LMFDB - Embedded newform 325.6.b.f.274.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,6,Mod(274,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.274"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-268,0,-104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 326x^{10} + 40401x^{8} + 2365264x^{6} + 65636064x^{4} + 738923264x^{2} + 2250553600 $ x^12 + 326x^10 + 40401x^8 + 2365264x^6 + 65636064x^4 + 738923264*x^2 + 2250553600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.7
Root			$2.16876i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.f.274.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.16876i q^{2} -2.56930i q^{3} +27.2965 q^{4} +5.57220 q^{6} +75.5966i q^{7} +128.600i q^{8} +236.399 q^{9} +624.896 q^{11} -70.1329i q^{12} -169.000i q^{13} -163.951 q^{14} +594.585 q^{16} -2344.47i q^{17} +512.692i q^{18} +283.916 q^{19} +194.231 q^{21} +1355.25i q^{22} +2043.60i q^{23} +330.412 q^{24} +366.520 q^{26} -1231.72i q^{27} +2063.52i q^{28} -6172.51 q^{29} +687.967 q^{31} +5404.71i q^{32} -1605.55i q^{33} +5084.60 q^{34} +6452.85 q^{36} +2797.24i q^{37} +615.745i q^{38} -434.212 q^{39} +8236.40 q^{41} +421.239i q^{42} -13268.3i q^{43} +17057.5 q^{44} -4432.07 q^{46} +15489.4i q^{47} -1527.67i q^{48} +11092.1 q^{49} -6023.66 q^{51} -4613.11i q^{52} -9603.36i q^{53} +2671.30 q^{54} -9721.71 q^{56} -729.465i q^{57} -13386.7i q^{58} -40189.6 q^{59} +32587.1 q^{61} +1492.03i q^{62} +17870.9i q^{63} +7305.22 q^{64} +3482.05 q^{66} -15914.0i q^{67} -63995.9i q^{68} +5250.62 q^{69} +60206.9 q^{71} +30400.8i q^{72} -35469.1i q^{73} -6066.54 q^{74} +7749.90 q^{76} +47240.1i q^{77} -941.701i q^{78} -65026.6 q^{79} +54280.2 q^{81} +17862.8i q^{82} +86013.0i q^{83} +5301.81 q^{84} +28775.7 q^{86} +15859.1i q^{87} +80361.5i q^{88} +139114. q^{89} +12775.8 q^{91} +55783.0i q^{92} -1767.59i q^{93} -33592.8 q^{94} +13886.3 q^{96} +8013.02i q^{97} +24056.2i q^{98} +147725. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{4} - 104 q^{6} - 2068 q^{9} + 1600 q^{11} - 4216 q^{14} + 644 q^{16} - 10608 q^{19} + 2144 q^{21} - 9512 q^{24} - 676 q^{26} + 7328 q^{29} + 33328 q^{31} - 2760 q^{34} + 3636 q^{36} + 6760 q^{39}+ \cdots - 602528 q^{99}+O(q^{100}) $ 12 * q - 268 * q^4 - 104 * q^6 - 2068 * q^9 + 1600 * q^11 - 4216 * q^14 + 644 * q^16 - 10608 * q^19 + 2144 * q^21 - 9512 * q^24 - 676 * q^26 + 7328 * q^29 + 33328 * q^31 - 2760 * q^34 + 3636 * q^36 + 6760 * q^39 + 24872 * q^41 + 70248 * q^44 - 9112 * q^46 + 55684 * q^49 - 55328 * q^51 + 265824 * q^54 + 88840 * q^56 + 23792 * q^59 + 114464 * q^61 + 79828 * q^64 + 19816 * q^66 - 144808 * q^69 + 428208 * q^71 + 364208 * q^74 + 733272 * q^76 - 72048 * q^79 + 701028 * q^81 - 12048 * q^84 + 58408 * q^86 + 104056 * q^89 - 58136 * q^91 - 315144 * q^94 + 1095816 * q^96 - 602528 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

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

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 $

$2$	2.16876i	0.383386i	0.981455	+	0.191693i	$0.0613978\pi$
$2$	2.16876i	0.383386i	−0.981455	+	0.191693i	$0.938602\pi$
$3$	− 2.56930i	− 0.164821i	−0.996598	−	0.0824104i	$-0.973738\pi$
$3$	− 2.56930i	− 0.164821i	0.996598	−	0.0824104i	$-0.0262618\pi$
$4$	27.2965	0.853015
$5$	0	0
$5$	0	0
$6$	5.57220	0.0631900
$7$	75.5966i	0.583119i	0.956553	+	0.291560i	$0.0941742\pi$
$7$	75.5966i	0.583119i	−0.956553	+	0.291560i	$0.905826\pi$
$8$	128.600i	0.710420i
$9$	236.399	0.972834
$10$	0	0
$11$	624.896	1.55713	0.778567	−	0.627561i	$-0.215947\pi$
$11$	624.896	1.55713	0.778567	+	0.627561i	$0.215947\pi$
$12$	− 70.1329i	− 0.140595i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−163.951	−0.223560
$15$	0	0
$16$	594.585	0.580650
$17$	− 2344.47i	− 1.96754i	−0.179439	−	0.983769i	$-0.557428\pi$
$17$	− 2344.47i	− 1.96754i	0.179439	−	0.983769i	$-0.442572\pi$
$18$	512.692i	0.372971i
$19$	283.916	0.180429	0.0902143	−	0.995922i	$-0.471245\pi$
$19$	283.916	0.180429	0.0902143	+	0.995922i	$0.471245\pi$
$20$	0	0
$21$	194.231	0.0961102
$22$	1355.25i	0.596984i
$23$	2043.60i	0.805519i	0.915306	+	0.402760i	$0.131949\pi$
$23$	2043.60i	0.805519i	−0.915306	+	0.402760i	$0.868051\pi$
$24$	330.412	0.117092
$25$	0	0
$26$	366.520	0.106332
$27$	− 1231.72i	− 0.325164i
$28$	2063.52i	0.497410i
$29$	−6172.51	−1.36291	−0.681455	−	0.731860i	$-0.738652\pi$
$29$	−6172.51	−1.36291	−0.681455	+	0.731860i	$0.738652\pi$
$30$	0	0
$31$	687.967	0.128577	0.0642885	−	0.997931i	$-0.479522\pi$
$31$	687.967	0.128577	0.0642885	+	0.997931i	$0.479522\pi$
$32$	5404.71i	0.933033i
$33$	− 1605.55i	− 0.256648i
$34$	5084.60	0.754327
$35$	0	0
$36$	6452.85	0.829842
$37$	2797.24i	0.335912i	0.985795	+	0.167956i	$0.0537166\pi$
$37$	2797.24i	0.335912i	−0.985795	+	0.167956i	$0.946283\pi$
$38$	615.745i	0.0691738i
$39$	−434.212	−0.0457131
$40$	0	0
$41$	8236.40	0.765205	0.382602	−	0.923913i	$-0.375028\pi$
$41$	8236.40	0.765205	0.382602	+	0.923913i	$0.375028\pi$
$42$	421.239i	0.0368473i
$43$	− 13268.3i	− 1.09432i	−0.837029	−	0.547158i	$-0.815710\pi$
$43$	− 13268.3i	− 1.09432i	0.837029	−	0.547158i	$-0.184290\pi$
$44$	17057.5	1.32826
$45$	0	0
$46$	−4432.07	−0.308825
$47$	15489.4i	1.02280i	0.859343	+	0.511400i	$0.170873\pi$
$47$	15489.4i	1.02280i	−0.859343	+	0.511400i	$0.829127\pi$
$48$	− 1527.67i	− 0.0957032i
$49$	11092.1	0.659972
$50$	0	0
$51$	−6023.66	−0.324291
$52$	− 4613.11i	− 0.236584i
$53$	− 9603.36i	− 0.469606i	−0.972043	−	0.234803i	$-0.924556\pi$
$53$	− 9603.36i	− 0.469606i	0.972043	−	0.234803i	$-0.0754444\pi$
$54$	2671.30	0.124663
$55$	0	0
$56$	−9721.71	−0.414260
$57$	− 729.465i	− 0.0297384i
$58$	− 13386.7i	− 0.522521i
$59$	−40189.6	−1.50308	−0.751542	−	0.659685i	$-0.770690\pi$
$59$	−40189.6	−1.50308	−0.751542	+	0.659685i	$0.770690\pi$
$60$	0	0
