LMFDB - Embedded newform 775.2.b.d.249.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.d.249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{2} -3.23607i q^{3} -0.618034 q^{4} -5.23607 q^{6} -0.236068i q^{7} -2.23607i q^{8} -7.47214 q^{9} +O(q^{10})$	\(q-1.61803i q^{2} -3.23607i q^{3} -0.618034 q^{4} -5.23607 q^{6} -0.236068i q^{7} -2.23607i q^{8} -7.47214 q^{9} +2.00000 q^{11} +2.00000i q^{12} -3.23607i q^{13} -0.381966 q^{14} -4.85410 q^{16} -0.763932i q^{17} +12.0902i q^{18} +2.23607 q^{19} -0.763932 q^{21} -3.23607i q^{22} +5.70820i q^{23} -7.23607 q^{24} -5.23607 q^{26} +14.4721i q^{27} +0.145898i q^{28} -2.76393 q^{29} +1.00000 q^{31} +3.38197i q^{32} -6.47214i q^{33} -1.23607 q^{34} +4.61803 q^{36} +2.00000i q^{37} -3.61803i q^{38} -10.4721 q^{39} +7.00000 q^{41} +1.23607i q^{42} +1.23607i q^{43} -1.23607 q^{44} +9.23607 q^{46} -2.47214i q^{47} +15.7082i q^{48} +6.94427 q^{49} -2.47214 q^{51} +2.00000i q^{52} -10.4721i q^{53} +23.4164 q^{54} -0.527864 q^{56} -7.23607i q^{57} +4.47214i q^{58} -2.23607 q^{59} +8.18034 q^{61} -1.61803i q^{62} +1.76393i q^{63} -4.23607 q^{64} -10.4721 q^{66} -8.00000i q^{67} +0.472136i q^{68} +18.4721 q^{69} -9.18034 q^{71} +16.7082i q^{72} +8.47214i q^{73} +3.23607 q^{74} -1.38197 q^{76} -0.472136i q^{77} +16.9443i q^{78} +11.7082 q^{79} +24.4164 q^{81} -11.3262i q^{82} -14.9443i q^{83} +0.472136 q^{84} +2.00000 q^{86} +8.94427i q^{87} -4.47214i q^{88} -11.7082 q^{89} -0.763932 q^{91} -3.52786i q^{92} -3.23607i q^{93} -4.00000 q^{94} +10.9443 q^{96} +15.9443i q^{97} -11.2361i q^{98} -14.9443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9	$ 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} - 12 q^{21} - 20 q^{24} - 12 q^{26} - 20 q^{29} + 4 q^{31} + 4 q^{34} + 14 q^{36} - 24 q^{39} + 28 q^{41} + 4 q^{44} + 28 q^{46} - 8 q^{49} + 8 q^{51} + 40 q^{54} - 20 q^{56} - 12 q^{61} - 8 q^{64} - 24 q^{66} + 56 q^{69} + 8 q^{71} + 4 q^{74} - 10 q^{76} + 20 q^{79} + 44 q^{81} - 16 q^{84} + 8 q^{86} - 20 q^{89} - 12 q^{91} - 16 q^{94} + 8 q^{96} - 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 - 12 * q^21 - 20 * q^24 - 12 * q^26 - 20 * q^29 + 4 * q^31 + 4 * q^34 + 14 * q^36 - 24 * q^39 + 28 * q^41 + 4 * q^44 + 28 * q^46 - 8 * q^49 + 8 * q^51 + 40 * q^54 - 20 * q^56 - 12 * q^61 - 8 * q^64 - 24 * q^66 + 56 * q^69 + 8 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 44 * q^81 - 16 * q^84 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 16 * q^94 + 8 * q^96 - 24 * 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 $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−5.23607	−2.13762
$7$	− 0.236068i	− 0.0892253i	−0.999004	−	0.0446127i	$-0.985795\pi$
$7$	− 0.236068i	− 0.0892253i	0.999004	−	0.0446127i	$-0.0142054\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−7.47214	−2.49071
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000i	0.577350i
$13$	− 3.23607i	− 0.897524i	−0.893651	−	0.448762i	$-0.851865\pi$
$13$	− 3.23607i	− 0.897524i	0.893651	−	0.448762i	$-0.148135\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	−4.85410	−1.21353
$17$	− 0.763932i	− 0.185281i	−0.995700	−	0.0926404i	$-0.970469\pi$
$17$	− 0.763932i	− 0.185281i	0.995700	−	0.0926404i	$-0.0295307\pi$
$18$	12.0902i	2.84968i
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	−0.763932	−0.166704
$22$	− 3.23607i	− 0.689932i
$23$	5.70820i	1.19024i	0.803636	+	0.595121i	$0.202896\pi$
$23$	5.70820i	1.19024i	−0.803636	+	0.595121i	$0.797104\pi$
$24$	−7.23607	−1.47706
$25$	0	0
$26$	−5.23607	−1.02688
$27$	14.4721i	2.78516i
$28$	0.145898i	0.0275721i
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.38197i	0.597853i
$33$	− 6.47214i	− 1.12665i
$34$	−1.23607	−0.211984
$35$	0	0
$36$	4.61803	0.769672
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 3.61803i	− 0.586923i
$39$	−10.4721	−1.67688
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	1.23607i	0.190729i
$43$	1.23607i	0.188499i	0.995549	+	0.0942493i	$0.0300451\pi$
$43$	1.23607i	0.188499i	−0.995549	+	0.0942493i	$0.969955\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	9.23607	1.36178
$47$	− 2.47214i	− 0.360598i	−0.983612	−	0.180299i	$-0.942293\pi$
$47$	− 2.47214i	− 0.360598i	0.983612	−	0.180299i	$-0.0577065\pi$
$48$	15.7082i	2.26728i
$49$	6.94427	0.992039
$50$	0	0
$51$	−2.47214	−0.346168
$52$	2.00000i	0.277350i
$53$	− 10.4721i	− 1.43846i	−0.694773	−	0.719229i	$-0.744495\pi$
$53$	− 10.4721i	− 1.43846i	0.694773	−	0.719229i	$-0.255505\pi$
$54$	23.4164	3.18657
$55$	0	0
$56$	−0.527864	−0.0705388
$57$	− 7.23607i	− 0.958441i
$58$	4.47214i	0.587220i
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	8.18034	1.04739	0.523693	−	0.851907i	$-0.324554\pi$
$61$	8.18034	1.04739	0.523693	+	0.851907i	$0.324554\pi$
$62$	− 1.61803i	− 0.205491i
$63$	1.76393i	0.222235i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	−10.4721	−1.28903
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0.472136i	0.0572549i
$69$	18.4721	2.22378
$70$	0	0
$71$	−9.18034	−1.08951	−0.544753	−	0.838597i	$-0.683377\pi$
$71$	−9.18034	−1.08951	−0.544753	+	0.838597i	$0.683377\pi$
$72$	16.7082i	1.96908i
$73$	8.47214i	0.991589i	0.868440	+	0.495794i	$0.165123\pi$
$73$	8.47214i	0.991589i	−0.868440	+	0.495794i	$0.834877\pi$
$74$	3.23607	0.376185
$75$	0	0
$76$	−1.38197	−0.158522
$77$	− 0.472136i	− 0.0538049i
$78$	16.9443i	1.91856i
$79$	11.7082	1.31728	0.658638	−	0.752460i	$-0.271133\pi$
$79$	11.7082	1.31728	0.658638	+	0.752460i	$0.271133\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	− 11.3262i	− 1.25077i
