LMFDB - Embedded newform 350.6.c.f.99.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-32,0,64,0,0,358,0,-680] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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.f.99.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} +8.00000i q^{3} -16.0000 q^{4} +32.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +179.000 q^{9} -340.000 q^{11} -128.000i q^{12} -294.000i q^{13} +196.000 q^{14} +256.000 q^{16} -1226.00i q^{17} -716.000i q^{18} -2432.00 q^{19} -392.000 q^{21} +1360.00i q^{22} +2000.00i q^{23} -512.000 q^{24} -1176.00 q^{26} +3376.00i q^{27} -784.000i q^{28} +6746.00 q^{29} +8856.00 q^{31} -1024.00i q^{32} -2720.00i q^{33} -4904.00 q^{34} -2864.00 q^{36} -9182.00i q^{37} +9728.00i q^{38} +2352.00 q^{39} -14574.0 q^{41} +1568.00i q^{42} +8108.00i q^{43} +5440.00 q^{44} +8000.00 q^{46} +312.000i q^{47} +2048.00i q^{48} -2401.00 q^{49} +9808.00 q^{51} +4704.00i q^{52} -14634.0i q^{53} +13504.0 q^{54} -3136.00 q^{56} -19456.0i q^{57} -26984.0i q^{58} +27656.0 q^{59} +34338.0 q^{61} -35424.0i q^{62} +8771.00i q^{63} -4096.00 q^{64} -10880.0 q^{66} -12316.0i q^{67} +19616.0i q^{68} -16000.0 q^{69} +36920.0 q^{71} +11456.0i q^{72} -61718.0i q^{73} -36728.0 q^{74} +38912.0 q^{76} -16660.0i q^{77} -9408.00i q^{78} +64752.0 q^{79} +16489.0 q^{81} +58296.0i q^{82} -77056.0i q^{83} +6272.00 q^{84} +32432.0 q^{86} +53968.0i q^{87} -21760.0i q^{88} +8166.00 q^{89} +14406.0 q^{91} -32000.0i q^{92} +70848.0i q^{93} +1248.00 q^{94} +8192.00 q^{96} -20650.0i q^{97} +9604.00i q^{98} -60860.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39}+ \cdots - 121720 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 - 680 * q^11 + 392 * q^14 + 512 * q^16 - 4864 * q^19 - 784 * q^21 - 1024 * q^24 - 2352 * q^26 + 13492 * q^29 + 17712 * q^31 - 9808 * q^34 - 5728 * q^36 + 4704 * q^39 - 29148 * q^41 + 10880 * q^44 + 16000 * q^46 - 4802 * q^49 + 19616 * q^51 + 27008 * q^54 - 6272 * q^56 + 55312 * q^59 + 68676 * q^61 - 8192 * q^64 - 21760 * q^66 - 32000 * q^69 + 73840 * q^71 - 73456 * q^74 + 77824 * q^76 + 129504 * q^79 + 32978 * q^81 + 12544 * q^84 + 64864 * q^86 + 16332 * q^89 + 28812 * q^91 + 2496 * q^94 + 16384 * q^96 - 121720 * q^99

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
$3$	8.00000i	0.513200i	0.966518	+	0.256600i	$0.0826023\pi$
$3$	8.00000i	0.513200i	−0.966518	+	0.256600i	$0.917398\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	32.0000	0.362887
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	64.0000i	0.353553i
$9$	179.000	0.736626
$10$	0	0
$11$	−340.000	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$11$	−340.000	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$12$	− 128.000i	− 0.256600i
$13$	− 294.000i	− 0.482491i	−0.970464	−	0.241245i	$-0.922444\pi$
$13$	− 294.000i	− 0.482491i	0.970464	−	0.241245i	$-0.0775559\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	− 1226.00i	− 1.02889i	−0.857524	−	0.514444i	$-0.827998\pi$
$17$	− 1226.00i	− 1.02889i	0.857524	−	0.514444i	$-0.172002\pi$
$18$	− 716.000i	− 0.520873i
$19$	−2432.00	−1.54554	−0.772769	−	0.634688i	$-0.781129\pi$
$19$	−2432.00	−1.54554	−0.772769	+	0.634688i	$0.781129\pi$
$20$	0	0
$21$	−392.000	−0.193971
$22$	1360.00i	0.599076i
$23$	2000.00i	0.788334i	0.919039	+	0.394167i	$0.128967\pi$
$23$	2000.00i	0.788334i	−0.919039	+	0.394167i	$0.871033\pi$
$24$	−512.000	−0.181444
$25$	0	0
$26$	−1176.00	−0.341172
$27$	3376.00i	0.891237i
$28$	− 784.000i	− 0.188982i
$29$	6746.00	1.48954	0.744769	−	0.667323i	$-0.232560\pi$
$29$	6746.00	1.48954	0.744769	+	0.667323i	$0.232560\pi$
$30$	0	0
$31$	8856.00	1.65513	0.827567	−	0.561366i	$-0.189724\pi$
$31$	8856.00	1.65513	0.827567	+	0.561366i	$0.189724\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 2720.00i	− 0.434795i
$34$	−4904.00	−0.727534
$35$	0	0
$36$	−2864.00	−0.368313
$37$	− 9182.00i	− 1.10264i	−0.834295	−	0.551319i	$-0.814125\pi$
$37$	− 9182.00i	− 1.10264i	0.834295	−	0.551319i	$-0.185875\pi$
$38$	9728.00i	1.09286i
$39$	2352.00	0.247614
$40$	0	0
$41$	−14574.0	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$41$	−14574.0	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$42$	1568.00i	0.137159i
$43$	8108.00i	0.668717i	0.942446	+	0.334359i	$0.108520\pi$
$43$	8108.00i	0.668717i	−0.942446	+	0.334359i	$0.891480\pi$
$44$	5440.00	0.423611
$45$	0	0
$46$	8000.00	0.557437
$47$	312.000i	0.0206020i	0.999947	+	0.0103010i	$0.00327897\pi$
$47$	312.000i	0.0206020i	−0.999947	+	0.0103010i	$0.996721\pi$
$48$	2048.00i	0.128300i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	9808.00	0.528026
$52$	4704.00i	0.241245i
$53$	− 14634.0i	− 0.715605i	−0.933797	−	0.357803i	$-0.883526\pi$
$53$	− 14634.0i	− 0.715605i	0.933797	−	0.357803i	$-0.116474\pi$
$54$	13504.0	0.630199
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 19456.0i	− 0.793170i
$58$	− 26984.0i	− 1.05326i
$59$	27656.0	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$59$	27656.0	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$60$	0	0
$61$	34338.0	1.18155	0.590773	−	0.806838i	$-0.298823\pi$
$61$	34338.0	1.18155	0.590773	+	0.806838i	$0.298823\pi$
$62$	− 35424.0i	− 1.17036i
$63$	8771.00i	0.278418i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−10880.0	−0.307446
$67$	− 12316.0i	− 0.335184i	−0.985856	−	0.167592i	$-0.946401\pi$
$67$	− 12316.0i	− 0.335184i	0.985856	−	0.167592i	$-0.0535990\pi$
$68$	19616.0i	0.514444i
$69$	−16000.0	−0.404573
$70$	0	0
$71$	36920.0	0.869192	0.434596	−	0.900625i	$-0.356891\pi$
$71$	36920.0	0.869192	0.434596	+	0.900625i	$0.356891\pi$
$72$	11456.0i	0.260436i
$73$	− 61718.0i	− 1.35552i	−0.735285	−	0.677758i	$-0.762952\pi$
$73$	− 61718.0i	− 1.35552i	0.735285	−	0.677758i	$-0.237048\pi$