$61$	32587.1	1.12130	0.560649	−	0.828053i	$-0.310552\pi$
$61$	32587.1	1.12130	0.560649	+	0.828053i	$0.310552\pi$
$62$	1492.03i	0.0492946i
$63$	17870.9i	0.567278i
$64$	7305.22	0.222938
$65$	0	0
$66$	3482.05	0.0983953
$67$	− 15914.0i	− 0.433105i	−0.976271	−	0.216553i	$-0.930519\pi$
$67$	− 15914.0i	− 0.433105i	0.976271	−	0.216553i	$-0.0694813\pi$
$68$	− 63995.9i	− 1.67834i
$69$	5250.62	0.132766
$70$	0	0
$71$	60206.9	1.41743	0.708713	−	0.705496i	$-0.249276\pi$
$71$	60206.9	1.41743	0.708713	+	0.705496i	$0.249276\pi$
$72$	30400.8i	0.691121i
$73$	− 35469.1i	− 0.779010i	−0.921024	−	0.389505i	$-0.872646\pi$
$73$	− 35469.1i	− 0.779010i	0.921024	−	0.389505i	$-0.127354\pi$
$74$	−6066.54	−0.128784
$75$	0	0
$76$	7749.90	0.153908
$77$	47240.1i	0.907995i
$78$	− 941.701i	− 0.0175258i
$79$	−65026.6	−1.17226	−0.586129	−	0.810218i	$-0.699349\pi$
$79$	−65026.6	−1.17226	−0.586129	+	0.810218i	$0.699349\pi$
$80$	0	0
$81$	54280.2	0.919240
$82$	17862.8i	0.293369i
$83$	86013.0i	1.37047i	0.728323	+	0.685234i	$0.240300\pi$
$83$	86013.0i	1.37047i	−0.728323	+	0.685234i	$0.759700\pi$
$84$	5301.81	0.0819834
$85$	0	0
$86$	28775.7	0.419545
$87$	15859.1i	0.224636i
$88$	80361.5i	1.10622i
$89$	139114.	1.86163	0.930817	−	0.365485i	$-0.119097\pi$
$89$	139114.	1.86163	0.930817	+	0.365485i	$0.119097\pi$
$90$	0	0
$91$	12775.8	0.161728
$92$	55783.0i	0.687120i
$93$	− 1767.59i	− 0.0211922i
$94$	−33592.8	−0.392127
$95$	0	0
$96$	13886.3	0.153783
$97$	8013.02i	0.0864703i	0.999065	+	0.0432351i	$0.0137665\pi$
$97$	8013.02i	0.0864703i	−0.999065	+	0.0432351i	$0.986234\pi$
$98$	24056.2i	0.253024i
$99$	147725.	1.51483
$100$	0	0
$101$	127679.	1.24542	0.622709	−	0.782454i	$-0.286032\pi$
$101$	127679.	1.24542	0.622709	+	0.782454i	$0.286032\pi$
$102$	− 13063.9i	− 0.124329i
$103$	136390.i	1.26675i	0.773847	+	0.633373i	$0.218330\pi$
$103$	136390.i	1.26675i	−0.773847	+	0.633373i	$0.781670\pi$
$104$	21733.4	0.197035
$105$	0	0
$106$	20827.4	0.180040
$107$	− 33127.6i	− 0.279725i	−0.990171	−	0.139862i	$-0.955334\pi$
$107$	− 33127.6i	− 0.279725i	0.990171	−	0.139862i	$-0.0446660\pi$
$108$	− 33621.6i	− 0.277370i
$109$	−53441.1	−0.430834	−0.215417	−	0.976522i	$-0.569111\pi$
$109$	−53441.1	−0.430834	−0.215417	+	0.976522i	$0.569111\pi$
$110$	0	0
$111$	7186.95	0.0553652
$112$	44948.7i	0.338588i
$113$	− 181691.i	− 1.33855i	−0.743013	−	0.669277i	$-0.766604\pi$
$113$	− 181691.i	− 1.33855i	0.743013	−	0.669277i	$-0.233396\pi$
$114$	1582.03	0.0114013
$115$	0	0
$116$	−168488.	−1.16258
$117$	− 39951.4i	− 0.269816i
$118$	− 87161.5i	− 0.576262i
$119$	177234.	1.14731
$120$	0	0
$121$	229444.	1.42467
$122$	70673.6i	0.429890i
$123$	− 21161.8i	− 0.126122i
$124$	18779.1	0.109678
$125$	0	0
$126$	−38757.8	−0.217487
$127$	290716.i	1.59941i	0.600395	+	0.799704i	$0.295010\pi$
$127$	290716.i	1.59941i	−0.600395	+	0.799704i	$0.704990\pi$
$128$	188794.i	1.01850i
$129$	−34090.2	−0.180366
$130$	0	0
$131$	−216572.	−1.10262	−0.551308	−	0.834302i	$-0.685871\pi$
$131$	−216572.	−1.10262	−0.551308	+	0.834302i	$0.685871\pi$
$132$	− 43825.8i	− 0.218925i
$133$	21463.1i	0.105211i
$134$	34513.7	0.166047
$135$	0	0
$136$	301499.	1.39778
$137$	200864.i	0.914326i	0.889383	+	0.457163i	$0.151134\pi$
$137$	200864.i	0.914326i	−0.889383	+	0.457163i	$0.848866\pi$
$138$	11387.3i	0.0509008i
$139$	11515.0	0.0505508	0.0252754	−	0.999681i	$-0.491954\pi$
$139$	11515.0	0.0505508	0.0252754	+	0.999681i	$0.491954\pi$
$140$	0	0
$141$	39797.0	0.168579
$142$	130574.i	0.543422i
$143$	− 105607.i	− 0.431871i
$144$	140559.	0.564876
$145$	0	0
$146$	76924.0	0.298662
$147$	− 28499.1i	− 0.108777i
$148$	76354.7i	0.286538i
$149$	−386747.	−1.42712	−0.713561	−	0.700593i	$-0.752919\pi$
$149$	−386747.	−1.42712	−0.713561	+	0.700593i	$0.752919\pi$
$150$	0	0
$151$	−22668.6	−0.0809062	−0.0404531	−	0.999181i	$-0.512880\pi$
$151$	−22668.6	−0.0809062	−0.0404531	+	0.999181i	$0.512880\pi$
$152$	36511.5i	0.128180i
$153$	− 554231.i	− 1.91409i
$154$	−102452.	−0.348113
$155$	0	0
$156$	−11852.5	−0.0389939
$157$	237735.i	0.769741i	0.922971	+	0.384870i	$0.125754\pi$
$157$	237735.i	0.769741i	−0.922971	+	0.384870i	$0.874246\pi$
$158$	− 141027.i	− 0.449427i
$159$	−24673.9	−0.0774008
$160$	0	0
$161$	−154489.	−0.469714
$162$	117721.i	0.352424i
$163$	508182.i	1.49813i	0.662495	+	0.749066i	$0.269498\pi$
$163$	508182.i	1.49813i	−0.662495	+	0.749066i	$0.730502\pi$
$164$	224825.	0.652731
$165$	0	0
$166$	−186541.	−0.525418
$167$	− 307248.i	− 0.852507i	−0.904604	−	0.426253i	$-0.859833\pi$
$167$	− 307248.i	− 0.852507i	0.904604	−	0.426253i	$-0.140167\pi$
$168$	24978.0i	0.0682786i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	67117.3	0.175527
$172$	− 362177.i	− 0.933468i
$173$	− 4598.14i	− 0.0116807i	−0.999983	−	0.00584033i	$-0.998141\pi$
$173$	− 4598.14i	− 0.0116807i	0.999983	−	0.00584033i	$-0.00185905\pi$
$174$	−34394.5	−0.0861223
$175$	0	0
$176$	371554.	0.904150
$177$	103259.i	0.247740i
$178$	301704.i	0.713725i
$179$	−557239.	−1.29990	−0.649948	−	0.759978i	$-0.725209\pi$
$179$	−557239.	−1.29990	−0.649948	+	0.759978i	$0.725209\pi$
$180$	0	0
$181$	−410955.	−0.932389	−0.466195	−	0.884682i	$-0.654375\pi$
$181$	−410955.	−0.932389	−0.466195	+	0.884682i	$0.654375\pi$
$182$	27707.7i	0.0620043i
$183$	− 83726.1i	− 0.184813i
$184$	−262806.	−0.572257
$185$	0	0
$186$	3833.49	0.00812478
$187$	− 1.46505e6i	− 3.06372i
$188$	422807.i	0.872464i
$189$	93113.9	0.189609
$190$	0	0
$191$	−361321.	−0.716655	−0.358327	−	0.933596i	$-0.616653\pi$
$191$	−361321.	−0.716655	−0.358327	+	0.933596i	$0.616653\pi$
$192$	− 18769.3i	− 0.0367448i
$193$	− 149297.i	− 0.288509i	−0.989541	−	0.144254i	$-0.953922\pi$
$193$	− 149297.i	− 0.288509i	0.989541	−	0.144254i	$-0.0460784\pi$
$194$	−17378.3	−0.0331515
$195$	0	0
$196$	302777.	0.562966
$197$	− 139691.i	− 0.256449i	−0.991745	−	0.128225i	$-0.959072\pi$