$83$	− 14.9443i	− 1.64035i	−0.572115	−	0.820173i	$-0.693877\pi$
$83$	− 14.9443i	− 1.64035i	0.572115	−	0.820173i	$-0.306123\pi$
$84$	0.472136	0.0515143
$85$	0	0
$86$	2.00000	0.215666
$87$	8.94427i	0.958927i
$88$	− 4.47214i	− 0.476731i
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	− 3.52786i	− 0.367805i
$93$	− 3.23607i	− 0.335565i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	10.9443	1.11700
$97$	15.9443i	1.61890i	0.587192	+	0.809448i	$0.300233\pi$
$97$	15.9443i	1.61890i	−0.587192	+	0.809448i	$0.699767\pi$
$98$	− 11.2361i	− 1.13501i
$99$	−14.9443	−1.50196
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	4.00000i	0.396059i
$103$	6.23607i	0.614458i	0.951636	+	0.307229i	$0.0994018\pi$
$103$	6.23607i	0.614458i	−0.951636	+	0.307229i	$0.900598\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	−16.9443	−1.64577
$107$	− 5.76393i	− 0.557220i	−0.960404	−	0.278610i	$-0.910126\pi$
$107$	− 5.76393i	− 0.557220i	0.960404	−	0.278610i	$-0.0898737\pi$
$108$	− 8.94427i	− 0.860663i
$109$	13.9443	1.33562	0.667810	−	0.744332i	$-0.267232\pi$
$109$	13.9443	1.33562	0.667810	+	0.744332i	$0.267232\pi$
$110$	0	0
$111$	6.47214	0.614308
$112$	1.14590i	0.108277i
$113$	3.47214i	0.326631i	0.986574	+	0.163316i	$0.0522189\pi$
$113$	3.47214i	0.326631i	−0.986574	+	0.163316i	$0.947781\pi$
$114$	−11.7082	−1.09657
$115$	0	0
$116$	1.70820	0.158603
$117$	24.1803i	2.23547i
$118$	3.61803i	0.333067i
$119$	−0.180340	−0.0165317
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 13.2361i	− 1.19834i
$123$	− 22.6525i	− 2.04250i
$124$	−0.618034	−0.0555011
$125$	0	0
$126$	2.85410	0.254264
$127$	− 12.4721i	− 1.10672i	−0.832941	−	0.553362i	$-0.813345\pi$
$127$	− 12.4721i	− 1.10672i	0.832941	−	0.553362i	$-0.186655\pi$
$128$	13.6180i	1.20368i
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000i	0.348155i
$133$	− 0.527864i	− 0.0457716i
$134$	−12.9443	−1.11821
$135$	0	0
$136$	−1.70820	−0.146477
$137$	− 6.29180i	− 0.537544i	−0.963204	−	0.268772i	$-0.913382\pi$
$137$	− 6.29180i	− 0.537544i	0.963204	−	0.268772i	$-0.0866179\pi$
$138$	− 29.8885i	− 2.54428i
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	14.8541i	1.24653i
$143$	− 6.47214i	− 0.541227i
$144$	36.2705	3.02254
$145$	0	0
$146$	13.7082	1.13450
$147$	− 22.4721i	− 1.85347i
$148$	− 1.23607i	− 0.101604i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−14.1803	−1.15398	−0.576990	−	0.816751i	$-0.695773\pi$
$151$	−14.1803	−1.15398	−0.576990	+	0.816751i	$0.695773\pi$
$152$	− 5.00000i	− 0.405554i
$153$	5.70820i	0.461481i
$154$	−0.763932	−0.0615594
$155$	0	0
$156$	6.47214	0.518186
$157$	− 20.8885i	− 1.66709i	−0.552454	−	0.833544i	$-0.686308\pi$
$157$	− 20.8885i	− 1.66709i	0.552454	−	0.833544i	$-0.313692\pi$
$158$	− 18.9443i	− 1.50713i
$159$	−33.8885	−2.68754
$160$	0	0
$161$	1.34752	0.106200
$162$	− 39.5066i	− 3.10393i
$163$	10.7082i	0.838731i	0.907817	+	0.419366i	$0.137747\pi$
$163$	10.7082i	0.838731i	−0.907817	+	0.419366i	$0.862253\pi$
$164$	−4.32624	−0.337822
$165$	0	0
$166$	−24.1803	−1.87676
$167$	6.47214i	0.500829i	0.968139	+	0.250414i	$0.0805669\pi$
$167$	6.47214i	0.500829i	−0.968139	+	0.250414i	$0.919433\pi$
$168$	1.70820i	0.131791i
$169$	2.52786	0.194451
$170$	0	0
$171$	−16.7082	−1.27771
$172$	− 0.763932i	− 0.0582493i
$173$	2.94427i	0.223849i	0.993717	+	0.111924i	$0.0357015\pi$
$173$	2.94427i	0.223849i	−0.993717	+	0.111924i	$0.964299\pi$
$174$	14.4721	1.09713
$175$	0	0
$176$	−9.70820	−0.731783
$177$	7.23607i	0.543896i
$178$	18.9443i	1.41993i
$179$	−1.70820	−0.127677	−0.0638386	−	0.997960i	$-0.520334\pi$
$179$	−1.70820	−0.127677	−0.0638386	+	0.997960i	$0.520334\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	1.23607i	0.0916235i
$183$	− 26.4721i	− 1.95688i
$184$	12.7639	0.940970
$185$	0	0
$186$	−5.23607	−0.383927
$187$	− 1.52786i	− 0.111728i
$188$	1.52786i	0.111431i
$189$	3.41641	0.248507
$190$	0	0
$191$	−19.1803	−1.38784	−0.693920	−	0.720052i	$-0.744118\pi$
$191$	−19.1803	−1.38784	−0.693920	+	0.720052i	$0.744118\pi$
$192$	13.7082i	0.989304i
$193$	3.47214i	0.249930i	0.992161	+	0.124965i	$0.0398818\pi$
$193$	3.47214i	0.249930i	−0.992161	+	0.124965i	$0.960118\pi$
$194$	25.7984	1.85222
$195$	0	0
$196$	−4.29180	−0.306557
$197$	− 11.4164i	− 0.813385i	−0.913565	−	0.406693i	$-0.866682\pi$
$197$	− 11.4164i	− 0.813385i	0.913565	−	0.406693i	$-0.133318\pi$
$198$	24.1803i	1.71842i
$199$	18.9443	1.34292	0.671462	−	0.741039i	$-0.265667\pi$
$199$	18.9443	1.34292	0.671462	+	0.741039i	$0.265667\pi$
$200$	0	0
$201$	−25.8885	−1.82604
$202$	4.85410i	0.341533i
$203$	0.652476i	0.0457948i
$204$	1.52786	0.106972
$205$	0	0
$206$	10.0902	0.703015
$207$	− 42.6525i	− 2.96455i
$208$	15.7082i	1.08917i
$209$	4.47214	0.309344
$210$	0	0
$211$	23.1803	1.59580	0.797900	−	0.602790i	$-0.205944\pi$
$211$	23.1803	1.59580	0.797900	+	0.602790i	$0.205944\pi$
$212$	6.47214i	0.444508i
$213$	29.7082i	2.03557i
$214$	−9.32624	−0.637528
$215$	0	0
$216$	32.3607	2.20187
$217$	− 0.236068i	− 0.0160253i
$218$	− 22.5623i	− 1.52811i
$219$	27.4164	1.85263
$220$	0	0
$221$	−2.47214	−0.166294
$222$	− 10.4721i	− 0.702844i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	0.798374	0.0533436
$225$	0	0
$226$	5.61803	0.373706
$227$	6.47214i	0.429571i	0.976661	+	0.214785i	$0.0689052\pi$
$227$	6.47214i	0.429571i	−0.976661	+	0.214785i	$0.931095\pi$
$228$	4.47214i	0.296174i
$229$	13.4164	0.886581	0.443291	−	0.896378i	$-0.353811\pi$
$229$	13.4164	0.886581	0.443291	+	0.896378i	$0.353811\pi$
$230$	0	0
$231$	−1.52786	−0.100526
$232$	6.18034i	0.405759i
$233$	17.9443i	1.17557i	0.809018	+	0.587784i	$0.200000\pi$
