LMFDB - Embedded newform 475.2.b.b.324.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.4
Root			$2.65109i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.b.324.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27389i q^{2} +1.65109i q^{3} +0.377203 q^{4} -2.10331 q^{6} +3.65109i q^{7} +3.02830i q^{8} +0.273891 q^{9} +O(q^{10})$	\(q+1.27389i q^{2} +1.65109i q^{3} +0.377203 q^{4} -2.10331 q^{6} +3.65109i q^{7} +3.02830i q^{8} +0.273891 q^{9} +2.65109 q^{11} +0.622797i q^{12} -6.13161i q^{13} -4.65109 q^{14} -3.10331 q^{16} +2.34891i q^{17} +0.348907i q^{18} -1.00000 q^{19} -6.02830 q^{21} +3.37720i q^{22} -5.48052i q^{23} -5.00000 q^{24} +7.81100 q^{26} +5.40550i q^{27} +1.37720i q^{28} -0.651093 q^{29} -6.67939 q^{31} +2.10331i q^{32} +4.37720i q^{33} -2.99225 q^{34} +0.103312 q^{36} -8.70769i q^{37} -1.27389i q^{38} +10.1239 q^{39} +1.93273 q^{41} -7.67939i q^{42} +2.65884i q^{43} +1.00000 q^{44} +6.98158 q^{46} +3.71836i q^{47} -5.12386i q^{48} -6.33048 q^{49} -3.87826 q^{51} -2.31286i q^{52} -13.7544i q^{53} -6.88601 q^{54} -11.0566 q^{56} -1.65109i q^{57} -0.829422i q^{58} +7.84997 q^{59} -1.92498 q^{61} -8.50881i q^{62} +1.00000i q^{63} -8.88601 q^{64} -5.57608 q^{66} -4.44447i q^{67} +0.886014i q^{68} +9.04884 q^{69} +3.54778 q^{71} +0.829422i q^{72} +2.48052i q^{73} +11.0926 q^{74} -0.377203 q^{76} +9.67939i q^{77} +12.8967i q^{78} +15.1599 q^{79} -8.10331 q^{81} +2.46209i q^{82} +14.7282i q^{83} -2.27389 q^{84} -3.38708 q^{86} -1.07502i q^{87} +8.02830i q^{88} +5.06727 q^{89} +22.3871 q^{91} -2.06727i q^{92} -11.0283i q^{93} -4.73678 q^{94} -3.47277 q^{96} +3.22717i q^{97} -8.06434i q^{98} +0.726109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) $ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9	$ 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) $ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.27389i	0.900777i	0.892833	+	0.450388i	$0.148714\pi$
$2$	1.27389i	0.900777i	−0.892833	+	0.450388i	$0.851286\pi$
$3$	1.65109i	0.953259i	0.879104	+	0.476630i	$0.158142\pi$
$3$	1.65109i	0.953259i	−0.879104	+	0.476630i	$0.841858\pi$
$4$	0.377203	0.188601
$5$	0	0
$5$	0	0
$6$	−2.10331	−0.858674
$7$	3.65109i	1.37998i	0.723817	+	0.689992i	$0.242386\pi$
$7$	3.65109i	1.37998i	−0.723817	+	0.689992i	$0.757614\pi$
$8$	3.02830i	1.07066i
$9$	0.273891	0.0912969
$10$	0	0
$11$	2.65109	0.799335	0.399667	−	0.916660i	$-0.369126\pi$
$11$	2.65109	0.799335	0.399667	+	0.916660i	$0.369126\pi$
$12$	0.622797i	0.179786i
$13$	− 6.13161i	− 1.70060i	−0.526297	−	0.850301i	$-0.676420\pi$
$13$	− 6.13161i	− 1.70060i	0.526297	−	0.850301i	$-0.323580\pi$
$14$	−4.65109	−1.24306
$15$	0	0
$16$	−3.10331	−0.775828
$17$	2.34891i	0.569694i	0.958573	+	0.284847i	$0.0919427\pi$
$17$	2.34891i	0.569694i	−0.958573	+	0.284847i	$0.908057\pi$
$18$	0.348907i	0.0822381i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−6.02830	−1.31548
$22$	3.37720i	0.720022i
$23$	− 5.48052i	− 1.14277i	−0.820683	−	0.571383i	$-0.806407\pi$
$23$	− 5.48052i	− 1.14277i	0.820683	−	0.571383i	$-0.193593\pi$
$24$	−5.00000	−1.02062
$25$	0	0
$26$	7.81100	1.53186
$27$	5.40550i	1.04029i
$28$	1.37720i	0.260267i
$29$	−0.651093	−0.120905	−0.0604525	−	0.998171i	$-0.519254\pi$
$29$	−0.651093	−0.120905	−0.0604525	+	0.998171i	$0.519254\pi$
$30$	0	0
$31$	−6.67939	−1.19965	−0.599827	−	0.800130i	$-0.704764\pi$
$31$	−6.67939	−1.19965	−0.599827	+	0.800130i	$0.704764\pi$
$32$	2.10331i	0.371817i
$33$	4.37720i	0.761973i
$34$	−2.99225	−0.513167
$35$	0	0
$36$	0.103312	0.0172187
$37$	− 8.70769i	− 1.43153i	−0.698339	−	0.715767i	$-0.746077\pi$
$37$	− 8.70769i	− 1.43153i	0.698339	−	0.715767i	$-0.253923\pi$
$38$	− 1.27389i	− 0.206652i
$39$	10.1239	1.62111
$40$	0	0
$41$	1.93273	0.301842	0.150921	−	0.988546i	$-0.451776\pi$
$41$	1.93273	0.301842	0.150921	+	0.988546i	$0.451776\pi$
$42$	− 7.67939i	− 1.18496i
$43$	2.65884i	0.405470i	0.979234	+	0.202735i	$0.0649830\pi$
$43$	2.65884i	0.405470i	−0.979234	+	0.202735i	$0.935017\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.98158	1.02938
$47$	3.71836i	0.542378i	0.962526	+	0.271189i	$0.0874169\pi$
$47$	3.71836i	0.542378i	−0.962526	+	0.271189i	$0.912583\pi$
$48$	− 5.12386i	− 0.739565i
$49$	−6.33048	−0.904355
$50$	0	0
$51$	−3.87826	−0.543066
$52$	− 2.31286i	− 0.320736i
$53$	− 13.7544i	− 1.88931i	−0.328061	−	0.944656i	$-0.606395\pi$
$53$	− 13.7544i	− 1.88931i	0.328061	−	0.944656i	$-0.393605\pi$
$54$	−6.88601	−0.937068
$55$	0	0
$56$	−11.0566	−1.47750
$57$	− 1.65109i	− 0.218693i
$58$	− 0.829422i	− 0.108908i
$59$	7.84997	1.02198	0.510989	−	0.859587i	$-0.329279\pi$
$59$	7.84997	1.02198	0.510989	+	0.859587i	$0.329279\pi$
$60$	0	0
$61$	−1.92498	−0.246469	−0.123234	−	0.992378i	$-0.539327\pi$
$61$	−1.92498	−0.246469	−0.123234	+	0.992378i	$0.539327\pi$
$62$	− 8.50881i	− 1.08062i
$63$	1.00000i	0.125988i
$64$	−8.88601	−1.11075
$65$	0	0
$66$	−5.57608	−0.686368
$67$	− 4.44447i	− 0.542978i	−0.962442	−	0.271489i	$-0.912484\pi$
$67$	− 4.44447i	− 0.542978i	0.962442	−	0.271489i	$-0.0875161\pi$
$68$	0.886014i	0.107445i
$69$	9.04884	1.08935
$70$	0	0
$71$	3.54778	0.421044	0.210522	−	0.977589i	$-0.432484\pi$
$71$	3.54778	0.421044	0.210522	+	0.977589i	$0.432484\pi$
$72$	0.829422i	0.0977483i
$73$	2.48052i	0.290322i	0.989408	+	0.145161i	$0.0463701\pi$
$73$	2.48052i	0.290322i	−0.989408	+	0.145161i	$0.953630\pi$
$74$	11.0926	1.28949
$75$	0	0
$76$	−0.377203	−0.0432681
$77$	9.67939i	1.10307i
$78$	12.8967i	1.46026i
