LMFDB - Embedded newform 350.4.c.c.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,4,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, 4, names="a")

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
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:	$20.6506685020$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.4.c.c.99.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +4.00000i q^{3} -4.00000 q^{4} -8.00000 q^{6} -7.00000i q^{7} -8.00000i q^{8} +11.0000 q^{9} +5.00000 q^{11} -16.0000i q^{12} -82.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} -12.0000i q^{17} +22.0000i q^{18} +42.0000 q^{19} +28.0000 q^{21} +10.0000i q^{22} -175.000i q^{23} +32.0000 q^{24} +164.000 q^{26} +152.000i q^{27} +28.0000i q^{28} -1.00000 q^{29} +226.000 q^{31} +32.0000i q^{32} +20.0000i q^{33} +24.0000 q^{34} -44.0000 q^{36} -19.0000i q^{37} +84.0000i q^{38} +328.000 q^{39} +16.0000 q^{41} +56.0000i q^{42} -281.000i q^{43} -20.0000 q^{44} +350.000 q^{46} +334.000i q^{47} +64.0000i q^{48} -49.0000 q^{49} +48.0000 q^{51} +328.000i q^{52} +398.000i q^{53} -304.000 q^{54} -56.0000 q^{56} +168.000i q^{57} -2.00000i q^{58} -106.000 q^{59} +48.0000 q^{61} +452.000i q^{62} -77.0000i q^{63} -64.0000 q^{64} -40.0000 q^{66} +483.000i q^{67} +48.0000i q^{68} +700.000 q^{69} -15.0000 q^{71} -88.0000i q^{72} -1044.00i q^{73} +38.0000 q^{74} -168.000 q^{76} -35.0000i q^{77} +656.000i q^{78} +1253.00 q^{79} -311.000 q^{81} +32.0000i q^{82} -758.000i q^{83} -112.000 q^{84} +562.000 q^{86} -4.00000i q^{87} -40.0000i q^{88} -86.0000 q^{89} -574.000 q^{91} +700.000i q^{92} +904.000i q^{93} -668.000 q^{94} -128.000 q^{96} +710.000i q^{97} -98.0000i q^{98} +55.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} - 16 q^{6} + 22 q^{9} + 10 q^{11} + 28 q^{14} + 32 q^{16} + 84 q^{19} + 56 q^{21} + 64 q^{24} + 328 q^{26} - 2 q^{29} + 452 q^{31} + 48 q^{34} - 88 q^{36} + 656 q^{39} + 32 q^{41} - 40 q^{44}+ \cdots + 110 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 - 16 * q^6 + 22 * q^9 + 10 * q^11 + 28 * q^14 + 32 * q^16 + 84 * q^19 + 56 * q^21 + 64 * q^24 + 328 * q^26 - 2 * q^29 + 452 * q^31 + 48 * q^34 - 88 * q^36 + 656 * q^39 + 32 * q^41 - 40 * q^44 + 700 * q^46 - 98 * q^49 + 96 * q^51 - 608 * q^54 - 112 * q^56 - 212 * q^59 + 96 * q^61 - 128 * q^64 - 80 * q^66 + 1400 * q^69 - 30 * q^71 + 76 * q^74 - 336 * q^76 + 2506 * q^79 - 622 * q^81 - 224 * q^84 + 1124 * q^86 - 172 * q^89 - 1148 * q^91 - 1336 * q^94 - 256 * q^96 + 110 * 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

)

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

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−8.00000	−0.544331
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 8.00000i	− 0.353553i
$9$	11.0000	0.407407
$10$	0	0
$11$	5.00000	0.137051	0.0685253	−	0.997649i	$-0.478171\pi$
$11$	5.00000	0.137051	0.0685253	+	0.997649i	$0.478171\pi$
$12$	− 16.0000i	− 0.384900i
$13$	− 82.0000i	− 1.74944i	−0.484629	−	0.874720i	$-0.661046\pi$
$13$	− 82.0000i	− 1.74944i	0.484629	−	0.874720i	$-0.338954\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	− 12.0000i	− 0.171202i	−0.996330	−	0.0856008i	$-0.972719\pi$
$17$	− 12.0000i	− 0.171202i	0.996330	−	0.0856008i	$-0.0272810\pi$
$18$	22.0000i	0.288081i
$19$	42.0000	0.507130	0.253565	−	0.967318i	$-0.418397\pi$
$19$	42.0000	0.507130	0.253565	+	0.967318i	$0.418397\pi$
$20$	0	0
$21$	28.0000	0.290957
$22$	10.0000i	0.0969094i
$23$	− 175.000i	− 1.58652i	−0.608881	−	0.793261i	$-0.708381\pi$
$23$	− 175.000i	− 1.58652i	0.608881	−	0.793261i	$-0.291619\pi$
$24$	32.0000	0.272166
$25$	0	0
$26$	164.000	1.23704
$27$	152.000i	1.08342i
$28$	28.0000i	0.188982i
$29$	−1.00000	−0.00640329	−0.00320164	−	0.999995i	$-0.501019\pi$
$29$	−1.00000	−0.00640329	−0.00320164	+	0.999995i	$0.501019\pi$
$30$	0	0
$31$	226.000	1.30938	0.654690	−	0.755897i	$-0.272799\pi$
$31$	226.000	1.30938	0.654690	+	0.755897i	$0.272799\pi$
$32$	32.0000i	0.176777i
$33$	20.0000i	0.105502i
$34$	24.0000	0.121058
$35$	0	0
$36$	−44.0000	−0.203704
$37$	− 19.0000i	− 0.0844211i	−0.999109	−	0.0422106i	$-0.986560\pi$
$37$	− 19.0000i	− 0.0844211i	0.999109	−	0.0422106i	$-0.0134400\pi$
$38$	84.0000i	0.358595i
$39$	328.000	1.34672
$40$	0	0
$41$	16.0000	0.0609459	0.0304729	−	0.999536i	$-0.490299\pi$
$41$	16.0000	0.0609459	0.0304729	+	0.999536i	$0.490299\pi$
$42$	56.0000i	0.205738i
$43$	− 281.000i	− 0.996560i	−0.867016	−	0.498280i	$-0.833965\pi$
$43$	− 281.000i	− 0.996560i	0.867016	−	0.498280i	$-0.166035\pi$
$44$	−20.0000	−0.0685253
$45$	0	0
$46$	350.000	1.12184
$47$	334.000i	1.03657i	0.855207	+	0.518286i	$0.173430\pi$
$47$	334.000i	1.03657i	−0.855207	+	0.518286i	$0.826570\pi$
$48$	64.0000i	0.192450i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	48.0000	0.131791
$52$	328.000i	0.874720i
$53$	398.000i	1.03150i	0.856739	+	0.515750i	$0.172487\pi$
$53$	398.000i	1.03150i	−0.856739	+	0.515750i	$0.827513\pi$
$54$	−304.000	−0.766096
$55$	0	0
$56$	−56.0000	−0.133631
$57$	168.000i	0.390388i
$58$	− 2.00000i	− 0.00452781i
$59$	−106.000	−0.233899	−0.116949	−	0.993138i	$-0.537312\pi$
$59$	−106.000	−0.233899	−0.116949	+	0.993138i	$0.537312\pi$
$60$	0	0
$61$	48.0000	0.100750	0.0503752	−	0.998730i	$-0.483958\pi$
$61$	48.0000	0.100750	0.0503752	+	0.998730i	$0.483958\pi$