$233$	17.9443i	1.17557i	−0.809018	+	0.587784i	$0.800000\pi$
$234$	39.1246	2.55766
$235$	0	0
$236$	1.38197	0.0899583
$237$	− 37.8885i	− 2.46113i
$238$	0.291796i	0.0189143i
$239$	11.7082	0.757341	0.378670	−	0.925532i	$-0.376381\pi$
$239$	11.7082	0.757341	0.378670	+	0.925532i	$0.376381\pi$
$240$	0	0
$241$	14.3607	0.925053	0.462526	−	0.886606i	$-0.346943\pi$
$241$	14.3607	0.925053	0.462526	+	0.886606i	$0.346943\pi$
$242$	11.3262i	0.728078i
$243$	− 35.5967i	− 2.28353i
$244$	−5.05573	−0.323660
$245$	0	0
$246$	−36.6525	−2.33688
$247$	− 7.23607i	− 0.460420i
$248$	− 2.23607i	− 0.141990i
$249$	−48.3607	−3.06473
$250$	0	0
$251$	−1.81966	−0.114856	−0.0574280	−	0.998350i	$-0.518290\pi$
$251$	−1.81966	−0.114856	−0.0574280	+	0.998350i	$0.518290\pi$
$252$	− 1.09017i	− 0.0686743i
$253$	11.4164i	0.717743i
$254$	−20.1803	−1.26623
$255$	0	0
$256$	13.5623	0.847644
$257$	− 1.94427i	− 0.121280i	−0.998160	−	0.0606402i	$-0.980686\pi$
$257$	− 1.94427i	− 0.121280i	0.998160	−	0.0606402i	$-0.0193142\pi$
$258$	− 6.47214i	− 0.402938i
$259$	0.472136	0.0293371
$260$	0	0
$261$	20.6525	1.27836
$262$	− 19.4164i	− 1.19955i
$263$	− 23.2361i	− 1.43280i	−0.697691	−	0.716399i	$-0.745789\pi$
$263$	− 23.2361i	− 1.43280i	0.697691	−	0.716399i	$-0.254211\pi$
$264$	−14.4721	−0.890698
$265$	0	0
$266$	−0.854102	−0.0523684
$267$	37.8885i	2.31874i
$268$	4.94427i	0.302019i
$269$	11.0557	0.674080	0.337040	−	0.941490i	$-0.390574\pi$
$269$	11.0557	0.674080	0.337040	+	0.941490i	$0.390574\pi$
$270$	0	0
$271$	−14.1803	−0.861394	−0.430697	−	0.902497i	$-0.641732\pi$
$271$	−14.1803	−0.861394	−0.430697	+	0.902497i	$0.641732\pi$
$272$	3.70820i	0.224843i
$273$	2.47214i	0.149620i
$274$	−10.1803	−0.615017
$275$	0	0
$276$	−11.4164	−0.687187
$277$	12.6525i	0.760214i	0.924943	+	0.380107i	$0.124113\pi$
$277$	12.6525i	0.760214i	−0.924943	+	0.380107i	$0.875887\pi$
$278$	21.7082i	1.30197i
$279$	−7.47214	−0.447345
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	12.9443i	0.770820i
$283$	− 13.8885i	− 0.825588i	−0.910824	−	0.412794i	$-0.864553\pi$
$283$	− 13.8885i	− 0.825588i	0.910824	−	0.412794i	$-0.135447\pi$
$284$	5.67376	0.336676
$285$	0	0
$286$	−10.4721	−0.619230
$287$	− 1.65248i	− 0.0975426i
$288$	− 25.2705i	− 1.48908i
$289$	16.4164	0.965671
$290$	0	0
$291$	51.5967	3.02465
$292$	− 5.23607i	− 0.306418i
$293$	− 0.472136i	− 0.0275825i	−0.999905	−	0.0137912i	$-0.995610\pi$
$293$	− 0.472136i	− 0.0275825i	0.999905	−	0.0137912i	$-0.00439003\pi$
$294$	−36.3607	−2.12060
$295$	0	0
$296$	4.47214	0.259938
$297$	28.9443i	1.67952i
$298$	16.1803i	0.937302i
$299$	18.4721	1.06827
$300$	0	0
$301$	0.291796	0.0168188
$302$	22.9443i	1.32029i
$303$	9.70820i	0.557722i
$304$	−10.8541	−0.622525
$305$	0	0
$306$	9.23607	0.527991
$307$	28.7082i	1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$307$	28.7082i	1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$308$	0.291796i	0.0166266i
$309$	20.1803	1.14802
$310$	0	0
$311$	−29.1803	−1.65467	−0.827333	−	0.561712i	$-0.810143\pi$
$311$	−29.1803	−1.65467	−0.827333	+	0.561712i	$0.810143\pi$
$312$	23.4164i	1.32569i
$313$	16.7639i	0.947553i	0.880645	+	0.473777i	$0.157110\pi$
$313$	16.7639i	0.947553i	−0.880645	+	0.473777i	$0.842890\pi$
$314$	−33.7984	−1.90735
$315$	0	0
$316$	−7.23607	−0.407061
$317$	− 4.05573i	− 0.227792i	−0.993493	−	0.113896i	$-0.963667\pi$
$317$	− 4.05573i	− 0.227792i	0.993493	−	0.113896i	$-0.0363331\pi$
$318$	54.8328i	3.07487i
$319$	−5.52786	−0.309501
$320$	0	0
$321$	−18.6525	−1.04108
$322$	− 2.18034i	− 0.121506i
$323$	− 1.70820i	− 0.0950470i
$324$	−15.0902	−0.838343
$325$	0	0
$326$	17.3262	0.959612
$327$	− 45.1246i	− 2.49540i
$328$	− 15.6525i	− 0.864263i
$329$	−0.583592	−0.0321745
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	9.23607i	0.506895i
$333$	− 14.9443i	− 0.818941i
$334$	10.4721	0.573010
$335$	0	0
$336$	3.70820	0.202299
$337$	14.7639i	0.804243i	0.915586	+	0.402121i	$0.131727\pi$
$337$	14.7639i	0.804243i	−0.915586	+	0.402121i	$0.868273\pi$
$338$	− 4.09017i	− 0.222476i
$339$	11.2361	0.610259
$340$	0	0
$341$	2.00000	0.108306
$342$	27.0344i	1.46186i
$343$	− 3.29180i	− 0.177740i
$344$	2.76393	0.149021
$345$	0	0
$346$	4.76393	0.256111
$347$	− 24.1803i	− 1.29807i	−0.760759	−	0.649034i	$-0.775173\pi$
$347$	− 24.1803i	− 1.29807i	0.760759	−	0.649034i	$-0.224827\pi$
$348$	− 5.52786i	− 0.296325i
$349$	7.88854	0.422264	0.211132	−	0.977458i	$-0.432285\pi$
$349$	7.88854	0.422264	0.211132	+	0.977458i	$0.432285\pi$
$350$	0	0
$351$	46.8328	2.49975
$352$	6.76393i	0.360519i
$353$	7.41641i	0.394736i	0.980330	+	0.197368i	$0.0632393\pi$
$353$	7.41641i	0.394736i	−0.980330	+	0.197368i	$0.936761\pi$
$354$	11.7082	0.622284
$355$	0	0
$356$	7.23607	0.383511
$357$	0.583592i	0.0308870i
$358$	2.76393i	0.146078i
$359$	−22.2361	−1.17357	−0.586787	−	0.809741i	$-0.699608\pi$
$359$	−22.2361	−1.17357	−0.586787	+	0.809741i	$0.699608\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	6.76393i	0.355504i
$363$	22.6525i	1.18895i
$364$	0.472136	0.0247466
$365$	0	0
$366$	−42.8328	−2.23891
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 27.7082i	− 1.44439i
$369$	−52.3050	−2.72289
$370$	0	0
$371$	−2.47214	−0.128347
$372$	2.00000i	0.103695i
$373$	19.0000i	0.983783i	0.870657	+	0.491891i	$0.163694\pi$
$373$	19.0000i	0.983783i	−0.870657	+	0.491891i	$0.836306\pi$
$374$	−2.47214	−0.127831
$375$	0	0
$376$	−5.52786	−0.285078
$377$	8.94427i	0.460653i
$378$	− 5.52786i	− 0.284323i
$379$	2.11146	0.108458	0.0542291	−	0.998529i	$-0.482730\pi$
$379$	2.11146	0.108458	0.0542291	+	0.998529i	$0.482730\pi$
$380$	0	0
$381$	−40.3607	−2.06774
$382$	31.0344i	1.58786i
