LMFDB - Embedded newform 650.2.b.b.599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	650.b (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:	$5.19027613138$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.2.b.b.599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -10.0000 q^{21} +3.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} +5.00000i q^{28} -9.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +10.0000i q^{42} +2.00000i q^{43} +3.00000 q^{44} +6.00000 q^{46} +9.00000i q^{47} -2.00000i q^{48} -18.0000 q^{49} -6.00000 q^{51} -1.00000i q^{52} -9.00000i q^{53} -4.00000 q^{54} +5.00000 q^{56} -8.00000i q^{57} +9.00000i q^{58} +9.00000 q^{59} -1.00000 q^{61} -5.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} -5.00000i q^{67} +3.00000i q^{68} +12.0000 q^{69} -1.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +15.0000i q^{77} -2.00000i q^{78} +16.0000 q^{79} -11.0000 q^{81} -15.0000i q^{83} +10.0000 q^{84} +2.00000 q^{86} +18.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} +5.00000 q^{91} -6.00000i q^{92} -10.0000i q^{93} +9.00000 q^{94} -2.00000 q^{96} -8.00000i q^{97} +18.0000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} - 10 q^{14} + 2 q^{16} + 8 q^{19} - 20 q^{21} + 4 q^{24} + 2 q^{26} - 18 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 4 q^{39} + 6 q^{44} + 12 q^{46} - 36 q^{49} - 12 q^{51} - 8 q^{54} + 10 q^{56} + 18 q^{59} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} - 4 q^{74} - 8 q^{76} + 32 q^{79} - 22 q^{81} + 20 q^{84} + 4 q^{86} + 12 q^{89} + 10 q^{91} + 18 q^{94} - 4 q^{96} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 - 10 * q^14 + 2 * q^16 + 8 * q^19 - 20 * q^21 + 4 * q^24 + 2 * q^26 - 18 * q^29 + 10 * q^31 - 6 * q^34 + 2 * q^36 + 4 * q^39 + 6 * q^44 + 12 * q^46 - 36 * q^49 - 12 * q^51 - 8 * q^54 + 10 * q^56 + 18 * q^59 - 2 * q^61 - 2 * q^64 + 12 * q^66 + 24 * q^69 - 4 * q^74 - 8 * q^76 + 32 * q^79 - 22 * q^81 + 20 * q^84 + 4 * q^86 + 12 * q^89 + 10 * q^91 + 18 * q^94 - 4 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 5.00000i	− 1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	− 5.00000i	− 1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000i	0.577350i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−10.0000	−2.18218
$22$	3.00000i	0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	1.00000	0.196116
$27$	− 4.00000i	− 0.769800i
$28$	5.00000i	0.944911i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	6.00000i	1.04447i
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	10.0000i	1.54303i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 1.00000i	− 0.138675i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	5.00000	0.668153
$57$	− 8.00000i	− 1.05963i
$58$	9.00000i	1.18176i
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	− 5.00000i	− 0.635001i
$63$	5.00000i	0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	3.00000i	0.363803i
$69$	12.0000	1.44463
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	15.0000i	1.70941i
$78$	− 2.00000i	− 0.226455i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	10.0000	1.09109
$85$	0	0
$86$	2.00000	0.215666
$87$	18.0000i	1.92980i
$88$	− 3.00000i	− 0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	− 6.00000i	− 0.625543i
$93$	− 10.0000i	− 1.03695i
$94$	9.00000	0.928279
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	18.0000i	1.81827i
$99$	3.00000	0.301511
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	6.00000i	0.594089i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	4.00000i	0.384900i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 5.00000i	− 0.472456i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	9.00000	0.835629
$117$	− 1.00000i	− 0.0924500i
$118$	− 9.00000i	− 0.828517i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−2.00000	−0.181818
$122$	1.00000i	0.0905357i
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	5.00000	0.445435
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	− 6.00000i	− 0.522233i
$133$	− 20.0000i	− 1.73422i
