LMFDB - Embedded newform 350.6.c.e.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.e.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +3.00000i q^{3} -16.0000 q^{4} +12.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +234.000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +3.00000i q^{3} -16.0000 q^{4} +12.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +234.000 q^{9} +405.000 q^{11} -48.0000i q^{12} +391.000i q^{13} +196.000 q^{14} +256.000 q^{16} +999.000i q^{17} -936.000i q^{18} -2342.00 q^{19} -147.000 q^{21} -1620.00i q^{22} -2430.00i q^{23} -192.000 q^{24} +1564.00 q^{26} +1431.00i q^{27} -784.000i q^{28} -8259.00 q^{29} +4016.00 q^{31} -1024.00i q^{32} +1215.00i q^{33} +3996.00 q^{34} -3744.00 q^{36} -7042.00i q^{37} +9368.00i q^{38} -1173.00 q^{39} +3336.00 q^{41} +588.000i q^{42} +23518.0i q^{43} -6480.00 q^{44} -9720.00 q^{46} +10317.0i q^{47} +768.000i q^{48} -2401.00 q^{49} -2997.00 q^{51} -6256.00i q^{52} -3084.00i q^{53} +5724.00 q^{54} -3136.00 q^{56} -7026.00i q^{57} +33036.0i q^{58} +18816.0 q^{59} +21668.0 q^{61} -16064.0i q^{62} +11466.0i q^{63} -4096.00 q^{64} +4860.00 q^{66} +52124.0i q^{67} -15984.0i q^{68} +7290.00 q^{69} -28560.0 q^{71} +14976.0i q^{72} +70342.0i q^{73} -28168.0 q^{74} +37472.0 q^{76} +19845.0i q^{77} +4692.00i q^{78} -58823.0 q^{79} +52569.0 q^{81} -13344.0i q^{82} -756.000i q^{83} +2352.00 q^{84} +94072.0 q^{86} -24777.0i q^{87} +25920.0i q^{88} -135384. q^{89} -19159.0 q^{91} +38880.0i q^{92} +12048.0i q^{93} +41268.0 q^{94} +3072.00 q^{96} +110435. i q^{97} +9604.00i q^{98} +94770.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 24 q^{6} + 468 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 24 * q^6 + 468 * q^9	$ 2 q - 32 q^{4} + 24 q^{6} + 468 q^{9} + 810 q^{11} + 392 q^{14} + 512 q^{16} - 4684 q^{19} - 294 q^{21} - 384 q^{24} + 3128 q^{26} - 16518 q^{29} + 8032 q^{31} + 7992 q^{34} - 7488 q^{36} - 2346 q^{39} + 6672 q^{41} - 12960 q^{44} - 19440 q^{46} - 4802 q^{49} - 5994 q^{51} + 11448 q^{54} - 6272 q^{56} + 37632 q^{59} + 43336 q^{61} - 8192 q^{64} + 9720 q^{66} + 14580 q^{69} - 57120 q^{71} - 56336 q^{74} + 74944 q^{76} - 117646 q^{79} + 105138 q^{81} + 4704 q^{84} + 188144 q^{86} - 270768 q^{89} - 38318 q^{91} + 82536 q^{94} + 6144 q^{96} + 189540 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 24 * q^6 + 468 * q^9 + 810 * q^11 + 392 * q^14 + 512 * q^16 - 4684 * q^19 - 294 * q^21 - 384 * q^24 + 3128 * q^26 - 16518 * q^29 + 8032 * q^31 + 7992 * q^34 - 7488 * q^36 - 2346 * q^39 + 6672 * q^41 - 12960 * q^44 - 19440 * q^46 - 4802 * q^49 - 5994 * q^51 + 11448 * q^54 - 6272 * q^56 + 37632 * q^59 + 43336 * q^61 - 8192 * q^64 + 9720 * q^66 + 14580 * q^69 - 57120 * q^71 - 56336 * q^74 + 74944 * q^76 - 117646 * q^79 + 105138 * q^81 + 4704 * q^84 + 188144 * q^86 - 270768 * q^89 - 38318 * q^91 + 82536 * q^94 + 6144 * q^96 + 189540 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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
$3$	3.00000i	0.192450i	0.995360	+	0.0962250i	$0.0306768\pi$
$3$	3.00000i	0.192450i	−0.995360	+	0.0962250i	$0.969323\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	12.0000	0.136083
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	64.0000i	0.353553i
$9$	234.000	0.962963
$10$	0	0
$11$	405.000	1.00919	0.504595	−	0.863356i	$-0.331642\pi$
$11$	405.000	1.00919	0.504595	+	0.863356i	$0.331642\pi$
$12$	− 48.0000i	− 0.0962250i
$13$	391.000i	0.641680i	0.947133	+	0.320840i	$0.103965\pi$
$13$	391.000i	0.641680i	−0.947133	+	0.320840i	$0.896035\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	999.000i	0.838384i	0.907898	+	0.419192i	$0.137687\pi$
$17$	999.000i	0.838384i	−0.907898	+	0.419192i	$0.862313\pi$
$18$	− 936.000i	− 0.680918i
$19$	−2342.00	−1.48834	−0.744171	−	0.667989i	$-0.767155\pi$
$19$	−2342.00	−1.48834	−0.744171	+	0.667989i	$0.767155\pi$
$20$	0	0
$21$	−147.000	−0.0727393
$22$	− 1620.00i	− 0.713606i
$23$	− 2430.00i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$23$	− 2430.00i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$24$	−192.000	−0.0680414
$25$	0	0
$26$	1564.00	0.453736
$27$	1431.00i	0.377772i
$28$	− 784.000i	− 0.188982i
$29$	−8259.00	−1.82361	−0.911806	−	0.410621i	$-0.865312\pi$
$29$	−8259.00	−1.82361	−0.911806	+	0.410621i	$0.865312\pi$
$30$	0	0
$31$	4016.00	0.750567	0.375284	−	0.926910i	$-0.377545\pi$
$31$	4016.00	0.750567	0.375284	+	0.926910i	$0.377545\pi$
$32$	− 1024.00i	− 0.176777i
$33$	1215.00i	0.194219i
$34$	3996.00	0.592827
$35$	0	0
$36$	−3744.00	−0.481481
$37$	− 7042.00i	− 0.845652i	−0.906211	−	0.422826i	$-0.861038\pi$
$37$	− 7042.00i	− 0.845652i	0.906211	−	0.422826i	$-0.138962\pi$
$38$	9368.00i	1.05242i
$39$	−1173.00	−0.123491
$40$	0	0
$41$	3336.00	0.309932	0.154966	−	0.987920i	$-0.450473\pi$
$41$	3336.00	0.309932	0.154966	+	0.987920i	$0.450473\pi$
$42$	588.000i	0.0514344i
$43$	23518.0i	1.93968i	0.243750	+	0.969838i	$0.421622\pi$
$43$	23518.0i	1.93968i	−0.243750	+	0.969838i	$0.578378\pi$
$44$	−6480.00	−0.504595
$45$	0	0
$46$	−9720.00	−0.677285
$47$	10317.0i	0.681254i	0.940199	+	0.340627i	$0.110639\pi$
$47$	10317.0i	0.681254i	−0.940199	+	0.340627i	$0.889361\pi$
$48$	768.000i	0.0481125i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−2997.00	−0.161347
$52$	− 6256.00i	− 0.320840i
$53$	− 3084.00i	− 0.150808i	−0.997153	−	0.0754041i	$-0.975975\pi$
$53$	− 3084.00i	− 0.150808i	0.997153	−	0.0754041i	$-0.0240247\pi$
$54$	5724.00	0.267125
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 7026.00i	− 0.286432i
$58$	33036.0i	1.28949i
$59$	18816.0	0.703716	0.351858	−	0.936053i	$-0.385550\pi$
$59$	18816.0	0.703716	0.351858	+	0.936053i	$0.385550\pi$
$60$	0	0
$61$	21668.0	0.745580	0.372790	−	0.927916i	$-0.378401\pi$
$61$	21668.0	0.745580	0.372790	+	0.927916i	$0.378401\pi$