$197$	− 139691.i	− 0.256449i	0.991745	−	0.128225i	$-0.0409278\pi$
$198$	320379.i	0.580766i
$199$	433871.	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$199$	433871.	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$200$	0	0
$201$	−40888.0	−0.0713848
$202$	276904.i	0.477476i
$203$	− 466621.i	− 0.794739i
$204$	−164425.	−0.276625
$205$	0	0
$206$	−295797.	−0.485653
$207$	483104.i	0.783636i
$208$	− 100485.i	− 0.161043i
$209$	177418.	0.280952
$210$	0	0
$211$	392100.	0.606305	0.303153	−	0.952942i	$-0.401961\pi$
$211$	392100.	0.606305	0.303153	+	0.952942i	$0.401961\pi$
$212$	− 262138.i	− 0.400581i
$213$	− 154690.i	− 0.233621i
$214$	71845.8	0.107243
$215$	0	0
$216$	158399.	0.231003
$217$	52008.0i	0.0749757i
$218$	− 115901.i	− 0.165176i
$219$	−91130.9	−0.128397
$220$	0	0
$221$	−396216.	−0.545697
$222$	15586.8i	0.0212263i
$223$	− 594720.i	− 0.800848i	−0.916330	−	0.400424i	$-0.868863\pi$
$223$	− 594720.i	− 0.800848i	0.916330	−	0.400424i	$-0.131137\pi$
$224$	−408578.	−0.544070
$225$	0	0
$226$	394043.	0.513183
$227$	− 473048.i	− 0.609313i	−0.952462	−	0.304657i	$-0.901458\pi$
$227$	− 473048.i	− 0.609313i	0.952462	−	0.304657i	$-0.0985417\pi$
$228$	− 19911.8i	− 0.0253673i
$229$	905716.	1.14131	0.570655	−	0.821190i	$-0.306689\pi$
$229$	905716.	1.14131	0.570655	+	0.821190i	$0.306689\pi$
$230$	0	0
$231$	121374.	0.149656
$232$	− 793784.i	− 0.968239i
$233$	− 746250.i	− 0.900522i	−0.892897	−	0.450261i	$-0.851331\pi$
$233$	− 746250.i	− 0.900522i	0.892897	−	0.450261i	$-0.148669\pi$
$234$	86644.9	0.103444
$235$	0	0
$236$	−1.09703e6	−1.28215
$237$	167073.i	0.193212i
$238$	384379.i	0.439863i
$239$	−909186.	−1.02958	−0.514788	−	0.857318i	$-0.672129\pi$
$239$	−909186.	−1.02958	−0.514788	+	0.857318i	$0.672129\pi$
$240$	0	0
$241$	−575680.	−0.638467	−0.319233	−	0.947676i	$-0.603425\pi$
$241$	−575680.	−0.638467	−0.319233	+	0.947676i	$0.603425\pi$
$242$	497609.i	0.546198i
$243$	− 438770.i	− 0.476674i
$244$	889513.	0.956484
$245$	0	0
$246$	45894.8	0.0483533
$247$	− 47981.8i	− 0.0500419i
$248$	88472.4i	0.0913437i
$249$	220993.	0.225882
$250$	0	0
$251$	−1.77777e6	−1.78111	−0.890555	−	0.454876i	$-0.849683\pi$
$251$	−1.77777e6	−1.78111	−0.890555	+	0.454876i	$0.849683\pi$
$252$	487814.i	0.483897i
$253$	1.27704e6i	1.25430i
$254$	−630492.	−0.613191
$255$	0	0
$256$	−175681.	−0.167543
$257$	− 1.60384e6i	− 1.51470i	−0.653006	−	0.757352i	$-0.726493\pi$
$257$	− 1.60384e6i	− 1.51470i	0.653006	−	0.757352i	$-0.273507\pi$
$258$	− 73933.3i	− 0.0691498i
$259$	−211462.	−0.195877
$260$	0	0
$261$	−1.45917e6	−1.32589
$262$	− 469693.i	− 0.422728i
$263$	1.67245e6i	1.49095i	0.666532	+	0.745477i	$0.267778\pi$
$263$	1.67245e6i	1.49095i	−0.666532	+	0.745477i	$0.732222\pi$
$264$	206473.	0.182328
$265$	0	0
$266$	−46548.2	−0.0403366
$267$	− 357425.i	− 0.306836i
$268$	− 434397.i	− 0.369445i
$269$	64944.5	0.0547220	0.0273610	−	0.999626i	$-0.491290\pi$
$269$	64944.5	0.0547220	0.0273610	+	0.999626i	$0.491290\pi$
$270$	0	0
$271$	1.37951e6	1.14104	0.570519	−	0.821284i	$-0.306742\pi$
$271$	1.37951e6	1.14104	0.570519	+	0.821284i	$0.306742\pi$
$272$	− 1.39399e6i	− 1.14245i
$273$	− 32825.0i	− 0.0266562i
$274$	−435626.	−0.350540
$275$	0	0
$276$	143323.	0.113252
$277$	2.40558e6i	1.88374i	0.335978	+	0.941870i	$0.390933\pi$
$277$	2.40558e6i	1.88374i	−0.335978	+	0.941870i	$0.609067\pi$
$278$	24973.3i	0.0193805i
$279$	162634.	0.125084
$280$	0	0
$281$	−1.70763e6	−1.29011	−0.645055	−	0.764136i	$-0.723166\pi$
$281$	−1.70763e6	−1.29011	−0.645055	+	0.764136i	$0.723166\pi$
$282$	86310.2i	0.0646307i
$283$	513628.i	0.381226i	0.981665	+	0.190613i	$0.0610476\pi$
$283$	513628.i	0.381226i	−0.981665	+	0.190613i	$0.938952\pi$
$284$	1.64344e6	1.20909
$285$	0	0
$286$	229037.	0.165574
$287$	622644.i	0.446206i
$288$	1.27767e6i	0.907687i
$289$	−4.07670e6	−2.87121
$290$	0	0
$291$	20587.9	0.0142521
$292$	− 968182.i	− 0.664508i
$293$	− 2.29105e6i	− 1.55907i	−0.626359	−	0.779535i	$-0.715456\pi$
$293$	− 2.29105e6i	− 1.55907i	0.626359	−	0.779535i	$-0.284544\pi$
$294$	61807.6	0.0417036
$295$	0	0
$296$	−359724.	−0.238638
$297$	− 769697.i	− 0.506324i
$298$	− 838761.i	− 0.547139i
$299$	345368.	0.223411
$300$	0	0
$301$	1.00304e6	0.638117
$302$	− 49162.7i	− 0.0310183i
$303$	− 328045.i	− 0.205271i
$304$	168812.	0.104766
$305$	0	0
$306$	1.20199e6	0.733835
$307$	− 244483.i	− 0.148048i	−0.997256	−	0.0740240i	$-0.976416\pi$
$307$	− 244483.i	− 0.148048i	0.997256	−	0.0740240i	$-0.0235841\pi$
$308$	1.28949e6i	0.774534i
$309$	350427.	0.208786
$310$	0	0
$311$	1.21897e6	0.714648	0.357324	−	0.933981i	$-0.383689\pi$
$311$	1.21897e6	0.714648	0.357324	+	0.933981i	$0.383689\pi$
$312$	− 55839.6i	− 0.0324755i
$313$	− 1.21980e6i	− 0.703766i	−0.936044	−	0.351883i	$-0.885542\pi$
$313$	− 1.21980e6i	− 0.703766i	0.936044	−	0.351883i	$-0.114458\pi$
$314$	−515591.	−0.295108
$315$	0	0
$316$	−1.77500e6	−0.999954
$317$	− 2.20055e6i	− 1.22994i	−0.788551	−	0.614970i	$-0.789168\pi$
$317$	− 2.20055e6i	− 1.22994i	0.788551	−	0.614970i	$-0.210832\pi$
$318$	− 53511.8i	− 0.0296744i
$319$	−3.85718e6	−2.12223
$320$	0	0
$321$	−85114.9	−0.0461044
$322$	− 335050.i	− 0.180082i
$323$	− 665633.i	− 0.355000i
$324$	1.48166e6	0.784126
$325$	0	0
$326$	−1.10212e6	−0.574363
$327$	137306.i	0.0710103i
$328$	1.05920e6i	0.543617i
$329$	−1.17095e6	−0.596414
$330$	0	0
$331$	−755217.	−0.378880	−0.189440	−	0.981892i	$-0.560667\pi$
$331$	−755217.	−0.378880	−0.189440	+	0.981892i	$0.560667\pi$
$332$	2.34785e6i	1.16903i
$333$	661263.i	0.326786i
$334$	666347.	0.326839
$335$	0	0
$336$	115487.	0.0558064
$337$	1.10179e6i	0.528475i	0.964458	+	0.264238i	$0.0851203\pi$
$337$	1.10179e6i	0.528475i	−0.964458	+	0.264238i	$0.914880\pi$
$338$	− 61941.9i	− 0.0294912i
$339$	−466818.	−0.220622
$340$	0	0
$341$	429908.	0.200212
