LMFDB - Embedded newform 550.6.b.g.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(199,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, 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("550.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	550.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:	$88.2111008971$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	550.199
Dual form			550.6.b.g.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +21.0000i q^{3} -16.0000 q^{4} +84.0000 q^{6} +98.0000i q^{7} +64.0000i q^{8} -198.000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +21.0000i q^{3} -16.0000 q^{4} +84.0000 q^{6} +98.0000i q^{7} +64.0000i q^{8} -198.000 q^{9} +121.000 q^{11} -336.000i q^{12} -824.000i q^{13} +392.000 q^{14} +256.000 q^{16} +978.000i q^{17} +792.000i q^{18} +2140.00 q^{19} -2058.00 q^{21} -484.000i q^{22} -3699.00i q^{23} -1344.00 q^{24} -3296.00 q^{26} +945.000i q^{27} -1568.00i q^{28} -3480.00 q^{29} -7813.00 q^{31} -1024.00i q^{32} +2541.00i q^{33} +3912.00 q^{34} +3168.00 q^{36} -13597.0i q^{37} -8560.00i q^{38} +17304.0 q^{39} +6492.00 q^{41} +8232.00i q^{42} -14234.0i q^{43} -1936.00 q^{44} -14796.0 q^{46} -20352.0i q^{47} +5376.00i q^{48} +7203.00 q^{49} -20538.0 q^{51} +13184.0i q^{52} +366.000i q^{53} +3780.00 q^{54} -6272.00 q^{56} +44940.0i q^{57} +13920.0i q^{58} -9825.00 q^{59} +26132.0 q^{61} +31252.0i q^{62} -19404.0i q^{63} -4096.00 q^{64} +10164.0 q^{66} +17093.0i q^{67} -15648.0i q^{68} +77679.0 q^{69} -23583.0 q^{71} -12672.0i q^{72} +35176.0i q^{73} -54388.0 q^{74} -34240.0 q^{76} +11858.0i q^{77} -69216.0i q^{78} +42490.0 q^{79} -67959.0 q^{81} -25968.0i q^{82} -22674.0i q^{83} +32928.0 q^{84} -56936.0 q^{86} -73080.0i q^{87} +7744.00i q^{88} +17145.0 q^{89} +80752.0 q^{91} +59184.0i q^{92} -164073. i q^{93} -81408.0 q^{94} +21504.0 q^{96} -30727.0i q^{97} -28812.0i q^{98} -23958.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9	$ 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9} + 242 q^{11} + 784 q^{14} + 512 q^{16} + 4280 q^{19} - 4116 q^{21} - 2688 q^{24} - 6592 q^{26} - 6960 q^{29} - 15626 q^{31} + 7824 q^{34} + 6336 q^{36} + 34608 q^{39} + 12984 q^{41} - 3872 q^{44} - 29592 q^{46} + 14406 q^{49} - 41076 q^{51} + 7560 q^{54} - 12544 q^{56} - 19650 q^{59} + 52264 q^{61} - 8192 q^{64} + 20328 q^{66} + 155358 q^{69} - 47166 q^{71} - 108776 q^{74} - 68480 q^{76} + 84980 q^{79} - 135918 q^{81} + 65856 q^{84} - 113872 q^{86} + 34290 q^{89} + 161504 q^{91} - 162816 q^{94} + 43008 q^{96} - 47916 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9 + 242 * q^11 + 784 * q^14 + 512 * q^16 + 4280 * q^19 - 4116 * q^21 - 2688 * q^24 - 6592 * q^26 - 6960 * q^29 - 15626 * q^31 + 7824 * q^34 + 6336 * q^36 + 34608 * q^39 + 12984 * q^41 - 3872 * q^44 - 29592 * q^46 + 14406 * q^49 - 41076 * q^51 + 7560 * q^54 - 12544 * q^56 - 19650 * q^59 + 52264 * q^61 - 8192 * q^64 + 20328 * q^66 + 155358 * q^69 - 47166 * q^71 - 108776 * q^74 - 68480 * q^76 + 84980 * q^79 - 135918 * q^81 + 65856 * q^84 - 113872 * q^86 + 34290 * q^89 + 161504 * q^91 - 162816 * q^94 + 43008 * q^96 - 47916 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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
$3$	21.0000i	1.34715i	0.739119	+	0.673575i	$0.235242\pi$
$3$	21.0000i	1.34715i	−0.739119	+	0.673575i	$0.764758\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	84.0000	0.952579
$7$	98.0000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	98.0000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	64.0000i	0.353553i
$9$	−198.000	−0.814815
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	− 336.000i	− 0.673575i
$13$	− 824.000i	− 1.35229i	−0.736770	−	0.676143i	$-0.763650\pi$
$13$	− 824.000i	− 1.35229i	0.736770	−	0.676143i	$-0.236350\pi$
$14$	392.000	0.534522
$15$	0	0
$16$	256.000	0.250000
$17$	978.000i	0.820761i	0.911914	+	0.410380i	$0.134604\pi$
$17$	978.000i	0.820761i	−0.911914	+	0.410380i	$0.865396\pi$
$18$	792.000i	0.576161i
$19$	2140.00	1.35997	0.679986	−	0.733225i	$-0.261986\pi$
$19$	2140.00	1.35997	0.679986	+	0.733225i	$0.261986\pi$
$20$	0	0
$21$	−2058.00	−1.01835
$22$	− 484.000i	− 0.213201i
$23$	− 3699.00i	− 1.45802i	−0.684501	−	0.729012i	$-0.739980\pi$
$23$	− 3699.00i	− 1.45802i	0.684501	−	0.729012i	$-0.260020\pi$
$24$	−1344.00	−0.476290
$25$	0	0
$26$	−3296.00	−0.956211
$27$	945.000i	0.249472i
$28$	− 1568.00i	− 0.377964i
$29$	−3480.00	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−3480.00	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	−7813.00	−1.46020	−0.730102	−	0.683338i	$-0.760528\pi$
$31$	−7813.00	−1.46020	−0.730102	+	0.683338i	$0.760528\pi$
$32$	− 1024.00i	− 0.176777i
$33$	2541.00i	0.406181i
$34$	3912.00	0.580365
$35$	0	0
$36$	3168.00	0.407407
$37$	− 13597.0i	− 1.63282i	−0.577471	−	0.816411i	$-0.695961\pi$
$37$	− 13597.0i	− 1.63282i	0.577471	−	0.816411i	$-0.304039\pi$
$38$	− 8560.00i	− 0.961645i
$39$	17304.0	1.82173
$40$	0	0
$41$	6492.00	0.603141	0.301571	−	0.953444i	$-0.402489\pi$
$41$	6492.00	0.603141	0.301571	+	0.953444i	$0.402489\pi$
$42$	8232.00i	0.720082i
$43$	− 14234.0i	− 1.17397i	−0.809599	−	0.586983i	$-0.800315\pi$
$43$	− 14234.0i	− 1.17397i	0.809599	−	0.586983i	$-0.199685\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−14796.0	−1.03098
$47$	− 20352.0i	− 1.34389i	−0.740603	−	0.671943i	$-0.765460\pi$
$47$	− 20352.0i	− 1.34389i	0.740603	−	0.671943i	$-0.234540\pi$
$48$	5376.00i	0.336788i
$49$	7203.00	0.428571
$50$	0	0
$51$	−20538.0	−1.10569
$52$	13184.0i	0.676143i
$53$	366.000i	0.0178975i	0.999960	+	0.00894873i	$0.00284851\pi$
$53$	366.000i	0.0178975i	−0.999960	+	0.00894873i	$0.997151\pi$
$54$	3780.00	0.176404
$55$	0	0
$56$	−6272.00	−0.267261
$57$	44940.0i	1.83209i
$58$	13920.0i	0.543337i
$59$	−9825.00	−0.367454	−0.183727	−	0.982977i	$-0.558816\pi$
$59$	−9825.00	−0.367454	−0.183727	+	0.982977i	$0.558816\pi$