$74$	−36728.0	−0.779683
$75$	0	0
$76$	38912.0	0.772769
$77$	− 16660.0i	− 0.320220i
$78$	− 9408.00i	− 0.175090i
$79$	64752.0	1.16731	0.583654	−	0.812002i	$-0.301622\pi$
$79$	64752.0	1.16731	0.583654	+	0.812002i	$0.301622\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	58296.0i	0.957424i
$83$	− 77056.0i	− 1.22775i	−0.789402	−	0.613877i	$-0.789609\pi$
$83$	− 77056.0i	− 1.22775i	0.789402	−	0.613877i	$-0.210391\pi$
$84$	6272.00	0.0969857
$85$	0	0
$86$	32432.0	0.472855
$87$	53968.0i	0.764431i
$88$	− 21760.0i	− 0.299538i
$89$	8166.00	0.109278	0.0546392	−	0.998506i	$-0.482599\pi$
$89$	8166.00	0.109278	0.0546392	+	0.998506i	$0.482599\pi$
$90$	0	0
$91$	14406.0	0.182364
$92$	− 32000.0i	− 0.394167i
$93$	70848.0i	0.849416i
$94$	1248.00	0.0145678
$95$	0	0
$96$	8192.00	0.0907218
$97$	− 20650.0i	− 0.222839i	−0.993773	−	0.111419i	$-0.964460\pi$
$97$	− 20650.0i	− 0.222839i	0.993773	−	0.111419i	$-0.0355397\pi$
$98$	9604.00i	0.101015i
$99$	−60860.0	−0.624085
$100$	0	0
$101$	186250.	1.81674	0.908370	−	0.418167i	$-0.137327\pi$
$101$	186250.	1.81674	0.908370	+	0.418167i	$0.137327\pi$
$102$	− 39232.0i	− 0.373371i
$103$	− 60064.0i	− 0.557855i	−0.960312	−	0.278927i	$-0.910021\pi$
$103$	− 60064.0i	− 0.557855i	0.960312	−	0.278927i	$-0.0899789\pi$
$104$	18816.0	0.170586
$105$	0	0
$106$	−58536.0	−0.506009
$107$	− 47892.0i	− 0.404393i	−0.979345	−	0.202196i	$-0.935192\pi$
$107$	− 47892.0i	− 0.404393i	0.979345	−	0.202196i	$-0.0648080\pi$
$108$	− 54016.0i	− 0.445618i
$109$	−22102.0	−0.178183	−0.0890913	−	0.996023i	$-0.528396\pi$
$109$	−22102.0	−0.178183	−0.0890913	+	0.996023i	$0.528396\pi$
$110$	0	0
$111$	73456.0	0.565874
$112$	12544.0i	0.0944911i
$113$	− 245054.i	− 1.80537i	−0.430304	−	0.902684i	$-0.641594\pi$
$113$	− 245054.i	− 1.80537i	0.430304	−	0.902684i	$-0.358406\pi$
$114$	−77824.0	−0.560856
$115$	0	0
$116$	−107936.	−0.744769
$117$	− 52626.0i	− 0.355415i
$118$	− 110624.i	− 0.731382i
$119$	60074.0	0.388883
$120$	0	0
$121$	−45451.0	−0.282215
$122$	− 137352.i	− 0.835479i
$123$	− 116592.i	− 0.694874i
$124$	−141696.	−0.827567
$125$	0	0
$126$	35084.0	0.196871
$127$	96696.0i	0.531985i	0.963975	+	0.265992i	$0.0856996\pi$
$127$	96696.0i	0.531985i	−0.963975	+	0.265992i	$0.914300\pi$
$128$	16384.0i	0.0883883i
$129$	−64864.0	−0.343186
$130$	0	0
$131$	134368.	0.684097	0.342048	−	0.939682i	$-0.388879\pi$
$131$	134368.	0.684097	0.342048	+	0.939682i	$0.388879\pi$
$132$	43520.0i	0.217397i
$133$	− 119168.i	− 0.584158i
$134$	−49264.0	−0.237011
$135$	0	0
$136$	78464.0	0.363767
$137$	294662.i	1.34129i	0.741778	+	0.670645i	$0.233983\pi$
$137$	294662.i	1.34129i	−0.741778	+	0.670645i	$0.766017\pi$
$138$	64000.0i	0.286077i
$139$	−314944.	−1.38260	−0.691300	−	0.722568i	$-0.742962\pi$
$139$	−314944.	−1.38260	−0.691300	+	0.722568i	$0.742962\pi$
$140$	0	0
$141$	−2496.00	−0.0105730
$142$	− 147680.i	− 0.614612i
$143$	99960.0i	0.408777i
$144$	45824.0	0.184156
$145$	0	0
$146$	−246872.	−0.958495
$147$	− 19208.0i	− 0.0733143i
$148$	146912.i	0.551319i
$149$	−113622.	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$149$	−113622.	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$150$	0	0
$151$	408208.	1.45693	0.728466	−	0.685082i	$-0.240234\pi$
$151$	408208.	1.45693	0.728466	+	0.685082i	$0.240234\pi$
$152$	− 155648.i	− 0.546430i
$153$	− 219454.i	− 0.757905i
$154$	−66640.0	−0.226430
$155$	0	0
$156$	−37632.0	−0.123807
$157$	− 293546.i	− 0.950445i	−0.879866	−	0.475223i	$-0.842368\pi$
$157$	− 293546.i	− 0.950445i	0.879866	−	0.475223i	$-0.157632\pi$
$158$	− 259008.i	− 0.825411i
$159$	117072.	0.367249
$160$	0	0
$161$	−98000.0	−0.297962
$162$	− 65956.0i	− 0.197454i
$163$	− 317116.i	− 0.934866i	−0.884029	−	0.467433i	$-0.845179\pi$
$163$	− 317116.i	− 0.934866i	0.884029	−	0.467433i	$-0.154821\pi$
$164$	233184.	0.677001
$165$	0	0
$166$	−308224.	−0.868153
$167$	− 141568.i	− 0.392802i	−0.980524	−	0.196401i	$-0.937075\pi$
$167$	− 141568.i	− 0.392802i	0.980524	−	0.196401i	$-0.0629255\pi$
$168$	− 25088.0i	− 0.0685793i
$169$	284857.	0.767203
$170$	0	0
$171$	−435328.	−1.13848
$172$	− 129728.i	− 0.334359i
$173$	− 71222.0i	− 0.180925i	−0.995900	−	0.0904626i	$-0.971165\pi$
$173$	− 71222.0i	− 0.180925i	0.995900	−	0.0904626i	$-0.0288345\pi$
$174$	215872.	0.540534
$175$	0	0
$176$	−87040.0	−0.211805
$177$	221248.i	0.530819i
$178$	− 32664.0i	− 0.0772715i
$179$	−485628.	−1.13285	−0.566423	−	0.824114i	$-0.691673\pi$
$179$	−485628.	−1.13285	−0.566423	+	0.824114i	$0.691673\pi$
$180$	0	0
$181$	657090.	1.49083	0.745416	−	0.666600i	$-0.232251\pi$
$181$	657090.	1.49083	0.745416	+	0.666600i	$0.232251\pi$
$182$	− 57624.0i	− 0.128951i
$183$	274704.i	0.606369i
$184$	−128000.	−0.278718
$185$	0	0
$186$	283392.	0.600628
$187$	416840.i	0.871697i
$188$	− 4992.00i	− 0.0103010i
$189$	−165424.	−0.336856
$190$	0	0
$191$	68304.0	0.135476	0.0677381	−	0.997703i	$-0.478422\pi$
$191$	68304.0	0.135476	0.0677381	+	0.997703i	$0.478422\pi$
$192$	− 32768.0i	− 0.0641500i
$193$	352754.i	0.681677i	0.940122	+	0.340839i	$0.110711\pi$
$193$	352754.i	0.681677i	−0.940122	+	0.340839i	$0.889289\pi$
$194$	−82600.0	−0.157571
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 196982.i	− 0.361627i	−0.983517	−	0.180814i	$-0.942127\pi$
$197$	− 196982.i	− 0.361627i	0.983517	−	0.180814i	$-0.0578731\pi$
$198$	243440.i	0.441295i
$199$	1.10392e6	1.97608	0.988041	−	0.154192i	$-0.0492775\pi$
$199$	1.10392e6	1.97608	0.988041	+	0.154192i	$0.0492775\pi$
$200$	0	0
$201$	98528.0	0.172016
$202$	− 745000.i	− 1.28463i
$203$	330554.i	0.562992i
$204$	−156928.	−0.264013
$205$	0	0
$206$	−240256.	−0.394463
$207$	358000.i	0.580707i
$208$	− 75264.0i	− 0.120623i
$209$	826880.	1.30941
$210$	0	0
$211$	−103444.	−0.159955	−0.0799777	−	0.996797i	$-0.525485\pi$