$342$	145561.i	0.0672947i
$343$	2.10908e6i	0.967962i
$344$	1.70629e6	0.777424
$345$	0	0
$346$	9972.27	0.00447820
$347$	− 1.25898e6i	− 0.561301i	−0.959810	−	0.280650i	$-0.909450\pi$
$347$	− 1.25898e6i	− 0.561301i	0.959810	−	0.280650i	$-0.0905502\pi$
$348$	432896.i	0.191618i
$349$	674977.	0.296637	0.148318	−	0.988940i	$-0.452614\pi$
$349$	674977.	0.296637	0.148318	+	0.988940i	$0.452614\pi$
$350$	0	0
$351$	−208161.	−0.0901843
$352$	3.37738e6i	1.45286i
$353$	1.05470e6i	0.450496i	0.974301	+	0.225248i	$0.0723193\pi$
$353$	1.05470e6i	0.450496i	−0.974301	+	0.225248i	$0.927681\pi$
$354$	−223944.	−0.0949799
$355$	0	0
$356$	3.79731e6	1.58800
$357$	− 455369.i	− 0.189100i
$358$	− 1.20852e6i	− 0.498362i
$359$	−1.94312e6	−0.795728	−0.397864	−	0.917444i	$-0.630248\pi$
$359$	−1.94312e6	−0.795728	−0.397864	+	0.917444i	$0.630248\pi$
$360$	0	0
$361$	−2.39549e6	−0.967446
$362$	− 891262.i	− 0.357465i
$363$	− 589511.i	− 0.234815i
$364$	348735.	0.137957
$365$	0	0
$366$	181582.	0.0708548
$367$	3.34298e6i	1.29559i	0.761813	+	0.647797i	$0.224310\pi$
$367$	3.34298e6i	1.29559i	−0.761813	+	0.647797i	$0.775690\pi$
$368$	1.21509e6i	0.467725i
$369$	1.94707e6	0.744417
$370$	0	0
$371$	725981.	0.273836
$372$	− 48249.1i	− 0.0180772i
$373$	− 1.66367e6i	− 0.619149i	−0.950875	−	0.309575i	$-0.899813\pi$
$373$	− 1.66367e6i	− 0.619149i	0.950875	−	0.309575i	$-0.100187\pi$
$374$	3.17735e6	1.17459
$375$	0	0
$376$	−1.99194e6	−0.726618
$377$	1.04315e6i	0.378003i
$378$	201942.i	0.0726936i
$379$	5.46932e6	1.95585	0.977924	−	0.208962i	$-0.0670086\pi$
$379$	5.46932e6	1.95585	0.977924	+	0.208962i	$0.0670086\pi$
$380$	0	0
$381$	746936.	0.263616
$382$	− 783619.i	− 0.274756i
$383$	− 1.39999e6i	− 0.487674i	−0.969816	−	0.243837i	$-0.921594\pi$
$383$	− 1.39999e6i	− 0.487674i	0.969816	−	0.243837i	$-0.0784062\pi$
$384$	485068.	0.167871
$385$	0	0
$386$	323790.	0.110610
$387$	− 3.13660e6i	− 1.06459i
$388$	218727.i	0.0737605i
$389$	296544.	0.0993607	0.0496803	−	0.998765i	$-0.484180\pi$
$389$	296544.	0.0993607	0.0496803	+	0.998765i	$0.484180\pi$
$390$	0	0
$391$	4.79116e6	1.58489
$392$	1.42645e6i	0.468857i
$393$	556439.i	0.181734i
$394$	302955.	0.0983191
$395$	0	0
$396$	4.03236e6	1.29218
$397$	1.62560e6i	0.517653i	0.965924	+	0.258826i	$0.0833357\pi$
$397$	1.62560e6i	0.517653i	−0.965924	+	0.258826i	$0.916664\pi$
$398$	940962.i	0.297759i
$399$	55145.1	0.0173410
$400$	0	0
$401$	870992.	0.270491	0.135246	−	0.990812i	$-0.456818\pi$
$401$	870992.	0.270491	0.135246	+	0.990812i	$0.456818\pi$
$402$	− 88676.2i	− 0.0273679i
$403$	− 116266.i	− 0.0356608i
$404$	3.48518e6	1.06236
$405$	0	0
$406$	1.01199e6	0.304692
$407$	1.74798e6i	0.523060i
$408$	− 774642.i	− 0.230383i
$409$	−478324.	−0.141388	−0.0706942	−	0.997498i	$-0.522521\pi$
$409$	−478324.	−0.141388	−0.0706942	+	0.997498i	$0.522521\pi$
$410$	0	0
$411$	516081.	0.150700
$412$	3.72297e6i	1.08055i
$413$	− 3.03820e6i	− 0.876477i
$414$	−1.04774e6	−0.300435
$415$	0	0
$416$	913395.	0.258777
$417$	− 29585.6i	− 0.00833182i
$418$	384777.i	0.107713i
$419$	−1.58332e6	−0.440589	−0.220295	−	0.975433i	$-0.570702\pi$
$419$	−1.58332e6	−0.440589	−0.220295	+	0.975433i	$0.570702\pi$
$420$	0	0
$421$	−2.63883e6	−0.725614	−0.362807	−	0.931864i	$-0.618182\pi$
$421$	−2.63883e6	−0.725614	−0.362807	+	0.931864i	$0.618182\pi$
$422$	850371.i	0.232449i
$423$	3.66168e6i	0.995015i
$424$	1.23499e6	0.333617
$425$	0	0
$426$	335485.	0.0895672
$427$	2.46348e6i	0.653851i
$428$	− 904268.i	− 0.238609i
$429$	−271337.	−0.0711814
$430$	0	0
$431$	−2.05662e6	−0.533287	−0.266644	−	0.963795i	$-0.585915\pi$
$431$	−2.05662e6	−0.533287	−0.266644	+	0.963795i	$0.585915\pi$
$432$	− 732363.i	− 0.188806i
$433$	− 3.35571e6i	− 0.860131i	−0.902798	−	0.430066i	$-0.858490\pi$
$433$	− 3.35571e6i	− 0.860131i	0.902798	−	0.430066i	$-0.141510\pi$
$434$	−112793.	−0.0287447
$435$	0	0
$436$	−1.45876e6	−0.367507
$437$	580210.i	0.145339i
$438$	− 197641.i	− 0.0492257i
$439$	−594295.	−0.147177	−0.0735886	−	0.997289i	$-0.523445\pi$
$439$	−594295.	−0.147177	−0.0735886	+	0.997289i	$0.523445\pi$
$440$	0	0
$441$	2.62217e6	0.642043
$442$	− 859298.i	− 0.209213i
$443$	− 32710.8i	− 0.00791921i	−0.999992	−	0.00395960i	$-0.998740\pi$
$443$	− 32710.8i	− 0.00791921i	0.999992	−	0.00395960i	$-0.00126038\pi$
$444$	196178.	0.0472274
$445$	0	0
$446$	1.28980e6	0.307034
$447$	993669.i	0.235219i
$448$	552250.i	0.129999i
$449$	−1.14362e6	−0.267711	−0.133856	−	0.991001i	$-0.542736\pi$
$449$	−1.14362e6	−0.267711	−0.133856	+	0.991001i	$0.542736\pi$
$450$	0	0
$451$	5.14689e6	1.19153
$452$	− 4.95951e6i	− 1.14181i
$453$	58242.4i	0.0133350i
$454$	1.02593e6	0.233602
$455$	0	0
$456$	93809.1	0.0211268
$457$	1.26412e6i	0.283139i	0.989928	+	0.141569i	$0.0452148\pi$
$457$	1.26412e6i	0.283139i	−0.989928	+	0.141569i	$0.954785\pi$
$458$	1.96428e6i	0.437562i
$459$	−2.88774e6	−0.639773
$460$	0	0
$461$	2.38521e6	0.522727	0.261364	−	0.965240i	$-0.415828\pi$
$461$	2.38521e6	0.522727	0.261364	+	0.965240i	$0.415828\pi$
$462$	263231.i	0.0573762i
$463$	− 5.27680e6i	− 1.14398i	−0.820261	−	0.571990i	$-0.806172\pi$
$463$	− 5.27680e6i	− 1.14398i	0.820261	−	0.571990i	$-0.193828\pi$
$464$	−3.67009e6	−0.791373
$465$	0	0
$466$	1.61844e6	0.345248
$467$	5.46999e6i	1.16063i	0.814391	+	0.580316i	$0.197071\pi$
$467$	5.46999e6i	1.16063i	−0.814391	+	0.580316i	$0.802929\pi$
$468$	− 1.09053e6i	− 0.230157i
$469$	1.20305e6	0.252552
$470$	0	0
$471$	610814.	0.126869
$472$	− 5.16837e6i	− 1.06782i
$473$	− 8.29128e6i	− 1.70400i
$474$	−362341.	−0.0740750
$475$	0	0
$476$	4.83788e6	0.978672
$477$	− 2.27022e6i	− 0.456848i
$478$	− 1.97181e6i	− 0.394725i
$479$	−8.31612e6	−1.65608	−0.828042	−	0.560666i	$-0.810545\pi$
$479$	−8.31612e6	−1.65608	−0.828042	+	0.560666i	$0.810545\pi$
$480$	0	0
$481$	472733.	0.0931651
$482$	− 1.24851e6i	− 0.244779i
