LMFDB - Embedded newform 825.4.c.o.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9935104.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 16x^{2} - 8x + 2 $ x^6 - 2x^5 + 2x^4 + 6x^3 + 16x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.91004 - 1.91004i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.o.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82009i q^{2} +3.00000i q^{3} +0.0470959 q^{4} +8.46027 q^{6} -7.12434i q^{7} -22.6935i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-2.82009i q^{2} +3.00000i q^{3} +0.0470959 q^{4} +8.46027 q^{6} -7.12434i q^{7} -22.6935i q^{8} -9.00000 q^{9} -11.0000 q^{11} +0.141288i q^{12} -66.8837i q^{13} -20.0913 q^{14} -63.6210 q^{16} -40.9051i q^{17} +25.3808i q^{18} -4.85136 q^{19} +21.3730 q^{21} +31.0210i q^{22} +128.183i q^{23} +68.0806 q^{24} -188.618 q^{26} -27.0000i q^{27} -0.335527i q^{28} +224.689 q^{29} -267.207 q^{31} -2.13129i q^{32} -33.0000i q^{33} -115.356 q^{34} -0.423863 q^{36} +418.418i q^{37} +13.6813i q^{38} +200.651 q^{39} -496.282 q^{41} -60.2738i q^{42} +90.9549i q^{43} -0.518055 q^{44} +361.486 q^{46} -203.640i q^{47} -190.863i q^{48} +292.244 q^{49} +122.715 q^{51} -3.14995i q^{52} -219.416i q^{53} -76.1424 q^{54} -161.676 q^{56} -14.5541i q^{57} -633.644i q^{58} -585.745 q^{59} +156.973 q^{61} +753.547i q^{62} +64.1191i q^{63} -514.979 q^{64} -93.0629 q^{66} -638.333i q^{67} -1.92646i q^{68} -384.548 q^{69} -961.243 q^{71} +204.242i q^{72} -223.078i q^{73} +1179.98 q^{74} -0.228479 q^{76} +78.3678i q^{77} -565.854i q^{78} -415.648 q^{79} +81.0000 q^{81} +1399.56i q^{82} -44.7029i q^{83} +1.00658 q^{84} +256.501 q^{86} +674.068i q^{87} +249.629i q^{88} -809.758 q^{89} -476.503 q^{91} +6.03688i q^{92} -801.620i q^{93} -574.282 q^{94} +6.39387 q^{96} -429.786i q^{97} -824.154i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{6} - 54 q^{9}+O(q^{10}) $ 6 * q + 10 * q^4 - 6 * q^6 - 54 * q^9	$ 6 q + 10 q^{4} - 6 q^{6} - 54 q^{9} - 66 q^{11} + 152 q^{14} - 170 q^{16} + 560 q^{19} + 96 q^{21} - 18 q^{24} - 628 q^{26} + 580 q^{29} - 784 q^{31} - 620 q^{34} - 90 q^{36} - 252 q^{39} - 1324 q^{41} - 110 q^{44} + 1456 q^{46} - 1462 q^{49} + 204 q^{51} + 54 q^{54} + 408 q^{56} + 1224 q^{59} - 1164 q^{61} - 694 q^{64} + 66 q^{66} - 672 q^{69} - 3232 q^{71} + 1284 q^{74} + 1760 q^{76} - 248 q^{79} + 486 q^{81} - 1680 q^{84} - 2000 q^{86} - 1676 q^{89} - 3472 q^{91} - 4864 q^{94} + 138 q^{96} + 594 q^{99}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^6 - 54 * q^9 - 66 * q^11 + 152 * q^14 - 170 * q^16 + 560 * q^19 + 96 * q^21 - 18 * q^24 - 628 * q^26 + 580 * q^29 - 784 * q^31 - 620 * q^34 - 90 * q^36 - 252 * q^39 - 1324 * q^41 - 110 * q^44 + 1456 * q^46 - 1462 * q^49 + 204 * q^51 + 54 * q^54 + 408 * q^56 + 1224 * q^59 - 1164 * q^61 - 694 * q^64 + 66 * q^66 - 672 * q^69 - 3232 * q^71 + 1284 * q^74 + 1760 * q^76 - 248 * q^79 + 486 * q^81 - 1680 * q^84 - 2000 * q^86 - 1676 * q^89 - 3472 * q^91 - 4864 * q^94 + 138 * q^96 + 594 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.82009i	− 0.997052i	−0.866875	−	0.498526i	$-0.833875\pi$
$2$	− 2.82009i	− 0.997052i	0.866875	−	0.498526i	$-0.166125\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0.0470959	0.00588699
$5$	0	0
$5$	0	0
$6$	8.46027	0.575648
$7$	− 7.12434i	− 0.384678i	−0.981329	−	0.192339i	$-0.938393\pi$
$7$	− 7.12434i	− 0.384678i	0.981329	−	0.192339i	$-0.0616074\pi$
$8$	− 22.6935i	− 1.00292i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0.141288i	0.00339886i
$13$	− 66.8837i	− 1.42694i	−0.700686	−	0.713470i	$-0.747123\pi$
$13$	− 66.8837i	− 1.42694i	0.700686	−	0.713470i	$-0.252877\pi$
$14$	−20.0913	−0.383544
$15$	0	0
$16$	−63.6210	−0.994078
$17$	− 40.9051i	− 0.583585i	−0.956482	−	0.291793i	$-0.905748\pi$
$17$	− 40.9051i	− 0.583585i	0.956482	−	0.291793i	$-0.0942517\pi$
$18$	25.3808i	0.332351i
$19$	−4.85136	−0.0585779	−0.0292889	−	0.999571i	$-0.509324\pi$
$19$	−4.85136	−0.0585779	−0.0292889	+	0.999571i	$0.509324\pi$
$20$	0	0
$21$	21.3730	0.222094
$22$	31.0210i	0.300623i
$23$	128.183i	1.16208i	0.813874	+	0.581042i	$0.197355\pi$
$23$	128.183i	1.16208i	−0.813874	+	0.581042i	$0.802645\pi$
$24$	68.0806	0.579037
$25$	0	0
$26$	−188.618	−1.42273
$27$	− 27.0000i	− 0.192450i
$28$	− 0.335527i	− 0.00226460i
$29$	224.689	1.43875	0.719375	−	0.694622i	$-0.244428\pi$
$29$	224.689	1.43875	0.719375	+	0.694622i	$0.244428\pi$
$30$	0	0
$31$	−267.207	−1.54812	−0.774060	−	0.633112i	$-0.781777\pi$
$31$	−267.207	−1.54812	−0.774060	+	0.633112i	$0.781777\pi$
$32$	− 2.13129i	− 0.0117738i
$33$	− 33.0000i	− 0.174078i
$34$	−115.356	−0.581865
$35$	0	0
$36$	−0.423863	−0.00196233
$37$	418.418i	1.85912i	0.368669	+	0.929561i	$0.379813\pi$
$37$	418.418i	1.85912i	−0.368669	+	0.929561i	$0.620187\pi$
$38$	13.6813i	0.0584052i
$39$	200.651	0.823844
$40$	0	0
$41$	−496.282	−1.89040	−0.945198	−	0.326498i	$-0.894131\pi$
$41$	−496.282	−1.89040	−0.945198	+	0.326498i	$0.894131\pi$
$42$	− 60.2738i	− 0.221439i
$43$	90.9549i	0.322570i	0.986908	+	0.161285i	$0.0515638\pi$
$43$	90.9549i	0.322570i	−0.986908	+	0.161285i	$0.948436\pi$
$44$	−0.518055	−0.00177499
$45$	0	0
$46$	361.486	1.15866
$47$	− 203.640i	− 0.631998i	−0.948760	−	0.315999i	$-0.897660\pi$
$47$	− 203.640i	− 0.631998i	0.948760	−	0.315999i	$-0.102340\pi$
$48$	− 190.863i	− 0.573931i
$49$	292.244	0.852023
$50$	0	0
$51$	122.715	0.336933
$52$	− 3.14995i	− 0.00840038i
$53$	− 219.416i	− 0.568662i	−0.958726	−	0.284331i	$-0.908229\pi$
$53$	− 219.416i	− 0.568662i	0.958726	−	0.284331i	$-0.0917714\pi$
$54$	−76.1424	−0.191883
$55$	0	0
$56$	−161.676	−0.385802
$57$	− 14.5541i	− 0.0338199i
$58$	− 633.644i	− 1.43451i
$59$	−585.745	−1.29250	−0.646250	−	0.763125i	$-0.723664\pi$
