LMFDB - Embedded newform 920.2.e.c.369.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (of order $2$, degree $1$, 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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.16
Root			$-0.945903 - 0.945903i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.c.369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.20935i q^{3} +(2.02615 - 0.945903i) q^{5} -4.54713i q^{7} -7.29996 q^{9} +O(q^{10})$	$q+3.20935i q^{3} +(2.02615 - 0.945903i) q^{5} -4.54713i q^{7} -7.29996 q^{9} +4.59577 q^{11} -5.17990i q^{13} +(3.03574 + 6.50262i) q^{15} +3.33965i q^{17} +4.43893 q^{19} +14.5934 q^{21} -1.00000i q^{23} +(3.21054 - 3.83307i) q^{25} -13.8001i q^{27} -4.13029 q^{29} +7.82535 q^{31} +14.7494i q^{33} +(-4.30114 - 9.21315i) q^{35} -1.03174i q^{37} +16.6241 q^{39} -5.75780 q^{41} -1.67719i q^{43} +(-14.7908 + 6.90505i) q^{45} +6.20177i q^{47} -13.6764 q^{49} -10.7181 q^{51} +7.35094i q^{53} +(9.31170 - 4.34715i) q^{55} +14.2461i q^{57} -1.83951 q^{59} +0.524294 q^{61} +33.1939i q^{63} +(-4.89968 - 10.4952i) q^{65} -2.55477i q^{67} +3.20935 q^{69} -6.58764 q^{71} +2.41579i q^{73} +(12.3017 + 10.3038i) q^{75} -20.8975i q^{77} +9.85496 q^{79} +22.3895 q^{81} +10.7875i q^{83} +(3.15898 + 6.76662i) q^{85} -13.2556i q^{87} -5.13994 q^{89} -23.5537 q^{91} +25.1143i q^{93} +(8.99393 - 4.19880i) q^{95} -3.36751i q^{97} -33.5489 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 22 * q^9	$ 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			3.20935i			1.85292i	0.376391	+	0.926461i	$0.377165\pi$
$3$			3.20935i			1.85292i	−0.376391	+	0.926461i	$0.622835\pi$
$4$		0			0
$5$	2.02615	−	0.945903i	0.906120	−	0.423020i
$5$	2.02615	−	0.945903i	0.906120	−	0.423020i
$6$		0			0
$7$		−	4.54713i		−	1.71865i	−0.511427	−	0.859327i	$-0.670883\pi$
$7$		−	4.54713i		−	1.71865i	0.511427	−	0.859327i	$-0.329117\pi$
$8$		0			0
$9$	−7.29996			−2.43332
$10$		0			0
$11$	4.59577			1.38568			0.692838	−	0.721093i	$-0.256360\pi$
$11$	4.59577			1.38568			0.692838	+	0.721093i	$0.256360\pi$
$12$		0			0
$13$		−	5.17990i		−	1.43665i	−0.695710	−	0.718323i	$-0.744910\pi$
$13$		−	5.17990i		−	1.43665i	0.695710	−	0.718323i	$-0.255090\pi$
$14$		0			0
$15$	3.03574	+	6.50262i	0.783824	+	1.67897i
$16$		0			0
$17$			3.33965i			0.809984i	0.914320	+	0.404992i	$0.132726\pi$
$17$			3.33965i			0.809984i	−0.914320	+	0.404992i	$0.867274\pi$
$18$		0			0
$19$	4.43893			1.01836			0.509181	−	0.860660i	$-0.329949\pi$
$19$	4.43893			1.01836			0.509181	+	0.860660i	$0.329949\pi$
$20$		0			0
$21$	14.5934			3.18453
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	3.21054	−	3.83307i	0.642107	−	0.766615i
$26$		0			0
$27$		−	13.8001i		−	2.65583i
$28$		0			0
$29$	−4.13029			−0.766976			−0.383488	−	0.923546i	$-0.625277\pi$
$29$	−4.13029			−0.766976			−0.383488	+	0.923546i	$0.625277\pi$
$30$		0			0
$31$	7.82535			1.40547			0.702737	−	0.711450i	$-0.251961\pi$
$31$	7.82535			1.40547			0.702737	+	0.711450i	$0.251961\pi$
$32$		0			0
$33$			14.7494i			2.56755i
$34$		0			0
$35$	−4.30114	−	9.21315i	−0.727025	−	1.55731i
$36$		0			0
$37$		−	1.03174i		−	0.169618i	−0.996397	−	0.0848089i	$-0.972972\pi$
$37$		−	1.03174i		−	0.169618i	0.996397	−	0.0848089i	$-0.0270280\pi$
$38$		0			0
$39$	16.6241			2.66199
$40$		0			0
$41$	−5.75780			−0.899218			−0.449609	−	0.893225i	$-0.648437\pi$
$41$	−5.75780			−0.899218			−0.449609	+	0.893225i	$0.648437\pi$
$42$		0			0
$43$		−	1.67719i		−	0.255770i	−0.991789	−	0.127885i	$-0.959181\pi$
$43$		−	1.67719i		−	0.255770i	0.991789	−	0.127885i	$-0.0408188\pi$
$44$		0			0
$45$	−14.7908	+	6.90505i	−2.20488	+	1.02934i
$46$		0			0
$47$			6.20177i			0.904621i	0.891861	+	0.452310i	$0.149400\pi$
$47$			6.20177i			0.904621i	−0.891861	+	0.452310i	$0.850600\pi$
$48$		0			0
$49$	−13.6764			−1.95377
$50$		0			0
$51$	−10.7181			−1.50084
$52$		0			0
$53$			7.35094i			1.00973i	0.863198	+	0.504865i	$0.168458\pi$
$53$			7.35094i			1.00973i	−0.863198	+	0.504865i	$0.831542\pi$
$54$		0			0
$55$	9.31170	−	4.34715i	1.25559	−	0.586169i
$56$		0			0
$57$			14.2461i			1.88694i
$58$		0			0
$59$	−1.83951			−0.239484			−0.119742	−	0.992805i	$-0.538207\pi$
$59$	−1.83951			−0.239484			−0.119742	+	0.992805i	$0.538207\pi$
$60$		0			0
$61$	0.524294			0.0671289			0.0335645	−	0.999437i	$-0.489314\pi$
$61$	0.524294			0.0671289			0.0335645	+	0.999437i	$0.489314\pi$
$62$		0			0
$63$			33.1939i			4.18203i
$64$		0			0
$65$	−4.89968	−	10.4952i	−0.607730	−	1.30177i
$66$		0			0
$67$		−	2.55477i		−	0.312115i	−0.987748	−	0.156057i	$-0.950122\pi$
$67$		−	2.55477i		−	0.312115i	0.987748	−	0.156057i	$-0.0498785\pi$
$68$		0			0
$69$	3.20935			0.386361
$70$		0			0
$71$	−6.58764			−0.781809			−0.390904	−	0.920431i	$-0.627838\pi$
$71$	−6.58764			−0.781809			−0.390904	+	0.920431i	$0.627838\pi$
$72$		0			0
$73$			2.41579i			0.282747i	0.989956	+	0.141374i	$0.0451519\pi$
$73$			2.41579i			0.282747i	−0.989956	+	0.141374i	$0.954848\pi$
$74$		0			0
$75$	12.3017	+	10.3038i	1.42048	+	1.18977i
$76$		0			0
$77$		−	20.8975i		−	2.38150i
$78$		0			0
$79$	9.85496			1.10877			0.554385	−	0.832261i	$-0.312954\pi$
$79$	9.85496			1.10877			0.554385	+	0.832261i	$0.312954\pi$
$80$		0			0
$81$	22.3895			2.48772
$82$		0			0
$83$			10.7875i			1.18408i	0.805908	+	0.592040i	$0.201677\pi$
$83$			10.7875i			1.18408i	−0.805908	+	0.592040i	$0.798323\pi$
$84$		0			0