$483$	396929.i	0.0774186i
$484$	6.26302e6	1.21526
$485$	0	0
$486$	951587.	0.182750
$487$	− 1.23823e6i	− 0.236581i	−0.992979	−	0.118291i	$-0.962259\pi$
$487$	− 1.23823e6i	− 0.236581i	0.992979	−	0.118291i	$-0.0377415\pi$
$488$	4.19069e6i	0.796593i
$489$	1.30567e6	0.246923
$490$	0	0
$491$	472497.	0.0884494	0.0442247	−	0.999022i	$-0.485918\pi$
$491$	472497.	0.0884494	0.0442247	+	0.999022i	$0.485918\pi$
$492$	− 577643.i	− 0.107584i
$493$	1.44713e7i	2.68158i
$494$	104061.	0.0191854
$495$	0	0
$496$	409055.	0.0746582
$497$	4.55144e6i	0.826529i
$498$	479281.i	0.0865998i
$499$	−3.43591e6	−0.617718	−0.308859	−	0.951108i	$-0.599947\pi$
$499$	−3.43591e6	−0.617718	−0.308859	+	0.951108i	$0.599947\pi$
$500$	0	0
$501$	−789413.	−0.140511
$502$	− 3.85555e6i	− 0.682853i
$503$	− 9.25786e6i	− 1.63151i	−0.578396	−	0.815756i	$-0.696321\pi$
$503$	− 9.25786e6i	− 1.63151i	0.578396	−	0.815756i	$-0.303679\pi$
$504$	−2.29820e6	−0.403006
$505$	0	0
$506$	−2.76959e6	−0.480882
$507$	73381.8i	0.0126785i
$508$	7.93552e6i	1.36432i
$509$	−3.39691e6	−0.581152	−0.290576	−	0.956852i	$-0.593847\pi$
$509$	−3.39691e6	−0.581152	−0.290576	+	0.956852i	$0.593847\pi$
$510$	0	0
$511$	2.68135e6	0.454256
$512$	5.66039e6i	0.954271i
$513$	− 349705.i	− 0.0586689i
$514$	3.47834e6	0.580717
$515$	0	0
$516$	−930541.	−0.153855
$517$	9.67928e6i	1.59264i
$518$	− 458610.i	− 0.0750964i
$519$	−11814.0	−0.00192522
$520$	0	0
$521$	7.22852e6	1.16669	0.583344	−	0.812225i	$-0.301744\pi$
$521$	7.22852e6	1.16669	0.583344	+	0.812225i	$0.301744\pi$
$522$	− 3.16460e6i	− 0.508326i
$523$	− 5.69535e6i	− 0.910471i	−0.890371	−	0.455236i	$-0.849555\pi$
$523$	− 5.69535e6i	− 0.910471i	0.890371	−	0.455236i	$-0.150445\pi$
$524$	−5.91166e6	−0.940549
$525$	0	0
$526$	−3.62714e6	−0.571611
$527$	− 1.61292e6i	− 0.252980i
$528$	− 954635.i	− 0.149023i
$529$	2.26005e6	0.351139
$530$	0	0
$531$	−9.50076e6	−1.46225
$532$	585866.i	0.0897469i
$533$	− 1.39195e6i	− 0.212230i
$534$	775168.	0.117637
$535$	0	0
$536$	2.04654e6	0.307687
$537$	1.43171e6i	0.214250i
$538$	140849.i	0.0209796i
$539$	6.93144e6	1.02767
$540$	0	0
$541$	−3.69650e6	−0.542998	−0.271499	−	0.962439i	$-0.587519\pi$
$541$	−3.69650e6	−0.542998	−0.271499	+	0.962439i	$0.587519\pi$
$542$	2.99181e6i	0.437458i
$543$	1.05587e6i	0.153677i
$544$	1.26712e7	1.83578
$545$	0	0
$546$	71189.5	0.0102196
$547$	− 1.94896e6i	− 0.278507i	−0.990257	−	0.139253i	$-0.955530\pi$
$547$	− 1.94896e6i	− 0.278507i	0.990257	−	0.139253i	$-0.0444702\pi$
$548$	5.48288e6i	0.779934i
$549$	7.70355e6	1.09084
$550$	0	0
$551$	−1.75247e6	−0.245908
$552$	675229.i	0.0943199i
$553$	− 4.91579e6i	− 0.683566i
$554$	−5.21713e6	−0.722200
$555$	0	0
$556$	314320.	0.0431206
$557$	508289.i	0.0694180i	0.999397	+	0.0347090i	$0.0110504\pi$
$557$	508289.i	0.0694180i	−0.999397	+	0.0347090i	$0.988950\pi$
$558$	352715.i	0.0479555i
$559$	−2.24234e6	−0.303509
$560$	0	0
$561$	−3.76416e6	−0.504965
$562$	− 3.70343e6i	− 0.494611i
$563$	− 1.76345e6i	− 0.234473i	−0.993104	−	0.117237i	$-0.962596\pi$
$563$	− 1.76345e6i	− 0.234473i	0.993104	−	0.117237i	$-0.0374036\pi$
$564$	1.08632e6	0.143800
$565$	0	0
$566$	−1.11394e6	−0.146157
$567$	4.10340e6i	0.536027i
$568$	7.74260e6i	1.00697i
$569$	−1.08114e7	−1.39992	−0.699960	−	0.714182i	$-0.746799\pi$
$569$	−1.08114e7	−1.39992	−0.699960	+	0.714182i	$0.746799\pi$
$570$	0	0
$571$	−8.48510e6	−1.08910	−0.544549	−	0.838729i	$-0.683299\pi$
$571$	−8.48510e6	−1.08910	−0.544549	+	0.838729i	$0.683299\pi$
$572$	− 2.88271e6i	− 0.368393i
$573$	928343.i	0.118120i
$574$	−1.35037e6	−0.171069
$575$	0	0
$576$	1.72695e6	0.216881
$577$	− 8.97573e6i	− 1.12235i	−0.827695	−	0.561177i	$-0.810349\pi$
$577$	− 8.97573e6i	− 1.12235i	0.827695	−	0.561177i	$-0.189651\pi$
$578$	− 8.84139e6i	− 1.10078i
$579$	−383590.	−0.0475523
$580$	0	0
$581$	−6.50229e6	−0.799146
$582$	44650.1i	0.00546406i
$583$	− 6.00110e6i	− 0.731239i
$584$	4.56132e6	0.553425
$585$	0	0
$586$	4.96874e6	0.597726
$587$	− 9.29097e6i	− 1.11293i	−0.830873	−	0.556463i	$-0.812158\pi$
$587$	− 9.29097e6i	− 1.11293i	0.830873	−	0.556463i	$-0.187842\pi$
$588$	− 777925.i	− 0.0927885i
$589$	195325.	0.0231990
$590$	0	0
$591$	−358907.	−0.0422682
$592$	1.66320e6i	0.195047i
$593$	6.88126e6i	0.803584i	0.915731	+	0.401792i	$0.131613\pi$
$593$	6.88126e6i	0.803584i	−0.915731	+	0.401792i	$0.868387\pi$
$594$	1.66929e6	0.194118
$595$	0	0
$596$	−1.05568e7	−1.21736
$597$	− 1.11475e6i	− 0.128009i
$598$	749020.i	0.0856526i
$599$	−2.36690e6	−0.269533	−0.134767	−	0.990877i	$-0.543028\pi$
$599$	−2.36690e6	−0.269533	−0.134767	+	0.990877i	$0.543028\pi$
$600$	0	0
$601$	−1.66780e7	−1.88346	−0.941731	−	0.336368i	$-0.890801\pi$
$601$	−1.66780e7	−1.88346	−0.941731	+	0.336368i	$0.890801\pi$
$602$	2.17534e6i	0.244645i
$603$	− 3.76206e6i	− 0.421340i
$604$	−618772.	−0.0690142
$605$	0	0
$606$	711451.	0.0786979
$607$	− 148356.i	− 0.0163430i	−0.999967	−	0.00817152i	$-0.997399\pi$
$607$	− 148356.i	− 0.0163430i	0.999967	−	0.00817152i	$-0.00260110\pi$
$608$	1.53448e6i	0.168346i
$609$	−1.19889e6	−0.130989
$610$	0	0
$611$	2.61771e6	0.283674
$612$	− 1.51285e7i	− 1.63275i
$613$	1.61218e7i	1.73286i	0.499301	+	0.866429i	$0.333590\pi$
$613$	1.61218e7i	1.73286i	−0.499301	+	0.866429i	$0.666410\pi$
$614$	530225.	0.0567595
$615$	0	0
$616$	−6.07506e6	−0.645058
$617$	9.27027e6i	0.980346i	0.871625	+	0.490173i	$0.163066\pi$
$617$	9.27027e6i	0.980346i	−0.871625	+	0.490173i	$0.836934\pi$
$618$	759992.i	0.0800457i
$619$	2.39889e6	0.251643	0.125821	−	0.992053i	$-0.459843\pi$
$619$	2.39889e6	0.251643	0.125821	+	0.992053i	$0.459843\pi$
$620$	0	0
$621$	2.51714e6	0.261926
$622$	2.64365e6i	0.273986i
$623$	1.05165e7i	1.08555i
$624$	−258176.	−0.0265433
$625$	0	0
$626$	2.64546e6	0.269814
$627$	− 455840.i	− 0.0463067i
$628$	6.48934e6i	0.656600i
$629$	6.55805e6	0.660919
