LMFDB - Embedded newform 4100.2.d.c.1149.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 4x^{5} + 14x^{4} - 14x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 4x^5 + 14x^4 - 14x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 164)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.4
Root			$-1.20518 + 1.20518i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.c.1149.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0950939i q^{3} +3.14501i q^{7} +2.99096 q^{9} +O(q^{10})$	$q-0.0950939i q^{3} +3.14501i q^{7} +2.99096 q^{9} -1.67570 q^{11} -6.63052i q^{13} +5.16120i q^{17} +4.72562 q^{19} +0.299072 q^{21} +8.82071i q^{23} -0.569704i q^{27} +1.80981 q^{29} -1.65951 q^{31} +0.159349i q^{33} -1.99096i q^{37} -0.630522 q^{39} -1.00000 q^{41} -1.46932i q^{43} -8.53543i q^{47} -2.89112 q^{49} +0.490799 q^{51} +9.35139i q^{53} -0.449377i q^{57} -8.82071 q^{59} +12.6305 q^{61} +9.40660i q^{63} +9.67570i q^{67} +0.838796 q^{69} -0.776081 q^{71} -8.33145i q^{73} -5.27009i q^{77} -0.915804 q^{79} +8.91870 q^{81} +10.0998i q^{83} -0.172102i q^{87} +6.44033 q^{89} +20.8531 q^{91} +0.157809i q^{93} -8.32241i q^{97} -5.01193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.0950939i	− 0.0549025i	−0.999623	−	0.0274513i	$-0.991261\pi$
$3$	− 0.0950939i	− 0.0549025i	0.999623	−	0.0274513i	$-0.00873910\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.14501i	1.18870i	0.804205	+	0.594352i	$0.202591\pi$
$7$	3.14501i	1.18870i	−0.804205	+	0.594352i	$0.797409\pi$
$8$	0	0
$9$	2.99096	0.996986
$10$	0	0
$11$	−1.67570	−0.505241	−0.252621	−	0.967565i	$-0.581292\pi$
$11$	−1.67570	−0.505241	−0.252621	+	0.967565i	$0.581292\pi$
$12$	0	0
$13$	− 6.63052i	− 1.83898i	−0.393118	−	0.919488i	$-0.628604\pi$
$13$	− 6.63052i	− 1.83898i	0.393118	−	0.919488i	$-0.371396\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.16120i	1.25178i	0.779913	+	0.625888i	$0.215263\pi$
$17$	5.16120i	1.25178i	−0.779913	+	0.625888i	$0.784737\pi$
$18$	0	0
$19$	4.72562	1.08413	0.542065	−	0.840336i	$-0.317643\pi$
$19$	4.72562	1.08413	0.542065	+	0.840336i	$0.317643\pi$
$20$	0	0
$21$	0.299072	0.0652628
$22$	0	0
$23$	8.82071i	1.83925i	0.392803	+	0.919623i	$0.371505\pi$
$23$	8.82071i	1.83925i	−0.392803	+	0.919623i	$0.628495\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 0.569704i	− 0.109640i
$28$	0	0
$29$	1.80981	0.336074	0.168037	−	0.985781i	$-0.446257\pi$
$29$	1.80981	0.336074	0.168037	+	0.985781i	$0.446257\pi$
$30$	0	0
$31$	−1.65951	−0.298056	−0.149028	−	0.988833i	$-0.547614\pi$
$31$	−1.65951	−0.298056	−0.149028	+	0.988833i	$0.547614\pi$
$32$	0	0
$33$	0.159349i	0.0277390i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.99096i	− 0.327311i	−0.986518	−	0.163656i	$-0.947671\pi$
$37$	− 1.99096i	− 0.327311i	0.986518	−	0.163656i	$-0.0523286\pi$
$38$	0	0
$39$	−0.630522	−0.100964
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	− 1.46932i	− 0.224069i	−0.993704	−	0.112034i	$-0.964263\pi$
$43$	− 1.46932i	− 0.224069i	0.993704	−	0.112034i	$-0.0357367\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.53543i	− 1.24502i	−0.782612	−	0.622510i	$-0.786113\pi$
$47$	− 8.53543i	− 1.24502i	0.782612	−	0.622510i	$-0.213887\pi$
$48$	0	0
$49$	−2.89112	−0.413017
$50$	0	0
$51$	0.490799	0.0687256
$52$	0	0
$53$	9.35139i	1.28451i	0.766490	+	0.642256i	$0.222001\pi$
$53$	9.35139i	1.28451i	−0.766490	+	0.642256i	$0.777999\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.449377i	− 0.0595215i
$58$	0	0
$59$	−8.82071	−1.14836	−0.574179	−	0.818730i	$-0.694679\pi$
$59$	−8.82071	−1.14836	−0.574179	+	0.818730i	$0.694679\pi$
$60$	0	0
$61$	12.6305	1.61717	0.808586	−	0.588378i	$-0.200233\pi$
$61$	12.6305	1.61717	0.808586	+	0.588378i	$0.200233\pi$
$62$	0	0