$383$	− 23.8885i	− 1.22065i	−0.792152	−	0.610324i	$-0.791039\pi$
$383$	− 23.8885i	− 1.22065i	0.792152	−	0.610324i	$-0.208961\pi$
$384$	44.0689	2.24888
$385$	0	0
$386$	5.61803	0.285950
$387$	− 9.23607i	− 0.469496i
$388$	− 9.85410i	− 0.500266i
$389$	17.8885	0.906985	0.453493	−	0.891260i	$-0.350178\pi$
$389$	17.8885	0.906985	0.453493	+	0.891260i	$0.350178\pi$
$390$	0	0
$391$	4.36068	0.220529
$392$	− 15.5279i	− 0.784276i
$393$	− 38.8328i	− 1.95886i
$394$	−18.4721	−0.930613
$395$	0	0
$396$	9.23607	0.464130
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	− 30.6525i	− 1.53647i
$399$	−1.70820	−0.0855172
$400$	0	0
$401$	38.1803	1.90664	0.953318	−	0.301969i	$-0.0976441\pi$
$401$	38.1803	1.90664	0.953318	+	0.301969i	$0.0976441\pi$
$402$	41.8885i	2.08921i
$403$	− 3.23607i	− 0.161200i
$404$	1.85410	0.0922450
$405$	0	0
$406$	1.05573	0.0523949
$407$	4.00000i	0.198273i
$408$	5.52786i	0.273670i
$409$	3.81966	0.188870	0.0944350	−	0.995531i	$-0.469896\pi$
$409$	3.81966	0.188870	0.0944350	+	0.995531i	$0.469896\pi$
$410$	0	0
$411$	−20.3607	−1.00432
$412$	− 3.85410i	− 0.189878i
$413$	0.527864i	0.0259745i
$414$	−69.0132	−3.39181
$415$	0	0
$416$	10.9443	0.536587
$417$	43.4164i	2.12611i
$418$	− 7.23607i	− 0.353928i
$419$	10.1246	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$419$	10.1246	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$420$	0	0
$421$	29.3607	1.43095	0.715476	−	0.698637i	$-0.246210\pi$
$421$	29.3607	1.43095	0.715476	+	0.698637i	$0.246210\pi$
$422$	− 37.5066i	− 1.82579i
$423$	18.4721i	0.898146i
$424$	−23.4164	−1.13720
$425$	0	0
$426$	48.0689	2.32895
$427$	− 1.93112i	− 0.0934533i
$428$	3.56231i	0.172191i
$429$	−20.9443	−1.01120
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 70.2492i	− 3.37987i
$433$	10.1803i	0.489236i	0.969620	+	0.244618i	$0.0786625\pi$
$433$	10.1803i	0.489236i	−0.969620	+	0.244618i	$0.921337\pi$
$434$	−0.381966	−0.0183350
$435$	0	0
$436$	−8.61803	−0.412729
$437$	12.7639i	0.610582i
$438$	− 44.3607i	− 2.11964i
$439$	1.18034	0.0563345	0.0281673	−	0.999603i	$-0.491033\pi$
$439$	1.18034	0.0563345	0.0281673	+	0.999603i	$0.491033\pi$
$440$	0	0
$441$	−51.8885	−2.47088
$442$	4.00000i	0.190261i
$443$	30.7082i	1.45899i	0.683986	+	0.729495i	$0.260245\pi$
$443$	30.7082i	1.45899i	−0.683986	+	0.729495i	$0.739755\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	6.47214	0.306465
$447$	32.3607i	1.53061i
$448$	1.00000i	0.0472456i
$449$	31.3050	1.47737	0.738686	−	0.674050i	$-0.235447\pi$
$449$	31.3050	1.47737	0.738686	+	0.674050i	$0.235447\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	− 2.14590i	− 0.100935i
$453$	45.8885i	2.15603i
$454$	10.4721	0.491482
$455$	0	0
$456$	−16.1803	−0.757714
$457$	3.05573i	0.142941i	0.997443	+	0.0714705i	$0.0227692\pi$
$457$	3.05573i	0.142941i	−0.997443	+	0.0714705i	$0.977231\pi$
$458$	− 21.7082i	− 1.01436i
$459$	11.0557	0.516037
$460$	0	0
$461$	34.3607	1.60034	0.800168	−	0.599776i	$-0.204744\pi$
$461$	34.3607	1.60034	0.800168	+	0.599776i	$0.204744\pi$
$462$	2.47214i	0.115014i
$463$	− 2.58359i	− 0.120070i	−0.998196	−	0.0600349i	$-0.980879\pi$
$463$	− 2.58359i	− 0.120070i	0.998196	−	0.0600349i	$-0.0191212\pi$
$464$	13.4164	0.622841
$465$	0	0
$466$	29.0344	1.34499
$467$	− 4.70820i	− 0.217870i	−0.994049	−	0.108935i	$-0.965256\pi$
$467$	− 4.70820i	− 0.217870i	0.994049	−	0.108935i	$-0.0347440\pi$
$468$	− 14.9443i	− 0.690799i
$469$	−1.88854	−0.0872049
$470$	0	0
$471$	−67.5967	−3.11469
$472$	5.00000i	0.230144i
$473$	2.47214i	0.113669i
$474$	−61.3050	−2.81583
$475$	0	0
$476$	0.111456	0.00510859
$477$	78.2492i	3.58279i
$478$	− 18.9443i	− 0.866491i
$479$	−23.2918	−1.06423	−0.532115	−	0.846672i	$-0.678602\pi$
$479$	−23.2918	−1.06423	−0.532115	+	0.846672i	$0.678602\pi$
$480$	0	0
$481$	6.47214	0.295104
$482$	− 23.2361i	− 1.05837i
$483$	− 4.36068i	− 0.198418i
$484$	4.32624	0.196647
$485$	0	0
$486$	−57.5967	−2.61264
$487$	19.2361i	0.871669i	0.900027	+	0.435835i	$0.143547\pi$
$487$	19.2361i	0.871669i	−0.900027	+	0.435835i	$0.856453\pi$
$488$	− 18.2918i	− 0.828031i
$489$	34.6525	1.56704
$490$	0	0
$491$	4.36068	0.196795	0.0983974	−	0.995147i	$-0.468628\pi$
$491$	4.36068	0.196795	0.0983974	+	0.995147i	$0.468628\pi$
$492$	14.0000i	0.631169i
$493$	2.11146i	0.0950952i
$494$	−11.7082	−0.526777
$495$	0	0
$496$	−4.85410	−0.217956
$497$	2.16718i	0.0972115i
$498$	78.2492i	3.50643i
$499$	6.58359	0.294722	0.147361	−	0.989083i	$-0.452922\pi$
$499$	6.58359	0.294722	0.147361	+	0.989083i	$0.452922\pi$
$500$	0	0
$501$	20.9443	0.935721
$502$	2.94427i	0.131409i
$503$	29.6525i	1.32214i	0.750325	+	0.661069i	$0.229897\pi$
$503$	29.6525i	1.32214i	−0.750325	+	0.661069i	$0.770103\pi$
$504$	3.94427	0.175692
$505$	0	0
$506$	18.4721	0.821187
$507$	− 8.18034i	− 0.363302i
$508$	7.70820i	0.341996i
$509$	−29.5967	−1.31185	−0.655926	−	0.754825i	$-0.727722\pi$
$509$	−29.5967	−1.31185	−0.655926	+	0.754825i	$0.727722\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	5.29180i	0.233867i
$513$	32.3607i	1.42876i
$514$	−3.14590	−0.138760
$515$	0	0
$516$	−2.47214	−0.108830
$517$	− 4.94427i	− 0.217449i
$518$	− 0.763932i	− 0.0335652i
$519$	9.52786	0.418227
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	− 33.4164i	− 1.46260i
$523$	− 17.7082i	− 0.774326i	−0.922011	−	0.387163i	$-0.873455\pi$
$523$	− 17.7082i	− 0.774326i	0.922011	−	0.387163i	$-0.126545\pi$
$524$	−7.41641	−0.323987
$525$	0	0
$526$	−37.5967	−1.63930
$527$	− 0.763932i	− 0.0332774i
$528$	31.4164i	1.36722i
$529$	−9.58359	−0.416678
$530$	0	0
$531$	16.7082	0.725074
$532$	0.326238i	0.0141442i
$533$	− 22.6525i	− 0.981188i