$60$	0	0
$61$	26132.0	0.899183	0.449591	−	0.893234i	$-0.351570\pi$
$61$	26132.0	0.899183	0.449591	+	0.893234i	$0.351570\pi$
$62$	31252.0i	1.03252i
$63$	− 19404.0i	− 0.615942i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	10164.0	0.287213
$67$	17093.0i	0.465191i	0.972574	+	0.232595i	$0.0747218\pi$
$67$	17093.0i	0.465191i	−0.972574	+	0.232595i	$0.925278\pi$
$68$	− 15648.0i	− 0.410380i
$69$	77679.0	1.96418
$70$	0	0
$71$	−23583.0	−0.555205	−0.277602	−	0.960696i	$-0.589540\pi$
$71$	−23583.0	−0.555205	−0.277602	+	0.960696i	$0.589540\pi$
$72$	− 12672.0i	− 0.288081i
$73$	35176.0i	0.772573i	0.922379	+	0.386286i	$0.126242\pi$
$73$	35176.0i	0.772573i	−0.922379	+	0.386286i	$0.873758\pi$
$74$	−54388.0	−1.15458
$75$	0	0
$76$	−34240.0	−0.679986
$77$	11858.0i	0.227921i
$78$	− 69216.0i	− 1.28816i
$79$	42490.0	0.765983	0.382991	−	0.923752i	$-0.374894\pi$
$79$	42490.0	0.765983	0.382991	+	0.923752i	$0.374894\pi$
$80$	0	0
$81$	−67959.0	−1.15089
$82$	− 25968.0i	− 0.426485i
$83$	− 22674.0i	− 0.361271i	−0.983550	−	0.180635i	$-0.942185\pi$
$83$	− 22674.0i	− 0.361271i	0.983550	−	0.180635i	$-0.0578154\pi$
$84$	32928.0	0.509175
$85$	0	0
$86$	−56936.0	−0.830120
$87$	− 73080.0i	− 1.03514i
$88$	7744.00i	0.106600i
$89$	17145.0	0.229436	0.114718	−	0.993398i	$-0.463403\pi$
$89$	17145.0	0.229436	0.114718	+	0.993398i	$0.463403\pi$
$90$	0	0
$91$	80752.0	1.02223
$92$	59184.0i	0.729012i
$93$	− 164073.i	− 1.96711i
$94$	−81408.0	−0.950271
$95$	0	0
$96$	21504.0	0.238145
$97$	− 30727.0i	− 0.331582i	−0.986161	−	0.165791i	$-0.946982\pi$
$97$	− 30727.0i	− 0.331582i	0.986161	−	0.165791i	$-0.0530177\pi$
$98$	− 28812.0i	− 0.303046i
$99$	−23958.0	−0.245676
$100$	0	0
$101$	138102.	1.34709	0.673545	−	0.739146i	$-0.264771\pi$
$101$	138102.	1.34709	0.673545	+	0.739146i	$0.264771\pi$
$102$	82152.0i	0.781840i
$103$	− 11864.0i	− 0.110189i	−0.998481	−	0.0550945i	$-0.982454\pi$
$103$	− 11864.0i	− 0.110189i	0.998481	−	0.0550945i	$-0.0175460\pi$
$104$	52736.0	0.478106
$105$	0	0
$106$	1464.00	0.0126554
$107$	16998.0i	0.143529i	0.997422	+	0.0717643i	$0.0228629\pi$
$107$	16998.0i	0.143529i	−0.997422	+	0.0717643i	$0.977137\pi$
$108$	− 15120.0i	− 0.124736i
$109$	221830.	1.78836	0.894178	−	0.447711i	$-0.147761\pi$
$109$	221830.	1.78836	0.894178	+	0.447711i	$0.147761\pi$
$110$	0	0
$111$	285537.	2.19966
$112$	25088.0i	0.188982i
$113$	196671.i	1.44892i	0.689317	+	0.724460i	$0.257911\pi$
$113$	196671.i	1.44892i	−0.689317	+	0.724460i	$0.742089\pi$
$114$	179760.	1.29548
$115$	0	0
$116$	55680.0	0.384197
$117$	163152.i	1.10186i
$118$	39300.0i	0.259829i
$119$	−95844.0	−0.620437
$120$	0	0
$121$	14641.0	0.0909091
$122$	− 104528.i	− 0.635818i
$123$	136332.i	0.812522i
$124$	125008.	0.730102
$125$	0	0
$126$	−77616.0	−0.435537
$127$	120548.i	0.663209i	0.943418	+	0.331605i	$0.107590\pi$
$127$	120548.i	0.663209i	−0.943418	+	0.331605i	$0.892410\pi$
$128$	16384.0i	0.0883883i
$129$	298914.	1.58151
$130$	0	0
$131$	68442.0	0.348453	0.174227	−	0.984706i	$-0.444257\pi$
$131$	68442.0	0.348453	0.174227	+	0.984706i	$0.444257\pi$
$132$	− 40656.0i	− 0.203091i
$133$	209720.i	1.02804i
$134$	68372.0	0.328940
$135$	0	0
$136$	−62592.0	−0.290183
$137$	− 373647.i	− 1.70083i	−0.526115	−	0.850413i	$-0.676352\pi$
$137$	− 373647.i	− 1.70083i	0.526115	−	0.850413i	$-0.323648\pi$
$138$	− 310716.i	− 1.38888i
$139$	60610.0	0.266077	0.133038	−	0.991111i	$-0.457527\pi$
$139$	60610.0	0.266077	0.133038	+	0.991111i	$0.457527\pi$
$140$	0	0
$141$	427392.	1.81042
$142$	94332.0i	0.392589i
$143$	− 99704.0i	− 0.407730i
$144$	−50688.0	−0.203704
$145$	0	0
$146$	140704.	0.546291
$147$	151263.i	0.577350i
$148$	217552.i	0.816411i
$149$	438030.	1.61636	0.808180	−	0.588935i	$-0.200453\pi$
$149$	438030.	1.61636	0.808180	+	0.588935i	$0.200453\pi$
$150$	0	0
$151$	−239398.	−0.854433	−0.427217	−	0.904149i	$-0.640506\pi$
$151$	−239398.	−0.854433	−0.427217	+	0.904149i	$0.640506\pi$
$152$	136960.i	0.480822i
$153$	− 193644.i	− 0.668768i
$154$	47432.0	0.161165
$155$	0	0
$156$	−276864.	−0.910867
$157$	62153.0i	0.201239i	0.994925	+	0.100620i	$0.0320825\pi$
$157$	62153.0i	0.201239i	−0.994925	+	0.100620i	$0.967917\pi$
$158$	− 169960.i	− 0.541632i
$159$	−7686.00	−0.0241106
$160$	0	0
$161$	362502.	1.10216
$162$	271836.i	0.813803i
$163$	− 298724.i	− 0.880645i	−0.897840	−	0.440323i	$-0.854864\pi$
$163$	− 298724.i	− 0.880645i	0.897840	−	0.440323i	$-0.145136\pi$
$164$	−103872.	−0.301571
$165$	0	0
$166$	−90696.0	−0.255457
$167$	82728.0i	0.229542i	0.993392	+	0.114771i	$0.0366134\pi$
$167$	82728.0i	0.229542i	−0.993392	+	0.114771i	$0.963387\pi$
$168$	− 131712.i	− 0.360041i
$169$	−307683.	−0.828680
$170$	0	0
$171$	−423720.	−1.10812
$172$	227744.i	0.586983i
$173$	− 135834.i	− 0.345059i	−0.985004	−	0.172529i	$-0.944806\pi$
$173$	− 135834.i	− 0.345059i	0.985004	−	0.172529i	$-0.0551940\pi$
$174$	−292320.	−0.731957
$175$	0	0
$176$	30976.0	0.0753778
$177$	− 206325.i	− 0.495015i
$178$	− 68580.0i	− 0.162236i
$179$	−112725.	−0.262959	−0.131479	−	0.991319i	$-0.541973\pi$
$179$	−112725.	−0.262959	−0.131479	+	0.991319i	$0.541973\pi$
$180$	0	0
$181$	593807.	1.34725	0.673626	−	0.739072i	$-0.264736\pi$
$181$	593807.	1.34725	0.673626	+	0.739072i	$0.264736\pi$
$182$	− 323008.i	− 0.722828i
$183$	548772.i	1.21133i
$184$	236736.	0.515489
$185$	0	0
$186$	−656292.	−1.39096
$187$	118338.i	0.247469i
$188$	325632.i	0.671943i
$189$	−92610.0	−0.188583
$190$	0	0
$191$	652557.	1.29430	0.647150	−	0.762363i	$-0.275961\pi$
$191$	652557.	1.29430	0.647150	+	0.762363i	$0.275961\pi$
$192$	− 86016.0i	− 0.168394i
$193$	− 402164.i	− 0.777159i	−0.921415	−	0.388580i	$-0.872966\pi$
$193$	− 402164.i	− 0.777159i	0.921415	−	0.388580i	$-0.127034\pi$
$194$	−122908.	−0.234464
$195$	0	0
$196$	−115248.	−0.214286
$197$	− 268482.i	− 0.492890i	−0.969157	−	0.246445i	$-0.920738\pi$
$197$	− 268482.i	− 0.492890i	0.969157	−	0.246445i	$-0.0792624\pi$