$630$	0	0
$631$	5.93145e6	0.593045	0.296523	−	0.955026i	$-0.404173\pi$
$631$	5.93145e6	0.593045	0.296523	+	0.955026i	$0.404173\pi$
$632$	− 8.36240e6i	− 0.832796i
$633$	− 1.00742e6i	− 0.0999317i
$634$	4.77247e6	0.471542
$635$	0	0
$636$	−673511.	−0.0660240
$637$	− 1.87457e6i	− 0.183043i
$638$	− 8.36530e6i	− 0.813635i
$639$	1.42328e7	1.37892
$640$	0	0
$641$	−1.85865e7	−1.78670	−0.893351	−	0.449360i	$-0.851652\pi$
$641$	−1.85865e7	−1.78670	−0.893351	+	0.449360i	$0.851652\pi$
$642$	− 184594.i	− 0.0176758i
$643$	1.36174e7i	1.29887i	0.760416	+	0.649437i	$0.224995\pi$
$643$	1.36174e7i	1.29887i	−0.760416	+	0.649437i	$0.775005\pi$
$644$	−4.21701e6	−0.400673
$645$	0	0
$646$	1.44360e6	0.136102
$647$	− 1.49003e6i	− 0.139937i	−0.997549	−	0.0699686i	$-0.977710\pi$
$647$	− 1.49003e6i	− 0.139937i	0.997549	−	0.0699686i	$-0.0222899\pi$
$648$	6.98043e6i	0.653047i
$649$	−2.51143e7	−2.34050
$650$	0	0
$651$	133624.	0.0123576
$652$	1.38716e7i	1.27793i
$653$	− 1.02829e7i	− 0.943698i	−0.881679	−	0.471849i	$-0.843587\pi$
$653$	− 1.02829e7i	− 0.943698i	0.881679	−	0.471849i	$-0.156413\pi$
$654$	−297785.	−0.0272244
$655$	0	0
$656$	4.89724e6	0.444316
$657$	− 8.38485e6i	− 0.757848i
$658$	− 2.53951e6i	− 0.228657i
$659$	−1.29392e7	−1.16063	−0.580313	−	0.814393i	$-0.697070\pi$
$659$	−1.29392e7	−1.16063	−0.580313	+	0.814393i	$0.697070\pi$
$660$	0	0
$661$	1.78038e7	1.58492	0.792461	−	0.609922i	$-0.208799\pi$
$661$	1.78038e7	1.58492	0.792461	+	0.609922i	$0.208799\pi$
$662$	− 1.63788e6i	− 0.145257i
$663$	1.01800e6i	0.0899422i
$664$	−1.10613e7	−0.973608
$665$	0	0
$666$	−1.43412e6	−0.125285
$667$	− 1.26141e7i	− 1.09785i
$668$	− 8.38679e6i	− 0.727201i
$669$	−1.52801e6	−0.131996
$670$	0	0
$671$	2.03636e7	1.74601
$672$	1.04976e6i	0.0896740i
$673$	5.55797e6i	0.473019i	0.971629	+	0.236509i	$0.0760034\pi$
$673$	5.55797e6i	0.473019i	−0.971629	+	0.236509i	$0.923997\pi$
$674$	−2.38952e6	−0.202610
$675$	0	0
$676$	−779615.	−0.0656165
$677$	− 1.41700e7i	− 1.18822i	−0.804384	−	0.594110i	$-0.797504\pi$
$677$	− 1.41700e7i	− 1.18822i	0.804384	−	0.594110i	$-0.202496\pi$
$678$	− 1.01242e6i	− 0.0845833i
$679$	−605757.	−0.0504225
$680$	0	0
$681$	−1.21540e6	−0.100427
$682$	932367.i	0.0767584i
$683$	− 2.01906e7i	− 1.65614i	−0.560621	−	0.828072i	$-0.689438\pi$
$683$	− 2.01906e7i	− 1.65614i	0.560621	−	0.828072i	$-0.310562\pi$
$684$	1.83207e6	0.149727
$685$	0	0
$686$	−4.57409e6	−0.371103
$687$	− 2.32706e6i	− 0.188112i
$688$	− 7.88911e6i	− 0.635414i
$689$	−1.62297e6	−0.130245
$690$	0	0
$691$	−8.18523e6	−0.652132	−0.326066	−	0.945347i	$-0.605723\pi$
$691$	−8.18523e6	−0.652132	−0.326066	+	0.945347i	$0.605723\pi$
$692$	− 125513.i	− 0.00996378i
$693$	1.11675e7i	0.883329i
$694$	2.73043e6	0.215195
$695$	0	0
$696$	−2.03947e6	−0.159586
$697$	− 1.93100e7i	− 1.50557i
$698$	1.46386e6i	0.113726i
$699$	−1.91734e6	−0.148425
$700$	0	0
$701$	−6.38587e6	−0.490823	−0.245411	−	0.969419i	$-0.578923\pi$
$701$	−6.38587e6	−0.490823	−0.245411	+	0.969419i	$0.578923\pi$
$702$	− 451450.i	− 0.0345754i
$703$	794180.i	0.0606081i
$704$	4.56501e6	0.347144
$705$	0	0
$706$	−2.28739e6	−0.172714
$707$	9.65208e6i	0.726227i
$708$	2.81861e6i	0.211326i
$709$	−1.36770e7	−1.02182	−0.510911	−	0.859633i	$-0.670692\pi$
$709$	−1.36770e7	−1.02182	−0.510911	+	0.859633i	$0.670692\pi$
$710$	0	0
$711$	−1.53722e7	−1.14041
$712$	1.78900e7i	1.32254i
$713$	1.40593e6i	0.103571i
$714$	987585.	0.0724985
$715$	0	0
$716$	−1.52107e7	−1.10883
$717$	2.33597e6i	0.169695i
$718$	− 4.21417e6i	− 0.305071i
$719$	−1.17397e6	−0.0846907	−0.0423453	−	0.999103i	$-0.513483\pi$
$719$	−1.17397e6	−0.0846907	−0.0423453	+	0.999103i	$0.513483\pi$
$720$	0	0
$721$	−1.03106e7	−0.738664
$722$	− 5.19524e6i	− 0.370905i
$723$	1.47910e6i	0.105233i
$724$	−1.12176e7	−0.795342
$725$	0	0
$726$	1.27851e6	0.0900248
$727$	− 2.09397e7i	− 1.46938i	−0.678402	−	0.734691i	$-0.737327\pi$
$727$	− 2.09397e7i	− 1.46938i	0.678402	−	0.734691i	$-0.262673\pi$
$728$	1.64297e6i	0.114895i
$729$	1.20628e7	0.840675
$730$	0	0
$731$	−3.11071e7	−2.15311
$732$	− 2.28543e6i	− 0.157648i
$733$	− 8.07907e6i	− 0.555394i	−0.960669	−	0.277697i	$-0.910429\pi$
$733$	− 8.07907e6i	− 0.555394i	0.960669	−	0.277697i	$-0.0895712\pi$
$734$	−7.25013e6	−0.496713
$735$	0	0
$736$	−1.10450e7	−0.751576
$737$	− 9.94462e6i	− 0.674403i
$738$	4.22273e6i	0.285399i
$739$	1.66515e7	1.12161	0.560806	−	0.827947i	$-0.310491\pi$
$739$	1.66515e7	1.12161	0.560806	+	0.827947i	$0.310491\pi$
$740$	0	0
$741$	−123280.	−0.00824794
$742$	1.57448e6i	0.104985i
$743$	6.99037e6i	0.464545i	0.972651	+	0.232273i	$0.0746162\pi$
$743$	6.99037e6i	0.464545i	−0.972651	+	0.232273i	$0.925384\pi$
$744$	227312.	0.0150553
$745$	0	0
$746$	3.60810e6	0.237373
$747$	2.03334e7i	1.33324i
$748$	− 3.99908e7i	− 2.61340i
$749$	2.50434e6	0.163113
$750$	0	0
$751$	1.16465e7	0.753520	0.376760	−	0.926311i	$-0.377038\pi$
$751$	1.16465e7	0.753520	0.376760	+	0.926311i	$0.377038\pi$
$752$	9.20979e6i	0.593889i
$753$	4.56762e6i	0.293564i
$754$	−2.26235e6	−0.144921
$755$	0	0
$756$	2.54168e6	0.161740
$757$	905942.i	0.0574593i	0.999587	+	0.0287297i	$0.00914619\pi$
$757$	905942.i	0.0574593i	−0.999587	+	0.0287297i	$0.990854\pi$
$758$	1.18616e7i	0.749845i
$759$	3.28109e6	0.206735
$760$	0	0
$761$	1.01329e7	0.634267	0.317134	−	0.948381i	$-0.397280\pi$
$761$	1.01329e7	0.634267	0.317134	+	0.948381i	$0.397280\pi$
$762$	1.61993e6i	0.101067i
$763$	− 4.03997e6i	− 0.251227i
$764$	−9.86280e6	−0.611317
$765$	0	0
$766$	3.03625e6	0.186967
$767$	6.79204e6i	0.416881i
$768$	451378.i	0.0276145i
$769$	−1.25661e7	−0.766274	−0.383137	−	0.923692i	$-0.625156\pi$
$769$	−1.25661e7	−0.766274	−0.383137	+	0.923692i	$0.625156\pi$
$770$	0	0
$771$	−4.12075e6	−0.249655
$772$	− 4.07529e6i	− 0.246102i
$773$	− 3.34145e6i	− 0.201135i	−0.994930	−	0.100567i	$-0.967934\pi$
