LMFDB - Embedded newform 750.6.c.c.499.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.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:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.289444000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 39x^{6} + 541x^{4} + 3084x^{2} + 5776 $ x^8 + 39x^6 + 541x^4 + 3084*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.1
Root			$1.91466i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.c.499.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -124.927i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -124.927i q^{7} +64.0000i q^{8} -81.0000 q^{9} -240.403 q^{11} -144.000i q^{12} -457.858i q^{13} -499.708 q^{14} +256.000 q^{16} +1604.83i q^{17} +324.000i q^{18} +2172.36 q^{19} +1124.34 q^{21} +961.612i q^{22} +550.861i q^{23} -576.000 q^{24} -1831.43 q^{26} -729.000i q^{27} +1998.83i q^{28} +4721.82 q^{29} -786.542 q^{31} -1024.00i q^{32} -2163.63i q^{33} +6419.33 q^{34} +1296.00 q^{36} +13932.7i q^{37} -8689.43i q^{38} +4120.72 q^{39} -18357.8 q^{41} -4497.37i q^{42} +3021.67i q^{43} +3846.45 q^{44} +2203.44 q^{46} -20006.4i q^{47} +2304.00i q^{48} +1200.25 q^{49} -14443.5 q^{51} +7325.73i q^{52} -20498.2i q^{53} -2916.00 q^{54} +7995.33 q^{56} +19551.2i q^{57} -18887.3i q^{58} -33616.6 q^{59} +16561.7 q^{61} +3146.17i q^{62} +10119.1i q^{63} -4096.00 q^{64} -8654.50 q^{66} -12539.9i q^{67} -25677.3i q^{68} -4957.75 q^{69} -1372.75 q^{71} -5184.00i q^{72} +44929.9i q^{73} +55731.0 q^{74} -34757.7 q^{76} +30032.8i q^{77} -16482.9i q^{78} +104735. q^{79} +6561.00 q^{81} +73431.4i q^{82} +60213.5i q^{83} -17989.5 q^{84} +12086.7 q^{86} +42496.4i q^{87} -15385.8i q^{88} -19799.6 q^{89} -57198.8 q^{91} -8813.78i q^{92} -7078.88i q^{93} -80025.4 q^{94} +9216.00 q^{96} -105229. i q^{97} -4801.01i q^{98} +19472.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	− 124.927i	− 0.963632i	−0.876272	−	0.481816i	$-0.839977\pi$
$7$	− 124.927i	− 0.963632i	0.876272	−	0.481816i	$-0.160023\pi$
$8$	64.0000i	0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−240.403	−0.599043	−0.299522	−	0.954090i	$-0.596827\pi$
$11$	−240.403	−0.599043	−0.299522	+	0.954090i	$0.596827\pi$
$12$	− 144.000i	− 0.288675i
$13$	− 457.858i	− 0.751403i	−0.926741	−	0.375701i	$-0.877402\pi$
$13$	− 457.858i	− 0.751403i	0.926741	−	0.375701i	$-0.122598\pi$
$14$	−499.708	−0.681391
$15$	0	0
$16$	256.000	0.250000
$17$	1604.83i	1.34681i	0.739273	+	0.673406i	$0.235170\pi$
$17$	1604.83i	1.34681i	−0.739273	+	0.673406i	$0.764830\pi$
$18$	324.000i	0.235702i
$19$	2172.36	1.38054	0.690268	−	0.723554i	$-0.257493\pi$
$19$	2172.36	1.38054	0.690268	+	0.723554i	$0.257493\pi$
$20$	0	0
$21$	1124.34	0.556353
$22$	961.612i	0.423587i
$23$	550.861i	0.217131i	0.994089	+	0.108566i	$0.0346258\pi$
$23$	550.861i	0.217131i	−0.994089	+	0.108566i	$0.965374\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	−1831.43	−0.531322
$27$	− 729.000i	− 0.192450i
$28$	1998.83i	0.481816i
$29$	4721.82	1.04259	0.521296	−	0.853376i	$-0.325449\pi$
$29$	4721.82	1.04259	0.521296	+	0.853376i	$0.325449\pi$
$30$	0	0
$31$	−786.542	−0.147000	−0.0735001	−	0.997295i	$-0.523417\pi$
$31$	−786.542	−0.147000	−0.0735001	+	0.997295i	$0.523417\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 2163.63i	− 0.345858i
$34$	6419.33	0.952340
$35$	0	0
$36$	1296.00	0.166667
$37$	13932.7i	1.67314i	0.547860	+	0.836570i	$0.315443\pi$
$37$	13932.7i	1.67314i	−0.547860	+	0.836570i	$0.684557\pi$
$38$	− 8689.43i	− 0.976186i
$39$	4120.72	0.433822
$40$	0	0
$41$	−18357.8	−1.70554	−0.852770	−	0.522286i	$-0.825079\pi$
$41$	−18357.8	−1.70554	−0.852770	+	0.522286i	$0.825079\pi$
$42$	− 4497.37i	− 0.393401i
$43$	3021.67i	0.249216i	0.992206	+	0.124608i	$0.0397673\pi$
$43$	3021.67i	0.249216i	−0.992206	+	0.124608i	$0.960233\pi$
$44$	3846.45	0.299522
$45$	0	0
$46$	2203.44	0.153535
$47$	− 20006.4i	− 1.32106i	−0.750799	−	0.660531i	$-0.770331\pi$
$47$	− 20006.4i	− 1.32106i	0.750799	−	0.660531i	$-0.229669\pi$
$48$	2304.00i	0.144338i
$49$	1200.25	0.0714138
$50$	0	0
$51$	−14443.5	−0.777583
$52$	7325.73i	0.375701i
$53$	− 20498.2i	− 1.00236i	−0.865342	−	0.501182i	$-0.832899\pi$
$53$	− 20498.2i	− 1.00236i	0.865342	−	0.501182i	$-0.167101\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	7995.33	0.340695
$57$	19551.2i	0.797052i
$58$	− 18887.3i	− 0.737225i
$59$	−33616.6	−1.25725	−0.628627	−	0.777707i	$-0.716383\pi$
$59$	−33616.6	−1.25725	−0.628627	+	0.777707i	$0.716383\pi$
$60$	0	0
$61$	16561.7	0.569876	0.284938	−	0.958546i	$-0.408027\pi$
$61$	16561.7	0.569876	0.284938	+	0.958546i	$0.408027\pi$
$62$	3146.17i	0.103945i
$63$	10119.1i	0.321211i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−8654.50	−0.244558
$67$	− 12539.9i	− 0.341277i	−0.985334	−	0.170638i	$-0.945417\pi$
$67$	− 12539.9i	− 0.341277i	0.985334	−	0.170638i	$-0.0545830\pi$
$68$	− 25677.3i	− 0.673406i
$69$	−4957.75	−0.125361
$70$	0	0
$71$	−1372.75	−0.0323182	−0.0161591	−	0.999869i	$-0.505144\pi$
$71$	−1372.75	−0.0323182	−0.0161591	+	0.999869i	$0.505144\pi$
$72$	− 5184.00i	− 0.117851i
$73$	44929.9i	0.986798i	0.869803	+	0.493399i	$0.164246\pi$
$73$	44929.9i	0.986798i	−0.869803	+	0.493399i	$0.835754\pi$
$74$	55731.0	1.18309
$75$	0	0
$76$	−34757.7	−0.690268
$77$	30032.8i	0.577257i
$78$	− 16482.9i	− 0.306759i
$79$	104735.	1.88809	0.944047	−	0.329811i	$-0.106985\pi$
$79$	104735.	1.88809	0.944047	+	0.329811i	$0.106985\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	73431.4i	1.20600i
