LMFDB - Embedded newform 775.2.b.b.249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(249,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, 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("775.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.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:	$6.18840615665$
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:	no (minimal twist has level 155)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.b.249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +2.00000i q^{12} +4.00000 q^{14} -1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +8.00000 q^{21} +4.00000i q^{22} +2.00000i q^{23} -6.00000 q^{24} +4.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +1.00000 q^{31} +5.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} -1.00000 q^{36} +4.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +8.00000i q^{42} -6.00000i q^{43} +4.00000 q^{44} -2.00000 q^{46} -8.00000i q^{47} -2.00000i q^{48} -9.00000 q^{49} -16.0000 q^{51} -12.0000i q^{53} -4.00000 q^{54} +12.0000 q^{56} -8.00000i q^{57} +6.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -8.00000 q^{66} -8.00000i q^{67} +8.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -4.00000i q^{73} -4.00000 q^{74} -4.00000 q^{76} -16.0000i q^{77} -11.0000 q^{81} -6.00000i q^{82} +2.00000i q^{83} +8.00000 q^{84} +6.00000 q^{86} +12.0000i q^{87} +12.0000i q^{88} -14.0000 q^{89} +2.00000i q^{92} +2.00000i q^{93} +8.00000 q^{94} -10.0000 q^{96} +18.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{11} + 8 q^{14} - 2 q^{16} - 8 q^{19} + 16 q^{21} - 12 q^{24} + 12 q^{29} + 2 q^{31} - 16 q^{34} - 2 q^{36} - 12 q^{41} + 8 q^{44} - 4 q^{46} - 18 q^{49} - 32 q^{51} - 8 q^{54} + 24 q^{56} + 8 q^{59} + 20 q^{61} - 14 q^{64} - 16 q^{66} - 8 q^{69} - 8 q^{74} - 8 q^{76} - 22 q^{81} + 16 q^{84} + 12 q^{86} - 28 q^{89} + 16 q^{94} - 20 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 4 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 - 8 * q^19 + 16 * q^21 - 12 * q^24 + 12 * q^29 + 2 * q^31 - 16 * q^34 - 2 * q^36 - 12 * q^41 + 8 * q^44 - 4 * q^46 - 18 * q^49 - 32 * q^51 - 8 * q^54 + 24 * q^56 + 8 * q^59 + 20 * q^61 - 14 * q^64 - 16 * q^66 - 8 * q^69 - 8 * q^74 - 8 * q^76 - 22 * q^81 + 16 * q^84 + 12 * q^86 - 28 * q^89 + 16 * q^94 - 20 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	2.00000i	0.577350i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	8.00000i	1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$	8.00000i	1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	4.00000i	0.852803i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	− 4.00000i	− 0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.00000i	0.883883i
$33$	8.00000i	1.39262i
$34$	−8.00000	−1.37199
$35$	0	0
$36$	−1.00000	−0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	8.00000i	1.23443i
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−2.00000	−0.294884
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−16.0000	−2.24045
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	12.0000	1.60357
$57$	− 8.00000i	− 1.05963i
$58$	6.00000i	0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	1.00000i	0.127000i
$63$	4.00000i	0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−8.00000	−0.984732
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	8.00000i	0.970143i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 3.00000i	− 0.353553i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 6.00000i	− 0.662589i
$83$	2.00000i	0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$	2.00000i	0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	6.00000	0.646997
$87$	12.0000i	1.28654i
$88$	12.0000i	1.27920i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	2.00000i	0.208514i
$93$	2.00000i	0.207390i
$94$	8.00000	0.825137
$95$	0	0
$96$	−10.0000	−1.02062
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 9.00000i	− 0.909137i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	− 16.0000i	− 1.58424i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	4.00000i	0.384900i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	4.00000i	0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	32.0000	2.93344
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	− 12.0000i	− 1.08200i
$124$	1.00000	0.0898027
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	3.00000i	0.265165i
$129$	12.0000	1.05654