$63$	9.40660i	1.18512i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.67570i	1.18207i	0.806644	+	0.591037i	$0.201281\pi$
$67$	9.67570i	1.18207i	−0.806644	+	0.591037i	$0.798719\pi$
$68$	0	0
$69$	0.838796	0.100979
$70$	0	0
$71$	−0.776081	−0.0921039	−0.0460519	−	0.998939i	$-0.514664\pi$
$71$	−0.776081	−0.0921039	−0.0460519	+	0.998939i	$0.514664\pi$
$72$	0	0
$73$	− 8.33145i	− 0.975123i	−0.873089	−	0.487561i	$-0.837887\pi$
$73$	− 8.33145i	− 0.975123i	0.873089	−	0.487561i	$-0.162113\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.27009i	− 0.600582i
$78$	0	0
$79$	−0.915804	−0.103036	−0.0515180	−	0.998672i	$-0.516406\pi$
$79$	−0.915804	−0.103036	−0.0515180	+	0.998672i	$0.516406\pi$
$80$	0	0
$81$	8.91870	0.990966
$82$	0	0
$83$	10.0998i	1.10860i	0.832316	+	0.554301i	$0.187014\pi$
$83$	10.0998i	1.10860i	−0.832316	+	0.554301i	$0.812986\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.172102i	− 0.0184513i
$88$	0	0
$89$	6.44033	0.682674	0.341337	−	0.939941i	$-0.389120\pi$
$89$	6.44033	0.682674	0.341337	+	0.939941i	$0.389120\pi$
$90$	0	0
$91$	20.8531	2.18600
$92$	0	0
$93$	0.157809i	0.0163640i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.32241i	− 0.845012i	−0.906360	−	0.422506i	$-0.861150\pi$
$97$	− 8.32241i	− 0.845012i	0.906360	−	0.422506i	$-0.138850\pi$
$98$	0	0
$99$	−5.01193	−0.503718
$100$	0	0
$101$	9.79173	0.974313	0.487157	−	0.873315i	$-0.338034\pi$
$101$	9.79173	0.974313	0.487157	+	0.873315i	$0.338034\pi$
$102$	0	0
$103$	0.648608i	0.0639093i	0.999489	+	0.0319546i	$0.0101732\pi$
$103$	0.648608i	0.0639093i	−0.999489	+	0.0319546i	$0.989827\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8026i	1.62437i	0.583399	+	0.812186i	$0.301722\pi$
$107$	16.8026i	1.62437i	−0.583399	+	0.812186i	$0.698278\pi$
$108$	0	0
$109$	−3.84969	−0.368734	−0.184367	−	0.982857i	$-0.559023\pi$
$109$	−3.84969	−0.368734	−0.184367	+	0.982857i	$0.559023\pi$
$110$	0	0
$111$	−0.189328	−0.0179702
$112$	0	0
$113$	7.65046i	0.719695i	0.933011	+	0.359848i	$0.117171\pi$
$113$	7.65046i	0.719695i	−0.933011	+	0.359848i	$0.882829\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 19.8316i	− 1.83343i
$118$	0	0
$119$	−16.2321	−1.48799
$120$	0	0
$121$	−8.19204	−0.744731
$122$	0	0
$123$	0.0950939i	0.00857433i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.9206i	1.32398i	0.749511	+	0.661992i	$0.230289\pi$
$127$	14.9206i	1.32398i	−0.749511	+	0.661992i	$0.769711\pi$
$128$	0	0
$129$	−0.139723	−0.0123019
$130$	0	0
$131$	4.19019	0.366098	0.183049	−	0.983104i	$-0.441403\pi$
$131$	4.19019	0.366098	0.183049	+	0.983104i	$0.441403\pi$
$132$	0	0
$133$	14.8621i	1.28871i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.09984i	0.350273i	0.984544	+	0.175137i	$0.0560367\pi$
$137$	4.09984i	0.350273i	−0.984544	+	0.175137i	$0.943963\pi$
$138$	0	0
$139$	11.9819	1.01629	0.508146	−	0.861271i	$-0.330331\pi$
$139$	11.9819	1.01629	0.508146	+	0.861271i	$0.330331\pi$
$140$	0	0
$141$	−0.811668	−0.0683547
$142$	0	0
$143$	11.1107i	0.929127i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.274928i	0.0226756i
$148$	0	0
$149$	1.80981	0.148266	0.0741328	−	0.997248i	$-0.476381\pi$
$149$	1.80981	0.148266	0.0741328	+	0.997248i	$0.476381\pi$
$150$	0	0
$151$	−21.3561	−1.73794	−0.868969	−	0.494867i	$-0.835217\pi$
$151$	−21.3561	−1.73794	−0.868969	+	0.494867i	$0.835217\pi$
$152$	0	0
$153$	15.4369i	1.24800i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.1107i	1.04635i	0.852225	+	0.523175i	$0.175253\pi$
$157$	13.1107i	1.04635i	−0.852225	+	0.523175i	$0.824747\pi$
$158$	0	0
$159$	0.889261	0.0705230
$160$	0	0
$161$	−27.7413	−2.18632
$162$	0	0
$163$	− 10.7808i	− 0.844420i	−0.906498	−	0.422210i	$-0.861255\pi$
$163$	− 10.7808i	− 0.844420i	0.906498	−	0.422210i	$-0.138745\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.6757i	1.05826i	0.848542	+	0.529129i	$0.177481\pi$