$83$	60213.5i	0.959398i	0.877433	+	0.479699i	$0.159254\pi$
$83$	60213.5i	0.959398i	−0.877433	+	0.479699i	$0.840746\pi$
$84$	−17989.5	−0.278177
$85$	0	0
$86$	12086.7	0.176222
$87$	42496.4i	0.601941i
$88$	− 15385.8i	− 0.211794i
$89$	−19799.6	−0.264961	−0.132480	−	0.991186i	$-0.542294\pi$
$89$	−19799.6	−0.264961	−0.132480	+	0.991186i	$0.542294\pi$
$90$	0	0
$91$	−57198.8	−0.724075
$92$	− 8813.78i	− 0.108566i
$93$	− 7078.88i	− 0.0848706i
$94$	−80025.4	−0.934132
$95$	0	0
$96$	9216.00	0.102062
$97$	− 105229.i	− 1.13555i	−0.823184	−	0.567774i	$-0.807805\pi$
$97$	− 105229.i	− 1.13555i	0.823184	−	0.567774i	$-0.192195\pi$
$98$	− 4801.01i	− 0.0504972i
$99$	19472.6	0.199681
$100$	0	0
$101$	103706.	1.01158	0.505788	−	0.862658i	$-0.331202\pi$
$101$	103706.	1.01158	0.505788	+	0.862658i	$0.331202\pi$
$102$	57773.9i	0.549834i
$103$	− 86595.2i	− 0.804268i	−0.915581	−	0.402134i	$-0.868269\pi$
$103$	− 86595.2i	− 0.804268i	0.915581	−	0.402134i	$-0.131731\pi$
$104$	29302.9	0.265661
$105$	0	0
$106$	−81992.6	−0.708778
$107$	− 184178.i	− 1.55517i	−0.628778	−	0.777585i	$-0.716445\pi$
$107$	− 184178.i	− 1.55517i	0.628778	−	0.777585i	$-0.283555\pi$
$108$	11664.0i	0.0962250i
$109$	62055.3	0.500280	0.250140	−	0.968210i	$-0.419523\pi$
$109$	62055.3	0.500280	0.250140	+	0.968210i	$0.419523\pi$
$110$	0	0
$111$	−125395.	−0.965988
$112$	− 31981.3i	− 0.240908i
$113$	− 177158.i	− 1.30517i	−0.757717	−	0.652583i	$-0.773685\pi$
$113$	− 177158.i	− 1.30517i	0.757717	−	0.652583i	$-0.226315\pi$
$114$	78204.9	0.563601
$115$	0	0
$116$	−75549.2	−0.521296
$117$	37086.5i	0.250468i
$118$	134466.i	0.889013i
$119$	200487.	1.29783
$120$	0	0
$121$	−103257.	−0.641147
$122$	− 66246.9i	− 0.402964i
$123$	− 165221.i	− 0.984694i
$124$	12584.7	0.0735001
$125$	0	0
$126$	40476.3	0.227130
$127$	− 77111.2i	− 0.424237i	−0.977244	−	0.212118i	$-0.931964\pi$
$127$	− 77111.2i	− 0.424237i	0.977244	−	0.212118i	$-0.0680363\pi$
$128$	16384.0i	0.0883883i
$129$	−27195.0	−0.143885
$130$	0	0
$131$	532.429	0.00271071	0.00135536	−	0.999999i	$-0.499569\pi$
$131$	532.429	0.00271071	0.00135536	+	0.999999i	$0.499569\pi$
$132$	34618.0i	0.172929i
$133$	− 271386.i	− 1.33033i
$134$	−50159.5	−0.241319
$135$	0	0
$136$	−102709.	−0.476170
$137$	152405.i	0.693743i	0.937913	+	0.346872i	$0.112756\pi$
$137$	152405.i	0.693743i	−0.937913	+	0.346872i	$0.887244\pi$
$138$	19831.0i	0.0886435i
$139$	202247.	0.887862	0.443931	−	0.896061i	$-0.353583\pi$
$139$	202247.	0.887862	0.443931	+	0.896061i	$0.353583\pi$
$140$	0	0
$141$	180057.	0.762716
$142$	5491.01i	0.0228524i
$143$	110070.i	0.450122i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	179720.	0.697772
$147$	10802.3i	0.0412308i
$148$	− 222924.i	− 0.836570i
$149$	78018.1	0.287892	0.143946	−	0.989586i	$-0.454021\pi$
$149$	78018.1	0.287892	0.143946	+	0.989586i	$0.454021\pi$
$150$	0	0
$151$	253610.	0.905159	0.452579	−	0.891724i	$-0.350504\pi$
$151$	253610.	0.905159	0.452579	+	0.891724i	$0.350504\pi$
$152$	139031.i	0.488093i
$153$	− 129991.i	− 0.448938i
$154$	120131.	0.408182
$155$	0	0
$156$	−65931.6	−0.216911
$157$	182618.i	0.591282i	0.955299	+	0.295641i	$0.0955332\pi$
$157$	182618.i	0.591282i	−0.955299	+	0.295641i	$0.904467\pi$
$158$	− 418940.i	− 1.33508i
$159$	184483.	0.578715
$160$	0	0
$161$	68817.4	0.209235
$162$	− 26244.0i	− 0.0785674i
$163$	− 461074.i	− 1.35926i	−0.733557	−	0.679628i	$-0.762141\pi$
$163$	− 461074.i	− 1.35926i	0.733557	−	0.679628i	$-0.237859\pi$
$164$	293726.	0.852770
$165$	0	0
$166$	240854.	0.678397
$167$	− 496943.i	− 1.37885i	−0.724359	−	0.689423i	$-0.757864\pi$
$167$	− 496943.i	− 1.37885i	0.724359	−	0.689423i	$-0.242136\pi$
$168$	71957.9i	0.196701i
$169$	161659.	0.435394
$170$	0	0
$171$	−175961.	−0.460178
$172$	− 48346.7i	− 0.124608i
$173$	− 560827.i	− 1.42467i	−0.701841	−	0.712334i	$-0.747638\pi$
$173$	− 560827.i	− 1.42467i	0.701841	−	0.712334i	$-0.252362\pi$
$174$	169986.	0.425637
$175$	0	0
$176$	−61543.1	−0.149761
$177$	− 302549.i	− 0.725876i
$178$	79198.4i	0.187355i
$179$	−15137.1	−0.0353110	−0.0176555	−	0.999844i	$-0.505620\pi$
$179$	−15137.1	−0.0353110	−0.0176555	+	0.999844i	$0.505620\pi$
$180$	0	0
$181$	428400.	0.971971	0.485986	−	0.873967i	$-0.338461\pi$
$181$	428400.	0.971971	0.485986	+	0.873967i	$0.338461\pi$
$182$	228795.i	0.511999i
$183$	149055.i	0.329018i
$184$	−35255.1	−0.0767675
$185$	0	0
$186$	−28315.5	−0.0600125
$187$	− 385806.i	− 0.806799i
$188$	320102.i	0.660531i
$189$	−91071.8	−0.185451
$190$	0	0
$191$	375103.	0.743990	0.371995	−	0.928235i	$-0.378674\pi$
$191$	375103.	0.743990	0.371995	+	0.928235i	$0.378674\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	− 415062.i	− 0.802083i	−0.916060	−	0.401042i	$-0.868648\pi$
$193$	− 415062.i	− 0.802083i	0.916060	−	0.401042i	$-0.131352\pi$
$194$	−420915.	−0.802954
$195$	0	0
$196$	−19204.0	−0.0357069
$197$	46456.6i	0.0852869i	0.999090	+	0.0426434i	$0.0135779\pi$
$197$	46456.6i	0.0852869i	−0.999090	+	0.0426434i	$0.986422\pi$
$198$	− 77890.5i	− 0.141196i
$199$	−800653.	−1.43322	−0.716608	−	0.697476i	$-0.754306\pi$
$199$	−800653.	−1.43322	−0.716608	+	0.697476i	$0.754306\pi$
$200$	0	0
$201$	112859.	0.197036
$202$	− 414822.i	− 0.715293i
$203$	− 589883.i	− 1.00468i
$204$	231096.	0.388791
$205$	0	0
$206$	−346381.	−0.568703
$207$	− 44619.7i	− 0.0723771i
$208$	− 117212.i	− 0.187851i
$209$	−522241.	−0.827000
$210$	0	0
$211$	−181403.	−0.280503	−0.140251	−	0.990116i	$-0.544791\pi$
$211$	−181403.	−0.280503	−0.140251	+	0.990116i	$0.544791\pi$
$212$	327970.i	0.501182i