$85$	3.15898	+	6.76662i	0.342640	+	0.733942i
$86$		0			0
$87$		−	13.2556i		−	1.42115i
$88$		0			0
$89$	−5.13994			−0.544832			−0.272416	−	0.962180i	$-0.587823\pi$
$89$	−5.13994			−0.544832			−0.272416	+	0.962180i	$0.587823\pi$
$90$		0			0
$91$	−23.5537			−2.46909
$92$		0			0
$93$			25.1143i			2.60423i
$94$		0			0
$95$	8.99393	−	4.19880i	0.922758	−	0.430788i
$96$		0			0
$97$		−	3.36751i		−	0.341919i	−0.985278	−	0.170959i	$-0.945313\pi$
$97$		−	3.36751i		−	0.341919i	0.985278	−	0.170959i	$-0.0546867\pi$
$98$		0			0
$99$	−33.5489			−3.37179
$100$		0			0
$101$	−4.18121			−0.416046			−0.208023	−	0.978124i	$-0.566703\pi$
$101$	−4.18121			−0.416046			−0.208023	+	0.978124i	$0.566703\pi$
$102$		0			0
$103$		−	9.92121i		−	0.977566i	−0.872405	−	0.488783i	$-0.837441\pi$
$103$		−	9.92121i		−	0.977566i	0.872405	−	0.488783i	$-0.162559\pi$
$104$		0			0
$105$	29.5683	−	13.8039i	2.88557	−	1.34712i
$106$		0			0
$107$			11.6829i			1.12943i	0.825287	+	0.564713i	$0.191013\pi$
$107$			11.6829i			1.12943i	−0.825287	+	0.564713i	$0.808987\pi$
$108$		0			0
$109$	5.58937			0.535365			0.267682	−	0.963507i	$-0.413742\pi$
$109$	5.58937			0.535365			0.267682	+	0.963507i	$0.413742\pi$
$110$		0			0
$111$	3.31123			0.314288
$112$		0			0
$113$			4.02597i			0.378731i	0.981907	+	0.189366i	$0.0606431\pi$
$113$			4.02597i			0.378731i	−0.981907	+	0.189366i	$0.939357\pi$
$114$		0			0
$115$	−0.945903	−	2.02615i	−0.0882059	−	0.188939i
$116$		0			0
$117$			37.8130i			3.49582i
$118$		0			0
$119$	15.1858			1.39208
$120$		0			0
$121$	10.1211			0.920098
$122$		0			0
$123$		−	18.4788i		−	1.66618i
$124$		0			0
$125$	2.87930	−	10.8032i	0.257533	−	0.966270i
$126$		0			0
$127$			15.8137i			1.40324i	0.712553	+	0.701618i	$0.247539\pi$
$127$			15.8137i			1.40324i	−0.712553	+	0.701618i	$0.752461\pi$
$128$		0			0
$129$	5.38271			0.473921
$130$		0			0
$131$	15.5004			1.35428			0.677139	−	0.735855i	$-0.263220\pi$
$131$	15.5004			1.35428			0.677139	+	0.735855i	$0.263220\pi$
$132$		0			0
$133$		−	20.1844i		−	1.75021i
$134$		0			0
$135$	−13.0535	−	27.9610i	−1.12347	−	2.40650i
$136$		0			0
$137$		−	4.94249i		−	0.422265i	−0.977457	−	0.211133i	$-0.932285\pi$
$137$		−	4.94249i		−	0.422265i	0.977457	−	0.211133i	$-0.0677152\pi$
$138$		0			0
$139$	−23.0723			−1.95697			−0.978483	−	0.206326i	$-0.933849\pi$
$139$	−23.0723			−1.95697			−0.978483	+	0.206326i	$0.933849\pi$
$140$		0			0
$141$	−19.9037			−1.67619
$142$		0			0
$143$		−	23.8056i		−	1.99072i
$144$		0			0
$145$	−8.36858	+	3.90685i	−0.694973	+	0.324447i
$146$		0			0
$147$		−	43.8924i		−	3.62018i
$148$		0			0
$149$	14.7115			1.20521			0.602606	−	0.798039i	$-0.294129\pi$
$149$	14.7115			1.20521			0.602606	+	0.798039i	$0.294129\pi$
$150$		0			0
$151$	−11.5461			−0.939606			−0.469803	−	0.882771i	$-0.655675\pi$
$151$	−11.5461			−0.939606			−0.469803	+	0.882771i	$0.655675\pi$
$152$		0			0
$153$		−	24.3793i		−	1.97095i
$154$		0			0
$155$	15.8553	−	7.40202i	1.27353	−	0.594544i
$156$		0			0
$157$			7.02677i			0.560797i	0.959884	+	0.280399i	$0.0904667\pi$
$157$			7.02677i			0.560797i	−0.959884	+	0.280399i	$0.909533\pi$
$158$		0			0
$159$	−23.5918			−1.87095
$160$		0			0
$161$	−4.54713			−0.358364
$162$		0			0
$163$		−	13.4203i		−	1.05116i	−0.850744	−	0.525580i	$-0.823848\pi$
$163$		−	13.4203i		−	1.05116i	0.850744	−	0.525580i	$-0.176152\pi$
$164$		0			0
$165$	13.9515	+	29.8845i	1.08613	+	2.32651i
$166$		0			0
$167$			8.77046i			0.678679i	0.940664	+	0.339339i	$0.110203\pi$
$167$			8.77046i			0.678679i	−0.940664	+	0.339339i	$0.889797\pi$
$168$		0			0
$169$	−13.8313			−1.06395
$170$		0			0
$171$	−32.4040			−2.47800
$172$		0			0
$173$		−	20.5416i		−	1.56175i	−0.624690	−	0.780873i	$-0.714775\pi$
$173$		−	20.5416i		−	1.56175i	0.624690	−	0.780873i	$-0.285225\pi$
$174$		0			0
$175$	−17.4295	−	14.5987i	−1.31754	−	1.10356i
$176$		0			0
$177$		−	5.90365i		−	0.443745i
$178$		0			0
$179$	15.4880			1.15763			0.578814	−	0.815460i	$-0.303516\pi$
$179$	15.4880			1.15763			0.578814	+	0.815460i	$0.303516\pi$
$180$		0			0
$181$	−3.31887			−0.246690			−0.123345	−	0.992364i	$-0.539362\pi$
$181$	−3.31887			−0.246690			−0.123345	+	0.992364i	$0.539362\pi$
$182$		0			0
$183$			1.68264i			0.124385i
$184$		0			0
$185$	−0.975930	−	2.09046i	−0.0717518	−	0.153694i
$186$		0			0
$187$			15.3482i			1.12237i
$188$		0			0
$189$	−62.7508			−4.56445
$190$		0			0
$191$	−5.46525			−0.395451			−0.197726	−	0.980257i	$-0.563356\pi$
$191$	−5.46525			−0.395451			−0.197726	+	0.980257i	$0.563356\pi$
$192$		0			0
$193$			1.88244i			0.135501i	0.997702	+	0.0677506i	$0.0215822\pi$
$193$			1.88244i			0.135501i	−0.997702	+	0.0677506i	$0.978418\pi$
$194$		0			0
$195$	33.6829	−	15.7248i	2.41208	−	1.12608i
$196$		0			0
$197$			7.27284i			0.518169i	0.965855	+	0.259084i	$0.0834208\pi$
$197$			7.27284i			0.518169i	−0.965855	+	0.259084i	$0.916579\pi$
$198$		0			0
$199$	7.84296			0.555972			0.277986	−	0.960585i	$-0.410333\pi$
$199$	7.84296			0.555972			0.277986	+	0.960585i	$0.410333\pi$
$200$		0			0
$201$	8.19916			0.578324
$202$		0			0
$203$			18.7810i			1.31817i
$204$		0			0
$205$	−11.6662	+	5.44632i	−0.814800	+	0.380388i
$206$		0			0
$207$			7.29996i			0.507382i
$208$		0			0
$209$	20.4003			1.41112
$210$		0			0
$211$	6.03882			0.415730			0.207865	−	0.978158i	$-0.433349\pi$
$211$	6.03882			0.415730			0.207865	+	0.978158i	$0.433349\pi$
$212$		0			0
$213$		−	21.1421i		−	1.44863i