$773$	− 3.34145e6i	− 0.201135i	0.994930	−	0.100567i	$-0.0320658\pi$
$774$	6.80253e6	0.408148
$775$	0	0
$776$	−1.03047e6	−0.0614302
$777$	543309.i	0.0322845i
$778$	643132.i	0.0380935i
$779$	2.33844e6	0.138065
$780$	0	0
$781$	3.76231e7	2.20712
$782$	1.03909e7i	0.607625i
$783$	7.60281e6i	0.443169i
$784$	6.59523e6	0.383213
$785$	0	0
$786$	−1.20678e6	−0.0696743
$787$	− 1.19270e7i	− 0.686424i	−0.939258	−	0.343212i	$-0.888485\pi$
$787$	− 1.19270e7i	− 0.686424i	0.939258	−	0.343212i	$-0.111515\pi$
$788$	− 3.81306e6i	− 0.218755i
$789$	4.29703e6	0.245740
$790$	0	0
$791$	1.37352e7	0.780537
$792$	1.89974e7i	1.07617i
$793$	− 5.50722e6i	− 0.310992i
$794$	−3.52554e6	−0.198461
$795$	0	0
$796$	1.18432e7	0.662498
$797$	1.42516e7i	0.794729i	0.917661	+	0.397365i	$0.130075\pi$
$797$	1.42516e7i	0.794729i	−0.917661	+	0.397365i	$0.869925\pi$
$798$	119596.i	0.00664831i
$799$	3.63146e7	2.01240
$800$	0	0
$801$	3.28863e7	1.81106
$802$	1.88897e6i	0.103703i
$803$	− 2.21645e7i	− 1.21302i
$804$	−1.11610e6	−0.0608923
$805$	0	0
$806$	252154.	0.0136719
$807$	− 166862.i	− 0.00901932i
$808$	1.64194e7i	0.884770i
$809$	2.13109e7	1.14480	0.572400	−	0.819974i	$-0.306012\pi$
$809$	2.13109e7	1.14480	0.572400	+	0.819974i	$0.306012\pi$
$810$	0	0
$811$	1.86425e7	0.995294	0.497647	−	0.867380i	$-0.334198\pi$
$811$	1.86425e7	0.995294	0.497647	+	0.867380i	$0.334198\pi$
$812$	− 1.27371e7i	− 0.677924i
$813$	− 3.54437e6i	− 0.188067i
$814$	−3.79095e6	−0.200534
$815$	0	0
$816$	−3.58158e6	−0.188300
$817$	− 3.76707e6i	− 0.197446i
$818$	− 1.03737e6i	− 0.0542064i
$819$	3.02019e6	0.157335
$820$	0	0
$821$	−3.17151e7	−1.64213	−0.821066	−	0.570833i	$-0.806620\pi$
$821$	−3.17151e7	−1.64213	−0.821066	+	0.570833i	$0.806620\pi$
$822$	1.11925e6i	0.0577762i
$823$	3.21047e7i	1.65222i	0.563506	+	0.826112i	$0.309452\pi$
$823$	3.21047e7i	1.65222i	−0.563506	+	0.826112i	$0.690548\pi$
$824$	−1.75397e7	−0.899922
$825$	0	0
$826$	6.58912e6	0.336029
$827$	3.37394e7i	1.71543i	0.514123	+	0.857717i	$0.328118\pi$
$827$	3.37394e7i	1.71543i	−0.514123	+	0.857717i	$0.671882\pi$
$828$	1.31870e7i	0.668454i
$829$	−7.35563e6	−0.371735	−0.185868	−	0.982575i	$-0.559510\pi$
$829$	−7.35563e6	−0.371735	−0.185868	+	0.982575i	$0.559510\pi$
$830$	0	0
$831$	6.18067e6	0.310480
$832$	− 1.23458e6i	− 0.0618318i
$833$	− 2.60053e7i	− 1.29852i
$834$	64164.0	0.00319430
$835$	0	0
$836$	4.84288e6	0.239656
$837$	− 847383.i	− 0.0418086i
$838$	− 3.43384e6i	− 0.168916i
$839$	1.82258e7	0.893885	0.446942	−	0.894563i	$-0.352513\pi$
$839$	1.82258e7	0.893885	0.446942	+	0.894563i	$0.352513\pi$
$840$	0	0
$841$	1.75888e7	0.857523
$842$	− 5.72298e6i	− 0.278190i
$843$	4.38741e6i	0.212637i
$844$	1.07030e7	0.517187
$845$	0	0
$846$	−7.94130e6	−0.381475
$847$	1.73452e7i	0.830751i
$848$	− 5.71001e6i	− 0.272676i
$849$	1.31967e6	0.0628340
$850$	0	0
$851$	−5.71643e6	−0.270583
$852$	− 4.22249e6i	− 0.199283i
$853$	1.85516e7i	0.872987i	0.899707	+	0.436494i	$0.143780\pi$
$853$	1.85516e7i	0.872987i	−0.899707	+	0.436494i	$0.856220\pi$
$854$	−5.34269e6	−0.250677
$855$	0	0
$856$	4.26021e6	0.198722
$857$	710710.i	0.0330552i	0.999863	+	0.0165276i	$0.00526115\pi$
$857$	710710.i	0.0330552i	−0.999863	+	0.0165276i	$0.994739\pi$
$858$	− 588466.i	− 0.0272900i
$859$	2.15616e7	0.997006	0.498503	−	0.866888i	$-0.333883\pi$
$859$	2.15616e7	0.997006	0.498503	+	0.866888i	$0.333883\pi$
$860$	0	0
$861$	1.59976e6	0.0735440
$862$	− 4.46032e6i	− 0.204455i
$863$	− 1.43049e7i	− 0.653817i	−0.945056	−	0.326909i	$-0.893993\pi$
$863$	− 1.43049e7i	− 0.653817i	0.945056	−	0.326909i	$-0.106007\pi$
$864$	6.65708e6	0.303389
$865$	0	0
$866$	7.27773e6	0.329762
$867$	1.04743e7i	0.473235i
$868$	1.41964e6i	0.0639554i
$869$	−4.06349e7	−1.82536
$870$	0	0
$871$	−2.68947e6	−0.120122
$872$	− 6.87252e6i	− 0.306073i
$873$	1.89427e6i	0.0841212i
$874$	−1.25833e6	−0.0557208
$875$	0	0
$876$	−2.48755e6	−0.109525
$877$	− 3.64324e7i	− 1.59952i	−0.600322	−	0.799759i	$-0.704961\pi$
$877$	− 3.64324e7i	− 1.59952i	0.600322	−	0.799759i	$-0.295039\pi$
$878$	− 1.28888e6i	− 0.0564257i
$879$	−5.88640e6	−0.256967
$880$	0	0
$881$	−1.29195e7	−0.560797	−0.280398	−	0.959884i	$-0.590467\pi$
$881$	−1.29195e7	−0.560797	−0.280398	+	0.959884i	$0.590467\pi$
$882$	5.68685e6i	0.246150i
$883$	1.74399e7i	0.752735i	0.926471	+	0.376367i	$0.122827\pi$
$883$	1.74399e7i	0.752735i	−0.926471	+	0.376367i	$0.877173\pi$
$884$	−1.08153e7	−0.465488
$885$	0	0
$886$	70941.9	0.00303612
$887$	− 5.45819e6i	− 0.232938i	−0.993194	−	0.116469i	$-0.962843\pi$
$887$	− 5.45819e6i	− 0.232938i	0.993194	−	0.116469i	$-0.0371575\pi$
$888$	924240.i	0.0393326i
$889$	−2.19771e7	−0.932645
$890$	0	0
$891$	3.39195e7	1.43138
$892$	− 1.62338e7i	− 0.683136i
$893$	4.39769e6i	0.184542i
$894$	−2.15503e6	−0.0901798
$895$	0	0
$896$	−1.42722e7	−0.593910
$897$	− 887355.i	− 0.0368227i
$898$	− 2.48024e6i	− 0.102637i
$899$	−4.24649e6	−0.175239
$900$	0	0
$901$	−2.25148e7	−0.923967
$902$	1.11624e7i	0.456815i
$903$	− 2.57710e6i	− 0.105175i
$904$	2.33654e7	0.950937
$905$	0	0
$906$	−126314.	−0.00511246
$907$	3.84724e7i	1.55285i	0.630207	+	0.776427i	$0.282970\pi$
$907$	3.84724e7i	1.55285i	−0.630207	+	0.776427i	$0.717030\pi$
$908$	− 1.29125e7i	− 0.519753i
$909$	3.01831e7	1.21158
$910$	0	0
$911$	2.86336e7	1.14309	0.571543	−	0.820572i	$-0.306345\pi$
$911$	2.86336e7	1.14309	0.571543	+	0.820572i	$0.306345\pi$
$912$	− 433729.i	− 0.0172676i
$913$	5.37492e7i	2.13400i
$914$	−2.74158e6	−0.108551
$915$	0	0
$916$	2.47229e7	0.973555
$917$	− 1.63721e7i	− 0.642957i
$918$	− 6.26280e6i	− 0.245280i
$919$	2.14158e7	0.836460	0.418230	−	0.908341i	$-0.362651\pi$
$919$	2.14158e7	0.836460	0.418230	+	0.908341i	$0.362651\pi$
$920$	0	0
$921$	−628150.	−0.0244014
$922$	5.17296e6i	0.200406i
$923$	− 1.01750e7i	− 0.393123i
$924$	3.31308e6	0.127659
$925$	0	0