$62$	− 16064.0i	− 0.530731i
$63$	11466.0i	0.363966i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	4860.00	0.137333
$67$	52124.0i	1.41857i	0.704922	+	0.709285i	$0.250982\pi$
$67$	52124.0i	1.41857i	−0.704922	+	0.709285i	$0.749018\pi$
$68$	− 15984.0i	− 0.419192i
$69$	7290.00	0.184334
$70$	0	0
$71$	−28560.0	−0.672376	−0.336188	−	0.941795i	$-0.609138\pi$
$71$	−28560.0	−0.672376	−0.336188	+	0.941795i	$0.609138\pi$
$72$	14976.0i	0.340459i
$73$	70342.0i	1.54493i	0.635060	+	0.772463i	$0.280975\pi$
$73$	70342.0i	1.54493i	−0.635060	+	0.772463i	$0.719025\pi$
$74$	−28168.0	−0.597966
$75$	0	0
$76$	37472.0	0.744171
$77$	19845.0i	0.381438i
$78$	4692.00i	0.0873216i
$79$	−58823.0	−1.06042	−0.530212	−	0.847865i	$-0.677888\pi$
$79$	−58823.0	−1.06042	−0.530212	+	0.847865i	$0.677888\pi$
$80$	0	0
$81$	52569.0	0.890261
$82$	− 13344.0i	− 0.219155i
$83$	− 756.000i	− 0.0120455i	−0.999982	−	0.00602277i	$-0.998083\pi$
$83$	− 756.000i	− 0.0120455i	0.999982	−	0.00602277i	$-0.00191712\pi$
$84$	2352.00	0.0363696
$85$	0	0
$86$	94072.0	1.37156
$87$	− 24777.0i	− 0.350954i
$88$	25920.0i	0.356803i
$89$	−135384.	−1.81173	−0.905863	−	0.423572i	$-0.860776\pi$
$89$	−135384.	−1.81173	−0.905863	+	0.423572i	$0.860776\pi$
$90$	0	0
$91$	−19159.0	−0.242532
$92$	38880.0i	0.478913i
$93$	12048.0i	0.144447i
$94$	41268.0	0.481719
$95$	0	0
$96$	3072.00	0.0340207
$97$	110435.i	1.19173i	0.803085	+	0.595864i	$0.203190\pi$
$97$	110435.i	1.19173i	−0.803085	+	0.595864i	$0.796810\pi$
$98$	9604.00i	0.101015i
$99$	94770.0	0.971813
$100$	0	0
$101$	33450.0	0.326282	0.163141	−	0.986603i	$-0.447838\pi$
$101$	33450.0	0.326282	0.163141	+	0.986603i	$0.447838\pi$
$102$	11988.0i	0.114090i
$103$	110311.i	1.02453i	0.858827	+	0.512266i	$0.171194\pi$
$103$	110311.i	1.02453i	−0.858827	+	0.512266i	$0.828806\pi$
$104$	−25024.0	−0.226868
$105$	0	0
$106$	−12336.0	−0.106637
$107$	35358.0i	0.298558i	0.988795	+	0.149279i	$0.0476952\pi$
$107$	35358.0i	0.298558i	−0.988795	+	0.149279i	$0.952305\pi$
$108$	− 22896.0i	− 0.188886i
$109$	151183.	1.21881	0.609406	−	0.792858i	$-0.291408\pi$
$109$	151183.	1.21881	0.609406	+	0.792858i	$0.291408\pi$
$110$	0	0
$111$	21126.0	0.162746
$112$	12544.0i	0.0944911i
$113$	133686.i	0.984895i	0.870342	+	0.492447i	$0.163898\pi$
$113$	133686.i	0.984895i	−0.870342	+	0.492447i	$0.836102\pi$
$114$	−28104.0	−0.202538
$115$	0	0
$116$	132144.	0.911806
$117$	91494.0i	0.617914i
$118$	− 75264.0i	− 0.497602i
$119$	−48951.0	−0.316880
$120$	0	0
$121$	2974.00	0.0184662
$122$	− 86672.0i	− 0.527205i
$123$	10008.0i	0.0596464i
$124$	−64256.0	−0.375284
$125$	0	0
$126$	45864.0	0.257363
$127$	− 283984.i	− 1.56237i	−0.624298	−	0.781186i	$-0.714615\pi$
$127$	− 283984.i	− 1.56237i	0.624298	−	0.781186i	$-0.285385\pi$
$128$	16384.0i	0.0883883i
$129$	−70554.0	−0.373291
$130$	0	0
$131$	261438.	1.33104	0.665519	−	0.746381i	$-0.268210\pi$
$131$	261438.	1.33104	0.665519	+	0.746381i	$0.268210\pi$
$132$	− 19440.0i	− 0.0971094i
$133$	− 114758.i	− 0.562541i
$134$	208496.	1.00308
$135$	0	0
$136$	−63936.0	−0.296414
$137$	39672.0i	0.180585i	0.995915	+	0.0902927i	$0.0287803\pi$
$137$	39672.0i	0.180585i	−0.995915	+	0.0902927i	$0.971220\pi$
$138$	− 29160.0i	− 0.130344i
$139$	182626.	0.801725	0.400863	−	0.916138i	$-0.368710\pi$
$139$	182626.	0.801725	0.400863	+	0.916138i	$0.368710\pi$
$140$	0	0
$141$	−30951.0	−0.131107
$142$	114240.i	0.475442i
$143$	158355.i	0.647577i
$144$	59904.0	0.240741
$145$	0	0
$146$	281368.	1.09243
$147$	− 7203.00i	− 0.0274929i
$148$	112672.i	0.422826i
$149$	12078.0	0.0445686	0.0222843	−	0.999752i	$-0.492906\pi$
$149$	12078.0	0.0445686	0.0222843	+	0.999752i	$0.492906\pi$
$150$	0	0
$151$	−208417.	−0.743859	−0.371930	−	0.928261i	$-0.621304\pi$
$151$	−208417.	−0.743859	−0.371930	+	0.928261i	$0.621304\pi$
$152$	− 149888.i	− 0.526209i
$153$	233766.i	0.807333i
$154$	79380.0	0.269718
$155$	0	0
$156$	18768.0	0.0617457
$157$	364094.i	1.17887i	0.807817	+	0.589433i	$0.200649\pi$
$157$	364094.i	1.17887i	−0.807817	+	0.589433i	$0.799351\pi$
$158$	235292.i	0.749833i
$159$	9252.00	0.0290230
$160$	0	0
$161$	119070.	0.362024
$162$	− 210276.i	− 0.629509i
$163$	− 626.000i	− 0.00184546i	−1.00000		0.000922731i	$-0.999706\pi$
$163$	− 626.000i	− 0.00184546i	1.00000		0.000922731i	$-0.000293715\pi$
$164$	−53376.0	−0.154966
$165$	0	0
$166$	−3024.00	−0.00851749
$167$	445617.i	1.23643i	0.786008	+	0.618216i	$0.212144\pi$
$167$	445617.i	1.23643i	−0.786008	+	0.618216i	$0.787856\pi$
$168$	− 9408.00i	− 0.0257172i
$169$	218412.	0.588247
$170$	0	0
$171$	−548028.	−1.43322
$172$	− 376288.i	− 0.969838i
$173$	− 643467.i	− 1.63460i	−0.576214	−	0.817299i	$-0.695470\pi$
$173$	− 643467.i	− 1.63460i	0.576214	−	0.817299i	$-0.304530\pi$
$174$	−99108.0	−0.248162
$175$	0	0
$176$	103680.	0.252298
$177$	56448.0i	0.135430i
$178$	541536.i	1.28108i
$179$	−245148.	−0.571868	−0.285934	−	0.958249i	$-0.592304\pi$
$179$	−245148.	−0.571868	−0.285934	+	0.958249i	$0.592304\pi$
$180$	0	0
$181$	686180.	1.55683	0.778416	−	0.627749i	$-0.216024\pi$
$181$	686180.	1.55683	0.778416	+	0.627749i	$0.216024\pi$
$182$	76636.0i	0.171496i
$183$	65004.0i	0.143487i
$184$	155520.	0.338643
$185$	0	0
$186$	48192.0	0.102139
$187$	404595.i	0.846090i
$188$	− 165072.i	− 0.340627i
$189$	−70119.0	−0.142785
$190$	0	0
$191$	−527031.	−1.04533	−0.522664	−	0.852539i	$-0.675062\pi$
$191$	−527031.	−1.04533	−0.522664	+	0.852539i	$0.675062\pi$
$192$	− 12288.0i	− 0.0240563i
$193$	− 143216.i	− 0.276757i	−0.990379	−	0.138378i	$-0.955811\pi$
$193$	− 143216.i	− 0.276757i	0.990379	−	0.138378i	$-0.0441890\pi$
$194$	441740.	0.842679
$195$	0	0
$196$	38416.0	0.0714286
$197$	348468.i	0.639731i	0.947463	+	0.319865i	$0.103638\pi$
$197$	348468.i	0.639731i	−0.947463	+	0.319865i	$0.896362\pi$
$198$	− 379080.i	− 0.687176i
$199$	−754520.	−1.35064	−0.675318	−	0.737527i	$-0.735993\pi$
$199$	−754520.	−1.35064	−0.675318	+	0.737527i	$0.735993\pi$