$167$	13.6757i	1.05826i	−0.848542	+	0.529129i	$0.822519\pi$
$168$	0	0
$169$	−30.9638	−2.38183
$170$	0	0
$171$	14.1341	1.08086
$172$	0	0
$173$	− 10.5801i	− 0.804387i	−0.915555	−	0.402193i	$-0.868248\pi$
$173$	− 10.5801i	− 0.804387i	0.915555	−	0.402193i	$-0.131752\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.838796i	0.0630478i
$178$	0	0
$179$	1.23536	0.0923352	0.0461676	−	0.998934i	$-0.485299\pi$
$179$	1.23536	0.0923352	0.0461676	+	0.998934i	$0.485299\pi$
$180$	0	0
$181$	−3.90965	−0.290602	−0.145301	−	0.989387i	$-0.546415\pi$
$181$	−3.90965	−0.290602	−0.145301	+	0.989387i	$0.546415\pi$
$182$	0	0
$183$	− 1.20109i	− 0.0887868i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.64861i	− 0.632449i
$188$	0	0
$189$	1.79173	0.130329
$190$	0	0
$191$	−7.31712	−0.529448	−0.264724	−	0.964324i	$-0.585281\pi$
$191$	−7.31712	−0.529448	−0.264724	+	0.964324i	$0.585281\pi$
$192$	0	0
$193$	13.3009i	0.957422i	0.877973	+	0.478711i	$0.158896\pi$
$193$	13.3009i	0.957422i	−0.877973	+	0.478711i	$0.841104\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.5692i	− 1.10926i	−0.832098	−	0.554628i	$-0.812860\pi$
$197$	− 15.5692i	− 1.10926i	0.832098	−	0.554628i	$-0.187140\pi$
$198$	0	0
$199$	27.8972	1.97758	0.988789	−	0.149319i	$-0.0477080\pi$
$199$	27.8972	1.97758	0.988789	+	0.149319i	$0.0477080\pi$
$200$	0	0
$201$	0.920100	0.0648989
$202$	0	0
$203$	5.69189i	0.399492i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.3824i	1.83370i
$208$	0	0
$209$	−7.91870	−0.547748
$210$	0	0
$211$	18.1484	1.24939	0.624694	−	0.780870i	$-0.285224\pi$
$211$	18.1484	1.24939	0.624694	+	0.780870i	$0.285224\pi$
$212$	0	0
$213$	0.0738006i	0.00505673i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.21917i	− 0.354300i
$218$	0	0
$219$	−0.792270	−0.0535367
$220$	0	0
$221$	34.2215	2.30199
$222$	0	0
$223$	6.55826i	0.439174i	0.975593	+	0.219587i	$0.0704709\pi$
$223$	6.55826i	0.439174i	−0.975593	+	0.219587i	$0.929529\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.4884i	1.55898i	0.626416	+	0.779489i	$0.284521\pi$
$227$	23.4884i	1.55898i	−0.626416	+	0.779489i	$0.715479\pi$
$228$	0	0
$229$	−12.1503	−0.802915	−0.401457	−	0.915878i	$-0.631496\pi$
$229$	−12.1503	−0.802915	−0.401457	+	0.915878i	$0.631496\pi$
$230$	0	0
$231$	−0.501153	−0.0329735
$232$	0	0
$233$	21.8422i	1.43093i	0.698649	+	0.715465i	$0.253785\pi$
$233$	21.8422i	1.43093i	−0.698649	+	0.715465i	$0.746215\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.0870874i	0.00565694i
$238$	0	0
$239$	−21.5968	−1.39698	−0.698490	−	0.715620i	$-0.746144\pi$
$239$	−21.5968	−1.39698	−0.698490	+	0.715620i	$0.746144\pi$
$240$	0	0
$241$	12.6305	0.813603	0.406802	−	0.913516i	$-0.366644\pi$
$241$	12.6305	0.813603	0.406802	+	0.913516i	$0.366644\pi$
$242$	0	0
$243$	− 2.55723i	− 0.164046i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 31.3333i	− 1.99369i
$248$	0	0
$249$	0.960434	0.0608650
$250$	0	0
$251$	20.7702	1.31101	0.655503	−	0.755192i	$-0.272457\pi$
$251$	20.7702	1.31101	0.655503	+	0.755192i	$0.272457\pi$
$252$	0	0
$253$	− 14.7808i	− 0.929263i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.8207i	0.924491i	0.886752	+	0.462245i	$0.152956\pi$
$257$	14.8207i	0.924491i	−0.886752	+	0.462245i	$0.847044\pi$
$258$	0	0
$259$	6.26159	0.389076
$260$	0	0
$261$	5.41307	0.335061
$262$	0	0
$263$	− 18.4175i	− 1.13567i	−0.823142	−	0.567836i	$-0.807781\pi$
$263$	− 18.4175i	− 1.13567i	0.823142	−	0.567836i	$-0.192219\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.612437i	− 0.0374805i
$268$	0	0
$269$	25.8916	1.57864	0.789318	−	0.613984i	$-0.210434\pi$
$269$	25.8916	1.57864	0.789318	+	0.613984i	$0.210434\pi$
$270$	0	0
$271$	−17.5017	−1.06315	−0.531576	−	0.847010i	$-0.678400\pi$