$198$	95832.0i	0.173719i
$199$	581200.	1.04038	0.520191	−	0.854050i	$-0.325861\pi$
$199$	581200.	1.04038	0.520191	+	0.854050i	$0.325861\pi$
$200$	0	0
$201$	−358953.	−0.626682
$202$	− 552408.i	− 0.952536i
$203$	− 341040.i	− 0.580852i
$204$	328608.	0.552844
$205$	0	0
$206$	−47456.0	−0.0779154
$207$	732402.i	1.18802i
$208$	− 210944.i	− 0.338072i
$209$	258940.	0.410047
$210$	0	0
$211$	−183988.	−0.284501	−0.142250	−	0.989831i	$-0.545434\pi$
$211$	−183988.	−0.284501	−0.142250	+	0.989831i	$0.545434\pi$
$212$	− 5856.00i	− 0.00894873i
$213$	− 495243.i	− 0.747944i
$214$	67992.0	0.101490
$215$	0	0
$216$	−60480.0	−0.0882018
$217$	− 765674.i	− 1.10381i
$218$	− 887320.i	− 1.26456i
$219$	−738696.	−1.04077
$220$	0	0
$221$	805872.	1.10990
$222$	− 1.14215e6i	− 1.55539i
$223$	− 631679.i	− 0.850617i	−0.905048	−	0.425309i	$-0.860166\pi$
$223$	− 631679.i	− 0.850617i	0.905048	−	0.425309i	$-0.139834\pi$
$224$	100352.	0.133631
$225$	0	0
$226$	786684.	1.02454
$227$	1398.00i	0.00180070i	1.00000		0.000900352i	$0.000286591\pi$
$227$	1398.00i	0.00180070i	−1.00000		0.000900352i	$0.999713\pi$
$228$	− 719040.i	− 0.916043i
$229$	−206135.	−0.259754	−0.129877	−	0.991530i	$-0.541458\pi$
$229$	−206135.	−0.259754	−0.129877	+	0.991530i	$0.541458\pi$
$230$	0	0
$231$	−249018.	−0.307044
$232$	− 222720.i	− 0.271668i
$233$	− 640704.i	− 0.773157i	−0.922257	−	0.386578i	$-0.873657\pi$
$233$	− 640704.i	− 0.773157i	0.922257	−	0.386578i	$-0.126343\pi$
$234$	652608.	0.779135
$235$	0	0
$236$	157200.	0.183727
$237$	892290.i	1.03189i
$238$	383376.i	0.438715i
$239$	−128250.	−0.145232	−0.0726161	−	0.997360i	$-0.523135\pi$
$239$	−128250.	−0.145232	−0.0726161	+	0.997360i	$0.523135\pi$
$240$	0	0
$241$	−1.69749e6	−1.88263	−0.941313	−	0.337535i	$-0.890407\pi$
$241$	−1.69749e6	−1.88263	−0.941313	+	0.337535i	$0.890407\pi$
$242$	− 58564.0i	− 0.0642824i
$243$	− 1.19750e6i	− 1.30095i
$244$	−418112.	−0.449591
$245$	0	0
$246$	545328.	0.574540
$247$	− 1.76336e6i	− 1.83907i
$248$	− 500032.i	− 0.516260i
$249$	476154.	0.486686
$250$	0	0
$251$	−325323.	−0.325935	−0.162967	−	0.986631i	$-0.552107\pi$
$251$	−325323.	−0.325935	−0.162967	+	0.986631i	$0.552107\pi$
$252$	310464.i	0.307971i
$253$	− 447579.i	− 0.439611i
$254$	482192.	0.468960
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.64948e6i	1.55781i	0.627144	+	0.778904i	$0.284224\pi$
$257$	1.64948e6i	1.55781i	−0.627144	+	0.778904i	$0.715776\pi$
$258$	− 1.19566e6i	− 1.11830i
$259$	1.33251e6	1.23430
$260$	0	0
$261$	689040.	0.626099
$262$	− 273768.i	− 0.246394i
$263$	1.37653e6i	1.22714i	0.789639	+	0.613571i	$0.210268\pi$
$263$	1.37653e6i	1.22714i	−0.789639	+	0.613571i	$0.789732\pi$
$264$	−162624.	−0.143607
$265$	0	0
$266$	838880.	0.726935
$267$	360045.i	0.309085i
$268$	− 273488.i	− 0.232595i
$269$	−75450.0	−0.0635739	−0.0317869	−	0.999495i	$-0.510120\pi$
$269$	−75450.0	−0.0635739	−0.0317869	+	0.999495i	$0.510120\pi$
$270$	0	0
$271$	−360568.	−0.298239	−0.149119	−	0.988819i	$-0.547644\pi$
$271$	−360568.	−0.298239	−0.149119	+	0.988819i	$0.547644\pi$
$272$	250368.i	0.205190i
$273$	1.69579e6i	1.37710i
$274$	−1.49459e6	−1.20267
$275$	0	0
$276$	−1.24286e6	−0.982089
$277$	− 418522.i	− 0.327732i	−0.986483	−	0.163866i	$-0.947604\pi$
$277$	− 418522.i	− 0.327732i	0.986483	−	0.163866i	$-0.0523965\pi$
$278$	− 242440.i	− 0.188145i
$279$	1.54697e6	1.18980
$280$	0	0
$281$	−794298.	−0.600092	−0.300046	−	0.953925i	$-0.597002\pi$
$281$	−794298.	−0.600092	−0.300046	+	0.953925i	$0.597002\pi$
$282$	− 1.70957e6i	− 1.28016i
$283$	− 1.80796e6i	− 1.34191i	−0.741498	−	0.670955i	$-0.765884\pi$
$283$	− 1.80796e6i	− 1.34191i	0.741498	−	0.670955i	$-0.234116\pi$
$284$	377328.	0.277602
$285$	0	0
$286$	−398816.	−0.288309
$287$	636216.i	0.455932i
$288$	202752.i	0.144040i
$289$	463373.	0.326352
$290$	0	0
$291$	645267.	0.446691
$292$	− 562816.i	− 0.386286i
$293$	875436.i	0.595738i	0.954607	+	0.297869i	$0.0962759\pi$
$293$	875436.i	0.595738i	−0.954607	+	0.297869i	$0.903724\pi$
$294$	605052.	0.408248
$295$	0	0
$296$	870208.	0.577290
$297$	114345.i	0.0752187i
$298$	− 1.75212e6i	− 1.14294i
$299$	−3.04798e6	−1.97167
$300$	0	0
$301$	1.39493e6	0.887436
$302$	957592.i	0.604176i
$303$	2.90014e6i	1.81473i
$304$	547840.	0.339993
$305$	0	0
$306$	−774576.	−0.472890
$307$	137468.i	0.0832445i	0.999133	+	0.0416223i	$0.0132526\pi$
$307$	137468.i	0.0832445i	−0.999133	+	0.0416223i	$0.986747\pi$
$308$	− 189728.i	− 0.113961i
$309$	249144.	0.148441
$310$	0	0
$311$	1.22629e6	0.718940	0.359470	−	0.933157i	$-0.382957\pi$
$311$	1.22629e6	0.718940	0.359470	+	0.933157i	$0.382957\pi$
$312$	1.10746e6i	0.644080i
$313$	3.13692e6i	1.80985i	0.425570	+	0.904925i	$0.360073\pi$
$313$	3.13692e6i	1.80985i	−0.425570	+	0.904925i	$0.639927\pi$
$314$	248612.	0.142298
$315$	0	0
$316$	−679840.	−0.382991
$317$	− 1.69119e6i	− 0.945243i	−0.881266	−	0.472622i	$-0.843308\pi$
$317$	− 1.69119e6i	− 0.945243i	0.881266	−	0.472622i	$-0.156692\pi$
$318$	30744.0i	0.0170488i
$319$	−421080.	−0.231680
$320$	0	0
$321$	−356958.	−0.193355
$322$	− 1.45001e6i	− 0.779347i
$323$	2.09292e6i	1.11621i
$324$	1.08734e6	0.575446
$325$	0	0
$326$	−1.19490e6	−0.622710
$327$	4.65843e6i	2.40919i
$328$	415488.i	0.213243i
$329$	1.99450e6	1.01588
$330$	0	0
$331$	−2.13901e6	−1.07311	−0.536554	−	0.843866i	$-0.680274\pi$
$331$	−2.13901e6	−1.07311	−0.536554	+	0.843866i	$0.680274\pi$
$332$	362784.i	0.180635i
$333$	2.69221e6i	1.33045i
$334$	330912.	0.162310
$335$	0	0
$336$	−526848.	−0.254588
$337$	553598.i	0.265534i	0.991147	+	0.132767i	$0.0423862\pi$
$337$	553598.i	0.265534i	−0.991147	+	0.132767i	$0.957614\pi$
$338$	1.23073e6i	0.585965i
$339$	−4.13009e6	−1.95191
$340$	0	0
$341$	−945373.	−0.440268
$342$	1.69488e6i	0.783563i
$343$	2.35298e6i	1.07990i
$344$	910976.	0.415060
$345$	0	0
$346$	−543336.	−0.243993
$347$	167208.i	0.0745475i	0.999305	+	0.0372738i	$0.0118674\pi$
$347$	167208.i	0.0745475i	−0.999305	+	0.0372738i	$0.988133\pi$