$62$	452.000i	0.925872i
$63$	− 77.0000i	− 0.153986i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−40.0000	−0.0746009
$67$	483.000i	0.880714i	0.897823	+	0.440357i	$0.145148\pi$
$67$	483.000i	0.880714i	−0.897823	+	0.440357i	$0.854852\pi$
$68$	48.0000i	0.0856008i
$69$	700.000	1.22131
$70$	0	0
$71$	−15.0000	−0.0250729	−0.0125364	−	0.999921i	$-0.503991\pi$
$71$	−15.0000	−0.0250729	−0.0125364	+	0.999921i	$0.503991\pi$
$72$	− 88.0000i	− 0.144040i
$73$	− 1044.00i	− 1.67385i	−0.547319	−	0.836924i	$-0.684351\pi$
$73$	− 1044.00i	− 1.67385i	0.547319	−	0.836924i	$-0.315649\pi$
$74$	38.0000	0.0596947
$75$	0	0
$76$	−168.000	−0.253565
$77$	− 35.0000i	− 0.0518003i
$78$	656.000i	0.952274i
$79$	1253.00	1.78447	0.892237	−	0.451567i	$-0.149135\pi$
$79$	1253.00	1.78447	0.892237	+	0.451567i	$0.149135\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	32.0000i	0.0430952i
$83$	− 758.000i	− 1.00243i	−0.865324	−	0.501213i	$-0.832887\pi$
$83$	− 758.000i	− 1.00243i	0.865324	−	0.501213i	$-0.167113\pi$
$84$	−112.000	−0.145479
$85$	0	0
$86$	562.000	0.704675
$87$	− 4.00000i	− 0.00492925i
$88$	− 40.0000i	− 0.0484547i
$89$	−86.0000	−0.102427	−0.0512134	−	0.998688i	$-0.516309\pi$
$89$	−86.0000	−0.102427	−0.0512134	+	0.998688i	$0.516309\pi$
$90$	0	0
$91$	−574.000	−0.661226
$92$	700.000i	0.793261i
$93$	904.000i	1.00796i
$94$	−668.000	−0.732967
$95$	0	0
$96$	−128.000	−0.136083
$97$	710.000i	0.743192i	0.928395	+	0.371596i	$0.121189\pi$
$97$	710.000i	0.743192i	−0.928395	+	0.371596i	$0.878811\pi$
$98$	− 98.0000i	− 0.101015i
$99$	55.0000	0.0558354
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	96.0000i	0.0931904i
$103$	838.000i	0.801656i	0.916153	+	0.400828i	$0.131278\pi$
$103$	838.000i	0.801656i	−0.916153	+	0.400828i	$0.868722\pi$
$104$	−656.000	−0.618520
$105$	0	0
$106$	−796.000	−0.729381
$107$	− 2164.00i	− 1.95516i	−0.210572	−	0.977578i	$-0.567533\pi$
$107$	− 2164.00i	− 1.95516i	0.210572	−	0.977578i	$-0.432467\pi$
$108$	− 608.000i	− 0.541711i
$109$	1827.00	1.60546	0.802729	−	0.596344i	$-0.203381\pi$
$109$	1827.00	1.60546	0.802729	+	0.596344i	$0.203381\pi$
$110$	0	0
$111$	76.0000	0.0649874
$112$	− 112.000i	− 0.0944911i
$113$	93.0000i	0.0774222i	0.999250	+	0.0387111i	$0.0123252\pi$
$113$	93.0000i	0.0774222i	−0.999250	+	0.0387111i	$0.987675\pi$
$114$	−336.000	−0.276046
$115$	0	0
$116$	4.00000	0.00320164
$117$	− 902.000i	− 0.712734i
$118$	− 212.000i	− 0.165391i
$119$	−84.0000	−0.0647081
$120$	0	0
$121$	−1306.00	−0.981217
$122$	96.0000i	0.0712412i
$123$	64.0000i	0.0469161i
$124$	−904.000	−0.654690
$125$	0	0
$126$	154.000	0.108884
$127$	− 1433.00i	− 1.00125i	−0.865666	−	0.500623i	$-0.833104\pi$
$127$	− 1433.00i	− 1.00125i	0.865666	−	0.500623i	$-0.166896\pi$
$128$	− 128.000i	− 0.0883883i
$129$	1124.00	0.767153
$130$	0	0
$131$	868.000	0.578912	0.289456	−	0.957191i	$-0.406526\pi$
$131$	868.000	0.578912	0.289456	+	0.957191i	$0.406526\pi$
$132$	− 80.0000i	− 0.0527508i
$133$	− 294.000i	− 0.191677i
$134$	−966.000	−0.622759
$135$	0	0
$136$	−96.0000	−0.0605289
$137$	− 2646.00i	− 1.65010i	−0.565063	−	0.825048i	$-0.691148\pi$
$137$	− 2646.00i	− 1.65010i	0.565063	−	0.825048i	$-0.308852\pi$
$138$	1400.00i	0.863594i
$139$	−896.000	−0.546746	−0.273373	−	0.961908i	$-0.588139\pi$
$139$	−896.000	−0.546746	−0.273373	+	0.961908i	$0.588139\pi$
$140$	0	0
$141$	−1336.00	−0.797954
$142$	− 30.0000i	− 0.0177292i
$143$	− 410.000i	− 0.239762i
$144$	176.000	0.101852
$145$	0	0
$146$	2088.00	1.18359
$147$	− 196.000i	− 0.109971i
$148$	76.0000i	0.0422106i
$149$	−2523.00	−1.38720	−0.693598	−	0.720362i	$-0.743976\pi$
$149$	−2523.00	−1.38720	−0.693598	+	0.720362i	$0.743976\pi$
$150$	0	0
$151$	2433.00	1.31122	0.655612	−	0.755098i	$-0.272411\pi$
$151$	2433.00	1.31122	0.655612	+	0.755098i	$0.272411\pi$
$152$	− 336.000i	− 0.179297i
$153$	− 132.000i	− 0.0697488i
$154$	70.0000	0.0366283
$155$	0	0
$156$	−1312.00	−0.673359
$157$	− 2572.00i	− 1.30744i	−0.756737	−	0.653720i	$-0.773208\pi$
$157$	− 2572.00i	− 1.30744i	0.756737	−	0.653720i	$-0.226792\pi$
$158$	2506.00i	1.26181i
$159$	−1592.00	−0.794049
$160$	0	0
$161$	−1225.00	−0.599649
$162$	− 622.000i	− 0.301660i
$163$	− 2488.00i	− 1.19555i	−0.801663	−	0.597777i	$-0.796051\pi$
$163$	− 2488.00i	− 1.19555i	0.801663	−	0.597777i	$-0.203949\pi$
$164$	−64.0000	−0.0304729
$165$	0	0
$166$	1516.00	0.708822
$167$	3054.00i	1.41512i	0.706652	+	0.707562i	$0.250205\pi$
$167$	3054.00i	1.41512i	−0.706652	+	0.707562i	$0.749795\pi$
$168$	− 224.000i	− 0.102869i
$169$	−4527.00	−2.06054
$170$	0	0
$171$	462.000	0.206608
$172$	1124.00i	0.498280i
$173$	3894.00i	1.71130i	0.517553	+	0.855651i	$0.326843\pi$
$173$	3894.00i	1.71130i	−0.517553	+	0.855651i	$0.673157\pi$
$174$	8.00000	0.00348551
$175$	0	0
$176$	80.0000	0.0342627
$177$	− 424.000i	− 0.180055i
$178$	− 172.000i	− 0.0724267i
$179$	1108.00	0.462658	0.231329	−	0.972876i	$-0.425693\pi$
$179$	1108.00	0.462658	0.231329	+	0.972876i	$0.425693\pi$
$180$	0	0
$181$	−1920.00	−0.788467	−0.394233	−	0.919010i	$-0.628990\pi$
$181$	−1920.00	−0.788467	−0.394233	+	0.919010i	$0.628990\pi$
$182$	− 1148.00i	− 0.467557i
$183$	192.000i	0.0775576i
$184$	−1400.00	−0.560920
$185$	0	0
$186$	−1808.00	−0.712737
$187$	− 60.0000i	− 0.0234633i
$188$	− 1336.00i	− 0.518286i
$189$	1064.00	0.409495
