LMFDB - Embedded newform 750.6.c.c.499.6

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			499.6
Root			$3.05722i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.c.499.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -89.9244i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -89.9244i q^{7} -64.0000i q^{8} -81.0000 q^{9} -360.249 q^{11} +144.000i q^{12} -616.499i q^{13} +359.698 q^{14} +256.000 q^{16} +1269.26i q^{17} -324.000i q^{18} +2596.01 q^{19} -809.320 q^{21} -1441.00i q^{22} +2927.99i q^{23} -576.000 q^{24} +2466.00 q^{26} +729.000i q^{27} +1438.79i q^{28} +3850.31 q^{29} +1795.93 q^{31} +1024.00i q^{32} +3242.24i q^{33} -5077.02 q^{34} +1296.00 q^{36} -6798.00i q^{37} +10384.0i q^{38} -5548.49 q^{39} +8114.32 q^{41} -3237.28i q^{42} -13157.6i q^{43} +5763.99 q^{44} -11712.0 q^{46} -7289.83i q^{47} -2304.00i q^{48} +8720.60 q^{49} +11423.3 q^{51} +9863.99i q^{52} +2492.63i q^{53} -2916.00 q^{54} -5755.16 q^{56} -23364.1i q^{57} +15401.3i q^{58} +6220.19 q^{59} -31298.8 q^{61} +7183.73i q^{62} +7283.88i q^{63} -4096.00 q^{64} -12969.0 q^{66} -39997.1i q^{67} -20308.1i q^{68} +26351.9 q^{69} -53563.0 q^{71} +5184.00i q^{72} -11950.4i q^{73} +27192.0 q^{74} -41536.2 q^{76} +32395.2i q^{77} -22194.0i q^{78} +83964.3 q^{79} +6561.00 q^{81} +32457.3i q^{82} +102769. i q^{83} +12949.1 q^{84} +52630.4 q^{86} -34652.8i q^{87} +23056.0i q^{88} -141653. q^{89} -55438.3 q^{91} -46847.8i q^{92} -16163.4i q^{93} +29159.3 q^{94} +9216.00 q^{96} +111454. i q^{97} +34882.4i q^{98} +29180.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 89.9244i	− 0.693637i	−0.937932	−	0.346819i	$-0.887262\pi$
$7$	− 89.9244i	− 0.693637i	0.937932	−	0.346819i	$-0.112738\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−360.249	−0.897680	−0.448840	−	0.893612i	$-0.648163\pi$
$11$	−360.249	−0.897680	−0.448840	+	0.893612i	$0.648163\pi$
$12$	144.000i	0.288675i
$13$	− 616.499i	− 1.01175i	−0.862606	−	0.505876i	$-0.831169\pi$
$13$	− 616.499i	− 1.01175i	0.862606	−	0.505876i	$-0.168831\pi$
$14$	359.698	0.490476
$15$	0	0
$16$	256.000	0.250000
$17$	1269.26i	1.06519i	0.846370	+	0.532595i	$0.178783\pi$
$17$	1269.26i	1.06519i	−0.846370	+	0.532595i	$0.821217\pi$
$18$	− 324.000i	− 0.235702i
$19$	2596.01	1.64977	0.824883	−	0.565304i	$-0.191241\pi$
$19$	2596.01	1.64977	0.824883	+	0.565304i	$0.191241\pi$
$20$	0	0
$21$	−809.320	−0.400472
$22$	− 1441.00i	− 0.634755i
$23$	2927.99i	1.15412i	0.816703	+	0.577058i	$0.195799\pi$
$23$	2927.99i	1.15412i	−0.816703	+	0.577058i	$0.804201\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	2466.00	0.715417
$27$	729.000i	0.192450i
$28$	1438.79i	0.346819i
$29$	3850.31	0.850161	0.425081	−	0.905156i	$-0.360246\pi$
$29$	3850.31	0.850161	0.425081	+	0.905156i	$0.360246\pi$
$30$	0	0
$31$	1795.93	0.335649	0.167825	−	0.985817i	$-0.446326\pi$
$31$	1795.93	0.335649	0.167825	+	0.985817i	$0.446326\pi$
$32$	1024.00i	0.176777i
$33$	3242.24i	0.518276i
$34$	−5077.02	−0.753203
$35$	0	0
$36$	1296.00	0.166667
$37$	− 6798.00i	− 0.816351i	−0.912904	−	0.408175i	$-0.866165\pi$
$37$	− 6798.00i	− 0.816351i	0.912904	−	0.408175i	$-0.133835\pi$
$38$	10384.0i	1.16656i
$39$	−5548.49	−0.584135
$40$	0	0
$41$	8114.32	0.753863	0.376931	−	0.926241i	$-0.376979\pi$
$41$	8114.32	0.753863	0.376931	+	0.926241i	$0.376979\pi$
$42$	− 3237.28i	− 0.283176i
$43$	− 13157.6i	− 1.08519i	−0.839995	−	0.542594i	$-0.817442\pi$
$43$	− 13157.6i	− 1.08519i	0.839995	−	0.542594i	$-0.182558\pi$
$44$	5763.99	0.448840
$45$	0	0
$46$	−11712.0	−0.816084
$47$	− 7289.83i	− 0.481363i	−0.970604	−	0.240681i	$-0.922629\pi$
$47$	− 7289.83i	− 0.481363i	0.970604	−	0.240681i	$-0.0773709\pi$
$48$	− 2304.00i	− 0.144338i
$49$	8720.60	0.518867
$50$	0	0
$51$	11423.3	0.614987
$52$	9863.99i	0.505876i
$53$	2492.63i	0.121890i	0.998141	+	0.0609449i	$0.0194114\pi$
$53$	2492.63i	0.121890i	−0.998141	+	0.0609449i	$0.980589\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	−5755.16	−0.245238
$57$	− 23364.1i	− 0.952493i
$58$	15401.3i	0.601155i
$59$	6220.19	0.232634	0.116317	−	0.993212i	$-0.462891\pi$
$59$	6220.19	0.232634	0.116317	+	0.993212i	$0.462891\pi$
$60$	0	0
$61$	−31298.8	−1.07697	−0.538484	−	0.842636i	$-0.681003\pi$
$61$	−31298.8	−1.07697	−0.538484	+	0.842636i	$0.681003\pi$
$62$	7183.73i	0.237340i
$63$	7283.88i	0.231212i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−12969.0	−0.366476
$67$	− 39997.1i	− 1.08853i	−0.838912	−	0.544267i	$-0.816808\pi$
$67$	− 39997.1i	− 1.08853i	0.838912	−	0.544267i	$-0.183192\pi$
$68$	− 20308.1i	− 0.532595i
$69$	26351.9	0.666330
$70$	0	0
$71$	−53563.0	−1.26101	−0.630506	−	0.776185i	$-0.717152\pi$
$71$	−53563.0	−1.26101	−0.630506	+	0.776185i	$0.717152\pi$
$72$	5184.00i	0.117851i
$73$	− 11950.4i	− 0.262468i	−0.991351	−	0.131234i	$-0.958106\pi$
$73$	− 11950.4i	− 0.262468i	0.991351	−	0.131234i	$-0.0418939\pi$
$74$	27192.0	0.577247
$75$	0	0
$76$	−41536.2	−0.824883
$77$	32395.2i	0.622664i
$78$	− 22194.0i	− 0.413046i
$79$	83964.3	1.51365	0.756827	−	0.653615i	$-0.226748\pi$
$79$	83964.3	1.51365	0.756827	+	0.653615i	$0.226748\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	32457.3i	0.533061i
$83$	102769.i	1.63745i	0.574184	+	0.818727i	$0.305320\pi$
$83$	102769.i	1.63745i	−0.574184	+	0.818727i	$0.694680\pi$
$84$	12949.1	0.200236
$85$	0	0
$86$	52630.4	0.767344
$87$	− 34652.8i	− 0.490841i
$88$	23056.0i	0.317378i
$89$	−141653.	−1.89561	−0.947807	−	0.318845i	$-0.896705\pi$
$89$	−141653.	−1.89561	−0.947807	+	0.318845i	$0.896705\pi$