$211$	−103444.	−0.159955	−0.0799777	+	0.996797i	$0.525485\pi$
$212$	234144.i	0.357803i
$213$	295360.i	0.446070i
$214$	−191568.	−0.285949
$215$	0	0
$216$	−216064.	−0.315100
$217$	433944.i	0.625582i
$218$	88408.0i	0.125994i
$219$	493744.	0.695651
$220$	0	0
$221$	−360444.	−0.496429
$222$	− 293824.i	− 0.400133i
$223$	307328.i	0.413847i	0.978357	+	0.206924i	$0.0663451\pi$
$223$	307328.i	0.413847i	−0.978357	+	0.206924i	$0.933655\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−980216.	−1.27659
$227$	891792.i	1.14868i	0.818617	+	0.574340i	$0.194741\pi$
$227$	891792.i	1.14868i	−0.818617	+	0.574340i	$0.805259\pi$
$228$	311296.i	0.396585i
$229$	−276706.	−0.348682	−0.174341	−	0.984685i	$-0.555780\pi$
$229$	−276706.	−0.348682	−0.174341	+	0.984685i	$0.555780\pi$
$230$	0	0
$231$	133280.	0.164337
$232$	431744.i	0.526631i
$233$	1.47943e6i	1.78528i	0.450772	+	0.892639i	$0.351149\pi$
$233$	1.47943e6i	1.78528i	−0.450772	+	0.892639i	$0.648851\pi$
$234$	−210504.	−0.251316
$235$	0	0
$236$	−442496.	−0.517165
$237$	518016.i	0.599063i
$238$	− 240296.i	− 0.274982i
$239$	−1.00034e6	−1.13280	−0.566402	−	0.824129i	$-0.691665\pi$
$239$	−1.00034e6	−1.13280	−0.566402	+	0.824129i	$0.691665\pi$
$240$	0	0
$241$	1.35833e6	1.50648	0.753239	−	0.657747i	$-0.228490\pi$
$241$	1.35833e6	1.50648	0.753239	+	0.657747i	$0.228490\pi$
$242$	181804.i	0.199556i
$243$	952280.i	1.03454i
$244$	−549408.	−0.590773
$245$	0	0
$246$	−466368.	−0.491350
$247$	715008.i	0.745708i
$248$	566784.i	0.585179i
$249$	616448.	0.630083
$250$	0	0
$251$	−177408.	−0.177742	−0.0888708	−	0.996043i	$-0.528326\pi$
$251$	−177408.	−0.177742	−0.0888708	+	0.996043i	$0.528326\pi$
$252$	− 140336.i	− 0.139209i
$253$	− 680000.i	− 0.667894i
$254$	386784.	0.376170
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 326658.i	− 0.308504i	−0.988032	−	0.154252i	$-0.950703\pi$
$257$	− 326658.i	− 0.308504i	0.988032	−	0.154252i	$-0.0492967\pi$
$258$	259456.i	0.242669i
$259$	449918.	0.416758
$260$	0	0
$261$	1.20753e6	1.09723
$262$	− 537472.i	− 0.483730i
$263$	− 34920.0i	− 0.0311304i	−0.999879	−	0.0155652i	$-0.995045\pi$
$263$	− 34920.0i	− 0.0311304i	0.999879	−	0.0155652i	$-0.00495476\pi$
$264$	174080.	0.153723
$265$	0	0
$266$	−476672.	−0.413062
$267$	65328.0i	0.0560817i
$268$	197056.i	0.167592i
$269$	−716458.	−0.603685	−0.301842	−	0.953358i	$-0.597602\pi$
$269$	−716458.	−0.603685	−0.301842	+	0.953358i	$0.597602\pi$
$270$	0	0
$271$	−953376.	−0.788571	−0.394286	−	0.918988i	$-0.629008\pi$
$271$	−953376.	−0.788571	−0.394286	+	0.918988i	$0.629008\pi$
$272$	− 313856.i	− 0.257222i
$273$	115248.i	0.0935894i
$274$	1.17865e6	0.948435
$275$	0	0
$276$	256000.	0.202287
$277$	1.84729e6i	1.44656i	0.690556	+	0.723279i	$0.257366\pi$
$277$	1.84729e6i	1.44656i	−0.690556	+	0.723279i	$0.742634\pi$
$278$	1.25978e6i	0.977645i
$279$	1.58522e6	1.21921
$280$	0	0
$281$	−1.99601e6	−1.50798	−0.753991	−	0.656885i	$-0.771874\pi$
$281$	−1.99601e6	−1.50798	−0.753991	+	0.656885i	$0.771874\pi$
$282$	9984.00i	0.00747622i
$283$	234088.i	0.173745i	0.996219	+	0.0868726i	$0.0276873\pi$
$283$	234088.i	0.173745i	−0.996219	+	0.0868726i	$0.972313\pi$
$284$	−590720.	−0.434596
$285$	0	0
$286$	399840.	0.289049
$287$	− 714126.i	− 0.511764i
$288$	− 183296.i	− 0.130218i
$289$	−83219.0	−0.0586108
$290$	0	0
$291$	165200.	0.114361
$292$	987488.i	0.677758i
$293$	− 2.50081e6i	− 1.70181i	−0.525320	−	0.850905i	$-0.676054\pi$
$293$	− 2.50081e6i	− 1.70181i	0.525320	−	0.850905i	$-0.323946\pi$
$294$	−76832.0	−0.0518411
$295$	0	0
$296$	587648.	0.389841
$297$	− 1.14784e6i	− 0.755075i
$298$	454488.i	0.296471i
$299$	588000.	0.380364
$300$	0	0
$301$	−397292.	−0.252751
$302$	− 1.63283e6i	− 1.03021i
$303$	1.49000e6i	0.932352i
$304$	−622592.	−0.386384
$305$	0	0
$306$	−877816.	−0.535920
$307$	− 2.34203e6i	− 1.41823i	−0.705092	−	0.709115i	$-0.749095\pi$
$307$	− 2.34203e6i	− 1.41823i	0.705092	−	0.709115i	$-0.250905\pi$
$308$	266560.i	0.160110i
$309$	480512.	0.286291
$310$	0	0
$311$	−163064.	−0.0955998	−0.0477999	−	0.998857i	$-0.515221\pi$
$311$	−163064.	−0.0955998	−0.0477999	+	0.998857i	$0.515221\pi$
$312$	150528.i	0.0875449i
$313$	1.73965e6i	1.00369i	0.864957	+	0.501847i	$0.167346\pi$
$313$	1.73965e6i	1.00369i	−0.864957	+	0.501847i	$0.832654\pi$
$314$	−1.17418e6	−0.672066
$315$	0	0
$316$	−1.03603e6	−0.583654
$317$	1.79771e6i	1.00478i	0.864640	+	0.502392i	$0.167546\pi$
$317$	1.79771e6i	1.00478i	−0.864640	+	0.502392i	$0.832454\pi$
$318$	− 468288.i	− 0.259684i
$319$	−2.29364e6	−1.26197
$320$	0	0
$321$	383136.	0.207535
$322$	392000.i	0.210691i
$323$	2.98163e6i	1.59019i
$324$	−263824.	−0.139621
$325$	0	0
$326$	−1.26846e6	−0.661050
$327$	− 176816.i	− 0.0914434i
$328$	− 932736.i	− 0.478712i
$329$	−15288.0	−0.00778683
$330$	0	0
$331$	−2.47541e6	−1.24187	−0.620937	−	0.783861i	$-0.713248\pi$
$331$	−2.47541e6	−1.24187	−0.620937	+	0.783861i	$0.713248\pi$
$332$	1.23290e6i	0.613877i
$333$	− 1.64358e6i	− 0.812231i
$334$	−566272.	−0.277753
$335$	0	0
$336$	−100352.	−0.0484929
$337$	− 89154.0i	− 0.0427628i	−0.999771	−	0.0213814i	$-0.993194\pi$
$337$	− 89154.0i	− 0.0427628i	0.999771	−	0.0213814i	$-0.00680643\pi$
$338$	− 1.13943e6i	− 0.542494i
$339$	1.96043e6	0.926515
$340$	0	0
$341$	−3.01104e6	−1.40227
$342$	1.74131e6i	0.805029i
$343$	− 117649.i	− 0.0539949i
$344$	−518912.	−0.236427
$345$	0	0
$346$	−284888.	−0.127933
$347$	− 938556.i	− 0.418443i	−0.977868	−	0.209222i	$-0.932907\pi$
$347$	− 938556.i	− 0.418443i	0.977868	−	0.209222i	$-0.0670930\pi$
$348$	− 863488.i	− 0.382215i
$349$	−3.34268e6	−1.46903	−0.734516	−	0.678591i	$-0.762591\pi$
$349$	−3.34268e6	−1.46903	−0.734516	+	0.678591i	$0.762591\pi$
$350$	0	0
$351$	992544.	0.430013
$352$	348160.i	0.149769i
$353$	− 3.76606e6i	− 1.60861i	−0.594217	−	0.804305i	$-0.702538\pi$