$190$	0	0
$191$	−936.000	−0.354589	−0.177295	−	0.984158i	$-0.556735\pi$
$191$	−936.000	−0.354589	−0.177295	+	0.984158i	$0.556735\pi$
$192$	− 256.000i	− 0.0962250i
$193$	− 3163.00i	− 1.17968i	−0.807521	−	0.589839i	$-0.799191\pi$
$193$	− 3163.00i	− 1.17968i	0.807521	−	0.589839i	$-0.200809\pi$
$194$	−1420.00	−0.525516
$195$	0	0
$196$	196.000	0.0714286
$197$	− 349.000i	− 0.126219i	−0.998007	−	0.0631097i	$-0.979898\pi$
$197$	− 349.000i	− 0.126219i	0.998007	−	0.0631097i	$-0.0201018\pi$
$198$	110.000i	0.0394816i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−1932.00	−0.677974
$202$	0	0
$203$	7.00000i	0.00242022i
$204$	−192.000	−0.0658955
$205$	0	0
$206$	−1676.00	−0.566857
$207$	− 1925.00i	− 0.646361i
$208$	− 1312.00i	− 0.437360i
$209$	210.000	0.0695024
$210$	0	0
$211$	1676.00	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$211$	1676.00	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$212$	− 1592.00i	− 0.515750i
$213$	− 60.0000i	− 0.0193011i
$214$	4328.00	1.38250
$215$	0	0
$216$	1216.00	0.383048
$217$	− 1582.00i	− 0.494899i
$218$	3654.00i	1.13523i
$219$	4176.00	1.28853
$220$	0	0
$221$	−984.000	−0.299507
$222$	152.000i	0.0459530i
$223$	− 1176.00i	− 0.353143i	−0.984288	−	0.176571i	$-0.943499\pi$
$223$	− 1176.00i	− 0.353143i	0.984288	−	0.176571i	$-0.0565006\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−186.000	−0.0547457
$227$	2234.00i	0.653197i	0.945163	+	0.326599i	$0.105903\pi$
$227$	2234.00i	0.653197i	−0.945163	+	0.326599i	$0.894097\pi$
$228$	− 672.000i	− 0.195194i
$229$	−4274.00	−1.23334	−0.616668	−	0.787223i	$-0.711518\pi$
$229$	−4274.00	−1.23334	−0.616668	+	0.787223i	$0.711518\pi$
$230$	0	0
$231$	140.000	0.0398759
$232$	8.00000i	0.00226390i
$233$	5427.00i	1.52590i	0.646458	+	0.762950i	$0.276250\pi$
$233$	5427.00i	1.52590i	−0.646458	+	0.762950i	$0.723750\pi$
$234$	1804.00	0.503979
$235$	0	0
$236$	424.000	0.116949
$237$	5012.00i	1.37369i
$238$	− 168.000i	− 0.0457556i
$239$	−3516.00	−0.951595	−0.475797	−	0.879555i	$-0.657840\pi$
$239$	−3516.00	−0.951595	−0.475797	+	0.879555i	$0.657840\pi$
$240$	0	0
$241$	−3890.00	−1.03974	−0.519869	−	0.854246i	$-0.674019\pi$
$241$	−3890.00	−1.03974	−0.519869	+	0.854246i	$0.674019\pi$
$242$	− 2612.00i	− 0.693825i
$243$	2860.00i	0.755017i
$244$	−192.000	−0.0503752
$245$	0	0
$246$	−128.000	−0.0331747
$247$	− 3444.00i	− 0.887192i
$248$	− 1808.00i	− 0.462936i
$249$	3032.00	0.771667
$250$	0	0
$251$	2512.00	0.631697	0.315849	−	0.948810i	$-0.397711\pi$
$251$	2512.00	0.631697	0.315849	+	0.948810i	$0.397711\pi$
$252$	308.000i	0.0769928i
$253$	− 875.000i	− 0.217434i
$254$	2866.00	0.707988
$255$	0	0
$256$	256.000	0.0625000
$257$	6934.00i	1.68300i	0.540257	+	0.841500i	$0.318327\pi$
$257$	6934.00i	1.68300i	−0.540257	+	0.841500i	$0.681673\pi$
$258$	2248.00i	0.542459i
$259$	−133.000	−0.0319082
$260$	0	0
$261$	−11.0000	−0.00260875
$262$	1736.00i	0.409353i
$263$	− 5505.00i	− 1.29070i	−0.763889	−	0.645348i	$-0.776713\pi$
$263$	− 5505.00i	− 1.29070i	0.763889	−	0.645348i	$-0.223287\pi$
$264$	160.000	0.0373005
$265$	0	0
$266$	588.000	0.135536
$267$	− 344.000i	− 0.0788482i
$268$	− 1932.00i	− 0.440357i
$269$	8698.00	1.97147	0.985737	−	0.168294i	$-0.0538258\pi$
$269$	8698.00	1.97147	0.985737	+	0.168294i	$0.0538258\pi$
$270$	0	0
$271$	2054.00	0.460412	0.230206	−	0.973142i	$-0.426060\pi$
$271$	2054.00	0.460412	0.230206	+	0.973142i	$0.426060\pi$
$272$	− 192.000i	− 0.0428004i
$273$	− 2296.00i	− 0.509012i
$274$	5292.00	1.16679
$275$	0	0
$276$	−2800.00	−0.610653
$277$	8450.00i	1.83289i	0.400158	+	0.916446i	$0.368955\pi$
$277$	8450.00i	1.83289i	−0.400158	+	0.916446i	$0.631045\pi$
$278$	− 1792.00i	− 0.386608i
$279$	2486.00	0.533451
$280$	0	0
$281$	4469.00	0.948748	0.474374	−	0.880323i	$-0.342674\pi$
$281$	4469.00	0.948748	0.474374	+	0.880323i	$0.342674\pi$
$282$	− 2672.00i	− 0.564239i
$283$	794.000i	0.166779i	0.996517	+	0.0833894i	$0.0265745\pi$
$283$	794.000i	0.166779i	−0.996517	+	0.0833894i	$0.973425\pi$
$284$	60.0000	0.0125364
$285$	0	0
$286$	820.000	0.169537
$287$	− 112.000i	− 0.0230354i
$288$	352.000i	0.0720201i
$289$	4769.00	0.970690
$290$	0	0
$291$	−2840.00	−0.572109
$292$	4176.00i	0.836924i
$293$	6462.00i	1.28844i	0.764839	+	0.644222i	$0.222819\pi$
$293$	6462.00i	1.28844i	−0.764839	+	0.644222i	$0.777181\pi$
$294$	392.000	0.0777616
$295$	0	0
$296$	−152.000	−0.0298474
$297$	760.000i	0.148484i
$298$	− 5046.00i	− 0.980896i
$299$	−14350.0	−2.77552
$300$	0	0
$301$	−1967.00	−0.376664
$302$	4866.00i	0.927175i
$303$	0	0
$304$	672.000	0.126782
$305$	0	0
$306$	264.000	0.0493199
$307$	56.0000i	0.0104107i	0.999986	+	0.00520536i	$0.00165692\pi$
$307$	56.0000i	0.0104107i	−0.999986	+	0.00520536i	$0.998343\pi$
$308$	140.000i	0.0259001i
$309$	−3352.00	−0.617115
$310$	0	0
$311$	10656.0	1.94291	0.971457	−	0.237215i	$-0.0762347\pi$
$311$	10656.0	1.94291	0.971457	+	0.237215i	$0.0762347\pi$
$312$	− 2624.00i	− 0.476137i
$313$	2010.00i	0.362977i	0.983393	+	0.181489i	$0.0580916\pi$
$313$	2010.00i	0.362977i	−0.983393	+	0.181489i	$0.941908\pi$
$314$	5144.00	0.924499
$315$	0	0
$316$	−5012.00	−0.892237
$317$	3903.00i	0.691528i	0.938322	+	0.345764i	$0.112380\pi$
$317$	3903.00i	0.691528i	−0.938322	+	0.345764i	$0.887620\pi$
$318$	− 3184.00i	− 0.561478i
$319$	−5.00000	−0.000877574 0
$320$	0	0
