510622i
$449$	1.91186e6	0.447549	0.223775	−	0.974641i	$-0.428162\pi$
$449$	1.91186e6	0.447549	0.223775	+	0.974641i	$0.428162\pi$
$450$	0	0
$451$	4.50350e6	1.04258
$452$	2.12766e6i	0.489843i
$453$	− 203430.i	− 0.0465767i
$454$	−3.52381e6	−0.802367
$455$	0	0
$456$	−703060.	−0.158336
$457$	7.83658e6i	1.75524i	0.479360	+	0.877619i	$0.340869\pi$
$457$	7.83658e6i	1.75524i	−0.479360	+	0.877619i	$0.659131\pi$
$458$	− 2.48830e6i	− 0.554294i
$459$	−8.88135e6	−1.96765
$460$	0	0
$461$	−3.95695e6	−0.867178	−0.433589	−	0.901111i	$-0.642753\pi$
$461$	−3.95695e6	−0.867178	−0.433589	+	0.901111i	$0.642753\pi$
$462$	826045.i	0.180052i
$463$	− 2.70615e6i	− 0.586677i	−0.956009	−	0.293338i	$-0.905234\pi$
$463$	− 2.70615e6i	− 0.586677i	0.956009	−	0.293338i	$-0.0947663\pi$
$464$	243076.	0.0524139
$465$	0	0
$466$	4.37303e6	0.932862
$467$	− 5.13059e6i	− 1.08862i	−0.838885	−	0.544308i	$-0.816792\pi$
$467$	− 5.13059e6i	− 1.08862i	0.838885	−	0.544308i	$-0.183208\pi$
$468$	− 45293.2i	− 0.00955912i
$469$	−2.30810e6	−0.484532
$470$	0	0
$471$	6.36174e6	1.32137
$472$	− 20923.7i	− 0.00432298i
$473$	− 2.61535e6i	− 0.537498i
$474$	4.58880e6	0.938108
$475$	0	0
$476$	−1.92612e6	−0.389643
$477$	270984.i	0.0545314i
$478$	612809.i	0.122675i
$479$	−2.22034e6	−0.442162	−0.221081	−	0.975255i	$-0.570958\pi$
$479$	−2.22034e6	−0.442162	−0.221081	+	0.975255i	$0.570958\pi$
$480$	0	0
$481$	161288.	0.0317863
$482$	− 4.96108e6i	− 0.972656i
$483$	− 1.57194e6i	− 0.306597i
$484$	1.50146e6	0.291341
$485$	0	0
$486$	−506678.	−0.0973065
$487$	− 5.01311e6i	− 0.957822i	−0.877863	−	0.478911i	$-0.841032\pi$
$487$	− 5.01311e6i	− 0.957822i	0.877863	−	0.478911i	$-0.158968\pi$
$488$	− 2.99612e6i	− 0.569522i
$489$	−1.00585e7	−1.90222
$490$	0	0
$491$	−3.79096e6	−0.709652	−0.354826	−	0.934932i	$-0.615460\pi$
$491$	−3.79096e6	−0.709652	−0.354826	+	0.934932i	$0.615460\pi$
$492$	− 4.18070e6i	− 0.778639i
$493$	− 2.15839e6i	− 0.399957i
$494$	−493703.	−0.0910224
$495$	0	0
$496$	57684.7	0.0105283
$497$	2.74036e6i	0.497641i
$498$	1.81386e6i	0.327741i
$499$	−7.59448e6	−1.36536	−0.682679	−	0.730718i	$-0.739185\pi$
$499$	−7.59448e6	−1.36536	−0.682679	+	0.730718i	$0.739185\pi$
$500$	0	0
$501$	4.93489e6	0.878381
$502$	6.10614e6i	1.08145i
$503$	− 2.64250e6i	− 0.465688i	−0.972514	−	0.232844i	$-0.925197\pi$
$503$	− 2.64250e6i	− 0.465688i	0.972514	−	0.232844i	$-0.0748032\pi$
$504$	−56772.9	−0.00995553
$505$	0	0
$506$	2.04636e6	0.355309
$507$	429603.i	0.0742245i
$508$	− 861741.i	− 0.148155i
$509$	−9.84097e6	−1.68362	−0.841809	−	0.539776i	$-0.818509\pi$
$509$	−9.84097e6	−1.68362	−0.841809	+	0.539776i	$0.818509\pi$
$510$	0	0
$511$	−704100.	−0.119284
$512$	− 262144.i	− 0.0441942i
$513$	2.85344e6i	0.478713i
$514$	5.10095e6	0.851614
$515$	0	0
$516$	−2.42789e6	−0.401425
$517$	1.57361e6i	0.258923i
$518$	− 202167.i	− 0.0331044i
$519$	−932709.	−0.151994
$520$	0	0
$521$	994770.	0.160557	0.0802783	−	0.996772i	$-0.474419\pi$
$521$	994770.	0.160557	0.0802783	+	0.996772i	$0.474419\pi$
$522$	− 63619.1i	− 0.0102191i