$271$	−17.5017	−1.06315	−0.531576	+	0.847010i	$0.678400\pi$
$272$	0	0
$273$	− 1.98300i	− 0.120017i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.77179i	0.527045i	0.964653	+	0.263523i	$0.0848844\pi$
$277$	8.77179i	0.527045i	−0.964653	+	0.263523i	$0.915116\pi$
$278$	0	0
$279$	−4.96351	−0.297158
$280$	0	0
$281$	16.2215	0.967692	0.483846	−	0.875153i	$-0.339239\pi$
$281$	16.2215	0.967692	0.483846	+	0.875153i	$0.339239\pi$
$282$	0	0
$283$	11.9819i	0.712251i	0.934438	+	0.356125i	$0.115902\pi$
$283$	11.9819i	0.712251i	−0.934438	+	0.356125i	$0.884098\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.14501i	− 0.185644i
$288$	0	0
$289$	−9.63803	−0.566943
$290$	0	0
$291$	−0.791411	−0.0463933
$292$	0	0
$293$	11.2516i	0.657323i	0.944448	+	0.328661i	$0.106597\pi$
$293$	11.2516i	0.657323i	−0.944448	+	0.328661i	$0.893403\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.954650i	0.0553944i
$298$	0	0
$299$	58.4859	3.38233
$300$	0	0
$301$	4.62103	0.266352
$302$	0	0
$303$	− 0.931134i	− 0.0534922i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.61244i	0.491538i	0.969328	+	0.245769i	$0.0790404\pi$
$307$	8.61244i	0.491538i	−0.969328	+	0.245769i	$0.920960\pi$
$308$	0	0
$309$	0.0616787	0.00350878
$310$	0	0
$311$	31.7575	1.80080	0.900400	−	0.435063i	$-0.143274\pi$
$311$	31.7575	1.80080	0.900400	+	0.435063i	$0.143274\pi$
$312$	0	0
$313$	− 2.21777i	− 0.125356i	−0.998034	−	0.0626778i	$-0.980036\pi$
$313$	− 2.21777i	− 0.125356i	0.998034	−	0.0626778i	$-0.0199641\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.3947i	− 1.03315i	−0.856243	−	0.516574i	$-0.827207\pi$
$317$	− 18.3947i	− 1.03315i	0.856243	−	0.516574i	$-0.172793\pi$
$318$	0	0
$319$	−3.03269	−0.169798
$320$	0	0
$321$	1.59783	0.0891820
$322$	0	0
$323$	24.3899i	1.35709i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.366083i	0.0202444i
$328$	0	0
$329$	26.8440	1.47996
$330$	0	0
$331$	10.1160	0.556027	0.278014	−	0.960577i	$-0.410324\pi$
$331$	10.1160	0.556027	0.278014	+	0.960577i	$0.410324\pi$
$332$	0	0
$333$	− 5.95487i	− 0.326325i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.81135i	0.425511i	0.977105	+	0.212756i	$0.0682439\pi$
$337$	7.81135i	0.425511i	−0.977105	+	0.212756i	$0.931756\pi$
$338$	0	0
$339$	0.727513	0.0395131
$340$	0	0
$341$	2.78083	0.150590
$342$	0	0
$343$	12.9225i	0.697749i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.6281i	− 0.838959i	−0.907765	−	0.419480i	$-0.862213\pi$
$347$	− 15.6281i	− 0.838959i	0.907765	−	0.419480i	$-0.137787\pi$
$348$	0	0
$349$	−7.49265	−0.401073	−0.200536	−	0.979686i	$-0.564268\pi$
$349$	−7.49265	−0.401073	−0.200536	+	0.979686i	$0.564268\pi$
$350$	0	0
$351$	−3.77743	−0.201624
$352$	0	0
$353$	− 22.4313i	− 1.19390i	−0.802279	−	0.596949i	$-0.796380\pi$
$353$	− 22.4313i	− 1.19390i	0.802279	−	0.596949i	$-0.203620\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.54357i	0.0816944i
$358$	0	0
$359$	15.6196	0.824372	0.412186	−	0.911100i	$-0.364765\pi$
$359$	15.6196	0.824372	0.412186	+	0.911100i	$0.364765\pi$
$360$	0	0
$361$	3.33145	0.175340
$362$	0	0
$363$	0.779014i	0.0408876i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.82790i	− 0.199815i	−0.994997	−	0.0999073i	$-0.968145\pi$
$367$	− 3.82790i	− 0.199815i	0.994997	−	0.0999073i	$-0.0318546\pi$
$368$	0	0
$369$	−2.99096	−0.155703
$370$	0	0
$371$	−29.4103	−1.52690
$372$	0	0
$373$	18.1997i	0.942344i	0.882041	+	0.471172i	$0.156169\pi$
$373$	18.1997i	0.942344i	−0.882041	+	0.471172i	$0.843831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	14.7627	0.758311	0.379156	−	0.925333i	$-0.376215\pi$
$379$	14.7627	0.758311	0.379156	+	0.925333i	$0.376215\pi$
$380$	0	0
$381$	1.41885	0.0726901
$382$	0	0
$383$	− 4.75406i	− 0.242921i	−0.992596	−	0.121460i	$-0.961242\pi$