$200$	0	0
$201$	−156372.	−0.273004
$202$	− 133800.i	− 0.230716i
$203$	− 404691.i	− 0.689261i
$204$	47952.0	0.0806736
$205$	0	0
$206$	441244.	0.724454
$207$	− 568620.i	− 0.922351i
$208$	100096.i	0.160420i
$209$	−948510.	−1.50202
$210$	0	0
$211$	−590749.	−0.913475	−0.456738	−	0.889601i	$-0.650982\pi$
$211$	−590749.	−0.913475	−0.456738	+	0.889601i	$0.650982\pi$
$212$	49344.0i	0.0754041i
$213$	− 85680.0i	− 0.129399i
$214$	141432.	0.211112
$215$	0	0
$216$	−91584.0	−0.133563
$217$	196784.i	0.283688i
$218$	− 604732.i	− 0.861830i
$219$	−211026.	−0.297321
$220$	0	0
$221$	−390609.	−0.537974
$222$	− 84504.0i	− 0.115079i
$223$	396103.i	0.533391i	0.963781	+	0.266696i	$0.0859319\pi$
$223$	396103.i	0.533391i	−0.963781	+	0.266696i	$0.914068\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	534744.	0.696426
$227$	9537.00i	0.0122842i	0.999981	+	0.00614210i	$0.00195510\pi$
$227$	9537.00i	0.0122842i	−0.999981	+	0.00614210i	$0.998045\pi$
$228$	112416.i	0.143216i
$229$	−705056.	−0.888454	−0.444227	−	0.895914i	$-0.646522\pi$
$229$	−705056.	−0.888454	−0.444227	+	0.895914i	$0.646522\pi$
$230$	0	0
$231$	−59535.0	−0.0734078
$232$	− 528576.i	− 0.644744i
$233$	− 534216.i	− 0.644655i	−0.946628	−	0.322327i	$-0.895535\pi$
$233$	− 534216.i	− 0.644655i	0.946628	−	0.322327i	$-0.104465\pi$
$234$	365976.	0.436931
$235$	0	0
$236$	−301056.	−0.351858
$237$	− 176469.i	− 0.204079i
$238$	195804.i	0.224068i
$239$	901221.	1.02056	0.510278	−	0.860010i	$-0.329543\pi$
$239$	901221.	1.02056	0.510278	+	0.860010i	$0.329543\pi$
$240$	0	0
$241$	−952390.	−1.05626	−0.528132	−	0.849162i	$-0.677107\pi$
$241$	−952390.	−1.05626	−0.528132	+	0.849162i	$0.677107\pi$
$242$	− 11896.0i	− 0.0130576i
$243$	505440.i	0.549103i
$244$	−346688.	−0.372790
$245$	0	0
$246$	40032.0	0.0421764
$247$	− 915722.i	− 0.955039i
$248$	257024.i	0.265366i
$249$	2268.00	0.00231817
$250$	0	0
$251$	−1.10024e6	−1.10231	−0.551153	−	0.834404i	$-0.685812\pi$
$251$	−1.10024e6	−1.10231	−0.551153	+	0.834404i	$0.685812\pi$
$252$	− 183456.i	− 0.181983i
$253$	− 984150.i	− 0.966629i
$254$	−1.13594e6	−1.10476
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.08230e6i	− 1.02215i	−0.859537	−	0.511074i	$-0.829248\pi$
$257$	− 1.08230e6i	− 1.02215i	0.859537	−	0.511074i	$-0.170752\pi$
$258$	282216.i	0.263957i
$259$	345058.	0.319626
$260$	0	0
$261$	−1.93261e6	−1.75607
$262$	− 1.04575e6i	− 0.941186i
$263$	− 82950.0i	− 0.0739481i	−0.999316	−	0.0369740i	$-0.988228\pi$
$263$	− 82950.0i	− 0.0739481i	0.999316	−	0.0369740i	$-0.0117719\pi$
$264$	−77760.0	−0.0686667
$265$	0	0
$266$	−459032.	−0.397776
$267$	− 406152.i	− 0.348667i
$268$	− 833984.i	− 0.709285i
$269$	633822.	0.534056	0.267028	−	0.963689i	$-0.413958\pi$
$269$	633822.	0.534056	0.267028	+	0.963689i	$0.413958\pi$
$270$	0	0
$271$	−278956.	−0.230734	−0.115367	−	0.993323i	$-0.536804\pi$
$271$	−278956.	−0.230734	−0.115367	+	0.993323i	$0.536804\pi$
$272$	255744.i	0.209596i
$273$	− 57477.0i	− 0.0466753i
$274$	158688.	0.127693
$275$	0	0
$276$	−116640.	−0.0921669
$277$	2.17523e6i	1.70336i	0.524064	+	0.851679i	$0.324415\pi$
$277$	2.17523e6i	1.70336i	−0.524064	+	0.851679i	$0.675585\pi$
$278$	− 730504.i	− 0.566905i
$279$	939744.	0.722768
$280$	0	0
$281$	−692901.	−0.523486	−0.261743	−	0.965138i	$-0.584297\pi$
$281$	−692901.	−0.523486	−0.261743	+	0.965138i	$0.584297\pi$
$282$	123804.i	0.0927069i
$283$	− 1.04021e6i	− 0.772065i	−0.922485	−	0.386032i	$-0.873845\pi$
$283$	− 1.04021e6i	− 0.772065i	0.922485	−	0.386032i	$-0.126155\pi$
$284$	456960.	0.336188
$285$	0	0
$286$	633420.	0.457906
$287$	163464.i	0.117143i
$288$	− 239616.i	− 0.170229i
$289$	421856.	0.297112
$290$	0	0
$291$	−331305.	−0.229348
$292$	− 1.12547e6i	− 0.772463i
$293$	1.08565e6i	0.738789i	0.929273	+	0.369394i	$0.120435\pi$
$293$	1.08565e6i	0.738789i	−0.929273	+	0.369394i	$0.879565\pi$
$294$	−28812.0	−0.0194404
$295$	0	0
$296$	450688.	0.298983
$297$	579555.i	0.381244i
$298$	− 48312.0i	− 0.0315148i
$299$	950130.	0.614618
$300$	0	0
$301$	−1.15238e6	−0.733129
$302$	833668.i	0.525988i
$303$	100350.i	0.0627929i
$304$	−599552.	−0.372086
$305$	0	0
$306$	935064.	0.570871
$307$	1463.00i	0 0.000885928i	1.00000		0.000442964i	$0.000141000\pi$
$307$	1463.00i	0 0.000885928i	−1.00000		0.000442964i	$0.999859\pi$
$308$	− 317520.i	− 0.190719i
$309$	−330933.	−0.197171
$310$	0	0
$311$	3.11977e6	1.82903	0.914515	−	0.404551i	$-0.132572\pi$
$311$	3.11977e6	1.82903	0.914515	+	0.404551i	$0.132572\pi$
$312$	− 75072.0i	− 0.0436608i
$313$	− 831425.i	− 0.479692i	−0.970811	−	0.239846i	$-0.922903\pi$
$313$	− 831425.i	− 0.479692i	0.970811	−	0.239846i	$-0.0770969\pi$
$314$	1.45638e6	0.833584
$315$	0	0
$316$	941168.	0.530212
$317$	− 1.25851e6i	− 0.703408i	−0.936111	−	0.351704i	$-0.885602\pi$
$317$	− 1.25851e6i	− 0.703408i	0.936111	−	0.351704i	$-0.114398\pi$
$318$	− 37008.0i	− 0.0205224i
$319$	−3.34489e6	−1.84037
$320$	0	0
$321$	−106074.	−0.0574575
$322$	− 476280.i	− 0.255990i
$323$	− 2.33966e6i	− 1.24780i
$324$	−841104.	−0.445130
$325$	0	0
$326$	−2504.00	−0.00130494
$327$	453549.i	0.234560i
$328$	213504.i	0.109578i
$329$	−505533.	−0.257490
$330$	0	0
$331$	−2.30465e6	−1.15621	−0.578103	−	0.815964i	$-0.696207\pi$
$331$	−2.30465e6	−1.15621	−0.578103	+	0.815964i	$0.696207\pi$
$332$	12096.0i	0.00602277i
$333$	− 1.64783e6i	− 0.814332i
$334$	1.78247e6	0.874290
$335$	0	0
$336$	−37632.0	−0.0181848
$337$	769166.i	0.368931i	0.982839	+	0.184466i	$0.0590554\pi$
$337$	769166.i	0.368931i	−0.982839	+	0.184466i	$0.940945\pi$
$338$	− 873648.i	− 0.415953i
$339$	−401058.	−0.189543
$340$	0	0
$341$	1.62648e6	0.757465
$342$	2.19211e6i	1.01344i
$343$	− 117649.i	− 0.0539949i
$344$	−1.50515e6	−0.685779
$345$	0	0
$346$	−2.57387e6	−1.15584
$347$	382074.i	0.170343i	0.996366	+	0.0851714i	$0.0271438\pi$
$347$	382074.i	0.170343i	−0.996366	+	0.0851714i	$0.972856\pi$