$348$	1.16928e6i	0.517572i
$349$	−469490.	−0.206330	−0.103165	−	0.994664i	$-0.532897\pi$
$349$	−469490.	−0.206330	−0.103165	+	0.994664i	$0.532897\pi$
$350$	0	0
$351$	778680.	0.337358
$352$	− 123904.i	− 0.0533002i
$353$	− 2.82154e6i	− 1.20517i	−0.798054	−	0.602586i	$-0.794137\pi$
$353$	− 2.82154e6i	− 1.20517i	0.798054	−	0.602586i	$-0.205863\pi$
$354$	−825300.	−0.350029
$355$	0	0
$356$	−274320.	−0.114718
$357$	− 2.01272e6i	− 0.835822i
$358$	450900.i	0.185940i
$359$	−1.95696e6	−0.801394	−0.400697	−	0.916211i	$-0.631232\pi$
$359$	−1.95696e6	−0.801394	−0.400697	+	0.916211i	$0.631232\pi$
$360$	0	0
$361$	2.10350e6	0.849522
$362$	− 2.37523e6i	− 0.952651i
$363$	307461.i	0.122468i
$364$	−1.29203e6	−0.511116
$365$	0	0
$366$	2.19509e6	0.856543
$367$	− 2.28592e6i	− 0.885922i	−0.896541	−	0.442961i	$-0.853928\pi$
$367$	− 2.28592e6i	− 0.885922i	0.896541	−	0.442961i	$-0.146072\pi$
$368$	− 946944.i	− 0.364506i
$369$	−1.28542e6	−0.491448
$370$	0	0
$371$	−35868.0	−0.0135292
$372$	2.62517e6i	0.983557i
$373$	− 583274.i	− 0.217070i	−0.994093	−	0.108535i	$-0.965384\pi$
$373$	− 583274.i	− 0.217070i	0.994093	−	0.108535i	$-0.0346160\pi$
$374$	473352.	0.174987
$375$	0	0
$376$	1.30253e6	0.475135
$377$	2.86752e6i	1.03909i
$378$	370440.i	0.133349i
$379$	−2.83629e6	−1.01427	−0.507135	−	0.861867i	$-0.669295\pi$
$379$	−2.83629e6	−1.01427	−0.507135	+	0.861867i	$0.669295\pi$
$380$	0	0
$381$	−2.53151e6	−0.893443
$382$	− 2.61023e6i	− 0.915208i
$383$	3.52202e6i	1.22686i	0.789749	+	0.613430i	$0.210211\pi$
$383$	3.52202e6i	1.22686i	−0.789749	+	0.613430i	$0.789789\pi$
$384$	−344064.	−0.119072
$385$	0	0
$386$	−1.60866e6	−0.549534
$387$	2.81833e6i	0.956566i
$388$	491632.i	0.165791i
$389$	1.81358e6	0.607661	0.303831	−	0.952726i	$-0.401734\pi$
$389$	1.81358e6	0.607661	0.303831	+	0.952726i	$0.401734\pi$
$390$	0	0
$391$	3.61762e6	1.19669
$392$	460992.i	0.151523i
$393$	1.43728e6i	0.469419i
$394$	−1.07393e6	−0.348526
$395$	0	0
$396$	383328.	0.122838
$397$	3.42076e6i	1.08930i	0.838665	+	0.544648i	$0.183337\pi$
$397$	3.42076e6i	1.08930i	−0.838665	+	0.544648i	$0.816663\pi$
$398$	− 2.32480e6i	− 0.735661i
$399$	−4.40412e6	−1.38493
$400$	0	0
$401$	398442.	0.123738	0.0618692	−	0.998084i	$-0.480294\pi$
$401$	398442.	0.123738	0.0618692	+	0.998084i	$0.480294\pi$
$402$	1.43581e6i	0.443131i
$403$	6.43791e6i	1.97461i
$404$	−2.20963e6	−0.673545
$405$	0	0
$406$	−1.36416e6	−0.410724
$407$	− 1.64524e6i	− 0.492314i
$408$	− 1.31443e6i	− 0.390920i
$409$	−1.40957e6	−0.416657	−0.208328	−	0.978059i	$-0.566802\pi$
$409$	−1.40957e6	−0.416657	−0.208328	+	0.978059i	$0.566802\pi$
$410$	0	0
$411$	7.84659e6	2.29127
$412$	189824.i	0.0550945i
$413$	− 962850.i	− 0.277769i
$414$	2.92961e6	0.840057
$415$	0	0
$416$	−843776.	−0.239053
$417$	1.27281e6i	0.358446i
$418$	− 1.03576e6i	− 0.289947i
$419$	1.86618e6	0.519300	0.259650	−	0.965703i	$-0.416393\pi$
$419$	1.86618e6	0.519300	0.259650	+	0.965703i	$0.416393\pi$
$420$	0	0
$421$	2.07774e6	0.571329	0.285665	−	0.958330i	$-0.407786\pi$
$421$	2.07774e6	0.571329	0.285665	+	0.958330i	$0.407786\pi$
$422$	735952.i	0.201172i
$423$	4.02970e6i	1.09502i
$424$	−23424.0	−0.00632771
$425$	0	0
$426$	−1.98097e6	−0.528877
$427$	2.56094e6i	0.679718i
$428$	− 271968.i	− 0.0717643i
$429$	2.09378e6	0.549273
$430$	0	0
$431$	−6.28436e6	−1.62955	−0.814775	−	0.579777i	$-0.803140\pi$
$431$	−6.28436e6	−1.62955	−0.814775	+	0.579777i	$0.803140\pi$
$432$	241920.i	0.0623681i
$433$	− 3.26559e6i	− 0.837031i	−0.908210	−	0.418516i	$-0.862550\pi$
$433$	− 3.26559e6i	− 0.837031i	0.908210	−	0.418516i	$-0.137450\pi$
$434$	−3.06270e6	−0.780512
$435$	0	0
$436$	−3.54928e6	−0.894178
$437$	− 7.91586e6i	− 1.98287i
$438$	2.95478e6i	0.735937i
$439$	2.64568e6	0.655203	0.327602	−	0.944816i	$-0.393760\pi$
$439$	2.64568e6	0.655203	0.327602	+	0.944816i	$0.393760\pi$
$440$	0	0
$441$	−1.42619e6	−0.349206
$442$	− 3.22349e6i	− 0.784821i
$443$	5.63531e6i	1.36430i	0.731214	+	0.682148i	$0.238954\pi$
$443$	5.63531e6i	1.36430i	−0.731214	+	0.682148i	$0.761046\pi$
$444$	−4.56859e6	−1.09983
$445$	0	0
$446$	−2.52672e6	−0.601477
$447$	9.19863e6i	2.17748i
$448$	− 401408.i	− 0.0944911i
$449$	370005.	0.0866147	0.0433074	−	0.999062i	$-0.486211\pi$
$449$	370005.	0.0866147	0.0433074	+	0.999062i	$0.486211\pi$
$450$	0	0
$451$	785532.	0.181854
$452$	− 3.14674e6i	− 0.724460i
$453$	− 5.02736e6i	− 1.15105i
$454$	5592.00	0.00127329
$455$	0	0
$456$	−2.87616e6	−0.647740
$457$	3.31891e6i	0.743369i	0.928359	+	0.371685i	$0.121220\pi$
$457$	3.31891e6i	0.743369i	−0.928359	+	0.371685i	$0.878780\pi$
$458$	824540.i	0.183674i
$459$	−924210.	−0.204757
$460$	0	0
$461$	8.77021e6	1.92202	0.961010	−	0.276515i	$-0.0891794\pi$
$461$	8.77021e6	1.92202	0.961010	+	0.276515i	$0.0891794\pi$
$462$	996072.i	0.217113i
$463$	− 224249.i	− 0.0486159i	−0.999705	−	0.0243079i	$-0.992262\pi$
$463$	− 224249.i	− 0.0486159i	0.999705	−	0.0243079i	$-0.00773822\pi$
$464$	−890880.	−0.192099
$465$	0	0
$466$	−2.56282e6	−0.546704
$467$	− 2.55573e6i	− 0.542278i	−0.962540	−	0.271139i	$-0.912600\pi$
$467$	− 2.55573e6i	− 0.542278i	0.962540	−	0.271139i	$-0.0874004\pi$
$468$	− 2.61043e6i	− 0.550932i
$469$	−1.67511e6	−0.351651
$470$	0	0
$471$	−1.30521e6	−0.271100
$472$	− 628800.i	− 0.129914i
$473$	− 1.72231e6i	− 0.353964i
$474$	3.56916e6	0.729659
$475$	0	0
$476$	1.53350e6	0.310218
$477$	− 72468.0i	− 0.0145831i
$478$	513000.i	0.102695i
$479$	−5.36664e6	−1.06872	−0.534360	−	0.845257i	$-0.679447\pi$
$479$	−5.36664e6	−1.06872	−0.534360	+	0.845257i	$0.679447\pi$
$480$	0	0
$481$	−1.12039e7	−2.20804
$482$	6.78995e6i	1.33122i
$483$	7.61254e6i	1.48478i
$484$	−234256.	−0.0454545
$485$	0	0
$486$	−4.79002e6	−0.919912
$487$	− 8.17528e6i	− 1.56200i	−0.624533	−	0.780998i	$-0.714711\pi$
$487$	− 8.17528e6i	− 1.56200i	0.624533	−	0.780998i	$-0.285289\pi$
$488$	1.67245e6i	0.317909i
$489$	6.27320e6	1.18636
$490$	0	0