$523$	− 5.82741e6i	− 0.931582i	−0.884895	−	0.465791i	$-0.845770\pi$
$523$	− 5.82741e6i	− 0.931582i	0.884895	−	0.465791i	$-0.154230\pi$
$524$	−2.01304e6	−0.320276
$525$	0	0
$526$	−3.76818e6	−0.593838
$527$	− 512212.i	− 0.0803384i
$528$	998272.i	0.155835i
$529$	2.54218e6	0.394972
$530$	0	0
$531$	−5476.26	−0.000842845 0
$532$	618833.i	0.0947970i
$533$	− 2.93577e6i	− 0.447614i
$534$	−4.01737e6	−0.609661
$535$	0	0
$536$	−2.78933e6	−0.419361
$537$	1.65799e6i	0.248111i
$538$	3.22435e6i	0.480271i
$539$	−3.63009e6	−0.538203
$540$	0	0
$541$	−4.57878e6	−0.672599	−0.336300	−	0.941755i	$-0.609175\pi$
$541$	−4.57878e6	−0.672599	−0.336300	+	0.941755i	$0.609175\pi$
$542$	− 2.40035e6i	− 0.350975i
$543$	5.58607e6i	0.813030i
$544$	−2.32771e6	−0.337234
$545$	0	0
$546$	538487.	0.0773026
$547$	− 7.15998e6i	− 1.02316i	−0.859236	−	0.511580i	$-0.829060\pi$
$547$	− 7.15998e6i	− 1.02316i	0.859236	−	0.511580i	$-0.170940\pi$
$548$	889751.i	0.126566i
$549$	−784162.	−0.111039
$550$	0	0
$551$	−693458.	−0.0973064
$552$	− 1.89968e6i	− 0.265359i
$553$	− 4.03906e6i	− 0.561652i
$554$	6.50877e6	0.901000
$555$	0	0
$556$	−3.20042e6	−0.439056
$557$	− 2.24322e6i	− 0.306361i	−0.988198	−	0.153180i	$-0.951048\pi$
$557$	− 2.24322e6i	− 0.306361i	0.988198	−	0.153180i	$-0.0489516\pi$
$558$	− 15097.6i	− 0.00205268i
$559$	−1.70491e6	−0.230766
$560$	0	0
$561$	8.86417e6	1.18913
$562$	913512.i	0.122004i
$563$	− 9.93603e6i	− 1.32112i	−0.750774	−	0.660560i	$-0.770319\pi$
$563$	− 9.93603e6i	− 1.32112i	0.750774	−	0.660560i	$-0.229681\pi$
$564$	1.46082e6	0.193374
$565$	0	0
$566$	5.09615e6	0.668653
$567$	− 2.89672e6i	− 0.378398i
$568$	3.31171e6i	0.430707i
$569$	1.17630e7	1.52313	0.761567	−	0.648086i	$-0.224430\pi$
$569$	1.17630e7	1.52313	0.761567	+	0.648086i	$0.224430\pi$
$570$	0	0
$571$	−9.88308e6	−1.26853	−0.634267	−	0.773114i	$-0.718698\pi$
$571$	−9.88308e6	−1.26853	−0.634267	+	0.773114i	$0.718698\pi$
$572$	701006.i	0.0895843i
$573$	1.46451e7i	1.86340i
$574$	−3.67985e6	−0.466177
$575$	0	0
$576$	−68609.8	−0.00861648
$577$	− 6.86460e6i	− 0.858372i	−0.903216	−	0.429186i	$-0.858800\pi$
$577$	− 6.86460e6i	− 0.858372i	0.903216	−	0.429186i	$-0.141200\pi$
$578$	1.49895e7i	1.86624i
$579$	269806.	0.0334468
$580$	0	0
$581$	1.59656e6	0.196221
$582$	− 9.11071e6i	− 1.11492i
$583$	− 4.19403e6i	− 0.511047i
$584$	−850902.	−0.103240
$585$	0	0
$586$	1.13651e7	1.36719
$587$	− 3.09692e6i	− 0.370966i	−0.982647	−	0.185483i	$-0.940615\pi$
$587$	− 3.09692e6i	− 0.370966i	0.982647	−	0.185483i	$-0.0593850\pi$
$588$	3.36990e6i	0.401951i
$589$	−164566.	−0.0195457
$590$	0	0
$591$	5.96075e6	0.701992
$592$	− 244318.i	− 0.0286518i
$593$	5.60192e6i	0.654184i	0.944992	+	0.327092i	$0.106069\pi$
$593$	5.60192e6i	0.654184i	−0.944992	+	0.327092i	$0.893931\pi$
$594$	4.05159e6	0.471150
$595$	0	0
$596$	−2.01611e6	−0.232487
$597$	− 5.63594e6i	− 0.647188i
$598$	− 1.33399e6i	− 0.152546i
$599$	−1.17534e7	−1.33843	−0.669214	−	0.743070i	$-0.733369\pi$
$599$	−1.17534e7	−1.33843	−0.669214	+	0.743070i	$0.733369\pi$
$600$	0	0
$601$	−8.93997e6	−1.00960	−0.504800	−	0.863236i	$-0.668434\pi$