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	8.00000i	0.696311i
$133$	16.0000i	1.38738i
$134$	8.00000	0.691095
$135$	0	0
$136$	−24.0000	−2.05798
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 4.00000i	− 0.340503i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	− 18.0000i	− 1.48461i
$148$	4.00000i	0.328798i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 12.0000i	− 0.973329i
$153$	− 8.00000i	− 0.646762i
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	24.0000	1.90332
$160$	0	0
$161$	8.00000	0.630488
$162$	− 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	$-0.215557\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−2.00000	−0.155230
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	24.0000i	1.85164i
$169$	13.0000	1.00000
$170$	0	0
$171$	4.00000	0.305888
$172$	− 6.00000i	− 0.457496i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	−4.00000	−0.301511
$177$	8.00000i	0.601317i
$178$	− 14.0000i	− 1.04934i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	20.0000i	1.47844i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−2.00000	−0.146647
$187$	32.0000i	2.34007i
$188$	− 8.00000i	− 0.583460i
$189$	16.0000	1.16383
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 14.0000i	− 1.01036i
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−9.00000	−0.642857
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	− 4.00000i	− 0.284268i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	10.0000i	0.703598i
$203$	− 24.0000i	− 1.68447i
$204$	−16.0000	−1.12022
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 12.0000i	− 0.824163i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−12.0000	−0.816497
$217$	− 4.00000i	− 0.271538i
$218$	− 10.0000i	− 0.677285i
$219$	8.00000	0.540590
$220$	0	0
$221$	0	0
$222$	− 8.00000i	− 0.536925i
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	18.0000	1.19734
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	− 8.00000i	− 0.529813i
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	32.0000	2.10545
$232$	18.0000i	1.18176i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	32.0000i	2.07425i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5.00000i	0.321412i
$243$	− 10.0000i	− 0.641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	12.0000	0.765092
$247$	0	0
$248$	3.00000i	0.190500i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	4.00000i	0.251976i
$253$	8.00000i	0.502956i
$254$	2.00000	0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	12.0000i	0.747087i
$259$	16.0000	0.994192
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 4.00000i	− 0.247121i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	−24.0000	−1.47710
$265$	0	0
$266$	−16.0000	−0.981023
$267$	− 28.0000i	− 1.71357i
$268$	− 8.00000i	− 0.488678i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	−4.00000	−0.240772
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	16.0000i	0.952786i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	− 5.00000i	− 0.294628i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	−36.0000	−2.11036
$292$	− 4.00000i	− 0.234082i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	−12.0000	−0.697486
$297$	16.0000i	0.928414i
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	20.0000i	1.14897i
$304$	4.00000	0.229416
$305$	0	0
$306$	8.00000	0.457330
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	− 16.0000i	− 0.911685i
$309$	−16.0000	−0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	24.0000i	1.34585i
$319$	24.0000	1.34374
$320$	0	0
$321$	−8.00000	−0.446516
$322$	8.00000i	0.445823i
$323$	− 32.0000i	− 1.78053i
$324$	−11.0000	−0.611111
$325$	0	0
$326$	16.0000	0.886158
$327$	− 20.0000i	− 1.10600i
$328$	− 18.0000i	− 0.993884i
$329$	−32.0000	−1.76422
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	2.00000i	0.109764i
$333$	− 4.00000i	− 0.219199i
$334$	14.0000	0.766046
$335$	0	0
$336$	−8.00000	−0.436436
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	13.0000i	0.707107i
$339$	36.0000	1.95525
$340$	0	0
$341$	4.00000	0.216612
$342$	4.00000i	0.216295i
$343$	8.00000i	0.431959i
$344$	18.0000	0.970495
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	12.0000i	0.643268i
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	20.0000i	1.06600i
$353$	− 16.0000i	− 0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	− 16.0000i	− 0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−14.0000	−0.741999
$357$	64.0000i	3.38724i