$214$		0			0
$215$	−1.58646	−	3.39824i	−0.108196	−	0.231758i
$216$		0			0
$217$		−	35.5829i		−	2.41552i
$218$		0			0
$219$	−7.75314			−0.523908
$220$		0			0
$221$	17.2990			1.16366
$222$		0			0
$223$			13.7931i			0.923654i	0.886970	+	0.461827i	$0.152806\pi$
$223$			13.7931i			0.923654i	−0.886970	+	0.461827i	$0.847194\pi$
$224$		0			0
$225$	−23.4368	+	27.9813i	−1.56245	+	1.86542i
$226$		0			0
$227$			18.2668i			1.21241i	0.795309	+	0.606204i	$0.207308\pi$
$227$			18.2668i			1.21241i	−0.795309	+	0.606204i	$0.792692\pi$
$228$		0			0
$229$	−15.5985			−1.03078			−0.515390	−	0.856956i	$-0.672353\pi$
$229$	−15.5985			−1.03078			−0.515390	+	0.856956i	$0.672353\pi$
$230$		0			0
$231$	67.0676			4.41273
$232$		0			0
$233$			26.9422i			1.76504i	0.470275	+	0.882520i	$0.344155\pi$
$233$			26.9422i			1.76504i	−0.470275	+	0.882520i	$0.655845\pi$
$234$		0			0
$235$	5.86627	+	12.5657i	0.382673	+	0.819695i
$236$		0			0
$237$			31.6281i			2.05446i
$238$		0			0
$239$	4.91116			0.317677			0.158838	−	0.987305i	$-0.449225\pi$
$239$	4.91116			0.317677			0.158838	+	0.987305i	$0.449225\pi$
$240$		0			0
$241$	20.3100			1.30828			0.654140	−	0.756373i	$-0.273031\pi$
$241$	20.3100			1.30828			0.654140	+	0.756373i	$0.273031\pi$
$242$		0			0
$243$			30.4556i			1.95373i
$244$		0			0
$245$	−27.7103	+	12.9365i	−1.77035	+	0.826484i
$246$		0			0
$247$		−	22.9932i		−	1.46302i
$248$		0			0
$249$	−34.6209			−2.19401
$250$		0			0
$251$	−15.1382			−0.955514			−0.477757	−	0.878492i	$-0.658550\pi$
$251$	−15.1382			−0.955514			−0.477757	+	0.878492i	$0.658550\pi$
$252$		0			0
$253$		−	4.59577i		−	0.288933i
$254$		0			0
$255$	−21.7165	+	10.1383i	−1.35994	+	0.634885i
$256$		0			0
$257$		−	21.7394i		−	1.35607i	−0.735031	−	0.678033i	$-0.762832\pi$
$257$		−	21.7394i		−	1.35607i	0.735031	−	0.678033i	$-0.237168\pi$
$258$		0			0
$259$	−4.69147			−0.291514
$260$		0			0
$261$	30.1510			1.86630
$262$		0			0
$263$			20.5960i			1.27000i	0.772511	+	0.635002i	$0.219001\pi$
$263$			20.5960i			1.27000i	−0.772511	+	0.635002i	$0.780999\pi$
$264$		0			0
$265$	6.95327	+	14.8941i	0.427136	+	0.914936i
$266$		0			0
$267$		−	16.4959i		−	1.00953i
$268$		0			0
$269$	−15.8810			−0.968284			−0.484142	−	0.874989i	$-0.660868\pi$
$269$	−15.8810			−0.968284			−0.484142	+	0.874989i	$0.660868\pi$
$270$		0			0
$271$	−3.55030			−0.215666			−0.107833	−	0.994169i	$-0.534391\pi$
$271$	−3.55030			−0.215666			−0.107833	+	0.994169i	$0.534391\pi$
$272$		0			0
$273$		−	75.5921i		−	4.57504i
$274$		0			0
$275$	14.7549	−	17.6159i	0.889753	−	1.06228i
$276$		0			0
$277$			23.3134i			1.40077i	0.713767	+	0.700383i	$0.246987\pi$
$277$			23.3134i			1.40077i	−0.713767	+	0.700383i	$0.753013\pi$
$278$		0			0
$279$	−57.1247			−3.41997
$280$		0			0
$281$	−8.93301			−0.532899			−0.266449	−	0.963849i	$-0.585850\pi$
$281$	−8.93301			−0.532899			−0.266449	+	0.963849i	$0.585850\pi$
$282$		0			0
$283$		−	14.6498i		−	0.870843i	−0.900227	−	0.435421i	$-0.856599\pi$
$283$		−	14.6498i		−	0.870843i	0.900227	−	0.435421i	$-0.143401\pi$
$284$		0			0
$285$	13.4754	+	28.8647i	0.798216	+	1.70980i
$286$		0			0
$287$			26.1815i			1.54544i
$288$		0			0
$289$	5.84675			0.343927
$290$		0			0
$291$	10.8075			0.633549
$292$		0			0
$293$		−	27.4766i		−	1.60520i	−0.596516	−	0.802601i	$-0.703449\pi$
$293$		−	27.4766i		−	1.60520i	0.596516	−	0.802601i	$-0.296551\pi$
$294$		0			0
$295$	−3.72712	+	1.74000i	−0.217001	+	0.101307i
$296$		0			0
$297$		−	63.4220i		−	3.68012i
$298$		0			0
$299$	−5.17990			−0.299561
$300$		0			0
$301$	−7.62641			−0.439579
$302$		0			0
$303$		−	13.4190i		−	0.770900i
$304$		0			0
$305$	1.06230	−	0.495931i	0.0608269	−	0.0283969i
$306$		0			0
$307$			19.1369i			1.09220i	0.837719	+	0.546101i	$0.183888\pi$
$307$			19.1369i			1.09220i	−0.837719	+	0.546101i	$0.816112\pi$
$308$		0			0
$309$	31.8407			1.81135
$310$		0			0
$311$	15.3439			0.870074			0.435037	−	0.900413i	$-0.356735\pi$
$311$	15.3439			0.870074			0.435037	+	0.900413i	$0.356735\pi$
$312$		0			0
$313$		−	8.41254i		−	0.475505i	−0.971326	−	0.237753i	$-0.923589\pi$
$313$		−	8.41254i		−	0.475505i	0.971326	−	0.237753i	$-0.0764107\pi$
$314$		0			0
$315$	31.3982	+	67.2556i	1.76909	+	3.78942i
$316$		0			0
$317$		−	16.5135i		−	0.927493i	−0.885968	−	0.463747i	$-0.846505\pi$
$317$		−	16.5135i		−	0.927493i	0.885968	−	0.463747i	$-0.153495\pi$
$318$		0			0
$319$	−18.9819			−1.06278
$320$		0			0
$321$	−37.4945			−2.09274
$322$		0			0
$323$			14.8245i			0.824856i
$324$		0			0
$325$	−19.8549	−	16.6303i	−1.10135	−	0.922480i
$326$		0			0
$327$			17.9383i			0.991989i
$328$		0			0
$329$	28.2002			1.55473
$330$		0			0
$331$	−2.78794			−0.153239			−0.0766196	−	0.997060i	$-0.524413\pi$
$331$	−2.78794			−0.153239			−0.0766196	+	0.997060i	$0.524413\pi$
$332$		0			0
$333$			7.53169i			0.412734i
$334$		0			0
$335$	−2.41656	−	5.17633i	−0.132031	−	0.282813i
$336$		0			0
$337$			8.70062i			0.473953i	0.971515	+	0.236977i	$0.0761565\pi$
$337$			8.70062i			0.473953i	−0.971515	+	0.236977i	$0.923844\pi$
$338$		0			0
$339$	−12.9208			−0.701759
$340$		0			0
$341$	35.9635			1.94753
$342$		0			0
$343$			30.3584i			1.63920i
$344$		0			0
$345$	6.50262	−	3.03574i	0.350089	−	0.163439i
$346$		0			0
$347$		−	1.77786i		−	0.0954405i	−0.998861	−	0.0477203i	$-0.984804\pi$
$347$		−	1.77786i		−	0.0954405i	0.998861	−	0.0477203i	$-0.0151956\pi$
$348$		0			0