$79$	15.1599	1.70562	0.852811	−	0.522219i	$-0.174896\pi$
$79$	15.1599	1.70562	0.852811	+	0.522219i	$0.174896\pi$
$80$	0	0
$81$	−8.10331	−0.900368
$82$	2.46209i	0.271893i
$83$	14.7282i	1.61663i	0.588748	+	0.808317i	$0.299621\pi$
$83$	14.7282i	1.61663i	−0.588748	+	0.808317i	$0.700379\pi$
$84$	−2.27389	−0.248102
$85$	0	0
$86$	−3.38708	−0.365238
$87$	− 1.07502i	− 0.115254i
$88$	8.02830i	0.855819i
$89$	5.06727	0.537129	0.268565	−	0.963262i	$-0.413451\pi$
$89$	5.06727	0.537129	0.268565	+	0.963262i	$0.413451\pi$
$90$	0	0
$91$	22.3871	2.34680
$92$	− 2.06727i	− 0.215527i
$93$	− 11.0283i	− 1.14358i
$94$	−4.73678	−0.488562
$95$	0	0
$96$	−3.47277	−0.354438
$97$	3.22717i	0.327670i	0.986488	+	0.163835i	$0.0523864\pi$
$97$	3.22717i	0.327670i	−0.986488	+	0.163835i	$0.947614\pi$
$98$	− 8.06434i	− 0.814622i
$99$	0.726109	0.0729767
$100$	0	0
$101$	16.8032	1.67199	0.835993	−	0.548740i	$-0.184892\pi$
$101$	16.8032	1.67199	0.835993	+	0.548740i	$0.184892\pi$
$102$	− 4.94048i	− 0.489181i
$103$	9.85772i	0.971310i	0.874151	+	0.485655i	$0.161419\pi$
$103$	9.85772i	0.971310i	−0.874151	+	0.485655i	$0.838581\pi$
$104$	18.5683	1.82077
$105$	0	0
$106$	17.5216	1.70185
$107$	− 5.13936i	− 0.496841i	−0.968652	−	0.248420i	$-0.920089\pi$
$107$	− 5.13936i	− 0.496841i	0.968652	−	0.248420i	$-0.0799114\pi$
$108$	2.03897i	0.196200i
$109$	−13.9738	−1.33845	−0.669225	−	0.743060i	$-0.733374\pi$
$109$	−13.9738	−1.33845	−0.669225	+	0.743060i	$0.733374\pi$
$110$	0	0
$111$	14.3772	1.36462
$112$	− 11.3305i	− 1.07063i
$113$	0.527235i	0.0495981i	0.999692	+	0.0247990i	$0.00789459\pi$
$113$	0.527235i	0.0495981i	−0.999692	+	0.0247990i	$0.992105\pi$
$114$	2.10331	0.196993
$115$	0	0
$116$	−0.245594	−0.0228029
$117$	− 1.67939i	− 0.155260i
$118$	10.0000i	0.920575i
$119$	−8.57608	−0.786168
$120$	0	0
$121$	−3.97170	−0.361064
$122$	− 2.45222i	− 0.222013i
$123$	3.19112i	0.287734i
$124$	−2.51948	−0.226256
$125$	0	0
$126$	−1.27389	−0.113487
$127$	− 11.8217i	− 1.04900i	−0.851409	−	0.524502i	$-0.824252\pi$
$127$	− 11.8217i	− 1.04900i	0.851409	−	0.524502i	$-0.175748\pi$
$128$	− 7.11319i	− 0.628723i
$129$	−4.39000	−0.386518
$130$	0	0
$131$	8.12386	0.709785	0.354892	−	0.934907i	$-0.384517\pi$
$131$	8.12386	0.709785	0.354892	+	0.934907i	$0.384517\pi$
$132$	1.65109i	0.143709i
$133$	− 3.65109i	− 0.316590i
$134$	5.66177	0.489102
$135$	0	0
$136$	−7.11319	−0.609951
$137$	17.5761i	1.50163i	0.660515	+	0.750813i	$0.270338\pi$
$137$	17.5761i	1.50163i	−0.660515	+	0.750813i	$0.729662\pi$
$138$	11.5272i	0.981263i
$139$	−18.4154	−1.56197	−0.780986	−	0.624549i	$-0.785283\pi$
$139$	−18.4154	−1.56197	−0.780986	+	0.624549i	$0.785283\pi$
$140$	0	0
$141$	−6.13936	−0.517027
$142$	4.51948i	0.379267i
$143$	− 16.2555i	− 1.35935i
$144$	−0.849968	−0.0708307
$145$	0	0
$146$	−3.15990	−0.261516
$147$	− 10.4522i	− 0.862084i
$148$	− 3.28456i	− 0.269989i
$149$	17.2915	1.41658	0.708288	−	0.705924i	$-0.249468\pi$
$149$	17.2915	1.41658	0.708288	+	0.705924i	$0.249468\pi$
$150$	0	0
$151$	2.58383	0.210269	0.105134	−	0.994458i	$-0.466473\pi$
$151$	2.58383	0.210269	0.105134	+	0.994458i	$0.466473\pi$
$152$	− 3.02830i	− 0.245627i
$153$	0.643343i	0.0520112i
$154$	−12.3305	−0.993619
$155$	0	0
$156$	3.81875	0.305745
$157$	− 10.9738i	− 0.875807i	−0.899022	−	0.437903i	$-0.855721\pi$
$157$	− 10.9738i	− 0.875807i	0.899022	−	0.437903i	$-0.144279\pi$
$158$	19.3121i	1.53638i
$159$	22.7098	1.80100
$160$	0	0
$161$	20.0099	1.57700
$162$	− 10.3227i	− 0.811031i
$163$	− 22.6794i	− 1.77639i	−0.459470	−	0.888193i	$-0.651961\pi$
$163$	− 22.6794i	− 1.77639i	0.459470	−	0.888193i	$-0.348039\pi$
$164$	0.729033	0.0569279
$165$	0	0
$166$	−18.7622	−1.45623
$167$	− 5.16283i	− 0.399512i	−0.979846	−	0.199756i	$-0.935985\pi$
$167$	− 5.16283i	− 0.399512i	0.979846	−	0.199756i	$-0.0640149\pi$
$168$	− 18.2555i	− 1.40844i
$169$	−24.5966	−1.89205
$170$	0	0
$171$	−0.273891	−0.0209449
$172$	1.00292i	0.0764722i
$173$	17.2165i	1.30895i	0.756085	+	0.654473i	$0.227109\pi$
$173$	17.2165i	1.30895i	−0.756085	+	0.654473i	$0.772891\pi$
$174$	1.36945	0.103818
$175$	0	0
$176$	−8.22717	−0.620146
$177$	12.9610i	0.974211i
$178$	6.45514i	0.483833i
$179$	−10.6999	−0.799751	−0.399875	−	0.916570i	$-0.630947\pi$
$179$	−10.6999	−0.799751	−0.399875	+	0.916570i	$0.630947\pi$
$180$	0	0
$181$	−16.7720	−1.24666	−0.623328	−	0.781961i	$-0.714220\pi$
$181$	−16.7720	−1.24666	−0.623328	+	0.781961i	$0.714220\pi$
$182$	28.5187i	2.11395i
$183$	− 3.17833i	− 0.234949i
$184$	16.5966	1.22352
$185$	0	0
$186$	14.0488	1.03011
$187$	6.22717i	0.455376i
$188$	1.40258i	0.102293i
$189$	−19.7360	−1.43558
$190$	0	0
$191$	−13.8598	−1.00286	−0.501431	−	0.865197i	$-0.667193\pi$
$191$	−13.8598	−1.00286	−0.501431	+	0.865197i	$0.667193\pi$
$192$	− 14.6716i	− 1.05883i
$193$	− 21.0694i	− 1.51661i	−0.651901	−	0.758304i	$-0.726028\pi$
$193$	− 21.0694i	− 1.51661i	0.651901	−	0.758304i	$-0.273972\pi$
$194$	−4.11106	−0.295157
$195$	0	0
$196$	−2.38788	−0.170563
$197$	21.2555i	1.51439i	0.653189	+	0.757195i	$0.273431\pi$
$197$	21.2555i	1.51439i	−0.653189	+	0.757195i	$0.726569\pi$
$198$	0.924984i	0.0657357i
$199$	7.69006	0.545134	0.272567	−	0.962137i	$-0.412127\pi$
$199$	7.69006	0.545134	0.272567	+	0.962137i	$0.412127\pi$
$200$	0	0
$201$	7.33823	0.517599
$202$	21.4055i	1.50609i
$203$	− 2.37720i	− 0.166847i
$204$	−1.46289	−0.102423
$205$	0	0
$206$	−12.5577	−0.874933
$207$	− 1.50106i	− 0.104331i
$208$	19.0283i	1.31937i
$209$	−2.65109	−0.183380
$210$	0	0