$321$	8656.00	1.50508
$322$	− 2450.00i	− 0.424016i
$323$	− 504.000i	− 0.0868214i
$324$	1244.00	0.213306
$325$	0	0
$326$	4976.00	0.845384
$327$	7308.00i	1.23588i
$328$	− 128.000i	− 0.0215476i
$329$	2338.00	0.391788
$330$	0	0
$331$	−3277.00	−0.544170	−0.272085	−	0.962273i	$-0.587713\pi$
$331$	−3277.00	−0.544170	−0.272085	+	0.962273i	$0.587713\pi$
$332$	3032.00i	0.501213i
$333$	− 209.000i	− 0.0343938i
$334$	−6108.00	−1.00064
$335$	0	0
$336$	448.000	0.0727393
$337$	6462.00i	1.04453i	0.852782	+	0.522266i	$0.174913\pi$
$337$	6462.00i	1.04453i	−0.852782	+	0.522266i	$0.825087\pi$
$338$	− 9054.00i	− 1.45702i
$339$	−372.000	−0.0595996
$340$	0	0
$341$	1130.00	0.179451
$342$	924.000i	0.146094i
$343$	343.000i	0.0539949i
$344$	−2248.00	−0.352337
$345$	0	0
$346$	−7788.00	−1.21007
$347$	− 3787.00i	− 0.585870i	−0.956132	−	0.292935i	$-0.905368\pi$
$347$	− 3787.00i	− 0.585870i	0.956132	−	0.292935i	$-0.0946319\pi$
$348$	16.0000i	0.00246463i
$349$	−7398.00	−1.13469	−0.567344	−	0.823481i	$-0.692029\pi$
$349$	−7398.00	−1.13469	−0.567344	+	0.823481i	$0.692029\pi$
$350$	0	0
$351$	12464.0	1.89538
$352$	160.000i	0.0242274i
$353$	9654.00i	1.45561i	0.685784	+	0.727805i	$0.259460\pi$
$353$	9654.00i	1.45561i	−0.685784	+	0.727805i	$0.740540\pi$
$354$	848.000	0.127318
$355$	0	0
$356$	344.000	0.0512134
$357$	− 336.000i	− 0.0498123i
$358$	2216.00i	0.327149i
$359$	7691.00	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$359$	7691.00	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$360$	0	0
$361$	−5095.00	−0.742820
$362$	− 3840.00i	− 0.557530i
$363$	− 5224.00i	− 0.755341i
$364$	2296.00	0.330613
$365$	0	0
$366$	−384.000	−0.0548415
$367$	− 6836.00i	− 0.972306i	−0.873874	−	0.486153i	$-0.838400\pi$
$367$	− 6836.00i	− 0.972306i	0.873874	−	0.486153i	$-0.161600\pi$
$368$	− 2800.00i	− 0.396631i
$369$	176.000	0.0248298
$370$	0	0
$371$	2786.00	0.389870
$372$	− 3616.00i	− 0.503981i
$373$	− 12123.0i	− 1.68286i	−0.540370	−	0.841428i	$-0.681716\pi$
$373$	− 12123.0i	− 1.68286i	0.540370	−	0.841428i	$-0.318284\pi$
$374$	120.000	0.0165910
$375$	0	0
$376$	2672.00	0.366484
$377$	82.0000i	0.0112022i
$378$	2128.00i	0.289557i
$379$	−10811.0	−1.46523	−0.732617	−	0.680641i	$-0.761701\pi$
$379$	−10811.0	−1.46523	−0.732617	+	0.680641i	$0.761701\pi$
$380$	0	0
$381$	5732.00	0.770759
$382$	− 1872.00i	− 0.250733i
$383$	12778.0i	1.70477i	0.522918	+	0.852383i	$0.324843\pi$
$383$	12778.0i	1.70477i	−0.522918	+	0.852383i	$0.675157\pi$
$384$	512.000	0.0680414
$385$	0	0
$386$	6326.00	0.834158
$387$	− 3091.00i	− 0.406006i
$388$	− 2840.00i	− 0.371596i
$389$	383.000	0.0499200	0.0249600	−	0.999688i	$-0.492054\pi$
$389$	383.000	0.0499200	0.0249600	+	0.999688i	$0.492054\pi$
$390$	0	0
$391$	−2100.00	−0.271615
$392$	392.000i	0.0505076i
$393$	3472.00i	0.445647i
$394$	698.000	0.0892506
$395$	0	0
$396$	−220.000	−0.0279177
$397$	12050.0i	1.52336i	0.647956	+	0.761678i	$0.275624\pi$
$397$	12050.0i	1.52336i	−0.647956	+	0.761678i	$0.724376\pi$
$398$	0	0
$399$	1176.00	0.147553
$400$	0	0
$401$	−7523.00	−0.936860	−0.468430	−	0.883501i	$-0.655180\pi$
$401$	−7523.00	−0.936860	−0.468430	+	0.883501i	$0.655180\pi$
$402$	− 3864.00i	− 0.479400i
$403$	− 18532.0i	− 2.29068i
$404$	0	0
$405$	0	0
$406$	−14.0000	−0.00171135
$407$	− 95.0000i	− 0.0115700i
$408$	− 384.000i	− 0.0465952i
$409$	4534.00	0.548146	0.274073	−	0.961709i	$-0.411629\pi$
$409$	4534.00	0.548146	0.274073	+	0.961709i	$0.411629\pi$
$410$	0	0
$411$	10584.0	1.27024
$412$	− 3352.00i	− 0.400828i
$413$	742.000i	0.0884054i
$414$	3850.00	0.457046
$415$	0	0
$416$	2624.00	0.309260
$417$	− 3584.00i	− 0.420885i
$418$	420.000i	0.0491456i
$419$	−8494.00	−0.990356	−0.495178	−	0.868792i	$-0.664897\pi$
$419$	−8494.00	−0.990356	−0.495178	+	0.868792i	$0.664897\pi$
$420$	0	0
$421$	−8765.00	−1.01468	−0.507340	−	0.861746i	$-0.669371\pi$
$421$	−8765.00	−1.01468	−0.507340	+	0.861746i	$0.669371\pi$
$422$	3352.00i	0.386665i
$423$	3674.00i	0.422307i
$424$	3184.00	0.364690
$425$	0	0
$426$	120.000	0.0136479
$427$	− 336.000i	− 0.0380800i
$428$	8656.00i	0.977578i
$429$	1640.00	0.184569
$430$	0	0
$431$	−6696.00	−0.748341	−0.374170	−	0.927360i	$-0.622073\pi$
$431$	−6696.00	−0.748341	−0.374170	+	0.927360i	$0.622073\pi$
$432$	2432.00i	0.270856i
$433$	4148.00i	0.460370i	0.973147	+	0.230185i	$0.0739331\pi$
$433$	4148.00i	0.460370i	−0.973147	+	0.230185i	$0.926067\pi$
$434$	3164.00	0.349947
$435$	0	0
$436$	−7308.00	−0.802729
$437$	− 7350.00i	− 0.804572i
$438$	8352.00i	0.911128i
$439$	−6182.00	−0.672097	−0.336049	−	0.941845i	$-0.609091\pi$
$439$	−6182.00	−0.672097	−0.336049	+	0.941845i	$0.609091\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	− 1968.00i	− 0.211783i
$443$	− 616.000i	− 0.0660656i	−0.999454	−	0.0330328i	$-0.989483\pi$
$443$	− 616.000i	− 0.0660656i	0.999454	−	0.0330328i	$-0.0105166\pi$
$444$	−304.000	−0.0324937
$445$	0	0
$446$	2352.00	0.249709
$447$	− 10092.0i	− 1.06786i
$448$	448.000i	0.0472456i
$449$	8653.00	0.909488	0.454744	−	0.890622i	$-0.349731\pi$
$449$	8653.00	0.909488	0.454744	+	0.890622i	$0.349731\pi$
$450$	0	0
$451$	80.0000	0.00835267
$452$	− 372.000i	− 0.0387111i
$453$	9732.00i	1.00938i
$454$	−4468.00	−0.461880
$455$	0	0
$456$	1344.00	0.138023
$457$	− 7785.00i	− 0.796864i	−0.917198	−	0.398432i	$-0.869554\pi$