$491$	3.78007e6	0.707614	0.353807	−	0.935318i	$-0.384887\pi$
$491$	3.78007e6	0.707614	0.353807	+	0.935318i	$0.384887\pi$
$492$	− 2.18131e6i	− 0.406261i
$493$	− 3.40344e6i	− 0.630668i
$494$	−7.05344e6	−1.30042
$495$	0	0
$496$	−2.00013e6	−0.365051
$497$	− 2.31113e6i	− 0.419695i
$498$	− 1.90462e6i	− 0.344139i
$499$	−3.98186e6	−0.715871	−0.357935	−	0.933746i	$-0.616519\pi$
$499$	−3.98186e6	−0.715871	−0.357935	+	0.933746i	$0.616519\pi$
$500$	0	0
$501$	−1.73729e6	−0.309227
$502$	1.30129e6i	0.230471i
$503$	− 1.04569e7i	− 1.84281i	−0.388601	−	0.921406i	$-0.627042\pi$
$503$	− 1.04569e7i	− 1.84281i	0.388601	−	0.921406i	$-0.372958\pi$
$504$	1.24186e6	0.217768
$505$	0	0
$506$	−1.79032e6	−0.310852
$507$	− 6.46134e6i	− 1.11636i
$508$	− 1.92877e6i	− 0.331605i
$509$	6.05507e6	1.03592	0.517958	−	0.855406i	$-0.326692\pi$
$509$	6.05507e6	1.03592	0.517958	+	0.855406i	$0.326692\pi$
$510$	0	0
$511$	−3.44725e6	−0.584010
$512$	− 262144.i	− 0.0441942i
$513$	2.02230e6i	0.339275i
$514$	6.59791e6	1.10154
$515$	0	0
$516$	−4.78262e6	−0.790755
$517$	− 2.46259e6i	− 0.405197i
$518$	− 5.33002e6i	− 0.872780i
$519$	2.85251e6	0.464846
$520$	0	0
$521$	1.05053e7	1.69556	0.847780	−	0.530348i	$-0.177939\pi$
$521$	1.05053e7	1.69556	0.847780	+	0.530348i	$0.177939\pi$
$522$	− 2.75616e6i	− 0.442719i
$523$	− 6.49492e6i	− 1.03829i	−0.854685	−	0.519146i	$-0.826250\pi$
$523$	− 6.49492e6i	− 1.03829i	0.854685	−	0.519146i	$-0.173750\pi$
$524$	−1.09507e6	−0.174227
$525$	0	0
$526$	5.50610e6	0.867721
$527$	− 7.64111e6i	− 1.19848i
$528$	650496.i	0.101545i
$529$	−7.24626e6	−1.12583
$530$	0	0
$531$	1.94535e6	0.299407
$532$	− 3.35552e6i	− 0.514021i
$533$	− 5.34941e6i	− 0.815620i
$534$	1.44018e6	0.218556
$535$	0	0
$536$	−1.09395e6	−0.164470
$537$	− 2.36722e6i	− 0.354245i
$538$	301800.i	0.0449535i
$539$	871563.	0.129219
$540$	0	0
$541$	5.66349e6	0.831938	0.415969	−	0.909379i	$-0.363442\pi$
$541$	5.66349e6	0.831938	0.415969	+	0.909379i	$0.363442\pi$
$542$	1.44227e6i	0.210887i
$543$	1.24699e7i	1.81495i
$544$	1.00147e6	0.145091
$545$	0	0
$546$	6.78317e6	0.973758
$547$	− 1.33609e7i	− 1.90927i	−0.297775	−	0.954636i	$-0.596245\pi$
$547$	− 1.33609e7i	− 1.90927i	0.297775	−	0.954636i	$-0.403755\pi$
$548$	5.97835e6i	0.850413i
$549$	−5.17414e6	−0.732668
$550$	0	0
$551$	−7.44720e6	−1.04499
$552$	4.97146e6i	0.694442i
$553$	4.16402e6i	0.579029i
$554$	−1.67409e6	−0.231742
$555$	0	0
$556$	−969760.	−0.133038
$557$	− 1.00947e7i	− 1.37866i	−0.724447	−	0.689330i	$-0.757905\pi$
$557$	− 1.00947e7i	− 1.37866i	0.724447	−	0.689330i	$-0.242095\pi$
$558$	− 6.18790e6i	− 0.841313i
$559$	−1.17288e7	−1.58754
$560$	0	0
$561$	−2.48510e6	−0.333378
$562$	3.17719e6i	0.424329i
$563$	− 5.58692e6i	− 0.742851i	−0.928463	−	0.371426i	$-0.878869\pi$
$563$	− 5.58692e6i	− 0.742851i	0.928463	−	0.371426i	$-0.121131\pi$
$564$	−6.83827e6	−0.905208
$565$	0	0
$566$	−7.23186e6	−0.948874
$567$	− 6.65998e6i	− 0.869992i
$568$	− 1.50931e6i	− 0.196295i
$569$	−2.20884e6	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$569$	−2.20884e6	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$570$	0	0
$571$	−1.05324e7	−1.35188	−0.675940	−	0.736957i	$-0.736262\pi$
$571$	−1.05324e7	−1.35188	−0.675940	+	0.736957i	$0.736262\pi$
$572$	1.59526e6i	0.203865i
$573$	1.37037e7i	1.74362i
$574$	2.54486e6	0.322392
$575$	0	0
$576$	811008.	0.101852
$577$	1.84319e6i	0.230479i	0.993338	+	0.115239i	$0.0367635\pi$
$577$	1.84319e6i	0.230479i	−0.993338	+	0.115239i	$0.963236\pi$
$578$	− 1.85349e6i	− 0.230766i
$579$	8.44544e6	1.04695
$580$	0	0
$581$	2.22205e6	0.273095
$582$	− 2.58107e6i	− 0.315858i
$583$	44286.0i	0.00539629i
$584$	−2.25126e6	−0.273146
$585$	0	0
$586$	3.50174e6	0.421250
$587$	1.16959e7i	1.40100i	0.713652	+	0.700501i	$0.247040\pi$
$587$	1.16959e7i	1.40100i	−0.713652	+	0.700501i	$0.752960\pi$
$588$	− 2.42021e6i	− 0.288675i
$589$	−1.67198e7	−1.98584
$590$	0	0
$591$	5.63812e6	0.663996
$592$	− 3.48083e6i	− 0.408205i
$593$	− 5.81252e6i	− 0.678778i	−0.940646	−	0.339389i	$-0.889780\pi$
$593$	− 5.81252e6i	− 0.678778i	0.940646	−	0.339389i	$-0.110220\pi$
$594$	457380.	0.0531877
$595$	0	0
$596$	−7.00848e6	−0.808180
$597$	1.22052e7i	1.40155i
$598$	1.21919e7i	1.39418i
$599$	−7.85604e6	−0.894616	−0.447308	−	0.894380i	$-0.647617\pi$
$599$	−7.85604e6	−0.894616	−0.447308	+	0.894380i	$0.647617\pi$
$600$	0	0
$601$	1.09429e7	1.23579	0.617895	−	0.786261i	$-0.287986\pi$
$601$	1.09429e7	1.23579	0.617895	+	0.786261i	$0.287986\pi$
$602$	− 5.57973e6i	− 0.627512i
$603$	− 3.38441e6i	− 0.379045i
$604$	3.83037e6	0.427217
$605$	0	0
$606$	1.16006e7	1.28321
$607$	185018.i	0.0203818i	0.999948	+	0.0101909i	$0.00324392\pi$
$607$	185018.i	0.0203818i	−0.999948	+	0.0101909i	$0.996756\pi$
$608$	− 2.19136e6i	− 0.240411i
$609$	7.16184e6	0.782495
$610$	0	0
$611$	−1.67700e7	−1.81732
$612$	3.09830e6i	0.334384i
$613$	− 1.77449e7i	− 1.90731i	−0.300901	−	0.953655i	$-0.597287\pi$
$613$	− 1.77449e7i	− 1.90731i	0.300901	−	0.953655i	$-0.402713\pi$
$614$	549872.	0.0588628
$615$	0	0
$616$	−758912.	−0.0805823
$617$	1.53912e7i	1.62765i	0.581113	+	0.813823i	$0.302618\pi$
$617$	1.53912e7i	1.62765i	−0.581113	+	0.813823i	$0.697382\pi$
$618$	− 996576.i	− 0.104964i
$619$	1.75502e7	1.84101	0.920504	−	0.390734i	$-0.127779\pi$
$619$	1.75502e7	1.84101	0.920504	+	0.390734i	$0.127779\pi$
$620$	0	0
$621$	3.49556e6	0.363737
$622$	− 4.90517e6i	− 0.508368i
$623$	1.68021e6i	0.173438i
$624$	4.42982e6	0.455434
$625$	0	0
$626$	1.25477e7	1.27976
$627$	5.43774e6i	0.552395i
$628$	− 994448.i	− 0.100620i
$629$	1.32979e7	1.34016
$630$	0	0
$631$	−5.64613e6	−0.564518	−0.282259	−	0.959338i	$-0.591084\pi$
$631$	−5.64613e6	−0.564518	−0.282259	+	0.959338i	$0.591084\pi$
$632$	2.71936e6i	0.270816i
$633$	− 3.86375e6i	− 0.383265i
$634$	−6.76475e6	−0.668388
$635$	0	0
$636$	122976.	0.0120553
$637$	− 5.93527e6i	− 0.579552i
$638$	1.68432e6i	0.163822i
$639$	4.66943e6	0.452389
$640$	0	0
$641$	−4.46307e6	−0.429031	−0.214516	−	0.976721i	$-0.568817\pi$