$134$	−5.00000	−0.431934
$135$	0	0
$136$	3.00000	0.257248
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 12.0000i	− 1.02151i
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	18.0000	1.51587
$142$	0	0
$143$	− 3.00000i	− 0.250873i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	36.0000i	2.96923i
$148$	2.00000i	0.164399i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	4.00000i	0.324443i
$153$	3.00000i	0.242536i
$154$	15.0000	1.20873
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 17.0000i	− 1.35675i	−0.734717	−	0.678374i	$-0.762685\pi$
$157$	− 17.0000i	− 1.35675i	0.734717	−	0.678374i	$-0.237315\pi$
$158$	− 16.0000i	− 1.27289i
$159$	−18.0000	−1.42749
$160$	0	0
$161$	30.0000	2.36433
$162$	11.0000i	0.864242i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	−15.0000	−1.16423
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	− 10.0000i	− 0.771517i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	− 2.00000i	− 0.152499i
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	18.0000	1.36458
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 18.0000i	− 1.35296i
$178$	− 6.00000i	− 0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	− 5.00000i	− 0.370625i
$183$	2.00000i	0.147844i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−10.0000	−0.733236
$187$	9.00000i	0.658145i
$188$	− 9.00000i	− 0.656392i
$189$	−20.0000	−1.45479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	2.00000i	0.144338i
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	18.0000	1.28571
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	− 3.00000i	− 0.213201i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	3.00000i	0.211079i
$203$	45.0000i	3.15838i
$204$	6.00000	0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	− 6.00000i	− 0.417029i
$208$	1.00000i	0.0693375i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	9.00000i	0.618123i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	4.00000	0.272166
$217$	− 25.0000i	− 1.69711i
$218$	2.00000i	0.135457i
$219$	28.0000	1.89206
$220$	0	0
$221$	3.00000	0.201802
$222$	4.00000i	0.268462i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−6.00000	−0.399114
$227$	9.00000i	0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$	9.00000i	0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$	8.00000i	0.529813i
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	30.0000	1.97386
$232$	− 9.00000i	− 0.590879i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−9.00000	−0.585850
$237$	− 32.0000i	− 2.07862i
$238$	15.0000i	0.972306i
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	2.00000i	0.128565i
$243$	10.0000i	0.641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	5.00000i	0.317500i
$249$	−30.0000	−1.90117
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	− 5.00000i	− 0.314970i
$253$	− 18.0000i	− 1.13165i
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	− 4.00000i	− 0.249029i
$259$	−10.0000	−0.621370
$260$	0	0
$261$	9.00000	0.557086
$262$	6.00000i	0.370681i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	−20.0000	−1.22628
$267$	− 12.0000i	− 0.734388i
$268$	5.00000i	0.305424i
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	− 3.00000i	− 0.181902i
$273$	− 10.0000i	− 0.605228i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−12.0000	−0.722315
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 10.0000i	− 0.599760i
$279$	−5.00000	−0.299342
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	− 18.0000i	− 1.07188i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−16.0000	−0.937937
$292$	− 14.0000i	− 0.819288i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	36.0000	2.09956
$295$	0	0
$296$	2.00000	0.116248
$297$	12.0000i	0.696311i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	10.0000	0.576390
$302$	7.00000i	0.402805i
$303$	6.00000i	0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	3.00000	0.171499
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	− 15.0000i	− 0.854704i
$309$	−32.0000	−1.82042