$349$	−23.9187			−1.28034			−0.640170	−	0.768233i	$-0.721136\pi$
$349$	−23.9187			−1.28034			−0.640170	+	0.768233i	$0.721136\pi$
$350$		0			0
$351$	−71.4831			−3.81548
$352$		0			0
$353$			8.55855i			0.455526i	0.973717	+	0.227763i	$0.0731411\pi$
$353$			8.55855i			0.455526i	−0.973717	+	0.227763i	$0.926859\pi$
$354$		0			0
$355$	−13.3475	+	6.23126i	−0.708413	+	0.330721i
$356$		0			0
$357$			48.7367i			2.57942i
$358$		0			0
$359$	−28.6262			−1.51083			−0.755417	−	0.655245i	$-0.772565\pi$
$359$	−28.6262			−1.51083			−0.755417	+	0.655245i	$0.772565\pi$
$360$		0			0
$361$	0.704135			0.0370597
$362$		0			0
$363$			32.4821i			1.70487i
$364$		0			0
$365$	2.28510	+	4.89475i	0.119608	+	0.256203i
$366$		0			0
$367$		−	25.3094i		−	1.32114i	−0.750764	−	0.660570i	$-0.770315\pi$
$367$		−	25.3094i		−	1.32114i	0.750764	−	0.660570i	$-0.229685\pi$
$368$		0			0
$369$	42.0317			2.18808
$370$		0			0
$371$	33.4257			1.73537
$372$		0			0
$373$		−	24.0094i		−	1.24316i	−0.783351	−	0.621579i	$-0.786491\pi$
$373$		−	24.0094i		−	1.24316i	0.783351	−	0.621579i	$-0.213509\pi$
$374$		0			0
$375$	34.6714	+	9.24070i	1.79042	+	0.477188i
$376$		0			0
$377$			21.3945i			1.10187i
$378$		0			0
$379$	25.6358			1.31682			0.658411	−	0.752659i	$-0.271229\pi$
$379$	25.6358			1.31682			0.658411	+	0.752659i	$0.271229\pi$
$380$		0			0
$381$	−50.7517			−2.60009
$382$		0			0
$383$		−	17.7477i		−	0.906865i	−0.891291	−	0.453433i	$-0.850199\pi$
$383$		−	17.7477i		−	0.906865i	0.891291	−	0.453433i	$-0.149801\pi$
$384$		0			0
$385$	−19.7670	−	42.3415i	−1.00742	−	2.15792i
$386$		0			0
$387$			12.2434i			0.622369i
$388$		0			0
$389$	−20.4043			−1.03454			−0.517269	−	0.855823i	$-0.673051\pi$
$389$	−20.4043			−1.03454			−0.517269	+	0.855823i	$0.673051\pi$
$390$		0			0
$391$	3.33965			0.168893
$392$		0			0
$393$			49.7463i			2.50937i
$394$		0			0
$395$	19.9676	−	9.32183i	1.00468	−	0.469032i
$396$		0			0
$397$			22.2458i			1.11648i	0.829678	+	0.558242i	$0.188524\pi$
$397$			22.2458i			1.11648i	−0.829678	+	0.558242i	$0.811476\pi$
$398$		0			0
$399$	64.7789			3.24300
$400$		0			0
$401$	−8.44952			−0.421949			−0.210975	−	0.977492i	$-0.567664\pi$
$401$	−8.44952			−0.421949			−0.210975	+	0.977492i	$0.567664\pi$
$402$		0			0
$403$		−	40.5345i		−	2.01917i
$404$		0			0
$405$	45.3644	−	21.1783i	2.25418	−	1.05236i
$406$		0			0
$407$		−	4.74166i		−	0.235035i
$408$		0			0
$409$	−3.18182			−0.157331			−0.0786654	−	0.996901i	$-0.525066\pi$
$409$	−3.18182			−0.157331			−0.0786654	+	0.996901i	$0.525066\pi$
$410$		0			0
$411$	15.8622			0.782425
$412$		0			0
$413$			8.36450i			0.411590i
$414$		0			0
$415$	10.2039	+	21.8570i	0.500890	+	1.07292i
$416$		0			0
$417$		−	74.0472i		−	3.62611i
$418$		0			0
$419$	−33.7228			−1.64747			−0.823733	−	0.566977i	$-0.808113\pi$
$419$	−33.7228			−1.64747			−0.823733	+	0.566977i	$0.808113\pi$
$420$		0			0
$421$	−23.6641			−1.15332			−0.576659	−	0.816985i	$-0.695644\pi$
$421$	−23.6641			−1.15332			−0.576659	+	0.816985i	$0.695644\pi$
$422$		0			0
$423$		−	45.2726i		−	2.20123i
$424$		0			0
$425$	12.8011	+	10.7221i	0.620945	+	0.520096i
$426$		0			0
$427$		−	2.38403i		−	0.115371i
$428$		0			0
$429$	76.4006			3.68866
$430$		0			0
$431$	36.4849			1.75742			0.878708	−	0.477359i	$-0.158406\pi$
$431$	36.4849			1.75742			0.878708	+	0.477359i	$0.158406\pi$
$432$		0			0
$433$			28.1736i			1.35393i	0.736013	+	0.676967i	$0.236706\pi$
$433$			28.1736i			1.35393i	−0.736013	+	0.676967i	$0.763294\pi$
$434$		0			0
$435$	−12.5385	−	26.8577i	−0.601174	−	1.28773i
$436$		0			0
$437$		−	4.43893i		−	0.212343i
$438$		0			0
$439$	26.6445			1.27167			0.635837	−	0.771824i	$-0.280655\pi$
$439$	26.6445			1.27167			0.635837	+	0.771824i	$0.280655\pi$
$440$		0			0
$441$	99.8370			4.75414
$442$		0			0
$443$		−	5.98936i		−	0.284563i	−0.989826	−	0.142281i	$-0.954556\pi$
$443$		−	5.98936i		−	0.284563i	0.989826	−	0.142281i	$-0.0454438\pi$
$444$		0			0
$445$	−10.4143	+	4.86188i	−0.493684	+	0.230475i
$446$		0			0
$447$			47.2144i			2.23316i
$448$		0			0
$449$	−35.0989			−1.65642			−0.828209	−	0.560420i	$-0.810640\pi$
$449$	−35.0989			−1.65642			−0.828209	+	0.560420i	$0.810640\pi$
$450$		0			0
$451$	−26.4615			−1.24602
$452$		0			0
$453$		−	37.0555i		−	1.74102i
$454$		0			0
$455$	−47.7232	+	22.2795i	−2.23730	+	1.04448i
$456$		0			0
$457$			5.90460i			0.276205i	0.990418	+	0.138103i	$0.0441004\pi$
$457$			5.90460i			0.276205i	−0.990418	+	0.138103i	$0.955900\pi$
$458$		0			0
$459$	46.0875			2.15118
$460$		0			0
$461$	−6.25369			−0.291263			−0.145632	−	0.989339i	$-0.546521\pi$
$461$	−6.25369			−0.291263			−0.145632	+	0.989339i	$0.546521\pi$
$462$		0			0
$463$		−	31.7798i		−	1.47693i	−0.674291	−	0.738466i	$-0.735550\pi$
$463$		−	31.7798i		−	1.47693i	0.674291	−	0.738466i	$-0.264450\pi$
$464$		0			0
$465$	23.7557	+	50.8853i	1.10164	+	2.35975i
$466$		0			0
$467$		−	9.63128i		−	0.445682i	−0.974855	−	0.222841i	$-0.928467\pi$
$467$		−	9.63128i		−	0.445682i	0.974855	−	0.222841i	$-0.0715331\pi$
$468$		0			0
$469$	−11.6169			−0.536417
$470$		0			0
$471$	−22.5514			−1.03911
$472$		0			0
$473$		−	7.70799i		−	0.354414i
$474$		0			0
$475$	14.2514	−	17.0148i	0.653897	−	0.780691i
$476$		0			0
$477$		−	53.6616i		−	2.45699i
$478$		0			0
$479$	−0.656246			−0.0299846			−0.0149923	−	0.999888i	$-0.504772\pi$
$479$	−0.656246			−0.0299846			−0.0149923	+	0.999888i	$0.504772\pi$
$480$		0			0
$481$	−5.34433			−0.243680