$213$	− 12354.8i	− 0.0186589i
$214$	−736711.	−1.09967
$215$	0	0
$216$	46656.0	0.0680414
$217$	98260.3i	0.141654i
$218$	− 248221.i	− 0.353751i
$219$	−404369.	−0.569728
$220$	0	0
$221$	734785.	1.01200
$222$	501579.i	0.683057i
$223$	− 542646.i	− 0.730726i	−0.930865	−	0.365363i	$-0.880945\pi$
$223$	− 542646.i	− 0.730726i	0.930865	−	0.365363i	$-0.119055\pi$
$224$	−127925.	−0.170348
$225$	0	0
$226$	−708634.	−0.922892
$227$	906165.i	1.16719i	0.812044	+	0.583597i	$0.198355\pi$
$227$	906165.i	1.16719i	−0.812044	+	0.583597i	$0.801645\pi$
$228$	− 312820.i	− 0.398526i
$229$	686335.	0.864863	0.432431	−	0.901667i	$-0.357656\pi$
$229$	686335.	0.864863	0.432431	+	0.901667i	$0.357656\pi$
$230$	0	0
$231$	−270295.	−0.333279
$232$	302197.i	0.368612i
$233$	− 818792.i	− 0.988061i	−0.869445	−	0.494030i	$-0.835523\pi$
$233$	− 818792.i	− 0.988061i	0.869445	−	0.494030i	$-0.164477\pi$
$234$	148346.	0.177107
$235$	0	0
$236$	537865.	0.628627
$237$	942614.i	1.09009i
$238$	− 801947.i	− 0.917705i
$239$	−996516.	−1.12847	−0.564234	−	0.825615i	$-0.690829\pi$
$239$	−996516.	−1.12847	−0.564234	+	0.825615i	$0.690829\pi$
$240$	0	0
$241$	−563238.	−0.624668	−0.312334	−	0.949972i	$-0.601111\pi$
$241$	−563238.	−0.624668	−0.312334	+	0.949972i	$0.601111\pi$
$242$	413030.i	0.453360i
$243$	59049.0i	0.0641500i
$244$	−264987.	−0.284938
$245$	0	0
$246$	−660882.	−0.696284
$247$	− 994632.i	− 1.03734i
$248$	− 50338.7i	− 0.0519724i
$249$	−541922.	−0.553909
$250$	0	0
$251$	1.84528e6	1.84875	0.924376	−	0.381482i	$-0.124586\pi$
$251$	1.84528e6	1.84875	0.924376	+	0.381482i	$0.124586\pi$
$252$	− 161905.i	− 0.160605i
$253$	− 132429.i	− 0.130071i
$254$	−308445.	−0.299981
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.14291e6i	1.07939i	0.841861	+	0.539695i	$0.181460\pi$
$257$	1.14291e6i	1.07939i	−0.841861	+	0.539695i	$0.818540\pi$
$258$	108780.i	0.101742i
$259$	1.74058e6	1.61229
$260$	0	0
$261$	−382468.	−0.347531
$262$	− 2129.72i	− 0.00191676i
$263$	− 883440.i	− 0.787568i	−0.919203	−	0.393784i	$-0.871166\pi$
$263$	− 883440.i	− 0.787568i	0.919203	−	0.393784i	$-0.128834\pi$
$264$	138472.	0.122279
$265$	0	0
$266$	−1.08554e6	−0.940684
$267$	− 178196.i	− 0.152975i
$268$	200638.i	0.170638i
$269$	−1.00260e6	−0.844783	−0.422391	−	0.906414i	$-0.638809\pi$
$269$	−1.00260e6	−0.844783	−0.422391	+	0.906414i	$0.638809\pi$
$270$	0	0
$271$	869979.	0.719590	0.359795	−	0.933031i	$-0.382847\pi$
$271$	869979.	0.719590	0.359795	+	0.933031i	$0.382847\pi$
$272$	410837.i	0.336703i
$273$	− 514790.i	− 0.418045i
$274$	609621.	0.490550
$275$	0	0
$276$	79324.0	0.0626804
$277$	− 2.27336e6i	− 1.78020i	−0.455763	−	0.890101i	$-0.650634\pi$
$277$	− 2.27336e6i	− 1.78020i	0.455763	−	0.890101i	$-0.349366\pi$
$278$	− 808989.i	− 0.627813i
$279$	63709.9	0.0490000
$280$	0	0
$281$	−1.83097e6	−1.38329	−0.691647	−	0.722236i	$-0.743115\pi$
$281$	−1.83097e6	−1.38329	−0.691647	+	0.722236i	$0.743115\pi$
$282$	− 720229.i	− 0.539321i
$283$	− 1.68384e6i	− 1.24978i	−0.780711	−	0.624892i	$-0.785143\pi$
$283$	− 1.68384e6i	− 1.24978i	0.780711	−	0.624892i	$-0.214857\pi$
$284$	21964.1	0.0161591
$285$	0	0
$286$	440282.	0.318285
$287$	2.29339e6i	1.64351i
$288$	82944.0i	0.0589256i
$289$	−1.15563e6	−0.813904
$290$	0	0
$291$	947060.	0.655609
$292$	− 718878.i	− 0.493399i
$293$	− 1.02071e6i	− 0.694597i	−0.937755	−	0.347298i	$-0.887099\pi$
$293$	− 1.02071e6i	− 0.694597i	0.937755	−	0.347298i	$-0.112901\pi$
$294$	43209.1	0.0291546
$295$	0	0
$296$	−891696.	−0.591545
$297$	175254.i	0.115286i
$298$	− 312072.i	− 0.203570i
$299$	252216.	0.163153
$300$	0	0
$301$	377488.	0.240152
$302$	− 1.01444e6i	− 0.640044i
$303$	933351.i	0.584034i
$304$	556124.	0.345134
$305$	0	0
$306$	−519965.	−0.317447
$307$	− 1.35784e6i	− 0.822250i	−0.911579	−	0.411125i	$-0.865136\pi$
$307$	− 1.35784e6i	− 0.822250i	0.911579	−	0.411125i	$-0.134864\pi$
$308$	− 480525.i	− 0.288628i
$309$	779357.	0.464344
$310$	0	0
$311$	236818.	0.138840	0.0694199	−	0.997588i	$-0.477885\pi$
$311$	236818.	0.138840	0.0694199	+	0.997588i	$0.477885\pi$
$312$	263726.i	0.153379i
$313$	1.26133e6i	0.727728i	0.931452	+	0.363864i	$0.118543\pi$
$313$	1.26133e6i	0.727728i	−0.931452	+	0.363864i	$0.881457\pi$
$314$	730472.	0.418099
$315$	0	0
$316$	−1.67576e6	−0.944047
$317$	− 209340.i	− 0.117005i	−0.998287	−	0.0585024i	$-0.981367\pi$
$317$	− 209340.i	− 0.117005i	0.998287	−	0.0585024i	$-0.0186325\pi$
$318$	− 737934.i	− 0.409213i
$319$	−1.13514e6	−0.624558
$320$	0	0
$321$	1.65760e6	0.897878
$322$	− 275270.i	− 0.147951i
$323$	3.48627e6i	1.85932i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−1.84429e6	−0.961139
$327$	558498.i	0.288837i
$328$	− 1.17490e6i	− 0.603000i
$329$	−2.49933e6	−1.27302
$330$	0	0
$331$	−706376.	−0.354377	−0.177189	−	0.984177i	$-0.556700\pi$
$331$	−706376.	−0.354377	−0.177189	+	0.984177i	$0.556700\pi$
$332$	− 963417.i	− 0.479699i
$333$	− 1.12855e6i	− 0.557714i
$334$	−1.98777e6	−0.974991
$335$	0	0
$336$	287832.	0.139088
$337$	984215.i	0.472080i	0.971743	+	0.236040i	$0.0758496\pi$
$337$	984215.i	0.472080i	−0.971743	+	0.236040i	$0.924150\pi$
$338$	− 646635.i	− 0.307870i
$339$	1.59443e6	0.753538
$340$	0	0
$341$	189087.	0.0880594
$342$	703844.i	0.325395i
$343$	− 2.24959e6i	− 1.03245i
$344$	−193387.	−0.0881110
$345$	0	0
$346$	−2.24331e6	−1.00739
$347$	515675.i	0.229907i	0.993371	+	0.114954i	$0.0366719\pi$
$347$	515675.i	0.229907i	−0.993371	+	0.114954i	$0.963328\pi$
$348$	− 679943.i	− 0.300971i
$349$	1.10336e6	0.484902	0.242451	−	0.970164i	$-0.422049\pi$