$211$	−1.69781	−0.116882	−0.0584411	−	0.998291i	$-0.518613\pi$
$211$	−1.69781	−0.116882	−0.0584411	+	0.998291i	$0.518613\pi$
$212$	− 5.18820i	− 0.356327i
$213$	5.85772i	0.401364i
$214$	6.54698	0.447542
$215$	0	0
$216$	−16.3695	−1.11380
$217$	− 24.3871i	− 1.65550i
$218$	− 17.8011i	− 1.20564i
$219$	−4.09556	−0.276752
$220$	0	0
$221$	14.4026	0.968822
$222$	18.3150i	1.22922i
$223$	25.2632i	1.69175i	0.533381	+	0.845875i	$0.320921\pi$
$223$	25.2632i	1.69175i	−0.533381	+	0.845875i	$0.679079\pi$
$224$	−7.67939	−0.513101
$225$	0	0
$226$	−0.671640	−0.0446768
$227$	− 16.8217i	− 1.11649i	−0.829675	−	0.558247i	$-0.811474\pi$
$227$	− 16.8217i	− 1.11649i	0.829675	−	0.558247i	$-0.188526\pi$
$228$	− 0.622797i	− 0.0412457i
$229$	14.6249	0.966442	0.483221	−	0.875498i	$-0.339467\pi$
$229$	14.6249	0.966442	0.483221	+	0.875498i	$0.339467\pi$
$230$	0	0
$231$	−15.9816	−1.05151
$232$	− 1.97170i	− 0.129449i
$233$	− 8.91431i	− 0.583996i	−0.956419	−	0.291998i	$-0.905680\pi$
$233$	− 8.91431i	− 0.583996i	0.956419	−	0.291998i	$-0.0943200\pi$
$234$	2.13936	0.139854
$235$	0	0
$236$	2.96103	0.192747
$237$	25.0304i	1.62590i
$238$	− 10.9250i	− 0.708162i
$239$	24.6015	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$239$	24.6015	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$240$	0	0
$241$	−14.6150	−0.941438	−0.470719	−	0.882283i	$-0.656005\pi$
$241$	−14.6150	−0.941438	−0.470719	+	0.882283i	$0.656005\pi$
$242$	− 5.05952i	− 0.325238i
$243$	2.83717i	0.182005i
$244$	−0.726109	−0.0464844
$245$	0	0
$246$	−4.06514	−0.259184
$247$	6.13161i	0.390145i
$248$	− 20.2272i	− 1.28443i
$249$	−24.3177	−1.54107
$250$	0	0
$251$	−13.3382	−0.841902	−0.420951	−	0.907083i	$-0.638304\pi$
$251$	−13.3382	−0.841902	−0.420951	+	0.907083i	$0.638304\pi$
$252$	0.377203i	0.0237615i
$253$	− 14.5294i	− 0.913453i
$254$	15.0595	0.944918
$255$	0	0
$256$	−8.71061	−0.544413
$257$	− 3.35103i	− 0.209031i	−0.994523	−	0.104516i	$-0.966671\pi$
$257$	− 3.35103i	− 0.209031i	0.994523	−	0.104516i	$-0.0333293\pi$
$258$	− 5.59238i	− 0.348166i
$259$	31.7926	1.97549
$260$	0	0
$261$	−0.178328	−0.0110382
$262$	10.3489i	0.639358i
$263$	− 10.8860i	− 0.671260i	−0.941994	−	0.335630i	$-0.891051\pi$
$263$	− 10.8860i	− 0.671260i	0.941994	−	0.335630i	$-0.108949\pi$
$264$	−13.2555	−0.815818
$265$	0	0
$266$	4.65109	0.285177
$267$	8.36653i	0.512023i
$268$	− 1.67647i	− 0.102406i
$269$	−18.4338	−1.12393	−0.561964	−	0.827162i	$-0.689954\pi$
$269$	−18.4338	−1.12393	−0.561964	+	0.827162i	$0.689954\pi$
$270$	0	0
$271$	−20.4076	−1.23967	−0.619837	−	0.784730i	$-0.712801\pi$
$271$	−20.4076	−1.23967	−0.619837	+	0.784730i	$0.712801\pi$
$272$	− 7.28939i	− 0.441984i
$273$	36.9632i	2.23711i
$274$	−22.3900	−1.35263
$275$	0	0
$276$	3.41325	0.205453
$277$	− 2.69994i	− 0.162223i	−0.996705	−	0.0811117i	$-0.974153\pi$
$277$	− 2.69994i	− 0.162223i	0.996705	−	0.0811117i	$-0.0258471\pi$
$278$	− 23.4592i	− 1.40699i
$279$	−1.82942	−0.109525
$280$	0	0
$281$	−15.2242	−0.908202	−0.454101	−	0.890950i	$-0.650040\pi$
$281$	−15.2242	−0.908202	−0.454101	+	0.890950i	$0.650040\pi$
$282$	− 7.82087i	− 0.465726i
$283$	− 6.18045i	− 0.367390i	−0.982983	−	0.183695i	$-0.941194\pi$
$283$	− 6.18045i	− 0.367390i	0.982983	−	0.183695i	$-0.0588058\pi$
$284$	1.33823	0.0794095
$285$	0	0
$286$	20.7077	1.22447
$287$	7.05659i	0.416537i
$288$	0.576077i	0.0339457i
$289$	11.4826	0.675449
$290$	0	0
$291$	−5.32836	−0.312354
$292$	0.935657i	0.0547552i
$293$	− 30.6893i	− 1.79289i	−0.443159	−	0.896443i	$-0.646142\pi$
$293$	− 30.6893i	− 1.79289i	0.443159	−	0.896443i	$-0.353858\pi$
$294$	13.3150	0.776546
$295$	0	0
$296$	26.3695	1.53269
$297$	14.3305i	0.831539i
$298$	22.0275i	1.27602i
$299$	−33.6044	−1.94339
$300$	0	0
$301$	−9.70769	−0.559542
$302$	3.29151i	0.189405i
$303$	27.7437i	1.59384i
$304$	3.10331	0.177987
$305$	0	0
$306$	−0.819549	−0.0468505
$307$	− 14.7827i	− 0.843693i	−0.906667	−	0.421847i	$-0.861382\pi$
$307$	− 14.7827i	− 0.843693i	0.906667	−	0.421847i	$-0.138618\pi$
$308$	3.65109i	0.208040i
$309$	−16.2760	−0.925910
$310$	0	0
$311$	−26.9992	−1.53098	−0.765492	−	0.643445i	$-0.777504\pi$
$311$	−26.9992	−1.53098	−0.765492	+	0.643445i	$0.777504\pi$
$312$	30.6580i	1.73567i
$313$	9.74666i	0.550914i	0.961313	+	0.275457i	$0.0888291\pi$
$313$	9.74666i	0.550914i	−0.961313	+	0.275457i	$0.911171\pi$
$314$	13.9795	0.788906
$315$	0	0
$316$	5.71836	0.321683
$317$	19.7261i	1.10793i	0.832540	+	0.553964i	$0.186886\pi$
$317$	19.7261i	1.10793i	−0.832540	+	0.553964i	$0.813114\pi$
$318$	28.9298i	1.62230i
$319$	−1.72611	−0.0966436
$320$	0	0
$321$	8.48556	0.473618
$322$	25.4904i	1.42052i
$323$	− 2.34891i	− 0.130697i
$324$	−3.05659	−0.169811
$325$	0	0
$326$	28.8911	1.60013
$327$	− 23.0721i	− 1.27589i
$328$	5.85289i	0.323172i
$329$	−13.5761	−0.748473
$330$	0	0
$331$	19.9426	1.09614	0.548072	−	0.836431i	$-0.315362\pi$
$331$	19.9426	1.09614	0.548072	+	0.836431i	$0.315362\pi$
$332$	5.55553i	0.304899i
$333$	− 2.38495i	− 0.130695i
$334$	6.57688	0.359871
$335$	0	0
$336$	18.7077	1.02059
$337$	2.05952i	0.112189i	0.998425	+	0.0560945i	$0.0178648\pi$
$337$	2.05952i	0.112189i	−0.998425	+	0.0560945i	$0.982135\pi$
$338$	− 31.3334i	− 1.70431i
$339$	−0.870514	−0.0472798
$340$	0	0
$341$	−17.7077	−0.958925
$342$	− 0.348907i	− 0.0188667i
$343$	2.44447i	0.131989i
$344$	−8.05177	−0.434122
$345$	0	0
$346$	−21.9319	−1.17907
$347$	− 10.5011i	− 0.563727i	−0.959455	−	0.281863i	$-0.909048\pi$
$347$	− 10.5011i	− 0.563727i	0.959455	−	0.281863i	$-0.0909525\pi$