$601$	−8.93997e6	−1.00960	−0.504800	+	0.863236i	$0.668434\pi$
$602$	2.13702e6i	0.240336i
$603$	730038.i	0.0817622i
$604$	−216392.	−0.0241351
$605$	0	0
$606$	8.73606e6	0.966349
$607$	− 1.73574e7i	− 1.91211i	−0.293194	−	0.956053i	$-0.594718\pi$
$607$	− 1.73574e7i	− 1.91211i	0.293194	−	0.956053i	$-0.405282\pi$
$608$	747857.i	0.0820464i
$609$	756363.	0.0826394
$610$	0	0
$611$	1.02581e6	0.111164
$612$	609221.i	0.0657501i
$613$	1.63664e7i	1.75915i	0.475760	+	0.879575i	$0.342173\pi$
$613$	1.63664e7i	1.75915i	−0.475760	+	0.879575i	$0.657827\pi$
$614$	−1.81146e6	−0.193914
$615$	0	0
$616$	878679.	0.0932993
$617$	− 6.81573e6i	− 0.720775i	−0.932803	−	0.360387i	$-0.882645\pi$
$617$	− 6.81573e6i	− 0.720775i	0.932803	−	0.360387i	$-0.117355\pi$
$618$	− 4.30335e6i	− 0.453247i
$619$	3.32032e6	0.348300	0.174150	−	0.984719i	$-0.444282\pi$
$619$	3.32032e6	0.348300	0.174150	+	0.984719i	$0.444282\pi$
$620$	0	0
$621$	−7.71005e6	−0.802284
$622$	5.73553e6i	0.594426i
$623$	3.53608e6i	0.365008i
$624$	650760.	0.0669051
$625$	0	0
$626$	661122.	0.0674289
$627$	− 2.84792e6i	− 0.289307i
$628$	− 6.76709e6i	− 0.684704i
$629$	−2.16942e6	−0.218634
$630$	0	0
$631$	−1.25729e7	−1.25708	−0.628541	−	0.777777i	$-0.716347\pi$
$631$	−1.25729e7	−1.25708	−0.628541	+	0.777777i	$0.716347\pi$
$632$	− 4.88118e6i	− 0.486107i
$633$	− 1.00191e7i	− 0.993844i
$634$	−6.14321e6	−0.606977
$635$	0	0
$636$	−3.89341e6	−0.381670
$637$	2.36641e6i	0.231069i
$638$	984639.i	0.0957690i
$639$	866759.	0.0839743
$640$	0	0
$641$	5.56432e6	0.534893	0.267446	−	0.963573i	$-0.413820\pi$
$641$	5.56432e6	0.534893	0.267446	+	0.963573i	$0.413820\pi$
$642$	− 5.61693e6i	− 0.537850i
$643$	1.05294e7i	1.00433i	0.864772	+	0.502165i	$0.167463\pi$
$643$	1.05294e7i	1.00433i	−0.864772	+	0.502165i	$0.832537\pi$
$644$	−1.67210e6	−0.158872
$645$	0	0
$646$	6.64060e6	0.626075
$647$	1.29138e7i	1.21281i	0.795155	+	0.606406i	$0.207389\pi$
$647$	1.29138e7i	1.21281i	−0.795155	+	0.606406i	$0.792611\pi$
$648$	− 3.50068e6i	− 0.327502i
$649$	84756.6	0.00789881
$650$	0	0
$651$	179494.	0.0165996
$652$	1.06994e7i	0.985687i
$653$	1.36537e7i	1.25305i	0.779403	+	0.626523i	$0.215523\pi$
$653$	1.36537e7i	1.25305i	−0.779403	+	0.626523i	$0.784477\pi$
$654$	6.15942e6	0.563113
$655$	0	0
$656$	−4.44708e6	−0.403474
$657$	222703.i	0.0201285i
$658$	− 1.28581e6i	− 0.115774i
$659$	612348.	0.0549268	0.0274634	−	0.999623i	$-0.491257\pi$
$659$	612348.	0.0549268	0.0274634	+	0.999623i	$0.491257\pi$
$660$	0	0
$661$	1.10774e7	0.986129	0.493064	−	0.869993i	$-0.335877\pi$
$661$	1.10774e7	0.986129	0.493064	+	0.869993i	$0.335877\pi$
$662$	6.31034e6i	0.559639i
$663$	− 5.77843e6i	− 0.510535i
$664$	1.92943e6	0.169828
$665$	0	0
$666$	−63944.3	−0.00558619
$667$	− 1.87374e6i	− 0.163078i
$668$	− 5.24932e6i	− 0.455158i
$669$	1.26412e7	1.09200
$670$	0	0
$671$	1.21365e7	1.04061
$672$	− 815697.i	− 0.0696796i
$673$	− 1.39729e6i	− 0.118918i	−0.998231	−	0.0594590i	$-0.981062\pi$
$673$	− 1.39729e6i	− 0.118918i	0.998231	−	0.0594590i	$-0.0189376\pi$
$674$	−1.21567e6	−0.103078