$358$	− 4.00000i	− 0.211407i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 18.0000i	− 0.946059i
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	−20.0000	−1.04542
$367$	− 22.0000i	− 1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$	− 22.0000i	− 1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$	− 2.00000i	− 0.104257i
$369$	6.00000	0.312348
$370$	0	0
$371$	−48.0000	−2.49204
$372$	2.00000i	0.103695i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	−32.0000	−1.65468
$375$	0	0
$376$	24.0000	1.23771
$377$	0	0
$378$	16.0000i	0.822951i
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	8.00000i	0.409316i
$383$	18.0000i	0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$	18.0000i	0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	18.0000	0.916176
$387$	6.00000i	0.304997i
$388$	18.0000i	0.913812i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 27.0000i	− 1.36371i
$393$	− 8.00000i	− 0.403547i
$394$	−8.00000	−0.403034
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	20.0000i	1.00251i
$399$	−32.0000	−1.60200
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	16.0000i	0.798007i
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	24.0000	1.19110
$407$	16.0000i	0.793091i
$408$	− 48.0000i	− 2.37635i
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	8.00000i	0.394132i
$413$	− 16.0000i	− 0.787309i
$414$	2.00000	0.0982946
$415$	0	0
$416$	0	0
$417$	0	0
$418$	− 16.0000i	− 0.782586i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	20.0000i	0.973585i
$423$	8.00000i	0.388973i
$424$	36.0000	1.74831
$425$	0	0
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	− 4.00000i	− 0.192450i
$433$	24.0000i	1.15337i	0.816968	+	0.576683i	$0.195653\pi$
$433$	24.0000i	1.15337i	−0.816968	+	0.576683i	$0.804347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 8.00000i	− 0.382692i
$438$	8.00000i	0.382255i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	26.0000	1.23114
$447$	− 12.0000i	− 0.567581i
$448$	28.0000i	1.32288i
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	24.0000	1.12390
$457$	20.0000i	0.935561i	0.883845	+	0.467780i	$0.154946\pi$
$457$	20.0000i	0.935561i	−0.883845	+	0.467780i	$0.845054\pi$
$458$	− 18.0000i	− 0.841085i
$459$	−32.0000	−1.49363
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	32.0000i	1.48877i
$463$	2.00000i	0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$	2.00000i	0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	28.0000	1.29017
$472$	12.0000i	0.552345i
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	0	0
$476$	32.0000	1.46672
$477$	12.0000i	0.549442i
$478$	− 20.0000i	− 0.914779i
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	− 10.0000i	− 0.455488i
$483$	16.0000i	0.728025i
$484$	5.00000	0.227273
$485$	0	0
$486$	10.0000	0.453609
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	30.0000i	1.35804i
$489$	32.0000	1.44709
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 12.0000i	− 0.541002i
$493$	48.0000i	2.16181i
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	28.0000	1.25095
$502$	16.0000i	0.714115i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−12.0000	−0.534522
$505$	0	0
$506$	−8.00000	−0.355643
$507$	26.0000i	1.15470i
$508$	− 2.00000i	− 0.0887357i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	− 11.0000i	− 0.486136i
$513$	− 16.0000i	− 0.706417i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	12.0000	0.528271
$517$	− 32.0000i	− 1.40736i
$518$	16.0000i	0.703000i
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	− 6.00000i	− 0.262613i
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−6.00000	−0.261612
$527$	8.00000i	0.348485i
$528$	− 8.00000i	− 0.348155i
$529$	19.0000	0.826087
$530$	0	0
$531$	−4.00000	−0.173585
$532$	16.0000i	0.693688i
$533$	0	0
$534$	28.0000	1.21168
$535$	0	0
$536$	24.0000	1.03664
$537$	− 8.00000i	− 0.345225i
$538$	− 6.00000i	− 0.258678i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	− 4.00000i	− 0.171815i
$543$	− 36.0000i	− 1.54491i
$544$	−40.0000	−1.71499
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 12.0000i	− 0.512615i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−24.0000	−1.02243
$552$	− 12.0000i	− 0.510754i
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	0	0
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	0	0
$560$	0	0
$561$	−64.0000	−2.70208
$562$	6.00000i	0.253095i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	16.0000	0.673722
$565$	0	0
$566$	−16.0000	−0.672530