$90$	0	0
$91$	−55438.3	−0.701789
$92$	− 46847.8i	− 0.577058i
$93$	− 16163.4i	− 0.193787i
$94$	29159.3	0.340375
$95$	0	0
$96$	9216.00	0.102062
$97$	111454.i	1.20272i	0.798978	+	0.601360i	$0.205374\pi$
$97$	111454.i	1.20272i	−0.798978	+	0.601360i	$0.794626\pi$
$98$	34882.4i	0.366895i
$99$	29180.2	0.299227
$100$	0	0
$101$	−81460.6	−0.794592	−0.397296	−	0.917691i	$-0.630051\pi$
$101$	−81460.6	−0.794592	−0.397296	+	0.917691i	$0.630051\pi$
$102$	45693.2i	0.434862i
$103$	− 22572.7i	− 0.209648i	−0.994491	−	0.104824i	$-0.966572\pi$
$103$	− 22572.7i	− 0.209648i	0.994491	−	0.104824i	$-0.0334279\pi$
$104$	−39456.0	−0.357708
$105$	0	0
$106$	−9970.51	−0.0861891
$107$	− 112608.i	− 0.950849i	−0.879757	−	0.475425i	$-0.842294\pi$
$107$	− 112608.i	− 0.950849i	0.879757	−	0.475425i	$-0.157706\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	7765.29	0.0626025	0.0313012	−	0.999510i	$-0.490035\pi$
$109$	7765.29	0.0626025	0.0313012	+	0.999510i	$0.490035\pi$
$110$	0	0
$111$	−61182.0	−0.471320
$112$	− 23020.6i	− 0.173409i
$113$	− 132827.i	− 0.978563i	−0.872126	−	0.489282i	$-0.837259\pi$
$113$	− 132827.i	− 0.978563i	0.872126	−	0.489282i	$-0.162741\pi$
$114$	93456.4	0.673514
$115$	0	0
$116$	−61605.0	−0.425081
$117$	49936.4i	0.337251i
$118$	24880.8i	0.164497i
$119$	114137.	0.738855
$120$	0	0
$121$	−31271.4	−0.194171
$122$	− 125195.i	− 0.761531i
$123$	− 73028.8i	− 0.435243i
$124$	−28734.9	−0.167825
$125$	0	0
$126$	−29135.5	−0.163492
$127$	− 275485.i	− 1.51562i	−0.652478	−	0.757808i	$-0.726270\pi$
$127$	− 275485.i	− 1.51562i	0.652478	−	0.757808i	$-0.273730\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−118418.	−0.626534
$130$	0	0
$131$	−203396.	−1.03553	−0.517767	−	0.855522i	$-0.673237\pi$
$131$	−203396.	−1.03553	−0.517767	+	0.855522i	$0.673237\pi$
$132$	− 51875.9i	− 0.259138i
$133$	− 233445.i	− 1.14434i
$134$	159989.	0.769710
$135$	0	0
$136$	81232.4	0.376601
$137$	− 212517.i	− 0.967369i	−0.875242	−	0.483685i	$-0.839298\pi$
$137$	− 212517.i	− 0.967369i	0.875242	−	0.483685i	$-0.160702\pi$
$138$	105408.i	0.471166i
$139$	−96377.9	−0.423098	−0.211549	−	0.977367i	$-0.567851\pi$
$139$	−96377.9	−0.423098	−0.211549	+	0.977367i	$0.567851\pi$
$140$	0	0
$141$	−65608.5	−0.277915
$142$	− 214252.i	− 0.891670i
$143$	222093.i	0.908230i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	47801.7	0.185593
$147$	− 78485.4i	− 0.299568i
$148$	108768.i	0.408175i
$149$	−128293.	−0.473410	−0.236705	−	0.971582i	$-0.576068\pi$
$149$	−128293.	−0.473410	−0.236705	+	0.971582i	$0.576068\pi$
$150$	0	0
$151$	254367.	0.907859	0.453929	−	0.891038i	$-0.350022\pi$
$151$	254367.	0.907859	0.453929	+	0.891038i	$0.350022\pi$
$152$	− 166145.i	− 0.583280i
$153$	− 102810.i	− 0.355063i
$154$	−129581.	−0.440290
$155$	0	0
$156$	88775.9	0.292068
$157$	− 298604.i	− 0.966822i	−0.875393	−	0.483411i	$-0.839398\pi$
$157$	− 298604.i	− 0.966822i	0.875393	−	0.483411i	$-0.160602\pi$
$158$	335857.i	1.07032i
$159$	22433.6	0.0703731
$160$	0	0
$161$	263298.	0.800538
$162$	26244.0i	0.0785674i
$163$	− 442956.i	− 1.30584i	−0.757425	−	0.652922i	$-0.773543\pi$
$163$	− 442956.i	− 1.30584i	0.757425	−	0.652922i	$-0.226457\pi$
$164$	−129829.	−0.376931
$165$	0	0
$166$	−411078.	−1.15785
$167$	74320.9i	0.206215i	0.994670	+	0.103107i	$0.0328785\pi$
$167$	74320.9i	0.206215i	−0.994670	+	0.103107i	$0.967121\pi$
$168$	51796.5i	0.141588i
$169$	−8778.35	−0.0236426
$170$	0	0
$171$	−210277.	−0.549922
$172$	210521.i	0.542594i
$173$	− 452596.i	− 1.14973i	−0.818249	−	0.574864i	$-0.805055\pi$
$173$	− 452596.i	− 1.14973i	0.818249	−	0.574864i	$-0.194945\pi$
$174$	138611.	0.347077
$175$	0	0
$176$	−92223.8	−0.224420
$177$	− 55981.7i	− 0.134311i
$178$	− 566611.i	− 1.34040i
$179$	11167.5	0.0260510	0.0130255	−	0.999915i	$-0.495854\pi$
$179$	11167.5	0.0260510	0.0130255	+	0.999915i	$0.495854\pi$
$180$	0	0
$181$	326476.	0.740721	0.370361	−	0.928888i	$-0.379234\pi$
$181$	326476.	0.740721	0.370361	+	0.928888i	$0.379234\pi$
$182$	− 221753.i	− 0.496240i
$183$	281689.i	0.621788i
$184$	187391.	0.408042
$185$	0	0
$186$	64653.6	0.137028
$187$	− 457249.i	− 0.956199i
$188$	116637.i	0.240681i
$189$	65554.9	0.133491
$190$	0	0
$191$	−227786.	−0.451797	−0.225899	−	0.974151i	$-0.572532\pi$
$191$	−227786.	−0.451797	−0.225899	+	0.974151i	$0.572532\pi$
$192$	36864.0i	0.0721688i
$193$	108146.i	0.208985i	0.994526	+	0.104493i	$0.0333218\pi$
$193$	108146.i	0.208985i	−0.994526	+	0.104493i	$0.966678\pi$
$194$	−445814.	−0.850452
$195$	0	0
$196$	−139530.	−0.259434
$197$	− 790807.i	− 1.45179i	−0.687804	−	0.725897i	$-0.741425\pi$
$197$	− 790807.i	− 1.45179i	0.687804	−	0.725897i	$-0.258575\pi$
$198$	116721.i	0.211585i
$199$	−402302.	−0.720144	−0.360072	−	0.932925i	$-0.617248\pi$
$199$	−402302.	−0.720144	−0.360072	+	0.932925i	$0.617248\pi$
$200$	0	0
$201$	−359974.	−0.628465
$202$	− 325842.i	− 0.561861i
$203$	− 346237.i	− 0.589703i
$204$	−182773.	−0.307494
$205$	0	0
$206$	90290.7	0.148243
$207$	− 237167.i	− 0.384706i
$208$	− 157824.i	− 0.252938i
$209$	−935211.	−1.48096
$210$	0	0
$211$	−1.23234e6	−1.90557	−0.952786	−	0.303643i	$-0.901797\pi$
$211$	−1.23234e6	−1.90557	−0.952786	+	0.303643i	$0.901797\pi$
$212$	− 39882.0i	− 0.0609449i
$213$	482067.i	0.728045i
$214$	450434.	0.672352
$215$	0	0
$216$	46656.0	0.0680414
$217$	− 161498.i	− 0.232819i
$218$	31061.2i	0.0442666i
$219$	−107554.	−0.151536
$220$	0	0
$221$	782495.	1.07771
$222$	− 244728.i	− 0.333274i
$223$	1.20987e6i	1.62921i	0.580019	+	0.814603i	$0.303045\pi$