$348$	− 0.405499i	− 0.0217370i
$349$	−16.5059	−0.883540	−0.441770	−	0.897128i	$-0.645649\pi$
$349$	−16.5059	−0.883540	−0.441770	+	0.897128i	$0.645649\pi$
$350$	0	0
$351$	33.1444	1.76912
$352$	5.57608i	0.297206i
$353$	− 15.8860i	− 0.845527i	−0.906240	−	0.422764i	$-0.861060\pi$
$353$	− 15.8860i	− 0.845527i	0.906240	−	0.422764i	$-0.138940\pi$
$354$	−16.5109	−0.877546
$355$	0	0
$356$	1.91139	0.101303
$357$	− 14.1599i	− 0.749422i
$358$	− 13.6305i	− 0.720397i
$359$	−1.28376	−0.0677544	−0.0338772	−	0.999426i	$-0.510786\pi$
$359$	−1.28376	−0.0677544	−0.0338772	+	0.999426i	$0.510786\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 21.3657i	− 1.12296i
$363$	− 6.55765i	− 0.344188i
$364$	8.44447	0.442610
$365$	0	0
$366$	4.04884	0.211636
$367$	− 19.8804i	− 1.03775i	−0.854851	−	0.518874i	$-0.826351\pi$
$367$	− 19.8804i	− 1.03775i	0.854851	−	0.518874i	$-0.173649\pi$
$368$	17.0078i	0.886590i
$369$	0.529358	0.0275573
$370$	0	0
$371$	50.2186	2.60722
$372$	− 4.15990i	− 0.215681i
$373$	16.4359i	0.851020i	0.904954	+	0.425510i	$0.139905\pi$
$373$	16.4359i	0.851020i	−0.904954	+	0.425510i	$0.860095\pi$
$374$	−7.93273	−0.410192
$375$	0	0
$376$	−11.2603	−0.580705
$377$	3.99225i	0.205611i
$378$	− 25.1415i	− 1.29314i
$379$	−0.338233	−0.0173739	−0.00868694	−	0.999962i	$-0.502765\pi$
$379$	−0.338233	−0.0173739	−0.00868694	+	0.999962i	$0.502765\pi$
$380$	0	0
$381$	19.5187	0.999972
$382$	− 17.6559i	− 0.903355i
$383$	13.7339i	0.701767i	0.936419	+	0.350884i	$0.114119\pi$
$383$	13.7339i	0.701767i	−0.936419	+	0.350884i	$0.885881\pi$
$384$	11.7445	0.599336
$385$	0	0
$386$	26.8401	1.36612
$387$	0.728232i	0.0370181i
$388$	1.21730i	0.0617989i
$389$	2.26109	0.114642	0.0573210	−	0.998356i	$-0.481744\pi$
$389$	2.26109	0.114642	0.0573210	+	0.998356i	$0.481744\pi$
$390$	0	0
$391$	12.8732	0.651027
$392$	− 19.1706i	− 0.968260i
$393$	13.4132i	0.676609i
$394$	−27.0771	−1.36413
$395$	0	0
$396$	0.273891	0.0137635
$397$	− 7.82942i	− 0.392947i	−0.980509	−	0.196474i	$-0.937051\pi$
$397$	− 7.82942i	− 0.392947i	0.980509	−	0.196474i	$-0.0629490\pi$
$398$	9.79630i	0.491044i
$399$	6.02830	0.301792
$400$	0	0
$401$	−35.0510	−1.75036	−0.875181	−	0.483796i	$-0.839258\pi$
$401$	−35.0510	−1.75036	−0.875181	+	0.483796i	$0.839258\pi$
$402$	9.34811i	0.466241i
$403$	40.9554i	2.04013i
$404$	6.33823	0.315339
$405$	0	0
$406$	3.02830	0.150292
$407$	− 23.0849i	− 1.14428i
$408$	− 11.7445i	− 0.581441i
$409$	8.34811	0.412787	0.206394	−	0.978469i	$-0.433827\pi$
$409$	8.34811	0.412787	0.206394	+	0.978469i	$0.433827\pi$
$410$	0	0
$411$	−29.0197	−1.43144
$412$	3.71836i	0.183190i
$413$	28.6610i	1.41031i
$414$	1.91219	0.0939789
$415$	0	0
$416$	12.8967	0.632312
$417$	− 30.4055i	− 1.48896i
$418$	− 3.37720i	− 0.165184i
$419$	19.8705	0.970738	0.485369	−	0.874309i	$-0.338685\pi$
$419$	19.8705	0.970738	0.485369	+	0.874309i	$0.338685\pi$
$420$	0	0
$421$	−0.400672	−0.0195276	−0.00976379	−	0.999952i	$-0.503108\pi$
$421$	−0.400672	−0.0195276	−0.00976379	+	0.999952i	$0.503108\pi$
$422$	− 2.16283i	− 0.105285i
$423$	1.01842i	0.0495174i
$424$	41.6524	2.02282
$425$	0	0
$426$	−7.46209	−0.361540
$427$	− 7.02830i	− 0.340123i
$428$	− 1.93858i	− 0.0937048i
$429$	26.8393	1.29581
$430$	0	0
$431$	−14.2533	−0.686559	−0.343280	−	0.939233i	$-0.611538\pi$
$431$	−14.2533	−0.686559	−0.343280	+	0.939233i	$0.611538\pi$
$432$	− 16.7750i	− 0.807085i
$433$	− 12.8393i	− 0.617017i	−0.951222	−	0.308509i	$-0.900170\pi$
$433$	− 12.8393i	− 0.617017i	0.951222	−	0.308509i	$-0.0998298\pi$
$434$	31.0665	1.49124
$435$	0	0
$436$	−5.27097	−0.252434
$437$	5.48052i	0.262169i
$438$	− 5.21730i	− 0.249292i
$439$	13.2555	0.632649	0.316324	−	0.948651i	$-0.397551\pi$
$439$	13.2555	0.632649	0.316324	+	0.948651i	$0.397551\pi$
$440$	0	0
$441$	−1.73386	−0.0825647
$442$	18.3473i	0.872692i
$443$	− 5.72611i	− 0.272056i	−0.990705	−	0.136028i	$-0.956566\pi$
$443$	− 5.72611i	− 0.272056i	0.990705	−	0.136028i	$-0.0434337\pi$
$444$	5.42312	0.257370
$445$	0	0
$446$	−32.1826	−1.52389
$447$	28.5499i	1.35036i
$448$	− 32.4437i	− 1.53282i
$449$	−3.11399	−0.146958	−0.0734790	−	0.997297i	$-0.523410\pi$
$449$	−3.11399	−0.146958	−0.0734790	+	0.997297i	$0.523410\pi$
$450$	0	0
$451$	5.12386	0.241273
$452$	0.198875i	0.00935427i
$453$	4.26614i	0.200441i
$454$	21.4290	1.00571
$455$	0	0
$456$	5.00000	0.234146
$457$	27.4535i	1.28422i	0.766611	+	0.642111i	$0.221941\pi$
$457$	27.4535i	1.28422i	−0.766611	+	0.642111i	$0.778059\pi$
$458$	18.6305i	0.870548i
$459$	−12.6970	−0.592646
$460$	0	0
$461$	13.9837	0.651286	0.325643	−	0.945493i	$-0.394419\pi$
$461$	13.9837	0.651286	0.325643	+	0.945493i	$0.394419\pi$
$462$	− 20.3588i	− 0.947176i
$463$	− 17.9992i	− 0.836494i	−0.908333	−	0.418247i	$-0.862645\pi$
$463$	− 17.9992i	− 0.836494i	0.908333	−	0.418247i	$-0.137355\pi$
$464$	2.02055	0.0938015
$465$	0	0
$466$	11.3559	0.526050
$467$	12.6871i	0.587091i	0.955945	+	0.293545i	$0.0948352\pi$
$467$	12.6871i	0.587091i	−0.955945	+	0.293545i	$0.905165\pi$
$468$	− 0.633471i	− 0.0292822i
$469$	16.2272	0.749301
$470$	0	0
$471$	18.1188	0.834871
$472$	23.7720i	1.09420i
$473$	7.04884i	0.324106i
$474$	−31.8860	−1.46457
$475$	0	0
$476$	−3.23492	−0.148272
$477$	− 3.76720i	− 0.172488i
$478$	31.3396i	1.43344i
$479$	−24.7544	−1.13106	−0.565529	−	0.824729i	$-0.691328\pi$
$479$	−24.7544	−1.13106	−0.565529	+	0.824729i	$0.691328\pi$
$480$	0	0
$481$	−53.3921	−2.43447
$482$	− 18.6180i	− 0.848025i
$483$	33.0382i	1.50329i
$484$	−1.49814	−0.0680972
$485$	0	0