$353$	− 3.76606e6i	− 1.60861i	0.594217	−	0.804305i	$-0.297462\pi$
$354$	884992.	0.375345
$355$	0	0
$356$	−130656.	−0.0546392
$357$	480592.i	0.199575i
$358$	1.94251e6i	0.801044i
$359$	1.53934e6	0.630376	0.315188	−	0.949029i	$-0.397932\pi$
$359$	1.53934e6	0.630376	0.315188	+	0.949029i	$0.397932\pi$
$360$	0	0
$361$	3.43852e6	1.38869
$362$	− 2.62836e6i	− 1.05418i
$363$	− 363608.i	− 0.144833i
$364$	−230496.	−0.0911822
$365$	0	0
$366$	1.09882e6	0.428768
$367$	859312.i	0.333032i	0.986039	+	0.166516i	$0.0532517\pi$
$367$	859312.i	0.333032i	−0.986039	+	0.166516i	$0.946748\pi$
$368$	512000.i	0.197084i
$369$	−2.60875e6	−0.997392
$370$	0	0
$371$	717066.	0.270473
$372$	− 1.13357e6i	− 0.424708i
$373$	− 976586.i	− 0.363445i	−0.983350	−	0.181722i	$-0.941833\pi$
$373$	− 976586.i	− 0.363445i	0.983350	−	0.181722i	$-0.0581672\pi$
$374$	1.66736e6	0.616383
$375$	0	0
$376$	−19968.0	−0.00728392
$377$	− 1.98332e6i	− 0.718688i
$378$	661696.i	0.238193i
$379$	−106444.	−0.0380648	−0.0190324	−	0.999819i	$-0.506059\pi$
$379$	−106444.	−0.0380648	−0.0190324	+	0.999819i	$0.506059\pi$
$380$	0	0
$381$	−773568.	−0.273015
$382$	− 273216.i	− 0.0957961i
$383$	− 2.00634e6i	− 0.698889i	−0.936957	−	0.349445i	$-0.886370\pi$
$383$	− 2.00634e6i	− 0.698889i	0.936957	−	0.349445i	$-0.113630\pi$
$384$	−131072.	−0.0453609
$385$	0	0
$386$	1.41102e6	0.482018
$387$	1.45133e6i	0.492594i
$388$	330400.i	0.111419i
$389$	684002.	0.229184	0.114592	−	0.993413i	$-0.463444\pi$
$389$	684002.	0.229184	0.114592	+	0.993413i	$0.463444\pi$
$390$	0	0
$391$	2.45200e6	0.811108
$392$	− 153664.i	− 0.0505076i
$393$	1.07494e6i	0.351079i
$394$	−787928.	−0.255709
$395$	0	0
$396$	973760.	0.312043
$397$	222870.i	0.0709701i	0.999370	+	0.0354850i	$0.0112976\pi$
$397$	222870.i	0.0709701i	−0.999370	+	0.0354850i	$0.988702\pi$
$398$	− 4.41568e6i	− 1.39730i
$399$	953344.	0.299790
$400$	0	0
$401$	1.90072e6	0.590279	0.295140	−	0.955454i	$-0.404634\pi$
$401$	1.90072e6	0.590279	0.295140	+	0.955454i	$0.404634\pi$
$402$	− 394112.i	− 0.121634i
$403$	− 2.60366e6i	− 0.798587i
$404$	−2.98000e6	−0.908370
$405$	0	0
$406$	1.32222e6	0.398096
$407$	3.12188e6i	0.934179i
$408$	627712.i	0.186685i
$409$	−1.77715e6	−0.525311	−0.262656	−	0.964890i	$-0.584598\pi$
$409$	−1.77715e6	−0.525311	−0.262656	+	0.964890i	$0.584598\pi$
$410$	0	0
$411$	−2.35730e6	−0.688350
$412$	961024.i	0.278927i
$413$	1.35514e6i	0.390940i
$414$	1.43200e6	0.410622
$415$	0	0
$416$	−301056.	−0.0852931
$417$	− 2.51955e6i	− 0.709550i
$418$	− 3.30752e6i	− 0.925895i
$419$	−28056.0	−0.00780712	−0.00390356	−	0.999992i	$-0.501243\pi$
$419$	−28056.0	−0.00780712	−0.00390356	+	0.999992i	$0.501243\pi$
$420$	0	0
$421$	−2.70897e6	−0.744902	−0.372451	−	0.928052i	$-0.621482\pi$
$421$	−2.70897e6	−0.744902	−0.372451	+	0.928052i	$0.621482\pi$
$422$	413776.i	0.113106i
$423$	55848.0i	0.0151760i
$424$	936576.	0.253005
$425$	0	0
$426$	1.18144e6	0.315419
$427$	1.68256e6i	0.446582i
$428$	766272.i	0.202196i
$429$	−799680.	−0.209784
$430$	0	0
$431$	5.53898e6	1.43627	0.718136	−	0.695902i	$-0.244995\pi$
$431$	5.53898e6	1.43627	0.718136	+	0.695902i	$0.244995\pi$
$432$	864256.i	0.222809i
$433$	− 868294.i	− 0.222560i	−0.993789	−	0.111280i	$-0.964505\pi$
$433$	− 868294.i	− 0.222560i	0.993789	−	0.111280i	$-0.0354950\pi$
$434$	1.73578e6	0.442353
$435$	0	0
$436$	353632.	0.0890913
$437$	− 4.86400e6i	− 1.21840i
$438$	− 1.97498e6i	− 0.491900i
$439$	1.13767e6	0.281745	0.140872	−	0.990028i	$-0.455009\pi$
$439$	1.13767e6	0.281745	0.140872	+	0.990028i	$0.455009\pi$
$440$	0	0
$441$	−429779.	−0.105232
$442$	1.44178e6i	0.351028i
$443$	1.75399e6i	0.424636i	0.977201	+	0.212318i	$0.0681013\pi$
$443$	1.75399e6i	0.424636i	−0.977201	+	0.212318i	$0.931899\pi$
$444$	−1.17530e6	−0.282937
$445$	0	0
$446$	1.22931e6	0.292634
$447$	− 908976.i	− 0.215171i
$448$	− 200704.i	− 0.0472456i
$449$	−2.41674e6	−0.565736	−0.282868	−	0.959159i	$-0.591286\pi$
$449$	−2.41674e6	−0.565736	−0.282868	+	0.959159i	$0.591286\pi$
$450$	0	0
$451$	4.95516e6	1.14714
$452$	3.92086e6i	0.902684i
$453$	3.26566e6i	0.747698i
$454$	3.56717e6	0.812239
$455$	0	0
$456$	1.24518e6	0.280428
$457$	127430.i	0.0285418i	0.999898	+	0.0142709i	$0.00454272\pi$
$457$	127430.i	0.0285418i	−0.999898	+	0.0142709i	$0.995457\pi$
$458$	1.10682e6i	0.246556i
$459$	4.13898e6	0.916983
$460$	0	0
$461$	−128198.	−0.0280950	−0.0140475	−	0.999901i	$-0.504472\pi$
$461$	−128198.	−0.0280950	−0.0140475	+	0.999901i	$0.504472\pi$
$462$	− 533120.i	− 0.116204i
$463$	− 4.01653e6i	− 0.870760i	−0.900247	−	0.435380i	$-0.856614\pi$
$463$	− 4.01653e6i	− 0.870760i	0.900247	−	0.435380i	$-0.143386\pi$
$464$	1.72698e6	0.372384
$465$	0	0
$466$	5.91774e6	1.26238
$467$	− 8.67246e6i	− 1.84014i	−0.391757	−	0.920069i	$-0.628133\pi$
$467$	− 8.67246e6i	− 1.84014i	0.391757	−	0.920069i	$-0.371867\pi$
$468$	842016.i	0.177707i
$469$	603484.	0.126687
$470$	0	0
$471$	2.34837e6	0.487769
$472$	1.76998e6i	0.365691i
$473$	− 2.75672e6i	− 0.566552i
$474$	2.07206e6	0.423601
$475$	0	0
$476$	−961184.	−0.194442
$477$	− 2.61949e6i	− 0.527133i
$478$	4.00138e6i	0.801013i
$479$	−8.28946e6	−1.65077	−0.825387	−	0.564567i	$-0.809043\pi$
$479$	−8.28946e6	−1.65077	−0.825387	+	0.564567i	$0.809043\pi$
$480$	0	0
$481$	−2.69951e6	−0.532013
$482$	− 5.43332e6i	− 1.06524i
$483$	− 784000.i	− 0.152914i
$484$	727216.	0.141107
$485$	0	0
$486$	3.80912e6	0.731533
$487$	8.91770e6i	1.70385i	0.523667	+	0.851923i	$0.324563\pi$
$487$	8.91770e6i	1.70385i	−0.523667	+	0.851923i	$0.675437\pi$
$488$	2.19763e6i	0.417739i
$489$	2.53693e6	0.479773
$490$	0	0
$491$	−5.71537e6	−1.06989	−0.534947	−	0.844886i	$-0.679668\pi$
$491$	−5.71537e6	−1.06989	−0.534947	+	0.844886i	$0.679668\pi$
$492$	1.86547e6i	0.347437i
$493$	− 8.27060e6i	− 1.53257i
$494$	2.86003e6	0.527295
$495$	0	0