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	2.00000i	0.113228i
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	−17.0000	−0.959366
$315$	0	0
$316$	−16.0000	−0.900070
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	18.0000i	1.00939i
$319$	27.0000	1.51171
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 30.0000i	− 1.67183i
$323$	− 12.0000i	− 0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−16.0000	−0.886158
$327$	4.00000i	0.221201i
$328$	0	0
$329$	45.0000	2.48093
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	15.0000i	0.823232i
$333$	2.00000i	0.109599i
$334$	24.0000	1.31322
$335$	0	0
$336$	−10.0000	−0.545545
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	1.00000i	0.0543928i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−15.0000	−0.812296
$342$	4.00000i	0.216295i
$343$	55.0000i	2.96972i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−3.00000	−0.161281
$347$	− 30.0000i	− 1.61048i	−0.592946	−	0.805242i	$-0.702035\pi$
$347$	− 30.0000i	− 1.61048i	0.592946	−	0.805242i	$-0.297965\pi$
$348$	− 18.0000i	− 0.964901i
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	3.00000i	0.159901i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	−18.0000	−0.956689
$355$	0	0
$356$	−6.00000	−0.317999
$357$	30.0000i	1.58777i
$358$	0	0
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 11.0000i	− 0.578147i
$363$	4.00000i	0.209946i
$364$	−5.00000	−0.262071
$365$	0	0
$366$	2.00000	0.104542
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	−45.0000	−2.33628
$372$	10.0000i	0.518476i
$373$	29.0000i	1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$	29.0000i	1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	− 9.00000i	− 0.463524i
$378$	20.0000i	1.02869i
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	−40.0000	−2.04926
$382$	− 12.0000i	− 0.613973i
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	20.0000	1.01797
$387$	− 2.00000i	− 0.101666i
$388$	8.00000i	0.406138i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 18.0000i	− 0.909137i
$393$	12.0000i	0.605320i
$394$	24.0000	1.20910
$395$	0	0
$396$	−3.00000	−0.150756
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 16.0000i	− 0.802008i
$399$	−40.0000	−2.00250
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	10.0000i	0.498755i
$403$	5.00000i	0.249068i
$404$	3.00000	0.149256
$405$	0	0
$406$	45.0000	2.23331
$407$	6.00000i	0.297409i
$408$	− 6.00000i	− 0.297044i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	16.0000i	0.788263i
$413$	− 45.0000i	− 2.21431i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	1.00000	0.0490290
$417$	− 20.0000i	− 0.979404i
$418$	12.0000i	0.586939i
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	− 14.0000i	− 0.681509i
$423$	− 9.00000i	− 0.437595i
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	5.00000i	0.241967i
$428$	6.00000i	0.290021i
$429$	−6.00000	−0.289683
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	− 4.00000i	− 0.192450i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−25.0000	−1.20004
$435$	0	0
$436$	2.00000	0.0957826
$437$	24.0000i	1.14808i
$438$	− 28.0000i	− 1.33789i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	− 3.00000i	− 0.142695i
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	5.00000i	0.236228i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	14.0000i	0.657777i
$454$	9.00000	0.422391
$455$	0	0
$456$	8.00000	0.374634
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	− 16.0000i	− 0.747631i
$459$	−12.0000	−0.560112
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	− 30.0000i	− 1.39573i
$463$	− 1.00000i	− 0.0464739i	−0.999730	−	0.0232370i	$-0.992603\pi$
$463$	− 1.00000i	− 0.0464739i	0.999730	−	0.0232370i	$-0.00739722\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	6.00000	0.277945
$467$	− 30.0000i	− 1.38823i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	− 30.0000i	− 1.38823i	0.719862	−	0.694117i	$-0.244205\pi$
$468$	1.00000i	0.0462250i
$469$	−25.0000	−1.15439
$470$	0	0
$471$	−34.0000	−1.56664
$472$	9.00000i	0.414259i
$473$	− 6.00000i	− 0.275880i