$534$	61.3050	2.65292
$535$	0	0
$536$	−17.8885	−0.772667
$537$	5.52786i	0.238545i
$538$	− 17.8885i	− 0.771230i
$539$	13.8885	0.598222
$540$	0	0
$541$	−25.3607	−1.09034	−0.545170	−	0.838325i	$-0.683535\pi$
$541$	−25.3607	−1.09034	−0.545170	+	0.838325i	$0.683535\pi$
$542$	22.9443i	0.985541i
$543$	13.5279i	0.580536i
$544$	2.58359	0.110771
$545$	0	0
$546$	4.00000	0.171184
$547$	12.1246i	0.518411i	0.965822	+	0.259205i	$0.0834607\pi$
$547$	12.1246i	0.518411i	−0.965822	+	0.259205i	$0.916539\pi$
$548$	3.88854i	0.166110i
$549$	−61.1246	−2.60873
$550$	0	0
$551$	−6.18034	−0.263291
$552$	− 41.3050i	− 1.75806i
$553$	− 2.76393i	− 0.117534i
$554$	20.4721	0.869778
$555$	0	0
$556$	8.29180	0.351650
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	12.0902i	0.511818i
$559$	4.00000	0.169182
$560$	0	0
$561$	−4.94427	−0.208747
$562$	− 27.5066i	− 1.16029i
$563$	27.5410i	1.16072i	0.814362	+	0.580358i	$0.197087\pi$
$563$	27.5410i	1.16072i	−0.814362	+	0.580358i	$0.802913\pi$
$564$	4.94427	0.208191
$565$	0	0
$566$	−22.4721	−0.944574
$567$	− 5.76393i	− 0.242062i
$568$	20.5279i	0.861330i
$569$	−5.52786	−0.231740	−0.115870	−	0.993264i	$-0.536966\pi$
$569$	−5.52786	−0.231740	−0.115870	+	0.993264i	$0.536966\pi$
$570$	0	0
$571$	28.1803	1.17931	0.589655	−	0.807655i	$-0.299264\pi$
$571$	28.1803	1.17931	0.589655	+	0.807655i	$0.299264\pi$
$572$	4.00000i	0.167248i
$573$	62.0689i	2.59296i
$574$	−2.67376	−0.111601
$575$	0	0
$576$	31.6525	1.31885
$577$	28.8328i	1.20033i	0.799878	+	0.600163i	$0.204898\pi$
$577$	28.8328i	1.20033i	−0.799878	+	0.600163i	$0.795102\pi$
$578$	− 26.5623i	− 1.10485i
$579$	11.2361	0.466955
$580$	0	0
$581$	−3.52786	−0.146360
$582$	− 83.4853i	− 3.46058i
$583$	− 20.9443i	− 0.867423i
$584$	18.9443	0.783920
$585$	0	0
$586$	−0.763932	−0.0315577
$587$	6.47214i	0.267134i	0.991040	+	0.133567i	$0.0426431\pi$
$587$	6.47214i	0.267134i	−0.991040	+	0.133567i	$0.957357\pi$
$588$	13.8885i	0.572754i
$589$	2.23607	0.0921356
$590$	0	0
$591$	−36.9443	−1.51968
$592$	− 9.70820i	− 0.399005i
$593$	− 6.52786i	− 0.268067i	−0.990977	−	0.134034i	$-0.957207\pi$
$593$	− 6.52786i	− 0.268067i	0.990977	−	0.134034i	$-0.0427930\pi$
$594$	46.8328	1.92157
$595$	0	0
$596$	6.18034	0.253157
$597$	− 61.3050i	− 2.50904i
$598$	− 29.8885i	− 1.22223i
$599$	14.5967	0.596407	0.298203	−	0.954502i	$-0.403613\pi$
$599$	14.5967	0.596407	0.298203	+	0.954502i	$0.403613\pi$
$600$	0	0
$601$	30.5410	1.24579	0.622897	−	0.782304i	$-0.285956\pi$
$601$	30.5410	1.24579	0.622897	+	0.782304i	$0.285956\pi$
$602$	− 0.472136i	− 0.0192428i
$603$	59.7771i	2.43431i
$604$	8.76393	0.356599
$605$	0	0
$606$	15.7082	0.638102
$607$	− 22.4721i	− 0.912116i	−0.889950	−	0.456058i	$-0.849261\pi$
$607$	− 22.4721i	− 0.912116i	0.889950	−	0.456058i	$-0.150739\pi$
$608$	7.56231i	0.306692i
$609$	2.11146	0.0855605
$610$	0	0
$611$	−8.00000	−0.323645
$612$	− 3.52786i	− 0.142605i
$613$	− 43.8885i	− 1.77264i	−0.463072	−	0.886321i	$-0.653253\pi$
$613$	− 43.8885i	− 1.77264i	0.463072	−	0.886321i	$-0.346747\pi$
$614$	46.4508	1.87460
$615$	0	0
$616$	−1.05573	−0.0425365
$617$	− 32.4721i	− 1.30728i	−0.756806	−	0.653639i	$-0.773241\pi$
$617$	− 32.4721i	− 1.30728i	0.756806	−	0.653639i	$-0.226759\pi$
$618$	− 32.6525i	− 1.31348i
$619$	−6.18034	−0.248409	−0.124204	−	0.992257i	$-0.539638\pi$
$619$	−6.18034	−0.248409	−0.124204	+	0.992257i	$0.539638\pi$
$620$	0	0
$621$	−82.6099	−3.31502
$622$	47.2148i	1.89314i
$623$	2.76393i	0.110735i
$624$	50.8328	2.03494
$625$	0	0
$626$	27.1246	1.08412
$627$	− 14.4721i	− 0.577961i
$628$	12.9098i	0.515158i
$629$	1.52786	0.0609199
$630$	0	0
$631$	34.3607	1.36788	0.683939	−	0.729540i	$-0.260266\pi$
$631$	34.3607	1.36788	0.683939	+	0.729540i	$0.260266\pi$
$632$	− 26.1803i	− 1.04140i
$633$	− 75.0132i	− 2.98151i
$634$	−6.56231	−0.260622
$635$	0	0
$636$	20.9443	0.830494
$637$	− 22.4721i	− 0.890378i
$638$	8.94427i	0.354107i
$639$	68.5967	2.71365
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	30.1803i	1.19112i
$643$	19.5279i	0.770104i	0.922895	+	0.385052i	$0.125816\pi$
$643$	19.5279i	0.770104i	−0.922895	+	0.385052i	$0.874184\pi$
$644$	−0.832816	−0.0328175
$645$	0	0
$646$	−2.76393	−0.108745
$647$	0.944272i	0.0371232i	0.999828	+	0.0185616i	$0.00590867\pi$
$647$	0.944272i	0.0371232i	−0.999828	+	0.0185616i	$0.994091\pi$
$648$	− 54.5967i	− 2.14476i
$649$	−4.47214	−0.175547
$650$	0	0
$651$	−0.763932	−0.0299409
$652$	− 6.61803i	− 0.259182i
$653$	− 47.3050i	− 1.85119i	−0.378521	−	0.925593i	$-0.623567\pi$
$653$	− 47.3050i	− 1.85119i	0.378521	−	0.925593i	$-0.376433\pi$
$654$	−73.0132	−2.85504
$655$	0	0
$656$	−33.9787	−1.32665
$657$	− 63.3050i	− 2.46976i
$658$	0.944272i	0.0368116i
$659$	−25.6525	−0.999279	−0.499639	−	0.866234i	$-0.666534\pi$
$659$	−25.6525	−0.999279	−0.499639	+	0.866234i	$0.666534\pi$
$660$	0	0
$661$	−0.639320	−0.0248667	−0.0124333	−	0.999923i	$-0.503958\pi$
$661$	−0.639320	−0.0248667	−0.0124333	+	0.999923i	$0.503958\pi$
$662$	− 3.23607i	− 0.125773i
$663$	8.00000i	0.310694i
$664$	−33.4164	−1.29681
$665$	0	0
$666$	−24.1803	−0.936969
$667$	− 15.7771i	− 0.610891i
$668$	− 4.00000i	− 0.154765i
$669$	12.9443	0.500454
$670$	0	0
$671$	16.3607	0.631597
$672$	− 2.58359i	− 0.0996642i
$673$	− 29.0132i	− 1.11837i	−0.829041	−	0.559187i	$-0.811113\pi$
$673$	− 29.0132i	− 1.11837i	0.829041	−	0.559187i	$-0.188887\pi$
$674$	23.8885	0.920152
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	46.7214i	1.79565i	0.440355	+	0.897824i	$0.354853\pi$
$677$	46.7214i	1.79565i	−0.440355	+	0.897824i	$0.645147\pi$
$678$	− 18.1803i	− 0.698212i
$679$	3.76393	0.144446