$349$	1.10336e6	0.484902	0.242451	+	0.970164i	$0.422049\pi$
$350$	0	0
$351$	−333779.	−0.144607
$352$	246173.i	0.105897i
$353$	− 2.71743e6i	− 1.16071i	−0.814365	−	0.580353i	$-0.802915\pi$
$353$	− 2.71743e6i	− 1.16071i	0.814365	−	0.580353i	$-0.197085\pi$
$354$	−1.21020e6	−0.513272
$355$	0	0
$356$	316793.	0.132480
$357$	1.80438e6i	0.749303i
$358$	60548.4i	0.0249687i
$359$	−2.94263e6	−1.20504	−0.602518	−	0.798105i	$-0.705836\pi$
$359$	−2.94263e6	−1.20504	−0.602518	+	0.798105i	$0.705836\pi$
$360$	0	0
$361$	2.24304e6	0.905877
$362$	− 1.71360e6i	− 0.687288i
$363$	− 929317.i	− 0.370167i
$364$	915181.	0.362038
$365$	0	0
$366$	596222.	0.232651
$367$	− 1.64927e6i	− 0.639187i	−0.947555	−	0.319593i	$-0.896454\pi$
$367$	− 1.64927e6i	− 0.639187i	0.947555	−	0.319593i	$-0.103546\pi$
$368$	141020.i	0.0542828i
$369$	1.48699e6	0.568514
$370$	0	0
$371$	−2.56077e6	−0.965909
$372$	113262.i	0.0424353i
$373$	1.53003e6i	0.569415i	0.958614	+	0.284708i	$0.0918965\pi$
$373$	1.53003e6i	0.569415i	−0.958614	+	0.284708i	$0.908103\pi$
$374$	−1.54322e6	−0.570493
$375$	0	0
$376$	1.28041e6	0.467066
$377$	− 2.16193e6i	− 0.783407i
$378$	364287.i	0.131134i
$379$	2863.37	0.00102395	0.000511976	−	1.00000i	$-0.499837\pi$
$379$	2863.37	0.00102395	0.000511976		1.00000i	$0.499837\pi$
$380$	0	0
$381$	694001.	0.244933
$382$	− 1.50041e6i	− 0.526081i
$383$	− 3.45716e6i	− 1.20427i	−0.798396	−	0.602133i	$-0.794318\pi$
$383$	− 3.45716e6i	− 1.20427i	0.798396	−	0.602133i	$-0.205682\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	−1.66025e6	−0.567158
$387$	− 244755.i	− 0.0830719i
$388$	1.68366e6i	0.567774i
$389$	573847.	0.192275	0.0961373	−	0.995368i	$-0.469351\pi$
$389$	573847.	0.192275	0.0961373	+	0.995368i	$0.469351\pi$
$390$	0	0
$391$	−884039.	−0.292435
$392$	76816.1i	0.0252486i
$393$	4791.86i	0.00156503i
$394$	185827.	0.0603069
$395$	0	0
$396$	−311562.	−0.0998405
$397$	4.56917e6i	1.45499i	0.686111	+	0.727496i	$0.259316\pi$
$397$	4.56917e6i	1.45499i	−0.686111	+	0.727496i	$0.740684\pi$
$398$	3.20261e6i	1.01344i
$399$	2.44248e6	0.768065
$400$	0	0
$401$	3.94763e6	1.22596	0.612979	−	0.790099i	$-0.289971\pi$
$401$	3.94763e6	1.22596	0.612979	+	0.790099i	$0.289971\pi$
$402$	− 451436.i	− 0.139326i
$403$	360125.i	0.110456i
$404$	−1.65929e6	−0.505788
$405$	0	0
$406$	−2.35953e6	−0.710413
$407$	− 3.34947e6i	− 1.00228i
$408$	− 924383.i	− 0.274917i
$409$	5.07330e6	1.49962	0.749812	−	0.661651i	$-0.230144\pi$
$409$	5.07330e6	1.49962	0.749812	+	0.661651i	$0.230144\pi$
$410$	0	0
$411$	−1.37165e6	−0.400533
$412$	1.38552e6i	0.402134i
$413$	4.19961e6i	1.21153i
$414$	−178479.	−0.0511783
$415$	0	0
$416$	−468847.	−0.132830
$417$	1.82023e6i	0.512607i
$418$	2.08896e6i	0.584777i
$419$	−1.82100e6	−0.506728	−0.253364	−	0.967371i	$-0.581537\pi$
$419$	−1.82100e6	−0.506728	−0.253364	+	0.967371i	$0.581537\pi$
$420$	0	0
$421$	4.25877e6	1.17106	0.585530	−	0.810651i	$-0.300886\pi$
$421$	4.25877e6	1.17106	0.585530	+	0.810651i	$0.300886\pi$
$422$	725610.i	0.198345i
$423$	1.62051e6i	0.440354i
$424$	1.31188e6	0.354389
$425$	0	0
$426$	−49419.1	−0.0131938
$427$	− 2.06900e6i	− 0.549151i
$428$	2.94684e6i	0.777585i
$429$	−990634.	−0.259878
$430$	0	0
$431$	5.60721e6	1.45396	0.726982	−	0.686657i	$-0.240922\pi$
$431$	5.60721e6	1.45396	0.726982	+	0.686657i	$0.240922\pi$
$432$	− 186624.i	− 0.0481125i
$433$	− 6.47035e6i	− 1.65847i	−0.558899	−	0.829236i	$-0.688776\pi$
$433$	− 6.47035e6i	− 1.65847i	0.558899	−	0.829236i	$-0.311224\pi$
$434$	393041.	0.100164
$435$	0	0
$436$	−992885.	−0.250140
$437$	1.19667e6i	0.299757i
$438$	1.61748e6i	0.402859i
$439$	−5.20026e6	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$439$	−5.20026e6	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$440$	0	0
$441$	−97220.4	−0.0238046
$442$	− 2.93914e6i	− 0.715591i
$443$	− 1.18394e6i	− 0.286630i	−0.989677	−	0.143315i	$-0.954224\pi$
$443$	− 1.18394e6i	− 0.286630i	0.989677	−	0.143315i	$-0.0457762\pi$
$444$	2.00632e6	0.482994
$445$	0	0
$446$	−2.17058e6	−0.516701
$447$	702163.i	0.166214i
$448$	511701.i	0.120454i
$449$	−5.08864e6	−1.19120	−0.595601	−	0.803280i	$-0.703086\pi$
$449$	−5.08864e6	−1.19120	−0.595601	+	0.803280i	$0.703086\pi$
$450$	0	0
$451$	4.41328e6	1.02169
$452$	2.83453e6i	0.652583i
$453$	2.28249e6i	0.522594i
$454$	3.62466e6	0.825330
$455$	0	0
$456$	−1.25128e6	−0.281801
$457$	− 4.91205e6i	− 1.10020i	−0.835099	−	0.550100i	$-0.814589\pi$
$457$	− 4.91205e6i	− 1.10020i	0.835099	−	0.550100i	$-0.185411\pi$
$458$	− 2.74534e6i	− 0.611550i
$459$	1.16992e6	0.259194
$460$	0	0
$461$	−4.35320e6	−0.954017	−0.477009	−	0.878899i	$-0.658279\pi$
$461$	−4.35320e6	−0.954017	−0.477009	+	0.878899i	$0.658279\pi$
$462$	1.08118e6i	0.235664i
$463$	8.02160e6i	1.73904i	0.493902	+	0.869518i	$0.335570\pi$
$463$	8.02160e6i	1.73904i	−0.493902	+	0.869518i	$0.664430\pi$
$464$	1.20879e6	0.260648
$465$	0	0
$466$	−3.27517e6	−0.698664
$467$	2.11967e6i	0.449756i	0.974387	+	0.224878i	$0.0721984\pi$
$467$	2.11967e6i	0.449756i	−0.974387	+	0.224878i	$0.927802\pi$
$468$	− 593384.i	− 0.125234i
$469$	−1.56657e6	−0.328865
$470$	0	0
$471$	−1.64356e6	−0.341377
$472$	− 2.15146e6i	− 0.444507i
$473$	− 726417.i	− 0.149291i
$474$	3.77046e6	0.770811
$475$	0	0
$476$	−3.20779e6	−0.648916
$477$	1.66035e6i	0.334121i
$478$	3.98606e6i	0.797948i
$479$	1.89974e6	0.378317	0.189159	−	0.981947i	$-0.439424\pi$
$479$	1.89974e6	0.378317	0.189159	+	0.981947i	$0.439424\pi$
$480$	0	0
$481$	6.37922e6	1.25720
$482$	2.25295e6i	0.441707i
$483$	619357.i	0.120802i