$474$	−32.0000	−1.46981
$475$	0	0
$476$	15.0000	0.687524
$477$	9.00000i	0.412082i
$478$	− 15.0000i	− 0.686084i
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 8.00000i	− 0.364390i
$483$	− 60.0000i	− 2.73009i
$484$	2.00000	0.0909091
$485$	0	0
$486$	10.0000	0.453609
$487$	7.00000i	0.317200i	0.987343	+	0.158600i	$0.0506981\pi$
$487$	7.00000i	0.317200i	−0.987343	+	0.158600i	$0.949302\pi$
$488$	− 1.00000i	− 0.0452679i
$489$	−32.0000	−1.44709
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	27.0000i	1.21602i
$494$	4.00000	0.179969
$495$	0	0
$496$	5.00000	0.224507
$497$	0	0
$498$	30.0000i	1.34433i
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	48.0000	2.14448
$502$	24.0000i	1.07117i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−5.00000	−0.222718
$505$	0	0
$506$	−18.0000	−0.800198
$507$	2.00000i	0.0888231i
$508$	20.0000i	0.887357i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	70.0000	3.09662
$512$	− 1.00000i	− 0.0441942i
$513$	− 16.0000i	− 0.706417i
$514$	15.0000	0.661622
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 27.0000i	− 1.18746i
$518$	10.0000i	0.439375i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	− 9.00000i	− 0.393919i
$523$	− 40.0000i	− 1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$	− 40.0000i	− 1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	18.0000	0.784837
$527$	− 15.0000i	− 0.653410i
$528$	6.00000i	0.261116i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−9.00000	−0.390567
$532$	20.0000i	0.867110i
$533$	0	0
$534$	−12.0000	−0.519291
$535$	0	0
$536$	5.00000	0.215967
$537$	0	0
$538$	9.00000i	0.388018i
$539$	54.0000	2.32594
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	19.0000i	0.816120i
$543$	− 22.0000i	− 0.944110i
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−10.0000	−0.427960
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	6.00000i	0.256307i
$549$	1.00000	0.0426790
$550$	0	0
$551$	−36.0000	−1.53365
$552$	12.0000i	0.510754i
$553$	− 80.0000i	− 3.40195i
$554$	22.0000	0.934690
$555$	0	0
$556$	−10.0000	−0.424094
$557$	36.0000i	1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$	36.0000i	1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$	5.00000i	0.211667i
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	18.0000	0.759961
$562$	− 12.0000i	− 0.506189i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−18.0000	−0.757937
$565$	0	0
$566$	14.0000	0.588464
$567$	55.0000i	2.30978i
$568$	0	0
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	3.00000i	0.125436i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	− 8.00000i	− 0.332756i
$579$	40.0000	1.66234
$580$	0	0
$581$	−75.0000	−3.11152
$582$	16.0000i	0.663221i
$583$	27.0000i	1.11823i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	− 36.0000i	− 1.48461i
$589$	20.0000	0.824086
$590$	0	0
$591$	48.0000	1.97446
$592$	− 2.00000i	− 0.0821995i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	0	0
$597$	− 32.0000i	− 1.30967i
$598$	6.00000i	0.245358i
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	− 10.0000i	− 0.407570i
$603$	5.00000i	0.203616i
$604$	7.00000	0.284826
$605$	0	0
$606$	6.00000	0.243733
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	− 4.00000i	− 0.162221i
$609$	90.0000	3.64698
$610$	0	0
$611$	−9.00000	−0.364101
$612$	− 3.00000i	− 0.121268i
$613$	32.0000i	1.29247i	0.763139	+	0.646234i	$0.223657\pi$
$613$	32.0000i	1.29247i	−0.763139	+	0.646234i	$0.776343\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−15.0000	−0.604367
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	32.0000i	1.28723i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	18.0000i	0.721734i
$623$	− 30.0000i	− 1.20192i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−19.0000	−0.759393
$627$	24.0000i	0.958468i
$628$	17.0000i	0.678374i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	16.0000i	0.636446i
$633$	− 28.0000i	− 1.11290i
$634$	0	0
$635$	0	0
$636$	18.0000	0.713746
$637$	− 18.0000i	− 0.713186i
$638$	− 27.0000i	− 1.06894i
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	12.0000i	0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	−30.0000	−1.18217