$59$	−585.745	−1.29250	−0.646250	+	0.763125i	$0.723664\pi$
$60$	0	0
$61$	156.973	0.329481	0.164740	−	0.986337i	$-0.447321\pi$
$61$	156.973	0.329481	0.164740	+	0.986337i	$0.447321\pi$
$62$	753.547i	1.54356i
$63$	64.1191i	0.128226i
$64$	−514.979	−1.00582
$65$	0	0
$66$	−93.0629	−0.173565
$67$	− 638.333i	− 1.16395i	−0.813206	−	0.581976i	$-0.802280\pi$
$67$	− 638.333i	− 1.16395i	0.813206	−	0.581976i	$-0.197720\pi$
$68$	− 1.92646i	− 0.00343556i
$69$	−384.548	−0.670929
$70$	0	0
$71$	−961.243	−1.60674	−0.803371	−	0.595479i	$-0.796962\pi$
$71$	−961.243	−1.60674	−0.803371	+	0.595479i	$0.796962\pi$
$72$	204.242i	0.334307i
$73$	− 223.078i	− 0.357662i	−0.983880	−	0.178831i	$-0.942769\pi$
$73$	− 223.078i	− 0.357662i	0.983880	−	0.178831i	$-0.0572315\pi$
$74$	1179.98	1.85364
$75$	0	0
$76$	−0.228479	−0.000344847 0
$77$	78.3678i	0.115985i
$78$	− 565.854i	− 0.821415i
$79$	−415.648	−0.591950	−0.295975	−	0.955196i	$-0.595644\pi$
$79$	−415.648	−0.591950	−0.295975	+	0.955196i	$0.595644\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1399.56i	1.88482i
$83$	− 44.7029i	− 0.0591178i	−0.999563	−	0.0295589i	$-0.990590\pi$
$83$	− 44.7029i	− 0.0591178i	0.999563	−	0.0295589i	$-0.00941027\pi$
$84$	1.00658	0.00130747
$85$	0	0
$86$	256.501	0.321619
$87$	674.068i	0.830663i
$88$	249.629i	0.302392i
$89$	−809.758	−0.964429	−0.482214	−	0.876053i	$-0.660167\pi$
$89$	−809.758	−0.964429	−0.482214	+	0.876053i	$0.660167\pi$
$90$	0	0
$91$	−476.503	−0.548913
$92$	6.03688i	0.00684117i
$93$	− 801.620i	− 0.893808i
$94$	−574.282	−0.630135
$95$	0	0
$96$	6.39387	0.00679762
$97$	− 429.786i	− 0.449878i	−0.974373	−	0.224939i	$-0.927782\pi$
$97$	− 429.786i	− 0.449878i	0.974373	−	0.224939i	$-0.0722183\pi$
$98$	− 824.154i	− 0.849511i
$99$	99.0000	0.100504
$100$	0	0
$101$	−1758.18	−1.73213	−0.866067	−	0.499928i	$-0.833360\pi$
$101$	−1758.18	−1.73213	−0.866067	+	0.499928i	$0.833360\pi$
$102$	− 346.068i	− 0.335940i
$103$	− 959.852i	− 0.918223i	−0.888379	−	0.459112i	$-0.848168\pi$
$103$	− 959.852i	− 0.918223i	0.888379	−	0.459112i	$-0.151832\pi$
$104$	−1517.83	−1.43111
$105$	0	0
$106$	−618.772	−0.566985
$107$	− 1120.32i	− 1.01220i	−0.862476	−	0.506098i	$-0.831087\pi$
$107$	− 1120.32i	− 1.01220i	0.862476	−	0.506098i	$-0.168913\pi$
$108$	− 1.27159i	− 0.00113295i
$109$	1811.63	1.59195	0.795973	−	0.605331i	$-0.206959\pi$
$109$	1811.63	1.59195	0.795973	+	0.605331i	$0.206959\pi$
$110$	0	0
$111$	−1255.25	−1.07336
$112$	453.258i	0.382400i
$113$	961.487i	0.800435i	0.916420	+	0.400217i	$0.131065\pi$
$113$	961.487i	0.800435i	−0.916420	+	0.400217i	$0.868935\pi$
$114$	−41.0438	−0.0337202
$115$	0	0
$116$	10.5820	0.00846991
$117$	601.954i	0.475646i
$118$	1651.85i	1.28869i
$119$	−291.422	−0.224493
$120$	0	0
$121$	121.000	0.0909091
$122$	− 442.677i	− 0.328509i
$123$	− 1488.85i	− 1.09142i
$124$	−12.5843	−0.00911377
$125$	0	0
$126$	180.822	0.127848
$127$	− 1524.75i	− 1.06535i	−0.846320	−	0.532675i	$-0.821187\pi$
$127$	− 1524.75i	− 1.06535i	0.846320	−	0.532675i	$-0.178813\pi$
$128$	1435.24i	0.991079i
$129$	−272.865	−0.186236
$130$	0	0
$131$	−869.085	−0.579636	−0.289818	−	0.957082i	$-0.593595\pi$
$131$	−869.085	−0.579636	−0.289818	+	0.957082i	$0.593595\pi$
$132$	− 1.55417i	− 0.00102479i
$133$	34.5628i	0.0225336i
$134$	−1800.16	−1.16052
$135$	0	0
$136$	−928.281	−0.585290
$137$	1050.15i	0.654893i	0.944870	+	0.327447i	$0.106188\pi$
$137$	1050.15i	0.654893i	−0.944870	+	0.327447i	$0.893812\pi$
$138$	1084.46i	0.668951i
$139$	272.869	0.166507	0.0832534	−	0.996528i	$-0.473469\pi$
$139$	272.869	0.166507	0.0832534	+	0.996528i	$0.473469\pi$
$140$	0	0
$141$	610.919	0.364884
$142$	2710.79i	1.60200i
$143$	735.721i	0.430238i
$144$	572.589	0.331359
$145$	0	0
$146$	−629.100	−0.356607
$147$	876.731i	0.491915i
$148$	19.7058i	0.0109446i
$149$	199.929	0.109925	0.0549626	−	0.998488i	$-0.482496\pi$
$149$	199.929	0.109925	0.0549626	+	0.998488i	$0.482496\pi$
$150$	0	0
$151$	−2625.47	−1.41495	−0.707476	−	0.706737i	$-0.750166\pi$
$151$	−2625.47	−1.41495	−0.707476	+	0.706737i	$0.750166\pi$
$152$	110.095i	0.0587490i
$153$	368.146i	0.194528i
$154$	221.004	0.115643
$155$	0	0
$156$	9.44985	0.00484996
$157$	1581.68i	0.804023i	0.915635	+	0.402012i	$0.131689\pi$
$157$	1581.68i	0.804023i	−0.915635	+	0.402012i	$0.868311\pi$
$158$	1172.16i	0.590205i
$159$	658.247	0.328317
$160$	0	0
$161$	913.216	0.447028
$162$	− 228.427i	− 0.110784i
$163$	− 1104.98i	− 0.530974i	−0.964114	−	0.265487i	$-0.914467\pi$
$163$	− 1104.98i	− 0.530974i	0.964114	−	0.265487i	$-0.0855328\pi$
$164$	−23.3729	−0.0111287
$165$	0	0
$166$	−126.066	−0.0589436
$167$	− 200.333i	− 0.0928276i	−0.998922	−	0.0464138i	$-0.985221\pi$
$167$	− 200.333i	− 0.0928276i	0.998922	−	0.0464138i	$-0.0147793\pi$
$168$	− 485.029i	− 0.222743i
$169$	−2276.43	−1.03616
$170$	0	0
$171$	43.6623	0.0195260
$172$	4.28360i	0.00189896i
$173$	2070.79i	0.910052i	0.890478	+	0.455026i	$0.150370\pi$
$173$	2070.79i	0.910052i	−0.890478	+	0.455026i	$0.849630\pi$
$174$	1900.93	0.828214
$175$	0	0
$176$	699.831	0.299726
$177$	− 1757.24i	− 0.746226i
$178$	2283.59i	0.961586i
$179$	−2060.68	−0.860462	−0.430231	−	0.902719i	$-0.641568\pi$
$179$	−2060.68	−0.860462	−0.430231	+	0.902719i	$0.641568\pi$
$180$	0	0
$181$	4113.93	1.68943	0.844713	−	0.535220i	$-0.179771\pi$
$181$	4113.93	1.68943	0.844713	+	0.535220i	$0.179771\pi$
$182$	1343.78i	0.547294i
$183$	470.918i	0.190226i
$184$	2908.91	1.16548
$185$	0	0
$186$	−2260.64	−0.891173
$187$	449.956i	0.175958i
$188$	− 9.59060i	− 0.00372056i
$189$	−192.357	−0.0740314
$190$	0	0
$191$	2771.96	1.05012	0.525058	−	0.851066i	$-0.324044\pi$
$191$	2771.96	1.05012	0.525058	+	0.851066i	$0.324044\pi$
$192$	− 1544.94i	− 0.580709i
$193$	− 219.884i	− 0.0820082i	−0.999159	−	0.0410041i	$-0.986944\pi$