$457$	− 7785.00i	− 0.796864i	0.917198	−	0.398432i	$-0.130446\pi$
$458$	− 8548.00i	− 0.872100i
$459$	1824.00	0.185484
$460$	0	0
$461$	−14718.0	−1.48695	−0.743477	−	0.668762i	$-0.766825\pi$
$461$	−14718.0	−1.48695	−0.743477	+	0.668762i	$0.766825\pi$
$462$	280.000i	0.0281965i
$463$	− 2864.00i	− 0.287476i	−0.989616	−	0.143738i	$-0.954088\pi$
$463$	− 2864.00i	− 0.287476i	0.989616	−	0.143738i	$-0.0459123\pi$
$464$	−16.0000	−0.00160082
$465$	0	0
$466$	−10854.0	−1.07897
$467$	− 1878.00i	− 0.186089i	−0.995662	−	0.0930444i	$-0.970340\pi$
$467$	− 1878.00i	− 0.186089i	0.995662	−	0.0930444i	$-0.0296599\pi$
$468$	3608.00i	0.356367i
$469$	3381.00	0.332879
$470$	0	0
$471$	10288.0	1.00647
$472$	848.000i	0.0826957i
$473$	− 1405.00i	− 0.136579i
$474$	−10024.0	−0.971345
$475$	0	0
$476$	336.000	0.0323541
$477$	4378.00i	0.420241i
$478$	− 7032.00i	− 0.672879i
$479$	9914.00	0.945683	0.472842	−	0.881147i	$-0.343228\pi$
$479$	9914.00	0.945683	0.472842	+	0.881147i	$0.343228\pi$
$480$	0	0
$481$	−1558.00	−0.147690
$482$	− 7780.00i	− 0.735206i
$483$	− 4900.00i	− 0.461610i
$484$	5224.00	0.490609
$485$	0	0
$486$	−5720.00	−0.533878
$487$	− 12883.0i	− 1.19874i	−0.800474	−	0.599368i	$-0.795419\pi$
$487$	− 12883.0i	− 1.19874i	0.800474	−	0.599368i	$-0.204581\pi$
$488$	− 384.000i	− 0.0356206i
$489$	9952.00	0.920337
$490$	0	0
$491$	−16567.0	−1.52273	−0.761363	−	0.648326i	$-0.775469\pi$
$491$	−16567.0	−1.52273	−0.761363	+	0.648326i	$0.775469\pi$
$492$	− 256.000i	− 0.0234581i
$493$	12.0000i	0.00109625i
$494$	6888.00	0.627340
$495$	0	0
$496$	3616.00	0.327345
$497$	105.000i	0.00947665i
$498$	6064.00i	0.545651i
$499$	−2044.00	−0.183371	−0.0916854	−	0.995788i	$-0.529225\pi$
$499$	−2044.00	−0.183371	−0.0916854	+	0.995788i	$0.529225\pi$
$500$	0	0
$501$	−12216.0	−1.08936
$502$	5024.00i	0.446677i
$503$	14610.0i	1.29508i	0.762029	+	0.647542i	$0.224203\pi$
$503$	14610.0i	1.29508i	−0.762029	+	0.647542i	$0.775797\pi$
$504$	−616.000	−0.0544421
$505$	0	0
$506$	1750.00	0.153749
$507$	− 18108.0i	− 1.58620i
$508$	5732.00i	0.500623i
$509$	−18074.0	−1.57390	−0.786951	−	0.617016i	$-0.788341\pi$
$509$	−18074.0	−1.57390	−0.786951	+	0.617016i	$0.788341\pi$
$510$	0	0
$511$	−7308.00	−0.632655
$512$	512.000i	0.0441942i
$513$	6384.00i	0.549436i
$514$	−13868.0	−1.19006
$515$	0	0
$516$	−4496.00	−0.383576
$517$	1670.00i	0.142063i
$518$	− 266.000i	− 0.0225625i
$519$	−15576.0	−1.31736
$520$	0	0
$521$	−13554.0	−1.13975	−0.569877	−	0.821730i	$-0.693009\pi$
$521$	−13554.0	−1.13975	−0.569877	+	0.821730i	$0.693009\pi$
$522$	− 22.0000i	− 0.00184466i
$523$	− 11984.0i	− 1.00196i	−0.865460	−	0.500979i	$-0.832973\pi$
$523$	− 11984.0i	− 1.00196i	0.865460	−	0.500979i	$-0.167027\pi$
$524$	−3472.00	−0.289456
$525$	0	0
$526$	11010.0	0.912659
$527$	− 2712.00i	− 0.224168i
$528$	320.000i	0.0263754i
$529$	−18458.0	−1.51705
$530$	0	0
$531$	−1166.00	−0.0952921
$532$	1176.00i	0.0958385i
$533$	− 1312.00i	− 0.106621i
$534$	688.000	0.0557541
$535$	0	0
$536$	3864.00	0.311379
$537$	4432.00i	0.356154i
$538$	17396.0i	1.39404i
$539$	−245.000	−0.0195787
$540$	0	0
$541$	3863.00	0.306993	0.153497	−	0.988149i	$-0.450947\pi$
$541$	3863.00	0.306993	0.153497	+	0.988149i	$0.450947\pi$
$542$	4108.00i	0.325560i
$543$	− 7680.00i	− 0.606962i
$544$	384.000	0.0302645
$545$	0	0
$546$	4592.00	0.359926
$547$	19579.0i	1.53042i	0.643783	+	0.765208i	$0.277364\pi$
$547$	19579.0i	1.53042i	−0.643783	+	0.765208i	$0.722636\pi$
$548$	10584.0i	0.825048i
$549$	528.000	0.0410464
$550$	0	0
$551$	−42.0000	−0.00324730
$552$	− 5600.00i	− 0.431797i
$553$	− 8771.00i	− 0.674468i
$554$	−16900.0	−1.29605
$555$	0	0
$556$	3584.00	0.273373
$557$	− 7437.00i	− 0.565738i	−0.959159	−	0.282869i	$-0.908714\pi$
$557$	− 7437.00i	− 0.565738i	0.959159	−	0.282869i	$-0.0912861\pi$
$558$	4972.00i	0.377207i
$559$	−23042.0	−1.74342
$560$	0	0
$561$	240.000	0.0180620
$562$	8938.00i	0.670866i
$563$	5690.00i	0.425941i	0.977059	+	0.212971i	$0.0683139\pi$
$563$	5690.00i	0.425941i	−0.977059	+	0.212971i	$0.931686\pi$
$564$	5344.00	0.398977
$565$	0	0
$566$	−1588.00	−0.117930
$567$	2177.00i	0.161244i
$568$	120.000i	0.00886459i
$569$	24771.0	1.82505	0.912526	−	0.409019i	$-0.134129\pi$
$569$	24771.0	1.82505	0.912526	+	0.409019i	$0.134129\pi$
$570$	0	0
$571$	−8633.00	−0.632714	−0.316357	−	0.948640i	$-0.602460\pi$
$571$	−8633.00	−0.632714	−0.316357	+	0.948640i	$0.602460\pi$
$572$	1640.00i	0.119881i
$573$	− 3744.00i	− 0.272963i
$574$	224.000	0.0162885
$575$	0	0
$576$	−704.000	−0.0509259
$577$	1666.00i	0.120202i	0.998192	+	0.0601009i	$0.0191423\pi$
$577$	1666.00i	0.120202i	−0.998192	+	0.0601009i	$0.980858\pi$
$578$	9538.00i	0.686381i
$579$	12652.0	0.908116
$580$	0	0
$581$	−5306.00	−0.378881
$582$	− 5680.00i	− 0.404542i
$583$	1990.00i	0.141368i
$584$	−8352.00	−0.591795
$585$	0	0
$586$	−12924.0	−0.911067
$587$	− 7536.00i	− 0.529888i	−0.964264	−	0.264944i	$-0.914647\pi$
$587$	− 7536.00i	− 0.529888i	0.964264	−	0.264944i	$-0.0853534\pi$
$588$	784.000i	0.0549857i
$589$	9492.00	0.664026
$590$	0	0
$591$	1396.00	0.0971637
$592$	− 304.000i	− 0.0211053i
$593$	− 3568.00i	− 0.247083i	−0.992339	−	0.123541i	$-0.960575\pi$
$593$	− 3568.00i	− 0.247083i	0.992339	−	0.123541i	$-0.0394252\pi$
$594$	−1520.00	−0.104994
$595$	0	0
$596$	10092.0	0.693598
$597$	0	0
$598$	− 28700.0i	− 1.96259i