$486$	−3.61425	−0.163946
$487$	− 4.08277i	− 0.185008i	−0.995712	−	0.0925039i	$-0.970513\pi$
$487$	− 4.08277i	− 0.185008i	0.995712	−	0.0925039i	$-0.0294871\pi$
$488$	− 5.82942i	− 0.263886i
$489$	37.4458	1.69336
$490$	0	0
$491$	29.2547	1.32024	0.660122	−	0.751158i	$-0.270504\pi$
$491$	29.2547	1.32024	0.660122	+	0.751158i	$0.270504\pi$
$492$	1.20370i	0.0542670i
$493$	− 1.52936i	− 0.0688788i
$494$	−7.81100	−0.351433
$495$	0	0
$496$	20.7282	0.930725
$497$	12.9533i	0.581034i
$498$	− 30.9781i	− 1.38816i
$499$	−1.57315	−0.0704240	−0.0352120	−	0.999380i	$-0.511211\pi$
$499$	−1.57315	−0.0704240	−0.0352120	+	0.999380i	$0.511211\pi$
$500$	0	0
$501$	8.52431	0.380838
$502$	− 16.9914i	− 0.758365i
$503$	− 20.7819i	− 0.926619i	−0.886196	−	0.463310i	$-0.846662\pi$
$503$	− 20.7819i	− 0.926619i	0.886196	−	0.463310i	$-0.153338\pi$
$504$	−3.02830	−0.134891
$505$	0	0
$506$	18.5088	0.822817
$507$	− 40.6113i	− 1.80361i
$508$	− 4.45917i	− 0.197844i
$509$	31.8238	1.41056	0.705282	−	0.708926i	$-0.250820\pi$
$509$	31.8238	1.41056	0.705282	+	0.708926i	$0.250820\pi$
$510$	0	0
$511$	−9.05659	−0.400640
$512$	− 25.3227i	− 1.11912i
$513$	− 5.40550i	− 0.238659i
$514$	4.26884	0.188291
$515$	0	0
$516$	−1.65592	−0.0728978
$517$	9.85772i	0.433542i
$518$	40.5003i	1.77948i
$519$	−28.4260	−1.24776
$520$	0	0
$521$	−27.4720	−1.20357	−0.601784	−	0.798659i	$-0.705543\pi$
$521$	−27.4720	−1.20357	−0.601784	+	0.798659i	$0.705543\pi$
$522$	− 0.227171i	− 0.00994300i
$523$	10.4466i	0.456798i	0.973568	+	0.228399i	$0.0733490\pi$
$523$	10.4466i	0.456798i	−0.973568	+	0.228399i	$0.926651\pi$
$524$	3.06434	0.133866
$525$	0	0
$526$	13.8676	0.604656
$527$	− 15.6893i	− 0.683435i
$528$	− 13.5838i	− 0.591160i
$529$	−7.03605	−0.305915
$530$	0	0
$531$	2.15003	0.0933034
$532$	− 1.37720i	− 0.0597093i
$533$	− 11.8508i	− 0.513314i
$534$	−10.6580	−0.461219
$535$	0	0
$536$	13.4592	0.581348
$537$	− 17.6666i	− 0.762370i
$538$	− 23.4826i	− 1.01241i
$539$	−16.7827	−0.722882
$540$	0	0
$541$	13.4989	0.580365	0.290182	−	0.956971i	$-0.406284\pi$
$541$	13.4989	0.580365	0.290182	+	0.956971i	$0.406284\pi$
$542$	− 25.9971i	− 1.11667i
$543$	− 27.6922i	− 1.18839i
$544$	−4.94048	−0.211822
$545$	0	0
$546$	−47.0870	−2.01514
$547$	8.54698i	0.365442i	0.983165	+	0.182721i	$0.0584906\pi$
$547$	8.54698i	0.365442i	−0.983165	+	0.182721i	$0.941509\pi$
$548$	6.62975i	0.283209i
$549$	−0.527235	−0.0225018
$550$	0	0
$551$	0.651093	0.0277375
$552$	27.4026i	1.16633i
$553$	55.3502i	2.35373i
$554$	3.43942	0.146127
$555$	0	0
$556$	−6.94633	−0.294590
$557$	27.2378i	1.15410i	0.816707	+	0.577052i	$0.195797\pi$
$557$	27.2378i	1.15410i	−0.816707	+	0.577052i	$0.804203\pi$
$558$	− 2.33048i	− 0.0986572i
$559$	16.3030	0.689543
$560$	0	0
$561$	−10.2816	−0.434091
$562$	− 19.3940i	− 0.818088i
$563$	1.44235i	0.0607876i	0.999538	+	0.0303938i	$0.00967614\pi$
$563$	1.44235i	0.0607876i	−0.999538	+	0.0303938i	$0.990324\pi$
$564$	−2.31578	−0.0975121
$565$	0	0
$566$	7.87322	0.330936
$567$	− 29.5860i	− 1.24249i
$568$	10.7437i	0.450797i
$569$	14.2632	0.597945	0.298973	−	0.954262i	$-0.403356\pi$
$569$	14.2632	0.597945	0.298973	+	0.954262i	$0.403356\pi$
$570$	0	0
$571$	9.91723	0.415023	0.207512	−	0.978233i	$-0.433464\pi$
$571$	9.91723	0.415023	0.207512	+	0.978233i	$0.433464\pi$
$572$	− 6.13161i	− 0.256375i
$573$	− 22.8839i	− 0.955988i
$574$	−8.98933	−0.375207
$575$	0	0
$576$	−2.43380	−0.101408
$577$	− 8.23997i	− 0.343034i	−0.985181	−	0.171517i	$-0.945133\pi$
$577$	− 8.23997i	− 0.343034i	0.985181	−	0.171517i	$-0.0548669\pi$
$578$	14.6276i	0.608429i
$579$	34.7875	1.44572
$580$	0	0
$581$	−53.7742	−2.23093
$582$	− 6.78775i	− 0.281361i
$583$	− 36.4642i	− 1.51019i
$584$	−7.51173	−0.310838
$585$	0	0
$586$	39.0948	1.61499
$587$	36.9554i	1.52531i	0.646804	+	0.762656i	$0.276105\pi$
$587$	36.9554i	1.52531i	−0.646804	+	0.762656i	$0.723895\pi$
$588$	− 3.94261i	− 0.162590i
$589$	6.67939	0.275219
$590$	0	0
$591$	−35.0948	−1.44361
$592$	27.0227i	1.11062i
$593$	1.36170i	0.0559184i	0.999609	+	0.0279592i	$0.00890085\pi$
$593$	1.36170i	0.0559184i	−0.999609	+	0.0279592i	$0.991099\pi$
$594$	−18.2555	−0.749031
$595$	0	0
$596$	6.52241	0.267168
$597$	12.6970i	0.519654i
$598$	− 42.8083i	− 1.75056i
$599$	−33.5753	−1.37185	−0.685924	−	0.727673i	$-0.740602\pi$
$599$	−33.5753	−1.37185	−0.685924	+	0.727673i	$0.740602\pi$
$600$	0	0
$601$	12.6561	0.516255	0.258127	−	0.966111i	$-0.416895\pi$
$601$	12.6561	0.516255	0.258127	+	0.966111i	$0.416895\pi$
$602$	− 12.3665i	− 0.504022i
$603$	− 1.21730i	− 0.0495722i
$604$	0.974627	0.0396570
$605$	0	0
$606$	−35.3425	−1.43569
$607$	17.4047i	0.706435i	0.935541	+	0.353217i	$0.114912\pi$
$607$	17.4047i	0.706435i	−0.935541	+	0.353217i	$0.885088\pi$
$608$	− 2.10331i	− 0.0853006i
$609$	3.92498	0.159048
$610$	0	0
$611$	22.7995	0.922370
$612$	0.242671i	0.00980939i
$613$	37.2603i	1.50493i	0.658633	+	0.752465i	$0.271135\pi$
$613$	37.2603i	1.50493i	−0.658633	+	0.752465i	$0.728865\pi$
$614$	18.8315	0.759979
$615$	0	0
$616$	−29.3121	−1.18102
$617$	− 18.3764i	− 0.739806i	−0.929070	−	0.369903i	$-0.879391\pi$
$617$	− 18.3764i	− 0.739806i	0.929070	−	0.369903i	$-0.120609\pi$
$618$	− 20.7339i	− 0.834038i
$619$	−14.5526	−0.584919	−0.292459	−	0.956278i	$-0.594474\pi$
$619$	−14.5526	−0.584919	−0.292459	+	0.956278i	$0.594474\pi$
$620$	0	0
$621$	29.6249	1.18881
$622$	− 34.3940i	− 1.37907i
$623$	18.5011i	0.741229i
$624$	−31.4175	−1.25771
$625$	0	0
$626$	−12.4162	−0.496250
$627$	− 4.37720i	− 0.174809i