$675$	0	0
$676$	456976.	0.0384615
$677$	1.49879e7i	1.25681i	0.777886	+	0.628406i	$0.216292\pi$
$677$	1.49879e7i	1.25681i	−0.777886	+	0.628406i	$0.783708\pi$
$678$	8.00086e6i	0.668440i
$679$	−8.01924e6	−0.667512
$680$	0	0
$681$	−1.32509e7	−1.09491
$682$	233666.i	0.0192369i
$683$	5.44672e6i	0.446769i	0.974730	+	0.223384i	$0.0717105\pi$
$683$	5.44672e6i	0.446769i	−0.974730	+	0.223384i	$0.928289\pi$
$684$	195733.	0.0159965
$685$	0	0
$686$	6.52647e6	0.529502
$687$	− 9.35702e6i	− 0.756390i
$688$	2.58259e6i	0.208010i
$689$	−2.73403e6	−0.219410
$690$	0	0
$691$	1.20363e7	0.958951	0.479475	−	0.877555i	$-0.340827\pi$
$691$	1.20363e7	0.958951	0.479475	+	0.877555i	$0.340827\pi$
$692$	992138.i	0.0787602i
$693$	− 229973.i	− 0.0181904i
$694$	−1.31937e7	−1.03985
$695$	0	0
$696$	914062.	0.0715241
$697$	3.94879e7i	3.07881i
$698$	− 1.42401e6i	− 0.110630i
$699$	1.64443e7	1.27299
$700$	0	0
$701$	793102.	0.0609585	0.0304792	−	0.999535i	$-0.490297\pi$
$701$	793102.	0.0609585	0.0304792	+	0.999535i	$0.490297\pi$
$702$	− 2.64117e6i	− 0.202281i
$703$	697002.i	0.0531920i
$704$	1.06188e6	0.0807502
$705$	0	0
$706$	−1.49161e7	−1.12627
$707$	− 7.68947e6i	− 0.578560i
$708$	− 78681.4i	− 0.00589914i
$709$	1.94248e7	1.45125	0.725623	−	0.688092i	$-0.241552\pi$
$709$	1.94248e7	1.45125	0.725623	+	0.688092i	$0.241552\pi$
$710$	0	0
$711$	−1.27753e6	−0.0947757
$712$	4.27334e6i	0.315913i
$713$	− 444660.i	− 0.0327570i
$714$	−7.24299e6	−0.531707
$715$	0	0
$716$	1.76363e6	0.128566
$717$	2.30441e6i	0.167402i
$718$	− 9.10013e6i	− 0.658774i
$719$	5.30618e6	0.382789	0.191395	−	0.981513i	$-0.438699\pi$
$719$	5.30618e6	0.382789	0.191395	+	0.981513i	$0.438699\pi$
$720$	0	0
$721$	−3.78780e6	−0.271362
$722$	7.77087e6i	0.554788i
$723$	− 1.86557e7i	− 1.32729i
$724$	5.94200e6	0.421295
$725$	0	0
$726$	5.64611e6	0.397564
$727$	− 2.30593e7i	− 1.61812i	−0.587728	−	0.809059i	$-0.699977\pi$
$727$	− 2.30593e7i	− 1.61812i	0.587728	−	0.809059i	$-0.300023\pi$
$728$	− 572798.i	− 0.0400565i
$729$	−1.51969e7	−1.05910
$730$	0	0
$731$	2.29321e7	1.58727
$732$	− 1.12666e7i	− 0.777170i
$733$	− 1.84481e7i	− 1.26821i	−0.773248	−	0.634104i	$-0.781369\pi$
$733$	− 1.84481e7i	− 1.26821i	0.773248	−	0.634104i	$-0.218631\pi$
$734$	4.85641e6	0.332717
$735$	0	0
$736$	−2.02072e6	−0.137503
$737$	− 1.12989e7i	− 0.766242i
$738$	1.16392e6i	0.0786648i
$739$	6.14417e6	0.413859	0.206929	−	0.978356i	$-0.433653\pi$
$739$	6.14417e6	0.413859	0.206929	+	0.978356i	$0.433653\pi$
$740$	0	0
$741$	−1.85652e6	−0.124209
$742$	3.42698e6i	0.228508i
$743$	1.18784e7i	0.789383i	0.918814	+	0.394691i	$0.129148\pi$
$743$	1.18784e7i	0.789383i	−0.918814	+	0.394691i	$0.870852\pi$
$744$	216918.	0.0143669
$745$	0	0
$746$	6.26931e6	0.412451
$747$	− 504982.i	− 0.0331112i
$748$	− 9.42897e6i	− 0.616183i
$749$	−4.94402e6	−0.322014
$750$	0	0
$751$	−3.38234e6	−0.218835	−0.109418	−	0.993996i	$-0.534899\pi$
$751$	−3.38234e6	−0.218835	−0.109418	+	0.993996i	$0.534899\pi$
$752$	− 1.55390e6i	− 0.100202i
$753$	2.29615e7i	1.47575i
$754$	641872.	0.0411169
$755$	0	0