$193$	− 219.884i	− 0.0820082i	0.999159	−	0.0410041i	$-0.0130557\pi$
$194$	−1212.03	−0.448552
$195$	0	0
$196$	13.7635	0.00501585
$197$	2211.07i	0.799655i	0.916590	+	0.399827i	$0.130930\pi$
$197$	2211.07i	0.799655i	−0.916590	+	0.399827i	$0.869070\pi$
$198$	− 279.189i	− 0.100208i
$199$	471.758	0.168050	0.0840252	−	0.996464i	$-0.473222\pi$
$199$	471.758	0.168050	0.0840252	+	0.996464i	$0.473222\pi$
$200$	0	0
$201$	1915.00	0.672008
$202$	4958.23i	1.72703i
$203$	− 1600.76i	− 0.553456i
$204$	5.77939	0.00198352
$205$	0	0
$206$	−2706.87	−0.915517
$207$	− 1153.64i	− 0.387361i
$208$	4255.21i	1.41849i
$209$	53.3650	0.0176619
$210$	0	0
$211$	5623.68	1.83483	0.917417	−	0.397928i	$-0.130271\pi$
$211$	5623.68	1.83483	0.917417	+	0.397928i	$0.130271\pi$
$212$	− 10.3336i	− 0.00334771i
$213$	− 2883.73i	− 0.927652i
$214$	−3159.39	−1.00921
$215$	0	0
$216$	−612.725	−0.193012
$217$	1903.67i	0.595529i
$218$	− 5108.94i	− 1.58725i
$219$	669.234	0.206496
$220$	0	0
$221$	−2735.89	−0.832741
$222$	3539.93i	1.07020i
$223$	4421.93i	1.32787i	0.747792	+	0.663933i	$0.231114\pi$
$223$	4421.93i	1.32787i	−0.747792	+	0.663933i	$0.768886\pi$
$224$	−15.1840	−0.00452914
$225$	0	0
$226$	2711.48	0.798075
$227$	− 3489.83i	− 1.02039i	−0.860060	−	0.510194i	$-0.829574\pi$
$227$	− 3489.83i	− 1.02039i	0.860060	−	0.510194i	$-0.170426\pi$
$228$	− 0.685438i	0 0.000199098i
$229$	84.8948	0.0244978	0.0122489	−	0.999925i	$-0.496101\pi$
$229$	84.8948	0.0244978	0.0122489	+	0.999925i	$0.496101\pi$
$230$	0	0
$231$	−235.103	−0.0669639
$232$	− 5099.00i	− 1.44295i
$233$	− 4075.86i	− 1.14600i	−0.819555	−	0.573001i	$-0.805779\pi$
$233$	− 4075.86i	− 1.14600i	0.819555	−	0.573001i	$-0.194221\pi$
$234$	1697.56	0.474244
$235$	0	0
$236$	−27.5862	−0.00760894
$237$	− 1246.94i	− 0.341762i
$238$	821.836i	0.223831i
$239$	−763.901	−0.206747	−0.103374	−	0.994643i	$-0.532964\pi$
$239$	−763.901	−0.206747	−0.103374	+	0.994643i	$0.532964\pi$
$240$	0	0
$241$	2526.32	0.675248	0.337624	−	0.941281i	$-0.390377\pi$
$241$	2526.32	0.675248	0.337624	+	0.941281i	$0.390377\pi$
$242$	− 341.231i	− 0.0906411i
$243$	243.000i	0.0641500i
$244$	7.39278	0.00193965
$245$	0	0
$246$	−4198.68	−1.08820
$247$	324.477i	0.0835870i
$248$	6063.86i	1.55264i
$249$	134.109	0.0341317
$250$	0	0
$251$	963.988	0.242416	0.121208	−	0.992627i	$-0.461323\pi$
$251$	963.988	0.242416	0.121208	+	0.992627i	$0.461323\pi$
$252$	3.01975i	0 0.000754866i
$253$	− 1410.01i	− 0.350381i
$254$	−4299.92	−1.06221
$255$	0	0
$256$	−72.3368	−0.0176604
$257$	1517.07i	0.368218i	0.982906	+	0.184109i	$0.0589399\pi$
$257$	1517.07i	0.368218i	−0.982906	+	0.184109i	$0.941060\pi$
$258$	769.503i	0.185687i
$259$	2980.95	0.715164
$260$	0	0
$261$	−2022.20	−0.479584
$262$	2450.90i	0.577927i
$263$	1420.58i	0.333068i	0.986036	+	0.166534i	$0.0532575\pi$
$263$	1420.58i	0.333068i	−0.986036	+	0.166534i	$0.946742\pi$
$264$	−748.886	−0.174586
$265$	0	0
$266$	97.4701	0.0224672
$267$	− 2429.27i	− 0.556813i
$268$	− 30.0629i	− 0.00685217i
$269$	−6870.16	−1.55718	−0.778590	−	0.627533i	$-0.784065\pi$
$269$	−6870.16	−1.55718	−0.778590	+	0.627533i	$0.784065\pi$
$270$	0	0
$271$	2499.86	0.560353	0.280177	−	0.959948i	$-0.409607\pi$
$271$	2499.86	0.560353	0.280177	+	0.959948i	$0.409607\pi$
$272$	2602.42i	0.580129i
$273$	− 1429.51i	− 0.316915i
$274$	2961.52	0.652963
$275$	0	0
$276$	−18.1106	−0.00394975
$277$	− 1562.11i	− 0.338837i	−0.985544	−	0.169419i	$-0.945811\pi$
$277$	− 1562.11i	− 0.338837i	0.985544	−	0.169419i	$-0.0541890\pi$
$278$	− 769.515i	− 0.166016i
$279$	2404.86	0.516040
$280$	0	0
$281$	−7476.58	−1.58724	−0.793621	−	0.608412i	$-0.791807\pi$
$281$	−7476.58	−1.58724	−0.793621	+	0.608412i	$0.791807\pi$
$282$	− 1722.85i	− 0.363808i
$283$	− 2680.33i	− 0.563001i	−0.959561	−	0.281501i	$-0.909168\pi$
$283$	− 2680.33i	− 0.563001i	0.959561	−	0.281501i	$-0.0908322\pi$
$284$	−45.2706	−0.00945887
$285$	0	0
$286$	2074.80	0.428970
$287$	3535.68i	0.727194i
$288$	19.1816i	0.00392461i
$289$	3239.77	0.659428
$290$	0	0
$291$	1289.36	0.259737
$292$	− 10.5061i	− 0.00210555i
$293$	− 1320.86i	− 0.263363i	−0.991292	−	0.131682i	$-0.957962\pi$
$293$	− 1320.86i	− 0.263363i	0.991292	−	0.131682i	$-0.0420376\pi$
$294$	2472.46	0.490465
$295$	0	0
$296$	9495.38	1.86455
$297$	297.000i	0.0580259i
$298$	− 563.819i	− 0.109601i
$299$	8573.33	1.65822
$300$	0	0
$301$	647.994	0.124085
$302$	7404.06i	1.41078i
$303$	− 5274.54i	− 1.00005i
$304$	308.649	0.0582310
$305$	0	0
$306$	1038.20	0.193955
$307$	− 6644.48i	− 1.23525i	−0.786475	−	0.617623i	$-0.788096\pi$
$307$	− 6644.48i	− 1.23525i	0.786475	−	0.617623i	$-0.211904\pi$
$308$	3.69080i	0 0.000682802i
$309$	2879.56	0.530136
$310$	0	0
$311$	−6911.90	−1.26025	−0.630125	−	0.776493i	$-0.716996\pi$
$311$	−6911.90	−1.26025	−0.630125	+	0.776493i	$0.716996\pi$
$312$	− 4553.48i	− 0.826251i
$313$	3967.78i	0.716524i	0.933621	+	0.358262i	$0.116630\pi$
$313$	3967.78i	0.716524i	−0.933621	+	0.358262i	$0.883370\pi$
$314$	4460.47	0.801653
$315$	0	0
$316$	−19.5753	−0.00348480
$317$	− 3402.43i	− 0.602838i	−0.953492	−	0.301419i	$-0.902540\pi$
$317$	− 3402.43i	− 0.602838i	0.953492	−	0.301419i	$-0.0974602\pi$
$318$	− 1856.32i	− 0.327349i
$319$	−2471.58	−0.433800
$320$	0	0
$321$	3360.95	0.584392
$322$	− 2575.35i	− 0.445710i
$323$	198.446i	0.0341852i
$324$	3.81477	0.000654110 0
$325$	0	0
$326$	−3116.15	−0.529409
$327$	5434.88i	0.919111i
$328$	11262.4i	1.89592i
$329$	−1450.80	−0.243116
$330$	0	0
$331$	4518.53	0.750335	0.375167	−	0.926957i	$-0.377585\pi$
$331$	4518.53	0.750335	0.375167	+	0.926957i	$0.377585\pi$
$332$	− 2.10532i	0 0.000348026i
$333$	− 3765.76i	− 0.619707i
$334$	−564.956	−0.0925540
$335$	0	0
$336$	−1359.77	−0.220779
$337$	− 615.717i	− 0.0995260i	−0.998761	−	0.0497630i	$-0.984153\pi$