$223$	1.20987e6i	1.62921i	−0.580019	+	0.814603i	$0.696955\pi$
$224$	92082.6	0.122619
$225$	0	0
$226$	531306.	0.691949
$227$	186777.i	0.240579i	0.992739	+	0.120290i	$0.0383823\pi$
$227$	186777.i	0.240579i	−0.992739	+	0.120290i	$0.961618\pi$
$228$	373825.i	0.476246i
$229$	305063.	0.384416	0.192208	−	0.981354i	$-0.438435\pi$
$229$	305063.	0.384416	0.192208	+	0.981354i	$0.438435\pi$
$230$	0	0
$231$	291557.	0.359495
$232$	− 246420.i	− 0.300577i
$233$	686214.i	0.828076i	0.910260	+	0.414038i	$0.135882\pi$
$233$	686214.i	0.828076i	−0.910260	+	0.414038i	$0.864118\pi$
$234$	−199746.	−0.238472
$235$	0	0
$236$	−99523.1	−0.116317
$237$	− 755679.i	− 0.873909i
$238$	456548.i	0.522449i
$239$	−1.57933e6	−1.78846	−0.894230	−	0.447607i	$-0.852276\pi$
$239$	−1.57933e6	−1.78846	−0.894230	+	0.447607i	$0.852276\pi$
$240$	0	0
$241$	504072.	0.559049	0.279525	−	0.960139i	$-0.409823\pi$
$241$	504072.	0.559049	0.279525	+	0.960139i	$0.409823\pi$
$242$	− 125086.i	− 0.137300i
$243$	− 59049.0i	− 0.0641500i
$244$	500780.	0.538484
$245$	0	0
$246$	292115.	0.307763
$247$	− 1.60044e6i	− 1.66915i
$248$	− 114940.i	− 0.118670i
$249$	924925.	0.945384
$250$	0	0
$251$	271053.	0.271563	0.135781	−	0.990739i	$-0.456646\pi$
$251$	271053.	0.271563	0.135781	+	0.990739i	$0.456646\pi$
$252$	− 116542.i	− 0.115606i
$253$	− 1.05481e6i	− 1.03603i
$254$	1.10194e6	1.07170
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 786761.i	− 0.743037i	−0.928426	−	0.371518i	$-0.878837\pi$
$257$	− 786761.i	− 0.743037i	0.928426	−	0.371518i	$-0.121163\pi$
$258$	− 473673.i	− 0.443026i
$259$	−611306.	−0.566251
$260$	0	0
$261$	−311876.	−0.283387
$262$	− 813585.i	− 0.732233i
$263$	− 27858.5i	− 0.0248353i	−0.999923	−	0.0124176i	$-0.996047\pi$
$263$	− 27858.5i	− 0.0248353i	0.999923	−	0.0124176i	$-0.00395276\pi$
$264$	207504.	0.183238
$265$	0	0
$266$	933778.	0.809170
$267$	1.27487e6i	1.09443i
$268$	639954.i	0.544267i
$269$	222843.	0.187766	0.0938832	−	0.995583i	$-0.470072\pi$
$269$	222843.	0.187766	0.0938832	+	0.995583i	$0.470072\pi$
$270$	0	0
$271$	2.01747e6	1.66872	0.834362	−	0.551218i	$-0.185837\pi$
$271$	2.01747e6	1.66872	0.834362	+	0.551218i	$0.185837\pi$
$272$	324929.i	0.266297i
$273$	498945.i	0.405178i
$274$	850068.	0.684033
$275$	0	0
$276$	−421630.	−0.333165
$277$	− 1.05741e6i	− 0.828026i	−0.910271	−	0.414013i	$-0.864127\pi$
$277$	− 1.05741e6i	− 0.828026i	0.910271	−	0.414013i	$-0.135873\pi$
$278$	− 385512.i	− 0.299175i
$279$	−145471.	−0.111883
$280$	0	0
$281$	1.68285e6	1.27139	0.635696	−	0.771939i	$-0.280713\pi$
$281$	1.68285e6	1.27139	0.635696	+	0.771939i	$0.280713\pi$
$282$	− 262434.i	− 0.196516i
$283$	− 2.00362e6i	− 1.48713i	−0.668662	−	0.743567i	$-0.733133\pi$
$283$	− 2.00362e6i	− 1.48713i	0.668662	−	0.743567i	$-0.266867\pi$
$284$	857008.	0.630506
$285$	0	0
$286$	−888374.	−0.642215
$287$	− 729675.i	− 0.522907i
$288$	− 82944.0i	− 0.0589256i
$289$	−191153.	−0.134629
$290$	0	0
$291$	1.00308e6	0.694391
$292$	191207.i	0.131234i
$293$	227065.i	0.154519i	0.997011	+	0.0772594i	$0.0246170\pi$
$293$	227065.i	0.154519i	−0.997011	+	0.0772594i	$0.975383\pi$
$294$	313942.	0.211827
$295$	0	0
$296$	−435072.	−0.288624
$297$	− 262622.i	− 0.172759i
$298$	− 513173.i	− 0.334752i
$299$	1.80510e6	1.16768
$300$	0	0
$301$	−1.18319e6	−0.752727
$302$	1.01747e6i	0.641953i
$303$	733145.i	0.458758i
$304$	664578.	0.412441
$305$	0	0
$306$	411239.	0.251068
$307$	− 2.13475e6i	− 1.29271i	−0.763037	−	0.646355i	$-0.776293\pi$
$307$	− 2.13475e6i	− 1.29271i	0.763037	−	0.646355i	$-0.223707\pi$
$308$	− 518323.i	− 0.311332i
$309$	−203154.	−0.121040
$310$	0	0
$311$	2.16691e6	1.27040	0.635198	−	0.772349i	$-0.280918\pi$
$311$	2.16691e6	1.27040	0.635198	+	0.772349i	$0.280918\pi$
$312$	355104.i	0.206523i
$313$	365853.i	0.211079i	0.994415	+	0.105540i	$0.0336570\pi$
$313$	365853.i	0.211079i	−0.994415	+	0.105540i	$0.966343\pi$
$314$	1.19442e6	0.683647
$315$	0	0
$316$	−1.34343e6	−0.756827
$317$	− 3.50640e6i	− 1.95981i	−0.199466	−	0.979905i	$-0.563921\pi$
$317$	− 3.50640e6i	− 1.95981i	0.199466	−	0.979905i	$-0.436079\pi$
$318$	89734.6i	0.0497613i
$319$	−1.38707e6	−0.763172
$320$	0	0
$321$	−1.01348e6	−0.548973
$322$	1.05319e6i	0.566066i
$323$	3.29500e6i	1.75731i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.77182e6	0.923371
$327$	− 69887.6i	− 0.0361435i
$328$	− 519316.i	− 0.266531i
$329$	−655533.	−0.333891
$330$	0	0
$331$	−1.93044e6	−0.968472	−0.484236	−	0.874937i	$-0.660902\pi$
$331$	−1.93044e6	−0.968472	−0.484236	+	0.874937i	$0.660902\pi$
$332$	− 1.64431e6i	− 0.818727i
$333$	550638.i	0.272117i
$334$	−297284.	−0.145816
$335$	0	0
$336$	−207186.	−0.100118
$337$	3.92204e6i	1.88121i	0.339503	+	0.940605i	$0.389741\pi$
$337$	3.92204e6i	1.88121i	−0.339503	+	0.940605i	$0.610259\pi$
$338$	− 35113.4i	− 0.0167179i
$339$	−1.19544e6	−0.564974
$340$	0	0
$341$	−646984.	−0.301306
$342$	− 841107.i	− 0.388854i
$343$	− 2.29555e6i	− 1.05354i
$344$	−842086.	−0.383672
$345$	0	0
$346$	1.81038e6	0.812980
$347$	− 771922.i	− 0.344152i	−0.985084	−	0.172076i	$-0.944953\pi$
$347$	− 771922.i	− 0.344152i	0.985084	−	0.172076i	$-0.0550474\pi$
$348$	554445.i	0.245420i
$349$	−3.93074e6	−1.72747	−0.863735	−	0.503947i	$-0.831881\pi$
$349$	−3.93074e6	−1.72747	−0.863735	+	0.503947i	$0.831881\pi$
$350$	0	0
$351$	449428.	0.194712
$352$	− 368895.i	− 0.158689i
$353$	− 1.78726e6i	− 0.763396i	−0.924287	−	0.381698i	$-0.875339\pi$
$353$	− 1.78726e6i	− 0.763396i	0.924287	−	0.381698i	$-0.124661\pi$