$599$	24901.0	1.69854	0.849272	−	0.527956i	$-0.177042\pi$
$599$	24901.0	1.69854	0.849272	+	0.527956i	$0.177042\pi$
$600$	0	0
$601$	−23744.0	−1.61154	−0.805772	−	0.592226i	$-0.798249\pi$
$601$	−23744.0	−1.61154	−0.805772	+	0.592226i	$0.798249\pi$
$602$	− 3934.00i	− 0.266342i
$603$	5313.00i	0.358809i
$604$	−9732.00	−0.655612
$605$	0	0
$606$	0	0
$607$	− 22894.0i	− 1.53087i	−0.643513	−	0.765436i	$-0.722524\pi$
$607$	− 22894.0i	− 1.53087i	0.643513	−	0.765436i	$-0.277476\pi$
$608$	1344.00i	0.0896487i
$609$	−28.0000	−0.00186308
$610$	0	0
$611$	27388.0	1.81342
$612$	528.000i	0.0348744i
$613$	7811.00i	0.514655i	0.966324	+	0.257327i	$0.0828419\pi$
$613$	7811.00i	0.514655i	−0.966324	+	0.257327i	$0.917158\pi$
$614$	−112.000	−0.00736149
$615$	0	0
$616$	−280.000	−0.0183142
$617$	11833.0i	0.772089i	0.922480	+	0.386044i	$0.126159\pi$
$617$	11833.0i	0.772089i	−0.922480	+	0.386044i	$0.873841\pi$
$618$	− 6704.00i	− 0.436366i
$619$	4186.00	0.271809	0.135904	−	0.990722i	$-0.456606\pi$
$619$	4186.00	0.271809	0.135904	+	0.990722i	$0.456606\pi$
$620$	0	0
$621$	26600.0	1.71887
$622$	21312.0i	1.37385i
$623$	602.000i	0.0387137i
$624$	5248.00	0.336680
$625$	0	0
$626$	−4020.00	−0.256664
$627$	840.000i	0.0535030i
$628$	10288.0i	0.653720i
$629$	−228.000	−0.0144530
$630$	0	0
$631$	12079.0	0.762056	0.381028	−	0.924563i	$-0.375570\pi$
$631$	12079.0	0.762056	0.381028	+	0.924563i	$0.375570\pi$
$632$	− 10024.0i	− 0.630907i
$633$	6704.00i	0.420948i
$634$	−7806.00	−0.488984
$635$	0	0
$636$	6368.00	0.397025
$637$	4018.00i	0.249920i
$638$	− 10.0000i	0 0.000620539i
$639$	−165.000	−0.0102149
$640$	0	0
$641$	4017.00	0.247523	0.123761	−	0.992312i	$-0.460504\pi$
$641$	4017.00	0.247523	0.123761	+	0.992312i	$0.460504\pi$
$642$	17312.0i	1.06425i
$643$	− 13508.0i	− 0.828466i	−0.910171	−	0.414233i	$-0.864050\pi$
$643$	− 13508.0i	− 0.828466i	0.910171	−	0.414233i	$-0.135950\pi$
$644$	4900.00	0.299825
$645$	0	0
$646$	1008.00	0.0613920
$647$	− 1346.00i	− 0.0817878i	−0.999163	−	0.0408939i	$-0.986979\pi$
$647$	− 1346.00i	− 0.0817878i	0.999163	−	0.0408939i	$-0.0130206\pi$
$648$	2488.00i	0.150830i
$649$	−530.000	−0.0320560
$650$	0	0
$651$	6328.00	0.380974
$652$	9952.00i	0.597777i
$653$	10274.0i	0.615701i	0.951435	+	0.307850i	$0.0996096\pi$
$653$	10274.0i	0.615701i	−0.951435	+	0.307850i	$0.900390\pi$
$654$	−14616.0	−0.873900
$655$	0	0
$656$	256.000	0.0152365
$657$	− 11484.0i	− 0.681938i
$658$	4676.00i	0.277036i
$659$	13116.0	0.775306	0.387653	−	0.921805i	$-0.373286\pi$
$659$	13116.0	0.775306	0.387653	+	0.921805i	$0.373286\pi$
$660$	0	0
$661$	23070.0	1.35752	0.678759	−	0.734361i	$-0.262518\pi$
$661$	23070.0	1.35752	0.678759	+	0.734361i	$0.262518\pi$
$662$	− 6554.00i	− 0.384786i
$663$	− 3936.00i	− 0.230560i
$664$	−6064.00	−0.354411
$665$	0	0
$666$	418.000	0.0243201
$667$	175.000i	0.0101590i
$668$	− 12216.0i	− 0.707562i
$669$	4704.00	0.271849
$670$	0	0
$671$	240.000	0.0138079
$672$	896.000i	0.0514344i
$673$	− 24922.0i	− 1.42745i	−0.700427	−	0.713724i	$-0.747007\pi$
$673$	− 24922.0i	− 1.42745i	0.700427	−	0.713724i	$-0.252993\pi$
$674$	−12924.0	−0.738596
$675$	0	0
$676$	18108.0	1.03027
$677$	14294.0i	0.811467i	0.913991	+	0.405733i	$0.132984\pi$
$677$	14294.0i	0.811467i	−0.913991	+	0.405733i	$0.867016\pi$
$678$	− 744.000i	− 0.0421433i
$679$	4970.00	0.280900
$680$	0	0
$681$	−8936.00	−0.502832
$682$	2260.00i	0.126891i
$683$	4979.00i	0.278940i	0.990226	+	0.139470i	$0.0445399\pi$
$683$	4979.00i	0.278940i	−0.990226	+	0.139470i	$0.955460\pi$
$684$	−1848.00	−0.103304
$685$	0	0
$686$	−686.000	−0.0381802
$687$	− 17096.0i	− 0.949422i
$688$	− 4496.00i	− 0.249140i
$689$	32636.0	1.80455
$690$	0	0
$691$	6744.00	0.371279	0.185640	−	0.982618i	$-0.440564\pi$
$691$	6744.00	0.371279	0.185640	+	0.982618i	$0.440564\pi$
$692$	− 15576.0i	− 0.855651i
$693$	− 385.000i	− 0.0211038i
$694$	7574.00	0.414272
$695$	0	0
$696$	−32.0000	−0.00174275
$697$	− 192.000i	− 0.0104340i
$698$	− 14796.0i	− 0.802345i
$699$	−21708.0	−1.17464
$700$	0	0
$701$	−17682.0	−0.952696	−0.476348	−	0.879257i	$-0.658040\pi$
$701$	−17682.0	−0.952696	−0.476348	+	0.879257i	$0.658040\pi$
$702$	24928.0i	1.34024i
$703$	− 798.000i	− 0.0428124i
$704$	−320.000	−0.0171313
$705$	0	0
$706$	−19308.0	−1.02927
$707$	0	0
$708$	1696.00i	0.0900277i
$709$	3514.00	0.186137	0.0930684	−	0.995660i	$-0.470332\pi$
$709$	3514.00	0.186137	0.0930684	+	0.995660i	$0.470332\pi$
$710$	0	0
$711$	13783.0	0.727008
$712$	688.000i	0.0362133i
$713$	− 39550.0i	− 2.07736i
$714$	672.000	0.0352226
$715$	0	0
$716$	−4432.00	−0.231329
$717$	− 14064.0i	− 0.732538i
$718$	15382.0i	0.799514i
$719$	34132.0	1.77039	0.885194	−	0.465222i	$-0.154026\pi$
$719$	34132.0	1.77039	0.885194	+	0.465222i	$0.154026\pi$
$720$	0	0
$721$	5866.00	0.302998
$722$	− 10190.0i	− 0.525253i
$723$	− 15560.0i	− 0.800391i
$724$	7680.00	0.394233
$725$	0	0
$726$	10448.0	0.534107
$727$	22112.0i	1.12804i	0.825759	+	0.564022i	$0.190747\pi$
$727$	22112.0i	1.12804i	−0.825759	+	0.564022i	$0.809253\pi$
$728$	4592.00i	0.233779i
$729$	−19837.0	−1.00782
$730$	0	0
$731$	−3372.00	−0.170613
$732$	− 768.000i	− 0.0387788i
$733$	22180.0i	1.11765i	0.829286	+	0.558825i	$0.188748\pi$
$733$	22180.0i	1.11765i	−0.829286	+	0.558825i	$0.811252\pi$
$734$	13672.0	0.687524
$735$	0	0
$736$	5600.00	0.280460
$737$	2415.00i	0.120702i