$348$	396432.i	0.175477i
$349$	3.88710e6	1.70829	0.854146	−	0.520034i	$-0.174081\pi$
$349$	3.88710e6	1.70829	0.854146	+	0.520034i	$0.174081\pi$
$350$	0	0
$351$	−559521.	−0.242409
$352$	− 414720.i	− 0.178401i
$353$	366453.i	0.156524i	0.996933	+	0.0782621i	$0.0249371\pi$
$353$	366453.i	0.156524i	−0.996933	+	0.0782621i	$0.975063\pi$
$354$	225792.	0.0957636
$355$	0	0
$356$	2.16614e6	0.905863
$357$	− 146853.i	− 0.0609835i
$358$	980592.i	0.404372i
$359$	3.14858e6	1.28937	0.644687	−	0.764446i	$-0.276988\pi$
$359$	3.14858e6	1.28937	0.644687	+	0.764446i	$0.276988\pi$
$360$	0	0
$361$	3.00887e6	1.21516
$362$	− 2.74472e6i	− 1.10085i
$363$	8922.00i	0.00355382i
$364$	306544.	0.121266
$365$	0	0
$366$	260016.	0.101461
$367$	2.13740e6i	0.828362i	0.910195	+	0.414181i	$0.135932\pi$
$367$	2.13740e6i	0.828362i	−0.910195	+	0.414181i	$0.864068\pi$
$368$	− 622080.i	− 0.239457i
$369$	780624.	0.298453
$370$	0	0
$371$	151116.	0.0570001
$372$	− 192768.i	− 0.0722233i
$373$	205624.i	0.0765247i	0.999268	+	0.0382624i	$0.0121823\pi$
$373$	205624.i	0.0765247i	−0.999268	+	0.0382624i	$0.987818\pi$
$374$	1.61838e6	0.598276
$375$	0	0
$376$	−660288.	−0.240860
$377$	− 3.22927e6i	− 1.17018i
$378$	280476.i	0.100964i
$379$	−3.50536e6	−1.25353	−0.626766	−	0.779208i	$-0.715622\pi$
$379$	−3.50536e6	−1.25353	−0.626766	+	0.779208i	$0.715622\pi$
$380$	0	0
$381$	851952.	0.300679
$382$	2.10812e6i	0.739159i
$383$	− 1.12904e6i	− 0.393291i	−0.980475	−	0.196645i	$-0.936995\pi$
$383$	− 1.12904e6i	− 0.393291i	0.980475	−	0.196645i	$-0.0630048\pi$
$384$	−49152.0	−0.0170103
$385$	0	0
$386$	−572864.	−0.195697
$387$	5.50321e6i	1.86784i
$388$	− 1.76696e6i	− 0.595864i
$389$	1.20003e6	0.402084	0.201042	−	0.979583i	$-0.435567\pi$
$389$	1.20003e6	0.402084	0.201042	+	0.979583i	$0.435567\pi$
$390$	0	0
$391$	2.42757e6	0.803026
$392$	− 153664.i	− 0.0505076i
$393$	784314.i	0.256158i
$394$	1.39387e6	0.452358
$395$	0	0
$396$	−1.51632e6	−0.485907
$397$	− 4.41836e6i	− 1.40697i	−0.710709	−	0.703486i	$-0.751626\pi$
$397$	− 4.41836e6i	− 1.40697i	0.710709	−	0.703486i	$-0.248374\pi$
$398$	3.01808e6i	0.955043i
$399$	344274.	0.108261
$400$	0	0
$401$	−3.13278e6	−0.972903	−0.486451	−	0.873708i	$-0.661709\pi$
$401$	−3.13278e6	−0.972903	−0.486451	+	0.873708i	$0.661709\pi$
$402$	625488.i	0.193043i
$403$	1.57026e6i	0.481624i
$404$	−535200.	−0.163141
$405$	0	0
$406$	−1.61876e6	−0.487381
$407$	− 2.85201e6i	− 0.853424i
$408$	− 191808.i	− 0.0570448i
$409$	−861494.	−0.254650	−0.127325	−	0.991861i	$-0.540639\pi$
$409$	−861494.	−0.254650	−0.127325	+	0.991861i	$0.540639\pi$
$410$	0	0
$411$	−119016.	−0.0347537
$412$	− 1.76498e6i	− 0.512266i
$413$	921984.i	0.265980i
$414$	−2.27448e6	−0.652201
$415$	0	0
$416$	400384.	0.113434
$417$	547878.i	0.154292i
$418$	3.79404e6i	1.06209i
$419$	−4.65796e6	−1.29617	−0.648083	−	0.761570i	$-0.724429\pi$
$419$	−4.65796e6	−1.29617	−0.648083	+	0.761570i	$0.724429\pi$
$420$	0	0
$421$	6.99894e6	1.92454	0.962271	−	0.272093i	$-0.0877159\pi$
$421$	6.99894e6	1.92454	0.962271	+	0.272093i	$0.0877159\pi$
$422$	2.36300e6i	0.645925i
$423$	2.41418e6i	0.656022i
$424$	197376.	0.0533187
$425$	0	0
$426$	−342720.	−0.0914988
$427$	1.06173e6i	0.281803i
$428$	− 565728.i	− 0.149279i
$429$	−475065.	−0.124626
$430$	0	0
$431$	−227091.	−0.0588853	−0.0294426	−	0.999566i	$-0.509373\pi$
$431$	−227091.	−0.0588853	−0.0294426	+	0.999566i	$0.509373\pi$
$432$	366336.i	0.0944431i
$433$	7.09613e6i	1.81887i	0.415846	+	0.909435i	$0.363485\pi$
$433$	7.09613e6i	1.81887i	−0.415846	+	0.909435i	$0.636515\pi$
$434$	787136.	0.200597
$435$	0	0
$436$	−2.41893e6	−0.609406
$437$	5.69106e6i	1.42557i
$438$	844104.i	0.210238i
$439$	−593258.	−0.146920	−0.0734602	−	0.997298i	$-0.523404\pi$
$439$	−593258.	−0.146920	−0.0734602	+	0.997298i	$0.523404\pi$
$440$	0	0
$441$	−561834.	−0.137566
$442$	1.56244e6i	0.380405i
$443$	3.27692e6i	0.793334i	0.917963	+	0.396667i	$0.129833\pi$
$443$	3.27692e6i	0.793334i	−0.917963	+	0.396667i	$0.870167\pi$
$444$	−338016.	−0.0813729
$445$	0	0
$446$	1.58441e6	0.377165
$447$	36234.0i	0.00857724i
$448$	− 200704.i	− 0.0472456i
$449$	4.32930e6	1.01345	0.506724	−	0.862108i	$-0.330856\pi$
$449$	4.32930e6	1.01345	0.506724	+	0.862108i	$0.330856\pi$
$450$	0	0
$451$	1.35108e6	0.312781
$452$	− 2.13898e6i	− 0.492447i
$453$	− 625251.i	− 0.143156i
$454$	38148.0	0.00868625
$455$	0	0
$456$	449664.	0.101269
$457$	− 4.91638e6i	− 1.10117i	−0.834779	−	0.550586i	$-0.814404\pi$
$457$	− 4.91638e6i	− 1.10117i	0.834779	−	0.550586i	$-0.185596\pi$
$458$	2.82022e6i	0.628232i
$459$	−1.42957e6	−0.316718
$460$	0	0
$461$	7.02919e6	1.54047	0.770235	−	0.637761i	$-0.220139\pi$
$461$	7.02919e6	1.54047	0.770235	+	0.637761i	$0.220139\pi$
$462$	238140.i	0.0519072i
$463$	− 2.88559e6i	− 0.625579i	−0.949823	−	0.312789i	$-0.898737\pi$
$463$	− 2.88559e6i	− 0.625579i	0.949823	−	0.312789i	$-0.101263\pi$
$464$	−2.11430e6	−0.455903
$465$	0	0
$466$	−2.13686e6	−0.455840
$467$	− 6.00583e6i	− 1.27433i	−0.770729	−	0.637163i	$-0.780108\pi$
$467$	− 6.00583e6i	− 1.27433i	0.770729	−	0.637163i	$-0.219892\pi$
$468$	− 1.46390e6i	− 0.308957i
$469$	−2.55408e6	−0.536169
$470$	0	0
$471$	−1.09228e6	−0.226873
$472$	1.20422e6i	0.248801i
$473$	9.52479e6i	1.95750i
$474$	−705876.	−0.144305
$475$	0	0
$476$	783216.	0.158440
$477$	− 721656.i	− 0.145223i
$478$	− 3.60488e6i	− 0.721642i
$479$	−941094.	−0.187411	−0.0937053	−	0.995600i	$-0.529871\pi$
$479$	−941094.	−0.187411	−0.0937053	+	0.995600i	$0.529871\pi$
$480$	0	0
$481$	2.75342e6	0.542638
$482$	3.80956e6i	0.746891i
$483$	357210.i	0.0696716i
$484$	−47584.0	−0.00923310
$485$	0	0
$486$	2.02176e6	0.388275
$487$	1.91121e6i	0.365162i	0.983191	+	0.182581i	$0.0584451\pi$
$487$	1.91121e6i	0.365162i	−0.983191	+	0.182581i	$0.941555\pi$
$488$	1.38675e6i	0.263602i
$489$	1878.00	0.000355160 0
$490$	0	0
$491$	3.95490e6	0.740342	0.370171	−	0.928964i	$-0.379299\pi$