$756$	−3.31059e6	−0.210669
$757$	− 2.42823e6i	− 0.154010i	−0.997031	−	0.0770052i	$-0.975464\pi$
$757$	− 2.42823e6i	− 0.154010i	0.997031	−	0.0770052i	$-0.0245358\pi$
$758$	1.76129e7i	1.11341i
$759$	7.69513e6	0.484855
$760$	0	0
$761$	−2.18931e7	−1.37040	−0.685199	−	0.728356i	$-0.740285\pi$
$761$	−2.18931e7	−1.37040	−0.685199	+	0.728356i	$0.740285\pi$
$762$	− 3.24049e6i	− 0.202173i
$763$	− 5.42152e6i	− 0.337139i
$764$	1.55782e7	0.965573
$765$	0	0
$766$	1.38238e7	0.851249
$767$	− 55251.6i	− 0.00339122i
$768$	− 985766.i	− 0.0603074i
$769$	1.96376e7	1.19749	0.598747	−	0.800939i	$-0.295666\pi$
$769$	1.96376e7	1.19749	0.598747	+	0.800939i	$0.295666\pi$
$770$	0	0
$771$	1.91816e7	1.16211
$772$	− 286997.i	− 0.0173314i
$773$	− 1.21015e7i	− 0.728433i	−0.931314	−	0.364216i	$-0.881337\pi$
$773$	− 1.21015e7i	− 0.728433i	0.931314	−	0.364216i	$-0.118663\pi$
$774$	675928.	0.0405554
$775$	0	0
$776$	−9.69122e6	−0.577729
$777$	− 760229.i	− 0.0451743i
$778$	2.22648e6i	0.131877i
$779$	1.26869e7	0.749049
$780$	0	0
$781$	−1.34149e7	−0.786973
$782$	1.79430e7i	1.04925i
$783$	− 3.70981e6i	− 0.216246i
$784$	3.58462e6	0.208282
$785$	0	0
$786$	−7.56983e6	−0.437049
$787$	2.30778e7i	1.32818i	0.747651	+	0.664092i	$0.231182\pi$
$787$	2.30778e7i	1.32818i	−0.747651	+	0.664092i	$0.768818\pi$
$788$	− 6.34055e6i	− 0.363757i
$789$	−1.41699e7	−0.810351
$790$	0	0
$791$	7.04235e6	0.400199
$792$	− 277921.i	− 0.0157438i
$793$	− 7.91164e6i	− 0.446769i
$794$	−902017.	−0.0507766
$795$	0	0
$796$	−5.99505e6	−0.335359
$797$	8.88351e6i	0.495380i	0.968839	+	0.247690i	$0.0796715\pi$
$797$	8.88351e6i	0.495380i	−0.968839	+	0.247690i	$0.920328\pi$
$798$	2.32706e6i	0.129360i
$799$	−1.37978e7	−0.764617
$800$	0	0
$801$	1.11844e6	0.0615932
$802$	− 3.59317e6i	− 0.197261i
$803$	− 3.44679e6i	− 0.188637i
$804$	−1.04890e7	−0.572260
$805$	0	0
$806$	152324.	0.00825905
$807$	1.21248e7i	0.655378i
$808$	− 9.29270e6i	− 0.500741i
$809$	1.27959e7	0.687382	0.343691	−	0.939083i	$-0.388323\pi$
$809$	1.27959e7	0.687382	0.343691	+	0.939083i	$0.388323\pi$
$810$	0	0
$811$	3.55180e6	0.189626	0.0948128	−	0.995495i	$-0.469775\pi$
$811$	3.55180e6	0.189626	0.0948128	+	0.995495i	$0.469775\pi$
$812$	− 804557.i	− 0.0428220i
$813$	− 9.02627e6i	− 0.478941i
$814$	989671.	0.0523516
$815$	0	0
$816$	−8.75312e6	−0.460190
$817$	− 7.36773e6i	− 0.386170i
$818$	− 7.08744e6i	− 0.370345i
$819$	−149916.	−0.00780976
$820$	0	0
$821$	−4.71136e6	−0.243943	−0.121971	−	0.992534i	$-0.538922\pi$
$821$	−4.71136e6	−0.243943	−0.121971	+	0.992534i	$0.538922\pi$
$822$	3.34582e6i	0.172712i
$823$	4.15156e6i	0.213654i	0.994278	+	0.106827i	$0.0340691\pi$
$823$	4.15156e6i	0.213654i	−0.994278	+	0.106827i	$0.965931\pi$
$824$	−4.57754e6	−0.234863
$825$	0	0
$826$	−69255.3	−0.00353186
$827$	2.57396e7i	1.30869i	0.756195	+	0.654346i	$0.227056\pi$
$827$	2.57396e7i	1.30869i	−0.756195	+	0.654346i	$0.772944\pi$
$828$	528875.i	0.0268088i
$829$	1.95125e7	0.986114	0.493057	−	0.869997i	$-0.335879\pi$
$829$	1.95125e7	0.986114	0.493057	+	0.869997i	$0.335879\pi$
$830$	0	0
$831$	2.44756e7	1.22951