$496$	2.26714e6	0.413784
$497$	1.80908e6i	0.328524i
$498$	− 2.46579e6i	− 0.445536i
$499$	−125116.	−0.0224937	−0.0112469	−	0.999937i	$-0.503580\pi$
$499$	−125116.	−0.0224937	−0.0112469	+	0.999937i	$0.503580\pi$
$500$	0	0
$501$	1.13254e6	0.201586
$502$	709632.i	0.125682i
$503$	− 2.77116e6i	− 0.488362i	−0.969730	−	0.244181i	$-0.921481\pi$
$503$	− 2.77116e6i	− 0.488362i	0.969730	−	0.244181i	$-0.0785191\pi$
$504$	−561344.	−0.0984357
$505$	0	0
$506$	−2.72000e6	−0.472272
$507$	2.27886e6i	0.393729i
$508$	− 1.54714e6i	− 0.265992i
$509$	138534.	0.0237007	0.0118504	−	0.999930i	$-0.496228\pi$
$509$	138534.	0.0237007	0.0118504	+	0.999930i	$0.496228\pi$
$510$	0	0
$511$	3.02418e6	0.512337
$512$	− 262144.i	− 0.0441942i
$513$	− 8.21043e6i	− 1.37744i
$514$	−1.30663e6	−0.218145
$515$	0	0
$516$	1.03782e6	0.171593
$517$	− 106080.i	− 0.0174545i
$518$	− 1.79967e6i	− 0.294692i
$519$	569776.	0.0928508
$520$	0	0
$521$	−1.80281e6	−0.290976	−0.145488	−	0.989360i	$-0.546475\pi$
$521$	−1.80281e6	−0.290976	−0.145488	+	0.989360i	$0.546475\pi$
$522$	− 4.83014e6i	− 0.775860i
$523$	9.77247e6i	1.56225i	0.624375	+	0.781124i	$0.285354\pi$
$523$	9.77247e6i	1.56225i	−0.624375	+	0.781124i	$0.714646\pi$
$524$	−2.14989e6	−0.342048
$525$	0	0
$526$	−139680.	−0.0220125
$527$	− 1.08575e7i	− 1.70295i
$528$	− 696320.i	− 0.108699i
$529$	2.43634e6	0.378529
$530$	0	0
$531$	4.95042e6	0.761914
$532$	1.90669e6i	0.292079i
$533$	4.28476e6i	0.653293i
$534$	261312.	0.0396558
$535$	0	0
$536$	788224.	0.118505
$537$	− 3.88502e6i	− 0.581377i
$538$	2.86583e6i	0.426869i
$539$	816340.	0.121032
$540$	0	0
$541$	2.45504e6	0.360633	0.180316	−	0.983609i	$-0.442288\pi$
$541$	2.45504e6	0.360633	0.180316	+	0.983609i	$0.442288\pi$
$542$	3.81350e6i	0.557604i
$543$	5.25672e6i	0.765095i
$544$	−1.25542e6	−0.181883
$545$	0	0
$546$	460992.	0.0661777
$547$	− 1.32081e7i	− 1.88744i	−0.330743	−	0.943721i	$-0.607299\pi$
$547$	− 1.32081e7i	− 1.88744i	0.330743	−	0.943721i	$-0.392701\pi$
$548$	− 4.71459e6i	− 0.670645i
$549$	6.14650e6	0.870356
$550$	0	0
$551$	−1.64063e7	−2.30214
$552$	− 1.02400e6i	− 0.143038i
$553$	3.17285e6i	0.441201i
$554$	7.38916e6	1.02287
$555$	0	0
$556$	5.03910e6	0.691300
$557$	− 7.83293e6i	− 1.06976i	−0.844928	−	0.534880i	$-0.820357\pi$
$557$	− 7.83293e6i	− 1.06976i	0.844928	−	0.534880i	$-0.179643\pi$
$558$	− 6.34090e6i	− 0.862115i
$559$	2.38375e6	0.322650
$560$	0	0
$561$	−3.33472e6	−0.447355
$562$	7.98402e6i	1.06630i
$563$	3.57908e6i	0.475883i	0.971279	+	0.237942i	$0.0764727\pi$
$563$	3.57908e6i	0.475883i	−0.971279	+	0.237942i	$0.923527\pi$
$564$	39936.0	0.00528648
$565$	0	0
$566$	936352.	0.122856
$567$	807961.i	0.105544i
$568$	2.36288e6i	0.307306i
$569$	3.39581e6	0.439707	0.219853	−	0.975533i	$-0.429442\pi$
$569$	3.39581e6	0.439707	0.219853	+	0.975533i	$0.429442\pi$
$570$	0	0
$571$	−1.47695e6	−0.189572	−0.0947862	−	0.995498i	$-0.530217\pi$
$571$	−1.47695e6	−0.189572	−0.0947862	+	0.995498i	$0.530217\pi$
$572$	− 1.59936e6i	− 0.204388i
$573$	546432.i	0.0695264i
$574$	−2.85650e6	−0.361872
$575$	0	0
$576$	−733184.	−0.0920782
$577$	1.49961e7i	1.87516i	0.347771	+	0.937580i	$0.386939\pi$
$577$	1.49961e7i	1.87516i	−0.347771	+	0.937580i	$0.613061\pi$
$578$	332876.i	0.0414441i
$579$	−2.82203e6	−0.349837
$580$	0	0
$581$	3.77574e6	0.464047
$582$	− 660800.i	− 0.0808654i
$583$	4.97556e6i	0.606276i
$584$	3.94995e6	0.479247
$585$	0	0
$586$	−1.00032e7	−1.20336
$587$	3.29291e6i	0.394444i	0.980359	+	0.197222i	$0.0631919\pi$
$587$	3.29291e6i	0.394444i	−0.980359	+	0.197222i	$0.936808\pi$
$588$	307328.i	0.0366572i
$589$	−2.15378e7	−2.55807
$590$	0	0
$591$	1.57586e6	0.185587
$592$	− 2.35059e6i	− 0.275660i
$593$	− 1.17908e7i	− 1.37692i	−0.725275	−	0.688459i	$-0.758287\pi$
$593$	− 1.17908e7i	− 1.37692i	0.725275	−	0.688459i	$-0.241713\pi$
$594$	−4.59136e6	−0.533919
$595$	0	0
$596$	1.81795e6	0.209636
$597$	8.83136e6i	1.01413i
$598$	− 2.35200e6i	− 0.268958i
$599$	1.52642e6	0.173823	0.0869117	−	0.996216i	$-0.472300\pi$
$599$	1.52642e6	0.173823	0.0869117	+	0.996216i	$0.472300\pi$
$600$	0	0
$601$	−1.00142e7	−1.13092	−0.565458	−	0.824777i	$-0.691301\pi$
$601$	−1.00142e7	−1.13092	−0.565458	+	0.824777i	$0.691301\pi$
$602$	1.58917e6i	0.178722i
$603$	− 2.20456e6i	− 0.246905i
$604$	−6.53133e6	−0.728466
$605$	0	0
$606$	5.96000e6	0.659272
$607$	− 1.20660e7i	− 1.32920i	−0.747200	−	0.664599i	$-0.768602\pi$
$607$	− 1.20660e7i	− 1.32920i	0.747200	−	0.664599i	$-0.231398\pi$
$608$	2.49037e6i	0.273215i
$609$	−2.64443e6	−0.288928
$610$	0	0
$611$	91728.0	0.00994029
$612$	3.51126e6i	0.378953i
$613$	5.81950e6i	0.625511i	0.949834	+	0.312755i	$0.101252\pi$
$613$	5.81950e6i	0.625511i	−0.949834	+	0.312755i	$0.898748\pi$
$614$	−9.36813e6	−1.00284
$615$	0	0
$616$	1.06624e6	0.113215
$617$	4.16589e6i	0.440550i	0.975438	+	0.220275i	$0.0706955\pi$
$617$	4.16589e6i	0.440550i	−0.975438	+	0.220275i	$0.929305\pi$
$618$	− 1.92205e6i	− 0.202438i
$619$	8.08090e6	0.847683	0.423841	−	0.905736i	$-0.360681\pi$
$619$	8.08090e6	0.847683	0.423841	+	0.905736i	$0.360681\pi$
$620$	0	0
$621$	−6.75200e6	−0.702592
$622$	652256.i	0.0675993i
$623$	400134.i	0.0413034i
$624$	602112.	0.0619036
$625$	0	0
$626$	6.95860e6	0.709718
$627$	6.61504e6i	0.671991i
$628$	4.69674e6i	0.475223i
$629$	−1.12571e7	−1.13449
$630$	0	0
$631$	−8.40878e6	−0.840735	−0.420368	−	0.907354i	$-0.638099\pi$
$631$	−8.40878e6	−0.840735	−0.420368	+	0.907354i	$0.638099\pi$
$632$	4.14413e6i	0.412706i
$633$	− 827552.i	− 0.0820892i
$634$	7.19086e6	0.710489
$635$	0	0
$636$	−1.87315e6	−0.183624
$637$	705894.i	0.0689272i
$638$	9.17456e6i	0.892347i
$639$	6.60868e6	0.640269
$640$	0	0
$641$	6.29760e6	0.605383	0.302691	−	0.953089i	$-0.402115\pi$
$641$	6.29760e6	0.605383	0.302691	+	0.953089i	$0.402115\pi$
$642$	− 1.53254e6i	− 0.146749i