$484$	1.65212e6	0.320574
$485$	0	0
$486$	236196.	0.0453609
$487$	940193.i	0.179637i	0.995958	+	0.0898183i	$0.0286286\pi$
$487$	940193.i	0.179637i	−0.995958	+	0.0898183i	$0.971371\pi$
$488$	1.05995e6i	0.201482i
$489$	4.14966e6	0.784767
$490$	0	0
$491$	742531.	0.138999	0.0694994	−	0.997582i	$-0.477860\pi$
$491$	742531.	0.138999	0.0694994	+	0.997582i	$0.477860\pi$
$492$	2.64353e6i	0.492347i
$493$	7.57773e6i	1.40418i
$494$	−3.97853e6	−0.733508
$495$	0	0
$496$	−201355.	−0.0367500
$497$	171494.i	0.0311428i
$498$	2.16769e6i	0.391673i
$499$	1.01900e7	1.83198	0.915992	−	0.401198i	$-0.131406\pi$
$499$	1.01900e7	1.83198	0.915992	+	0.401198i	$0.131406\pi$
$500$	0	0
$501$	4.47249e6	0.796077
$502$	− 7.38113e6i	− 1.30727i
$503$	1.02717e7i	1.81018i	0.425225	+	0.905088i	$0.360195\pi$
$503$	1.02717e7i	1.81018i	−0.425225	+	0.905088i	$0.639805\pi$
$504$	−647621.	−0.113565
$505$	0	0
$506$	−529714.	−0.0919741
$507$	1.45493e6i	0.251375i
$508$	1.23378e6i	0.212118i
$509$	8.36906e6	1.43180	0.715900	−	0.698203i	$-0.246017\pi$
$509$	8.36906e6	1.43180	0.715900	+	0.698203i	$0.246017\pi$
$510$	0	0
$511$	5.61296e6	0.950910
$512$	− 262144.i	− 0.0441942i
$513$	− 1.58365e6i	− 0.265684i
$514$	4.57163e6	0.763243
$515$	0	0
$516$	435120.	0.0719424
$517$	4.80958e6i	0.791373i
$518$	− 6.96230e6i	− 1.14006i
$519$	5.04744e6	0.822532
$520$	0	0
$521$	−679178.	−0.109620	−0.0548100	−	0.998497i	$-0.517455\pi$
$521$	−679178.	−0.109620	−0.0548100	+	0.998497i	$0.517455\pi$
$522$	1.52987e6i	0.245742i
$523$	5.57749e6i	0.891629i	0.895125	+	0.445815i	$0.147086\pi$
$523$	5.57749e6i	0.891629i	−0.895125	+	0.445815i	$0.852914\pi$
$524$	−8518.87	−0.00135536
$525$	0	0
$526$	−3.53376e6	−0.556894
$527$	− 1.26227e6i	− 0.197982i
$528$	− 553888.i	− 0.0864644i
$529$	6.13290e6	0.952854
$530$	0	0
$531$	2.72294e6	0.419085
$532$	4.34218e6i	0.665164i
$533$	8.40529e6i	1.28155i
$534$	−712785.	−0.108170
$535$	0	0
$536$	802553.	0.120660
$537$	− 136234.i	− 0.0203868i
$538$	4.01038e6i	0.597352i
$539$	−288544.	−0.0427799
$540$	0	0
$541$	−1.87436e6	−0.275334	−0.137667	−	0.990479i	$-0.543960\pi$
$541$	−1.87436e6	−0.275334	−0.137667	+	0.990479i	$0.543960\pi$
$542$	− 3.47991e6i	− 0.508827i
$543$	3.85560e6i	0.561168i
$544$	1.64335e6	0.238085
$545$	0	0
$546$	−2.05916e6	−0.295603
$547$	− 1.57162e6i	− 0.224584i	−0.993675	−	0.112292i	$-0.964181\pi$
$547$	− 1.57162e6i	− 0.224584i	0.993675	−	0.112292i	$-0.0358193\pi$
$548$	− 2.43849e6i	− 0.346872i
$549$	−1.34150e6	−0.189959
$550$	0	0
$551$	1.02575e7	1.43934
$552$	− 317296.i	− 0.0443217i
$553$	− 1.30842e7i	− 1.81943i
$554$	−9.09345e6	−1.25879
$555$	0	0
$556$	−3.23596e6	−0.443931
$557$	3.61438e6i	0.493623i	0.969064	+	0.246811i	$0.0793828\pi$
$557$	3.61438e6i	0.493623i	−0.969064	+	0.246811i	$0.920617\pi$
$558$	− 254840.i	− 0.0346483i
$559$	1.38349e6	0.187261
$560$	0	0
$561$	3.47226e6	0.465805
$562$	7.32386e6i	0.978137i
$563$	9.92493e6i	1.31964i	0.751422	+	0.659821i	$0.229368\pi$
$563$	9.92493e6i	1.31964i	−0.751422	+	0.659821i	$0.770632\pi$
$564$	−2.88091e6	−0.381358
$565$	0	0
$566$	−6.73536e6	−0.883731
$567$	− 819646.i	− 0.107070i
$568$	− 87856.2i	− 0.0114262i
$569$	−1.36153e7	−1.76298	−0.881490	−	0.472202i	$-0.843459\pi$
$569$	−1.36153e7	−1.76298	−0.881490	+	0.472202i	$0.843459\pi$
$570$	0	0
$571$	9.30619e6	1.19449	0.597244	−	0.802060i	$-0.296262\pi$
$571$	9.30619e6	1.19449	0.597244	+	0.802060i	$0.296262\pi$
$572$	− 1.76113e6i	− 0.225061i
$573$	3.37593e6i	0.429543i
$574$	9.17356e6	1.16214
$575$	0	0
$576$	331776.	0.0416667
$577$	− 8.45398e6i	− 1.05711i	−0.848898	−	0.528557i	$-0.822733\pi$
$577$	− 8.45398e6i	− 1.05711i	0.848898	−	0.528557i	$-0.177267\pi$
$578$	4.62251e6i	0.575517i
$579$	3.73556e6	0.463083
$580$	0	0
$581$	7.52230e6	0.924507
$582$	− 3.78824e6i	− 0.463586i
$583$	4.92782e6i	0.600459i
$584$	−2.87551e6	−0.348886
$585$	0	0
$586$	−4.08283e6	−0.491154
$587$	1.31623e7i	1.57665i	0.615257	+	0.788327i	$0.289052\pi$
$587$	1.31623e7i	1.57665i	−0.615257	+	0.788327i	$0.710948\pi$
$588$	− 172836.i	− 0.0206154i
$589$	−1.70865e6	−0.202939
$590$	0	0
$591$	−418110.	−0.0492404
$592$	3.56678e6i	0.418285i
$593$	5.45158e6i	0.636628i	0.947985	+	0.318314i	$0.103117\pi$
$593$	5.45158e6i	0.636628i	−0.947985	+	0.318314i	$0.896883\pi$
$594$	701015.	0.0815194
$595$	0	0
$596$	−1.24829e6	−0.143946
$597$	− 7.20588e6i	− 0.827468i
$598$	− 1.00887e6i	− 0.115367i
$599$	−560962.	−0.0638802	−0.0319401	−	0.999490i	$-0.510169\pi$
$599$	−560962.	−0.0638802	−0.0319401	+	0.999490i	$0.510169\pi$
$600$	0	0
$601$	−1.66959e7	−1.88549	−0.942744	−	0.333516i	$-0.891765\pi$
$601$	−1.66959e7	−1.88549	−0.942744	+	0.333516i	$0.891765\pi$
$602$	− 1.50995e6i	− 0.169813i
$603$	1.01573e6i	0.113759i
$604$	−4.05777e6	−0.452579
$605$	0	0
$606$	3.73340e6	0.412974
$607$	7.07011e6i	0.778851i	0.921058	+	0.389426i	$0.127326\pi$
$607$	7.07011e6i	0.778851i	−0.921058	+	0.389426i	$0.872674\pi$
$608$	− 2.22449e6i	− 0.244046i
$609$	5.30895e6	0.580050
$610$	0	0
$611$	−9.16007e6	−0.992649
$612$	2.07986e6i	0.224469i
$613$	2.29070e6i	0.246216i	0.992393	+	0.123108i	$0.0392862\pi$
$613$	2.29070e6i	0.246216i	−0.992393	+	0.123108i	$0.960714\pi$
$614$	−5.43138e6	−0.581419
$615$	0	0
$616$	−1.92210e6	−0.204091
$617$	− 2.87154e6i	− 0.303670i	−0.988406	−	0.151835i	$-0.951482\pi$
$617$	− 2.87154e6i	− 0.303670i	0.988406	−	0.151835i	$-0.0485183\pi$
$618$	− 3.11743e6i	− 0.328341i
$619$	1.95944e6	0.205544	0.102772	−	0.994705i	$-0.467229\pi$
$619$	1.95944e6	0.205544	0.102772	+	0.994705i	$0.467229\pi$