$383$	− 4.75406i	− 0.242921i	0.992596	−	0.121460i	$-0.0387578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.39467i	− 0.223394i
$388$	0	0
$389$	−14.3804	−0.729114	−0.364557	−	0.931181i	$-0.618780\pi$
$389$	−14.3804	−0.729114	−0.364557	+	0.931181i	$0.618780\pi$
$390$	0	0
$391$	−45.5255	−2.30232
$392$	0	0
$393$	− 0.398461i	− 0.0200997i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.9311i	− 0.548618i	−0.961642	−	0.274309i	$-0.911551\pi$
$397$	− 10.9311i	− 0.548618i	0.961642	−	0.274309i	$-0.0884491\pi$
$398$	0	0
$399$	1.41330	0.0707534
$400$	0	0
$401$	30.4132	1.51876	0.759382	−	0.650646i	$-0.225502\pi$
$401$	30.4132	1.51876	0.759382	+	0.650646i	$0.225502\pi$
$402$	0	0
$403$	11.0034i	0.548118i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.33624i	0.165371i
$408$	0	0
$409$	−13.3100	−0.658136	−0.329068	−	0.944306i	$-0.606734\pi$
$409$	−13.3100	−0.658136	−0.329068	+	0.944306i	$0.606734\pi$
$410$	0	0
$411$	0.389870	0.0192309
$412$	0	0
$413$	− 27.7413i	− 1.36506i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.13941i	− 0.0557970i
$418$	0	0
$419$	−15.7917	−0.771476	−0.385738	−	0.922608i	$-0.626053\pi$
$419$	−15.7917	−0.771476	−0.385738	+	0.922608i	$0.626053\pi$
$420$	0	0
$421$	−34.4317	−1.67810	−0.839050	−	0.544054i	$-0.816889\pi$
$421$	−34.4317	−1.67810	−0.839050	+	0.544054i	$0.816889\pi$
$422$	0	0
$423$	− 25.5291i	− 1.24127i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	39.7232i	1.92234i
$428$	0	0
$429$	1.05656	0.0510114
$430$	0	0
$431$	5.69189	0.274168	0.137084	−	0.990559i	$-0.456227\pi$
$431$	5.69189	0.274168	0.137084	+	0.990559i	$0.456227\pi$
$432$	0	0
$433$	13.0034i	0.624903i	0.949934	+	0.312452i	$0.101150\pi$
$433$	13.0034i	0.624903i	−0.949934	+	0.312452i	$0.898850\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.6833i	1.99398i
$438$	0	0
$439$	−9.59593	−0.457989	−0.228994	−	0.973428i	$-0.573544\pi$
$439$	−9.59593	−0.457989	−0.228994	+	0.973428i	$0.573544\pi$
$440$	0	0
$441$	−8.64720	−0.411772
$442$	0	0
$443$	2.47072i	0.117388i	0.998276	+	0.0586938i	$0.0186936\pi$
$443$	2.47072i	0.117388i	−0.998276	+	0.0586938i	$0.981306\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 0.172102i	− 0.00814015i
$448$	0	0
$449$	−35.2324	−1.66272	−0.831359	−	0.555735i	$-0.812437\pi$
$449$	−35.2324	−1.66272	−0.831359	+	0.555735i	$0.812437\pi$
$450$	0	0
$451$	1.67570	0.0789054
$452$	0	0
$453$	2.03084i	0.0954172i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.0241i	− 1.73191i	−0.500118	−	0.865957i	$-0.666710\pi$
$457$	− 37.0241i	− 1.73191i	0.500118	−	0.865957i	$-0.333290\pi$
$458$	0	0
$459$	2.94036	0.137244
$460$	0	0
$461$	−27.6324	−1.28697	−0.643484	−	0.765460i	$-0.722512\pi$
$461$	−27.6324	−1.28697	−0.643484	+	0.765460i	$0.722512\pi$
$462$	0	0
$463$	− 23.1877i	− 1.07763i	−0.842425	−	0.538813i	$-0.818873\pi$
$463$	− 23.1877i	− 1.07763i	0.842425	−	0.538813i	$-0.181127\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.5235i	− 1.27364i	−0.771014	−	0.636818i	$-0.780250\pi$
$467$	− 27.5235i	− 1.27364i	0.771014	−	0.636818i	$-0.219750\pi$
$468$	0	0
$469$	−30.4302	−1.40514
$470$	0	0
$471$	1.24675	0.0574473
$472$	0	0
$473$	2.46213i	0.113209i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	27.9696i	1.28064i
$478$	0	0
$479$	−11.1088	−0.507576	−0.253788	−	0.967260i	$-0.581677\pi$
$479$	−11.1088	−0.507576	−0.253788	+	0.967260i	$0.581677\pi$
$480$	0	0
$481$	−13.2011	−0.601918
$482$	0	0
$483$	2.63803i	0.120034i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 31.3933i	− 1.42256i	−0.702906	−	0.711282i	$-0.748115\pi$
$487$	− 31.3933i	− 1.42256i	0.702906	−	0.711282i	$-0.251885\pi$
$488$	0	0
$489$	−1.02519	−0.0463608
$490$	0	0
$491$	39.9590	1.80333	0.901663	−	0.432440i	$-0.142347\pi$
$491$	39.9590	1.80333	0.901663	+	0.432440i	$0.142347\pi$
$492$	0	0