$354$	223927.	0.0949726
$355$	0	0
$356$	2.26644e6	0.947807
$357$	− 1.02723e6i	− 0.426578i
$358$	44670.1i	0.0184208i
$359$	1.23619e6	0.506229	0.253115	−	0.967436i	$-0.418545\pi$
$359$	1.23619e6	0.506229	0.253115	+	0.967436i	$0.418545\pi$
$360$	0	0
$361$	4.26317e6	1.72173
$362$	1.30590e6i	0.523769i
$363$	281443.i	0.112105i
$364$	887013.	0.350895
$365$	0	0
$366$	−1.12676e6	−0.439670
$367$	2.06191e6i	0.799106i	0.916710	+	0.399553i	$0.130835\pi$
$367$	2.06191e6i	0.799106i	−0.916710	+	0.399553i	$0.869165\pi$
$368$	749565.i	0.288529i
$369$	−657260.	−0.251288
$370$	0	0
$371$	224148.	0.0845473
$372$	258614.i	0.0968936i
$373$	− 4.03115e6i	− 1.50023i	−0.661310	−	0.750113i	$-0.729999\pi$
$373$	− 4.03115e6i	− 1.50023i	0.661310	−	0.750113i	$-0.270001\pi$
$374$	1.82899e6	0.676135
$375$	0	0
$376$	−466549.	−0.170187
$377$	− 2.37372e6i	− 0.860152i
$378$	262220.i	0.0943921i
$379$	3.31522e6	1.18553	0.592767	−	0.805374i	$-0.298035\pi$
$379$	3.31522e6	1.18553	0.592767	+	0.805374i	$0.298035\pi$
$380$	0	0
$381$	−2.47937e6	−0.875041
$382$	− 911143.i	− 0.319469i
$383$	2.35490e6i	0.820306i	0.912017	+	0.410153i	$0.134525\pi$
$383$	2.35490e6i	0.820306i	−0.912017	+	0.410153i	$0.865475\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	−432582.	−0.147775
$387$	1.06576e6i	0.361730i
$388$	− 1.78326e6i	− 0.601360i
$389$	837303.	0.280549	0.140275	−	0.990113i	$-0.455201\pi$
$389$	837303.	0.280549	0.140275	+	0.990113i	$0.455201\pi$
$390$	0	0
$391$	−3.71637e6	−1.22935
$392$	− 558119.i	− 0.183447i
$393$	1.83057e6i	0.597866i
$394$	3.16323e6	1.02657
$395$	0	0
$396$	−466883.	−0.149613
$397$	− 3.52492e6i	− 1.12247i	−0.827658	−	0.561233i	$-0.810327\pi$
$397$	− 3.52492e6i	− 1.12247i	0.827658	−	0.561233i	$-0.189673\pi$
$398$	− 1.60921e6i	− 0.509219i
$399$	−2.10100e6	−0.660685
$400$	0	0
$401$	−2.64668e6	−0.821942	−0.410971	−	0.911648i	$-0.634810\pi$
$401$	−2.64668e6	−0.821942	−0.410971	+	0.911648i	$0.634810\pi$
$402$	− 1.43990e6i	− 0.444392i
$403$	− 1.10719e6i	− 0.339594i
$404$	1.30337e6	0.397296
$405$	0	0
$406$	1.38495e6	0.416983
$407$	2.44897e6i	0.732822i
$408$	− 731091.i	− 0.217431i
$409$	−868441.	−0.256704	−0.128352	−	0.991729i	$-0.540969\pi$
$409$	−868441.	−0.256704	−0.128352	+	0.991729i	$0.540969\pi$
$410$	0	0
$411$	−1.91265e6	−0.558511
$412$	361163.i	0.104824i
$413$	− 559347.i	− 0.161364i
$414$	948668.	0.272028
$415$	0	0
$416$	631295.	0.178854
$417$	867401.i	0.244276i
$418$	− 3.74084e6i	− 1.04720i
$419$	−2.03207e6	−0.565463	−0.282732	−	0.959199i	$-0.591241\pi$
$419$	−2.03207e6	−0.565463	−0.282732	+	0.959199i	$0.591241\pi$
$420$	0	0
$421$	−3.03094e6	−0.833434	−0.416717	−	0.909036i	$-0.636819\pi$
$421$	−3.03094e6	−0.833434	−0.416717	+	0.909036i	$0.636819\pi$
$422$	− 4.92937e6i	− 1.34744i
$423$	590476.i	0.160454i
$424$	159528.	0.0430946
$425$	0	0
$426$	−1.92827e6	−0.514806
$427$	2.81452e6i	0.747025i
$428$	1.80174e6i	0.475425i
$429$	1.99884e6	0.524367
$430$	0	0
$431$	5.47557e6	1.41983	0.709915	−	0.704287i	$-0.248733\pi$
$431$	5.47557e6	1.41983	0.709915	+	0.704287i	$0.248733\pi$
$432$	186624.i	0.0481125i
$433$	1.22740e6i	0.314605i	0.987550	+	0.157303i	$0.0502798\pi$
$433$	1.22740e6i	0.314605i	−0.987550	+	0.157303i	$0.949720\pi$
$434$	645993.	0.164628
$435$	0	0
$436$	−124245.	−0.0313012
$437$	7.60109e6i	1.90402i
$438$	− 430215.i	− 0.107152i
$439$	−1.08369e6	−0.268375	−0.134188	−	0.990956i	$-0.542842\pi$
$439$	−1.08369e6	−0.268375	−0.134188	+	0.990956i	$0.542842\pi$
$440$	0	0
$441$	−706369.	−0.172956
$442$	3.12998e6i	0.762055i
$443$	4.23673e6i	1.02570i	0.858477	+	0.512852i	$0.171411\pi$
$443$	4.23673e6i	1.02570i	−0.858477	+	0.512852i	$0.828589\pi$
$444$	978912.	0.235660
$445$	0	0
$446$	−4.83947e6	−1.15202
$447$	1.15464e6i	0.273324i
$448$	368330.i	0.0867047i
$449$	−5.63499e6	−1.31910	−0.659549	−	0.751661i	$-0.729253\pi$
$449$	−5.63499e6	−1.31910	−0.659549	+	0.751661i	$0.729253\pi$
$450$	0	0
$451$	−2.92318e6	−0.676727
$452$	2.12522e6i	0.489282i
$453$	− 2.28930e6i	− 0.524153i
$454$	−747107.	−0.170115
$455$	0	0
$456$	−1.49530e6	−0.336757
$457$	− 3.26607e6i	− 0.731534i	−0.930706	−	0.365767i	$-0.880807\pi$
$457$	− 3.26607e6i	− 0.731534i	0.930706	−	0.365767i	$-0.119193\pi$
$458$	1.22025e6i	0.271823i
$459$	−925288.	−0.204996
$460$	0	0
$461$	4.56075e6	0.999503	0.499751	−	0.866169i	$-0.333425\pi$
$461$	4.56075e6	0.999503	0.499751	+	0.866169i	$0.333425\pi$
$462$	1.16623e6i	0.254202i
$463$	4.12105e6i	0.893418i	0.894679	+	0.446709i	$0.147404\pi$
$463$	4.12105e6i	0.893418i	−0.894679	+	0.446709i	$0.852596\pi$
$464$	985681.	0.212540
$465$	0	0
$466$	−2.74486e6	−0.585538
$467$	8.82505e6i	1.87251i	0.351316	+	0.936257i	$0.385734\pi$
$467$	8.82505e6i	1.87251i	−0.351316	+	0.936257i	$0.614266\pi$
$468$	− 798983.i	− 0.168625i
$469$	−3.59672e6	−0.755048
$470$	0	0
$471$	−2.68744e6	−0.558195
$472$	− 398092.i	− 0.0822487i
$473$	4.74001e6i	0.974152i
$474$	3.02271e6	0.617947
$475$	0	0
$476$	−1.82619e6	−0.369428
$477$	− 201903.i	− 0.0406299i
$478$	− 6.31734e6i	− 1.26463i
$479$	−1.77143e6	−0.352765	−0.176382	−	0.984322i	$-0.556440\pi$
$479$	−1.77143e6	−0.352765	−0.176382	+	0.984322i	$0.556440\pi$
$480$	0	0
$481$	−4.19096e6	−0.825945
$482$	2.01629e6i	0.395308i
$483$	− 2.36968e6i	− 0.462191i
$484$	500343.	0.0970855
$485$	0	0
$486$	236196.	0.0453609
$487$	4.03669e6i	0.771263i	0.922653	+	0.385632i	$0.126016\pi$
$487$	4.03669e6i	0.771263i	−0.922653	+	0.385632i	$0.873984\pi$
$488$	2.00312e6i	0.380766i
$489$	−3.98660e6	−0.753930
$490$	0	0