$337$	− 615.717i	− 0.0995260i	0.998761	−	0.0497630i	$-0.0158466\pi$
$338$	6419.75i	1.03310i
$339$	−2884.46	−0.462131
$340$	0	0
$341$	2939.27	0.466776
$342$	− 123.132i	− 0.0194684i
$343$	− 4525.69i	− 0.712433i
$344$	2064.09	0.323512
$345$	0	0
$346$	5839.80	0.907369
$347$	− 8001.58i	− 1.23789i	−0.785435	−	0.618944i	$-0.787561\pi$
$347$	− 8001.58i	− 1.23789i	0.785435	−	0.618944i	$-0.212439\pi$
$348$	31.7459i	0.00489011i
$349$	−934.074	−0.143266	−0.0716330	−	0.997431i	$-0.522821\pi$
$349$	−934.074	−0.143266	−0.0716330	+	0.997431i	$0.522821\pi$
$350$	0	0
$351$	−1805.86	−0.274615
$352$	23.4442i	0.00354994i
$353$	− 11951.7i	− 1.80205i	−0.433764	−	0.901026i	$-0.642815\pi$
$353$	− 11951.7i	− 1.80205i	0.433764	−	0.901026i	$-0.357185\pi$
$354$	−4955.56	−0.744026
$355$	0	0
$356$	−38.1363	−0.00567758
$357$	− 874.266i	− 0.129611i
$358$	5811.31i	0.857926i
$359$	−2014.59	−0.296173	−0.148086	−	0.988974i	$-0.547311\pi$
$359$	−2014.59	−0.296173	−0.148086	+	0.988974i	$0.547311\pi$
$360$	0	0
$361$	−6835.46	−0.996569
$362$	− 11601.7i	− 1.68445i
$363$	363.000i	0.0524864i
$364$	−22.4413	−0.00323144
$365$	0	0
$366$	1328.03	0.189665
$367$	4133.31i	0.587893i	0.955822	+	0.293947i	$0.0949688\pi$
$367$	4133.31i	0.587893i	−0.955822	+	0.293947i	$0.905031\pi$
$368$	− 8155.10i	− 1.15520i
$369$	4466.54	0.630132
$370$	0	0
$371$	−1563.19	−0.218752
$372$	− 37.7530i	− 0.00526184i
$373$	− 3255.81i	− 0.451955i	−0.974133	−	0.225978i	$-0.927442\pi$
$373$	− 3255.81i	− 0.451955i	0.974133	−	0.225978i	$-0.0725576\pi$
$374$	1268.92	0.175439
$375$	0	0
$376$	−4621.30	−0.633844
$377$	− 15028.1i	− 2.05301i
$378$	542.465i	0.0738131i
$379$	3279.44	0.444469	0.222234	−	0.974993i	$-0.428665\pi$
$379$	3279.44	0.444469	0.222234	+	0.974993i	$0.428665\pi$
$380$	0	0
$381$	4574.24	0.615080
$382$	− 7817.18i	− 1.04702i
$383$	− 10824.2i	− 1.44410i	−0.691841	−	0.722050i	$-0.743200\pi$
$383$	− 10824.2i	− 1.44410i	0.691841	−	0.722050i	$-0.256800\pi$
$384$	−4305.71	−0.572200
$385$	0	0
$386$	−620.092	−0.0817664
$387$	− 818.594i	− 0.107523i
$388$	− 20.2412i	− 0.00264843i
$389$	8828.59	1.15071	0.575356	−	0.817903i	$-0.304863\pi$
$389$	8828.59	1.15071	0.575356	+	0.817903i	$0.304863\pi$
$390$	0	0
$391$	5243.32	0.678174
$392$	− 6632.04i	− 0.854512i
$393$	− 2607.25i	− 0.334653i
$394$	6235.41	0.797297
$395$	0	0
$396$	4.66250	0.000591665 0
$397$	− 8975.60i	− 1.13469i	−0.823480	−	0.567346i	$-0.807970\pi$
$397$	− 8975.60i	− 1.13469i	0.823480	−	0.567346i	$-0.192030\pi$
$398$	− 1330.40i	− 0.167555i
$399$	−103.688	−0.0130098
$400$	0	0
$401$	10084.8	1.25589	0.627947	−	0.778257i	$-0.283896\pi$
$401$	10084.8	1.25589	0.627947	+	0.778257i	$0.283896\pi$
$402$	− 5400.47i	− 0.670027i
$403$	17871.8i	2.20907i
$404$	−82.8031	−0.0101971
$405$	0	0
$406$	−4514.30	−0.551825
$407$	− 4602.60i	− 0.560546i
$408$	− 2784.84i	− 0.337917i
$409$	−7533.27	−0.910749	−0.455374	−	0.890300i	$-0.650495\pi$
$409$	−7533.27	−0.910749	−0.455374	+	0.890300i	$0.650495\pi$
$410$	0	0
$411$	−3150.45	−0.378103
$412$	− 45.2051i	− 0.00540557i
$413$	4173.05i	0.497197i
$414$	−3253.38	−0.386219
$415$	0	0
$416$	−142.549	−0.0168005
$417$	818.607i	0.0961327i
$418$	− 150.494i	− 0.0176098i
$419$	1987.86	0.231774	0.115887	−	0.993262i	$-0.463029\pi$
$419$	1987.86	0.231774	0.115887	+	0.993262i	$0.463029\pi$
$420$	0	0
$421$	13280.0	1.53736	0.768678	−	0.639635i	$-0.220915\pi$
$421$	13280.0	1.53736	0.768678	+	0.639635i	$0.220915\pi$
$422$	− 15859.3i	− 1.82942i
$423$	1832.76i	0.210666i
$424$	−4979.32	−0.570323
$425$	0	0
$426$	−8132.38	−0.924918
$427$	− 1118.33i	− 0.126744i
$428$	− 52.7623i	− 0.00595879i
$429$	−2207.16	−0.248398
$430$	0	0
$431$	−11935.3	−1.33388	−0.666939	−	0.745112i	$-0.732396\pi$
$431$	−11935.3	−1.33388	−0.666939	+	0.745112i	$0.732396\pi$
$432$	1717.77i	0.191310i
$433$	− 12937.5i	− 1.43588i	−0.696102	−	0.717942i	$-0.745084\pi$
$433$	− 12937.5i	− 1.43588i	0.696102	−	0.717942i	$-0.254916\pi$
$434$	5368.53	0.593773
$435$	0	0
$436$	85.3202	0.00937178
$437$	− 621.860i	− 0.0680723i
$438$	− 1887.30i	− 0.205887i
$439$	2181.67	0.237187	0.118594	−	0.992943i	$-0.462161\pi$
$439$	2181.67	0.237187	0.118594	+	0.992943i	$0.462161\pi$
$440$	0	0
$441$	−2630.19	−0.284008
$442$	7715.45i	0.830286i
$443$	8733.61i	0.936673i	0.883550	+	0.468337i	$0.155147\pi$
$443$	8733.61i	0.936673i	−0.883550	+	0.468337i	$0.844853\pi$
$444$	−59.1173	−0.00631888
$445$	0	0
$446$	12470.2	1.32395
$447$	599.788i	0.0634653i
$448$	3668.88i	0.386916i
$449$	10113.3	1.06298	0.531490	−	0.847065i	$-0.321632\pi$
$449$	10113.3	1.06298	0.531490	+	0.847065i	$0.321632\pi$
$450$	0	0
$451$	5459.10	0.569976
$452$	45.2821i	0.00471215i
$453$	− 7876.41i	− 0.816923i
$454$	−9841.62	−1.01738
$455$	0	0
$456$	−330.284	−0.0339188
$457$	18924.7i	1.93711i	0.248793	+	0.968557i	$0.419966\pi$
$457$	18924.7i	1.93711i	−0.248793	+	0.968557i	$0.580034\pi$
$458$	− 239.411i	− 0.0244256i
$459$	−1104.44	−0.112311
$460$	0	0
$461$	12279.2	1.24056	0.620280	−	0.784380i	$-0.287019\pi$
$461$	12279.2	1.24056	0.620280	+	0.784380i	$0.287019\pi$
$462$	663.012i	0.0667665i
$463$	2512.53i	0.252197i	0.992018	+	0.126098i	$0.0402455\pi$
$463$	2512.53i	0.252197i	−0.992018	+	0.126098i	$0.959755\pi$
$464$	−14295.0	−1.43023
$465$	0	0
$466$	−11494.3	−1.14262
$467$	13924.8i	1.37979i	0.723910	+	0.689894i	$0.242343\pi$
$467$	13924.8i	1.37979i	−0.723910	+	0.689894i	$0.757657\pi$
$468$	28.3496i	0.00280013i
$469$	−4547.70	−0.447747
$470$	0	0
$471$	−4745.03	−0.464203
$472$	13292.6i	1.29628i
$473$	− 1000.50i	− 0.0972584i
$474$	−3516.49	−0.340755
$475$	0	0
$476$	−13.7248	−0.00132159
$477$	1974.74i	0.189554i
$478$	2154.27i	0.206138i
$479$	17622.0	1.68094	0.840468	−	0.541862i	$-0.182280\pi$