$832$	− 692224.i	− 0.0346688i
$833$	− 3.18296e7i	− 1.58935i
$834$	−1.20349e7	−0.599137
$835$	0	0
$836$	−3.02938e6	−0.149913
$837$	− 880382.i	− 0.0434368i
$838$	2.46804e6i	0.121407i
$839$	1.56844e7	0.769243	0.384621	−	0.923074i	$-0.374332\pi$
$839$	1.56844e7	0.769243	0.384621	+	0.923074i	$0.374332\pi$
$840$	0	0
$841$	−1.96096e7	−0.956044
$842$	2.51195e7i	1.22104i
$843$	3.43517e6i	0.166487i
$844$	−1.06575e7	−0.514988
$845$	0	0
$846$	−406694.	−0.0195363
$847$	− 4.96970e6i	− 0.238024i
$848$	4.14149e6i	0.197773i
$849$	1.91635e7	0.912445
$850$	0	0
$851$	−1.88331e6	−0.0891454
$852$	1.24534e7i	0.587743i
$853$	2.35674e7i	1.10902i	0.832177	+	0.554510i	$0.187094\pi$
$853$	2.35674e7i	1.10902i	−0.832177	+	0.554510i	$0.812906\pi$
$854$	−9.91687e6	−0.465297
$855$	0	0
$856$	−5.97482e6	−0.278702
$857$	4.23900e6i	0.197157i	0.995129	+	0.0985783i	$0.0314295\pi$
$857$	4.23900e6i	0.197157i	−0.995129	+	0.0985783i	$0.968571\pi$
$858$	2.63606e6i	0.122247i
$859$	−1.30196e7	−0.602026	−0.301013	−	0.953620i	$-0.597325\pi$
$859$	−1.30196e7	−0.602026	−0.301013	+	0.953620i	$0.597325\pi$
$860$	0	0
$861$	−1.38377e7	−0.636145
$862$	− 8.12208e6i	− 0.372305i
$863$	− 3.41837e7i	− 1.56240i	−0.624280	−	0.781200i	$-0.714608\pi$
$863$	− 3.41837e7i	− 1.56240i	0.624280	−	0.781200i	$-0.285392\pi$
$864$	−4.00083e6	−0.182333
$865$	0	0
$866$	5.17640e6	0.234549
$867$	5.63665e7i	2.54667i
$868$	− 190931.i	− 0.00860155i
$869$	1.97724e7	0.888200
$870$	0	0
$871$	−7.36557e6	−0.328974
$872$	− 6.55188e6i	− 0.291793i
$873$	2.53644e6i	0.112639i
$874$	5.76482e6	0.255274
$875$	0	0
$876$	−3.19973e6	−0.140881
$877$	3.07617e7i	1.35055i	0.737566	+	0.675275i	$0.235975\pi$
$877$	3.07617e7i	1.35055i	−0.737566	+	0.675275i	$0.764025\pi$
$878$	− 1.01528e7i	− 0.444476i
$879$	4.27373e7	1.86567
$880$	0	0
$881$	8.83910e6	0.383679	0.191840	−	0.981426i	$-0.438555\pi$
$881$	8.83910e6	0.383679	0.191840	+	0.981426i	$0.438555\pi$
$882$	− 938186.i	− 0.0406085i
$883$	2.09708e7i	0.905136i	0.891730	+	0.452568i	$0.149492\pi$
$883$	2.09708e7i	0.905136i	−0.891730	+	0.452568i	$0.850508\pi$
$884$	−6.14661e6	−0.264548
$885$	0	0
$886$	−2.51071e7	−1.07452
$887$	3.98177e7i	1.69929i	0.527357	+	0.849644i	$0.323183\pi$
$887$	3.98177e7i	1.69929i	−0.527357	+	0.849644i	$0.676817\pi$
$888$	− 918733.i	− 0.0390982i
$889$	−2.85228e6	−0.121042
$890$	0	0
$891$	1.41804e7	0.598402
$892$	− 1.34466e7i	− 0.565851i
$893$	4.43303e6i	0.186025i
$894$	−7.58138e6	−0.317252
$895$	0	0
$896$	−867671.	−0.0361065
$897$	− 5.01635e6i	− 0.208165i
$898$	− 7.64745e6i	− 0.316465i
$899$	213955.	0.00882924
$900$	0	0
$901$	3.67744e7	1.50915
$902$	− 1.80140e7i	− 0.737215i
$903$	8.03607e6i	0.327962i
$904$	8.51065e6	0.346371
$905$	0	0
$906$	−813719.	−0.0329347
$907$	3.42787e6i	0.138358i	0.997604	+	0.0691792i	$0.0220380\pi$
$907$	3.42787e6i	0.138358i	−0.997604	+	0.0691792i	$0.977962\pi$
$908$	1.40953e7i	0.567359i
$909$	−2.43214e6	−0.0976288
$910$	0	0
$911$	−3.60531e7	−1.43929	−0.719643	−	0.694344i	$-0.755695\pi$
$911$	−3.60531e7	−1.43929	−0.719643	+	0.694344i	$0.755695\pi$