$491$	4.30507e6	0.805892	0.402946	−	0.915224i	$-0.367986\pi$
$491$	4.30507e6	0.805892	0.402946	+	0.915224i	$0.367986\pi$
$492$	1.16846e6i	0.217621i
$493$	4.88703e6i	0.905583i
$494$	6.40175e6	1.18027
$495$	0	0
$496$	459759.	0.0839124
$497$	4.81662e6i	0.874685i
$498$	3.69970e6i	0.668487i
$499$	−2.96275e6	−0.532652	−0.266326	−	0.963883i	$-0.585810\pi$
$499$	−2.96275e6	−0.532652	−0.266326	+	0.963883i	$0.585810\pi$
$500$	0	0
$501$	668888.	0.119058
$502$	1.08421e6i	0.192024i
$503$	− 2.97709e6i	− 0.524653i	−0.964979	−	0.262326i	$-0.915510\pi$
$503$	− 2.97709e6i	− 0.524653i	0.964979	−	0.262326i	$-0.0844897\pi$
$504$	466168.	0.0817459
$505$	0	0
$506$	4.21922e6	0.732582
$507$	79005.1i	0.0136501i
$508$	4.40776e6i	0.757808i
$509$	−1.01613e7	−1.73843	−0.869213	−	0.494437i	$-0.835374\pi$
$509$	−1.01613e7	−1.73843	−0.869213	+	0.494437i	$0.835374\pi$
$510$	0	0
$511$	−1.07464e6	−0.182058
$512$	262144.i	0.0441942i
$513$	1.89249e6i	0.317498i
$514$	3.14705e6	0.525406
$515$	0	0
$516$	1.89469e6	0.313267
$517$	2.62616e6i	0.432110i
$518$	− 2.44522e6i	− 0.400400i
$519$	−4.07336e6	−0.663796
$520$	0	0
$521$	2.19969e6	0.355032	0.177516	−	0.984118i	$-0.443194\pi$
$521$	2.19969e6	0.355032	0.177516	+	0.984118i	$0.443194\pi$
$522$	− 1.24750e6i	− 0.200385i
$523$	− 2.78087e6i	− 0.444556i	−0.974983	−	0.222278i	$-0.928651\pi$
$523$	− 2.78087e6i	− 0.444556i	0.974983	−	0.222278i	$-0.0713492\pi$
$524$	3.25434e6	0.517767
$525$	0	0
$526$	111434.	0.0175612
$527$	2.27950e6i	0.357530i
$528$	830014.i	0.129569i
$529$	−2.13677e6	−0.331985
$530$	0	0
$531$	−503836.	−0.0775448
$532$	3.73511e6i	0.572170i
$533$	− 5.00247e6i	− 0.762722i
$534$	−5.09950e6	−0.773881
$535$	0	0
$536$	−2.55982e6	−0.384855
$537$	− 100508.i	− 0.0150406i
$538$	891371.i	0.132771i
$539$	−3.14159e6	−0.465777
$540$	0	0
$541$	−8.13072e6	−1.19436	−0.597181	−	0.802107i	$-0.703713\pi$
$541$	−8.13072e6	−1.19436	−0.597181	+	0.802107i	$0.703713\pi$
$542$	8.06989e6i	1.17997i
$543$	− 2.93828e6i	− 0.427656i
$544$	−1.29972e6	−0.188301
$545$	0	0
$546$	−1.99578e6	−0.286504
$547$	4.84274e6i	0.692027i	0.938230	+	0.346013i	$0.112465\pi$
$547$	4.84274e6i	0.692027i	−0.938230	+	0.346013i	$0.887535\pi$
$548$	3.40027e6i	0.483685i
$549$	2.53520e6	0.358989
$550$	0	0
$551$	9.99545e6	1.40257
$552$	− 1.68652e6i	− 0.235583i
$553$	− 7.55044e6i	− 1.04993i
$554$	4.22964e6	0.585503
$555$	0	0
$556$	1.54205e6	0.211549
$557$	609055.i	0.0831799i	0.999135	+	0.0415900i	$0.0132423\pi$
$557$	609055.i	0.0831799i	−0.999135	+	0.0415900i	$0.986758\pi$
$558$	− 581882.i	− 0.0791133i
$559$	−8.11164e6	−1.09794
$560$	0	0
$561$	−4.11524e6	−0.552062
$562$	6.73140e6i	0.899010i
$563$	− 5.21882e6i	− 0.693907i	−0.937882	−	0.346954i	$-0.887216\pi$
$563$	− 5.21882e6i	− 0.693907i	0.937882	−	0.346954i	$-0.112784\pi$
$564$	1.04974e6	0.138958
$565$	0	0
$566$	8.01449e6	1.05156
$567$	− 589994.i	− 0.0770708i
$568$	3.42803e6i	0.445835i
$569$	1.26116e7	1.63302	0.816509	−	0.577333i	$-0.195907\pi$
$569$	1.26116e7	1.63302	0.816509	+	0.577333i	$0.195907\pi$
$570$	0	0
$571$	−1.50353e7	−1.92984	−0.964919	−	0.262548i	$-0.915437\pi$
$571$	−1.50353e7	−1.92984	−0.964919	+	0.262548i	$0.915437\pi$
$572$	− 3.55349e6i	− 0.454115i
$573$	2.05007e6i	0.260845i
$574$	2.91870e6	0.369751
$575$	0	0
$576$	331776.	0.0416667
$577$	− 4.53332e6i	− 0.566862i	−0.958993	−	0.283431i	$-0.908527\pi$
$577$	− 4.53332e6i	− 0.566862i	0.958993	−	0.283431i	$-0.0914726\pi$
$578$	− 764614.i	− 0.0951968i
$579$	973310.	0.120658
$580$	0	0
$581$	9.24148e6	1.13580
$582$	4.01233e6i	0.491008i
$583$	− 897967.i	− 0.109418i
$584$	−764827.	−0.0927965
$585$	0	0
$586$	−908261.	−0.109261
$587$	− 5.07512e6i	− 0.607926i	−0.952684	−	0.303963i	$-0.901690\pi$
$587$	− 5.07512e6i	− 0.607926i	0.952684	−	0.303963i	$-0.0983100\pi$
$588$	1.25577e6i	0.149784i
$589$	4.66226e6	0.553743
$590$	0	0
$591$	−7.11726e6	−0.838193
$592$	− 1.74029e6i	− 0.204088i
$593$	395983.i	0.0462424i	0.999733	+	0.0231212i	$0.00736036\pi$
$593$	395983.i	0.0462424i	−0.999733	+	0.0231212i	$0.992640\pi$
$594$	1.05049e6	0.122159
$595$	0	0
$596$	2.05269e6	0.236705
$597$	3.62072e6i	0.415775i
$598$	7.22041e6i	0.825675i
$599$	−8.62905e6	−0.982643	−0.491322	−	0.870978i	$-0.663486\pi$
$599$	−8.62905e6	−0.982643	−0.491322	+	0.870978i	$0.663486\pi$
$600$	0	0
$601$	−80994.1	−0.00914675	−0.00457338	−	0.999990i	$-0.501456\pi$
$601$	−80994.1	−0.00914675	−0.00457338	+	0.999990i	$0.501456\pi$
$602$	− 4.73275e6i	− 0.532259i
$603$	3.23977e6i	0.362845i
$604$	−4.06987e6	−0.453929
$605$	0	0
$606$	−2.93258e6	−0.324391
$607$	1.22328e6i	0.134758i	0.997727	+	0.0673790i	$0.0214637\pi$
$607$	1.22328e6i	0.134758i	−0.997727	+	0.0673790i	$0.978536\pi$
$608$	2.65831e6i	0.291640i
$609$	−3.11614e6	−0.340465
$610$	0	0
$611$	−4.49417e6	−0.487020
$612$	1.64496e6i	0.177532i
$613$	4.59365e6i	0.493750i	0.969047	+	0.246875i	$0.0794037\pi$
$613$	4.59365e6i	0.493750i	−0.969047	+	0.246875i	$0.920596\pi$
$614$	8.53900e6	0.914084
$615$	0	0
$616$	2.07329e6	0.220145
$617$	1.17378e7i	1.24129i	0.784093	+	0.620644i	$0.213129\pi$
$617$	1.17378e7i	1.24129i	−0.784093	+	0.620644i	$0.786871\pi$
$618$	− 812617.i	− 0.0855883i
$619$	−9.15580e6	−0.960439	−0.480219	−	0.877148i	$-0.659443\pi$
$619$	−9.15580e6	−0.960439	−0.480219	+	0.877148i	$0.659443\pi$
$620$	0	0
$621$	−2.13450e6	−0.222110
$622$	8.66763e6i	0.898306i
$623$	1.27380e7i	1.31487i
$624$	−1.42041e6	−0.146034
$625$	0	0
$626$	−1.46341e6	−0.149256
$627$	8.41690e6i	0.855033i
$628$	4.77767e6i	0.483411i
$629$	8.62840e6	0.869568