$738$	352.000i	0.0175573i
$739$	16279.0	0.810328	0.405164	−	0.914244i	$-0.367214\pi$
$739$	16279.0	0.810328	0.405164	+	0.914244i	$0.367214\pi$
$740$	0	0
$741$	13776.0	0.682961
$742$	5572.00i	0.275680i
$743$	− 29068.0i	− 1.43526i	−0.696422	−	0.717632i	$-0.745226\pi$
$743$	− 29068.0i	− 1.43526i	0.696422	−	0.717632i	$-0.254774\pi$
$744$	7232.00	0.356368
$745$	0	0
$746$	24246.0	1.18996
$747$	− 8338.00i	− 0.408396i
$748$	240.000i	0.0117316i
$749$	−15148.0	−0.738980
$750$	0	0
$751$	20.0000	0.000971785 0	0.000485892	−	1.00000i	$-0.499845\pi$
$751$	20.0000	0.000971785 0	0.000485892		1.00000i	$0.499845\pi$
$752$	5344.00i	0.259143i
$753$	10048.0i	0.486281i
$754$	−164.000	−0.00792112
$755$	0	0
$756$	−4256.00	−0.204748
$757$	22961.0i	1.10242i	0.834367	+	0.551210i	$0.185834\pi$
$757$	22961.0i	1.10242i	−0.834367	+	0.551210i	$0.814166\pi$
$758$	− 21622.0i	− 1.03608i
$759$	3500.00	0.167381
$760$	0	0
$761$	2832.00	0.134901	0.0674507	−	0.997723i	$-0.478513\pi$
$761$	2832.00	0.134901	0.0674507	+	0.997723i	$0.478513\pi$
$762$	11464.0i	0.545009i
$763$	− 12789.0i	− 0.606806i
$764$	3744.00	0.177295
$765$	0	0
$766$	−25556.0	−1.20545
$767$	8692.00i	0.409192i
$768$	1024.00i	0.0481125i
$769$	20204.0	0.947432	0.473716	−	0.880678i	$-0.342912\pi$
$769$	20204.0	0.947432	0.473716	+	0.880678i	$0.342912\pi$
$770$	0	0
$771$	−27736.0	−1.29557
$772$	12652.0i	0.589839i
$773$	27882.0i	1.29734i	0.761069	+	0.648671i	$0.224675\pi$
$773$	27882.0i	1.29734i	−0.761069	+	0.648671i	$0.775325\pi$
$774$	6182.00	0.287090
$775$	0	0
$776$	5680.00	0.262758
$777$	− 532.000i	− 0.0245629i
$778$	766.000i	0.0352988i
$779$	672.000	0.0309074
$780$	0	0
$781$	−75.0000	−0.00343625
$782$	− 4200.00i	− 0.192061i
$783$	− 152.000i	− 0.00693747i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	−6944.00	−0.315120
$787$	36304.0i	1.64434i	0.569240	+	0.822171i	$0.307238\pi$
$787$	36304.0i	1.64434i	−0.569240	+	0.822171i	$0.692762\pi$
$788$	1396.00i	0.0631097i
$789$	22020.0	0.993578
$790$	0	0
$791$	651.000	0.0292628
$792$	− 440.000i	− 0.0197408i
$793$	− 3936.00i	− 0.176257i
$794$	−24100.0	−1.07718
$795$	0	0
$796$	0	0
$797$	− 21096.0i	− 0.937589i	−0.883307	−	0.468795i	$-0.844688\pi$
$797$	− 21096.0i	− 0.937589i	0.883307	−	0.468795i	$-0.155312\pi$
$798$	2352.00i	0.104336i
$799$	4008.00	0.177463
$800$	0	0
$801$	−946.000	−0.0417294
$802$	− 15046.0i	− 0.662460i
$803$	− 5220.00i	− 0.229402i
$804$	7728.00	0.338987
$805$	0	0
$806$	37064.0	1.61976
$807$	34792.0i	1.51764i
$808$	0	0
$809$	−15879.0	−0.690081	−0.345041	−	0.938588i	$-0.612135\pi$
$809$	−15879.0	−0.690081	−0.345041	+	0.938588i	$0.612135\pi$
$810$	0	0
$811$	33402.0	1.44624	0.723121	−	0.690721i	$-0.242707\pi$
$811$	33402.0	1.44624	0.723121	+	0.690721i	$0.242707\pi$
$812$	− 28.0000i	− 0.00121011i
$813$	8216.00i	0.354425i
$814$	190.000	0.00818120
$815$	0	0
$816$	768.000	0.0329478
$817$	− 11802.0i	− 0.505385i
$818$	9068.00i	0.387598i
$819$	−6314.00	−0.269388
$820$	0	0
$821$	10506.0	0.446604	0.223302	−	0.974749i	$-0.428316\pi$
$821$	10506.0	0.446604	0.223302	+	0.974749i	$0.428316\pi$
$822$	21168.0i	0.898198i
$823$	− 1035.00i	− 0.0438370i	−0.999760	−	0.0219185i	$-0.993023\pi$
$823$	− 1035.00i	− 0.0438370i	0.999760	−	0.0219185i	$-0.00697743\pi$
$824$	6704.00	0.283428
$825$	0	0
$826$	−1484.00	−0.0625121
$827$	30137.0i	1.26719i	0.773665	+	0.633595i	$0.218421\pi$
$827$	30137.0i	1.26719i	−0.773665	+	0.633595i	$0.781579\pi$
$828$	7700.00i	0.323181i
$829$	−35464.0	−1.48578	−0.742892	−	0.669411i	$-0.766547\pi$
$829$	−35464.0	−1.48578	−0.742892	+	0.669411i	$0.766547\pi$
$830$	0	0
$831$	−33800.0	−1.41096
$832$	5248.00i	0.218680i
$833$	588.000i	0.0244574i
$834$	7168.00	0.297611
$835$	0	0
$836$	−840.000	−0.0347512
$837$	34352.0i	1.41861i
$838$	− 16988.0i	− 0.700287i
$839$	1492.00	0.0613940	0.0306970	−	0.999529i	$-0.490227\pi$
$839$	1492.00	0.0613940	0.0306970	+	0.999529i	$0.490227\pi$
$840$	0	0
$841$	−24388.0	−0.999959
$842$	− 17530.0i	− 0.717487i
$843$	17876.0i	0.730347i
$844$	−6704.00	−0.273414
$845$	0	0
$846$	−7348.00	−0.298616
$847$	9142.00i	0.370865i
$848$	6368.00i	0.257875i
$849$	−3176.00	−0.128386
$850$	0	0
$851$	−3325.00	−0.133936
$852$	240.000i	0.00965055i
$853$	− 19288.0i	− 0.774219i	−0.922034	−	0.387109i	$-0.873474\pi$
$853$	− 19288.0i	− 0.774219i	0.922034	−	0.387109i	$-0.126526\pi$
$854$	672.000	0.0269267
$855$	0	0
$856$	−17312.0	−0.691252
$857$	2656.00i	0.105866i	0.998598	+	0.0529330i	$0.0168570\pi$
$857$	2656.00i	0.105866i	−0.998598	+	0.0529330i	$0.983143\pi$
$858$	3280.00i	0.130510i
$859$	−3786.00	−0.150380	−0.0751901	−	0.997169i	$-0.523956\pi$
$859$	−3786.00	−0.150380	−0.0751901	+	0.997169i	$0.523956\pi$
$860$	0	0
$861$	448.000	0.0177326
$862$	− 13392.0i	− 0.529157i
$863$	45503.0i	1.79483i	0.441185	+	0.897416i	$0.354558\pi$
$863$	45503.0i	1.79483i	−0.441185	+	0.897416i	$0.645442\pi$
$864$	−4864.00	−0.191524
$865$	0	0
$866$	−8296.00	−0.325531
$867$	19076.0i	0.747238i
$868$	6328.00i	0.247450i
$869$	6265.00	0.244563
$870$	0	0
$871$	39606.0	1.54076
$872$	− 14616.0i	− 0.567615i
$873$	7810.00i	0.302782i
$874$	14700.0	0.568919
$875$	0	0
$876$	−16704.0	−0.644265
$877$	− 32626.0i	− 1.25622i	−0.778126	−	0.628108i	$-0.783830\pi$
$877$	− 32626.0i	− 1.25622i	0.778126	−	0.628108i	$-0.216170\pi$
$878$	− 12364.0i	− 0.475245i