$493$	9.34081i	0.420689i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.44079i	− 0.109484i
$498$	0	0
$499$	−16.0951	−0.720515	−0.360258	−	0.932853i	$-0.617311\pi$
$499$	−16.0951	−0.720515	−0.360258	+	0.932853i	$0.617311\pi$
$500$	0	0
$501$	1.30048	0.0581010
$502$	0	0
$503$	− 5.97717i	− 0.266509i	−0.991082	−	0.133254i	$-0.957457\pi$
$503$	− 5.97717i	− 0.266509i	0.991082	−	0.133254i	$-0.0425427\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.94447i	0.130769i
$508$	0	0
$509$	12.5726	0.557269	0.278634	−	0.960397i	$-0.410118\pi$
$509$	12.5726	0.557269	0.278634	+	0.960397i	$0.410118\pi$
$510$	0	0
$511$	26.2025	1.15913
$512$	0	0
$513$	− 2.69220i	− 0.118864i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.3028i	0.629036i
$518$	0	0
$519$	−1.00610	−0.0441629
$520$	0	0
$521$	−31.6233	−1.38544	−0.692722	−	0.721205i	$-0.743589\pi$
$521$	−31.6233	−1.38544	−0.692722	+	0.721205i	$0.743589\pi$
$522$	0	0
$523$	21.5405i	0.941900i	0.882160	+	0.470950i	$0.156089\pi$
$523$	21.5405i	0.941900i	−0.882160	+	0.470950i	$0.843911\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.56505i	− 0.373099i
$528$	0	0
$529$	−54.8049	−2.38282
$530$	0	0
$531$	−26.3824	−1.14490
$532$	0	0
$533$	6.63052i	0.287200i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.117475i	− 0.00506944i
$538$	0	0
$539$	4.84463	0.208673
$540$	0	0
$541$	31.4927	1.35397	0.676987	−	0.735995i	$-0.263285\pi$
$541$	31.4927	1.35397	0.676987	+	0.735995i	$0.263285\pi$
$542$	0	0
$543$	0.371784i	0.0159548i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 17.6861i	− 0.756201i	−0.925765	−	0.378100i	$-0.876577\pi$
$547$	− 17.6861i	− 0.756201i	0.925765	−	0.378100i	$-0.123423\pi$
$548$	0	0
$549$	37.7774	1.61230
$550$	0	0
$551$	8.55248	0.364348
$552$	0	0
$553$	− 2.88022i	− 0.122479i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.3296i	− 0.819021i	−0.912305	−	0.409511i	$-0.865699\pi$
$557$	− 19.3296i	− 0.819021i	0.912305	−	0.409511i	$-0.134301\pi$
$558$	0	0
$559$	−9.74235	−0.412057
$560$	0	0
$561$	−0.822430	−0.0347230
$562$	0	0
$563$	− 35.7299i	− 1.50583i	−0.658115	−	0.752917i	$-0.728646\pi$
$563$	− 35.7299i	− 1.50583i	0.658115	−	0.752917i	$-0.271354\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	28.0494i	1.17797i
$568$	0	0
$569$	−4.65187	−0.195016	−0.0975082	−	0.995235i	$-0.531087\pi$
$569$	−4.65187	−0.195016	−0.0975082	+	0.995235i	$0.531087\pi$
$570$	0	0
$571$	−20.6791	−0.865393	−0.432697	−	0.901540i	$-0.642438\pi$
$571$	−20.6791	−0.865393	−0.432697	+	0.901540i	$0.642438\pi$
$572$	0	0
$573$	0.695813i	0.0290680i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.7808i	− 1.03164i	−0.856697	−	0.515820i	$-0.827487\pi$
$577$	− 24.7808i	− 1.03164i	0.856697	−	0.515820i	$-0.172513\pi$
$578$	0	0
$579$	1.26484	0.0525649
$580$	0	0
$581$	−31.7641	−1.31780
$582$	0	0
$583$	− 15.6701i	− 0.648989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13.4027i	− 0.553187i	−0.960987	−	0.276594i	$-0.910794\pi$
$587$	− 13.4027i	− 0.553187i	0.960987	−	0.276594i	$-0.0892056\pi$
$588$	0	0
$589$	−7.84219	−0.323132
$590$	0	0
$591$	−1.48053	−0.0609010
$592$	0	0
$593$	1.83911i	0.0755233i	0.999287	+	0.0377616i	$0.0120228\pi$
$593$	1.83911i	0.0755233i	−0.999287	+	0.0377616i	$0.987977\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.65285i	− 0.108574i
$598$	0	0
$599$	35.9457	1.46870	0.734352	−	0.678769i	$-0.237486\pi$
$599$	35.9457	1.46870	0.734352	+	0.678769i	$0.237486\pi$
$600$	0	0
$601$	−29.8858	−1.21907	−0.609533	−	0.792760i	$-0.708643\pi$
$601$	−29.8858	−1.21907	−0.609533	+	0.792760i	$0.708643\pi$
$602$	0	0
$603$	28.9396i	1.17851i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.25015i	0.172508i	0.996273	+	0.0862541i	$0.0274897\pi$
$607$	4.25015i	0.172508i	−0.996273	+	0.0862541i	$0.972510\pi$
$608$	0	0