$482$		0			0
$483$		−	14.5934i		−	0.664020i
$484$		0			0
$485$	−3.18534	−	6.82307i	−0.144639	−	0.309820i
$486$		0			0
$487$		−	33.5157i		−	1.51874i	−0.650658	−	0.759371i	$-0.725507\pi$
$487$		−	33.5157i		−	1.51874i	0.650658	−	0.759371i	$-0.274493\pi$
$488$		0			0
$489$	43.0705			1.94772
$490$		0			0
$491$	14.7081			0.663768			0.331884	−	0.943320i	$-0.392316\pi$
$491$	14.7081			0.663768			0.331884	+	0.943320i	$0.392316\pi$
$492$		0			0
$493$		−	13.7937i		−	0.621238i
$494$		0			0
$495$	−67.9750	+	31.7340i	−3.05525	+	1.42634i
$496$		0			0
$497$			29.9548i			1.34366i
$498$		0			0
$499$	−11.8889			−0.532221			−0.266110	−	0.963943i	$-0.585739\pi$
$499$	−11.8889			−0.532221			−0.266110	+	0.963943i	$0.585739\pi$
$500$		0			0
$501$	−28.1475			−1.25754
$502$		0			0
$503$			27.2549i			1.21524i	0.794230	+	0.607618i	$0.207875\pi$
$503$			27.2549i			1.21524i	−0.794230	+	0.607618i	$0.792125\pi$
$504$		0			0
$505$	−8.47174	+	3.95502i	−0.376987	+	0.175996i
$506$		0			0
$507$		−	44.3897i		−	1.97142i
$508$		0			0
$509$	9.46671			0.419605			0.209802	−	0.977744i	$-0.432718\pi$
$509$	9.46671			0.419605			0.209802	+	0.977744i	$0.432718\pi$
$510$		0			0
$511$	10.9849			0.485944
$512$		0			0
$513$		−	61.2577i		−	2.70459i
$514$		0			0
$515$	−9.38450	−	20.1018i	−0.413530	−	0.885792i
$516$		0			0
$517$			28.5019i			1.25351i
$518$		0			0
$519$	65.9251			2.89379
$520$		0			0
$521$	4.85164			0.212554			0.106277	−	0.994337i	$-0.466107\pi$
$521$	4.85164			0.212554			0.106277	+	0.994337i	$0.466107\pi$
$522$		0			0
$523$			39.1914i			1.71372i	0.515549	+	0.856860i	$0.327588\pi$
$523$			39.1914i			1.71372i	−0.515549	+	0.856860i	$0.672412\pi$
$524$		0			0
$525$	46.8525	−	55.9374i	2.04481	−	2.44131i
$526$		0			0
$527$			26.1339i			1.13841i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	13.4284			0.582741
$532$		0			0
$533$			29.8248i			1.29186i
$534$		0			0
$535$	11.0509	+	23.6712i	0.477770	+	1.02340i
$536$		0			0
$537$			49.7065i			2.14499i
$538$		0			0
$539$	−62.8535			−2.70729
$540$		0			0
$541$	−35.9793			−1.54687			−0.773436	−	0.633875i	$-0.781464\pi$
$541$	−35.9793			−1.54687			−0.773436	+	0.633875i	$0.781464\pi$
$542$		0			0
$543$		−	10.6514i		−	0.457097i
$544$		0			0
$545$	11.3249	−	5.28700i	0.485105	−	0.226470i
$546$		0			0
$547$			11.0470i			0.472335i	0.971712	+	0.236167i	$0.0758914\pi$
$547$			11.0470i			0.472335i	−0.971712	+	0.236167i	$0.924109\pi$
$548$		0			0
$549$	−3.82732			−0.163346
$550$		0			0
$551$	−18.3341			−0.781059
$552$		0			0
$553$		−	44.8118i		−	1.90559i
$554$		0			0
$555$	6.70904	−	3.13210i	0.284783	−	0.132950i
$556$		0			0
$557$			2.20298i			0.0933433i	0.998910	+	0.0466716i	$0.0148614\pi$
$557$			2.20298i			0.0933433i	−0.998910	+	0.0466716i	$0.985139\pi$
$558$		0			0
$559$	−8.68769			−0.367450
$560$		0			0
$561$	−49.2580			−2.07967
$562$		0			0
$563$			44.9181i			1.89307i	0.322601	+	0.946535i	$0.395443\pi$
$563$			44.9181i			1.89307i	−0.322601	+	0.946535i	$0.604557\pi$
$564$		0			0
$565$	3.80817	+	8.15720i	0.160211	+	0.343176i
$566$		0			0
$567$		−	101.808i		−	4.27554i
$568$		0			0
$569$	37.4663			1.57067			0.785336	−	0.619070i	$-0.212490\pi$
$569$	37.4663			1.57067			0.785336	+	0.619070i	$0.212490\pi$
$570$		0			0
$571$	−23.8787			−0.999291			−0.499645	−	0.866230i	$-0.666536\pi$
$571$	−23.8787			−0.999291			−0.499645	+	0.866230i	$0.666536\pi$
$572$		0			0
$573$		−	17.5399i		−	0.732741i
$574$		0			0
$575$	−3.83307	−	3.21054i	−0.159850	−	0.133889i
$576$		0			0
$577$			32.6048i			1.35735i	0.734437	+	0.678677i	$0.237447\pi$
$577$			32.6048i			1.35735i	−0.734437	+	0.678677i	$0.762553\pi$
$578$		0			0
$579$	−6.04143			−0.251073
$580$		0			0
$581$	49.0521			2.03502
$582$		0			0
$583$			33.7832i			1.39916i
$584$		0			0
$585$	35.7675	+	76.6148i	1.47880	+	3.16763i
$586$		0			0
$587$			40.1563i			1.65743i	0.559673	+	0.828714i	$0.310927\pi$
$587$			40.1563i			1.65743i	−0.559673	+	0.828714i	$0.689073\pi$
$588$		0			0
$589$	34.7362			1.43128
$590$		0			0
$591$	−23.3411			−0.960126
$592$		0			0
$593$			4.08558i			0.167775i	0.996475	+	0.0838873i	$0.0267336\pi$
$593$			4.08558i			0.167775i	−0.996475	+	0.0838873i	$0.973266\pi$
$594$		0			0
$595$	30.7687	−	14.3643i	1.26139	−	0.588879i
$596$		0			0
$597$			25.1708i			1.03017i
$598$		0			0
$599$	2.18194			0.0891516			0.0445758	−	0.999006i	$-0.485806\pi$
$599$	2.18194			0.0891516			0.0445758	+	0.999006i	$0.485806\pi$
$600$		0			0
$601$	−17.9240			−0.731136			−0.365568	−	0.930785i	$-0.619125\pi$
$601$	−17.9240			−0.731136			−0.365568	+	0.930785i	$0.619125\pi$
$602$		0			0
$603$			18.6497i			0.759475i
$604$		0			0
$605$	20.5068	−	9.57355i	0.833719	−	0.389220i
$606$		0			0
$607$			17.4158i			0.706886i	0.935456	+	0.353443i	$0.114989\pi$
$607$			17.4158i			0.706886i	−0.935456	+	0.353443i	$0.885011\pi$
$608$		0			0
$609$	−60.2748			−2.44246
$610$		0			0
$611$	32.1245			1.29962
$612$		0			0
$613$		−	5.74163i		−	0.231902i	−0.993255	−	0.115951i	$-0.963008\pi$
$613$		−	5.74163i		−	0.231902i	0.993255	−	0.115951i	$-0.0369916\pi$
$614$		0			0
$615$	−17.4792	−	37.4408i	−0.704829	−	1.50976i
$616$		0			0
$617$			5.74232i			0.231177i	0.993297	+	0.115589i	$0.0368754\pi$
$617$			5.74232i			0.231177i	−0.993297	+	0.115589i	$0.963125\pi$
$618$		0			0
$619$	−9.23964			−0.371373			−0.185686	−	0.982609i	$-0.559451\pi$
$619$	−9.23964			−0.371373			−0.185686	+	0.982609i	$0.559451\pi$