$680$	0	0
$681$	20.9443	0.802586
$682$	− 3.23607i	− 0.123915i
$683$	5.18034i	0.198220i	0.995076	+	0.0991101i	$0.0315996\pi$
$683$	5.18034i	0.198220i	−0.995076	+	0.0991101i	$0.968400\pi$
$684$	10.3262	0.394834
$685$	0	0
$686$	−5.32624	−0.203357
$687$	− 43.4164i	− 1.65644i
$688$	− 6.00000i	− 0.228748i
$689$	−33.8885	−1.29105
$690$	0	0
$691$	3.18034	0.120986	0.0604929	−	0.998169i	$-0.480733\pi$
$691$	3.18034	0.120986	0.0604929	+	0.998169i	$0.480733\pi$
$692$	− 1.81966i	− 0.0691731i
$693$	3.52786i	0.134012i
$694$	−39.1246	−1.48515
$695$	0	0
$696$	20.0000	0.758098
$697$	− 5.34752i	− 0.202552i
$698$	− 12.7639i	− 0.483122i
$699$	58.0689	2.19637
$700$	0	0
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	− 75.7771i	− 2.86002i
$703$	4.47214i	0.168670i
$704$	−8.47214	−0.319306
$705$	0	0
$706$	12.0000	0.451626
$707$	0.708204i	0.0266348i
$708$	− 4.47214i	− 0.168073i
$709$	−25.5279	−0.958719	−0.479360	−	0.877619i	$-0.659131\pi$
$709$	−25.5279	−0.958719	−0.479360	+	0.877619i	$0.659131\pi$
$710$	0	0
$711$	−87.4853	−3.28095
$712$	26.1803i	0.981150i
$713$	5.70820i	0.213774i
$714$	0.944272	0.0353385
$715$	0	0
$716$	1.05573	0.0394544
$717$	− 37.8885i	− 1.41497i
$718$	35.9787i	1.34271i
$719$	13.8197	0.515386	0.257693	−	0.966227i	$-0.417038\pi$
$719$	13.8197	0.515386	0.257693	+	0.966227i	$0.417038\pi$
$720$	0	0
$721$	1.47214	0.0548252
$722$	22.6525i	0.843038i
$723$	− 46.4721i	− 1.72832i
$724$	2.58359	0.0960184
$725$	0	0
$726$	36.6525	1.36030
$727$	44.2361i	1.64062i	0.571916	+	0.820312i	$0.306200\pi$
$727$	44.2361i	1.64062i	−0.571916	+	0.820312i	$0.693800\pi$
$728$	1.70820i	0.0633102i
$729$	−41.9443	−1.55349
$730$	0	0
$731$	0.944272	0.0349252
$732$	16.3607i	0.604708i
$733$	3.47214i	0.128246i	0.997942	+	0.0641231i	$0.0204250\pi$
$733$	3.47214i	0.128246i	−0.997942	+	0.0641231i	$0.979575\pi$
$734$	−29.1246	−1.07501
$735$	0	0
$736$	−19.3050	−0.711590
$737$	− 16.0000i	− 0.589368i
$738$	84.6312i	3.11532i
$739$	−6.18034	−0.227347	−0.113674	−	0.993518i	$-0.536262\pi$
$739$	−6.18034	−0.227347	−0.113674	+	0.993518i	$0.536262\pi$
$740$	0	0
$741$	−23.4164	−0.860223
$742$	4.00000i	0.146845i
$743$	50.1803i	1.84094i	0.390815	+	0.920469i	$0.372193\pi$
$743$	50.1803i	1.84094i	−0.390815	+	0.920469i	$0.627807\pi$
$744$	−7.23607	−0.265287
$745$	0	0
$746$	30.7426	1.12557
$747$	111.666i	4.08563i
$748$	0.944272i	0.0345260i
$749$	−1.36068	−0.0497182
$750$	0	0
$751$	−21.5410	−0.786043	−0.393021	−	0.919529i	$-0.628570\pi$
$751$	−21.5410	−0.786043	−0.393021	+	0.919529i	$0.628570\pi$
$752$	12.0000i	0.437595i
$753$	5.88854i	0.214590i
$754$	14.4721	0.527044
$755$	0	0
$756$	−2.11146	−0.0767929
$757$	− 8.65248i	− 0.314480i	−0.987560	−	0.157240i	$-0.949740\pi$
$757$	− 8.65248i	− 0.314480i	0.987560	−	0.157240i	$-0.0502596\pi$
$758$	− 3.41641i	− 0.124090i
$759$	36.9443	1.34099
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	65.3050i	2.36575i
$763$	− 3.29180i	− 0.119171i
$764$	11.8541	0.428866
$765$	0	0
$766$	−38.6525	−1.39657
$767$	7.23607i	0.261279i
$768$	− 43.8885i	− 1.58369i
$769$	47.3607	1.70787	0.853935	−	0.520380i	$-0.174210\pi$
$769$	47.3607	1.70787	0.853935	+	0.520380i	$0.174210\pi$
$770$	0	0
$771$	−6.29180	−0.226594
$772$	− 2.14590i	− 0.0772326i
$773$	− 11.1246i	− 0.400124i	−0.979783	−	0.200062i	$-0.935886\pi$
$773$	− 11.1246i	− 0.400124i	0.979783	−	0.200062i	$-0.0641144\pi$
$774$	−14.9443	−0.537161
$775$	0	0
$776$	35.6525	1.27985
$777$	− 1.52786i	− 0.0548118i
$778$	− 28.9443i	− 1.03770i
$779$	15.6525	0.560808
$780$	0	0
$781$	−18.3607	−0.656997
$782$	− 7.05573i	− 0.252312i
$783$	− 40.0000i	− 1.42948i
$784$	−33.7082	−1.20386
$785$	0	0
$786$	−62.8328	−2.24117
$787$	− 7.34752i	− 0.261911i	−0.991388	−	0.130955i	$-0.958196\pi$
$787$	− 7.34752i	− 0.261911i	0.991388	−	0.130955i	$-0.0418045\pi$
$788$	7.05573i	0.251350i
$789$	−75.1935	−2.67696
$790$	0	0
$791$	0.819660	0.0291438
$792$	33.4164i	1.18740i
$793$	− 26.4721i	− 0.940053i
$794$	11.3262	0.401953
$795$	0	0
$796$	−11.7082	−0.414986
$797$	55.4164i	1.96295i	0.191591	+	0.981475i	$0.438635\pi$
$797$	55.4164i	1.96295i	−0.191591	+	0.981475i	$0.561365\pi$
$798$	2.76393i	0.0978421i
$799$	−1.88854	−0.0668119
$800$	0	0
$801$	87.4853	3.09114
$802$	− 61.7771i	− 2.18142i
$803$	16.9443i	0.597950i
$804$	16.0000	0.564276
$805$	0	0
$806$	−5.23607	−0.184433
$807$	− 35.7771i	− 1.25941i
$808$	6.70820i	0.235994i
$809$	−23.4164	−0.823277	−0.411639	−	0.911347i	$-0.635043\pi$
$809$	−23.4164	−0.823277	−0.411639	+	0.911347i	$0.635043\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	− 0.403252i	− 0.0141514i
$813$	45.8885i	1.60938i
$814$	6.47214	0.226848
$815$	0	0
$816$	12.0000	0.420084
$817$	2.76393i	0.0966977i
$818$	− 6.18034i	− 0.216091i
$819$	5.70820	0.199461
$820$	0	0
$821$	30.5410	1.06589	0.532944	−	0.846150i	$-0.321085\pi$
$821$	30.5410	1.06589	0.532944	+	0.846150i	$0.321085\pi$
$822$	32.9443i	1.14906i
$823$	− 14.2918i	− 0.498181i	−0.968480	−	0.249090i	$-0.919868\pi$
$823$	− 14.2918i	− 0.498181i	0.968480	−	0.249090i	$-0.0801316\pi$
$824$	13.9443	0.485772
$825$	0	0
$826$	0.854102	0.0297180
$827$	− 17.3475i	− 0.603233i	−0.953429	−	0.301616i	$-0.902474\pi$
$827$	− 17.3475i	− 0.603233i	0.953429	−	0.301616i	$-0.0975261\pi$
$828$	26.3607i	0.916097i
$829$	−16.8328	−0.584628	−0.292314	−	0.956322i	$-0.594425\pi$
$829$	−16.8328	−0.584628	−0.292314	+	0.956322i	$0.594425\pi$
$830$	0	0
$831$	40.9443	1.42034
$832$	13.7082i	0.475246i
$833$	− 5.30495i	− 0.183806i
$834$	70.2492	2.43253
$835$	0	0
$836$	−2.76393	−0.0955926
$837$	14.4721i	0.500230i
$838$	− 16.3820i	− 0.565906i