$628$	− 4.13936i	− 0.165178i
$629$	20.4535	0.815536
$630$	0	0
$631$	22.0304	0.877017	0.438509	−	0.898727i	$-0.355507\pi$
$631$	22.0304	0.877017	0.438509	+	0.898727i	$0.355507\pi$
$632$	45.9087i	1.82615i
$633$	− 2.80325i	− 0.111419i
$634$	−25.1289	−0.997996
$635$	0	0
$636$	8.56620	0.339672
$637$	38.8160i	1.53795i
$638$	− 2.19887i	− 0.0870543i
$639$	0.971704	0.0384400
$640$	0	0
$641$	−43.6765	−1.72512	−0.862558	−	0.505958i	$-0.831139\pi$
$641$	−43.6765	−1.72512	−0.862558	+	0.505958i	$0.831139\pi$
$642$	10.8097i	0.426624i
$643$	25.9263i	1.02243i	0.859452	+	0.511217i	$0.170805\pi$
$643$	25.9263i	1.02243i	−0.859452	+	0.511217i	$0.829195\pi$
$644$	7.54778	0.297424
$645$	0	0
$646$	2.99225	0.117728
$647$	− 42.1046i	− 1.65530i	−0.561242	−	0.827652i	$-0.689676\pi$
$647$	− 42.1046i	− 1.65530i	0.561242	−	0.827652i	$-0.310324\pi$
$648$	− 24.5392i	− 0.963992i
$649$	20.8110	0.816903
$650$	0	0
$651$	40.2653	1.57812
$652$	− 8.55473i	− 0.335029i
$653$	− 40.4671i	− 1.58360i	−0.610779	−	0.791801i	$-0.709144\pi$
$653$	− 40.4671i	− 1.58360i	0.610779	−	0.791801i	$-0.290856\pi$
$654$	29.3913	1.14929
$655$	0	0
$656$	−5.99788	−0.234178
$657$	0.679390i	0.0265055i
$658$	− 17.2944i	− 0.674207i
$659$	−33.4204	−1.30187	−0.650937	−	0.759131i	$-0.725624\pi$
$659$	−33.4204	−1.30187	−0.650937	+	0.759131i	$0.725624\pi$
$660$	0	0
$661$	19.4883	0.758006	0.379003	−	0.925396i	$-0.376267\pi$
$661$	19.4883	0.758006	0.379003	+	0.925396i	$0.376267\pi$
$662$	25.4047i	0.987382i
$663$	23.7800i	0.923539i
$664$	−44.6015	−1.73087
$665$	0	0
$666$	3.03817	0.117727
$667$	3.56833i	0.138166i
$668$	− 1.94743i	− 0.0753485i
$669$	−41.7119	−1.61268
$670$	0	0
$671$	−5.10331	−0.197011
$672$	− 12.6794i	− 0.489118i
$673$	0.747456i	0.0288123i	0.999896	+	0.0144062i	$0.00458578\pi$
$673$	0.747456i	0.0288123i	−0.999896	+	0.0144062i	$0.995414\pi$
$674$	−2.62360	−0.101057
$675$	0	0
$676$	−9.27792	−0.356843
$677$	25.0275i	0.961885i	0.876752	+	0.480942i	$0.159705\pi$
$677$	25.0275i	0.961885i	−0.876752	+	0.480942i	$0.840295\pi$
$678$	− 1.10894i	− 0.0425886i
$679$	−11.7827	−0.452179
$680$	0	0
$681$	27.7742	1.06431
$682$	− 22.5577i	− 0.863777i
$683$	36.7253i	1.40525i	0.711558	+	0.702627i	$0.247990\pi$
$683$	36.7253i	1.40525i	−0.711558	+	0.702627i	$0.752010\pi$
$684$	−0.103312	−0.00395024
$685$	0	0
$686$	−3.11399	−0.118893
$687$	24.1471i	0.921270i
$688$	− 8.25122i	− 0.314575i
$689$	−84.3366	−3.21297
$690$	0	0
$691$	−21.3177	−0.810963	−0.405482	−	0.914103i	$-0.632896\pi$
$691$	−21.3177	−0.810963	−0.405482	+	0.914103i	$0.632896\pi$
$692$	6.49411i	0.246869i
$693$	2.65109i	0.100707i
$694$	13.3772	0.507792
$695$	0	0
$696$	3.25547	0.123398
$697$	4.53981i	0.171958i
$698$	− 21.0267i	− 0.795872i
$699$	14.7184	0.556699
$700$	0	0
$701$	27.1161	1.02416	0.512081	−	0.858937i	$-0.328875\pi$
$701$	27.1161	1.02416	0.512081	+	0.858937i	$0.328875\pi$
$702$	42.2223i	1.59358i
$703$	8.70769i	0.328417i
$704$	−23.5577	−0.887862
$705$	0	0
$706$	20.2370	0.761631
$707$	61.3502i	2.30731i
$708$	4.88894i	0.183738i
$709$	10.1161	0.379918	0.189959	−	0.981792i	$-0.439164\pi$
$709$	10.1161	0.379918	0.189959	+	0.981792i	$0.439164\pi$
$710$	0	0
$711$	4.15215	0.155718
$712$	15.3452i	0.575085i
$713$	36.6065i	1.37092i
$714$	18.0382	0.675062
$715$	0	0
$716$	−4.03605	−0.150834
$717$	40.6193i	1.51696i
$718$	− 1.63537i	− 0.0610316i
$719$	−24.1471	−0.900535	−0.450268	−	0.892894i	$-0.648671\pi$
$719$	−24.1471	−0.900535	−0.450268	+	0.892894i	$0.648671\pi$
$720$	0	0
$721$	−35.9914	−1.34039
$722$	1.27389i	0.0474093i
$723$	− 24.1308i	− 0.897434i
$724$	−6.32646	−0.235121
$725$	0	0
$726$	8.35373	0.310036
$727$	− 19.8988i	− 0.738006i	−0.929428	−	0.369003i	$-0.879699\pi$
$727$	− 19.8988i	− 0.738006i	0.929428	−	0.369003i	$-0.120301\pi$
$728$	67.7947i	2.51264i
$729$	−28.9944	−1.07387
$730$	0	0
$731$	−6.24537	−0.230994
$732$	− 1.19887i	− 0.0443117i
$733$	− 19.9455i	− 0.736705i	−0.929686	−	0.368352i	$-0.879922\pi$
$733$	− 19.9455i	− 0.736705i	0.929686	−	0.368352i	$-0.120078\pi$
$734$	25.3254	0.934779
$735$	0	0
$736$	11.5272	0.424900
$737$	− 11.7827i	− 0.434021i
$738$	0.674344i	0.0248229i
$739$	38.2624	1.40751	0.703753	−	0.710445i	$-0.251506\pi$
$739$	38.2624	1.40751	0.703753	+	0.710445i	$0.251506\pi$
$740$	0	0
$741$	−10.1239	−0.371909
$742$	63.9730i	2.34852i
$743$	12.5547i	0.460588i	0.973121	+	0.230294i	$0.0739688\pi$
$743$	12.5547i	0.460588i	−0.973121	+	0.230294i	$0.926031\pi$
$744$	33.3969	1.22439
$745$	0	0
$746$	−20.9376	−0.766579
$747$	4.03392i	0.147594i
$748$	2.34891i	0.0858845i
$749$	18.7643	0.685632
$750$	0	0
$751$	23.2215	0.847366	0.423683	−	0.905810i	$-0.360737\pi$
$751$	23.2215	0.847366	0.423683	+	0.905810i	$0.360737\pi$
$752$	− 11.5392i	− 0.420792i
$753$	− 22.0227i	− 0.802551i
$754$	−5.08569	−0.185210
$755$	0	0
$756$	−7.44447	−0.270753
$757$	18.4105i	0.669143i	0.942370	+	0.334571i	$0.108592\pi$
$757$	18.4105i	0.669143i	−0.942370	+	0.334571i	$0.891408\pi$
$758$	− 0.430872i	− 0.0156500i
$759$	23.9893	0.870757
$760$	0	0
$761$	11.7982	0.427684	0.213842	−	0.976868i	$-0.431402\pi$
$761$	11.7982	0.427684	0.213842	+	0.976868i	$0.431402\pi$
$762$	24.8647i	0.900752i
$763$	− 51.0197i	− 1.84704i
$764$	−5.22797	−0.189141
$765$	0	0
$766$	−17.4954	−0.632136
$767$	− 48.1329i	− 1.73798i
$768$	− 14.3820i	− 0.518967i
$769$	−41.2653	−1.48807	−0.744033	−	0.668143i	$-0.767090\pi$
$769$	−41.2653	−1.48807	−0.744033	+	0.668143i	$0.767090\pi$
$770$	0	0
$771$	5.53286	0.199261
$772$	− 7.94743i	− 0.286034i