$620$		0			0
$621$	−13.8001			−0.553779
$622$		0			0
$623$			23.3720i			0.936378i
$624$		0			0
$625$	−4.38491	−	24.6124i	−0.175396	−	0.984498i
$626$		0			0
$627$			65.4718i			2.61469i
$628$		0			0
$629$	3.44566			0.137388
$630$		0			0
$631$	40.0251			1.59338			0.796688	−	0.604391i	$-0.206584\pi$
$631$	40.0251			1.59338			0.796688	+	0.604391i	$0.206584\pi$
$632$		0			0
$633$			19.3807i			0.770314i
$634$		0			0
$635$	14.9582	+	32.0408i	0.593597	+	1.27150i
$636$		0			0
$637$			70.8423i			2.80687i
$638$		0			0
$639$	48.0895			1.90239
$640$		0			0
$641$	32.5523			1.28574			0.642869	−	0.765977i	$-0.277744\pi$
$641$	32.5523			1.28574			0.642869	+	0.765977i	$0.277744\pi$
$642$		0			0
$643$			20.0972i			0.792555i	0.918131	+	0.396277i	$0.129698\pi$
$643$			20.0972i			0.792555i	−0.918131	+	0.396277i	$0.870302\pi$
$644$		0			0
$645$	10.9062	−	5.09152i	0.429429	−	0.200478i
$646$		0			0
$647$		−	0.167428i		−	0.00658228i	−0.999995	−	0.00329114i	$-0.998952\pi$
$647$		−	0.167428i		−	0.00658228i	0.999995	−	0.00329114i	$-0.00104760\pi$
$648$		0			0
$649$	−8.45397			−0.331847
$650$		0			0
$651$	114.198			4.47577
$652$		0			0
$653$		−	37.2262i		−	1.45677i	−0.685166	−	0.728387i	$-0.740270\pi$
$653$		−	37.2262i		−	1.45677i	0.685166	−	0.728387i	$-0.259730\pi$
$654$		0			0
$655$	31.4061	−	14.6619i	1.22714	−	0.572887i
$656$		0			0
$657$		−	17.6352i		−	0.688014i
$658$		0			0
$659$	−5.23004			−0.203734			−0.101867	−	0.994798i	$-0.532482\pi$
$659$	−5.23004			−0.203734			−0.101867	+	0.994798i	$0.532482\pi$
$660$		0			0
$661$	31.8801			1.23999			0.619996	−	0.784605i	$-0.287134\pi$
$661$	31.8801			1.23999			0.619996	+	0.784605i	$0.287134\pi$
$662$		0			0
$663$			55.5187i			2.15617i
$664$		0			0
$665$	−19.0925	−	40.8966i	−0.740375	−	1.58590i
$666$		0			0
$667$			4.13029i			0.159926i
$668$		0			0
$669$	−44.2670			−1.71146
$670$		0			0
$671$	2.40953			0.0930189
$672$		0			0
$673$		−	22.5996i		−	0.871151i	−0.900152	−	0.435576i	$-0.856545\pi$
$673$		−	22.5996i		−	0.871151i	0.900152	−	0.435576i	$-0.143455\pi$
$674$		0			0
$675$	−52.8968	−	44.3057i	−2.03600	−	1.70533i
$676$		0			0
$677$			11.6369i			0.447242i	0.974676	+	0.223621i	$0.0717878\pi$
$677$			11.6369i			0.447242i	−0.974676	+	0.223621i	$0.928212\pi$
$678$		0			0
$679$	−15.3125			−0.587640
$680$		0			0
$681$	−58.6245			−2.24650
$682$		0			0
$683$		−	24.6575i		−	0.943492i	−0.881735	−	0.471746i	$-0.843624\pi$
$683$		−	24.6575i		−	0.943492i	0.881735	−	0.471746i	$-0.156376\pi$
$684$		0			0
$685$	−4.67511	−	10.0142i	−0.178627	−	0.382623i
$686$		0			0
$687$		−	50.0612i		−	1.90995i
$688$		0			0
$689$	38.0771			1.45062
$690$		0			0
$691$	−49.7207			−1.89146			−0.945732	−	0.324949i	$-0.894653\pi$
$691$	−49.7207			−1.89146			−0.945732	+	0.324949i	$0.894653\pi$
$692$		0			0
$693$			152.551i			5.79494i
$694$		0			0
$695$	−46.7478	+	21.8241i	−1.77325	+	0.827837i
$696$		0			0
$697$		−	19.2290i		−	0.728352i
$698$		0			0
$699$	−86.4670			−3.27048
$700$		0			0
$701$	−36.2499			−1.36914			−0.684570	−	0.728947i	$-0.740010\pi$
$701$	−36.2499			−1.36914			−0.684570	+	0.728947i	$0.740010\pi$
$702$		0			0
$703$		−	4.57984i		−	0.172732i
$704$		0			0
$705$	−40.3277	+	18.8269i	−1.51883	+	0.709063i
$706$		0			0
$707$			19.0125i			0.715038i
$708$		0			0
$709$	−42.1097			−1.58146			−0.790732	−	0.612162i	$-0.790300\pi$
$709$	−42.1097			−1.58146			−0.790732	+	0.612162i	$0.790300\pi$
$710$		0			0
$711$	−71.9408			−2.69799
$712$		0			0
$713$		−	7.82535i		−	0.293062i
$714$		0			0
$715$	−22.5178	−	48.2336i	−0.842117	−	1.80384i
$716$		0			0
$717$			15.7617i			0.588630i
$718$		0			0
$719$	−27.9125			−1.04096			−0.520481	−	0.853873i	$-0.674247\pi$
$719$	−27.9125			−1.04096			−0.520481	+	0.853873i	$0.674247\pi$
$720$		0			0
$721$	−45.1130			−1.68010
$722$		0			0
$723$			65.1819i			2.42414i
$724$		0			0
$725$	−13.2605	+	15.8317i	−0.492481	+	0.587975i
$726$		0			0
$727$		−	3.14225i		−	0.116539i	−0.998301	−	0.0582697i	$-0.981442\pi$
$727$		−	3.14225i		−	0.116539i	0.998301	−	0.0582697i	$-0.0185583\pi$
$728$		0			0
$729$	−30.5744			−1.13239
$730$		0			0
$731$	5.60123			0.207169
$732$		0			0
$733$			43.2304i			1.59675i	0.602161	+	0.798375i	$0.294307\pi$
$733$			43.2304i			1.59675i	−0.602161	+	0.798375i	$0.705693\pi$
$734$		0			0
$735$	−41.5179	−	88.9323i	−1.53141	−	3.28032i
$736$		0			0
$737$		−	11.7411i		−	0.432490i
$738$		0			0
$739$	−21.2903			−0.783177			−0.391589	−	0.920140i	$-0.628074\pi$
$739$	−21.2903			−0.783177			−0.391589	+	0.920140i	$0.628074\pi$
$740$		0			0
$741$	73.7934			2.71087
$742$		0			0
$743$		−	23.6168i		−	0.866417i	−0.901294	−	0.433208i	$-0.857381\pi$
$743$		−	23.6168i		−	0.866417i	0.901294	−	0.433208i	$-0.142619\pi$
$744$		0			0
$745$	29.8076	−	13.9156i	1.09207	−	0.509830i
$746$		0			0
$747$		−	78.7482i		−	2.88125i
$748$		0			0
$749$	53.1235			1.94109
$750$		0			0
$751$	−24.1779			−0.882263			−0.441131	−	0.897443i	$-0.645423\pi$
$751$	−24.1779			−0.882263			−0.441131	+	0.897443i	$0.645423\pi$
$752$		0			0
$753$		−	48.5838i		−	1.77049i
$754$		0			0
$755$	−23.3940	+	10.9215i	−0.851396	+	0.397473i
$756$		0			0
$757$			17.9076i			0.650864i	0.945565	+	0.325432i	$0.105510\pi$
$757$			17.9076i			0.650864i	−0.945565	+	0.325432i	$0.894490\pi$
$758$		0			0
$759$	14.7494			0.535371
$760$		0			0
$761$	13.7932			0.500002			0.250001	−	0.968246i	$-0.419569\pi$