$643$	4.39762e6i	0.419460i	0.977759	+	0.209730i	$0.0672585\pi$
$643$	4.39762e6i	0.419460i	−0.977759	+	0.209730i	$0.932741\pi$
$644$	1.56800e6	0.148981
$645$	0	0
$646$	1.19265e7	1.12443
$647$	− 6.55397e6i	− 0.615522i	−0.951464	−	0.307761i	$-0.900420\pi$
$647$	− 6.55397e6i	− 0.615522i	0.951464	−	0.307761i	$-0.0995797\pi$
$648$	1.05530e6i	0.0987272i
$649$	−9.40304e6	−0.876308
$650$	0	0
$651$	−3.47155e6	−0.321049
$652$	5.07386e6i	0.467433i
$653$	3.79652e6i	0.348420i	0.984709	+	0.174210i	$0.0557371\pi$
$653$	3.79652e6i	0.348420i	−0.984709	+	0.174210i	$0.944263\pi$
$654$	−707264.	−0.0646602
$655$	0	0
$656$	−3.73094e6	−0.338500
$657$	− 1.10475e7i	− 0.998508i
$658$	61152.0i	0.00550612i
$659$	8.82684e6	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$659$	8.82684e6	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$660$	0	0
$661$	−341270.	−0.0303805	−0.0151902	−	0.999885i	$-0.504835\pi$
$661$	−341270.	−0.0303805	−0.0151902	+	0.999885i	$0.504835\pi$
$662$	9.90165e6i	0.878137i
$663$	− 2.88355e6i	− 0.254767i
$664$	4.93158e6	0.434076
$665$	0	0
$666$	−6.57431e6	−0.574334
$667$	1.34920e7i	1.17425i
$668$	2.26509e6i	0.196401i
$669$	−2.45862e6	−0.212386
$670$	0	0
$671$	−1.16749e7	−1.00103
$672$	401408.i	0.0342896i
$673$	4.41807e6i	0.376006i	0.982168	+	0.188003i	$0.0602014\pi$
$673$	4.41807e6i	0.376006i	−0.982168	+	0.188003i	$0.939799\pi$
$674$	−356616.	−0.0302379
$675$	0	0
$676$	−4.55771e6	−0.383601
$677$	− 1.63858e7i	− 1.37403i	−0.726644	−	0.687014i	$-0.758921\pi$
$677$	− 1.63858e7i	− 1.37403i	0.726644	−	0.687014i	$-0.241079\pi$
$678$	− 7.84173e6i	− 0.655145i
$679$	1.01185e6	0.0842251
$680$	0	0
$681$	−7.13434e6	−0.589503
$682$	1.20442e7i	0.991552i
$683$	− 1.75399e7i	− 1.43872i	−0.694638	−	0.719360i	$-0.744435\pi$
$683$	− 1.75399e7i	− 1.43872i	0.694638	−	0.719360i	$-0.255565\pi$
$684$	6.96525e6	0.569241
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 2.21365e6i	− 0.178944i
$688$	2.07565e6i	0.167179i
$689$	−4.30240e6	−0.345273
$690$	0	0
$691$	3.14638e6	0.250678	0.125339	−	0.992114i	$-0.459998\pi$
$691$	3.14638e6	0.250678	0.125339	+	0.992114i	$0.459998\pi$
$692$	1.13955e6i	0.0904626i
$693$	− 2.98214e6i	− 0.235882i
$694$	−3.75422e6	−0.295884
$695$	0	0
$696$	−3.45395e6	−0.270267
$697$	1.78677e7i	1.39312i
$698$	1.33707e7i	1.03876i
$699$	−1.18355e7	−0.916205
$700$	0	0
$701$	−1.90919e7	−1.46742	−0.733709	−	0.679464i	$-0.762212\pi$
$701$	−1.90919e7	−1.46742	−0.733709	+	0.679464i	$0.762212\pi$
$702$	− 3.97018e6i	− 0.304065i
$703$	2.23306e7i	1.70417i
$704$	1.39264e6	0.105903
$705$	0	0
$706$	−1.50642e7	−1.13746
$707$	9.12625e6i	0.686663i
$708$	− 3.53997e6i	− 0.265409i
$709$	−990974.	−0.0740366	−0.0370183	−	0.999315i	$-0.511786\pi$
$709$	−990974.	−0.0740366	−0.0370183	+	0.999315i	$0.511786\pi$
$710$	0	0
$711$	1.15906e7	0.859869
$712$	522624.i	0.0386358i
$713$	1.77120e7i	1.30480i
$714$	1.92237e6	0.141121
$715$	0	0
$716$	7.77005e6	0.566423
$717$	− 8.00275e6i	− 0.581355i
$718$	− 6.15738e6i	− 0.445743i
$719$	1.69014e7	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$719$	1.69014e7	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$720$	0	0
$721$	2.94314e6	0.210849
$722$	− 1.37541e7i	− 0.981950i
$723$	1.08666e7i	0.773125i
$724$	−1.05134e7	−0.745416
$725$	0	0
$726$	−1.45443e6	−0.102412
$727$	2.34302e7i	1.64414i	0.569384	+	0.822071i	$0.307182\pi$
$727$	2.34302e7i	1.64414i	−0.569384	+	0.822071i	$0.692818\pi$
$728$	921984.i	0.0644755i
$729$	−3.61141e6	−0.251686
$730$	0	0
$731$	9.94041e6	0.688035
$732$	− 4.39526e6i	− 0.303185i
$733$	975810.i	0.0670819i	0.999437	+	0.0335409i	$0.0106784\pi$
$733$	975810.i	0.0670819i	−0.999437	+	0.0335409i	$0.989322\pi$
$734$	3.43725e6	0.235489
$735$	0	0
$736$	2.04800e6	0.139359
$737$	4.18744e6i	0.283975i
$738$	1.04350e7i	0.705263i
$739$	6.30208e6	0.424495	0.212247	−	0.977216i	$-0.431922\pi$
$739$	6.30208e6	0.424495	0.212247	+	0.977216i	$0.431922\pi$
$740$	0	0
$741$	−5.72006e6	−0.382697
$742$	− 2.86826e6i	− 0.191253i
$743$	− 6.95698e6i	− 0.462326i	−0.972915	−	0.231163i	$-0.925747\pi$
$743$	− 6.95698e6i	− 0.462326i	0.972915	−	0.231163i	$-0.0742531\pi$
$744$	−4.53427e6	−0.300314
$745$	0	0
$746$	−3.90634e6	−0.256994
$747$	− 1.37930e7i	− 0.904395i
$748$	− 6.66944e6i	− 0.435848i
$749$	2.34671e6	0.152846
$750$	0	0
$751$	2.74535e7	1.77622	0.888112	−	0.459628i	$-0.152017\pi$
$751$	2.74535e7	1.77622	0.888112	+	0.459628i	$0.152017\pi$
$752$	79872.0i	0.00515051i
$753$	− 1.41926e6i	− 0.0912170i
$754$	−7.93330e6	−0.508189
$755$	0	0
$756$	2.64678e6	0.168428
$757$	1.96889e7i	1.24877i	0.781118	+	0.624384i	$0.214650\pi$
$757$	1.96889e7i	1.24877i	−0.781118	+	0.624384i	$0.785350\pi$
$758$	425776.i	0.0269159i
$759$	5.44000e6	0.342763
$760$	0	0
$761$	−2.82079e7	−1.76567	−0.882835	−	0.469684i	$-0.844368\pi$
$761$	−2.82079e7	−1.76567	−0.882835	+	0.469684i	$0.844368\pi$
$762$	3.09427e6i	0.193051i
$763$	− 1.08300e6i	− 0.0673467i
$764$	−1.09286e6	−0.0677381
$765$	0	0
$766$	−8.02538e6	−0.494189
$767$	− 8.13086e6i	− 0.499055i
$768$	524288.i	0.0320750i
$769$	1.38081e6	0.0842009	0.0421005	−	0.999113i	$-0.486595\pi$
$769$	1.38081e6	0.0842009	0.0421005	+	0.999113i	$0.486595\pi$
$770$	0	0
$771$	2.61326e6	0.158324
$772$	− 5.64406e6i	− 0.340839i
$773$	− 1.54347e7i	− 0.929074i	−0.885554	−	0.464537i	$-0.846221\pi$
$773$	− 1.54347e7i	− 0.929074i	0.885554	−	0.464537i	$-0.153779\pi$
$774$	5.80533e6	0.348317
$775$	0	0
$776$	1.32160e6	0.0787854
$777$	3.59934e6i	0.213880i
$778$	− 2.73601e6i	− 0.162057i
$779$	3.54440e7	2.09266
$780$	0	0
$781$	−1.25528e7	−0.736399
$782$	− 9.80800e6i	− 0.573540i
$783$	2.27745e7i	1.32753i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	4.29978e6	0.248250
$787$	7.10107e6i	0.408683i	0.978900	+	0.204342i	$0.0655054\pi$