$630$	0	0
$631$	−1.04864e7	−1.04847	−0.524233	−	0.851575i	$-0.675648\pi$
$631$	−1.04864e7	−1.04847	−0.524233	+	0.851575i	$0.675648\pi$
$632$	− 5.37371e6i	− 0.535158i
$633$	1.10911e7i	1.10018i
$634$	1.40256e7	1.38579
$635$	0	0
$636$	−358938.	−0.0351866
$637$	− 5.37625e6i	− 0.524965i
$638$	− 5.54829e6i	− 0.539644i
$639$	4.33860e6	0.420337
$640$	0	0
$641$	5.01085e6	0.481689	0.240844	−	0.970564i	$-0.422576\pi$
$641$	5.01085e6	0.481689	0.240844	+	0.970564i	$0.422576\pi$
$642$	− 4.05390e6i	− 0.388183i
$643$	− 4.51560e6i	− 0.430713i	−0.976536	−	0.215356i	$-0.930909\pi$
$643$	− 4.51560e6i	− 0.430713i	0.976536	−	0.215356i	$-0.0690913\pi$
$644$	−4.21276e6	−0.400269
$645$	0	0
$646$	−1.31800e7	−1.24261
$647$	9.07200e6i	0.852005i	0.904722	+	0.426003i	$0.140079\pi$
$647$	9.07200e6i	0.852005i	−0.904722	+	0.426003i	$0.859921\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−2.24082e6	−0.208831
$650$	0	0
$651$	−1.45348e6	−0.134418
$652$	7.08729e6i	0.652922i
$653$	− 1.57045e7i	− 1.44125i	−0.693323	−	0.720627i	$-0.743854\pi$
$653$	− 1.57045e7i	− 1.44125i	0.693323	−	0.720627i	$-0.256146\pi$
$654$	279550.	0.0255573
$655$	0	0
$656$	2.07726e6	0.188466
$657$	967985.i	0.0874893i
$658$	− 2.62213e6i	− 0.236097i
$659$	−2.73634e6	−0.245446	−0.122723	−	0.992441i	$-0.539163\pi$
$659$	−2.73634e6	−0.245446	−0.122723	+	0.992441i	$0.539163\pi$
$660$	0	0
$661$	1.84711e7	1.64433	0.822163	−	0.569252i	$-0.192767\pi$
$661$	1.84711e7	1.64433	0.822163	+	0.569252i	$0.192767\pi$
$662$	− 7.72177e6i	− 0.684813i
$663$	− 7.04246e6i	− 0.622215i
$664$	6.57725e6	0.578927
$665$	0	0
$666$	−2.20255e6	−0.192416
$667$	1.12737e7i	0.981185i
$668$	− 1.18913e6i	− 0.103107i
$669$	1.08888e7	0.940622
$670$	0	0
$671$	1.12754e7	0.966772
$672$	− 828743.i	− 0.0707941i
$673$	4.32011e6i	0.367669i	0.982957	+	0.183835i	$0.0588511\pi$
$673$	4.32011e6i	0.367669i	−0.982957	+	0.183835i	$0.941149\pi$
$674$	−1.56882e7	−1.33022
$675$	0	0
$676$	140454.	0.0118213
$677$	− 6.74627e6i	− 0.565707i	−0.959163	−	0.282854i	$-0.908719\pi$
$677$	− 6.74627e6i	− 0.565707i	0.959163	−	0.282854i	$-0.0912811\pi$
$678$	− 4.78176e6i	− 0.399497i
$679$	1.00224e7	0.834251
$680$	0	0
$681$	1.68099e6	0.138899
$682$	− 2.58793e6i	− 0.213055i
$683$	− 5.41921e6i	− 0.444512i	−0.974988	−	0.222256i	$-0.928658\pi$
$683$	− 5.41921e6i	− 0.444512i	0.974988	−	0.222256i	$-0.0713421\pi$
$684$	3.36443e6	0.274961
$685$	0	0
$686$	9.18222e6	0.744967
$687$	− 2.74557e6i	− 0.221942i
$688$	− 3.36834e6i	− 0.271297i
$689$	1.53670e6	0.123322
$690$	0	0
$691$	−3.15234e6	−0.251153	−0.125576	−	0.992084i	$-0.540078\pi$
$691$	−3.15234e6	−0.251153	−0.125576	+	0.992084i	$0.540078\pi$
$692$	7.24153e6i	0.574864i
$693$	− 2.62401e6i	− 0.207555i
$694$	3.08769e6	0.243352
$695$	0	0
$696$	−2.21778e6	−0.173538
$697$	1.02991e7i	0.803007i
$698$	− 1.57229e7i	− 1.22151i
$699$	6.17593e6	0.478090
$700$	0	0
$701$	1.95549e7	1.50301	0.751503	−	0.659730i	$-0.229329\pi$
$701$	1.95549e7	1.50301	0.751503	+	0.659730i	$0.229329\pi$
$702$	1.79771e6i	0.137682i
$703$	− 1.76477e7i	− 1.34679i
$704$	1.47558e6	0.112210
$705$	0	0
$706$	7.14902e6	0.539803
$707$	7.32530e6i	0.551159i
$708$	895708.i	0.0671557i
$709$	−7.45844e6	−0.557228	−0.278614	−	0.960403i	$-0.589875\pi$
$709$	−7.45844e6	−0.557228	−0.278614	+	0.960403i	$0.589875\pi$
$710$	0	0
$711$	−6.80111e6	−0.504552
$712$	9.06577e6i	0.670201i
$713$	5.25847e6i	0.387379i
$714$	4.10893e6	0.301636
$715$	0	0
$716$	−178680.	−0.0130255
$717$	1.42140e7i	1.03257i
$718$	4.94474e6i	0.357958i
$719$	−2.00300e7	−1.44497	−0.722485	−	0.691387i	$-0.757000\pi$
$719$	−2.00300e7	−1.44497	−0.722485	+	0.691387i	$0.757000\pi$
$720$	0	0
$721$	−2.02983e6	−0.145419
$722$	1.70527e7i	1.21745i
$723$	− 4.53665e6i	− 0.322767i
$724$	−5.22361e6	−0.370361
$725$	0	0
$726$	−1.12577e6	−0.0792700
$727$	− 1.89691e7i	− 1.33110i	−0.746354	−	0.665549i	$-0.768197\pi$
$727$	− 1.89691e7i	− 1.33110i	0.746354	−	0.665549i	$-0.231803\pi$
$728$	3.54805e6i	0.248120i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.67003e7	1.15593
$732$	− 4.50702e6i	− 0.310894i
$733$	1.77489e7i	1.22014i	0.792347	+	0.610071i	$0.208859\pi$
$733$	1.77489e7i	1.22014i	−0.792347	+	0.610071i	$0.791141\pi$
$734$	−8.24764e6	−0.565054
$735$	0	0
$736$	−2.99826e6	−0.204021
$737$	1.44089e7i	0.977155i
$738$	− 2.62904e6i	− 0.177687i
$739$	2.96257e7	1.99553	0.997763	−	0.0668485i	$-0.0212944\pi$
$739$	2.96257e7	1.99553	0.997763	+	0.0668485i	$0.0212944\pi$
$740$	0	0
$741$	−1.44039e7	−0.963687
$742$	896592.i	0.0597840i
$743$	− 2.90044e7i	− 1.92749i	−0.266829	−	0.963744i	$-0.585976\pi$
$743$	− 2.90044e7i	− 1.92749i	0.266829	−	0.963744i	$-0.414024\pi$
$744$	−1.03446e6	−0.0685141
$745$	0	0
$746$	1.61246e7	1.06082
$747$	− 8.32433e6i	− 0.545818i
$748$	7.31598e6i	0.478100i
$749$	−1.01262e7	−0.659544
$750$	0	0
$751$	5.78238e6	0.374116	0.187058	−	0.982349i	$-0.440105\pi$
$751$	5.78238e6	0.374116	0.187058	+	0.982349i	$0.440105\pi$
$752$	− 1.86620e6i	− 0.120341i
$753$	− 2.43948e6i	− 0.156787i
$754$	9.49487e6	0.608220
$755$	0	0
$756$	−1.04888e6	−0.0667453
$757$	− 1.38814e6i	− 0.0880427i	−0.999031	−	0.0440213i	$-0.985983\pi$
$757$	− 1.38814e6i	− 0.0880427i	0.999031	−	0.0440213i	$-0.0140170\pi$
$758$	1.32609e7i	0.838299i
$759$	−9.49325e6	−0.598151
$760$	0	0
$761$	2.91152e7	1.82246	0.911231	−	0.411896i	$-0.135133\pi$
$761$	2.91152e7	1.82246	0.911231	+	0.411896i	$0.135133\pi$
$762$	− 9.91747e6i	− 0.618747i
$763$	− 698289.i	− 0.0434234i
$764$	3.64457e6	0.225899
$765$	0	0