$491$	3.95490e6	0.740342	0.370171	+	0.928964i	$0.379299\pi$
$492$	− 160128.i	− 0.0298232i
$493$	− 8.25074e6i	− 1.52889i
$494$	−3.66289e6	−0.675315
$495$	0	0
$496$	1.02810e6	0.187642
$497$	− 1.39944e6i	− 0.254134i
$498$	− 9072.00i	− 0.00163919i
$499$	−7.09708e6	−1.27593	−0.637967	−	0.770063i	$-0.720225\pi$
$499$	−7.09708e6	−1.27593	−0.637967	+	0.770063i	$0.720225\pi$
$500$	0	0
$501$	−1.33685e6	−0.237952
$502$	4.40095e6i	0.779448i
$503$	9.15982e6i	1.61424i	0.590390	+	0.807118i	$0.298974\pi$
$503$	9.15982e6i	1.61424i	−0.590390	+	0.807118i	$0.701026\pi$
$504$	−733824.	−0.128681
$505$	0	0
$506$	−3.93660e6	−0.683510
$507$	655236.i	0.113208i
$508$	4.54374e6i	0.781186i
$509$	9.42509e6	1.61247	0.806234	−	0.591596i	$-0.201502\pi$
$509$	9.42509e6	1.61247	0.806234	+	0.591596i	$0.201502\pi$
$510$	0	0
$511$	−3.44676e6	−0.583927
$512$	− 262144.i	− 0.0441942i
$513$	− 3.35140e6i	− 0.562255i
$514$	−4.32919e6	−0.722768
$515$	0	0
$516$	1.12886e6	0.186645
$517$	4.17838e6i	0.687515i
$518$	− 1.38023e6i	− 0.226010i
$519$	1.93040e6	0.314579
$520$	0	0
$521$	−6.18917e6	−0.998938	−0.499469	−	0.866332i	$-0.666471\pi$
$521$	−6.18917e6	−0.998938	−0.499469	+	0.866332i	$0.666471\pi$
$522$	7.73042e6i	1.24173i
$523$	3.81497e6i	0.609870i	0.952373	+	0.304935i	$0.0986347\pi$
$523$	3.81497e6i	0.609870i	−0.952373	+	0.304935i	$0.901365\pi$
$524$	−4.18301e6	−0.665519
$525$	0	0
$526$	−331800.	−0.0522892
$527$	4.01198e6i	0.629264i
$528$	311040.i	0.0485547i
$529$	531443.	0.0825691
$530$	0	0
$531$	4.40294e6	0.677652
$532$	1.83613e6i	0.281270i
$533$	1.30438e6i	0.198877i
$534$	−1.62461e6	−0.246545
$535$	0	0
$536$	−3.33594e6	−0.501540
$537$	− 735444.i	− 0.110056i
$538$	− 2.53529e6i	− 0.377634i
$539$	−972405.	−0.144170
$540$	0	0
$541$	6.30404e6	0.926032	0.463016	−	0.886350i	$-0.346767\pi$
$541$	6.30404e6	0.926032	0.463016	+	0.886350i	$0.346767\pi$
$542$	1.11582e6i	0.163154i
$543$	2.05854e6i	0.299612i
$544$	1.02298e6	0.148207
$545$	0	0
$546$	−229908.	−0.0330044
$547$	− 8.48475e6i	− 1.21247i	−0.795286	−	0.606234i	$-0.792679\pi$
$547$	− 8.48475e6i	− 1.21247i	0.795286	−	0.606234i	$-0.207321\pi$
$548$	− 634752.i	− 0.0902927i
$549$	5.07031e6	0.717966
$550$	0	0
$551$	1.93426e7	2.71416
$552$	466560.i	0.0651718i
$553$	− 2.88233e6i	− 0.400802i
$554$	8.70092e6	1.20446
$555$	0	0
$556$	−2.92202e6	−0.400863
$557$	6.87794e6i	0.939335i	0.882843	+	0.469668i	$0.155626\pi$
$557$	6.87794e6i	0.939335i	−0.882843	+	0.469668i	$0.844374\pi$
$558$	− 3.75898e6i	− 0.511074i
$559$	−9.19554e6	−1.24465
$560$	0	0
$561$	−1.21378e6	−0.162830
$562$	2.77160e6i	0.370161i
$563$	1.02257e7i	1.35964i	0.733379	+	0.679820i	$0.237942\pi$
$563$	1.02257e7i	1.35964i	−0.733379	+	0.679820i	$0.762058\pi$
$564$	495216.	0.0655537
$565$	0	0
$566$	−4.16083e6	−0.545932
$567$	2.57588e6i	0.336487i
$568$	− 1.82784e6i	− 0.237721i
$569$	1.26751e7	1.64123	0.820614	−	0.571482i	$-0.193631\pi$
$569$	1.26751e7	1.64123	0.820614	+	0.571482i	$0.193631\pi$
$570$	0	0
$571$	−6.67155e6	−0.856321	−0.428160	−	0.903703i	$-0.640838\pi$
$571$	−6.67155e6	−0.856321	−0.428160	+	0.903703i	$0.640838\pi$
$572$	− 2.53368e6i	− 0.323789i
$573$	− 1.58109e6i	− 0.201174i
$574$	653856.	0.0828328
$575$	0	0
$576$	−958464.	−0.120370
$577$	− 3.36511e6i	− 0.420784i	−0.977617	−	0.210392i	$-0.932526\pi$
$577$	− 3.36511e6i	− 0.420784i	0.977617	−	0.210392i	$-0.0674741\pi$
$578$	− 1.68742e6i	− 0.210090i
$579$	429648.	0.0532619
$580$	0	0
$581$	37044.0	0.00455279
$582$	1.32522e6i	0.162174i
$583$	− 1.24902e6i	− 0.152194i
$584$	−4.50189e6	−0.546214
$585$	0	0
$586$	4.34260e6	0.522403
$587$	− 1.10055e7i	− 1.31830i	−0.752012	−	0.659150i	$-0.770916\pi$
$587$	− 1.10055e7i	− 1.31830i	0.752012	−	0.659150i	$-0.229084\pi$
$588$	115248.i	0.0137464i
$589$	−9.40547e6	−1.11710
$590$	0	0
$591$	−1.04540e6	−0.123116
$592$	− 1.80275e6i	− 0.211413i
$593$	− 1.40222e6i	− 0.163749i	−0.996643	−	0.0818747i	$-0.973909\pi$
$593$	− 1.40222e6i	− 0.163749i	0.996643	−	0.0818747i	$-0.0260907\pi$
$594$	2.31822e6	0.269581
$595$	0	0
$596$	−193248.	−0.0222843
$597$	− 2.26356e6i	− 0.259930i
$598$	− 3.80052e6i	− 0.434600i
$599$	−1.93034e6	−0.219820	−0.109910	−	0.993942i	$-0.535056\pi$
$599$	−1.93034e6	−0.219820	−0.109910	+	0.993942i	$0.535056\pi$
$600$	0	0
$601$	−1.82271e6	−0.205841	−0.102921	−	0.994690i	$-0.532819\pi$
$601$	−1.82271e6	−0.205841	−0.102921	+	0.994690i	$0.532819\pi$
$602$	4.60953e6i	0.518400i
$603$	1.21970e7i	1.36603i
$604$	3.33467e6	0.371930
$605$	0	0
$606$	401400.	0.0444013
$607$	− 1.36917e7i	− 1.50830i	−0.656704	−	0.754148i	$-0.728050\pi$
$607$	− 1.36917e7i	− 1.50830i	0.656704	−	0.754148i	$-0.271950\pi$
$608$	2.39821e6i	0.263104i
$609$	1.21407e6	0.132648
$610$	0	0
$611$	−4.03395e6	−0.437147
$612$	− 3.74026e6i	− 0.403667i
$613$	− 1.11975e7i	− 1.20357i	−0.798658	−	0.601785i	$-0.794456\pi$
$613$	− 1.11975e7i	− 1.20357i	0.798658	−	0.601785i	$-0.205544\pi$
$614$	5852.00	0.000626446 0
$615$	0	0
$616$	−1.27008e6	−0.134859
$617$	− 1.37060e7i	− 1.44944i	−0.689045	−	0.724718i	$-0.741970\pi$
$617$	− 1.37060e7i	− 1.44944i	0.689045	−	0.724718i	$-0.258030\pi$
$618$	1.32373e6i	0.139421i
$619$	7.93359e6	0.832230	0.416115	−	0.909312i	$-0.363391\pi$
$619$	7.93359e6	0.832230	0.416115	+	0.909312i	$0.363391\pi$
$620$	0	0
$621$	3.47733e6	0.361840
$622$	− 1.24791e7i	− 1.29332i
$623$	− 6.63382e6i	− 0.684768i
$624$	−300288.	−0.0308728
$625$	0	0
$626$	−3.32570e6	−0.339193
$627$	− 2.84553e6i	− 0.289064i
$628$	− 5.82550e6i	− 0.589433i
$629$	7.03496e6	0.708981
$630$	0	0
$631$	−1.31143e7	−1.31121	−0.655604	−	0.755105i	$-0.727586\pi$
$631$	−1.31143e7	−1.31121	−0.655604	+	0.755105i	$0.727586\pi$
$632$	− 3.76467e6i	− 0.374916i
$633$	− 1.77225e6i	− 0.175798i
$634$	−5.03402e6	−0.497384
$635$	0	0
$636$	−148032.	−0.0145115
$637$	− 938791.i	− 0.0916685i
$638$	1.33796e7i	1.30134i
$639$	−6.68304e6	−0.647473
$640$	0	0