$773$	− 5.51736i	− 0.198446i	−0.995065	−	0.0992229i	$-0.968364\pi$
$773$	− 5.51736i	− 0.198446i	0.995065	−	0.0992229i	$-0.0316357\pi$
$774$	−0.927688	−0.0333451
$775$	0	0
$776$	−9.77283	−0.350824
$777$	52.4925i	1.88316i
$778$	2.88039i	0.103267i
$779$	−1.93273	−0.0692474
$780$	0	0
$781$	9.40550	0.336555
$782$	16.3991i	0.586430i
$783$	− 3.51948i	− 0.125776i
$784$	19.6455	0.701624
$785$	0	0
$786$	−17.0870	−0.609474
$787$	16.7771i	0.598038i	0.954247	+	0.299019i	$0.0966594\pi$
$787$	16.7771i	0.598038i	−0.954247	+	0.299019i	$0.903341\pi$
$788$	8.01762i	0.285616i
$789$	17.9738	0.639885
$790$	0	0
$791$	−1.92498	−0.0684446
$792$	2.19887i	0.0781336i
$793$	11.8032i	0.419146i
$794$	9.97383	0.353958
$795$	0	0
$796$	2.90071	0.102813
$797$	− 29.4260i	− 1.04232i	−0.853458	−	0.521162i	$-0.825499\pi$
$797$	− 29.4260i	− 1.04232i	0.853458	−	0.521162i	$-0.174501\pi$
$798$	7.67939i	0.271847i
$799$	−8.73408	−0.308989
$800$	0	0
$801$	1.38788	0.0490382
$802$	− 44.6511i	− 1.57668i
$803$	6.57608i	0.232065i
$804$	2.76800	0.0976199
$805$	0	0
$806$	−52.1727	−1.83771
$807$	− 30.4359i	− 1.07140i
$808$	50.8852i	1.79014i
$809$	−5.48614	−0.192883	−0.0964413	−	0.995339i	$-0.530746\pi$
$809$	−5.48614	−0.192883	−0.0964413	+	0.995339i	$0.530746\pi$
$810$	0	0
$811$	16.3927	0.575626	0.287813	−	0.957687i	$-0.407072\pi$
$811$	16.3927	0.575626	0.287813	+	0.957687i	$0.407072\pi$
$812$	− 0.896688i	− 0.0314676i
$813$	− 33.6949i	− 1.18173i
$814$	29.4076	1.03074
$815$	0	0
$816$	12.0355	0.421326
$817$	− 2.65884i	− 0.0930212i
$818$	10.6346i	0.371829i
$819$	6.13161	0.214256
$820$	0	0
$821$	15.1628	0.529186	0.264593	−	0.964360i	$-0.414762\pi$
$821$	15.1628	0.529186	0.264593	+	0.964360i	$0.414762\pi$
$822$	− 36.9680i	− 1.28941i
$823$	− 16.6094i	− 0.578968i	−0.957183	−	0.289484i	$-0.906516\pi$
$823$	− 16.6094i	− 0.578968i	0.957183	−	0.289484i	$-0.0934837\pi$
$824$	−29.8521	−1.03995
$825$	0	0
$826$	−36.5109	−1.27038
$827$	15.2293i	0.529574i	0.964307	+	0.264787i	$0.0853017\pi$
$827$	15.2293i	0.529574i	−0.964307	+	0.264787i	$0.914698\pi$
$828$	− 0.566205i	− 0.0196770i
$829$	−47.4565	−1.64823	−0.824116	−	0.566422i	$-0.808327\pi$
$829$	−47.4565	−1.64823	−0.824116	+	0.566422i	$0.808327\pi$
$830$	0	0
$831$	4.45785	0.154641
$832$	54.4856i	1.88895i
$833$	− 14.8697i	− 0.515205i
$834$	38.7333	1.34122
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	− 36.1054i	− 1.24799i
$838$	25.3129i	0.874418i
$839$	−2.26614	−0.0782359	−0.0391179	−	0.999235i	$-0.512455\pi$
$839$	−2.26614	−0.0782359	−0.0391179	+	0.999235i	$0.512455\pi$
$840$	0	0
$841$	−28.5761	−0.985382
$842$	− 0.510413i	− 0.0175900i
$843$	− 25.1367i	− 0.865752i
$844$	−0.640420	−0.0220442
$845$	0	0
$846$	−1.29736	−0.0446042
$847$	− 14.5011i	− 0.498262i
$848$	42.6842i	1.46578i
$849$	10.2045	0.350218
$850$	0	0
$851$	−47.7226	−1.63591
$852$	2.20955i	0.0756979i
$853$	− 51.8753i	− 1.77618i	−0.459672	−	0.888089i	$-0.652033\pi$
$853$	− 51.8753i	− 1.77618i	0.459672	−	0.888089i	$-0.347967\pi$
$854$	8.95328	0.306375
$855$	0	0
$856$	15.5635	0.531949
$857$	− 6.17058i	− 0.210783i	−0.994431	−	0.105391i	$-0.966390\pi$
$857$	− 6.17058i	− 0.210783i	0.994431	−	0.105391i	$-0.0336096\pi$
$858$	34.1903i	1.16724i
$859$	31.0635	1.05987	0.529937	−	0.848037i	$-0.322215\pi$
$859$	31.0635	1.05987	0.529937	+	0.848037i	$0.322215\pi$
$860$	0	0
$861$	−11.6511	−0.397068
$862$	− 18.1572i	− 0.618437i
$863$	3.24772i	0.110554i	0.998471	+	0.0552768i	$0.0176041\pi$
$863$	3.24772i	0.110554i	−0.998471	+	0.0552768i	$0.982396\pi$
$864$	−11.3695	−0.386797
$865$	0	0
$866$	16.3559	0.555795
$867$	18.9589i	0.643878i
$868$	− 9.19887i	− 0.312230i
$869$	40.1903	1.36336
$870$	0	0
$871$	−27.2517	−0.923390
$872$	− 42.3169i	− 1.43303i
$873$	0.883892i	0.0299152i
$874$	−6.98158	−0.236155
$875$	0	0
$876$	−1.54486	−0.0521959
$877$	− 41.7042i	− 1.40825i	−0.710076	−	0.704125i	$-0.751339\pi$
$877$	− 41.7042i	− 1.40825i	0.710076	−	0.704125i	$-0.248661\pi$
$878$	16.8860i	0.569875i
$879$	50.6708	1.70908
$880$	0	0
$881$	7.42392	0.250118	0.125059	−	0.992149i	$-0.460088\pi$
$881$	7.42392	0.250118	0.125059	+	0.992149i	$0.460088\pi$
$882$	− 2.20875i	− 0.0743724i
$883$	− 32.0333i	− 1.07801i	−0.842303	−	0.539004i	$-0.818801\pi$
$883$	− 32.0333i	− 1.07801i	0.842303	−	0.539004i	$-0.181199\pi$
$884$	5.43269	0.182721
$885$	0	0
$886$	7.29444	0.245061
$887$	20.0275i	0.672457i	0.941780	+	0.336229i	$0.109152\pi$
$887$	20.0275i	0.672457i	−0.941780	+	0.336229i	$0.890848\pi$
$888$	43.5384i	1.46105i
$889$	43.1620	1.44761
$890$	0	0
$891$	−21.4826	−0.719695
$892$	9.52936i	0.319066i
$893$	− 3.71836i	− 0.124430i
$894$	−36.3695	−1.21638
$895$	0	0
$896$	25.9709	0.867627
$897$	− 55.4840i	− 1.85256i
$898$	− 3.96688i	− 0.132376i
$899$	4.34891	0.145044
$900$	0	0
$901$	32.3078	1.07633
$902$	6.52723i	0.217333i
$903$	− 16.0283i	− 0.533388i
$904$	−1.59662	−0.0531029
$905$	0	0
$906$	−5.43460	−0.180552
$907$	4.94823i	0.164303i	0.996620	+	0.0821517i	$0.0261792\pi$
$907$	4.94823i	0.164303i	−0.996620	+	0.0821517i	$0.973821\pi$
$908$	− 6.34518i	− 0.210572i
$909$	4.60225	0.152647
$910$	0	0
$911$	32.3014	1.07019	0.535096	−	0.844791i	$-0.320275\pi$
$911$	32.3014	1.07019	0.535096	+	0.844791i	$0.320275\pi$
$912$	5.12386i	0.169668i
$913$	39.0459i	1.29223i
$914$	−34.9728	−1.15680
$915$	0	0
$916$	5.51656	0.182272
$917$	29.6610i	0.979491i
$918$	− 16.1746i	− 0.533841i
$919$	−21.3072	−0.702861	−0.351430	−	0.936214i	$-0.614305\pi$
$919$	−21.3072	−0.702861	−0.351430	+	0.936214i	$0.614305\pi$