$766$	−9.41960e6	−0.580044
$767$	− 3.83474e6i	− 0.235368i
$768$	− 589824.i	− 0.0360844i
$769$	−9.44170e6	−0.575751	−0.287875	−	0.957668i	$-0.592949\pi$
$769$	−9.44170e6	−0.575751	−0.287875	+	0.957668i	$0.592949\pi$
$770$	0	0
$771$	−7.08085e6	−0.428992
$772$	− 1.73033e6i	− 0.104493i
$773$	− 2.30054e6i	− 0.138478i	−0.997600	−	0.0692392i	$-0.977943\pi$
$773$	− 2.30054e6i	− 0.138478i	0.997600	−	0.0692392i	$-0.0220572\pi$
$774$	−4.26306e6	−0.255781
$775$	0	0
$776$	7.13303e6	0.425226
$777$	5.50175e6i	0.326925i
$778$	3.34921e6i	0.198378i
$779$	2.10648e7	1.24370
$780$	0	0
$781$	1.92960e7	1.13198
$782$	− 1.48655e7i	− 0.869284i
$783$	2.80688e6i	0.163614i
$784$	2.23247e6	0.129717
$785$	0	0
$786$	−7.32226e6	−0.422755
$787$	1.31660e7i	0.757733i	0.925451	+	0.378867i	$0.123686\pi$
$787$	1.31660e7i	0.757733i	−0.925451	+	0.378867i	$0.876314\pi$
$788$	1.26529e7i	0.725897i
$789$	−250727.	−0.0143386
$790$	0	0
$791$	−1.19443e7	−0.678768
$792$	− 1.86753e6i	− 0.105793i
$793$	1.92957e7i	1.08962i
$794$	1.40997e7	0.793703
$795$	0	0
$796$	6.43683e6	0.360072
$797$	− 2.79799e7i	− 1.56027i	−0.625610	−	0.780136i	$-0.715150\pi$
$797$	− 2.79799e7i	− 1.56027i	0.625610	−	0.780136i	$-0.284850\pi$
$798$	− 8.40401e6i	− 0.467174i
$799$	9.25266e6	0.512743
$800$	0	0
$801$	1.14739e7	0.631871
$802$	− 1.05867e7i	− 0.581201i
$803$	4.30513e6i	0.235612i
$804$	5.75959e6	0.314233
$805$	0	0
$806$	4.42877e6	0.240129
$807$	− 2.00558e6i	− 0.108407i
$808$	5.21348e6i	0.280931i
$809$	2.01057e7	1.08006	0.540030	−	0.841646i	$-0.318413\pi$
$809$	2.01057e7	1.08006	0.540030	+	0.841646i	$0.318413\pi$
$810$	0	0
$811$	5.69966e6	0.304296	0.152148	−	0.988358i	$-0.451381\pi$
$811$	5.69966e6	0.304296	0.152148	+	0.988358i	$0.451381\pi$
$812$	5.53980e6i	0.294852i
$813$	− 1.81572e7i	− 0.963438i
$814$	−9.79590e6	−0.518183
$815$	0	0
$816$	2.92437e6	0.153747
$817$	− 3.41572e7i	− 1.79031i
$818$	− 3.47376e6i	− 0.181517i
$819$	4.49050e6	0.233930
$820$	0	0
$821$	−2.40117e7	−1.24327	−0.621635	−	0.783307i	$-0.713531\pi$
$821$	−2.40117e7	−1.24327	−0.621635	+	0.783307i	$0.713531\pi$
$822$	− 7.65061e6i	− 0.394927i
$823$	1.83366e7i	0.943669i	0.881687	+	0.471835i	$0.156408\pi$
$823$	1.83366e7i	0.943669i	−0.881687	+	0.471835i	$0.843592\pi$
$824$	−1.44465e6	−0.0741217
$825$	0	0
$826$	2.23739e6	0.114101
$827$	9.81878e6i	0.499222i	0.968346	+	0.249611i	$0.0803027\pi$
$827$	9.81878e6i	0.499222i	−0.968346	+	0.249611i	$0.919697\pi$
$828$	3.79467e6i	0.192353i
$829$	−5.89137e6	−0.297735	−0.148868	−	0.988857i	$-0.547563\pi$
$829$	−5.89137e6	−0.297735	−0.148868	+	0.988857i	$0.547563\pi$
$830$	0	0
$831$	−9.51669e6	−0.478061
$832$	2.52518e6i	0.126469i
$833$	1.10687e7i	0.552692i
$834$	−3.46961e6	−0.172729
$835$	0	0
$836$	1.49634e7	0.740481
$837$	1.30924e6i	0.0645958i
$838$	− 8.12829e6i	− 0.399843i
$839$	−1.23856e7	−0.607450	−0.303725	−	0.952760i	$-0.598230\pi$
$839$	−1.23856e7	−0.607450	−0.303725	+	0.952760i	$0.598230\pi$
$840$	0	0
$841$	−5.68622e6	−0.277226
$842$	− 1.21237e7i	− 0.589327i
$843$	− 1.51456e7i	− 0.734039i
$844$	1.97175e7	0.952786
$845$	0	0
$846$	−2.36190e6	−0.113458
$847$	2.81207e6i	0.134684i
$848$	638112.i	0.0304725i
$849$	−1.80326e7	−0.858597
$850$	0	0
$851$	1.99045e7	0.942164
$852$	− 7.71307e6i	− 0.364023i
$853$	1.59588e7i	0.750981i	0.926826	+	0.375490i	$0.122526\pi$
$853$	1.59588e7i	0.750981i	−0.926826	+	0.375490i	$0.877474\pi$
$854$	−1.12581e7	−0.528226
$855$	0	0
$856$	−7.20694e6	−0.336176
$857$	− 1.31794e7i	− 0.612976i	−0.951875	−	0.306488i	$-0.900846\pi$
$857$	− 1.31794e7i	− 0.612976i	0.951875	−	0.306488i	$-0.0991539\pi$
$858$	7.99536e6i	0.370783i
$859$	−1.79181e7	−0.828532	−0.414266	−	0.910156i	$-0.635962\pi$
$859$	−1.79181e7	−0.828532	−0.414266	+	0.910156i	$0.635962\pi$
$860$	0	0
$861$	−6.56707e6	−0.301901
$862$	2.19023e7i	1.00397i
$863$	1.13619e6i	0.0519308i	0.999663	+	0.0259654i	$0.00826598\pi$
$863$	1.13619e6i	0.0519308i	−0.999663	+	0.0259654i	$0.991734\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	−4.90959e6	−0.222459
$867$	1.72038e6i	0.0777279i
$868$	2.58397e6i	0.116409i
$869$	−3.02481e7	−1.35878
$870$	0	0
$871$	−2.46582e7	−1.10133
$872$	− 496979.i	− 0.0221333i
$873$	− 9.02774e6i	− 0.400907i
$874$	−3.04043e7	−1.34635
$875$	0	0
$876$	1.72086e6	0.0757680
$877$	− 2.53434e7i	− 1.11267i	−0.830958	−	0.556335i	$-0.812207\pi$
$877$	− 2.53434e7i	− 1.11267i	0.830958	−	0.556335i	$-0.187793\pi$
$878$	− 4.33475e6i	− 0.189770i
$879$	2.04359e6	0.0892115
$880$	0	0
$881$	4.26012e7	1.84919	0.924597	−	0.380946i	$-0.124402\pi$
$881$	4.26012e7	1.84919	0.924597	+	0.380946i	$0.124402\pi$
$882$	− 2.82548e6i	− 0.122298i
$883$	3.30386e7i	1.42600i	0.701164	+	0.713000i	$0.252664\pi$
$883$	3.30386e7i	1.42600i	−0.701164	+	0.713000i	$0.747336\pi$
$884$	−1.25199e7	−0.538854
$885$	0	0
$886$	−1.69469e7	−0.725282
$887$	− 2.00654e7i	− 0.856323i	−0.903702	−	0.428162i	$-0.859161\pi$
$887$	− 2.00654e7i	− 0.856323i	0.903702	−	0.428162i	$-0.140839\pi$
$888$	3.91565e6i	0.166637i
$889$	−2.47728e7	−1.05129
$890$	0	0
$891$	−2.36360e6	−0.0997422
$892$	− 1.93579e7i	− 0.814603i
$893$	− 1.89245e7i	− 0.794136i
$894$	−4.61855e6	−0.193269
$895$	0	0
$896$	−1.47332e6	−0.0613095
$897$	− 1.62459e7i	− 0.674160i
$898$	− 2.25400e7i	− 0.932744i
$899$	6.91491e6	0.285356
$900$	0	0
$901$	−3.16378e6	−0.129836
$902$	− 1.16927e7i	− 0.478518i
$903$	1.06487e7i	0.434587i
$904$	−8.50090e6	−0.345974
$905$	0	0
$906$	9.15721e6	0.370632
$907$	3.74228e6i	0.151049i	0.997144	+	0.0755245i	$0.0240631\pi$