$641$	−1.27270e7	−1.22344	−0.611719	−	0.791075i	$-0.709522\pi$
$641$	−1.27270e7	−1.22344	−0.611719	+	0.791075i	$0.709522\pi$
$642$	424296.i	0.0406286i
$643$	− 1.88399e7i	− 1.79701i	−0.438964	−	0.898505i	$-0.644655\pi$
$643$	− 1.88399e7i	− 1.79701i	0.438964	−	0.898505i	$-0.355345\pi$
$644$	−1.90512e6	−0.181012
$645$	0	0
$646$	−9.35863e6	−0.882330
$647$	− 944688.i	− 0.0887213i	−0.999016	−	0.0443606i	$-0.985875\pi$
$647$	− 944688.i	− 0.0887213i	0.999016	−	0.0443606i	$-0.0141251\pi$
$648$	3.36442e6i	0.314755i
$649$	7.62048e6	0.710184
$650$	0	0
$651$	−590352.	−0.0545957
$652$	10016.0i	0 0.000922731i
$653$	− 2.01024e7i	− 1.84486i	−0.386158	−	0.922432i	$-0.626198\pi$
$653$	− 2.01024e7i	− 1.84486i	0.386158	−	0.922432i	$-0.373802\pi$
$654$	1.81420e6	0.165859
$655$	0	0
$656$	854016.	0.0774830
$657$	1.64600e7i	1.48771i
$658$	2.02213e6i	0.182073i
$659$	1.97097e7	1.76793	0.883967	−	0.467549i	$-0.154863\pi$
$659$	1.97097e7	1.76793	0.883967	+	0.467549i	$0.154863\pi$
$660$	0	0
$661$	−227080.	−0.0202151	−0.0101075	−	0.999949i	$-0.503217\pi$
$661$	−227080.	−0.0202151	−0.0101075	+	0.999949i	$0.503217\pi$
$662$	9.21861e6i	0.817561i
$663$	− 1.17183e6i	− 0.103533i
$664$	48384.0	0.00425874
$665$	0	0
$666$	−6.59131e6	−0.575819
$667$	2.00694e7i	1.74670i
$668$	− 7.12987e6i	− 0.618216i
$669$	−1.18831e6	−0.102651
$670$	0	0
$671$	8.77554e6	0.752433
$672$	150528.i	0.0128586i
$673$	− 1.93220e7i	− 1.64443i	−0.569178	−	0.822214i	$-0.692739\pi$
$673$	− 1.93220e7i	− 1.64443i	0.569178	−	0.822214i	$-0.307261\pi$
$674$	3.07666e6	0.260874
$675$	0	0
$676$	−3.49459e6	−0.294124
$677$	3.35334e6i	0.281194i	0.990067	+	0.140597i	$0.0449021\pi$
$677$	3.35334e6i	0.281194i	−0.990067	+	0.140597i	$0.955098\pi$
$678$	1.60423e6i	0.134027i
$679$	−5.41132e6	−0.450431
$680$	0	0
$681$	−28611.0	−0.00236410
$682$	− 6.50592e6i	− 0.535609i
$683$	− 1.60555e7i	− 1.31696i	−0.752598	−	0.658481i	$-0.771199\pi$
$683$	− 1.60555e7i	− 1.31696i	0.752598	−	0.658481i	$-0.228801\pi$
$684$	8.76845e6	0.716609
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 2.11517e6i	− 0.170983i
$688$	6.02061e6i	0.484919i
$689$	1.20584e6	0.0967705
$690$	0	0
$691$	1.35824e7	1.08213	0.541066	−	0.840980i	$-0.318021\pi$
$691$	1.35824e7	1.08213	0.541066	+	0.840980i	$0.318021\pi$
$692$	1.02955e7i	0.817299i
$693$	4.64373e6i	0.367311i
$694$	1.52830e6	0.120451
$695$	0	0
$696$	1.58573e6	0.124081
$697$	3.33266e6i	0.259842i
$698$	− 1.55484e7i	− 1.20794i
$699$	1.60265e6	0.124064
$700$	0	0
$701$	2.05454e7	1.57913	0.789567	−	0.613664i	$-0.210305\pi$
$701$	2.05454e7	1.57913	0.789567	+	0.613664i	$0.210305\pi$
$702$	2.23808e6i	0.171409i
$703$	1.64924e7i	1.25862i
$704$	−1.65888e6	−0.126149
$705$	0	0
$706$	1.46581e6	0.110679
$707$	1.63905e6i	0.123323i
$708$	− 903168.i	− 0.0677151i
$709$	−2.57278e7	−1.92215	−0.961075	−	0.276287i	$-0.910896\pi$
$709$	−2.57278e7	−1.92215	−0.961075	+	0.276287i	$0.910896\pi$
$710$	0	0
$711$	−1.37646e7	−1.02115
$712$	− 8.66458e6i	− 0.640542i
$713$	− 9.75888e6i	− 0.718913i
$714$	−587412.	−0.0431218
$715$	0	0
$716$	3.92237e6	0.285934
$717$	2.70366e6i	0.196406i
$718$	− 1.25943e7i	− 0.911726i
$719$	−7.04806e6	−0.508449	−0.254225	−	0.967145i	$-0.581820\pi$
$719$	−7.04806e6	−0.508449	−0.254225	+	0.967145i	$0.581820\pi$
$720$	0	0
$721$	−5.40524e6	−0.387237
$722$	− 1.20355e7i	− 0.859250i
$723$	− 2.85717e6i	− 0.203278i
$724$	−1.09789e7	−0.778416
$725$	0	0
$726$	35688.0	0.00251293
$727$	− 1.90997e7i	− 1.34027i	−0.742240	−	0.670134i	$-0.766237\pi$
$727$	− 1.90997e7i	− 1.34027i	0.742240	−	0.670134i	$-0.233763\pi$
$728$	− 1.22618e6i	− 0.0857481i
$729$	1.12579e7	0.784586
$730$	0	0
$731$	−2.34945e7	−1.62619
$732$	− 1.04006e6i	− 0.0717435i
$733$	2.30424e6i	0.158404i	0.996859	+	0.0792021i	$0.0252373\pi$
$733$	2.30424e6i	0.158404i	−0.996859	+	0.0792021i	$0.974763\pi$
$734$	8.54959e6	0.585740
$735$	0	0
$736$	−2.48832e6	−0.169321
$737$	2.11102e7i	1.43161i
$738$	− 3.12250e6i	− 0.211038i
$739$	−3.62955e6	−0.244479	−0.122240	−	0.992501i	$-0.539008\pi$
$739$	−3.62955e6	−0.244479	−0.122240	+	0.992501i	$0.539008\pi$
$740$	0	0
$741$	2.74717e6	0.183797
$742$	− 604464.i	− 0.0403052i
$743$	9.73856e6i	0.647177i	0.946198	+	0.323588i	$0.104889\pi$
$743$	9.73856e6i	0.647177i	−0.946198	+	0.323588i	$0.895111\pi$
$744$	−771072.	−0.0510696
$745$	0	0
$746$	822496.	0.0541111
$747$	− 176904.i	− 0.0115994i
$748$	− 6.47352e6i	− 0.423045i
$749$	−1.73254e6	−0.112844
$750$	0	0
$751$	−2.48272e7	−1.60630	−0.803152	−	0.595774i	$-0.796845\pi$
$751$	−2.48272e7	−1.60630	−0.803152	+	0.595774i	$0.796845\pi$
$752$	2.64115e6i	0.170313i
$753$	− 3.30071e6i	− 0.212139i
$754$	−1.29171e7	−0.827439
$755$	0	0
$756$	1.12190e6	0.0713923
$757$	− 1.28400e7i	− 0.814376i	−0.913344	−	0.407188i	$-0.866509\pi$
$757$	− 1.28400e7i	− 0.814376i	0.913344	−	0.407188i	$-0.133491\pi$
$758$	1.40215e7i	0.886380i
$759$	2.95245e6	0.186028
$760$	0	0
$761$	2.89560e7	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	2.89560e7	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	− 3.40781e6i	− 0.212612i
$763$	7.40797e6i	0.460668i
$764$	8.43250e6	0.522664
$765$	0	0
$766$	−4.51618e6	−0.278099
$767$	7.35706e6i	0.451560i
$768$	196608.i	0.0120281i
$769$	1.78116e7	1.08614	0.543071	−	0.839687i	$-0.317261\pi$
$769$	1.78116e7	1.08614	0.543071	+	0.839687i	$0.317261\pi$
$770$	0	0
$771$	3.24689e6	0.196713
$772$	2.29146e6i	0.138378i
$773$	− 1.73536e7i	− 1.04458i	−0.852768	−	0.522290i	$-0.825078\pi$
$773$	− 1.73536e7i	− 1.04458i	0.852768	−	0.522290i	$-0.174922\pi$
$774$	2.20128e7	1.32076
$775$	0	0
$776$	−7.06784e6	−0.421340
$777$	1.03517e6i	0.0615121i
$778$	− 4.80011e6i	− 0.284316i
$779$	−7.81291e6	−0.461285
$780$	0	0
$781$	−1.15668e7	−0.678556
$782$	− 9.71028e6i	− 0.567825i
$783$	− 1.18186e7i	− 0.688910i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	3.13726e6	0.181131