$567$	44.0000i	1.84783i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	16.0000i	0.668410i
$574$	−24.0000	−1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	− 47.0000i	− 1.95494i
$579$	36.0000	1.49611
$580$	0	0
$581$	8.00000	0.331896
$582$	− 36.0000i	− 1.49225i
$583$	− 48.0000i	− 1.98796i
$584$	12.0000	0.496564
$585$	0	0
$586$	−22.0000	−0.908812
$587$	− 42.0000i	− 1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$	− 42.0000i	− 1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$	− 18.0000i	− 0.742307i
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−16.0000	−0.658152
$592$	− 4.00000i	− 0.164399i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	−6.00000	−0.245770
$597$	40.0000i	1.63709i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	− 24.0000i	− 0.978167i
$603$	8.00000i	0.325785i
$604$	0	0
$605$	0	0
$606$	−20.0000	−0.812444
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	− 20.0000i	− 0.811107i
$609$	48.0000	1.94506
$610$	0	0
$611$	0	0
$612$	− 8.00000i	− 0.323381i
$613$	− 8.00000i	− 0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$	− 8.00000i	− 0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	48.0000	1.93398
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	− 16.0000i	− 0.643614i
$619$	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
$621$	−8.00000	−0.321029
$622$	24.0000i	0.962312i
$623$	56.0000i	2.24359i
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	− 32.0000i	− 1.27796i
$628$	− 14.0000i	− 0.558661i
$629$	−32.0000	−1.27592
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	40.0000i	1.58986i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	24.0000	0.951662
$637$	0	0
$638$	24.0000i	0.950169i
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	− 8.00000i	− 0.315735i
$643$	− 26.0000i	− 1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$	− 26.0000i	− 1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	32.0000	1.25902
$647$	14.0000i	0.550397i	0.961387	+	0.275198i	$0.0887435\pi$
$647$	14.0000i	0.550397i	−0.961387	+	0.275198i	$0.911256\pi$
$648$	− 33.0000i	− 1.29636i
$649$	16.0000	0.628055
$650$	0	0
$651$	8.00000	0.313545
$652$	− 16.0000i	− 0.626608i
$653$	− 46.0000i	− 1.80012i	−0.435767	−	0.900060i	$-0.643523\pi$
$653$	− 46.0000i	− 1.80012i	0.435767	−	0.900060i	$-0.356477\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	6.00000	0.234261
$657$	4.00000i	0.156055i
$658$	− 32.0000i	− 1.24749i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	8.00000i	0.310929i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	4.00000	0.154997
$667$	12.0000i	0.464642i
$668$	− 14.0000i	− 0.541676i
$669$	52.0000	2.01044
$670$	0	0
$671$	40.0000	1.54418
$672$	40.0000i	1.54303i
$673$	28.0000i	1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$	28.0000i	1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	13.0000	0.500000
$677$	16.0000i	0.614930i	0.951559	+	0.307465i	$0.0994807\pi$
$677$	16.0000i	0.614930i	−0.951559	+	0.307465i	$0.900519\pi$
$678$	36.0000i	1.38257i
$679$	72.0000	2.76311
$680$	0	0
$681$	16.0000	0.613121
$682$	4.00000i	0.153168i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 36.0000i	− 1.37349i
$688$	6.00000i	0.228748i
$689$	0	0
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	6.00000i	0.228086i
$693$	16.0000i	0.607790i
$694$	6.00000	0.227757
$695$	0	0
$696$	−36.0000	−1.36458
$697$	− 48.0000i	− 1.81813i
$698$	− 22.0000i	− 0.832712i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 16.0000i	− 0.603451i
$704$	−28.0000	−1.05529
$705$	0	0
$706$	16.0000	0.602168
$707$	− 40.0000i	− 1.50435i
$708$	8.00000i	0.300658i
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	0	0
$712$	− 42.0000i	− 1.57402i
$713$	2.00000i	0.0749006i
$714$	−64.0000	−2.39514
$715$	0	0
$716$	−4.00000	−0.149487
$717$	− 40.0000i	− 1.49383i
$718$	8.00000i	0.298557i
$719$	52.0000	1.93927	0.969636	−	0.244551i	$-0.0786406\pi$
$719$	52.0000	1.93927	0.969636	+	0.244551i	$0.0786406\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	− 3.00000i	− 0.111648i
$723$	− 20.0000i	− 0.743808i
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−10.0000	−0.371135
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	20.0000i	0.739221i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−10.0000	−0.368605
$737$	− 32.0000i	− 1.17874i
$738$	6.00000i	0.220863i
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	− 48.0000i	− 1.76214i
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−14.0000	−0.512576
$747$	− 2.00000i	− 0.0731762i