$645$	0	0
$646$	−12.0000	−0.472134
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	− 11.0000i	− 0.432121i
$649$	−27.0000	−1.05984
$650$	0	0
$651$	−50.0000	−1.95965
$652$	16.0000i	0.626608i
$653$	33.0000i	1.29139i	0.763596	+	0.645695i	$0.223432\pi$
$653$	33.0000i	1.29139i	−0.763596	+	0.645695i	$0.776568\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	0	0
$657$	− 14.0000i	− 0.546192i
$658$	− 45.0000i	− 1.75428i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 20.0000i	− 0.777322i
$663$	− 6.00000i	− 0.233021i
$664$	15.0000	0.582113
$665$	0	0
$666$	2.00000	0.0774984
$667$	− 54.0000i	− 2.09089i
$668$	− 24.0000i	− 0.928588i
$669$	−8.00000	−0.309298
$670$	0	0
$671$	3.00000	0.115814
$672$	10.0000i	0.385758i
$673$	− 25.0000i	− 0.963679i	−0.876259	−	0.481840i	$-0.839969\pi$
$673$	− 25.0000i	− 0.963679i	0.876259	−	0.481840i	$-0.160031\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	12.0000i	0.460857i
$679$	−40.0000	−1.53506
$680$	0	0
$681$	18.0000	0.689761
$682$	15.0000i	0.574380i
$683$	− 15.0000i	− 0.573959i	−0.957937	−	0.286980i	$-0.907349\pi$
$683$	− 15.0000i	− 0.573959i	0.957937	−	0.286980i	$-0.0926512\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	55.0000	2.09991
$687$	− 32.0000i	− 1.22088i
$688$	2.00000i	0.0762493i
$689$	9.00000	0.342873
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	3.00000i	0.114043i
$693$	− 15.0000i	− 0.569803i
$694$	−30.0000	−1.13878
$695$	0	0
$696$	−18.0000	−0.682288
$697$	0	0
$698$	20.0000i	0.757011i
$699$	12.0000	0.453882
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	− 4.00000i	− 0.150970i
$703$	− 8.00000i	− 0.301726i
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	15.0000i	0.564133i
$708$	18.0000i	0.676481i
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	6.00000i	0.224860i
$713$	30.0000i	1.12351i
$714$	30.0000	1.12272
$715$	0	0
$716$	0	0
$717$	− 30.0000i	− 1.12037i
$718$	− 21.0000i	− 0.783713i
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−80.0000	−2.97936
$722$	3.00000i	0.111648i
$723$	− 16.0000i	− 0.595046i
$724$	−11.0000	−0.408812
$725$	0	0
$726$	4.00000	0.148454
$727$	− 20.0000i	− 0.741759i	−0.928681	−	0.370879i	$-0.879056\pi$
$727$	− 20.0000i	− 0.741759i	0.928681	−	0.370879i	$-0.120944\pi$
$728$	5.00000i	0.185312i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	6.00000	0.221918
$732$	− 2.00000i	− 0.0739221i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	15.0000i	0.552532i
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	45.0000i	1.65200i
$743$	− 33.0000i	− 1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$	− 33.0000i	− 1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	29.0000	1.06177
$747$	15.0000i	0.548821i
$748$	− 9.00000i	− 0.329073i
$749$	−30.0000	−1.09618
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	9.00000i	0.328196i
$753$	48.0000i	1.74922i
$754$	−9.00000	−0.327761
$755$	0	0
$756$	20.0000	0.727393
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	29.0000i	1.05333i
$759$	−36.0000	−1.30672
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	40.0000i	1.44905i
$763$	10.0000i	0.362024i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	36.0000	1.30073
$767$	9.00000i	0.324971i
$768$	− 2.00000i	− 0.0721688i
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	− 20.0000i	− 0.719816i
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	8.00000	0.287183
$777$	20.0000i	0.717496i
$778$	18.0000i	0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 18.0000i	− 0.643679i
$783$	36.0000i	1.28654i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	12.0000	0.428026
$787$	49.0000i	1.74666i	0.487128	+	0.873331i	$0.338045\pi$
$787$	49.0000i	1.74666i	−0.487128	+	0.873331i	$0.661955\pi$
$788$	− 24.0000i	− 0.854965i
$789$	36.0000	1.28163
$790$	0	0
$791$	−30.0000	−1.06668
$792$	3.00000i	0.106600i
$793$	− 1.00000i	− 0.0355110i
$794$	22.0000	0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 3.00000i	− 0.106265i	−0.998587	−	0.0531327i	$-0.983079\pi$