$926$	1.14441e7	0.438586
$927$	3.22424e7i	1.23233i
$928$	− 3.33606e7i	− 1.27164i
$929$	−1.93670e6	−0.0736245	−0.0368123	−	0.999322i	$-0.511720\pi$
$929$	−1.93670e6	−0.0736245	−0.0368123	+	0.999322i	$0.511720\pi$
$930$	0	0
$931$	3.14923e6	0.119078
$932$	− 2.03700e7i	− 0.768159i
$933$	− 3.13190e6i	− 0.117789i
$934$	−1.18631e7	−0.444970
$935$	0	0
$936$	5.13774e6	0.191683
$937$	− 5.93574e6i	− 0.220865i	−0.993884	−	0.110432i	$-0.964776\pi$
$937$	− 5.93574e6i	− 0.220865i	0.993884	−	0.110432i	$-0.0352235\pi$
$938$	2.60912e6i	0.0968249i
$939$	−3.13404e6	−0.115995
$940$	0	0
$941$	4.84523e7	1.78378	0.891888	−	0.452256i	$-0.149381\pi$
$941$	4.84523e7	1.78378	0.891888	+	0.452256i	$0.149381\pi$
$942$	1.32471e6i	0.0486399i
$943$	1.68319e7i	0.616387i
$944$	−2.38961e7	−0.872766
$945$	0	0
$946$	1.79818e7	0.653289
$947$	1.16568e7i	0.422382i	0.977445	+	0.211191i	$0.0677341\pi$
$947$	1.16568e7i	0.422382i	−0.977445	+	0.211191i	$0.932266\pi$
$948$	4.56050e6i	0.164813i
$949$	−5.99428e6	−0.216059
$950$	0	0
$951$	−5.65389e6	−0.202720
$952$	2.27923e7i	0.815072i
$953$	− 2.70508e7i	− 0.964825i	−0.875944	−	0.482413i	$-0.839761\pi$
$953$	− 2.70508e7i	− 0.964825i	0.875944	−	0.482413i	$-0.160239\pi$
$954$	4.92356e6	0.175149
$955$	0	0
$956$	−2.48176e7	−0.878243
$957$	9.91026e6i	0.349788i
$958$	− 1.80357e7i	− 0.634919i
$959$	−1.51847e7	−0.533161
$960$	0	0
$961$	−2.81559e7	−0.983468
$962$	1.02524e6i	0.0357182i
$963$	− 7.83133e6i	− 0.272126i
$964$	−1.57140e7	−0.544622
$965$	0	0
$966$	−860844.	−0.0296812
$967$	6.67822e6i	0.229665i	0.993385	+	0.114832i	$0.0366331\pi$
$967$	6.67822e6i	0.229665i	−0.993385	+	0.114832i	$0.963367\pi$
$968$	2.95065e7i	1.01211i
$969$	−1.71021e6	−0.0585114
$970$	0	0
$971$	−1.10643e7	−0.376597	−0.188298	−	0.982112i	$-0.560297\pi$
$971$	−1.10643e7	−0.376597	−0.188298	+	0.982112i	$0.560297\pi$
$972$	− 1.19769e7i	− 0.406610i
$973$	870497.i	0.0294771i
$974$	2.68543e6	0.0907020
$975$	0	0
$976$	1.93758e7	0.651082
$977$	4.37043e7i	1.46483i	0.680857	+	0.732416i	$0.261607\pi$
$977$	4.37043e7i	1.46483i	−0.680857	+	0.732416i	$0.738393\pi$
$978$	2.83169e6i	0.0946670i
$979$	8.69315e7	2.89882
$980$	0	0
$981$	−1.26334e7	−0.419130
$982$	1.02473e6i	0.0339103i
$983$	− 5.65161e7i	− 1.86547i	−0.360563	−	0.932735i	$-0.617415\pi$
$983$	− 5.65161e7i	− 1.86547i	0.360563	−	0.932735i	$-0.382585\pi$
$984$	2.72140e6	0.0895994
$985$	0	0
$986$	−3.13848e7	−1.02808
$987$	3.00852e6i	0.0983015i
$988$	− 1.30973e6i	− 0.0426865i
$989$	2.71150e7	0.881492
$990$	0	0
$991$	2.40368e7	0.777487	0.388744	−	0.921346i	$-0.372909\pi$
$991$	2.40368e7	0.777487	0.388744	+	0.921346i	$0.372909\pi$
$992$	3.71826e6i	0.119967i
$993$	1.94038e6i	0.0624473i
$994$	−9.87098e6	−0.316880
$995$	0	0
$996$	6.03234e6	0.192680
$997$	4.18542e7i	1.33353i	0.745270	+	0.666763i	$0.232321\pi$
$997$	4.18542e7i	1.33353i	−0.745270	+	0.666763i	$0.767679\pi$
$998$	− 7.45166e6i	− 0.236825i
$999$	3.44541e6	0.109226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.f.274.7		12
5.2	odd	4		65.6.a.e.1.3	✓	6
5.3	odd	4		325.6.a.f.1.4		6
5.4	even	2	inner	325.6.b.f.274.6		12
15.2	even	4		585.6.a.k.1.4		6
20.7	even	4		1040.6.a.r.1.4		6
65.12	odd	4		845.6.a.g.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.e.1.3	✓	6	5.2	odd	4
325.6.a.f.1.4		6	5.3	odd	4
325.6.b.f.274.6		12	5.4	even	2	inner
325.6.b.f.274.7		12	1.1	even	1	trivial
585.6.a.k.1.4		6	15.2	even	4
845.6.a.g.1.4		6	65.12	odd	4
1040.6.a.r.1.4		6	20.7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.16876i q^{2} -2.56930i q^{3} +27.2965 q^{4} +5.57220 q^{6} +75.5966i q^{7} +128.600i q^{8} +236.399 q^{9} +624.896 q^{11} -70.1329i q^{12} -169.000i q^{13} -163.951 q^{14} +594.585 q^{16} -2344.47i q^{17} +512.692i q^{18} +283.916 q^{19} +194.231 q^{21} +1355.25i q^{22} +2043.60i q^{23} +330.412 q^{24} +366.520 q^{26} -1231.72i q^{27} +2063.52i q^{28} -6172.51 q^{29} +687.967 q^{31} +5404.71i q^{32} -1605.55i q^{33} +5084.60 q^{34} +6452.85 q^{36} +2797.24i q^{37} +615.745i q^{38} -434.212 q^{39} +8236.40 q^{41} +421.239i q^{42} -13268.3i q^{43} +17057.5 q^{44} -4432.07 q^{46} +15489.4i q^{47} -1527.67i q^{48} +11092.1 q^{49} -6023.66 q^{51} -4613.11i q^{52} -9603.36i q^{53} +2671.30 q^{54} -9721.71 q^{56} -729.465i q^{57} -13386.7i q^{58} -40189.6 q^{59} +32587.1 q^{61} +1492.03i q^{62} +17870.9i q^{63} +7305.22 q^{64} +3482.05 q^{66} -15914.0i q^{67} -63995.9i q^{68} +5250.62 q^{69} +60206.9 q^{71} +30400.8i q^{72} -35469.1i q^{73} -6066.54 q^{74} +7749.90 q^{76} +47240.1i q^{77} -941.701i q^{78} -65026.6 q^{79} +54280.2 q^{81} +17862.8i q^{82} +86013.0i q^{83} +5301.81 q^{84} +28775.7 q^{86} +15859.1i q^{87} +80361.5i q^{88} +139114. q^{89} +12775.8 q^{91} +55783.0i q^{92} -1767.59i q^{93} -33592.8 q^{94} +13886.3 q^{96} +8013.02i q^{97} +24056.2i q^{98} +147725. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} - 104 q^{6} - 2068 q^{9} + 1600 q^{11} - 4216 q^{14} + 644 q^{16} - 10608 q^{19} + 2144 q^{21} - 9512 q^{24} - 676 q^{26} + 7328 q^{29} + 33328 q^{31} - 2760 q^{34} + 3636 q^{36} + 6760 q^{39}+ \cdots - 602528 q^{99}+O(q^{100}) \) 12 * q - 268 * q^4 - 104 * q^6 - 2068 * q^9 + 1600 * q^11 - 4216 * q^14 + 644 * q^16 - 10608 * q^19 + 2144 * q^21 - 9512 * q^24 - 676 * q^26 + 7328 * q^29 + 33328 * q^31 - 2760 * q^34 + 3636 * q^36 + 6760 * q^39 + 24872 * q^41 + 70248 * q^44 - 9112 * q^46 + 55684 * q^49 - 55328 * q^51 + 265824 * q^54 + 88840 * q^56 + 23792 * q^59 + 114464 * q^61 + 79828 * q^64 + 19816 * q^66 - 144808 * q^69 + 428208 * q^71 + 364208 * q^74 + 733272 * q^76 - 72048 * q^79 + 701028 * q^81 - 12048 * q^84 + 58408 * q^86 + 104056 * q^89 - 58136 * q^91 - 315144 * q^94 + 1095816 * q^96 - 602528 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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