$479$	17622.0	1.68094	0.840468	+	0.541862i	$0.182280\pi$
$480$	0	0
$481$	27985.4	2.65285
$482$	− 7124.46i	− 0.673257i
$483$	2739.65i	0.258092i
$484$	5.69861	0.000535181 0
$485$	0	0
$486$	685.282	0.0639609
$487$	− 6109.87i	− 0.568510i	−0.958749	−	0.284255i	$-0.908254\pi$
$487$	− 6109.87i	− 0.568510i	0.958749	−	0.284255i	$-0.0917463\pi$
$488$	− 3562.27i	− 0.330443i
$489$	3314.94	0.306558
$490$	0	0
$491$	4134.23	0.379990	0.189995	−	0.981785i	$-0.439153\pi$
$491$	4134.23	0.379990	0.189995	+	0.981785i	$0.439153\pi$
$492$	− 70.1186i	− 0.00642518i
$493$	− 9190.95i	− 0.839634i
$494$	915.055	0.0833406
$495$	0	0
$496$	17000.0	1.53895
$497$	6848.23i	0.618078i
$498$	− 378.199i	− 0.0340311i
$499$	−16556.0	−1.48527	−0.742633	−	0.669698i	$-0.766423\pi$
$499$	−16556.0	−1.48527	−0.742633	+	0.669698i	$0.766423\pi$
$500$	0	0
$501$	600.998	0.0535941
$502$	− 2718.53i	− 0.241701i
$503$	− 22181.6i	− 1.96626i	−0.182899	−	0.983132i	$-0.558548\pi$
$503$	− 22181.6i	− 1.96626i	0.182899	−	0.983132i	$-0.441452\pi$
$504$	1455.09	0.128601
$505$	0	0
$506$	−3976.35	−0.349348
$507$	− 6829.30i	− 0.598225i
$508$	− 71.8094i	− 0.00627171i
$509$	19760.4	1.72076	0.860380	−	0.509654i	$-0.170226\pi$
$509$	19760.4	1.72076	0.860380	+	0.509654i	$0.170226\pi$
$510$	0	0
$511$	−1589.28	−0.137585
$512$	11685.9i	1.00869i
$513$	130.987i	0.0112733i
$514$	4278.26	0.367132
$515$	0	0
$516$	−12.8508	−0.00109637
$517$	2240.04i	0.190555i
$518$	− 8406.55i	− 0.713055i
$519$	−6212.36	−0.525419
$520$	0	0
$521$	−10283.0	−0.864697	−0.432349	−	0.901707i	$-0.642315\pi$
$521$	−10283.0	−0.864697	−0.432349	+	0.901707i	$0.642315\pi$
$522$	5702.80i	0.478170i
$523$	1923.24i	0.160798i	0.996763	+	0.0803992i	$0.0256195\pi$
$523$	1923.24i	0.160798i	−0.996763	+	0.0803992i	$0.974380\pi$
$524$	−40.9304	−0.00341231
$525$	0	0
$526$	4006.17	0.332086
$527$	10930.1i	0.903460i
$528$	2099.49i	0.173047i
$529$	−4263.77	−0.350437
$530$	0	0
$531$	5271.71	0.430834
$532$	1.62777i	0 0.000132655i
$533$	33193.2i	2.69748i
$534$	−6850.77	−0.555172
$535$	0	0
$536$	−14486.0	−1.16735
$537$	− 6182.05i	− 0.496788i
$538$	19374.5i	1.55259i
$539$	−3214.68	−0.256894
$540$	0	0
$541$	−10243.2	−0.814027	−0.407014	−	0.913422i	$-0.633430\pi$
$541$	−10243.2	−0.814027	−0.407014	+	0.913422i	$0.633430\pi$
$542$	− 7049.83i	− 0.558701i
$543$	12341.8i	0.975391i
$544$	−87.1807	−0.00687103
$545$	0	0
$546$	−4031.34	−0.315981
$547$	− 483.940i	− 0.0378277i	−0.999821	−	0.0189139i	$-0.993979\pi$
$547$	− 483.940i	− 0.0378277i	0.999821	−	0.0189139i	$-0.00602083\pi$
$548$	49.4578i	0.00385535i
$549$	−1412.76	−0.109827
$550$	0	0
$551$	−1090.05	−0.0842789
$552$	8726.74i	0.672889i
$553$	2961.22i	0.227710i
$554$	−4405.28	−0.337839
$555$	0	0
$556$	12.8510	0.000980223 0
$557$	23789.2i	1.80966i	0.425771	+	0.904831i	$0.360003\pi$
$557$	23789.2i	1.80966i	−0.425771	+	0.904831i	$0.639997\pi$
$558$	− 6781.92i	− 0.514519i
$559$	6083.40	0.460287
$560$	0	0
$561$	−1349.87	−0.101589
$562$	21084.6i	1.58256i
$563$	− 11331.9i	− 0.848280i	−0.905597	−	0.424140i	$-0.860576\pi$
$563$	− 11331.9i	− 0.848280i	0.905597	−	0.424140i	$-0.139424\pi$
$564$	28.7718	0.00214807
$565$	0	0
$566$	−7558.78	−0.561342
$567$	− 577.072i	− 0.0427420i
$568$	21814.0i	1.61144i
$569$	−2432.39	−0.179211	−0.0896056	−	0.995977i	$-0.528561\pi$
$569$	−2432.39	−0.179211	−0.0896056	+	0.995977i	$0.528561\pi$
$570$	0	0
$571$	20528.2	1.50451	0.752257	−	0.658870i	$-0.228965\pi$
$571$	20528.2	1.50451	0.752257	+	0.658870i	$0.228965\pi$
$572$	34.6495i	0.00253281i
$573$	8315.89i	0.606285i
$574$	9970.94	0.725051
$575$	0	0
$576$	4634.81	0.335272
$577$	13961.9i	1.00735i	0.863893	+	0.503675i	$0.168019\pi$
$577$	13961.9i	1.00735i	−0.863893	+	0.503675i	$0.831981\pi$
$578$	− 9136.45i	− 0.657484i
$579$	659.651	0.0473475
$580$	0	0
$581$	−318.479	−0.0227413
$582$	− 3636.10i	− 0.258971i
$583$	2413.57i	0.171458i
$584$	−5062.43	−0.358707
$585$	0	0
$586$	−3724.94	−0.262587
$587$	− 10785.8i	− 0.758393i	−0.925316	−	0.379196i	$-0.876200\pi$
$587$	− 10785.8i	− 0.758393i	0.925316	−	0.379196i	$-0.123800\pi$
$588$	41.2905i	0.00289590i
$589$	1296.32	0.0906856
$590$	0	0
$591$	−6633.20	−0.461681
$592$	− 26620.2i	− 1.84811i
$593$	8493.60i	0.588179i	0.955778	+	0.294090i	$0.0950164\pi$
$593$	8493.60i	0.588179i	−0.955778	+	0.294090i	$0.904984\pi$
$594$	837.567	0.0578548
$595$	0	0
$596$	9.41586	0.000647128 0
$597$	1415.27i	0.0970239i
$598$	− 24177.6i	− 1.65333i
$599$	12058.7	0.822549	0.411274	−	0.911512i	$-0.365084\pi$
$599$	12058.7	0.822549	0.411274	+	0.911512i	$0.365084\pi$
$600$	0	0
$601$	−12006.7	−0.814914	−0.407457	−	0.913224i	$-0.633584\pi$
$601$	−12006.7	−0.814914	−0.407457	+	0.913224i	$0.633584\pi$
$602$	− 1827.40i	− 0.123720i
$603$	5744.99i	0.387984i
$604$	−123.649	−0.00832981
$605$	0	0
$606$	−14874.7	−0.997100
$607$	− 27404.7i	− 1.83249i	−0.400619	−	0.916245i	$-0.631205\pi$
$607$	− 27404.7i	− 1.83249i	0.400619	−	0.916245i	$-0.368795\pi$
$608$	10.3397i	0 0.000689686i
$609$	4802.29	0.319538
$610$	0	0
$611$	−13620.2	−0.901822
$612$	17.3382i	0.00114519i
$613$	19837.8i	1.30708i	0.756891	+	0.653541i	$0.226717\pi$
$613$	19837.8i	1.30708i	−0.756891	+	0.653541i	$0.773283\pi$
$614$	−18738.0	−1.23160
$615$	0	0
$616$	1778.44	0.116324
$617$	− 6884.10i	− 0.449179i	−0.974453	−	0.224589i	$-0.927896\pi$
$617$	− 6884.10i	− 0.449179i	0.974453	−	0.224589i	$-0.0721041\pi$
$618$	− 8120.60i	− 0.528574i
$619$	11903.8	0.772944	0.386472	−	0.922301i	$-0.373694\pi$
$619$	11903.8	0.772944	0.386472	+	0.922301i	$0.373694\pi$
$620$	0	0
$621$	3460.93	0.223643
$622$	19492.2i	1.25654i
$623$	5768.99i	0.370995i
$624$	−12765.6	−0.818965
$625$	0	0
$626$	11189.5	0.714411
$627$	160.095i	0.0101971i
$628$	74.4906i	0.00473328i
$629$	17115.4	1.08496
$630$	0	0