$839$	−28.9443	−0.999267	−0.499634	−	0.866237i	$-0.666532\pi$
$839$	−28.9443	−0.999267	−0.499634	+	0.866237i	$0.666532\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	− 47.5066i	− 1.63718i
$843$	− 55.0132i	− 1.89475i
$844$	−14.3262	−0.493129
$845$	0	0
$846$	29.8885	1.02759
$847$	1.65248i	0.0567797i
$848$	50.8328i	1.74561i
$849$	−44.9443	−1.54248
$850$	0	0
$851$	−11.4164	−0.391349
$852$	− 18.3607i	− 0.629027i
$853$	10.5836i	0.362375i	0.983449	+	0.181188i	$0.0579941\pi$
$853$	10.5836i	0.362375i	−0.983449	+	0.181188i	$0.942006\pi$
$854$	−3.12461	−0.106922
$855$	0	0
$856$	−12.8885	−0.440521
$857$	55.6656i	1.90150i	0.309956	+	0.950751i	$0.399686\pi$
$857$	55.6656i	1.90150i	−0.309956	+	0.950751i	$0.600314\pi$
$858$	33.8885i	1.15694i
$859$	−2.11146	−0.0720420	−0.0360210	−	0.999351i	$-0.511468\pi$
$859$	−2.11146	−0.0720420	−0.0360210	+	0.999351i	$0.511468\pi$
$860$	0	0
$861$	−5.34752	−0.182243
$862$	− 19.4164i	− 0.661325i
$863$	− 9.81966i	− 0.334265i	−0.985934	−	0.167133i	$-0.946549\pi$
$863$	− 9.81966i	− 0.334265i	0.985934	−	0.167133i	$-0.0534508\pi$
$864$	−48.9443	−1.66512
$865$	0	0
$866$	16.4721	0.559746
$867$	− 53.1246i	− 1.80421i
$868$	0.145898i	0.00495210i
$869$	23.4164	0.794347
$870$	0	0
$871$	−25.8885	−0.877200
$872$	− 31.1803i	− 1.05590i
$873$	− 119.138i	− 4.03220i
$874$	20.6525	0.698580
$875$	0	0
$876$	−16.9443	−0.572494
$877$	18.0557i	0.609699i	0.952401	+	0.304849i	$0.0986061\pi$
$877$	18.0557i	0.609699i	−0.952401	+	0.304849i	$0.901394\pi$
$878$	− 1.90983i	− 0.0644536i
$879$	−1.52786	−0.0515336
$880$	0	0
$881$	−20.3607	−0.685969	−0.342984	−	0.939341i	$-0.611438\pi$
$881$	−20.3607	−0.685969	−0.342984	+	0.939341i	$0.611438\pi$
$882$	83.9574i	2.82699i
$883$	− 31.7771i	− 1.06938i	−0.845047	−	0.534692i	$-0.820428\pi$
$883$	− 31.7771i	− 1.06938i	0.845047	−	0.534692i	$-0.179572\pi$
$884$	1.52786	0.0513876
$885$	0	0
$886$	49.6869	1.66926
$887$	− 27.0689i	− 0.908884i	−0.890776	−	0.454442i	$-0.849839\pi$
$887$	− 27.0689i	− 0.908884i	0.890776	−	0.454442i	$-0.150161\pi$
$888$	− 14.4721i	− 0.485653i
$889$	−2.94427	−0.0987477
$890$	0	0
$891$	48.8328	1.63596
$892$	− 2.47214i	− 0.0827732i
$893$	− 5.52786i	− 0.184983i
$894$	52.3607	1.75120
$895$	0	0
$896$	3.21478	0.107398
$897$	− 59.7771i	− 1.99590i
$898$	− 50.6525i	− 1.69030i
$899$	−2.76393	−0.0921823
$900$	0	0
$901$	−8.00000	−0.266519
$902$	− 22.6525i	− 0.754245i
$903$	− 0.944272i	− 0.0314234i
$904$	7.76393	0.258225
$905$	0	0
$906$	74.2492	2.46677
$907$	24.2361i	0.804745i	0.915476	+	0.402373i	$0.131814\pi$
$907$	24.2361i	0.804745i	−0.915476	+	0.402373i	$0.868186\pi$
$908$	− 4.00000i	− 0.132745i
$909$	22.4164	0.743505
$910$	0	0
$911$	18.1803	0.602342	0.301171	−	0.953570i	$-0.402623\pi$
$911$	18.1803	0.602342	0.301171	+	0.953570i	$0.402623\pi$
$912$	35.1246i	1.16309i
$913$	− 29.8885i	− 0.989166i
$914$	4.94427	0.163542
$915$	0	0
$916$	−8.29180	−0.273969
$917$	− 2.83282i	− 0.0935478i
$918$	− 17.8885i	− 0.590410i
$919$	14.4721	0.477392	0.238696	−	0.971094i	$-0.423280\pi$
$919$	14.4721	0.477392	0.238696	+	0.971094i	$0.423280\pi$
$920$	0	0
$921$	92.9017	3.06122
$922$	− 55.5967i	− 1.83098i
$923$	29.7082i	0.977857i
$924$	0.944272	0.0310643
$925$	0	0
$926$	−4.18034	−0.137374
$927$	− 46.5967i	− 1.53044i
$928$	− 9.34752i	− 0.306848i
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	− 11.0902i	− 0.363271i
$933$	94.4296i	3.09149i
$934$	−7.61803	−0.249270
$935$	0	0
$936$	54.0689	1.76730
$937$	− 9.05573i	− 0.295838i	−0.988999	−	0.147919i	$-0.952743\pi$
$937$	− 9.05573i	− 0.295838i	0.988999	−	0.147919i	$-0.0472575\pi$
$938$	3.05573i	0.0997731i
$939$	54.2492	1.77036
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	109.374i	3.56359i
$943$	39.9574i	1.30119i
$944$	10.8541	0.353271
$945$	0	0
$946$	4.00000	0.130051
$947$	13.0557i	0.424254i	0.977242	+	0.212127i	$0.0680391\pi$
$947$	13.0557i	0.424254i	−0.977242	+	0.212127i	$0.931961\pi$
$948$	23.4164i	0.760530i
$949$	27.4164	0.889974
$950$	0	0
$951$	−13.1246	−0.425595
$952$	0.403252i	0.0130695i
$953$	45.7082i	1.48063i	0.672258	+	0.740317i	$0.265325\pi$
$953$	45.7082i	1.48063i	−0.672258	+	0.740317i	$0.734675\pi$
$954$	126.610	4.09915
$955$	0	0
$956$	−7.23607	−0.234031
$957$	17.8885i	0.578254i
$958$	37.6869i	1.21761i
$959$	−1.48529	−0.0479626
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 10.4721i	− 0.337635i
$963$	43.0689i	1.38788i
$964$	−8.87539	−0.285857
$965$	0	0
$966$	−7.05573	−0.227014
$967$	− 60.3607i	− 1.94107i	−0.240963	−	0.970534i	$-0.577463\pi$
$967$	− 60.3607i	− 1.94107i	0.240963	−	0.970534i	$-0.422537\pi$
$968$	15.6525i	0.503090i
$969$	−5.52786	−0.177581
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	22.0000i	0.705650i
$973$	3.16718i	0.101535i
$974$	31.1246	0.997297
$975$	0	0
$976$	−39.7082	−1.27103
$977$	47.2492i	1.51164i	0.654781	+	0.755818i	$0.272761\pi$
$977$	47.2492i	1.51164i	−0.654781	+	0.755818i	$0.727239\pi$
$978$	− 56.0689i	− 1.79289i
$979$	−23.4164	−0.748392
$980$	0	0
$981$	−104.193	−3.32664
$982$	− 7.05573i	− 0.225157i
$983$	39.5279i	1.26074i	0.776294	+	0.630372i	$0.217097\pi$
$983$	39.5279i	1.26074i	−0.776294	+	0.630372i	$0.782903\pi$
$984$	−50.6525	−1.61474
$985$	0	0
$986$	3.41641	0.108801
$987$	1.88854i	0.0601130i
$988$	4.47214i	0.142278i
$989$	−7.05573	−0.224359
$990$	0	0
$991$	−16.5410	−0.525443	−0.262721	−	0.964872i	$-0.584620\pi$
$991$	−16.5410	−0.525443	−0.262721	+	0.964872i	$0.584620\pi$
$992$	3.38197i	0.107378i
$993$	− 6.47214i	− 0.205387i
$994$	3.50658	0.111222
$995$	0	0
$996$	29.8885	0.947055
$997$	29.3607i	0.929862i	0.885347	+	0.464931i	$0.153921\pi$