$620$	0	0
$621$	401578.	0.0417869
$622$	− 947273.i	− 0.0981746i
$623$	2.47350e6i	0.255324i
$624$	1.05491e6	0.108456
$625$	0	0
$626$	5.04533e6	0.514581
$627$	− 4.70017e6i	− 0.477469i
$628$	− 2.92189e6i	− 0.295641i
$629$	−2.23597e7	−2.25341
$630$	0	0
$631$	1.28670e7	1.28648	0.643239	−	0.765666i	$-0.277590\pi$
$631$	1.28670e7	1.28648	0.643239	+	0.765666i	$0.277590\pi$
$632$	6.70303e6i	0.667542i
$633$	− 1.63262e6i	− 0.161948i
$634$	−837360.	−0.0827349
$635$	0	0
$636$	−2.95173e6	−0.289357
$637$	− 549545.i	− 0.0536605i
$638$	4.54056e6i	0.441629i
$639$	111193.	0.0107727
$640$	0	0
$641$	6.57751e6	0.632290	0.316145	−	0.948711i	$-0.397611\pi$
$641$	6.57751e6	0.632290	0.316145	+	0.948711i	$0.397611\pi$
$642$	− 6.63040e6i	− 0.634895i
$643$	− 7.16065e6i	− 0.683007i	−0.939881	−	0.341503i	$-0.889064\pi$
$643$	− 7.16065e6i	− 0.683007i	0.939881	−	0.341503i	$-0.110936\pi$
$644$	−1.10108e6	−0.104617
$645$	0	0
$646$	1.39451e7	1.31474
$647$	240767.i	0.0226118i	0.999936	+	0.0113059i	$0.00359886\pi$
$647$	240767.i	0.0226118i	−0.999936	+	0.0113059i	$0.996401\pi$
$648$	419904.i	0.0392837i
$649$	8.08152e6	0.753149
$650$	0	0
$651$	−884343.	−0.0817840
$652$	7.37718e6i	0.679628i
$653$	2.14609e6i	0.196954i	0.995139	+	0.0984771i	$0.0313971\pi$
$653$	2.14609e6i	0.196954i	−0.995139	+	0.0984771i	$0.968603\pi$
$654$	2.23399e6	0.204238
$655$	0	0
$656$	−4.69961e6	−0.426385
$657$	− 3.63932e6i	− 0.328933i
$658$	9.99733e6i	0.900159i
$659$	−1.71363e7	−1.53710	−0.768551	−	0.639789i	$-0.779022\pi$
$659$	−1.71363e7	−1.53710	−0.768551	+	0.639789i	$0.779022\pi$
$660$	0	0
$661$	−1.89309e7	−1.68526	−0.842631	−	0.538492i	$-0.818994\pi$
$661$	−1.89309e7	−1.68526	−0.842631	+	0.538492i	$0.818994\pi$
$662$	2.82550e6i	0.250583i
$663$	6.61307e6i	0.584278i
$664$	−3.85367e6	−0.339199
$665$	0	0
$666$	−4.51421e6	−0.394363
$667$	2.60107e6i	0.226380i
$668$	7.95109e6i	0.689423i
$669$	4.88382e6	0.421885
$670$	0	0
$671$	−3.98148e6	−0.341381
$672$	− 1.15133e6i	− 0.0983503i
$673$	7.48322e6i	0.636870i	0.947945	+	0.318435i	$0.103157\pi$
$673$	7.48322e6i	0.636870i	−0.947945	+	0.318435i	$0.896843\pi$
$674$	3.93686e6	0.333811
$675$	0	0
$676$	−2.58654e6	−0.217697
$677$	− 1.88408e7i	− 1.57990i	−0.613174	−	0.789948i	$-0.710108\pi$
$677$	− 1.88408e7i	− 1.57990i	0.613174	−	0.789948i	$-0.289892\pi$
$678$	− 6.37770e6i	− 0.532832i
$679$	−1.31459e7	−1.09425
$680$	0	0
$681$	−8.15549e6	−0.673879
$682$	− 756348.i	− 0.0622674i
$683$	1.23715e6i	0.101478i	0.998712	+	0.0507389i	$0.0161576\pi$
$683$	1.23715e6i	0.101478i	−0.998712	+	0.0507389i	$0.983842\pi$
$684$	2.81538e6	0.230089
$685$	0	0
$686$	−8.99837e6	−0.730051
$687$	6.17701e6i	0.499329i
$688$	773546.i	0.0623039i
$689$	−9.38525e6	−0.753178
$690$	0	0
$691$	−5.29043e6	−0.421498	−0.210749	−	0.977540i	$-0.567590\pi$
$691$	−5.29043e6	−0.421498	−0.210749	+	0.977540i	$0.567590\pi$
$692$	8.97323e6i	0.712334i
$693$	− 2.43266e6i	− 0.192419i
$694$	2.06270e6	0.162569
$695$	0	0
$696$	−2.71977e6	−0.212818
$697$	− 2.94613e7i	− 2.29704i
$698$	− 4.41344e6i	− 0.342878i
$699$	7.36913e6	0.570457
$700$	0	0
$701$	8.47646e6	0.651507	0.325754	−	0.945455i	$-0.394382\pi$
$701$	8.47646e6	0.651507	0.325754	+	0.945455i	$0.394382\pi$
$702$	1.33511e6i	0.102253i
$703$	3.02669e7i	2.30983i
$704$	984690.	0.0748804
$705$	0	0
$706$	−1.08697e7	−0.820742
$707$	− 1.29556e7i	− 0.974788i
$708$	4.84078e6i	0.362938i
$709$	1.06026e7	0.792133	0.396067	−	0.918222i	$-0.370375\pi$
$709$	1.06026e7	0.792133	0.396067	+	0.918222i	$0.370375\pi$
$710$	0	0
$711$	−8.48353e6	−0.629365
$712$	− 1.26717e6i	− 0.0936777i
$713$	− 433275.i	− 0.0319183i
$714$	7.21752e6	0.529837
$715$	0	0
$716$	242194.	0.0176555
$717$	− 8.96864e6i	− 0.651522i
$718$	1.17705e7i	0.852089i
$719$	1.72151e6	0.124190	0.0620952	−	0.998070i	$-0.480222\pi$
$719$	1.72151e6	0.124190	0.0620952	+	0.998070i	$0.480222\pi$
$720$	0	0
$721$	−1.08181e7	−0.775018
$722$	− 8.97217e6i	− 0.640552i
$723$	− 5.06914e6i	− 0.360652i
$724$	−6.85441e6	−0.485986
$725$	0	0
$726$	−3.71727e6	−0.261747
$727$	8.46001e6i	0.593656i	0.954931	+	0.296828i	$0.0959288\pi$
$727$	8.46001e6i	0.593656i	−0.954931	+	0.296828i	$0.904071\pi$
$728$	− 3.66073e6i	− 0.255999i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−4.84926e6	−0.335647
$732$	− 2.38489e6i	− 0.164509i
$733$	− 2.76814e7i	− 1.90295i	−0.307724	−	0.951476i	$-0.599567\pi$
$733$	− 2.76814e7i	− 1.90295i	0.307724	−	0.951476i	$-0.400433\pi$
$734$	−6.59710e6	−0.451973
$735$	0	0
$736$	564082.	0.0383838
$737$	3.01462e6i	0.204439i
$738$	− 5.94794e6i	− 0.402000i
$739$	2.73673e7	1.84340	0.921702	−	0.387899i	$-0.126799\pi$
$739$	2.73673e7	1.84340	0.921702	+	0.387899i	$0.126799\pi$
$740$	0	0
$741$	8.95169e6	0.598907
$742$	1.02431e7i	0.683001i
$743$	1.65069e7i	1.09697i	0.836161	+	0.548484i	$0.184795\pi$
$743$	1.65069e7i	1.09697i	−0.836161	+	0.548484i	$0.815205\pi$
$744$	453048.	0.0300063
$745$	0	0
$746$	6.12014e6	0.402637
$747$	− 4.87730e6i	− 0.319799i
$748$	6.17290e6i	0.403399i
$749$	−2.30088e7	−1.49861
$750$	0	0
$751$	2.18866e7	1.41605	0.708024	−	0.706188i	$-0.249587\pi$
$751$	2.18866e7	1.41605	0.708024	+	0.706188i	$0.249587\pi$
$752$	− 5.12163e6i	− 0.330266i
$753$	1.66075e7i	1.06738i
$754$	−8.64770e6	−0.553952
$755$	0	0
$756$	1.45715e6	0.0927255
$757$	2.64242e7i	1.67595i	0.545706	+	0.837977i	$0.316261\pi$
$757$	2.64242e7i	1.67595i	−0.545706	+	0.837977i	$0.683739\pi$
$758$	− 11453.5i	0 0.000724043i
$759$	1.19186e6	0.0750965
$760$	0	0
$761$	2.44412e7	1.52989	0.764945	−	0.644096i	$-0.222766\pi$