$631$	−912.730	−0.0575836	−0.0287918	−	0.999585i	$-0.509166\pi$
$631$	−912.730	−0.0575836	−0.0287918	+	0.999585i	$0.509166\pi$
$632$	9432.51i	0.593679i
$633$	16871.0i	1.05934i
$634$	−9595.16	−0.601061
$635$	0	0
$636$	31.0008	0.00193280
$637$	− 19546.4i	− 1.21578i
$638$	6970.09i	0.432521i
$639$	8651.19	0.535580
$640$	0	0
$641$	17066.1	1.05159	0.525796	−	0.850610i	$-0.323767\pi$
$641$	17066.1	1.05159	0.525796	+	0.850610i	$0.323767\pi$
$642$	− 9478.17i	− 0.582669i
$643$	27311.7i	1.67507i	0.546384	+	0.837535i	$0.316004\pi$
$643$	27311.7i	1.67507i	−0.546384	+	0.837535i	$0.683996\pi$
$644$	43.0088	0.00263165
$645$	0	0
$646$	559.634	0.0340844
$647$	15190.8i	0.923050i	0.887127	+	0.461525i	$0.152697\pi$
$647$	15190.8i	0.923050i	−0.887127	+	0.461525i	$0.847303\pi$
$648$	− 1838.18i	− 0.111436i
$649$	6443.20	0.389704
$650$	0	0
$651$	−5711.02	−0.343829
$652$	− 52.0401i	− 0.00312584i
$653$	− 9474.30i	− 0.567776i	−0.958857	−	0.283888i	$-0.908376\pi$
$653$	− 9474.30i	− 0.567776i	0.958857	−	0.283888i	$-0.0916244\pi$
$654$	15326.8	0.916402
$655$	0	0
$656$	31574.0	1.87920
$657$	2007.70i	0.119221i
$658$	4091.38i	0.242399i
$659$	1220.71	0.0721582	0.0360791	−	0.999349i	$-0.488513\pi$
$659$	1220.71	0.0721582	0.0360791	+	0.999349i	$0.488513\pi$
$660$	0	0
$661$	−22686.2	−1.33494	−0.667468	−	0.744639i	$-0.732622\pi$
$661$	−22686.2	−1.33494	−0.667468	+	0.744639i	$0.732622\pi$
$662$	− 12742.7i	− 0.748123i
$663$	− 8207.66i	− 0.480783i
$664$	−1014.47	−0.0592906
$665$	0	0
$666$	−10619.8	−0.617880
$667$	28801.3i	1.67195i
$668$	− 9.43486i	0 0.000546475i
$669$	−13265.8	−0.766644
$670$	0	0
$671$	−1726.70	−0.0993421
$672$	− 45.5521i	− 0.00261490i
$673$	2761.82i	0.158188i	0.996867	+	0.0790938i	$0.0252027\pi$
$673$	2761.82i	0.158188i	−0.996867	+	0.0790938i	$0.974797\pi$
$674$	−1736.38	−0.0992326
$675$	0	0
$676$	−107.211	−0.00609984
$677$	16975.2i	0.963680i	0.876259	+	0.481840i	$0.160031\pi$
$677$	16975.2i	0.963680i	−0.876259	+	0.481840i	$0.839969\pi$
$678$	8134.44i	0.460769i
$679$	−3061.94	−0.173058
$680$	0	0
$681$	10469.5	0.589121
$682$	− 8289.02i	− 0.465400i
$683$	− 2447.72i	− 0.137130i	−0.997647	−	0.0685649i	$-0.978158\pi$
$683$	− 2447.72i	− 0.137130i	0.997647	−	0.0685649i	$-0.0218420\pi$
$684$	2.05632	0.000114949 0
$685$	0	0
$686$	−12762.9	−0.710333
$687$	254.684i	0.0141438i
$688$	− 5786.64i	− 0.320659i
$689$	−14675.3	−0.811446
$690$	0	0
$691$	−27460.5	−1.51179	−0.755894	−	0.654694i	$-0.772797\pi$
$691$	−27460.5	−1.51179	−0.755894	+	0.654694i	$0.772797\pi$
$692$	97.5256i	0.00535747i
$693$	− 705.310i	− 0.0386616i
$694$	−22565.2	−1.23424
$695$	0	0
$696$	15297.0	0.833090
$697$	20300.5i	1.10321i
$698$	2634.17i	0.142844i
$699$	12227.6	0.661645
$700$	0	0
$701$	−7980.51	−0.429986	−0.214993	−	0.976616i	$-0.568973\pi$
$701$	−7980.51	−0.429986	−0.214993	+	0.976616i	$0.568973\pi$
$702$	5092.69i	0.273805i
$703$	− 2029.90i	− 0.108903i
$704$	5664.76	0.303265
$705$	0	0
$706$	−33704.9	−1.79674
$707$	12525.9i	0.666314i
$708$	− 82.7586i	− 0.00439302i
$709$	−4867.53	−0.257834	−0.128917	−	0.991655i	$-0.541150\pi$
$709$	−4867.53	−0.257834	−0.128917	+	0.991655i	$0.541150\pi$
$710$	0	0
$711$	3740.83	0.197317
$712$	18376.3i	0.967247i
$713$	− 34251.2i	− 1.79905i
$714$	−2465.51	−0.129229
$715$	0	0
$716$	−97.0498	−0.00506553
$717$	− 2291.70i	− 0.119366i
$718$	5681.33i	0.295300i
$719$	−17353.3	−0.900094	−0.450047	−	0.893005i	$-0.648593\pi$
$719$	−17353.3	−0.900094	−0.450047	+	0.893005i	$0.648593\pi$
$720$	0	0
$721$	−6838.31	−0.353221
$722$	19276.6i	0.993631i
$723$	7578.97i	0.389855i
$724$	193.749	0.00994563
$725$	0	0
$726$	1023.69	0.0523317
$727$	− 37570.5i	− 1.91666i	−0.285663	−	0.958330i	$-0.592214\pi$
$727$	− 37570.5i	− 1.91666i	0.285663	−	0.958330i	$-0.407786\pi$
$728$	10813.5i	0.550516i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	3720.52	0.188247
$732$	22.1783i	0.00111986i
$733$	− 10035.5i	− 0.505689i	−0.967507	−	0.252845i	$-0.918634\pi$
$733$	− 10035.5i	− 0.505689i	0.967507	−	0.252845i	$-0.0813661\pi$
$734$	11656.3	0.586160
$735$	0	0
$736$	273.194	0.0136822
$737$	7021.66i	0.350945i
$738$	− 12596.0i	− 0.628274i
$739$	20918.8	1.04129	0.520643	−	0.853775i	$-0.325692\pi$
$739$	20918.8	1.04129	0.520643	+	0.853775i	$0.325692\pi$
$740$	0	0
$741$	−973.432	−0.0482590
$742$	4408.34i	0.218107i
$743$	− 25389.4i	− 1.25363i	−0.779168	−	0.626815i	$-0.784358\pi$
$743$	− 25389.4i	− 1.25363i	0.779168	−	0.626815i	$-0.215642\pi$
$744$	−18191.6	−0.896420
$745$	0	0
$746$	−9181.66	−0.450623
$747$	402.326i	0.0197059i
$748$	21.1911i	0.00103586i
$749$	−7981.51	−0.389370
$750$	0	0
$751$	14657.0	0.712170	0.356085	−	0.934454i	$-0.384111\pi$
$751$	14657.0	0.712170	0.356085	+	0.934454i	$0.384111\pi$
$752$	12955.8i	0.628255i
$753$	2891.96i	0.139959i
$754$	−42380.5	−2.04696
$755$	0	0
$756$	−9.05924	−0.000435822 0
$757$	7874.42i	0.378072i	0.981970	+	0.189036i	$0.0605363\pi$
$757$	7874.42i	0.378072i	−0.981970	+	0.189036i	$0.939464\pi$
$758$	− 9248.33i	− 0.443159i
$759$	4230.02	0.202293
$760$	0	0
$761$	−20257.8	−0.964975	−0.482487	−	0.875903i	$-0.660267\pi$
$761$	−20257.8	−0.964975	−0.482487	+	0.875903i	$0.660267\pi$
$762$	− 12899.8i	− 0.613267i
$763$	− 12906.6i	− 0.612387i
$764$	130.548	0.00618202
$765$	0	0
$766$	−30525.2	−1.43984
$767$	39176.8i	1.84432i
$768$	− 217.010i	− 0.0101962i
$769$	39900.7	1.87107	0.935536	−	0.353231i	$-0.114917\pi$
$769$	39900.7	1.87107	0.935536	+	0.353231i	$0.114917\pi$
$770$	0	0
$771$	−4551.20	−0.212591
$772$	− 10.3556i	0 0.000482781i
$773$	− 22980.3i	− 1.06927i	−0.845083	−	0.534635i	$-0.820449\pi$
$773$	− 22980.3i	− 1.06927i	0.845083	−	0.534635i	$-0.179551\pi$
$774$	−2308.51	−0.107206
$775$	0	0
$776$	−9753.36	−0.451192
$777$	8942.86i	0.412900i
$778$	− 24897.4i	− 1.14732i
$779$	2407.65	0.110735