$609$	0.541264	0.0219331
$610$	0	0
$611$	−56.5944	−2.28956
$612$	0	0
$613$	9.35184i	0.377717i	0.982004	+	0.188859i	$0.0604788\pi$
$613$	9.35184i	0.377717i	−0.982004	+	0.188859i	$0.939521\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.6019i	0.507332i	0.967292	+	0.253666i	$0.0816363\pi$
$617$	12.6019i	0.507332i	−0.967292	+	0.253666i	$0.918364\pi$
$618$	0	0
$619$	2.54018	0.102098	0.0510491	−	0.998696i	$-0.483743\pi$
$619$	2.54018	0.102098	0.0510491	+	0.998696i	$0.483743\pi$
$620$	0	0
$621$	5.02519	0.201654
$622$	0	0
$623$	20.2549i	0.811497i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.753020i	0.0300727i
$628$	0	0
$629$	10.2757	0.409720
$630$	0	0
$631$	48.0057	1.91108	0.955538	−	0.294867i	$-0.0952752\pi$
$631$	48.0057	1.91108	0.955538	+	0.294867i	$0.0952752\pi$
$632$	0	0
$633$	− 1.72580i	− 0.0685945i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.1696i	0.759528i
$638$	0	0
$639$	−2.32123	−0.0918262
$640$	0	0
$641$	1.02148	0.0403461	0.0201730	−	0.999797i	$-0.493578\pi$
$641$	1.02148	0.0403461	0.0201730	+	0.999797i	$0.493578\pi$
$642$	0	0
$643$	8.40606i	0.331503i	0.986168	+	0.165751i	$0.0530049\pi$
$643$	8.40606i	0.331503i	−0.986168	+	0.165751i	$0.946995\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.86778i	0.230686i	0.993326	+	0.115343i	$0.0367968\pi$
$647$	5.86778i	0.230686i	−0.993326	+	0.115343i	$0.963203\pi$
$648$	0	0
$649$	14.7808	0.580198
$650$	0	0
$651$	−0.496312	−0.0194520
$652$	0	0
$653$	− 37.5749i	− 1.47042i	−0.677841	−	0.735209i	$-0.737084\pi$
$653$	− 37.5749i	− 1.47042i	0.677841	−	0.735209i	$-0.262916\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 24.9190i	− 0.972183i
$658$	0	0
$659$	16.8360	0.655839	0.327919	−	0.944706i	$-0.393653\pi$
$659$	16.8360	0.655839	0.327919	+	0.944706i	$0.393653\pi$
$660$	0	0
$661$	4.84983	0.188637	0.0943183	−	0.995542i	$-0.469933\pi$
$661$	4.84983	0.188637	0.0943183	+	0.995542i	$0.469933\pi$
$662$	0	0
$663$	− 3.25426i	− 0.126385i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.9638i	0.618122i
$668$	0	0
$669$	0.623651	0.0241117
$670$	0	0
$671$	−21.1649	−0.817062
$672$	0	0
$673$	− 6.24065i	− 0.240559i	−0.992740	−	0.120280i	$-0.961621\pi$
$673$	− 6.24065i	− 0.240559i	0.992740	−	0.120280i	$-0.0383792\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29.3566i	− 1.12827i	−0.825683	−	0.564134i	$-0.809210\pi$
$677$	− 29.3566i	− 1.12827i	0.825683	−	0.564134i	$-0.190790\pi$
$678$	0	0
$679$	26.1741	1.00447
$680$	0	0
$681$	2.23360	0.0855918
$682$	0	0
$683$	25.6861i	0.982849i	0.870920	+	0.491425i	$0.163524\pi$
$683$	25.6861i	0.982849i	−0.870920	+	0.491425i	$0.836476\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.15542i	0.0440820i
$688$	0	0
$689$	62.0046	2.36219
$690$	0	0
$691$	8.77128	0.333675	0.166838	−	0.985984i	$-0.446644\pi$
$691$	8.77128	0.333675	0.166838	+	0.985984i	$0.446644\pi$
$692$	0	0
$693$	− 15.7626i	− 0.598772i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.16120i	− 0.195495i
$698$	0	0
$699$	2.07706	0.0785616
$700$	0	0
$701$	−15.1075	−0.570602	−0.285301	−	0.958438i	$-0.592094\pi$
$701$	−15.1075	−0.570602	−0.285301	+	0.958438i	$0.592094\pi$
$702$	0	0
$703$	− 9.40850i	− 0.354848i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.7951i	1.15817i
$708$	0	0
$709$	12.0132	0.451165	0.225583	−	0.974224i	$-0.427571\pi$
$709$	12.0132	0.451165	0.225583	+	0.974224i	$0.427571\pi$
$710$	0	0
$711$	−2.73913	−0.102725
$712$	0	0
$713$	− 14.6380i	− 0.548198i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.05372i	0.0766977i
$718$	0	0
$719$	2.27778	0.0849468	0.0424734	−	0.999098i	$-0.486476\pi$
$719$	2.27778	0.0849468	0.0424734	+	0.999098i	$0.486476\pi$
$720$	0	0
$721$	−2.03988	−0.0759692
$722$	0	0
$723$	− 1.20109i	− 0.0446689i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.71952i	0.286301i	0.989701	+	0.143151i	$0.0457233\pi$