$787$	812177.i	0.0467427i	0.999727	+	0.0233714i	$0.00744001\pi$
$787$	812177.i	0.0467427i	−0.999727	+	0.0233714i	$0.992560\pi$
$788$	− 5.57549e6i	− 0.319865i
$789$	248850.	0.0142313
$790$	0	0
$791$	−6.55061e6	−0.372255
$792$	6.06528e6i	0.343588i
$793$	8.47219e6i	0.478424i
$794$	−1.76735e7	−0.994879
$795$	0	0
$796$	1.20723e7	0.675318
$797$	− 8.58201e6i	− 0.478568i	−0.970950	−	0.239284i	$-0.923087\pi$
$797$	− 8.58201e6i	− 0.478568i	0.970950	−	0.239284i	$-0.0769126\pi$
$798$	− 1.37710e6i	− 0.0765521i
$799$	−1.03067e7	−0.571152
$800$	0	0
$801$	−3.16799e7	−1.74462
$802$	1.25311e7i	0.687946i
$803$	2.84885e7i	1.55912i
$804$	2.50195e6	0.136502
$805$	0	0
$806$	6.28102e6	0.340559
$807$	1.90147e6i	0.102779i
$808$	2.14080e6i	0.115358i
$809$	2.83000e6	0.152025	0.0760125	−	0.997107i	$-0.475781\pi$
$809$	2.83000e6	0.152025	0.0760125	+	0.997107i	$0.475781\pi$
$810$	0	0
$811$	−1.06484e7	−0.568504	−0.284252	−	0.958750i	$-0.591745\pi$
$811$	−1.06484e7	−0.568504	−0.284252	+	0.958750i	$0.591745\pi$
$812$	6.47506e6i	0.344630i
$813$	− 836868.i	− 0.0444049i
$814$	−1.14080e7	−0.603462
$815$	0	0
$816$	−767232.	−0.0403368
$817$	− 5.50792e7i	− 2.88690i
$818$	3.44598e6i	0.180065i
$819$	−4.48321e6	−0.233549
$820$	0	0
$821$	−2.59970e7	−1.34606	−0.673032	−	0.739613i	$-0.735008\pi$
$821$	−2.59970e7	−1.34606	−0.673032	+	0.739613i	$0.735008\pi$
$822$	476064.i	0.0245746i
$823$	2.03099e7i	1.04522i	0.852571	+	0.522611i	$0.175042\pi$
$823$	2.03099e7i	1.04522i	−0.852571	+	0.522611i	$0.824958\pi$
$824$	−7.05990e6	−0.362227
$825$	0	0
$826$	3.68794e6	0.188076
$827$	1.68001e6i	0.0854175i	0.999088	+	0.0427088i	$0.0135988\pi$
$827$	1.68001e6i	0.0854175i	−0.999088	+	0.0427088i	$0.986401\pi$
$828$	9.09792e6i	0.461176i
$829$	6.71070e6	0.339142	0.169571	−	0.985518i	$-0.445762\pi$
$829$	6.71070e6	0.339142	0.169571	+	0.985518i	$0.445762\pi$
$830$	0	0
$831$	−6.52569e6	−0.327811
$832$	− 1.60154e6i	− 0.0802100i
$833$	− 2.39860e6i	− 0.119769i
$834$	2.19151e6	0.109101
$835$	0	0
$836$	1.51762e7	0.751011
$837$	5.74690e6i	0.283543i
$838$	1.86318e7i	0.916527i
$839$	−2.60856e7	−1.27937	−0.639686	−	0.768637i	$-0.720935\pi$
$839$	−2.60856e7	−1.27937	−0.639686	+	0.768637i	$0.720935\pi$
$840$	0	0
$841$	4.76999e7	2.32556
$842$	− 2.79958e7i	− 1.36086i
$843$	− 2.07870e6i	− 0.100745i
$844$	9.45198e6	0.456738
$845$	0	0
$846$	9.65671e6	0.463878
$847$	145726.i	0.00697957i
$848$	− 789504.i	− 0.0377020i
$849$	3.12062e6	0.148584
$850$	0	0
$851$	−1.71121e7	−0.809988
$852$	1.37088e6i	0.0646994i
$853$	− 9.54873e6i	− 0.449338i	−0.974435	−	0.224669i	$-0.927870\pi$
$853$	− 9.54873e6i	− 0.449338i	0.974435	−	0.224669i	$-0.0721300\pi$
$854$	4.24693e6	0.199265
$855$	0	0
$856$	−2.26291e6	−0.105556
$857$	3.51377e7i	1.63426i	0.576453	+	0.817130i	$0.304436\pi$
$857$	3.51377e7i	1.63426i	−0.576453	+	0.817130i	$0.695564\pi$
$858$	1.90026e6i	0.0881241i
$859$	1.60428e7	0.741816	0.370908	−	0.928670i	$-0.379047\pi$
$859$	1.60428e7	0.741816	0.370908	+	0.928670i	$0.379047\pi$
$860$	0	0
$861$	−490392.	−0.0225442
$862$	908364.i	0.0416382i
$863$	2.77776e7i	1.26960i	0.772675	+	0.634802i	$0.218918\pi$
$863$	2.77776e7i	1.26960i	−0.772675	+	0.634802i	$0.781082\pi$
$864$	1.46534e6	0.0667814
$865$	0	0
$866$	2.83845e7	1.28614
$867$	1.26557e6i	0.0571792i
$868$	− 3.14854e6i	− 0.141844i
$869$	−2.38233e7	−1.07017
$870$	0	0
$871$	−2.03805e7	−0.910268
$872$	9.67571e6i	0.430915i
$873$	2.58418e7i	1.14759i
$874$	2.27642e7	1.00803
$875$	0	0
$876$	3.37642e6	0.148661
$877$	2.46748e7i	1.08332i	0.840599	+	0.541658i	$0.182203\pi$
$877$	2.46748e7i	1.08332i	−0.840599	+	0.541658i	$0.817797\pi$
$878$	2.37303e6i	0.103888i
$879$	−3.25695e6	−0.142180
$880$	0	0
$881$	−1.27792e7	−0.554707	−0.277353	−	0.960768i	$-0.589457\pi$
$881$	−1.27792e7	−0.554707	−0.277353	+	0.960768i	$0.589457\pi$
$882$	2.24734e6i	0.0972739i
$883$	2.63417e7i	1.13695i	0.822700	+	0.568476i	$0.192467\pi$
$883$	2.63417e7i	1.13695i	−0.822700	+	0.568476i	$0.807533\pi$
$884$	6.24974e6	0.268987
$885$	0	0
$886$	1.31077e7	0.560972
$887$	2.60037e7i	1.10975i	0.831932	+	0.554877i	$0.187235\pi$
$887$	2.60037e7i	1.10975i	−0.831932	+	0.554877i	$0.812765\pi$
$888$	1.35206e6i	0.0575393i
$889$	1.39152e7	0.590521
$890$	0	0
$891$	2.12904e7	0.898443
$892$	− 6.33765e6i	− 0.266696i
$893$	− 2.41624e7i	− 1.01394i
$894$	144936.	0.00606502
$895$	0	0
$896$	−802816.	−0.0334077
$897$	2.85039e6i	0.118283i
$898$	− 1.73172e7i	− 0.716616i
$899$	−3.31681e7	−1.36874
$900$	0	0
$901$	3.08092e6	0.126435
$902$	− 5.40432e6i	− 0.221169i
$903$	− 3.45715e6i	− 0.141091i
$904$	−8.55590e6	−0.348213
$905$	0	0
$906$	−2.50100e6	−0.101226
$907$	− 4.11852e7i	− 1.66235i	−0.556008	−	0.831177i	$-0.687668\pi$
$907$	− 4.11852e7i	− 1.66235i	0.556008	−	0.831177i	$-0.312332\pi$
$908$	− 152592.i	− 0.00614210i
$909$	7.82730e6	0.314197
$910$	0	0
$911$	−7.92211e6	−0.316261	−0.158130	−	0.987418i	$-0.550547\pi$
$911$	−7.92211e6	−0.316261	−0.158130	+	0.987418i	$0.550547\pi$
$912$	− 1.79866e6i	− 0.0716079i
$913$	− 306180.i	− 0.0121563i
$914$	−1.96655e7	−0.778646
$915$	0	0
$916$	1.12809e7	0.444227
$917$	1.28105e7i	0.503085i
$918$	5.71828e6i	0.223954i
$919$	−1.59154e7	−0.621624	−0.310812	−	0.950471i	$-0.600601\pi$
$919$	−1.59154e7	−0.621624	−0.310812	+	0.950471i	$0.600601\pi$
$920$	0	0
$921$	−4389.00	−0.000170497 0
$922$	− 2.81168e7i	− 1.08928i
$923$	− 1.11670e7i	− 0.431450i
$924$	952560.	0.0367039
$925$	0	0
$926$	−1.15424e7	−0.442351
$927$	2.58128e7i	0.986587i
$928$	8.45722e6i	0.322372i
$929$	3.37148e7	1.28169	0.640843	−	0.767672i	$-0.278585\pi$
$929$	3.37148e7	1.28169	0.640843	+	0.767672i	$0.278585\pi$
$930$	0	0
$931$	5.62314e6	0.212620
$932$	8.54746e6i	0.322327i
$933$	9.35930e6i	0.351997i
$934$	−2.40233e7	−0.901085
$935$	0	0
$936$	−5.85562e6	−0.218466
$937$	− 4.04362e7i	− 1.50460i	−0.658820	−	0.752300i	$-0.728944\pi$
$937$	− 4.04362e7i	− 1.50460i	0.658820	−	0.752300i	$-0.271056\pi$