$641$	−4.46307e6	−0.429031	−0.214516	+	0.976721i	$0.568817\pi$
$642$	1.42783e6i	0.136722i
$643$	3.24099e6i	0.309137i	0.987982	+	0.154568i	$0.0493987\pi$
$643$	3.24099e6i	0.309137i	−0.987982	+	0.154568i	$0.950601\pi$
$644$	−5.80003e6	−0.551081
$645$	0	0
$646$	8.37168e6	0.789280
$647$	1.01885e7i	0.956858i	0.878126	+	0.478429i	$0.158794\pi$
$647$	1.01885e7i	0.956858i	−0.878126	+	0.478429i	$0.841206\pi$
$648$	− 4.34938e6i	− 0.406902i
$649$	−1.18882e6	−0.110791
$650$	0	0
$651$	1.60792e7	1.48700
$652$	4.77958e6i	0.440323i
$653$	4.66760e6i	0.428362i	0.976794	+	0.214181i	$0.0687082\pi$
$653$	4.66760e6i	0.428362i	−0.976794	+	0.214181i	$0.931292\pi$
$654$	1.86337e7	1.70355
$655$	0	0
$656$	1.66195e6	0.150785
$657$	− 6.96485e6i	− 0.629504i
$658$	− 7.97798e6i	− 0.718337i
$659$	−1.29177e6	−0.115870	−0.0579351	−	0.998320i	$-0.518452\pi$
$659$	−1.29177e6	−0.115870	−0.0579351	+	0.998320i	$0.518452\pi$
$660$	0	0
$661$	−1.15658e7	−1.02960	−0.514802	−	0.857309i	$-0.672135\pi$
$661$	−1.15658e7	−1.02960	−0.514802	+	0.857309i	$0.672135\pi$
$662$	8.55605e6i	0.758802i
$663$	1.69233e7i	1.49521i
$664$	1.45114e6	0.127729
$665$	0	0
$666$	1.07688e7	0.940768
$667$	1.28725e7i	1.12034i
$668$	− 1.32365e6i	− 0.114771i
$669$	1.32653e7	1.14591
$670$	0	0
$671$	3.16197e6	0.271114
$672$	2.10739e6i	0.180021i
$673$	− 1.47706e7i	− 1.25707i	−0.777781	−	0.628535i	$-0.783655\pi$
$673$	− 1.47706e7i	− 1.25707i	0.777781	−	0.628535i	$-0.216345\pi$
$674$	2.21439e6	0.187761
$675$	0	0
$676$	4.92293e6	0.414340
$677$	− 3.09022e6i	− 0.259130i	−0.991571	−	0.129565i	$-0.958642\pi$
$677$	− 3.09022e6i	− 0.259130i	0.991571	−	0.129565i	$-0.0413581\pi$
$678$	1.65204e7i	1.38021i
$679$	3.01125e6	0.250652
$680$	0	0
$681$	−29358.0	−0.00242582
$682$	3.78149e6i	0.311317i
$683$	− 1.47394e7i	− 1.20900i	−0.796604	−	0.604502i	$-0.793372\pi$
$683$	− 1.47394e7i	− 1.20900i	0.796604	−	0.604502i	$-0.206628\pi$
$684$	6.77952e6	0.554062
$685$	0	0
$686$	9.41192e6	0.763604
$687$	− 4.32884e6i	− 0.349928i
$688$	− 3.64390e6i	− 0.293492i
$689$	301584.	0.0242025
$690$	0	0
$691$	2.36276e6	0.188245	0.0941226	−	0.995561i	$-0.469995\pi$
$691$	2.36276e6	0.188245	0.0941226	+	0.995561i	$0.469995\pi$
$692$	2.17334e6i	0.172529i
$693$	− 2.34788e6i	− 0.185714i
$694$	668832.	0.0527131
$695$	0	0
$696$	4.67712e6	0.365978
$697$	6.34918e6i	0.495034i
$698$	1.87796e6i	0.145897i
$699$	1.34548e7	1.04156
$700$	0	0
$701$	−1.78888e7	−1.37495	−0.687473	−	0.726210i	$-0.741280\pi$
$701$	−1.78888e7	−1.37495	−0.687473	+	0.726210i	$0.741280\pi$
$702$	− 3.11472e6i	− 0.238548i
$703$	− 2.90976e7i	− 2.22059i
$704$	−495616.	−0.0376889
$705$	0	0
$706$	−1.12862e7	−0.852186
$707$	1.35340e7i	1.01830i
$708$	3.30120e6i	0.247508i
$709$	−1.22735e7	−0.916962	−0.458481	−	0.888704i	$-0.651606\pi$
$709$	−1.22735e7	−0.916962	−0.458481	+	0.888704i	$0.651606\pi$
$710$	0	0
$711$	−8.41302e6	−0.624134
$712$	1.09728e6i	0.0811180i
$713$	2.89003e7i	2.12901i
$714$	−8.05090e6	−0.591015
$715$	0	0
$716$	1.80360e6	0.131479
$717$	− 2.69325e6i	− 0.195650i
$718$	7.82784e6i	0.566671i
$719$	7.35232e6	0.530399	0.265199	−	0.964194i	$-0.414562\pi$
$719$	7.35232e6	0.530399	0.265199	+	0.964194i	$0.414562\pi$
$720$	0	0
$721$	1.16267e6	0.0832950
$722$	− 8.41400e6i	− 0.600703i
$723$	− 3.56472e7i	− 2.53618i
$724$	−9.50091e6	−0.673626
$725$	0	0
$726$	1.22984e6	0.0865981
$727$	− 3.16762e6i	− 0.222278i	−0.993805	−	0.111139i	$-0.964550\pi$
$727$	− 3.16762e6i	− 0.222278i	0.993805	−	0.111139i	$-0.0354499\pi$
$728$	5.16813e6i	0.361414i
$729$	8.63355e6	0.601687
$730$	0	0
$731$	1.39209e7	0.963546
$732$	− 8.78035e6i	− 0.605667i
$733$	− 857924.i	− 0.0589778i	−0.999565	−	0.0294889i	$-0.990612\pi$
$733$	− 857924.i	− 0.0589778i	0.999565	−	0.0294889i	$-0.00938798\pi$
$734$	−9.14367e6	−0.626441
$735$	0	0
$736$	−3.78778e6	−0.257745
$737$	2.06825e6i	0.140260i
$738$	5.14166e6i	0.347506i
$739$	1.93551e7	1.30372	0.651860	−	0.758339i	$-0.273989\pi$
$739$	1.93551e7	1.30372	0.651860	+	0.758339i	$0.273989\pi$
$740$	0	0
$741$	3.70306e7	2.47751
$742$	143472.i	0.00956660i
$743$	2.80305e7i	1.86277i	0.364040	+	0.931383i	$0.381397\pi$
$743$	2.80305e7i	1.86277i	−0.364040	+	0.931383i	$0.618603\pi$
$744$	1.05007e7	0.695480
$745$	0	0
$746$	−2.33310e6	−0.153492
$747$	4.48945e6i	0.294369i
$748$	− 1.89341e6i	− 0.123734i
$749$	−1.66580e6	−0.108497
$750$	0	0
$751$	2.57014e7	1.66287	0.831434	−	0.555624i	$-0.187521\pi$
$751$	2.57014e7	1.66287	0.831434	+	0.555624i	$0.187521\pi$
$752$	− 5.21011e6i	− 0.335972i
$753$	− 6.83178e6i	− 0.439083i
$754$	1.14701e7	0.734747
$755$	0	0
$756$	1.48176e6	0.0942917
$757$	− 1.17223e7i	− 0.743489i	−0.928335	−	0.371745i	$-0.878760\pi$
$757$	− 1.17223e7i	− 0.743489i	0.928335	−	0.371745i	$-0.121240\pi$
$758$	1.13452e7i	0.717197i
$759$	9.39916e6	0.592222
$760$	0	0
$761$	5.77783e6	0.361662	0.180831	−	0.983514i	$-0.442121\pi$
$761$	5.77783e6	0.361662	0.180831	+	0.983514i	$0.442121\pi$
$762$	1.01260e7i	0.631760i
$763$	2.17393e7i	1.35187i
$764$	−1.04409e7	−0.647150
$765$	0	0
$766$	1.40881e7	0.867521
$767$	8.09580e6i	0.496903i
$768$	1.37626e6i	0.0841969i
$769$	2.04989e7	1.25001	0.625007	−	0.780619i	$-0.285096\pi$
$769$	2.04989e7	1.25001	0.625007	+	0.780619i	$0.285096\pi$
$770$	0	0
$771$	−3.46390e7	−2.09860
$772$	6.43462e6i	0.388580i
$773$	− 1.50298e7i	− 0.904697i	−0.891841	−	0.452348i	$-0.850586\pi$
$773$	− 1.50298e7i	− 0.904697i	0.891841	−	0.452348i	$-0.149414\pi$
$774$	1.12733e7	0.676394
$775$	0	0
$776$	1.96653e6	0.117232
$777$	2.79826e7i	1.66278i
$778$	− 7.25430e6i	− 0.429681i
$779$	1.38929e7	0.820255
$780$	0	0
$781$	−2.85354e6	−0.167401
$782$	− 1.44705e7i	− 0.846187i
$783$	− 3.28860e6i	− 0.191693i
$784$	1.84397e6	0.107143
$785$	0	0
$786$	5.74913e6	0.331929
$787$	2.05325e7i	1.18169i	0.806784	+	0.590847i	$0.201206\pi$
$787$	2.05325e7i	1.18169i	−0.806784	+	0.590847i	$0.798794\pi$
$788$	4.29571e6i	0.246445i
$789$	−2.89070e7	−1.65315