$761$	13.7932			0.500002			0.250001	+	0.968246i	$0.419569\pi$
$762$		0			0
$763$		−	25.4156i		−	0.920106i
$764$		0			0
$765$	−23.0604	−	49.3960i	−0.833752	−	1.78592i
$766$		0			0
$767$			9.52849i			0.344054i
$768$		0			0
$769$	−8.37214			−0.301907			−0.150954	−	0.988541i	$-0.548234\pi$
$769$	−8.37214			−0.301907			−0.150954	+	0.988541i	$0.548234\pi$
$770$		0			0
$771$	69.7695			2.51269
$772$		0			0
$773$			3.64496i			0.131100i	0.997849	+	0.0655501i	$0.0208802\pi$
$773$			3.64496i			0.131100i	−0.997849	+	0.0655501i	$0.979120\pi$
$774$		0			0
$775$	25.1236	−	29.9951i	0.902465	−	1.07746i
$776$		0			0
$777$		−	15.0566i		−	0.540153i
$778$		0			0
$779$	−25.5585			−0.915729
$780$		0			0
$781$	−30.2753			−1.08333
$782$		0			0
$783$			56.9984i			2.03696i
$784$		0			0
$785$	6.64664	+	14.2373i	0.237229	+	0.508150i
$786$		0			0
$787$		−	3.54385i		−	0.126325i	−0.998003	−	0.0631624i	$-0.979881\pi$
$787$		−	3.54385i		−	0.126325i	0.998003	−	0.0631624i	$-0.0201186\pi$
$788$		0			0
$789$	−66.0999			−2.35322
$790$		0			0
$791$	18.3066			0.650908
$792$		0			0
$793$		−	2.71579i		−	0.0964404i
$794$		0			0
$795$	−47.8004	+	22.3155i	−1.69531	+	0.791450i
$796$		0			0
$797$		−	9.76387i		−	0.345854i	−0.984935	−	0.172927i	$-0.944677\pi$
$797$		−	9.76387i		−	0.345854i	0.984935	−	0.172927i	$-0.0553225\pi$
$798$		0			0
$799$	−20.7117			−0.732728
$800$		0			0
$801$	37.5213			1.32575
$802$		0			0
$803$			11.1024i			0.391796i
$804$		0			0
$805$	−9.21315	+	4.30114i	−0.324721	+	0.151595i
$806$		0			0
$807$		−	50.9679i		−	1.79415i
$808$		0			0
$809$	−6.55235			−0.230368			−0.115184	−	0.993344i	$-0.536746\pi$
$809$	−6.55235			−0.230368			−0.115184	+	0.993344i	$0.536746\pi$
$810$		0			0
$811$	48.5149			1.70359			0.851795	−	0.523875i	$-0.175514\pi$
$811$	48.5149			1.70359			0.851795	+	0.523875i	$0.175514\pi$
$812$		0			0
$813$		−	11.3942i		−	0.399611i
$814$		0			0
$815$	−12.6943	−	27.1915i	−0.444662	−	0.952477i
$816$		0			0
$817$		−	7.44495i		−	0.260466i
$818$		0			0
$819$	171.941			6.00810
$820$		0			0
$821$	−32.5956			−1.13759			−0.568797	−	0.822478i	$-0.692591\pi$
$821$	−32.5956			−1.13759			−0.568797	+	0.822478i	$0.692591\pi$
$822$		0			0
$823$			7.63895i			0.266277i	0.991097	+	0.133139i	$0.0425055\pi$
$823$			7.63895i			0.266277i	−0.991097	+	0.133139i	$0.957494\pi$
$824$		0			0
$825$	56.5357	+	47.3536i	1.96832	+	1.64864i
$826$		0			0
$827$			2.35631i			0.0819370i	0.999160	+	0.0409685i	$0.0130443\pi$
$827$			2.35631i			0.0819370i	−0.999160	+	0.0409685i	$0.986956\pi$
$828$		0			0
$829$	36.1206			1.25452			0.627261	−	0.778809i	$-0.284176\pi$
$829$	36.1206			1.25452			0.627261	+	0.778809i	$0.284176\pi$
$830$		0			0
$831$	−74.8209			−2.59551
$832$		0			0
$833$		−	45.6743i		−	1.58252i
$834$		0			0
$835$	8.29600	+	17.7702i	0.287095	+	0.614964i
$836$		0			0
$837$		−	107.991i		−	3.73270i
$838$		0			0
$839$	0.821644			0.0283663			0.0141832	−	0.999899i	$-0.495485\pi$
$839$	0.821644			0.0283663			0.0141832	+	0.999899i	$0.495485\pi$
$840$		0			0
$841$	−11.9407			−0.411748
$842$		0			0
$843$		−	28.6692i		−	0.987420i
$844$		0			0
$845$	−28.0243	+	13.0831i	−0.964066	+	0.450072i
$846$		0			0
$847$		−	46.0218i		−	1.58133i
$848$		0			0
$849$	47.0166			1.61360
$850$		0			0
$851$	−1.03174			−0.0353677
$852$		0			0
$853$		−	4.87966i		−	0.167076i	−0.996505	−	0.0835382i	$-0.973378\pi$
$853$		−	4.87966i		−	0.167076i	0.996505	−	0.0835382i	$-0.0266220\pi$
$854$		0			0
$855$	−65.6553	+	30.6511i	−2.24536	+	1.04824i
$856$		0			0
$857$		−	10.5428i		−	0.360136i	−0.983654	−	0.180068i	$-0.942368\pi$
$857$		−	10.5428i		−	0.360136i	0.983654	−	0.180068i	$-0.0576317\pi$
$858$		0			0
$859$	46.8583			1.59878			0.799392	−	0.600810i	$-0.205155\pi$
$859$	46.8583			1.59878			0.799392	+	0.600810i	$0.205155\pi$
$860$		0			0
$861$	−84.0257			−2.86359
$862$		0			0
$863$		−	31.1007i		−	1.05868i	−0.848410	−	0.529340i	$-0.822440\pi$
$863$		−	31.1007i		−	1.05868i	0.848410	−	0.529340i	$-0.177560\pi$
$864$		0			0
$865$	−19.4303	−	41.6202i	−0.660650	−	1.41513i
$866$		0			0
$867$			18.7643i			0.637269i
$868$		0			0
$869$	45.2911			1.53639
$870$		0			0
$871$	−13.2334			−0.448398
$872$		0			0
$873$			24.5827i			0.831998i
$874$		0			0
$875$	−49.1236	−	13.0926i	−1.66068	−	0.442609i
$876$		0			0
$877$			37.8579i			1.27837i	0.769053	+	0.639185i	$0.220728\pi$
$877$			37.8579i			1.27837i	−0.769053	+	0.639185i	$0.779272\pi$
$878$		0			0
$879$	88.1823			2.97431
$880$		0			0
$881$	−15.8882			−0.535286			−0.267643	−	0.963518i	$-0.586245\pi$
$881$	−15.8882			−0.535286			−0.267643	+	0.963518i	$0.586245\pi$
$882$		0			0
$883$		−	31.7598i		−	1.06880i	−0.845231	−	0.534401i	$-0.820537\pi$
$883$		−	31.7598i		−	1.06880i	0.845231	−	0.534401i	$-0.179463\pi$
$884$		0			0
$885$	−5.58428	−	11.9617i	−0.187713	−	0.402087i
$886$		0			0
$887$		−	6.46849i		−	0.217191i	−0.994086	−	0.108595i	$-0.965365\pi$
$887$		−	6.46849i		−	0.217191i	0.994086	−	0.108595i	$-0.0346353\pi$
$888$		0			0
$889$	71.9068			2.41168
$890$		0			0
$891$	102.897			3.44718
$892$		0			0
$893$			27.5292i			0.921231i
$894$		0			0
$895$	31.3809	−	14.6501i	1.04895	−	0.489700i
$896$		0			0
$897$		−	16.6241i		−	0.555064i
$898$		0			0
$899$	−32.3210			−1.07796
$900$		0			0
$901$	−24.5496			−0.817864
$902$		0			0
$903$		−	24.4759i		−	0.814506i
$904$		0			0
$905$	−6.72452	+	3.13933i	−0.223531	+	0.104355i
$906$		0			0
$907$		−	35.2611i		−	1.17082i	−0.810736	−	0.585412i	$-0.800933\pi$