$912$	2.81224e6i	0.111961i
$913$	7.81565e6i	0.310305i
$914$	3.13463e7	1.24114
$915$	0	0
$916$	−9.95322e6	−0.391945
$917$	6.66296e6i	0.261664i
$918$	3.55254e7i	1.39134i
$919$	2.96521e7	1.15816	0.579078	−	0.815272i	$-0.303413\pi$
$919$	2.96521e7	1.15816	0.579078	+	0.815272i	$0.303413\pi$
$920$	0	0
$921$	−6.81182e6	−0.264615
$922$	1.58278e7i	0.613187i
$923$	8.74499e6i	0.337874i
$924$	3.30418e6	0.127316
$925$	0	0
$926$	−1.08246e7	−0.414843
$927$	1.19806e6i	0.0457909i
$928$	− 972303.i	− 0.0370622i
$929$	1.01663e7	0.386478	0.193239	−	0.981152i	$-0.438101\pi$
$929$	1.01663e7	0.386478	0.193239	+	0.981152i	$0.438101\pi$
$930$	0	0
$931$	−1.02264e7	−0.386676
$932$	− 1.74921e7i	− 0.659633i
$933$	2.15679e7i	0.811154i
$934$	−2.05223e7	−0.769768
$935$	0	0
$936$	−181173.	−0.00675932
$937$	− 3.22787e7i	− 1.20107i	−0.799599	−	0.600534i	$-0.794955\pi$
$937$	− 3.22787e7i	− 1.20107i	0.799599	−	0.600534i	$-0.205045\pi$
$938$	9.23240e6i	0.342616i
$939$	2.48608e6	0.0920135
$940$	0	0
$941$	2.53184e7	0.932100	0.466050	−	0.884758i	$-0.345677\pi$
$941$	2.53184e7	0.932100	0.466050	+	0.884758i	$0.345677\pi$
$942$	− 2.54470e7i	− 0.934348i
$943$	3.42801e7i	1.25535i
$944$	−83694.7	−0.00305681
$945$	0	0
$946$	−1.04614e7	−0.380069
$947$	4.70863e7i	1.70616i	0.521780	+	0.853080i	$0.325268\pi$
$947$	4.70863e7i	1.70616i	−0.521780	+	0.853080i	$0.674732\pi$
$948$	− 1.83552e7i	− 0.663343i
$949$	−2.24691e6	−0.0809880
$950$	0	0
$951$	−2.31009e7	−0.828282
$952$	7.70449e6i	0.275519i
$953$	− 6.95299e6i	− 0.247993i	−0.992283	−	0.123997i	$-0.960429\pi$
$953$	− 6.95299e6i	− 0.247993i	0.992283	−	0.123997i	$-0.0395712\pi$
$954$	1.08393e6	0.0385595
$955$	0	0
$956$	2.45124e6	0.0867442
$957$	3.70263e6i	0.130687i
$958$	8.88137e6i	0.312656i
$959$	2.94499e6	0.103404
$960$	0	0
$961$	−2.85784e7	−0.998226
$962$	− 645152.i	− 0.0224763i
$963$	1.56376e6i	0.0543382i
$964$	−1.98443e7	−0.687771
$965$	0	0
$966$	−6.28776e6	−0.216797
$967$	− 4.00072e7i	− 1.37585i	−0.725780	−	0.687927i	$-0.758521\pi$
$967$	− 4.00072e7i	− 1.37585i	0.725780	−	0.687927i	$-0.241479\pi$
$968$	− 6.00586e6i	− 0.206009i
$969$	2.49713e7	0.854343
$970$	0	0
$971$	5.12102e7	1.74305	0.871523	−	0.490355i	$-0.163133\pi$
$971$	5.12102e7	1.74305	0.871523	+	0.490355i	$0.163133\pi$
$972$	2.02671e6i	0.0688061i
$973$	1.05931e7i	0.358707i
$974$	−2.00524e7	−0.677283
$975$	0	0
$976$	−1.19845e7	−0.402713
$977$	7.74195e6i	0.259486i	0.991548	+	0.129743i	$0.0414152\pi$
$977$	7.74195e6i	0.259486i	−0.991548	+	0.129743i	$0.958585\pi$
$978$	4.02339e7i	1.34507i
$979$	−1.73102e7	−0.577226
$980$	0	0
$981$	−1.71479e6	−0.0568905
$982$	1.51638e7i	0.501800i
$983$	− 2.28293e7i	− 0.753544i	−0.926306	−	0.376772i	$-0.877034\pi$
$983$	− 2.28293e7i	− 0.753544i	0.926306	−	0.376772i	$-0.122966\pi$
$984$	−1.67228e7	−0.550581
$985$	0	0
$986$	−8.63357e6	−0.282812
$987$	− 4.83516e6i	− 0.157986i
$988$	1.97481e6i	0.0643625i
$989$	1.99077e7	0.647189
$990$	0	0
$991$	3.12601e7	1.01113	0.505564	−	0.862789i	$-0.331284\pi$
$991$	3.12601e7	1.01113	0.505564	+	0.862789i	$0.331284\pi$