$790$	0	0
$791$	−1.92738e7	−1.09528
$792$	− 1.53331e6i	− 0.0868596i
$793$	− 2.15328e7i	− 1.21595i
$794$	1.36830e7	0.770249
$795$	0	0
$796$	−9.29920e6	−0.520191
$797$	− 1.76247e6i	− 0.0982823i	−0.998792	−	0.0491411i	$-0.984352\pi$
$797$	− 1.76247e6i	− 0.0982823i	0.998792	−	0.0491411i	$-0.0156484\pi$
$798$	1.76165e7i	0.979291i
$799$	1.99043e7	1.10301
$800$	0	0
$801$	−3.39471e6	−0.186948
$802$	− 1.59377e6i	− 0.0874962i
$803$	4.25630e6i	0.232939i
$804$	5.74325e6	0.313341
$805$	0	0
$806$	2.57516e7	1.39626
$807$	− 1.58445e6i	− 0.0856436i
$808$	8.83853e6i	0.476268i
$809$	−2.00289e7	−1.07593	−0.537967	−	0.842966i	$-0.680808\pi$
$809$	−2.00289e7	−1.07593	−0.537967	+	0.842966i	$0.680808\pi$
$810$	0	0
$811$	−2.55409e7	−1.36359	−0.681796	−	0.731542i	$-0.738801\pi$
$811$	−2.55409e7	−1.36359	−0.681796	+	0.731542i	$0.738801\pi$
$812$	5.45664e6i	0.290426i
$813$	− 7.57193e6i	− 0.401772i
$814$	−6.58095e6	−0.348119
$815$	0	0
$816$	−5.25773e6	−0.276422
$817$	− 3.04608e7i	− 1.59656i
$818$	5.63828e6i	0.294621i
$819$	−1.59889e7	−0.832930
$820$	0	0
$821$	−6.73166e6	−0.348549	−0.174275	−	0.984697i	$-0.555758\pi$
$821$	−6.73166e6	−0.348549	−0.174275	+	0.984697i	$0.555758\pi$
$822$	− 3.13863e7i	− 1.62017i
$823$	− 1.44880e7i	− 0.745606i	−0.927911	−	0.372803i	$-0.878397\pi$
$823$	− 1.44880e7i	− 0.745606i	0.927911	−	0.372803i	$-0.121603\pi$
$824$	759296.	0.0389577
$825$	0	0
$826$	−3.85140e6	−0.196412
$827$	− 4.15879e6i	− 0.211448i	−0.994396	−	0.105724i	$-0.966284\pi$
$827$	− 4.15879e6i	− 0.211448i	0.994396	−	0.105724i	$-0.0337160\pi$
$828$	− 1.17184e7i	− 0.594010i
$829$	−1.02525e7	−0.518136	−0.259068	−	0.965859i	$-0.583415\pi$
$829$	−1.02525e7	−0.518136	−0.259068	+	0.965859i	$0.583415\pi$
$830$	0	0
$831$	8.78896e6	0.441504
$832$	3.37510e6i	0.169036i
$833$	7.04453e6i	0.351755i
$834$	5.09124e6	0.253459
$835$	0	0
$836$	−4.14304e6	−0.205023
$837$	− 7.38328e6i	− 0.364281i
$838$	− 7.46472e6i	− 0.367201i
$839$	−3.07705e7	−1.50914	−0.754571	−	0.656218i	$-0.772155\pi$
$839$	−3.07705e7	−1.50914	−0.754571	+	0.656218i	$0.772155\pi$
$840$	0	0
$841$	−8.40075e6	−0.409570
$842$	− 8.31097e6i	− 0.403991i
$843$	− 1.66803e7i	− 0.808414i
$844$	2.94381e6	0.142250
$845$	0	0
$846$	1.61188e7	0.774295
$847$	1.43482e6i	0.0687208i
$848$	93696.0i	0.00447437i
$849$	3.79672e7	1.80776
$850$	0	0
$851$	−5.02953e7	−2.38069
$852$	7.92389e6i	0.373972i
$853$	− 4.19232e7i	− 1.97280i	−0.164377	−	0.986398i	$-0.552561\pi$
$853$	− 4.19232e7i	− 1.97280i	0.164377	−	0.986398i	$-0.447439\pi$
$854$	1.02437e7	0.480634
$855$	0	0
$856$	−1.08787e6	−0.0507450
$857$	− 1.87686e7i	− 0.872929i	−0.899722	−	0.436464i	$-0.856231\pi$
$857$	− 1.87686e7i	− 0.872929i	0.899722	−	0.436464i	$-0.143769\pi$
$858$	− 8.37514e6i	− 0.388395i
$859$	3.95520e7	1.82888	0.914441	−	0.404720i	$-0.132631\pi$
$859$	3.95520e7	1.82888	0.914441	+	0.404720i	$0.132631\pi$
$860$	0	0
$861$	−1.33605e7	−0.614209
$862$	2.51374e7i	1.15227i
$863$	− 1.50569e7i	− 0.688191i	−0.938935	−	0.344095i	$-0.888186\pi$
$863$	− 1.50569e7i	− 0.688191i	0.938935	−	0.344095i	$-0.111814\pi$
$864$	967680.	0.0441009
$865$	0	0
$866$	−1.30624e7	−0.591871
$867$	9.73083e6i	0.439645i
$868$	1.22508e7i	0.551905i
$869$	5.14129e6	0.230952
$870$	0	0
$871$	1.40846e7	0.629072
$872$	1.41971e7i	0.632279i
$873$	6.08395e6i	0.270178i
$874$	−3.16634e7	−1.40210
$875$	0	0
$876$	1.18191e7	0.520386
$877$	1.58591e7i	0.696272i	0.937444	+	0.348136i	$0.113185\pi$
$877$	1.58591e7i	0.696272i	−0.937444	+	0.348136i	$0.886815\pi$
$878$	− 1.05827e7i	− 0.463299i
$879$	−1.83842e7	−0.802549
$880$	0	0
$881$	−2.29142e7	−0.994640	−0.497320	−	0.867567i	$-0.665682\pi$
$881$	−2.29142e7	−0.994640	−0.497320	+	0.867567i	$0.665682\pi$
$882$	5.70478e6i	0.246926i
$883$	− 1.06073e7i	− 0.457830i	−0.973446	−	0.228915i	$-0.926482\pi$
$883$	− 1.06073e7i	− 0.457830i	0.973446	−	0.228915i	$-0.0735178\pi$
$884$	−1.28940e7	−0.554952
$885$	0	0
$886$	2.25412e7	0.964703
$887$	2.28527e7i	0.975278i	0.873045	+	0.487639i	$0.162142\pi$
$887$	2.28527e7i	0.975278i	−0.873045	+	0.487639i	$0.837858\pi$
$888$	1.82744e7i	0.777696i
$889$	−1.18137e7	−0.501339
$890$	0	0
$891$	−8.22304e6	−0.347007
$892$	1.01069e7i	0.425309i
$893$	− 4.35533e7i	− 1.82765i
$894$	3.67945e7	1.53971
$895$	0	0
$896$	−1.60563e6	−0.0668153
$897$	− 6.40075e7i	− 2.65613i
$898$	− 1.48002e6i	− 0.0612459i
$899$	2.71892e7	1.12201
$900$	0	0
$901$	−357948.	−0.0146895
$902$	− 3.14213e6i	− 0.128590i
$903$	2.92936e7i	1.19551i
$904$	−1.25869e7	−0.512270
$905$	0	0
$906$	−2.01094e7	−0.813915
$907$	− 3.21947e7i	− 1.29947i	−0.760160	−	0.649736i	$-0.774880\pi$
$907$	− 3.21947e7i	− 1.29947i	0.760160	−	0.649736i	$-0.225120\pi$
$908$	− 22368.0i	0 0.000900352i
$909$	−2.73442e7	−1.09763
$910$	0	0
$911$	2.02254e7	0.807424	0.403712	−	0.914886i	$-0.367720\pi$
$911$	2.02254e7	0.807424	0.403712	+	0.914886i	$0.367720\pi$
$912$	1.15046e7i	0.458022i
$913$	− 2.74355e6i	− 0.108927i
$914$	1.32756e7	0.525642
$915$	0	0
$916$	3.29816e6	0.129877
$917$	6.70732e6i	0.263406i
$918$	3.69684e6i	0.144785i
$919$	2.30019e7	0.898411	0.449206	−	0.893428i	$-0.351707\pi$
$919$	2.30019e7	0.898411	0.449206	+	0.893428i	$0.351707\pi$
$920$	0	0
$921$	−2.88683e6	−0.112143
$922$	− 3.50808e7i	− 1.35907i
$923$	1.94324e7i	0.750796i
$924$	3.98429e6	0.153522
$925$	0	0
$926$	−896996.	−0.0343766
$927$	2.34907e6i	0.0897836i
$928$	3.56352e6i	0.135834i
$929$	1.64913e7	0.626924	0.313462	−	0.949601i	$-0.398511\pi$
$929$	1.64913e7	0.626924	0.313462	+	0.949601i	$0.398511\pi$
$930$	0	0
$931$	1.54144e7	0.582845
$932$	1.02513e7i	0.386578i
$933$	2.57521e7i	0.968521i
$934$	−1.02229e7	−0.383449
$935$	0	0
$936$	−1.04417e7	−0.389568
$937$	− 4.99402e7i	− 1.85824i	−0.369780	−	0.929119i	$-0.620567\pi$
$937$	− 4.99402e7i	− 1.85824i	0.369780	−	0.929119i	$-0.379433\pi$
$938$	6.70046e6i	0.248655i
$939$	−6.58753e7	−2.43814
$940$	0	0
$941$	4.21000e6	0.154992	0.0774958	−	0.996993i	$-0.475308\pi$