$780$	0	0
$781$	10573.7	0.484451
$782$	− 14786.6i	− 0.676175i
$783$	− 6066.61i	− 0.276888i
$784$	−18592.8	−0.846977
$785$	0	0
$786$	−7352.69	−0.333666
$787$	− 2717.00i	− 0.123063i	−0.998105	−	0.0615316i	$-0.980402\pi$
$787$	− 2717.00i	− 0.123063i	0.998105	−	0.0615316i	$-0.0195985\pi$
$788$	104.132i	0.00470756i
$789$	−4261.75	−0.192297
$790$	0	0
$791$	6849.96	0.307910
$792$	− 2246.66i	− 0.100797i
$793$	− 10498.9i	− 0.470149i
$794$	−25312.0	−1.13135
$795$	0	0
$796$	22.2179	0.000989311 0
$797$	− 29346.0i	− 1.30425i	−0.758111	−	0.652125i	$-0.773878\pi$
$797$	− 29346.0i	− 1.30425i	0.758111	−	0.652125i	$-0.226122\pi$
$798$	292.410i	0.0129714i
$799$	−8329.90	−0.368825
$800$	0	0
$801$	7287.82	0.321476
$802$	− 28440.2i	− 1.25219i
$803$	2453.86i	0.107839i
$804$	90.1886	0.00395610
$805$	0	0
$806$	50400.0	2.20256
$807$	− 20610.5i	− 0.899038i
$808$	39899.3i	1.73719i
$809$	19651.7	0.854037	0.427019	−	0.904243i	$-0.359564\pi$
$809$	19651.7	0.854037	0.427019	+	0.904243i	$0.359564\pi$
$810$	0	0
$811$	−33226.2	−1.43863	−0.719316	−	0.694683i	$-0.755544\pi$
$811$	−33226.2	−1.43863	−0.719316	+	0.694683i	$0.755544\pi$
$812$	− 75.3895i	− 0.00325819i
$813$	7499.58i	0.323520i
$814$	−12979.7	−0.558894
$815$	0	0
$816$	−7807.27	−0.334938
$817$	− 441.255i	− 0.0188954i
$818$	21244.5i	0.908064i
$819$	4288.52	0.182971
$820$	0	0
$821$	−6761.60	−0.287432	−0.143716	−	0.989619i	$-0.545905\pi$
$821$	−6761.60	−0.287432	−0.143716	+	0.989619i	$0.545905\pi$
$822$	8884.55i	0.376988i
$823$	− 30624.0i	− 1.29707i	−0.761186	−	0.648534i	$-0.775383\pi$
$823$	− 30624.0i	− 1.29707i	0.761186	−	0.648534i	$-0.224617\pi$
$824$	−21782.4	−0.920906
$825$	0	0
$826$	11768.4	0.495731
$827$	18267.5i	0.768106i	0.923311	+	0.384053i	$0.125472\pi$
$827$	18267.5i	0.768106i	−0.923311	+	0.384053i	$0.874528\pi$
$828$	− 54.3319i	− 0.00228039i
$829$	15153.2	0.634853	0.317427	−	0.948283i	$-0.397181\pi$
$829$	15153.2	0.634853	0.317427	+	0.948283i	$0.397181\pi$
$830$	0	0
$831$	4686.32	0.195628
$832$	34443.7i	1.43524i
$833$	− 11954.3i	− 0.497228i
$834$	2308.54	0.0958493
$835$	0	0
$836$	2.51327	0.000103975 0
$837$	7214.58i	0.297936i
$838$	− 5605.93i	− 0.231090i
$839$	−22328.6	−0.918794	−0.459397	−	0.888231i	$-0.651934\pi$
$839$	−22328.6	−0.918794	−0.459397	+	0.888231i	$0.651934\pi$
$840$	0	0
$841$	26096.3	1.07000
$842$	− 37450.8i	− 1.53283i
$843$	− 22429.7i	− 0.916395i
$844$	264.852	0.0108016
$845$	0	0
$846$	5168.54	0.210045
$847$	− 862.045i	− 0.0349708i
$848$	13959.5i	0.565294i
$849$	8041.00	0.325049
$850$	0	0
$851$	−53633.9	−2.16045
$852$	− 135.812i	− 0.00546108i
$853$	888.939i	0.0356819i	0.999841	+	0.0178410i	$0.00567926\pi$
$853$	888.939i	0.0356819i	−0.999841	+	0.0178410i	$0.994321\pi$
$854$	−3153.79	−0.126370
$855$	0	0
$856$	−25423.9	−1.01515
$857$	11712.7i	0.466861i	0.972374	+	0.233430i	$0.0749951\pi$
$857$	11712.7i	0.466861i	−0.972374	+	0.233430i	$0.925005\pi$
$858$	6224.40i	0.247666i
$859$	9943.60	0.394961	0.197480	−	0.980307i	$-0.436724\pi$
$859$	9943.60	0.394961	0.197480	+	0.980307i	$0.436724\pi$
$860$	0	0
$861$	−10607.0	−0.419846
$862$	33658.5i	1.32995i
$863$	− 21912.2i	− 0.864309i	−0.901800	−	0.432154i	$-0.857754\pi$
$863$	− 21912.2i	− 0.864309i	0.901800	−	0.432154i	$-0.142246\pi$
$864$	−57.5448	−0.00226587
$865$	0	0
$866$	−36485.0	−1.43165
$867$	9719.31i	0.380721i
$868$	89.6552i	0.00350587i
$869$	4572.13	0.178480
$870$	0	0
$871$	−42694.1	−1.66089
$872$	− 41112.2i	− 1.59660i
$873$	3868.07i	0.149959i
$874$	−1753.70	−0.0678717
$875$	0	0
$876$	31.5182	0.00121564
$877$	− 3185.48i	− 0.122652i	−0.998118	−	0.0613261i	$-0.980467\pi$
$877$	− 3185.48i	− 0.122652i	0.998118	−	0.0613261i	$-0.0195330\pi$
$878$	− 6152.50i	− 0.236488i
$879$	3962.58	0.152053
$880$	0	0
$881$	−2812.86	−0.107568	−0.0537842	−	0.998553i	$-0.517128\pi$
$881$	−2812.86	−0.107568	−0.0537842	+	0.998553i	$0.517128\pi$
$882$	7417.38i	0.283170i
$883$	− 32216.0i	− 1.22781i	−0.789380	−	0.613905i	$-0.789598\pi$
$883$	− 32216.0i	− 1.22781i	0.789380	−	0.613905i	$-0.210402\pi$
$884$	−128.849	−0.00490234
$885$	0	0
$886$	24629.6	0.933912
$887$	44471.4i	1.68343i	0.539921	+	0.841715i	$0.318454\pi$
$887$	44471.4i	1.68343i	−0.539921	+	0.841715i	$0.681546\pi$
$888$	28486.1i	1.07650i
$889$	−10862.8	−0.409817
$890$	0	0
$891$	−891.000	−0.0335013
$892$	208.255i	0.00781713i
$893$	987.930i	0.0370211i
$894$	1691.46	0.0632782
$895$	0	0
$896$	10225.1	0.381246
$897$	25720.0i	0.957375i
$898$	− 28520.5i	− 1.05985i
$899$	−60038.5	−2.22736
$900$	0	0
$901$	−8975.23	−0.331862
$902$	− 15395.2i	− 0.568296i
$903$	1943.98i	0.0716408i
$904$	21819.5	0.802773
$905$	0	0
$906$	−22212.2	−0.814515
$907$	− 21224.4i	− 0.777005i	−0.921448	−	0.388502i	$-0.872992\pi$
$907$	− 21224.4i	− 0.777005i	0.921448	−	0.388502i	$-0.127008\pi$
$908$	− 164.357i	− 0.00600701i
$909$	15823.6	0.577378
$910$	0	0
$911$	9599.56	0.349119	0.174560	−	0.984647i	$-0.444150\pi$
$911$	9599.56	0.349119	0.174560	+	0.984647i	$0.444150\pi$
$912$	925.946i	0.0336197i
$913$	491.732i	0.0178247i
$914$	53369.4	1.93140
$915$	0	0
$916$	3.99820	0.000144219 0
$917$	6191.66i	0.222973i
$918$	3114.61i	0.111980i
$919$	−51795.7	−1.85918	−0.929588	−	0.368600i	$-0.879837\pi$
$919$	−51795.7	−1.85918	−0.929588	+	0.368600i	$0.879837\pi$
$920$	0	0
$921$	19933.4	0.713169
$922$	− 34628.4i	− 1.23690i
$923$	64291.6i	2.29272i
$924$	−11.0724	−0.000394216 0
$925$	0	0
$926$	7085.55	0.251453
$927$	8638.67i	0.306074i
$928$	− 478.878i	− 0.0169396i
$929$	18821.8	0.664719	0.332360	−	0.943153i	$-0.392155\pi$
$929$	18821.8	0.664719	0.332360	+	0.943153i	$0.392155\pi$
$930$	0	0
$931$	−1417.78	−0.0499097
$932$	− 191.956i	− 0.00674650i
$933$	− 20735.7i	− 0.727606i
$934$	39269.1	1.37572
$935$	0	0
$936$	13660.5	0.477036