$748$	32.0000i	1.17004i
$749$	16.0000	0.584627
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	8.00000i	0.291730i
$753$	32.0000i	1.16614i
$754$	0	0
$755$	0	0
$756$	16.0000	0.581914
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	− 28.0000i	− 1.01701i
$759$	−16.0000	−0.580763
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	4.00000i	0.144905i
$763$	40.0000i	1.44810i
$764$	8.00000	0.289430
$765$	0	0
$766$	−18.0000	−0.650366
$767$	0	0
$768$	− 34.0000i	− 1.22687i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	− 18.0000i	− 0.647834i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	−54.0000	−1.93849
$777$	32.0000i	1.14799i
$778$	10.0000i	0.358517i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	− 16.0000i	− 0.572159i
$783$	24.0000i	0.857690i
$784$	9.00000	0.321429
$785$	0	0
$786$	8.00000	0.285351
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	8.00000i	0.284988i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−72.0000	−2.56003
$792$	− 12.0000i	− 0.426401i
$793$	0	0
$794$	26.0000	0.922705
$795$	0	0
$796$	20.0000	0.708881
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	− 32.0000i	− 1.13279i
$799$	64.0000	2.26416
$800$	0	0
$801$	14.0000	0.494666
$802$	2.00000i	0.0706225i
$803$	− 16.0000i	− 0.564628i
$804$	16.0000	0.564276
$805$	0	0
$806$	0	0
$807$	− 12.0000i	− 0.422420i
$808$	30.0000i	1.05540i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 8.00000i	− 0.280572i
$814$	−16.0000	−0.560800
$815$	0	0
$816$	16.0000	0.560112
$817$	24.0000i	0.839654i
$818$	30.0000i	1.04893i
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	24.0000i	0.837096i
$823$	− 38.0000i	− 1.32460i	−0.749240	−	0.662298i	$-0.769581\pi$
$823$	− 38.0000i	− 1.32460i	0.749240	−	0.662298i	$-0.230419\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	16.0000	0.556711
$827$	42.0000i	1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$	42.0000i	1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$	− 2.00000i	− 0.0695048i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	− 72.0000i	− 2.49465i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	4.00000i	0.138260i
$838$	− 4.00000i	− 0.138178i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 18.0000i	− 0.620321i
$843$	12.0000i	0.413302i
$844$	20.0000	0.688428
$845$	0	0
$846$	−8.00000	−0.275046
$847$	− 20.0000i	− 0.687208i
$848$	12.0000i	0.412082i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	34.0000i	1.16414i	0.813139	+	0.582069i	$0.197757\pi$
$853$	34.0000i	1.16414i	−0.813139	+	0.582069i	$0.802243\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−48.0000	−1.63584
$862$	40.0000i	1.36241i
$863$	− 38.0000i	− 1.29354i	−0.762687	−	0.646768i	$-0.776120\pi$
$863$	− 38.0000i	− 1.29354i	0.762687	−	0.646768i	$-0.223880\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	−24.0000	−0.815553
$867$	− 94.0000i	− 3.19241i
$868$	− 4.00000i	− 0.135769i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 30.0000i	− 1.01593i
$873$	− 18.0000i	− 0.609208i
$874$	8.00000	0.270604
$875$	0	0
$876$	8.00000	0.270295
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	−44.0000	−1.48408
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	9.00000i	0.303046i
$883$	− 26.0000i	− 0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$	− 26.0000i	− 0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$	0	0
$885$	0	0
$886$	4.00000	0.134383
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 24.0000i	− 0.805387i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−44.0000	−1.47406
$892$	− 26.0000i	− 0.870544i
$893$	32.0000i	1.07084i
$894$	12.0000	0.401340
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	22.0000i	0.734150i
$899$	6.00000	0.200111
$900$	0	0
$901$	96.0000	3.19822
$902$	− 24.0000i	− 0.799113i
$903$	− 48.0000i	− 1.59734i
$904$	54.0000	1.79601
$905$	0	0
$906$	0	0
$907$	− 36.0000i	− 1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$	− 36.0000i	− 1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$	− 8.00000i	− 0.265489i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	8.00000i	0.264906i
$913$	8.00000i	0.264761i
$914$	−20.0000	−0.661541
$915$	0	0
$916$	−18.0000	−0.594737
$917$	16.0000i	0.528367i
$918$	− 32.0000i	− 1.05616i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	56.0000	1.84526
$922$	− 18.0000i	− 0.592798i
$923$	0	0
$924$	32.0000	1.05272
$925$	0	0
$926$	−2.00000	−0.0657241
$927$	− 8.00000i	− 0.262754i
$928$	30.0000i	0.984798i
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	6.00000i	0.196537i