$941$	4.21000e6	0.154992	0.0774958	+	0.996993i	$0.475308\pi$
$942$	5.22085e6i	0.191696i
$943$	− 2.40139e7i	− 0.879394i
$944$	−2.51520e6	−0.0918634
$945$	0	0
$946$	−6.88926e6	−0.250291
$947$	2.76066e7i	1.00032i	0.865933	+	0.500159i	$0.166725\pi$
$947$	2.76066e7i	1.00032i	−0.865933	+	0.500159i	$0.833275\pi$
$948$	− 1.42766e7i	− 0.515947i
$949$	2.89850e7	1.04474
$950$	0	0
$951$	3.55149e7	1.27338
$952$	− 6.13402e6i	− 0.219358i
$953$	− 1.69297e7i	− 0.603832i	−0.953335	−	0.301916i	$-0.902374\pi$
$953$	− 1.69297e7i	− 0.603832i	0.953335	−	0.301916i	$-0.0976261\pi$
$954$	−289872.	−0.0103118
$955$	0	0
$956$	2.05200e6	0.0726161
$957$	− 8.84268e6i	− 0.312107i
$958$	2.14666e7i	0.755699i
$959$	3.66174e7	1.28570
$960$	0	0
$961$	3.24138e7	1.13220
$962$	4.48157e7i	1.56132i
$963$	− 3.36560e6i	− 0.116949i
$964$	2.71598e7	0.941313
$965$	0	0
$966$	3.04502e7	1.04990
$967$	1.08793e6i	0.0374140i	0.999825	+	0.0187070i	$0.00595497\pi$
$967$	1.08793e6i	0.0374140i	−0.999825	+	0.0187070i	$0.994045\pi$
$968$	937024.i	0.0321412i
$969$	−4.39513e7	−1.50370
$970$	0	0
$971$	7.15335e6	0.243479	0.121739	−	0.992562i	$-0.461153\pi$
$971$	7.15335e6	0.243479	0.121739	+	0.992562i	$0.461153\pi$
$972$	1.91601e7i	0.650476i
$973$	5.93978e6i	0.201135i
$974$	−3.27011e7	−1.10450
$975$	0	0
$976$	6.68979e6	0.224796
$977$	− 7.88535e6i	− 0.264292i	−0.991230	−	0.132146i	$-0.957813\pi$
$977$	− 7.88535e6i	− 0.264292i	0.991230	−	0.132146i	$-0.0421868\pi$
$978$	− 2.50928e7i	− 0.838885i
$979$	2.07454e6	0.0691777
$980$	0	0
$981$	−4.39223e7	−1.45718
$982$	− 1.51203e7i	− 0.500359i
$983$	− 7.58842e6i	− 0.250477i	−0.992127	−	0.125238i	$-0.960030\pi$
$983$	− 7.58842e6i	− 0.250477i	0.992127	−	0.125238i	$-0.0399696\pi$
$984$	−8.72525e6	−0.287270
$985$	0	0
$986$	−1.36138e7	−0.445950
$987$	4.18844e7i	1.36855i
$988$	2.82138e7i	0.919536i
$989$	−5.26516e7	−1.71167
$990$	0	0
$991$	8.80935e6	0.284944	0.142472	−	0.989799i	$-0.454495\pi$
$991$	8.80935e6	0.284944	0.142472	+	0.989799i	$0.454495\pi$
$992$	8.00051e6i	0.258130i
$993$	− 4.49193e7i	− 1.44564i
$994$	−9.24454e6	−0.296769
$995$	0	0
$996$	−7.61846e6	−0.243343
$997$	3.53833e7i	1.12735i	0.825995	+	0.563677i	$0.190614\pi$
$997$	3.53833e7i	1.12735i	−0.825995	+	0.563677i	$0.809386\pi$
$998$	1.59274e7i	0.506197i
$999$	1.28492e7	0.407344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.6.b.g.199.1		2
5.2	odd	4		550.6.a.g.1.1		1
5.3	odd	4		22.6.a.a.1.1	✓	1
5.4	even	2	inner	550.6.b.g.199.2		2
15.8	even	4		198.6.a.d.1.1		1
20.3	even	4		176.6.a.d.1.1		1
35.13	even	4		1078.6.a.b.1.1		1
40.3	even	4		704.6.a.b.1.1		1
40.13	odd	4		704.6.a.i.1.1		1
55.43	even	4		242.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.a.1.1	✓	1	5.3	odd	4
176.6.a.d.1.1		1	20.3	even	4
198.6.a.d.1.1		1	15.8	even	4
242.6.a.c.1.1		1	55.43	even	4
550.6.a.g.1.1		1	5.2	odd	4
550.6.b.g.199.1		2	1.1	even	1	trivial
550.6.b.g.199.2		2	5.4	even	2	inner
704.6.a.b.1.1		1	40.3	even	4
704.6.a.i.1.1		1	40.13	odd	4
1078.6.a.b.1.1		1	35.13	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:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	550.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +21.0000i q^{3} -16.0000 q^{4} +84.0000 q^{6} +98.0000i q^{7} +64.0000i q^{8} -198.000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +21.0000i q^{3} -16.0000 q^{4} +84.0000 q^{6} +98.0000i q^{7} +64.0000i q^{8} -198.000 q^{9} +121.000 q^{11} -336.000i q^{12} -824.000i q^{13} +392.000 q^{14} +256.000 q^{16} +978.000i q^{17} +792.000i q^{18} +2140.00 q^{19} -2058.00 q^{21} -484.000i q^{22} -3699.00i q^{23} -1344.00 q^{24} -3296.00 q^{26} +945.000i q^{27} -1568.00i q^{28} -3480.00 q^{29} -7813.00 q^{31} -1024.00i q^{32} +2541.00i q^{33} +3912.00 q^{34} +3168.00 q^{36} -13597.0i q^{37} -8560.00i q^{38} +17304.0 q^{39} +6492.00 q^{41} +8232.00i q^{42} -14234.0i q^{43} -1936.00 q^{44} -14796.0 q^{46} -20352.0i q^{47} +5376.00i q^{48} +7203.00 q^{49} -20538.0 q^{51} +13184.0i q^{52} +366.000i q^{53} +3780.00 q^{54} -6272.00 q^{56} +44940.0i q^{57} +13920.0i q^{58} -9825.00 q^{59} +26132.0 q^{61} +31252.0i q^{62} -19404.0i q^{63} -4096.00 q^{64} +10164.0 q^{66} +17093.0i q^{67} -15648.0i q^{68} +77679.0 q^{69} -23583.0 q^{71} -12672.0i q^{72} +35176.0i q^{73} -54388.0 q^{74} -34240.0 q^{76} +11858.0i q^{77} -69216.0i q^{78} +42490.0 q^{79} -67959.0 q^{81} -25968.0i q^{82} -22674.0i q^{83} +32928.0 q^{84} -56936.0 q^{86} -73080.0i q^{87} +7744.00i q^{88} +17145.0 q^{89} +80752.0 q^{91} +59184.0i q^{92} -164073. i q^{93} -81408.0 q^{94} +21504.0 q^{96} -30727.0i q^{97} -28812.0i q^{98} -23958.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9	\( 2 q - 32 q^{4} + 168 q^{6} - 396 q^{9} + 242 q^{11} + 784 q^{14} + 512 q^{16} + 4280 q^{19} - 4116 q^{21} - 2688 q^{24} - 6592 q^{26} - 6960 q^{29} - 15626 q^{31} + 7824 q^{34} + 6336 q^{36} + 34608 q^{39} + 12984 q^{41} - 3872 q^{44} - 29592 q^{46} + 14406 q^{49} - 41076 q^{51} + 7560 q^{54} - 12544 q^{56} - 19650 q^{59} + 52264 q^{61} - 8192 q^{64} + 20328 q^{66} + 155358 q^{69} - 47166 q^{71} - 108776 q^{74} - 68480 q^{76} + 84980 q^{79} - 135918 q^{81} + 65856 q^{84} - 113872 q^{86} + 34290 q^{89} + 161504 q^{91} - 162816 q^{94} + 43008 q^{96} - 47916 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 168 * q^6 - 396 * q^9 + 242 * q^11 + 784 * q^14 + 512 * q^16 + 4280 * q^19 - 4116 * q^21 - 2688 * q^24 - 6592 * q^26 - 6960 * q^29 - 15626 * q^31 + 7824 * q^34 + 6336 * q^36 + 34608 * q^39 + 12984 * q^41 - 3872 * q^44 - 29592 * q^46 + 14406 * q^49 - 41076 * q^51 + 7560 * q^54 - 12544 * q^56 - 19650 * q^59 + 52264 * q^61 - 8192 * q^64 + 20328 * q^66 + 155358 * q^69 - 47166 * q^71 - 108776 * q^74 - 68480 * q^76 + 84980 * q^79 - 135918 * q^81 + 65856 * q^84 - 113872 * q^86 + 34290 * q^89 + 161504 * q^91 - 162816 * q^94 + 43008 * q^96 - 47916 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more