$937$	− 13493.1i	− 0.470437i	−0.971943	−	0.235218i	$-0.924419\pi$
$937$	− 13493.1i	− 0.470437i	0.971943	−	0.235218i	$-0.0755805\pi$
$938$	12824.9i	0.446427i
$939$	−11903.3	−0.413685
$940$	0	0
$941$	19364.0	0.670826	0.335413	−	0.942071i	$-0.391124\pi$
$941$	19364.0	0.670826	0.335413	+	0.942071i	$0.391124\pi$
$942$	13381.4i	0.462835i
$943$	− 63614.7i	− 2.19680i
$944$	37265.7	1.28485
$945$	0	0
$946$	−2821.51	−0.0969717
$947$	3304.64i	0.113396i	0.998391	+	0.0566982i	$0.0180573\pi$
$947$	3304.64i	0.113396i	−0.998391	+	0.0566982i	$0.981943\pi$
$948$	− 58.7259i	− 0.00201195i
$949$	−14920.3	−0.510361
$950$	0	0
$951$	10207.3	0.348049
$952$	6613.39i	0.225148i
$953$	38598.0i	1.31197i	0.754772	+	0.655987i	$0.227748\pi$
$953$	38598.0i	1.31197i	−0.754772	+	0.655987i	$0.772252\pi$
$954$	5568.95	0.188995
$955$	0	0
$956$	−35.9766	−0.00121712
$957$	− 7414.75i	− 0.250454i
$958$	− 49695.5i	− 1.67598i
$959$	7481.63	0.251923
$960$	0	0
$961$	41608.4	1.39668
$962$	− 78921.2i	− 2.64503i
$963$	10082.8i	0.337399i
$964$	118.979	0.00397518
$965$	0	0
$966$	7726.06	0.257331
$967$	− 23257.1i	− 0.773421i	−0.922201	−	0.386710i	$-0.873611\pi$
$967$	− 23257.1i	− 0.773421i	0.922201	−	0.386710i	$-0.126389\pi$
$968$	− 2745.92i	− 0.0911747i
$969$	−595.337	−0.0197368
$970$	0	0
$971$	32050.7	1.05928	0.529638	−	0.848224i	$-0.322328\pi$
$971$	32050.7	1.05928	0.529638	+	0.848224i	$0.322328\pi$
$972$	11.4443i	0 0.000377651i
$973$	− 1944.01i	− 0.0640515i
$974$	−17230.4	−0.566835
$975$	0	0
$976$	−9986.77	−0.327529
$977$	− 19808.8i	− 0.648660i	−0.945944	−	0.324330i	$-0.894861\pi$
$977$	− 19808.8i	− 0.648660i	0.945944	−	0.324330i	$-0.105139\pi$
$978$	− 9348.44i	− 0.305655i
$979$	8907.34	0.290786
$980$	0	0
$981$	−16304.6	−0.530649
$982$	− 11658.9i	− 0.378870i
$983$	− 10806.7i	− 0.350640i	−0.984511	−	0.175320i	$-0.943904\pi$
$983$	− 10806.7i	− 0.350640i	0.984511	−	0.175320i	$-0.0560961\pi$
$984$	−33787.2	−1.09461
$985$	0	0
$986$	−25919.3	−0.837158
$987$	− 4352.40i	− 0.140363i
$988$	15.2816i	0 0.000492076i
$989$	−11658.8	−0.374853
$990$	0	0
$991$	−38387.8	−1.23050	−0.615251	−	0.788331i	$-0.710945\pi$
$991$	−38387.8	−1.23050	−0.615251	+	0.788331i	$0.710945\pi$
$992$	569.495i	0.0182273i
$993$	13555.6i	0.433206i
$994$	19312.6	0.616256
$995$	0	0
$996$	6.31597	0.000200933 0
$997$	− 45526.2i	− 1.44617i	−0.690761	−	0.723083i	$-0.742724\pi$
$997$	− 45526.2i	− 1.44617i	0.690761	−	0.723083i	$-0.257276\pi$
$998$	46689.4i	1.48089i
$999$	11297.3	0.357788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.o.199.2		6
5.2	odd	4		825.4.a.n.1.3		3
5.3	odd	4		165.4.a.f.1.1	✓	3
5.4	even	2	inner	825.4.c.o.199.5		6
15.2	even	4		2475.4.a.w.1.1		3
15.8	even	4		495.4.a.g.1.3		3
55.43	even	4		1815.4.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.f.1.1	✓	3	5.3	odd	4
495.4.a.g.1.3		3	15.8	even	4
825.4.a.n.1.3		3	5.2	odd	4
825.4.c.o.199.2		6	1.1	even	1	trivial
825.4.c.o.199.5		6	5.4	even	2	inner
1815.4.a.p.1.3		3	55.43	even	4
2475.4.a.w.1.1		3	15.2	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82009i q^{2} +3.00000i q^{3} +0.0470959 q^{4} +8.46027 q^{6} -7.12434i q^{7} -22.6935i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-2.82009i q^{2} +3.00000i q^{3} +0.0470959 q^{4} +8.46027 q^{6} -7.12434i q^{7} -22.6935i q^{8} -9.00000 q^{9} -11.0000 q^{11} +0.141288i q^{12} -66.8837i q^{13} -20.0913 q^{14} -63.6210 q^{16} -40.9051i q^{17} +25.3808i q^{18} -4.85136 q^{19} +21.3730 q^{21} +31.0210i q^{22} +128.183i q^{23} +68.0806 q^{24} -188.618 q^{26} -27.0000i q^{27} -0.335527i q^{28} +224.689 q^{29} -267.207 q^{31} -2.13129i q^{32} -33.0000i q^{33} -115.356 q^{34} -0.423863 q^{36} +418.418i q^{37} +13.6813i q^{38} +200.651 q^{39} -496.282 q^{41} -60.2738i q^{42} +90.9549i q^{43} -0.518055 q^{44} +361.486 q^{46} -203.640i q^{47} -190.863i q^{48} +292.244 q^{49} +122.715 q^{51} -3.14995i q^{52} -219.416i q^{53} -76.1424 q^{54} -161.676 q^{56} -14.5541i q^{57} -633.644i q^{58} -585.745 q^{59} +156.973 q^{61} +753.547i q^{62} +64.1191i q^{63} -514.979 q^{64} -93.0629 q^{66} -638.333i q^{67} -1.92646i q^{68} -384.548 q^{69} -961.243 q^{71} +204.242i q^{72} -223.078i q^{73} +1179.98 q^{74} -0.228479 q^{76} +78.3678i q^{77} -565.854i q^{78} -415.648 q^{79} +81.0000 q^{81} +1399.56i q^{82} -44.7029i q^{83} +1.00658 q^{84} +256.501 q^{86} +674.068i q^{87} +249.629i q^{88} -809.758 q^{89} -476.503 q^{91} +6.03688i q^{92} -801.620i q^{93} -574.282 q^{94} +6.39387 q^{96} -429.786i q^{97} -824.154i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{6} - 54 q^{9}+O(q^{10}) \) 6 * q + 10 * q^4 - 6 * q^6 - 54 * q^9	\( 6 q + 10 q^{4} - 6 q^{6} - 54 q^{9} - 66 q^{11} + 152 q^{14} - 170 q^{16} + 560 q^{19} + 96 q^{21} - 18 q^{24} - 628 q^{26} + 580 q^{29} - 784 q^{31} - 620 q^{34} - 90 q^{36} - 252 q^{39} - 1324 q^{41} - 110 q^{44} + 1456 q^{46} - 1462 q^{49} + 204 q^{51} + 54 q^{54} + 408 q^{56} + 1224 q^{59} - 1164 q^{61} - 694 q^{64} + 66 q^{66} - 672 q^{69} - 3232 q^{71} + 1284 q^{74} + 1760 q^{76} - 248 q^{79} + 486 q^{81} - 1680 q^{84} - 2000 q^{86} - 1676 q^{89} - 3472 q^{91} - 4864 q^{94} + 138 q^{96} + 594 q^{99}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^6 - 54 * q^9 - 66 * q^11 + 152 * q^14 - 170 * q^16 + 560 * q^19 + 96 * q^21 - 18 * q^24 - 628 * q^26 + 580 * q^29 - 784 * q^31 - 620 * q^34 - 90 * q^36 - 252 * q^39 - 1324 * q^41 - 110 * q^44 + 1456 * q^46 - 1462 * q^49 + 204 * q^51 + 54 * q^54 + 408 * q^56 + 1224 * q^59 - 1164 * q^61 - 694 * q^64 + 66 * q^66 - 672 * q^69 - 3232 * q^71 + 1284 * q^74 + 1760 * q^76 - 248 * q^79 + 486 * q^81 - 1680 * q^84 - 2000 * q^86 - 1676 * q^89 - 3472 * q^91 - 4864 * q^94 + 138 * q^96 + 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more