$797$	− 3.00000i	− 0.106265i	0.998587	−	0.0531327i	$-0.0169206\pi$
$798$	40.0000i	1.41598i
$799$	27.0000	0.955191
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 12.0000i	− 0.423735i
$803$	− 42.0000i	− 1.48215i
$804$	10.0000	0.352673
$805$	0	0
$806$	5.00000	0.176117
$807$	18.0000i	0.633630i
$808$	− 3.00000i	− 0.105540i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	− 45.0000i	− 1.57919i
$813$	38.0000i	1.33272i
$814$	6.00000	0.210300
$815$	0	0
$816$	−6.00000	−0.210042
$817$	8.00000i	0.279885i
$818$	− 22.0000i	− 0.769212i
$819$	−5.00000	−0.174714
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	12.0000i	0.418548i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−45.0000	−1.56575
$827$	− 9.00000i	− 0.312961i	−0.987681	−	0.156480i	$-0.949985\pi$
$827$	− 9.00000i	− 0.312961i	0.987681	−	0.156480i	$-0.0500148\pi$
$828$	6.00000i	0.208514i
$829$	−5.00000	−0.173657	−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000	−0.173657	−0.0868286	+	0.996223i	$0.527673\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	− 1.00000i	− 0.0346688i
$833$	54.0000i	1.87099i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	12.0000	0.415029
$837$	− 20.0000i	− 0.691301i
$838$	6.00000i	0.207267i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	4.00000i	0.137849i
$843$	− 24.0000i	− 0.826604i
$844$	−14.0000	−0.481900
$845$	0	0
$846$	−9.00000	−0.309426
$847$	10.0000i	0.343604i
$848$	− 9.00000i	− 0.309061i
$849$	28.0000	0.960958
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	5.00000	0.171096
$855$	0	0
$856$	6.00000	0.205076
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	6.00000i	0.204837i
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	0	0
$861$	0	0
$862$	− 36.0000i	− 1.22616i
$863$	− 45.0000i	− 1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$	− 45.0000i	− 1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 16.0000i	− 0.543388i
$868$	25.0000i	0.848555i
$869$	−48.0000	−1.62829
$870$	0	0
$871$	5.00000	0.169419
$872$	− 2.00000i	− 0.0677285i
$873$	8.00000i	0.270759i
$874$	24.0000	0.811812
$875$	0	0
$876$	−28.0000	−0.946032
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	20.0000i	0.674967i
$879$	0	0
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	− 18.0000i	− 0.606092i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	6.00000	0.201574
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 4.00000i	− 0.134231i
$889$	−100.000	−3.35389
$890$	0	0
$891$	33.0000	1.10554
$892$	4.00000i	0.133930i
$893$	36.0000i	1.20469i
$894$	0	0
$895$	0	0
$896$	5.00000	0.167038
$897$	12.0000i	0.400668i
$898$	6.00000i	0.200223i
$899$	−45.0000	−1.50083
$900$	0	0
$901$	−27.0000	−0.899500
$902$	0	0
$903$	− 20.0000i	− 0.665558i
$904$	6.00000	0.199557
$905$	0	0
$906$	14.0000	0.465119
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	− 9.00000i	− 0.298675i
$909$	3.00000	0.0995037
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	− 8.00000i	− 0.264906i
$913$	45.0000i	1.48928i
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−16.0000	−0.528655
$917$	30.0000i	0.990687i
$918$	12.0000i	0.396059i
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−40.0000	−1.31804
$922$	0	0
$923$	0	0
$924$	−30.0000	−0.986928
$925$	0	0
$926$	−1.00000	−0.0328620
$927$	16.0000i	0.525509i
$928$	9.00000i	0.295439i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−72.0000	−2.35970
$932$	− 6.00000i	− 0.196537i
$933$	36.0000i	1.17859i
$934$	−30.0000	−0.981630
$935$	0	0
$936$	1.00000	0.0326860
$937$	13.0000i	0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$	13.0000i	0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	25.0000i	0.816279i
$939$	−38.0000	−1.24008
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	34.0000i	1.10778i
$943$	0	0
$944$	9.00000	0.292925
$945$	0	0
$946$	−6.00000	−0.195077
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	32.0000i	1.03931i
$949$	−14.0000	−0.454459
$950$	0	0
$951$	0	0
$952$	− 15.0000i	− 0.486153i
$953$	− 27.0000i	− 0.874616i	−0.899312	−	0.437308i	$-0.855932\pi$