$761$	2.44412e7	1.52989	0.764945	+	0.644096i	$0.222766\pi$
$762$	− 2.77600e6i	− 0.173194i
$763$	− 7.75238e6i	− 0.482085i
$764$	−6.00165e6	−0.371995
$765$	0	0
$766$	−1.38286e7	−0.851545
$767$	1.53916e7i	0.944704i
$768$	589824.i	0.0360844i
$769$	−3.25703e7	−1.98612	−0.993061	−	0.117601i	$-0.962480\pi$
$769$	−3.25703e7	−1.98612	−0.993061	+	0.117601i	$0.962480\pi$
$770$	0	0
$771$	−1.02862e7	−0.623186
$772$	6.64099e6i	0.401042i
$773$	− 1.57649e7i	− 0.948951i	−0.880269	−	0.474475i	$-0.842638\pi$
$773$	− 1.57649e7i	− 0.948951i	0.880269	−	0.474475i	$-0.157362\pi$
$774$	−979020.	−0.0587407
$775$	0	0
$776$	6.73465e6	0.401477
$777$	1.56652e7i	0.930857i
$778$	− 2.29539e6i	− 0.135959i
$779$	−3.98798e7	−2.35456
$780$	0	0
$781$	330014.	0.0193600
$782$	3.53616e6i	0.206783i
$783$	− 3.44221e6i	− 0.200647i
$784$	307264.	0.0178534
$785$	0	0
$786$	19167.5	0.00110664
$787$	− 8.18281e6i	− 0.470940i	−0.971882	−	0.235470i	$-0.924337\pi$
$787$	− 8.18281e6i	− 0.470940i	0.971882	−	0.235470i	$-0.0756630\pi$
$788$	− 743306.i	− 0.0426434i
$789$	7.95096e6	0.454702
$790$	0	0
$791$	−2.21319e7	−1.25770
$792$	1.24625e6i	0.0705979i
$793$	− 7.58292e6i	− 0.428207i
$794$	1.82767e7	1.02884
$795$	0	0
$796$	1.28105e7	0.716608
$797$	− 8.70162e6i	− 0.485237i	−0.970122	−	0.242619i	$-0.921994\pi$
$797$	− 8.70162e6i	− 0.485237i	0.970122	−	0.242619i	$-0.0780064\pi$
$798$	− 9.76990e6i	− 0.543104i
$799$	3.21068e7	1.77922
$800$	0	0
$801$	1.60377e6	0.0883202
$802$	− 1.57905e7i	− 0.866883i
$803$	− 1.08013e7i	− 0.591134i
$804$	−1.80574e6	−0.0985181
$805$	0	0
$806$	1.44050e6	0.0781044
$807$	− 9.02336e6i	− 0.487736i
$808$	6.63716e6i	0.357646i
$809$	−7.79738e6	−0.418868	−0.209434	−	0.977823i	$-0.567162\pi$
$809$	−7.79738e6	−0.418868	−0.209434	+	0.977823i	$0.567162\pi$
$810$	0	0
$811$	−1.46460e7	−0.781930	−0.390965	−	0.920406i	$-0.627859\pi$
$811$	−1.46460e7	−0.781930	−0.390965	+	0.920406i	$0.627859\pi$
$812$	9.43813e6i	0.502338i
$813$	7.82981e6i	0.415456i
$814$	−1.33979e7	−0.708721
$815$	0	0
$816$	−3.69753e6	−0.194396
$817$	6.56414e6i	0.344051i
$818$	− 2.02932e7i	− 1.06039i
$819$	4.63311e6	0.241358
$820$	0	0
$821$	2.03112e7	1.05167	0.525833	−	0.850588i	$-0.323754\pi$
$821$	2.03112e7	1.05167	0.525833	+	0.850588i	$0.323754\pi$
$822$	5.48659e6i	0.283219i
$823$	− 1.31744e7i	− 0.678000i	−0.940786	−	0.339000i	$-0.889911\pi$
$823$	− 1.31744e7i	− 0.678000i	0.940786	−	0.339000i	$-0.110089\pi$
$824$	5.54209e6	0.284352
$825$	0	0
$826$	1.67985e7	0.856681
$827$	1.31547e7i	0.668832i	0.942425	+	0.334416i	$0.108539\pi$
$827$	1.31547e7i	0.668832i	−0.942425	+	0.334416i	$0.891461\pi$
$828$	713916.i	0.0361886i
$829$	565122.	0.0285598	0.0142799	−	0.999898i	$-0.495454\pi$
$829$	565122.	0.0285598	0.0142799	+	0.999898i	$0.495454\pi$
$830$	0	0
$831$	2.04603e7	1.02780
$832$	1.87539e6i	0.0939253i
$833$	1.92620e6i	0.0961810i
$834$	7.28090e6	0.362468
$835$	0	0
$836$	8.35586e6	0.413500
$837$	573389.i	0.0282902i
$838$	7.28401e6i	0.358311i
$839$	1.99251e7	0.977227	0.488614	−	0.872500i	$-0.337503\pi$
$839$	1.99251e7	0.977227	0.488614	+	0.872500i	$0.337503\pi$
$840$	0	0
$841$	1.78447e6	0.0870001
$842$	− 1.70351e7i	− 0.828065i
$843$	− 1.64787e7i	− 0.798645i
$844$	2.90244e6	0.140251
$845$	0	0
$846$	6.48206e6	0.311377
$847$	1.28996e7i	0.617830i
$848$	− 5.24753e6i	− 0.250591i
$849$	1.51546e7	0.721563
$850$	0	0
$851$	−7.67501e6	−0.363291
$852$	197677.i	0.00932945i
$853$	− 1.69196e7i	− 0.796193i	−0.917343	−	0.398097i	$-0.869671\pi$
$853$	− 1.69196e7i	− 0.796193i	0.917343	−	0.398097i	$-0.130329\pi$
$854$	−8.27602e6	−0.388308
$855$	0	0
$856$	1.17874e7	0.549835
$857$	− 2.58888e7i	− 1.20409i	−0.798461	−	0.602046i	$-0.794352\pi$
$857$	− 2.58888e7i	− 1.20409i	0.798461	−	0.602046i	$-0.205648\pi$
$858$	3.96254e6i	0.183762i
$859$	1.67301e7	0.773601	0.386800	−	0.922163i	$-0.373580\pi$
$859$	1.67301e7	0.773601	0.386800	+	0.922163i	$0.373580\pi$
$860$	0	0
$861$	−2.06405e7	−0.948883
$862$	− 2.24288e7i	− 1.02811i
$863$	− 2.45423e7i	− 1.12173i	−0.827907	−	0.560865i	$-0.810469\pi$
$863$	− 2.45423e7i	− 1.12173i	0.827907	−	0.560865i	$-0.189531\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	−2.58814e7	−1.17272
$867$	− 1.04006e7i	− 0.469908i
$868$	− 1.57216e6i	− 0.0708270i
$869$	−2.51786e7	−1.13105
$870$	0	0
$871$	−5.74149e6	−0.256436
$872$	3.97154e6i	0.176876i
$873$	8.52354e6i	0.378516i
$874$	4.78667e6	0.211961
$875$	0	0
$876$	6.46990e6	0.284864
$877$	2.34899e7i	1.03130i	0.856801	+	0.515648i	$0.172449\pi$
$877$	2.34899e7i	1.03130i	−0.856801	+	0.515648i	$0.827551\pi$
$878$	2.08010e7i	0.910643i
$879$	9.18638e6	0.401026
$880$	0	0
$881$	6.54659e6	0.284168	0.142084	−	0.989855i	$-0.454620\pi$
$881$	6.54659e6	0.284168	0.142084	+	0.989855i	$0.454620\pi$
$882$	388882.i	0.0168324i
$883$	− 2.60687e7i	− 1.12517i	−0.826740	−	0.562584i	$-0.809807\pi$
$883$	− 2.60687e7i	− 1.12517i	0.826740	−	0.562584i	$-0.190193\pi$
$884$	−1.17566e7	−0.505999
$885$	0	0
$886$	−4.73578e6	−0.202678
$887$	− 2.08169e7i	− 0.888395i	−0.895929	−	0.444198i	$-0.853489\pi$
$887$	− 2.08169e7i	− 0.888395i	0.895929	−	0.444198i	$-0.146511\pi$
$888$	− 8.02526e6i	− 0.341528i
$889$	−9.63327e6	−0.408808
$890$	0	0
$891$	−1.57728e6	−0.0665603
$892$	8.68234e6i	0.365363i
$893$	− 4.34610e7i	− 1.82377i
$894$	2.80865e6	0.117531
$895$	0	0
$896$	2.04680e6	0.0851738
$897$	2.26995e6i	0.0941964i
$898$	2.03545e7i	0.842307i
$899$	−3.71391e6	−0.153261
$900$	0	0
$901$	3.28961e7	1.35000
$902$	− 1.76531e7i	− 0.722445i
$903$	3.39739e6i	0.138652i
$904$	1.13381e7	0.461446