$933$	48.0000i	1.57145i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	− 32.0000i	− 1.04484i
$939$	16.0000	0.522140
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	28.0000i	0.912289i
$943$	− 12.0000i	− 0.390774i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	24.0000	0.780307
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.0000	−0.389127
$952$	96.0000i	3.11138i
$953$	36.0000i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$	36.0000i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−20.0000	−0.646846
$957$	48.0000i	1.55162i
$958$	− 16.0000i	− 0.516937i
$959$	−48.0000	−1.55000
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−16.0000	−0.514792
$967$	54.0000i	1.73652i	0.496107	+	0.868261i	$0.334762\pi$
$967$	54.0000i	1.73652i	−0.496107	+	0.868261i	$0.665238\pi$
$968$	15.0000i	0.482118i
$969$	64.0000	2.05598
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	− 10.0000i	− 0.320750i
$973$	0	0
$974$	−22.0000	−0.704925
$975$	0	0
$976$	−10.0000	−0.320092
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	32.0000i	1.02325i
$979$	−56.0000	−1.78977
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	− 10.0000i	− 0.318950i	−0.987202	−	0.159475i	$-0.949020\pi$
$983$	− 10.0000i	− 0.318950i	0.987202	−	0.159475i	$-0.0509802\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	−48.0000	−1.52863
$987$	− 64.0000i	− 2.03714i
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	5.00000i	0.158750i
$993$	16.0000i	0.507745i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	− 30.0000i	− 0.950110i	−0.879956	−	0.475055i	$-0.842428\pi$
$997$	− 30.0000i	− 0.950110i	0.879956	−	0.475055i	$-0.157572\pi$
$998$	− 28.0000i	− 0.886325i
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.b.249.2		2
5.2	odd	4		155.2.a.b.1.1	✓	1
5.3	odd	4		775.2.a.b.1.1		1
5.4	even	2	inner	775.2.b.b.249.1		2
15.2	even	4		1395.2.a.d.1.1		1
15.8	even	4		6975.2.a.d.1.1		1
20.7	even	4		2480.2.a.b.1.1		1
35.27	even	4		7595.2.a.c.1.1		1
40.27	even	4		9920.2.a.bd.1.1		1
40.37	odd	4		9920.2.a.g.1.1		1
155.92	even	4		4805.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.b.1.1	✓	1	5.2	odd	4
775.2.a.b.1.1		1	5.3	odd	4
775.2.b.b.249.1		2	5.4	even	2	inner
775.2.b.b.249.2		2	1.1	even	1	trivial
1395.2.a.d.1.1		1	15.2	even	4
2480.2.a.b.1.1		1	20.7	even	4
4805.2.a.d.1.1		1	155.92	even	4
6975.2.a.d.1.1		1	15.8	even	4
7595.2.a.c.1.1		1	35.27	even	4
9920.2.a.g.1.1		1	40.37	odd	4
9920.2.a.bd.1.1		1	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +2.00000i q^{12} +4.00000 q^{14} -1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +8.00000 q^{21} +4.00000i q^{22} +2.00000i q^{23} -6.00000 q^{24} +4.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +1.00000 q^{31} +5.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} -1.00000 q^{36} +4.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +8.00000i q^{42} -6.00000i q^{43} +4.00000 q^{44} -2.00000 q^{46} -8.00000i q^{47} -2.00000i q^{48} -9.00000 q^{49} -16.0000 q^{51} -12.0000i q^{53} -4.00000 q^{54} +12.0000 q^{56} -8.00000i q^{57} +6.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -8.00000 q^{66} -8.00000i q^{67} +8.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -4.00000i q^{73} -4.00000 q^{74} -4.00000 q^{76} -16.0000i q^{77} -11.0000 q^{81} -6.00000i q^{82} +2.00000i q^{83} +8.00000 q^{84} +6.00000 q^{86} +12.0000i q^{87} +12.0000i q^{88} -14.0000 q^{89} +2.00000i q^{92} +2.00000i q^{93} +8.00000 q^{94} -10.0000 q^{96} +18.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 4 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} - 2 q^{16} - 8 q^{19} + 16 q^{21} - 12 q^{24} + 12 q^{29} + 2 q^{31} - 16 q^{34} - 2 q^{36} - 12 q^{41} + 8 q^{44} - 4 q^{46} - 18 q^{49} - 32 q^{51} - 8 q^{54} + 24 q^{56} + 8 q^{59} + 20 q^{61} - 14 q^{64} - 16 q^{66} - 8 q^{69} - 8 q^{74} - 8 q^{76} - 22 q^{81} + 16 q^{84} + 12 q^{86} - 28 q^{89} + 16 q^{94} - 20 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 4 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 - 8 * q^19 + 16 * q^21 - 12 * q^24 + 12 * q^29 + 2 * q^31 - 16 * q^34 - 2 * q^36 - 12 * q^41 + 8 * q^44 - 4 * q^46 - 18 * q^49 - 32 * q^51 - 8 * q^54 + 24 * q^56 + 8 * q^59 + 20 * q^61 - 14 * q^64 - 16 * q^66 - 8 * q^69 - 8 * q^74 - 8 * q^76 - 22 * q^81 + 16 * q^84 + 12 * q^86 - 28 * q^89 + 16 * q^94 - 20 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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