$953$	− 27.0000i	− 0.874616i	0.899312	−	0.437308i	$-0.144068\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−15.0000	−0.485135
$957$	− 54.0000i	− 1.74557i
$958$	− 15.0000i	− 0.484628i
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 2.00000i	− 0.0644826i
$963$	6.00000i	0.193347i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	−60.0000	−1.93047
$967$	31.0000i	0.996893i	0.866921	+	0.498446i	$0.166096\pi$
$967$	31.0000i	0.996893i	−0.866921	+	0.498446i	$0.833904\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	− 10.0000i	− 0.320750i
$973$	− 50.0000i	− 1.60293i
$974$	7.00000	0.224294
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	32.0000i	1.02325i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 6.00000i	− 0.191468i
$983$	21.0000i	0.669796i	0.942254	+	0.334898i	$0.108702\pi$
$983$	21.0000i	0.669796i	−0.942254	+	0.334898i	$0.891298\pi$
$984$	0	0
$985$	0	0
$986$	27.0000	0.859855
$987$	− 90.0000i	− 2.86473i
$988$	− 4.00000i	− 0.127257i
$989$	−12.0000	−0.381578
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	− 5.00000i	− 0.158750i
$993$	− 40.0000i	− 1.26936i
$994$	0	0
$995$	0	0
$996$	30.0000	0.950586
$997$	1.00000i	0.0316703i	0.999875	+	0.0158352i	$0.00504070\pi$
$997$	1.00000i	0.0316703i	−0.999875	+	0.0158352i	$0.994959\pi$
$998$	− 25.0000i	− 0.791361i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.b.b.599.1		2
3.2	odd	2		5850.2.e.ba.5149.2		2
5.2	odd	4		650.2.a.h.1.1	yes	1
5.3	odd	4		650.2.a.f.1.1	✓	1
5.4	even	2	inner	650.2.b.b.599.2		2
15.2	even	4		5850.2.a.bb.1.1		1
15.8	even	4		5850.2.a.bc.1.1		1
15.14	odd	2		5850.2.e.ba.5149.1		2
20.3	even	4		5200.2.a.i.1.1		1
20.7	even	4		5200.2.a.bc.1.1		1
65.12	odd	4		8450.2.a.a.1.1		1
65.38	odd	4		8450.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.a.f.1.1	✓	1	5.3	odd	4
650.2.a.h.1.1	yes	1	5.2	odd	4
650.2.b.b.599.1		2	1.1	even	1	trivial
650.2.b.b.599.2		2	5.4	even	2	inner
5200.2.a.i.1.1		1	20.3	even	4
5200.2.a.bc.1.1		1	20.7	even	4
5850.2.a.bb.1.1		1	15.2	even	4
5850.2.a.bc.1.1		1	15.8	even	4
5850.2.e.ba.5149.1		2	15.14	odd	2
5850.2.e.ba.5149.2		2	3.2	odd	2
8450.2.a.a.1.1		1	65.12	odd	4
8450.2.a.x.1.1		1	65.38	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -10.0000 q^{21} +3.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} +5.00000i q^{28} -9.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} -3.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +10.0000i q^{42} +2.00000i q^{43} +3.00000 q^{44} +6.00000 q^{46} +9.00000i q^{47} -2.00000i q^{48} -18.0000 q^{49} -6.00000 q^{51} -1.00000i q^{52} -9.00000i q^{53} -4.00000 q^{54} +5.00000 q^{56} -8.00000i q^{57} +9.00000i q^{58} +9.00000 q^{59} -1.00000 q^{61} -5.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} -5.00000i q^{67} +3.00000i q^{68} +12.0000 q^{69} -1.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +15.0000i q^{77} -2.00000i q^{78} +16.0000 q^{79} -11.0000 q^{81} -15.0000i q^{83} +10.0000 q^{84} +2.00000 q^{86} +18.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} +5.00000 q^{91} -6.00000i q^{92} -10.0000i q^{93} +9.00000 q^{94} -2.00000 q^{96} -8.00000i q^{97} +18.0000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} - 10 q^{14} + 2 q^{16} + 8 q^{19} - 20 q^{21} + 4 q^{24} + 2 q^{26} - 18 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 4 q^{39} + 6 q^{44} + 12 q^{46} - 36 q^{49} - 12 q^{51} - 8 q^{54} + 10 q^{56} + 18 q^{59} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} - 4 q^{74} - 8 q^{76} + 32 q^{79} - 22 q^{81} + 20 q^{84} + 4 q^{86} + 12 q^{89} + 10 q^{91} + 18 q^{94} - 4 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 - 10 * q^14 + 2 * q^16 + 8 * q^19 - 20 * q^21 + 4 * q^24 + 2 * q^26 - 18 * q^29 + 10 * q^31 - 6 * q^34 + 2 * q^36 + 4 * q^39 + 6 * q^44 + 12 * q^46 - 36 * q^49 - 12 * q^51 - 8 * q^54 + 10 * q^56 + 18 * q^59 - 2 * q^61 - 2 * q^64 + 12 * q^66 + 24 * q^69 - 4 * q^74 - 8 * q^76 + 32 * q^79 - 22 * q^81 + 20 * q^84 + 4 * q^86 + 12 * q^89 + 10 * q^91 + 18 * q^94 - 4 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

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