$787$	7.10107e6i	0.408683i	−0.978900	+	0.204342i	$0.934495\pi$
$788$	3.15171e6i	0.180814i
$789$	279360.	0.0159761
$790$	0	0
$791$	1.20076e7	0.682365
$792$	− 3.89504e6i	− 0.220647i
$793$	− 1.00954e7i	− 0.570085i
$794$	891480.	0.0501834
$795$	0	0
$796$	−1.76627e7	−0.988041
$797$	− 6.48182e6i	− 0.361452i	−0.983533	−	0.180726i	$-0.942155\pi$
$797$	− 6.48182e6i	− 0.361452i	0.983533	−	0.180726i	$-0.0578448\pi$
$798$	− 3.81338e6i	− 0.211984i
$799$	382512.	0.0211972
$800$	0	0
$801$	1.46171e6	0.0804973
$802$	− 7.60289e6i	− 0.417391i
$803$	2.09841e7i	1.14842i
$804$	−1.57645e6	−0.0860081
$805$	0	0
$806$	−1.04147e7	−0.564686
$807$	− 5.73166e6i	− 0.309811i
$808$	1.19200e7i	0.642315i
$809$	−1.60578e7	−0.862610	−0.431305	−	0.902206i	$-0.641947\pi$
$809$	−1.60578e7	−0.862610	−0.431305	+	0.902206i	$0.641947\pi$
$810$	0	0
$811$	4.84775e6	0.258814	0.129407	−	0.991592i	$-0.458693\pi$
$811$	4.84775e6	0.258814	0.129407	+	0.991592i	$0.458693\pi$
$812$	− 5.28886e6i	− 0.281496i
$813$	− 7.62701e6i	− 0.404695i
$814$	1.24875e7	0.660564
$815$	0	0
$816$	2.51085e6	0.132006
$817$	− 1.97187e7i	− 1.03353i
$818$	7.10862e6i	0.371451i
$819$	2.57867e6	0.134334
$820$	0	0
$821$	2.17976e7	1.12863	0.564314	−	0.825560i	$-0.309141\pi$
$821$	2.17976e7	1.12863	0.564314	+	0.825560i	$0.309141\pi$
$822$	9.42918e6i	0.486737i
$823$	3.20206e7i	1.64790i	0.566665	+	0.823948i	$0.308233\pi$
$823$	3.20206e7i	1.64790i	−0.566665	+	0.823948i	$0.691767\pi$
$824$	3.84410e6	0.197231
$825$	0	0
$826$	5.42058e6	0.276436
$827$	− 2.19008e7i	− 1.11352i	−0.830675	−	0.556758i	$-0.812045\pi$
$827$	− 2.19008e7i	− 1.11352i	0.830675	−	0.556758i	$-0.187955\pi$
$828$	− 5.72800e6i	− 0.290354i
$829$	1.45999e7	0.737844	0.368922	−	0.929460i	$-0.379727\pi$
$829$	1.45999e7	0.737844	0.368922	+	0.929460i	$0.379727\pi$
$830$	0	0
$831$	−1.47783e7	−0.742374
$832$	1.20422e6i	0.0603113i
$833$	2.94363e6i	0.146984i
$834$	−1.00782e7	−0.501728
$835$	0	0
$836$	−1.32301e7	−0.654707
$837$	2.98979e7i	1.47512i
$838$	112224.i	0.00552047i
$839$	−4.60947e6	−0.226072	−0.113036	−	0.993591i	$-0.536058\pi$
$839$	−4.60947e6	−0.226072	−0.113036	+	0.993591i	$0.536058\pi$
$840$	0	0
$841$	2.49974e7	1.21872
$842$	1.08359e7i	0.526725i
$843$	− 1.59680e7i	− 0.773897i
$844$	1.65510e6	0.0799777
$845$	0	0
$846$	223392.	0.0107310
$847$	− 2.22710e6i	− 0.106667i
$848$	− 3.74630e6i	− 0.178901i
$849$	−1.87270e6	−0.0891661
$850$	0	0
$851$	1.83640e7	0.869247
$852$	− 4.72576e6i	− 0.223035i
$853$	− 1.98437e7i	− 0.933793i	−0.884312	−	0.466897i	$-0.845372\pi$
$853$	− 1.98437e7i	− 0.933793i	0.884312	−	0.466897i	$-0.154628\pi$
$854$	6.73025e6	0.315781
$855$	0	0
$856$	3.06509e6	0.142974
$857$	1.22960e6i	0.0571888i	0.999591	+	0.0285944i	$0.00910312\pi$
$857$	1.22960e6i	0.0571888i	−0.999591	+	0.0285944i	$0.990897\pi$
$858$	3.19872e6i	0.148340i
$859$	−3.33041e7	−1.53998	−0.769989	−	0.638058i	$-0.779738\pi$
$859$	−3.33041e7	−1.53998	−0.769989	+	0.638058i	$0.779738\pi$
$860$	0	0
$861$	5.71301e6	0.262638
$862$	− 2.21559e7i	− 1.01560i
$863$	− 2.36616e7i	− 1.08148i	−0.841191	−	0.540738i	$-0.818145\pi$
$863$	− 2.36616e7i	− 1.08148i	0.841191	−	0.540738i	$-0.181855\pi$
$864$	3.45702e6	0.157550
$865$	0	0
$866$	−3.47318e6	−0.157374
$867$	− 665752.i	− 0.0300791i
$868$	− 6.94310e6i	− 0.312791i
$869$	−2.20157e7	−0.988969
$870$	0	0
$871$	−3.62090e6	−0.161723
$872$	− 1.41453e6i	− 0.0629971i
$873$	− 3.69635e6i	− 0.164149i
$874$	−1.94560e7	−0.861539
$875$	0	0
$876$	−7.89990e6	−0.347826
$877$	2.37812e7i	1.04408i	0.852920	+	0.522042i	$0.174830\pi$
$877$	2.37812e7i	1.04408i	−0.852920	+	0.522042i	$0.825170\pi$
$878$	− 4.55069e6i	− 0.199224i
$879$	2.00064e7	0.873369
$880$	0	0
$881$	−1.41871e7	−0.615818	−0.307909	−	0.951416i	$-0.599629\pi$
$881$	−1.41871e7	−0.615818	−0.307909	+	0.951416i	$0.599629\pi$
$882$	1.71912e6i	0.0744104i
$883$	2.09281e7i	0.903293i	0.892197	+	0.451647i	$0.149163\pi$
$883$	2.09281e7i	0.903293i	−0.892197	+	0.451647i	$0.850837\pi$
$884$	5.76710e6	0.248214
$885$	0	0
$886$	7.01595e6	0.300263
$887$	7.98586e6i	0.340810i	0.985374	+	0.170405i	$0.0545076\pi$
$887$	7.98586e6i	0.340810i	−0.985374	+	0.170405i	$0.945492\pi$
$888$	4.70118e6i	0.200067i
$889$	−4.73810e6	−0.201071
$890$	0	0
$891$	−5.60626e6	−0.236581
$892$	− 4.91725e6i	− 0.206924i
$893$	− 758784.i	− 0.0318412i
$894$	−3.63590e6	−0.152149
$895$	0	0
$896$	−802816.	−0.0334077
$897$	4.70400e6i	0.195203i
$898$	9.66695e6i	0.400036i
$899$	5.97426e7	2.46538
$900$	0	0
$901$	−1.79413e7	−0.736278
$902$	− 1.98206e7i	− 0.811150i
$903$	− 3.17834e6i	− 0.129712i
$904$	1.56835e7	0.638294
$905$	0	0
$906$	1.30627e7	0.528702
$907$	2.31861e7i	0.935856i	0.883767	+	0.467928i	$0.154999\pi$
$907$	2.31861e7i	0.935856i	−0.883767	+	0.467928i	$0.845001\pi$
$908$	− 1.42687e7i	− 0.574340i
$909$	3.33388e7	1.33826
$910$	0	0
$911$	1.65299e7	0.659895	0.329948	−	0.943999i	$-0.392969\pi$
$911$	1.65299e7	0.659895	0.329948	+	0.943999i	$0.392969\pi$
$912$	− 4.98074e6i	− 0.198293i
$913$	2.61990e7i	1.04018i
$914$	509720.	0.0201821
$915$	0	0
$916$	4.42730e6	0.174341
$917$	6.58403e6i	0.258564i
$918$	− 1.65559e7i	− 0.648405i
$919$	−1.28087e7	−0.500283	−0.250142	−	0.968209i	$-0.580477\pi$
$919$	−1.28087e7	−0.500283	−0.250142	+	0.968209i	$0.580477\pi$
$920$	0	0
$921$	1.87363e7	0.727836
$922$	512792.i	0.0198662i
$923$	− 1.08545e7i	− 0.419377i
$924$	−2.13248e6	−0.0821684
$925$	0	0
$926$	−1.60661e7	−0.615720
$927$	− 1.07515e7i	− 0.410930i
$928$	− 6.90790e6i	− 0.263315i
$929$	−2.97319e7	−1.13027	−0.565136	−	0.824998i	$-0.691176\pi$
$929$	−2.97319e7	−1.13027	−0.565136	+	0.824998i	$0.691176\pi$
$930$	0	0
$931$	5.83923e6	0.220791
$932$	− 2.36709e7i	− 0.892639i
$933$	− 1.30451e6i	− 0.0490619i
$934$	−3.46899e7	−1.30117
$935$	0	0
$936$	3.36806e6	0.125658