$920$	0	0
$921$	24.4076	0.804258
$922$	17.8137i	0.586663i
$923$	− 21.7536i	− 0.716029i
$924$	−6.02830	−0.198316
$925$	0	0
$926$	22.9290	0.753494
$927$	2.69994i	0.0886775i
$928$	− 1.36945i	− 0.0449545i
$929$	−24.7848	−0.813164	−0.406582	−	0.913614i	$-0.633279\pi$
$929$	−24.7848	−0.813164	−0.406582	+	0.913614i	$0.633279\pi$
$930$	0	0
$931$	6.33048	0.207473
$932$	− 3.36250i	− 0.110142i
$933$	− 44.5782i	− 1.45942i
$934$	−16.1620	−0.528838
$935$	0	0
$936$	5.08569	0.166231
$937$	− 32.3687i	− 1.05744i	−0.848797	−	0.528719i	$-0.822673\pi$
$937$	− 32.3687i	− 1.05744i	0.848797	−	0.528719i	$-0.177327\pi$
$938$	20.6716i	0.674953i
$939$	−16.0926	−0.525163
$940$	0	0
$941$	−1.48264	−0.0483326	−0.0241663	−	0.999708i	$-0.507693\pi$
$941$	−1.48264	−0.0483326	−0.0241663	+	0.999708i	$0.507693\pi$
$942$	23.0814i	0.752032i
$943$	− 10.5924i	− 0.344935i
$944$	−24.3609	−0.792880
$945$	0	0
$946$	−8.97945	−0.291947
$947$	− 8.86064i	− 0.287932i	−0.989583	−	0.143966i	$-0.954014\pi$
$947$	− 8.86064i	− 0.287932i	0.989583	−	0.143966i	$-0.0459856\pi$
$948$	9.44155i	0.306647i
$949$	15.2095	0.493723
$950$	0	0
$951$	−32.5696	−1.05614
$952$	− 25.9709i	− 0.841722i
$953$	− 10.1239i	− 0.327944i	−0.986465	−	0.163972i	$-0.947569\pi$
$953$	− 10.1239i	− 0.327944i	0.986465	−	0.163972i	$-0.0524307\pi$
$954$	4.79900	0.155373
$955$	0	0
$956$	9.27974	0.300128
$957$	− 2.84997i	− 0.0921264i
$958$	− 31.5344i	− 1.01883i
$959$	−64.1719	−2.07222
$960$	0	0
$961$	13.6142	0.439169
$962$	− 68.0157i	− 2.19291i
$963$	− 1.40762i	− 0.0453600i
$964$	−5.51284	−0.177557
$965$	0	0
$966$	−42.0870	−1.35413
$967$	41.8139i	1.34465i	0.740258	+	0.672323i	$0.234703\pi$
$967$	41.8139i	1.34465i	−0.740258	+	0.672323i	$0.765297\pi$
$968$	− 12.0275i	− 0.386578i
$969$	3.87826	0.124588
$970$	0	0
$971$	−5.53016	−0.177471	−0.0887356	−	0.996055i	$-0.528283\pi$
$971$	−5.53016	−0.177471	−0.0887356	+	0.996055i	$0.528283\pi$
$972$	1.07019i	0.0343263i
$973$	− 67.2362i	− 2.15549i
$974$	5.20100	0.166651
$975$	0	0
$976$	5.97383	0.191218
$977$	− 31.9447i	− 1.02200i	−0.859580	−	0.511001i	$-0.829275\pi$
$977$	− 31.9447i	− 1.02200i	0.859580	−	0.511001i	$-0.170725\pi$
$978$	47.7018i	1.52534i
$979$	13.4338	0.429346
$980$	0	0
$981$	−3.82730	−0.122196
$982$	37.2672i	1.18925i
$983$	− 8.82862i	− 0.281589i	−0.990039	−	0.140795i	$-0.955034\pi$
$983$	− 8.82862i	− 0.281589i	0.990039	−	0.140795i	$-0.0449657\pi$
$984$	−9.66367	−0.308067
$985$	0	0
$986$	1.94823	0.0620444
$987$	− 22.4154i	− 0.713489i
$988$	2.31286i	0.0735819i
$989$	14.5718	0.463357
$990$	0	0
$991$	−43.7304	−1.38914	−0.694570	−	0.719425i	$-0.744405\pi$
$991$	−43.7304	−1.38914	−0.694570	+	0.719425i	$0.744405\pi$
$992$	− 14.0488i	− 0.446051i
$993$	32.9271i	1.04491i
$994$	−16.5011	−0.523382
$995$	0	0
$996$	−9.17270	−0.290648
$997$	− 46.6738i	− 1.47817i	−0.673611	−	0.739086i	$-0.735257\pi$
$997$	− 46.6738i	− 1.47817i	0.673611	−	0.739086i	$-0.264743\pi$
$998$	− 2.00403i	− 0.0634363i
$999$	47.0694	1.48921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.b.324.4		6
5.2	odd	4		475.2.a.e.1.2	✓	3
5.3	odd	4		475.2.a.g.1.2	yes	3
5.4	even	2	inner	475.2.b.b.324.3		6
15.2	even	4		4275.2.a.bm.1.2		3
15.8	even	4		4275.2.a.ba.1.2		3
20.3	even	4		7600.2.a.bh.1.3		3
20.7	even	4		7600.2.a.cc.1.1		3
95.18	even	4		9025.2.a.y.1.2		3
95.37	even	4		9025.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.2	✓	3	5.2	odd	4
475.2.a.g.1.2	yes	3	5.3	odd	4
475.2.b.b.324.3		6	5.4	even	2	inner
475.2.b.b.324.4		6	1.1	even	1	trivial
4275.2.a.ba.1.2		3	15.8	even	4
4275.2.a.bm.1.2		3	15.2	even	4
7600.2.a.bh.1.3		3	20.3	even	4
7600.2.a.cc.1.1		3	20.7	even	4
9025.2.a.y.1.2		3	95.18	even	4
9025.2.a.bc.1.2		3	95.37	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.27389i q^{2} +1.65109i q^{3} +0.377203 q^{4} -2.10331 q^{6} +3.65109i q^{7} +3.02830i q^{8} +0.273891 q^{9} +O(q^{10})\)	\(q+1.27389i q^{2} +1.65109i q^{3} +0.377203 q^{4} -2.10331 q^{6} +3.65109i q^{7} +3.02830i q^{8} +0.273891 q^{9} +2.65109 q^{11} +0.622797i q^{12} -6.13161i q^{13} -4.65109 q^{14} -3.10331 q^{16} +2.34891i q^{17} +0.348907i q^{18} -1.00000 q^{19} -6.02830 q^{21} +3.37720i q^{22} -5.48052i q^{23} -5.00000 q^{24} +7.81100 q^{26} +5.40550i q^{27} +1.37720i q^{28} -0.651093 q^{29} -6.67939 q^{31} +2.10331i q^{32} +4.37720i q^{33} -2.99225 q^{34} +0.103312 q^{36} -8.70769i q^{37} -1.27389i q^{38} +10.1239 q^{39} +1.93273 q^{41} -7.67939i q^{42} +2.65884i q^{43} +1.00000 q^{44} +6.98158 q^{46} +3.71836i q^{47} -5.12386i q^{48} -6.33048 q^{49} -3.87826 q^{51} -2.31286i q^{52} -13.7544i q^{53} -6.88601 q^{54} -11.0566 q^{56} -1.65109i q^{57} -0.829422i q^{58} +7.84997 q^{59} -1.92498 q^{61} -8.50881i q^{62} +1.00000i q^{63} -8.88601 q^{64} -5.57608 q^{66} -4.44447i q^{67} +0.886014i q^{68} +9.04884 q^{69} +3.54778 q^{71} +0.829422i q^{72} +2.48052i q^{73} +11.0926 q^{74} -0.377203 q^{76} +9.67939i q^{77} +12.8967i q^{78} +15.1599 q^{79} -8.10331 q^{81} +2.46209i q^{82} +14.7282i q^{83} -2.27389 q^{84} -3.38708 q^{86} -1.07502i q^{87} +8.02830i q^{88} +5.06727 q^{89} +22.3871 q^{91} -2.06727i q^{92} -11.0283i q^{93} -4.73678 q^{94} -3.47277 q^{96} +3.22717i q^{97} -8.06434i q^{98} +0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \) 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9	\( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) \) 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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