$905$	0	0
$906$	9.12998e6	0.369530
$907$	4.16015e7i	1.67916i	0.543239	+	0.839578i	$0.317198\pi$
$907$	4.16015e7i	1.67916i	−0.543239	+	0.839578i	$0.682802\pi$
$908$	− 1.44986e7i	− 0.583597i
$909$	−8.40015e6	−0.337192
$910$	0	0
$911$	−4.03543e7	−1.61099	−0.805496	−	0.592601i	$-0.798101\pi$
$911$	−4.03543e7	−1.61099	−0.805496	+	0.592601i	$0.798101\pi$
$912$	5.00511e6i	0.199263i
$913$	− 1.44755e7i	− 0.574721i
$914$	−1.96482e7	−0.777959
$915$	0	0
$916$	−1.09814e7	−0.432431
$917$	− 66514.8i	− 0.00261213i
$918$	− 4.67969e6i	− 0.183278i
$919$	−3.93108e6	−0.153540	−0.0767702	−	0.997049i	$-0.524461\pi$
$919$	−3.93108e6	−0.153540	−0.0767702	+	0.997049i	$0.524461\pi$
$920$	0	0
$921$	1.22206e7	0.474726
$922$	1.74128e7i	0.674592i
$923$	628527.i	0.0242840i
$924$	4.32472e6	0.166640
$925$	0	0
$926$	3.20864e7	1.22968
$927$	7.01421e6i	0.268089i
$928$	− 4.83515e6i	− 0.184306i
$929$	−2.60407e7	−0.989951	−0.494975	−	0.868907i	$-0.664823\pi$
$929$	−2.60407e7	−0.989951	−0.494975	+	0.868907i	$0.664823\pi$
$930$	0	0
$931$	2.60738e6	0.0985892
$932$	1.31007e7i	0.494030i
$933$	2.13136e6i	0.0801592i
$934$	8.47870e6	0.318025
$935$	0	0
$936$	−2.37354e6	−0.0885536
$937$	4.80926e7i	1.78949i	0.446578	+	0.894744i	$0.352642\pi$
$937$	4.80926e7i	1.78949i	−0.446578	+	0.894744i	$0.647358\pi$
$938$	6.26628e6i	0.232543i
$939$	−1.13520e7	−0.420154
$940$	0	0
$941$	−4.32518e7	−1.59232	−0.796160	−	0.605086i	$-0.793139\pi$
$941$	−4.32518e7	−1.59232	−0.796160	+	0.605086i	$0.793139\pi$
$942$	6.57425e6i	0.241390i
$943$	− 1.01126e7i	− 0.370326i
$944$	−8.60584e6	−0.314314
$945$	0	0
$946$	−2.90567e6	−0.105565
$947$	− 2.24095e7i	− 0.812003i	−0.913872	−	0.406002i	$-0.866923\pi$
$947$	− 2.24095e7i	− 0.812003i	0.913872	−	0.406002i	$-0.133077\pi$
$948$	− 1.50818e7i	− 0.545046i
$949$	2.05715e7	0.741482
$950$	0	0
$951$	1.88406e6	0.0675528
$952$	1.28312e7i	0.458853i
$953$	1.05506e7i	0.376308i	0.982140	+	0.188154i	$0.0602504\pi$
$953$	1.05506e7i	0.376308i	−0.982140	+	0.188154i	$0.939750\pi$
$954$	6.64140e6	0.236259
$955$	0	0
$956$	1.59443e7	0.564234
$957$	− 1.02163e7i	− 0.360589i
$958$	− 7.59897e6i	− 0.267511i
$959$	1.90395e7	0.668513
$960$	0	0
$961$	−2.80105e7	−0.978391
$962$	− 2.55169e7i	− 0.888976i
$963$	1.49184e7i	0.518390i
$964$	9.01180e6	0.312334
$965$	0	0
$966$	2.47743e6	0.0854197
$967$	2.21516e7i	0.761797i	0.924617	+	0.380898i	$0.124385\pi$
$967$	2.21516e7i	0.761797i	−0.924617	+	0.380898i	$0.875615\pi$
$968$	− 6.60848e6i	− 0.226680i
$969$	−3.13764e7	−1.07348
$970$	0	0
$971$	3.51751e7	1.19726	0.598628	−	0.801027i	$-0.295713\pi$
$971$	3.51751e7	1.19726	0.598628	+	0.801027i	$0.295713\pi$
$972$	− 944784.i	− 0.0320750i
$973$	− 2.52661e7i	− 0.855572i
$974$	3.76077e6	0.127022
$975$	0	0
$976$	4.23980e6	0.142469
$977$	2.24312e7i	0.751823i	0.926656	+	0.375911i	$0.122670\pi$
$977$	2.24312e7i	0.751823i	−0.926656	+	0.375911i	$0.877330\pi$
$978$	− 1.65986e7i	− 0.554914i
$979$	4.75988e6	0.158723
$980$	0	0
$981$	−5.02648e6	−0.166760
$982$	− 2.97012e6i	− 0.0982869i
$983$	1.34900e7i	0.445276i	0.974901	+	0.222638i	$0.0714667\pi$
$983$	1.34900e7i	0.445276i	−0.974901	+	0.222638i	$0.928533\pi$
$984$	1.05741e7	0.348142
$985$	0	0
$986$	3.03109e7	0.992903
$987$	− 2.24940e7i	− 0.734977i
$988$	1.59141e7i	0.518669i
$989$	−1.66452e6	−0.0541125
$990$	0	0
$991$	−1.29539e6	−0.0419003	−0.0209501	−	0.999781i	$-0.506669\pi$
$991$	−1.29539e6	−0.0419003	−0.0209501	+	0.999781i	$0.506669\pi$
$992$	805419.i	0.0259862i
$993$	− 6.35738e6i	− 0.204600i
$994$	685976.	0.0220213
$995$	0	0
$996$	8.67075e6	0.276954
$997$	3.42137e6i	0.109009i	0.998514	+	0.0545045i	$0.0173579\pi$
$997$	3.42137e6i	0.109009i	−0.998514	+	0.0545045i	$0.982642\pi$
$998$	− 4.07599e7i	− 1.29541i
$999$	1.01570e7	0.321996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.c.499.1		8
5.2	odd	4		750.6.a.h.1.4	yes	4
5.3	odd	4		750.6.a.a.1.1	✓	4
5.4	even	2	inner	750.6.c.c.499.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.a.1.1	✓	4	5.3	odd	4
750.6.a.h.1.4	yes	4	5.2	odd	4
750.6.c.c.499.1		8	1.1	even	1	trivial
750.6.c.c.499.8		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -124.927i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -124.927i q^{7} +64.0000i q^{8} -81.0000 q^{9} -240.403 q^{11} -144.000i q^{12} -457.858i q^{13} -499.708 q^{14} +256.000 q^{16} +1604.83i q^{17} +324.000i q^{18} +2172.36 q^{19} +1124.34 q^{21} +961.612i q^{22} +550.861i q^{23} -576.000 q^{24} -1831.43 q^{26} -729.000i q^{27} +1998.83i q^{28} +4721.82 q^{29} -786.542 q^{31} -1024.00i q^{32} -2163.63i q^{33} +6419.33 q^{34} +1296.00 q^{36} +13932.7i q^{37} -8689.43i q^{38} +4120.72 q^{39} -18357.8 q^{41} -4497.37i q^{42} +3021.67i q^{43} +3846.45 q^{44} +2203.44 q^{46} -20006.4i q^{47} +2304.00i q^{48} +1200.25 q^{49} -14443.5 q^{51} +7325.73i q^{52} -20498.2i q^{53} -2916.00 q^{54} +7995.33 q^{56} +19551.2i q^{57} -18887.3i q^{58} -33616.6 q^{59} +16561.7 q^{61} +3146.17i q^{62} +10119.1i q^{63} -4096.00 q^{64} -8654.50 q^{66} -12539.9i q^{67} -25677.3i q^{68} -4957.75 q^{69} -1372.75 q^{71} -5184.00i q^{72} +44929.9i q^{73} +55731.0 q^{74} -34757.7 q^{76} +30032.8i q^{77} -16482.9i q^{78} +104735. q^{79} +6561.00 q^{81} +73431.4i q^{82} +60213.5i q^{83} -17989.5 q^{84} +12086.7 q^{86} +42496.4i q^{87} -15385.8i q^{88} -19799.6 q^{89} -57198.8 q^{91} -8813.78i q^{92} -7078.88i q^{93} -80025.4 q^{94} +9216.00 q^{96} -105229. i q^{97} -4801.01i q^{98} +19472.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more