$938$	1.02163e7i	0.379129i
$939$	2.49428e6	0.0923167
$940$	0	0
$941$	3.62378e7	1.33410	0.667048	−	0.745015i	$-0.267557\pi$
$941$	3.62378e7	1.33410	0.667048	+	0.745015i	$0.267557\pi$
$942$	4.36913e6i	0.160423i
$943$	− 8.10648e6i	− 0.296861i
$944$	4.81690e6	0.175929
$945$	0	0
$946$	3.80992e7	1.38416
$947$	2.94238e7i	1.06616i	0.846064	+	0.533082i	$0.178966\pi$
$947$	2.94238e7i	1.06616i	−0.846064	+	0.533082i	$0.821034\pi$
$948$	2.82350e6i	0.102039i
$949$	−2.75037e7	−0.991348
$950$	0	0
$951$	3.77552e6	0.135371
$952$	− 3.13286e6i	− 0.112034i
$953$	− 3.59497e7i	− 1.28222i	−0.767449	−	0.641110i	$-0.778474\pi$
$953$	− 3.59497e7i	− 1.28222i	0.767449	−	0.641110i	$-0.221526\pi$
$954$	−2.88662e6	−0.102688
$955$	0	0
$956$	−1.44195e7	−0.510278
$957$	− 1.00347e7i	− 0.354180i
$958$	3.76438e6i	0.132519i
$959$	−1.94393e6	−0.0682549
$960$	0	0
$961$	−1.25009e7	−0.436649
$962$	− 1.10137e7i	− 0.383703i
$963$	8.27377e6i	0.287500i
$964$	1.52382e7	0.528132
$965$	0	0
$966$	1.42884e6	0.0492653
$967$	1.19506e6i	0.0410982i	0.999789	+	0.0205491i	$0.00654144\pi$
$967$	1.19506e6i	0.0410982i	−0.999789	+	0.0205491i	$0.993459\pi$
$968$	190336.i	0.00652879i
$969$	7.01897e6	0.240140
$970$	0	0
$971$	3.26221e7	1.11036	0.555180	−	0.831730i	$-0.312649\pi$
$971$	3.26221e7	1.11036	0.555180	+	0.831730i	$0.312649\pi$
$972$	− 8.08704e6i	− 0.274552i
$973$	8.94867e6i	0.303024i
$974$	7.64482e6	0.258208
$975$	0	0
$976$	5.54701e6	0.186395
$977$	5.36858e7i	1.79938i	0.436529	+	0.899690i	$0.356208\pi$
$977$	5.36858e7i	1.79938i	−0.436529	+	0.899690i	$0.643792\pi$
$978$	− 7512.00i	0 0.000251136i
$979$	−5.48305e7	−1.82838
$980$	0	0
$981$	3.53768e7	1.17367
$982$	− 1.58196e7i	− 0.523501i
$983$	3.31124e7i	1.09297i	0.837469	+	0.546484i	$0.184034\pi$
$983$	3.31124e7i	1.09297i	−0.837469	+	0.546484i	$0.815966\pi$
$984$	−640512.	−0.0210882
$985$	0	0
$986$	−3.30030e7	−1.08109
$987$	− 1.51660e6i	− 0.0495539i
$988$	1.46516e7i	0.477520i
$989$	5.71487e7	1.85787
$990$	0	0
$991$	−1.97082e7	−0.637475	−0.318738	−	0.947843i	$-0.603259\pi$
$991$	−1.97082e7	−0.637475	−0.318738	+	0.947843i	$0.603259\pi$
$992$	− 4.11238e6i	− 0.132683i
$993$	− 6.91396e6i	− 0.222512i
$994$	−5.59776e6	−0.179700
$995$	0	0
$996$	−36288.0	−0.00115908
$997$	3.31940e7i	1.05760i	0.848747	+	0.528800i	$0.177358\pi$
$997$	3.31940e7i	1.05760i	−0.848747	+	0.528800i	$0.822642\pi$
$998$	2.83883e7i	0.902222i
$999$	1.00771e7	0.319464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.e.99.1		2
5.2	odd	4		350.6.a.k.1.1		1
5.3	odd	4		70.6.a.c.1.1	✓	1
5.4	even	2	inner	350.6.c.e.99.2		2
15.8	even	4		630.6.a.n.1.1		1
20.3	even	4		560.6.a.d.1.1		1
35.13	even	4		490.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.c.1.1	✓	1	5.3	odd	4
350.6.a.k.1.1		1	5.2	odd	4
350.6.c.e.99.1		2	1.1	even	1	trivial
350.6.c.e.99.2		2	5.4	even	2	inner
490.6.a.e.1.1		1	35.13	even	4
560.6.a.d.1.1		1	20.3	even	4
630.6.a.n.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +3.00000i q^{3} -16.0000 q^{4} +12.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +234.000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +3.00000i q^{3} -16.0000 q^{4} +12.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +234.000 q^{9} +405.000 q^{11} -48.0000i q^{12} +391.000i q^{13} +196.000 q^{14} +256.000 q^{16} +999.000i q^{17} -936.000i q^{18} -2342.00 q^{19} -147.000 q^{21} -1620.00i q^{22} -2430.00i q^{23} -192.000 q^{24} +1564.00 q^{26} +1431.00i q^{27} -784.000i q^{28} -8259.00 q^{29} +4016.00 q^{31} -1024.00i q^{32} +1215.00i q^{33} +3996.00 q^{34} -3744.00 q^{36} -7042.00i q^{37} +9368.00i q^{38} -1173.00 q^{39} +3336.00 q^{41} +588.000i q^{42} +23518.0i q^{43} -6480.00 q^{44} -9720.00 q^{46} +10317.0i q^{47} +768.000i q^{48} -2401.00 q^{49} -2997.00 q^{51} -6256.00i q^{52} -3084.00i q^{53} +5724.00 q^{54} -3136.00 q^{56} -7026.00i q^{57} +33036.0i q^{58} +18816.0 q^{59} +21668.0 q^{61} -16064.0i q^{62} +11466.0i q^{63} -4096.00 q^{64} +4860.00 q^{66} +52124.0i q^{67} -15984.0i q^{68} +7290.00 q^{69} -28560.0 q^{71} +14976.0i q^{72} +70342.0i q^{73} -28168.0 q^{74} +37472.0 q^{76} +19845.0i q^{77} +4692.00i q^{78} -58823.0 q^{79} +52569.0 q^{81} -13344.0i q^{82} -756.000i q^{83} +2352.00 q^{84} +94072.0 q^{86} -24777.0i q^{87} +25920.0i q^{88} -135384. q^{89} -19159.0 q^{91} +38880.0i q^{92} +12048.0i q^{93} +41268.0 q^{94} +3072.00 q^{96} +110435. i q^{97} +9604.00i q^{98} +94770.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 24 q^{6} + 468 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 24 * q^6 + 468 * q^9	\( 2 q - 32 q^{4} + 24 q^{6} + 468 q^{9} + 810 q^{11} + 392 q^{14} + 512 q^{16} - 4684 q^{19} - 294 q^{21} - 384 q^{24} + 3128 q^{26} - 16518 q^{29} + 8032 q^{31} + 7992 q^{34} - 7488 q^{36} - 2346 q^{39} + 6672 q^{41} - 12960 q^{44} - 19440 q^{46} - 4802 q^{49} - 5994 q^{51} + 11448 q^{54} - 6272 q^{56} + 37632 q^{59} + 43336 q^{61} - 8192 q^{64} + 9720 q^{66} + 14580 q^{69} - 57120 q^{71} - 56336 q^{74} + 74944 q^{76} - 117646 q^{79} + 105138 q^{81} + 4704 q^{84} + 188144 q^{86} - 270768 q^{89} - 38318 q^{91} + 82536 q^{94} + 6144 q^{96} + 189540 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 24 * q^6 + 468 * q^9 + 810 * q^11 + 392 * q^14 + 512 * q^16 - 4684 * q^19 - 294 * q^21 - 384 * q^24 + 3128 * q^26 - 16518 * q^29 + 8032 * q^31 + 7992 * q^34 - 7488 * q^36 - 2346 * q^39 + 6672 * q^41 - 12960 * q^44 - 19440 * q^46 - 4802 * q^49 - 5994 * q^51 + 11448 * q^54 - 6272 * q^56 + 37632 * q^59 + 43336 * q^61 - 8192 * q^64 + 9720 * q^66 + 14580 * q^69 - 57120 * q^71 - 56336 * q^74 + 74944 * q^76 - 117646 * q^79 + 105138 * q^81 + 4704 * q^84 + 188144 * q^86 - 270768 * q^89 - 38318 * q^91 + 82536 * q^94 + 6144 * q^96 + 189540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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