$907$		−	35.2611i		−	1.17082i	0.810736	−	0.585412i	$-0.199067\pi$
$908$		0			0
$909$	30.5226			1.01237
$910$		0			0
$911$	16.2888			0.539673			0.269837	−	0.962906i	$-0.413030\pi$
$911$	16.2888			0.539673			0.269837	+	0.962906i	$0.413030\pi$
$912$		0			0
$913$			49.5768i			1.64075i
$914$		0			0
$915$	1.59162	+	3.40928i	0.0526172	+	0.112707i
$916$		0			0
$917$		−	70.4824i		−	2.32753i
$918$		0			0
$919$	42.9436			1.41658			0.708289	−	0.705923i	$-0.249467\pi$
$919$	42.9436			1.41658			0.708289	+	0.705923i	$0.249467\pi$
$920$		0			0
$921$	−61.4172			−2.02376
$922$		0			0
$923$			34.1233i			1.12318i
$924$		0			0
$925$	−3.95475	−	3.31245i	−0.130031	−	0.108913i
$926$		0			0
$927$			72.4244i			2.37873i
$928$		0			0
$929$	3.19236			0.104738			0.0523690	−	0.998628i	$-0.483323\pi$
$929$	3.19236			0.104738			0.0523690	+	0.998628i	$0.483323\pi$
$930$		0			0
$931$	−60.7086			−1.98964
$932$		0			0
$933$			49.2441i			1.61218i
$934$		0			0
$935$	14.5179	+	31.0978i	0.474788	+	1.01701i
$936$		0			0
$937$			5.68073i			0.185581i	0.995686	+	0.0927907i	$0.0295787\pi$
$937$			5.68073i			0.185581i	−0.995686	+	0.0927907i	$0.970421\pi$
$938$		0			0
$939$	26.9988			0.881074
$940$		0			0
$941$	54.0630			1.76240			0.881202	−	0.472739i	$-0.156735\pi$
$941$	54.0630			1.76240			0.881202	+	0.472739i	$0.156735\pi$
$942$		0			0
$943$			5.75780i			0.187500i
$944$		0			0
$945$	−127.142	+	59.3561i	−4.13594	+	1.93086i
$946$		0			0
$947$		−	19.3780i		−	0.629700i	−0.949141	−	0.314850i	$-0.898046\pi$
$947$		−	19.3780i		−	0.629700i	0.949141	−	0.314850i	$-0.101954\pi$
$948$		0			0
$949$	12.5136			0.406207
$950$		0			0
$951$	52.9978			1.71857
$952$		0			0
$953$		−	10.7090i		−	0.346898i	−0.984843	−	0.173449i	$-0.944509\pi$
$953$		−	10.7090i		−	0.346898i	0.984843	−	0.173449i	$-0.0554912\pi$
$954$		0			0
$955$	−11.0734	+	5.16959i	−0.358327	+	0.167284i
$956$		0			0
$957$		−	60.9195i		−	1.96925i
$958$		0			0
$959$	−22.4741			−0.725728
$960$		0			0
$961$	30.2360			0.975356
$962$		0			0
$963$		−	85.2845i		−	2.74825i
$964$		0			0
$965$	1.78061	+	3.81410i	0.0573198	+	0.122780i
$966$		0			0
$967$			32.6034i			1.04845i	0.851579	+	0.524227i	$0.175645\pi$
$967$			32.6034i			1.04845i	−0.851579	+	0.524227i	$0.824355\pi$
$968$		0			0
$969$	−47.5770			−1.52839
$970$		0			0
$971$	−42.8194			−1.37414			−0.687070	−	0.726591i	$-0.741104\pi$
$971$	−42.8194			−1.37414			−0.687070	+	0.726591i	$0.741104\pi$
$972$		0			0
$973$			104.913i			3.36335i
$974$		0			0
$975$	53.3724	−	63.7215i	1.70928	−	2.04072i
$976$		0			0
$977$		−	7.76025i		−	0.248272i	−0.992265	−	0.124136i	$-0.960384\pi$
$977$		−	7.76025i		−	0.248272i	0.992265	−	0.124136i	$-0.0396160\pi$
$978$		0			0
$979$	−23.6220			−0.754961
$980$		0			0
$981$	−40.8022			−1.30271
$982$		0			0
$983$			1.15071i			0.0367021i	0.999832	+	0.0183510i	$0.00584165\pi$
$983$			1.15071i			0.0367021i	−0.999832	+	0.0183510i	$0.994158\pi$
$984$		0			0
$985$	6.87940	+	14.7358i	0.219196	+	0.469523i
$986$		0			0
$987$			90.5046i			2.88079i
$988$		0			0
$989$	−1.67719			−0.0533316
$990$		0			0
$991$	31.0124			0.985142			0.492571	−	0.870272i	$-0.336057\pi$
$991$	31.0124			0.985142			0.492571	+	0.870272i	$0.336057\pi$
$992$		0			0
$993$		−	8.94750i		−	0.283940i
$994$		0			0
$995$	15.8910	−	7.41867i	0.503778	−	0.235188i
$996$		0			0
$997$		−	24.0635i		−	0.762097i	−0.924555	−	0.381049i	$-0.875563\pi$
$997$		−	24.0635i		−	0.762097i	0.924555	−	0.381049i	$-0.124437\pi$
$998$		0			0
$999$	−14.2382			−0.450476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.c.369.16	yes	16
4.3	odd	2		1840.2.e.h.369.1		16
5.2	odd	4		4600.2.a.bk.1.8		8
5.3	odd	4		4600.2.a.bj.1.1		8
5.4	even	2	inner	920.2.e.c.369.1	✓	16
20.3	even	4		9200.2.a.de.1.8		8
20.7	even	4		9200.2.a.dd.1.1		8
20.19	odd	2		1840.2.e.h.369.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.1	✓	16	5.4	even	2	inner
920.2.e.c.369.16	yes	16	1.1	even	1	trivial
1840.2.e.h.369.1		16	4.3	odd	2
1840.2.e.h.369.16		16	20.19	odd	2
4600.2.a.bj.1.1		8	5.3	odd	4
4600.2.a.bk.1.8		8	5.2	odd	4
9200.2.a.dd.1.1		8	20.7	even	4
9200.2.a.de.1.8		8	20.3	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.20935i q^{3} +(2.02615 - 0.945903i) q^{5} -4.54713i q^{7} -7.29996 q^{9} +O(q^{10})\)	\(q+3.20935i q^{3} +(2.02615 - 0.945903i) q^{5} -4.54713i q^{7} -7.29996 q^{9} +4.59577 q^{11} -5.17990i q^{13} +(3.03574 + 6.50262i) q^{15} +3.33965i q^{17} +4.43893 q^{19} +14.5934 q^{21} -1.00000i q^{23} +(3.21054 - 3.83307i) q^{25} -13.8001i q^{27} -4.13029 q^{29} +7.82535 q^{31} +14.7494i q^{33} +(-4.30114 - 9.21315i) q^{35} -1.03174i q^{37} +16.6241 q^{39} -5.75780 q^{41} -1.67719i q^{43} +(-14.7908 + 6.90505i) q^{45} +6.20177i q^{47} -13.6764 q^{49} -10.7181 q^{51} +7.35094i q^{53} +(9.31170 - 4.34715i) q^{55} +14.2461i q^{57} -1.83951 q^{59} +0.524294 q^{61} +33.1939i q^{63} +(-4.89968 - 10.4952i) q^{65} -2.55477i q^{67} +3.20935 q^{69} -6.58764 q^{71} +2.41579i q^{73} +(12.3017 + 10.3038i) q^{75} -20.8975i q^{77} +9.85496 q^{79} +22.3895 q^{81} +10.7875i q^{83} +(3.15898 + 6.76662i) q^{85} -13.2556i q^{87} -5.13994 q^{89} -23.5537 q^{91} +25.1143i q^{93} +(8.99393 - 4.19880i) q^{95} -3.36751i q^{97} -33.5489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 22 * q^9	\( 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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