$937$	− 1.10970e7i	− 0.412911i	−0.978456	−	0.206456i	$-0.933807\pi$
$937$	− 1.10970e7i	− 0.412911i	0.978456	−	0.206456i	$-0.0661929\pi$
$938$	− 2.41394e6i	− 0.0895816i
$939$	−1.39172e7	−0.515096
$940$	0	0
$941$	3.74313e7	1.37804	0.689019	−	0.724743i	$-0.258042\pi$
$941$	3.74313e7	1.37804	0.689019	+	0.724743i	$0.258042\pi$
$942$	− 9.39347e6i	− 0.344905i
$943$	− 2.91480e7i	− 1.06741i
$944$	7.07994e6	0.258583
$945$	0	0
$946$	−1.10269e7	−0.400613
$947$	− 1.50907e7i	− 0.546808i	−0.961899	−	0.273404i	$-0.911850\pi$
$947$	− 1.50907e7i	− 0.546808i	0.961899	−	0.273404i	$-0.0881496\pi$
$948$	− 8.28826e6i	− 0.299531i
$949$	−1.81451e7	−0.654024
$950$	0	0
$951$	−1.43817e7	−0.515655
$952$	3.84474e6i	0.137491i
$953$	− 2.15741e7i	− 0.769484i	−0.923024	−	0.384742i	$-0.874290\pi$
$953$	− 2.15741e7i	− 0.769484i	0.923024	−	0.384742i	$-0.125710\pi$
$954$	−1.04779e7	−0.372739
$955$	0	0
$956$	1.60055e7	0.566402
$957$	− 1.83491e7i	− 0.647643i
$958$	3.31579e7i	1.16727i
$959$	−1.44384e7	−0.506960
$960$	0	0
$961$	4.97996e7	1.73947
$962$	1.07980e7i	0.376190i
$963$	− 8.57267e6i	− 0.297886i
$964$	−2.17333e7	−0.753239
$965$	0	0
$966$	−3.13600e6	−0.108127
$967$	3.29467e7i	1.13304i	0.824048	+	0.566520i	$0.191711\pi$
$967$	3.29467e7i	1.13304i	−0.824048	+	0.566520i	$0.808289\pi$
$968$	− 2.90886e6i	− 0.0997781i
$969$	−2.38531e7	−0.816083
$970$	0	0
$971$	2.24599e7	0.764470	0.382235	−	0.924065i	$-0.375154\pi$
$971$	2.24599e7	0.764470	0.382235	+	0.924065i	$0.375154\pi$
$972$	− 1.52365e7i	− 0.517272i
$973$	− 1.54323e7i	− 0.522573i
$974$	3.56708e7	1.20480
$975$	0	0
$976$	8.79053e6	0.295386
$977$	5.16236e7i	1.73026i	0.501545	+	0.865132i	$0.332765\pi$
$977$	5.16236e7i	1.73026i	−0.501545	+	0.865132i	$0.667235\pi$
$978$	− 1.01477e7i	− 0.339251i
$979$	−2.77644e6	−0.0925831
$980$	0	0
$981$	−3.95626e6	−0.131254
$982$	2.28615e7i	0.756529i
$983$	− 1.10202e7i	− 0.363751i	−0.983322	−	0.181876i	$-0.941783\pi$
$983$	− 1.10202e7i	− 0.363751i	0.983322	−	0.181876i	$-0.0582169\pi$
$984$	7.46189e6	0.245675
$985$	0	0
$986$	−3.30824e7	−1.08369
$987$	− 122304.i	− 0.00399621i
$988$	− 1.14401e7i	− 0.372854i
$989$	−1.62160e7	−0.527173
$990$	0	0
$991$	3.21029e7	1.03839	0.519194	−	0.854656i	$-0.326232\pi$
$991$	3.21029e7	1.03839	0.519194	+	0.854656i	$0.326232\pi$
$992$	− 9.06854e6i	− 0.292589i
$993$	− 1.98033e7i	− 0.637330i
$994$	7.23632e6	0.232301
$995$	0	0
$996$	−9.86317e6	−0.315042
$997$	− 2.81772e7i	− 0.897759i	−0.893592	−	0.448879i	$-0.851823\pi$
$997$	− 2.81772e7i	− 0.897759i	0.893592	−	0.448879i	$-0.148177\pi$
$998$	500464.i	0.0159055i
$999$	3.09984e7	0.982711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.f.99.1		2
5.2	odd	4		14.6.a.b.1.1	✓	1
5.3	odd	4		350.6.a.b.1.1		1
5.4	even	2	inner	350.6.c.f.99.2		2
15.2	even	4		126.6.a.c.1.1		1
20.7	even	4		112.6.a.d.1.1		1
35.2	odd	12		98.6.c.a.67.1		2
35.12	even	12		98.6.c.b.67.1		2
35.17	even	12		98.6.c.b.79.1		2
35.27	even	4		98.6.a.b.1.1		1
35.32	odd	12		98.6.c.a.79.1		2
40.27	even	4		448.6.a.k.1.1		1
40.37	odd	4		448.6.a.f.1.1		1
60.47	odd	4		1008.6.a.n.1.1		1
105.62	odd	4		882.6.a.g.1.1		1
140.27	odd	4		784.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.b.1.1	✓	1	5.2	odd	4
98.6.a.b.1.1		1	35.27	even	4
98.6.c.a.67.1		2	35.2	odd	12
98.6.c.a.79.1		2	35.32	odd	12
98.6.c.b.67.1		2	35.12	even	12
98.6.c.b.79.1		2	35.17	even	12
112.6.a.d.1.1		1	20.7	even	4
126.6.a.c.1.1		1	15.2	even	4
350.6.a.b.1.1		1	5.3	odd	4
350.6.c.f.99.1		2	1.1	even	1	trivial
350.6.c.f.99.2		2	5.4	even	2	inner
448.6.a.f.1.1		1	40.37	odd	4
448.6.a.k.1.1		1	40.27	even	4
784.6.a.h.1.1		1	140.27	odd	4
882.6.a.g.1.1		1	105.62	odd	4
1008.6.a.n.1.1		1	60.47	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +8.00000i q^{3} -16.0000 q^{4} +32.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +179.000 q^{9} -340.000 q^{11} -128.000i q^{12} -294.000i q^{13} +196.000 q^{14} +256.000 q^{16} -1226.00i q^{17} -716.000i q^{18} -2432.00 q^{19} -392.000 q^{21} +1360.00i q^{22} +2000.00i q^{23} -512.000 q^{24} -1176.00 q^{26} +3376.00i q^{27} -784.000i q^{28} +6746.00 q^{29} +8856.00 q^{31} -1024.00i q^{32} -2720.00i q^{33} -4904.00 q^{34} -2864.00 q^{36} -9182.00i q^{37} +9728.00i q^{38} +2352.00 q^{39} -14574.0 q^{41} +1568.00i q^{42} +8108.00i q^{43} +5440.00 q^{44} +8000.00 q^{46} +312.000i q^{47} +2048.00i q^{48} -2401.00 q^{49} +9808.00 q^{51} +4704.00i q^{52} -14634.0i q^{53} +13504.0 q^{54} -3136.00 q^{56} -19456.0i q^{57} -26984.0i q^{58} +27656.0 q^{59} +34338.0 q^{61} -35424.0i q^{62} +8771.00i q^{63} -4096.00 q^{64} -10880.0 q^{66} -12316.0i q^{67} +19616.0i q^{68} -16000.0 q^{69} +36920.0 q^{71} +11456.0i q^{72} -61718.0i q^{73} -36728.0 q^{74} +38912.0 q^{76} -16660.0i q^{77} -9408.00i q^{78} +64752.0 q^{79} +16489.0 q^{81} +58296.0i q^{82} -77056.0i q^{83} +6272.00 q^{84} +32432.0 q^{86} +53968.0i q^{87} -21760.0i q^{88} +8166.00 q^{89} +14406.0 q^{91} -32000.0i q^{92} +70848.0i q^{93} +1248.00 q^{94} +8192.00 q^{96} -20650.0i q^{97} +9604.00i q^{98} -60860.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39}+ \cdots - 121720 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 - 680 * q^11 + 392 * q^14 + 512 * q^16 - 4864 * q^19 - 784 * q^21 - 1024 * q^24 - 2352 * q^26 + 13492 * q^29 + 17712 * q^31 - 9808 * q^34 - 5728 * q^36 + 4704 * q^39 - 29148 * q^41 + 10880 * q^44 + 16000 * q^46 - 4802 * q^49 + 19616 * q^51 + 27008 * q^54 - 6272 * q^56 + 55312 * q^59 + 68676 * q^61 - 8192 * q^64 - 21760 * q^66 - 32000 * q^69 + 73840 * q^71 - 73456 * q^74 + 77824 * q^76 + 129504 * q^79 + 32978 * q^81 + 12544 * q^84 + 64864 * q^86 + 16332 * q^89 + 28812 * q^91 + 2496 * q^94 + 16384 * q^96 - 121720 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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