$879$	−25848.0	−0.991845
$880$	0	0
$881$	−25754.0	−0.984874	−0.492437	−	0.870348i	$-0.663894\pi$
$881$	−25754.0	−0.984874	−0.492437	+	0.870348i	$0.663894\pi$
$882$	− 1078.00i	− 0.0411544i
$883$	9165.00i	0.349294i	0.984631	+	0.174647i	$0.0558785\pi$
$883$	9165.00i	0.349294i	−0.984631	+	0.174647i	$0.944122\pi$
$884$	3936.00	0.149753
$885$	0	0
$886$	1232.00	0.0467154
$887$	1472.00i	0.0557214i	0.999612	+	0.0278607i	$0.00886949\pi$
$887$	1472.00i	0.0557214i	−0.999612	+	0.0278607i	$0.991131\pi$
$888$	− 608.000i	− 0.0229765i
$889$	−10031.0	−0.378435
$890$	0	0
$891$	−1555.00	−0.0584674
$892$	4704.00i	0.176571i
$893$	14028.0i	0.525677i
$894$	20184.0	0.755094
$895$	0	0
$896$	−896.000	−0.0334077
$897$	− 57400.0i	− 2.13660i
$898$	17306.0i	0.643105i
$899$	−226.000	−0.00838434
$900$	0	0
$901$	4776.00	0.176594
$902$	160.000i	0.00590623i
$903$	− 7868.00i	− 0.289956i
$904$	744.000	0.0273729
$905$	0	0
$906$	−19464.0	−0.713740
$907$	22412.0i	0.820483i	0.911977	+	0.410242i	$0.134556\pi$
$907$	22412.0i	0.820483i	−0.911977	+	0.410242i	$0.865444\pi$
$908$	− 8936.00i	− 0.326599i
$909$	0	0
$910$	0	0
$911$	15163.0	0.551452	0.275726	−	0.961236i	$-0.411082\pi$
$911$	15163.0	0.551452	0.275726	+	0.961236i	$0.411082\pi$
$912$	2688.00i	0.0975971i
$913$	− 3790.00i	− 0.137383i
$914$	15570.0	0.563468
$915$	0	0
$916$	17096.0	0.616668
$917$	− 6076.00i	− 0.218808i
$918$	3648.00i	0.131157i
$919$	6501.00	0.233350	0.116675	−	0.993170i	$-0.462776\pi$
$919$	6501.00	0.233350	0.116675	+	0.993170i	$0.462776\pi$
$920$	0	0
$921$	−224.000	−0.00801417
$922$	− 29436.0i	− 1.05143i
$923$	1230.00i	0.0438634i
$924$	−560.000	−0.0199379
$925$	0	0
$926$	5728.00	0.203276
$927$	9218.00i	0.326601i
$928$	− 32.0000i	− 0.00113195i
$929$	−1464.00	−0.0517032	−0.0258516	−	0.999666i	$-0.508230\pi$
$929$	−1464.00	−0.0517032	−0.0258516	+	0.999666i	$0.508230\pi$
$930$	0	0
$931$	−2058.00	−0.0724471
$932$	− 21708.0i	− 0.762950i
$933$	42624.0i	1.49566i
$934$	3756.00	0.131585
$935$	0	0
$936$	−7216.00	−0.251990
$937$	− 14064.0i	− 0.490342i	−0.969480	−	0.245171i	$-0.921156\pi$
$937$	− 14064.0i	− 0.490342i	0.969480	−	0.245171i	$-0.0788442\pi$
$938$	6762.00i	0.235381i
$939$	−8040.00	−0.279420
$940$	0	0
$941$	−39114.0	−1.35503	−0.677513	−	0.735511i	$-0.736942\pi$
$941$	−39114.0	−1.35503	−0.677513	+	0.735511i	$0.736942\pi$
$942$	20576.0i	0.711680i
$943$	− 2800.00i	− 0.0966920i
$944$	−1696.00	−0.0584747
$945$	0	0
$946$	2810.00	0.0965761
$947$	− 15068.0i	− 0.517048i	−0.966005	−	0.258524i	$-0.916764\pi$
$947$	− 15068.0i	− 0.517048i	0.966005	−	0.258524i	$-0.0832361\pi$
$948$	− 20048.0i	− 0.686845i
$949$	−85608.0	−2.92830
$950$	0	0
$951$	−15612.0	−0.532338
$952$	672.000i	0.0228778i
$953$	− 27847.0i	− 0.946540i	−0.880917	−	0.473270i	$-0.843073\pi$
$953$	− 27847.0i	− 0.946540i	0.880917	−	0.473270i	$-0.156927\pi$
$954$	−8756.00	−0.297155
$955$	0	0
$956$	14064.0	0.475797
$957$	− 20.0000i	0 0.000675557i
$958$	19828.0i	0.668699i
$959$	−18522.0	−0.623677
$960$	0	0
$961$	21285.0	0.714478
$962$	− 3116.00i	− 0.104432i
$963$	− 23804.0i	− 0.796545i
$964$	15560.0	0.519869
$965$	0	0
$966$	9800.00	0.326408
$967$	− 33704.0i	− 1.12084i	−0.828210	−	0.560418i	$-0.810641\pi$
$967$	− 33704.0i	− 1.12084i	0.828210	−	0.560418i	$-0.189359\pi$
$968$	10448.0i	0.346913i
$969$	2016.00	0.0668351
$970$	0	0
$971$	−30006.0	−0.991698	−0.495849	−	0.868409i	$-0.665143\pi$
$971$	−30006.0	−0.991698	−0.495849	+	0.868409i	$0.665143\pi$
$972$	− 11440.0i	− 0.377508i
$973$	6272.00i	0.206651i
$974$	25766.0	0.847634
$975$	0	0
$976$	768.000	0.0251876
$977$	14457.0i	0.473409i	0.971582	+	0.236704i	$0.0760673\pi$
$977$	14457.0i	0.473409i	−0.971582	+	0.236704i	$0.923933\pi$
$978$	19904.0i	0.650777i
$979$	−430.000	−0.0140377
$980$	0	0
$981$	20097.0	0.654075
$982$	− 33134.0i	− 1.07673i
$983$	43742.0i	1.41928i	0.704564	+	0.709640i	$0.251143\pi$
$983$	43742.0i	1.41928i	−0.704564	+	0.709640i	$0.748857\pi$
$984$	512.000	0.0165874
$985$	0	0
$986$	−24.0000	−0.000775168 0
$987$	9352.00i	0.301598i
$988$	13776.0i	0.443596i
$989$	−49175.0	−1.58107
$990$	0	0
$991$	−8201.00	−0.262879	−0.131440	−	0.991324i	$-0.541960\pi$
$991$	−8201.00	−0.262879	−0.131440	+	0.991324i	$0.541960\pi$
$992$	7232.00i	0.231468i
$993$	− 13108.0i	− 0.418902i
$994$	−210.000	−0.00670100
$995$	0	0
$996$	−12128.0	−0.385834
$997$	4274.00i	0.135766i	0.997693	+	0.0678831i	$0.0216245\pi$
$997$	4274.00i	0.135766i	−0.997693	+	0.0678831i	$0.978376\pi$
$998$	− 4088.00i	− 0.129663i
$999$	2888.00	0.0914637

Display

a_p

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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

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.4.c.c.99.2		2
5.2	odd	4		350.4.a.h.1.1	✓	1
5.3	odd	4		350.4.a.n.1.1	yes	1
5.4	even	2	inner	350.4.c.c.99.1		2
35.13	even	4		2450.4.a.bk.1.1		1
35.27	even	4		2450.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.a.h.1.1	✓	1	5.2	odd	4
350.4.a.n.1.1	yes	1	5.3	odd	4
350.4.c.c.99.1		2	5.4	even	2	inner
350.4.c.c.99.2		2	1.1	even	1	trivial
2450.4.a.e.1.1		1	35.27	even	4
2450.4.a.bk.1.1		1	35.13	even	4

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	350.4.c.c.99.2

Level	$350$
Weight	$4$
Character	350.99
Analytic conductor	$20.651$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$