$992$	− 230739.i	− 0.00744461i
$993$	2.37294e7i	0.763684i
$994$	1.09614e7	0.351885
$995$	0	0
$996$	7.25544e6	0.231748
$997$	1.09102e7i	0.347611i	0.984780	+	0.173805i	$0.0556064\pi$
$997$	1.09102e7i	0.347611i	−0.984780	+	0.173805i	$0.944394\pi$
$998$	3.03779e7i	0.965454i
$999$	−3.72877e6	−0.118209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.f.599.1		4
5.2	odd	4		650.6.a.f.1.1		2
5.3	odd	4		130.6.a.c.1.2	✓	2
5.4	even	2	inner	650.6.b.f.599.4		4
20.3	even	4		1040.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.c.1.2	✓	2	5.3	odd	4
650.6.a.f.1.1		2	5.2	odd	4
650.6.b.f.599.1		4	1.1	even	1	trivial
650.6.b.f.599.4		4	5.4	even	2	inner
1040.6.a.c.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -15.0416i q^{3} -16.0000 q^{4} -60.1664 q^{6} +52.9584i q^{7} +64.0000i q^{8} +16.7504 q^{9} -259.248 q^{11} +240.666i q^{12} +169.000i q^{13} +211.834 q^{14} +256.000 q^{16} -2273.15i q^{17} -67.0017i q^{18} -730.329 q^{19} +796.579 q^{21} +1036.99i q^{22} -1973.36i q^{23} +962.662 q^{24} +676.000 q^{26} -3907.06i q^{27} -847.334i q^{28} +949.515 q^{29} +225.331 q^{31} -1024.00i q^{32} +3899.50i q^{33} -9092.62 q^{34} -268.007 q^{36} -954.367i q^{37} +2921.32i q^{38} +2542.03 q^{39} -17371.4 q^{41} -3186.32i q^{42} +10088.2i q^{43} +4147.97 q^{44} -7893.46 q^{46} -6069.90i q^{47} -3850.65i q^{48} +14002.4 q^{49} -34191.9 q^{51} -2704.00i q^{52} +16177.7i q^{53} -15628.2 q^{54} -3389.34 q^{56} +10985.3i q^{57} -3798.06i q^{58} -326.933 q^{59} -46814.4 q^{61} -901.324i q^{62} +887.076i q^{63} -4096.00 q^{64} +15598.0 q^{66} +43583.3i q^{67} +36370.5i q^{68} -29682.5 q^{69} +51745.5 q^{71} +1072.03i q^{72} +13295.3i q^{73} -3817.47 q^{74} +11685.3 q^{76} -13729.4i q^{77} -10168.1i q^{78} -76268.5 q^{79} -54698.1 q^{81} +69485.7i q^{82} -30147.4i q^{83} -12745.3 q^{84} +40352.9 q^{86} -14282.2i q^{87} -16591.9i q^{88} +66771.0 q^{89} -8949.97 q^{91} +31573.8i q^{92} -3389.34i q^{93} -24279.6 q^{94} -15402.6 q^{96} +151425. i q^{97} -56009.6i q^{98} -4342.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} - 48 q^{6} + 356 q^{9} + 408 q^{11} + 1040 q^{14} + 1024 q^{16} - 224 q^{19} + 200 q^{21} + 768 q^{24} + 2704 q^{26} + 13624 q^{29} - 640 q^{31} - 3232 q^{34} - 5696 q^{36} + 2028 q^{39}+ \cdots + 140712 q^{99}+O(q^{100}) \) 4 * q - 64 * q^4 - 48 * q^6 + 356 * q^9 + 408 * q^11 + 1040 * q^14 + 1024 * q^16 - 224 * q^19 + 200 * q^21 + 768 * q^24 + 2704 * q^26 + 13624 * q^29 - 640 * q^31 - 3232 * q^34 - 5696 * q^36 + 2028 * q^39 - 70160 * q^41 - 6528 * q^44 - 2096 * q^46 + 49748 * q^49 - 102184 * q^51 - 2016 * q^54 - 16640 * q^56 + 43776 * q^59 - 82544 * q^61 - 16384 * q^64 + 64704 * q^66 - 90312 * q^69 - 30960 * q^71 - 106208 * q^74 + 3584 * q^76 - 54416 * q^79 - 97124 * q^81 - 3200 * q^84 - 20272 * q^86 + 136264 * q^89 - 43940 * q^91 - 176304 * q^94 - 12288 * q^96 + 140712 * q^99

Introduction

L-functions

Modular forms

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Database

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