$997$	29.3607i	0.929862i	−0.885347	+	0.464931i	$0.846079\pi$
$998$	− 10.6525i	− 0.337198i
$999$	−28.9443	−0.915756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.d.249.1		4
5.2	odd	4		31.2.a.a.1.2	✓	2
5.3	odd	4		775.2.a.d.1.1		2
5.4	even	2	inner	775.2.b.d.249.4		4
15.2	even	4		279.2.a.a.1.1		2
15.8	even	4		6975.2.a.y.1.2		2
20.7	even	4		496.2.a.i.1.2		2
35.27	even	4		1519.2.a.a.1.2		2
40.27	even	4		1984.2.a.n.1.1		2
40.37	odd	4		1984.2.a.r.1.2		2
55.32	even	4		3751.2.a.b.1.1		2
60.47	odd	4		4464.2.a.bf.1.1		2
65.12	odd	4		5239.2.a.f.1.1		2
85.67	odd	4		8959.2.a.b.1.2		2
155.2	odd	20		961.2.d.d.531.1		4
155.7	odd	60		961.2.g.a.235.1		8
155.12	even	60		961.2.g.e.547.1		8
155.17	even	60		961.2.g.d.816.1		8
155.22	even	60		961.2.g.d.732.1		8
155.27	even	20		961.2.d.a.388.1		4
155.37	even	12		961.2.c.c.439.2		4
155.42	even	60		961.2.g.d.338.1		8
155.47	odd	20		961.2.d.d.628.1		4
155.52	even	60		961.2.g.e.844.1		8
155.57	even	12		961.2.c.c.521.2		4
155.67	odd	12		961.2.c.e.521.2		4
155.72	odd	60		961.2.g.h.844.1		8
155.77	even	20		961.2.d.g.628.1		4
155.82	odd	60		961.2.g.a.338.1		8
155.87	odd	12		961.2.c.e.439.2		4
155.92	even	4		961.2.a.f.1.2		2
155.97	odd	20		961.2.d.c.388.1		4
155.102	odd	60		961.2.g.a.732.1		8
155.107	odd	60		961.2.g.a.816.1		8
155.112	odd	60		961.2.g.h.547.1		8
155.117	even	60		961.2.g.d.235.1		8
155.122	even	20		961.2.d.g.531.1		4
155.127	even	60		961.2.g.e.846.1		8
155.132	odd	20		961.2.d.c.374.1		4
155.137	even	60		961.2.g.e.448.1		8
155.142	odd	60		961.2.g.h.448.1		8
155.147	even	20		961.2.d.a.374.1		4
155.152	odd	60		961.2.g.h.846.1		8
465.92	odd	4		8649.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.2	✓	2	5.2	odd	4
279.2.a.a.1.1		2	15.2	even	4
496.2.a.i.1.2		2	20.7	even	4
775.2.a.d.1.1		2	5.3	odd	4
775.2.b.d.249.1		4	1.1	even	1	trivial
775.2.b.d.249.4		4	5.4	even	2	inner
961.2.a.f.1.2		2	155.92	even	4
961.2.c.c.439.2		4	155.37	even	12
961.2.c.c.521.2		4	155.57	even	12
961.2.c.e.439.2		4	155.87	odd	12
961.2.c.e.521.2		4	155.67	odd	12
961.2.d.a.374.1		4	155.147	even	20
961.2.d.a.388.1		4	155.27	even	20
961.2.d.c.374.1		4	155.132	odd	20
961.2.d.c.388.1		4	155.97	odd	20
961.2.d.d.531.1		4	155.2	odd	20
961.2.d.d.628.1		4	155.47	odd	20
961.2.d.g.531.1		4	155.122	even	20
961.2.d.g.628.1		4	155.77	even	20
961.2.g.a.235.1		8	155.7	odd	60
961.2.g.a.338.1		8	155.82	odd	60
961.2.g.a.732.1		8	155.102	odd	60
961.2.g.a.816.1		8	155.107	odd	60
961.2.g.d.235.1		8	155.117	even	60
961.2.g.d.338.1		8	155.42	even	60
961.2.g.d.732.1		8	155.22	even	60
961.2.g.d.816.1		8	155.17	even	60
961.2.g.e.448.1		8	155.137	even	60
961.2.g.e.547.1		8	155.12	even	60
961.2.g.e.844.1		8	155.52	even	60
961.2.g.e.846.1		8	155.127	even	60
961.2.g.h.448.1		8	155.142	odd	60
961.2.g.h.547.1		8	155.112	odd	60
961.2.g.h.844.1		8	155.72	odd	60
961.2.g.h.846.1		8	155.152	odd	60
1519.2.a.a.1.2		2	35.27	even	4
1984.2.a.n.1.1		2	40.27	even	4
1984.2.a.r.1.2		2	40.37	odd	4
3751.2.a.b.1.1		2	55.32	even	4
4464.2.a.bf.1.1		2	60.47	odd	4
5239.2.a.f.1.1		2	65.12	odd	4
6975.2.a.y.1.2		2	15.8	even	4
8649.2.a.c.1.1		2	465.92	odd	4
8959.2.a.b.1.2		2	85.67	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -3.23607i q^{3} -0.618034 q^{4} -5.23607 q^{6} -0.236068i q^{7} -2.23607i q^{8} -7.47214 q^{9} +O(q^{10})\)	\(q-1.61803i q^{2} -3.23607i q^{3} -0.618034 q^{4} -5.23607 q^{6} -0.236068i q^{7} -2.23607i q^{8} -7.47214 q^{9} +2.00000 q^{11} +2.00000i q^{12} -3.23607i q^{13} -0.381966 q^{14} -4.85410 q^{16} -0.763932i q^{17} +12.0902i q^{18} +2.23607 q^{19} -0.763932 q^{21} -3.23607i q^{22} +5.70820i q^{23} -7.23607 q^{24} -5.23607 q^{26} +14.4721i q^{27} +0.145898i q^{28} -2.76393 q^{29} +1.00000 q^{31} +3.38197i q^{32} -6.47214i q^{33} -1.23607 q^{34} +4.61803 q^{36} +2.00000i q^{37} -3.61803i q^{38} -10.4721 q^{39} +7.00000 q^{41} +1.23607i q^{42} +1.23607i q^{43} -1.23607 q^{44} +9.23607 q^{46} -2.47214i q^{47} +15.7082i q^{48} +6.94427 q^{49} -2.47214 q^{51} +2.00000i q^{52} -10.4721i q^{53} +23.4164 q^{54} -0.527864 q^{56} -7.23607i q^{57} +4.47214i q^{58} -2.23607 q^{59} +8.18034 q^{61} -1.61803i q^{62} +1.76393i q^{63} -4.23607 q^{64} -10.4721 q^{66} -8.00000i q^{67} +0.472136i q^{68} +18.4721 q^{69} -9.18034 q^{71} +16.7082i q^{72} +8.47214i q^{73} +3.23607 q^{74} -1.38197 q^{76} -0.472136i q^{77} +16.9443i q^{78} +11.7082 q^{79} +24.4164 q^{81} -11.3262i q^{82} -14.9443i q^{83} +0.472136 q^{84} +2.00000 q^{86} +8.94427i q^{87} -4.47214i q^{88} -11.7082 q^{89} -0.763932 q^{91} -3.52786i q^{92} -3.23607i q^{93} -4.00000 q^{94} +10.9443 q^{96} +15.9443i q^{97} -11.2361i q^{98} -14.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9	\( 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} - 12 q^{21} - 20 q^{24} - 12 q^{26} - 20 q^{29} + 4 q^{31} + 4 q^{34} + 14 q^{36} - 24 q^{39} + 28 q^{41} + 4 q^{44} + 28 q^{46} - 8 q^{49} + 8 q^{51} + 40 q^{54} - 20 q^{56} - 12 q^{61} - 8 q^{64} - 24 q^{66} + 56 q^{69} + 8 q^{71} + 4 q^{74} - 10 q^{76} + 20 q^{79} + 44 q^{81} - 16 q^{84} + 8 q^{86} - 20 q^{89} - 12 q^{91} - 16 q^{94} + 8 q^{96} - 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 - 12 * q^21 - 20 * q^24 - 12 * q^26 - 20 * q^29 + 4 * q^31 + 4 * q^34 + 14 * q^36 - 24 * q^39 + 28 * q^41 + 4 * q^44 + 28 * q^46 - 8 * q^49 + 8 * q^51 + 40 * q^54 - 20 * q^56 - 12 * q^61 - 8 * q^64 - 24 * q^66 + 56 * q^69 + 8 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 44 * q^81 - 16 * q^84 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 16 * q^94 + 8 * q^96 - 24 * q^99

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