$907$	3.74228e6i	0.151049i	−0.997144	+	0.0755245i	$0.975937\pi$
$908$	− 2.98843e6i	− 0.120290i
$909$	6.59831e6	0.264864
$910$	0	0
$911$	−7.58815e6	−0.302928	−0.151464	−	0.988463i	$-0.548399\pi$
$911$	−7.58815e6	−0.302928	−0.151464	+	0.988463i	$0.548399\pi$
$912$	− 5.98121e6i	− 0.238123i
$913$	− 3.70226e7i	− 1.46991i
$914$	1.30643e7	0.517273
$915$	0	0
$916$	−4.88101e6	−0.192208
$917$	1.82903e7i	0.718285i
$918$	− 3.70115e6i	− 0.144954i
$919$	3.71165e7	1.44970	0.724850	−	0.688906i	$-0.241909\pi$
$919$	3.71165e7	1.44970	0.724850	+	0.688906i	$0.241909\pi$
$920$	0	0
$921$	−1.92127e7	−0.746346
$922$	1.82430e7i	0.706755i
$923$	3.30216e7i	1.27583i
$924$	−4.66491e6	−0.179748
$925$	0	0
$926$	−1.64842e7	−0.631742
$927$	1.82839e6i	0.0698826i
$928$	3.94272e6i	0.150289i
$929$	2.95329e7	1.12271	0.561354	−	0.827576i	$-0.310281\pi$
$929$	2.95329e7	1.12271	0.561354	+	0.827576i	$0.310281\pi$
$930$	0	0
$931$	2.26388e7	0.856010
$932$	− 1.09794e7i	− 0.414038i
$933$	− 1.95022e7i	− 0.733464i
$934$	−3.53002e7	−1.32407
$935$	0	0
$936$	3.19593e6	0.119236
$937$	− 4.05813e7i	− 1.51000i	−0.655724	−	0.755001i	$-0.727636\pi$
$937$	− 4.05813e7i	− 1.51000i	0.655724	−	0.755001i	$-0.272364\pi$
$938$	− 1.43869e7i	− 0.533899i
$939$	3.29268e6	0.121867
$940$	0	0
$941$	1.21780e7	0.448334	0.224167	−	0.974551i	$-0.428034\pi$
$941$	1.21780e7	0.448334	0.224167	+	0.974551i	$0.428034\pi$
$942$	− 1.07497e7i	− 0.394704i
$943$	2.37586e7i	0.870045i
$944$	1.59237e6	0.0581586
$945$	0	0
$946$	−1.89600e7	−0.688829
$947$	9.77356e6i	0.354142i	0.984198	+	0.177071i	$0.0566623\pi$
$947$	9.77356e6i	0.354142i	−0.984198	+	0.177071i	$0.943338\pi$
$948$	1.20909e7i	0.436955i
$949$	−7.36743e6	−0.265553
$950$	0	0
$951$	−3.15576e7	−1.13150
$952$	− 7.30477e6i	− 0.261225i
$953$	3.92931e7i	1.40147i	0.713422	+	0.700735i	$0.247144\pi$
$953$	3.92931e7i	1.40147i	−0.713422	+	0.700735i	$0.752856\pi$
$954$	807611.	0.0287297
$955$	0	0
$956$	2.52694e7	0.894230
$957$	1.24837e7i	0.440618i
$958$	− 7.08572e6i	− 0.249442i
$959$	−1.91105e7	−0.671003
$960$	0	0
$961$	−2.54038e7	−0.887339
$962$	− 1.67638e7i	− 0.584031i
$963$	9.12129e6i	0.316950i
$964$	−8.06516e6	−0.279525
$965$	0	0
$966$	9.47871e6	0.326818
$967$	− 2.17823e7i	− 0.749097i	−0.927207	−	0.374549i	$-0.877798\pi$
$967$	− 2.17823e7i	− 0.749097i	0.927207	−	0.374549i	$-0.122202\pi$
$968$	2.00137e6i	0.0686498i
$969$	2.96550e7	1.01459
$970$	0	0
$971$	−2.78603e7	−0.948281	−0.474141	−	0.880449i	$-0.657241\pi$
$971$	−2.78603e7	−0.948281	−0.474141	+	0.880449i	$0.657241\pi$
$972$	944784.i	0.0320750i
$973$	8.66673e6i	0.293476i
$974$	−1.61467e7	−0.545365
$975$	0	0
$976$	−8.01248e6	−0.269242
$977$	− 3.68504e7i	− 1.23511i	−0.786527	−	0.617555i	$-0.788123\pi$
$977$	− 3.68504e7i	− 1.23511i	0.786527	−	0.617555i	$-0.211877\pi$
$978$	− 1.59464e7i	− 0.533109i
$979$	5.10303e7	1.70165
$980$	0	0
$981$	−628988.	−0.0208675
$982$	1.72203e7i	0.569852i
$983$	− 2.83162e7i	− 0.934655i	−0.884084	−	0.467327i	$-0.845217\pi$
$983$	− 2.83162e7i	− 0.934655i	0.884084	−	0.467327i	$-0.154783\pi$
$984$	−4.67385e6	−0.153882
$985$	0	0
$986$	−1.95481e7	−0.640344
$987$	5.89980e6i	0.192772i
$988$	2.56070e7i	0.834577i
$989$	3.85253e7	1.25243
$990$	0	0
$991$	−5.14027e7	−1.66265	−0.831327	−	0.555784i	$-0.812418\pi$
$991$	−5.14027e7	−1.66265	−0.831327	+	0.555784i	$0.812418\pi$
$992$	1.83904e6i	0.0593350i
$993$	1.73740e7i	0.559148i
$994$	−1.92665e7	−0.618495
$995$	0	0
$996$	−1.47988e7	−0.472692
$997$	− 4.26816e7i	− 1.35989i	−0.733265	−	0.679943i	$-0.762004\pi$
$997$	− 4.26816e7i	− 1.35989i	0.733265	−	0.679943i	$-0.237996\pi$
$998$	− 1.18510e7i	− 0.376642i
$999$	4.95574e6	0.157107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.c.499.6		8
5.2	odd	4		750.6.a.a.1.3	✓	4
5.3	odd	4		750.6.a.h.1.2	yes	4
5.4	even	2	inner	750.6.c.c.499.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.a.1.3	✓	4	5.2	odd	4
750.6.a.h.1.2	yes	4	5.3	odd	4
750.6.c.c.499.3		8	5.4	even	2	inner
750.6.c.c.499.6		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -89.9244i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -89.9244i q^{7} -64.0000i q^{8} -81.0000 q^{9} -360.249 q^{11} +144.000i q^{12} -616.499i q^{13} +359.698 q^{14} +256.000 q^{16} +1269.26i q^{17} -324.000i q^{18} +2596.01 q^{19} -809.320 q^{21} -1441.00i q^{22} +2927.99i q^{23} -576.000 q^{24} +2466.00 q^{26} +729.000i q^{27} +1438.79i q^{28} +3850.31 q^{29} +1795.93 q^{31} +1024.00i q^{32} +3242.24i q^{33} -5077.02 q^{34} +1296.00 q^{36} -6798.00i q^{37} +10384.0i q^{38} -5548.49 q^{39} +8114.32 q^{41} -3237.28i q^{42} -13157.6i q^{43} +5763.99 q^{44} -11712.0 q^{46} -7289.83i q^{47} -2304.00i q^{48} +8720.60 q^{49} +11423.3 q^{51} +9863.99i q^{52} +2492.63i q^{53} -2916.00 q^{54} -5755.16 q^{56} -23364.1i q^{57} +15401.3i q^{58} +6220.19 q^{59} -31298.8 q^{61} +7183.73i q^{62} +7283.88i q^{63} -4096.00 q^{64} -12969.0 q^{66} -39997.1i q^{67} -20308.1i q^{68} +26351.9 q^{69} -53563.0 q^{71} +5184.00i q^{72} -11950.4i q^{73} +27192.0 q^{74} -41536.2 q^{76} +32395.2i q^{77} -22194.0i q^{78} +83964.3 q^{79} +6561.00 q^{81} +32457.3i q^{82} +102769. i q^{83} +12949.1 q^{84} +52630.4 q^{86} -34652.8i q^{87} +23056.0i q^{88} -141653. q^{89} -55438.3 q^{91} -46847.8i q^{92} -16163.4i q^{93} +29159.3 q^{94} +9216.00 q^{96} +111454. i q^{97} +34882.4i q^{98} +29180.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) \) 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

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