$727$	7.71952i	0.286301i	−0.989701	+	0.143151i	$0.954277\pi$
$728$	0	0
$729$	26.5129	0.981960
$730$	0	0
$731$	7.58345	0.280484
$732$	0	0
$733$	− 33.2753i	− 1.22905i	−0.788896	−	0.614526i	$-0.789347\pi$
$733$	− 33.2753i	− 1.22905i	0.788896	−	0.614526i	$-0.210653\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.2135i	− 0.597233i
$738$	0	0
$739$	−48.0057	−1.76592	−0.882959	−	0.469450i	$-0.844452\pi$
$739$	−48.0057	−1.76592	−0.882959	+	0.469450i	$0.844452\pi$
$740$	0	0
$741$	−2.97961	−0.109459
$742$	0	0
$743$	− 43.1345i	− 1.58245i	−0.611524	−	0.791226i	$-0.709443\pi$
$743$	− 43.1345i	− 1.58245i	0.611524	−	0.791226i	$-0.290557\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.2082i	1.10526i
$748$	0	0
$749$	−52.8445	−1.93090
$750$	0	0
$751$	−6.46883	−0.236051	−0.118025	−	0.993011i	$-0.537656\pi$
$751$	−6.46883	−0.236051	−0.118025	+	0.993011i	$0.537656\pi$
$752$	0	0
$753$	− 1.97512i	− 0.0719775i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 39.5531i	− 1.43758i	−0.695227	−	0.718790i	$-0.744696\pi$
$757$	− 39.5531i	− 1.43758i	0.695227	−	0.718790i	$-0.255304\pi$
$758$	0	0
$759$	−1.40557	−0.0510189
$760$	0	0
$761$	28.5130	1.03360	0.516799	−	0.856107i	$-0.327124\pi$
$761$	28.5130	1.03360	0.516799	+	0.856107i	$0.327124\pi$
$762$	0	0
$763$	− 12.1073i	− 0.438315i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58.4859i	2.11180i
$768$	0	0
$769$	−49.7184	−1.79289	−0.896445	−	0.443154i	$-0.853859\pi$
$769$	−49.7184	−1.79289	−0.896445	+	0.443154i	$0.853859\pi$
$770$	0	0
$771$	1.40936	0.0507569
$772$	0	0
$773$	24.2539i	0.872351i	0.899862	+	0.436175i	$0.143667\pi$
$773$	24.2539i	0.872351i	−0.899862	+	0.436175i	$0.856333\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.595439i	− 0.0213613i
$778$	0	0
$779$	−4.72562	−0.169313
$780$	0	0
$781$	1.30048	0.0465347
$782$	0	0
$783$	− 1.03106i	− 0.0368470i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.907346i	0.0323434i	0.999869	+	0.0161717i	$0.00514784\pi$
$787$	0.907346i	0.0323434i	−0.999869	+	0.0161717i	$0.994852\pi$
$788$	0	0
$789$	−1.75139	−0.0623512
$790$	0	0
$791$	−24.0608	−0.855504
$792$	0	0
$793$	− 83.7470i	− 2.97394i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32.4934i	− 1.15098i	−0.817810	−	0.575488i	$-0.804812\pi$
$797$	− 32.4934i	− 1.15098i	0.817810	−	0.575488i	$-0.195188\pi$
$798$	0	0
$799$	44.0531	1.55849
$800$	0	0
$801$	19.2628	0.680616
$802$	0	0
$803$	13.9610i	0.492672i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.46213i	− 0.0866711i
$808$	0	0
$809$	2.07804	0.0730602	0.0365301	−	0.999333i	$-0.488370\pi$
$809$	2.07804	0.0730602	0.0365301	+	0.999333i	$0.488370\pi$
$810$	0	0
$811$	−13.7518	−0.482893	−0.241446	−	0.970414i	$-0.577622\pi$
$811$	−13.7518	−0.482893	−0.241446	+	0.970414i	$0.577622\pi$
$812$	0	0
$813$	1.66431i	0.0583697i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.94344i	− 0.242920i
$818$	0	0
$819$	62.3707	2.17941
$820$	0	0
$821$	6.76979	0.236267	0.118134	−	0.992998i	$-0.462309\pi$
$821$	6.76979	0.236267	0.118134	+	0.992998i	$0.462309\pi$
$822$	0	0
$823$	− 22.3575i	− 0.779335i	−0.920956	−	0.389667i	$-0.872590\pi$
$823$	− 22.3575i	− 0.779335i	0.920956	−	0.389667i	$-0.127410\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.1160i	0.908143i	0.890965	+	0.454072i	$0.150029\pi$
$827$	26.1160i	0.908143i	−0.890965	+	0.454072i	$0.849971\pi$
$828$	0	0
$829$	19.7540	0.686085	0.343043	−	0.939320i	$-0.388542\pi$
$829$	19.7540	0.686085	0.343043	+	0.939320i	$0.388542\pi$
$830$	0	0
$831$	0.834144	0.0289361
$832$	0	0
$833$	− 14.9216i	− 0.517004i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.945427i	0.0326787i
$838$	0	0
$839$	−20.8188	−0.718745	−0.359373	−	0.933194i	$-0.617009\pi$
$839$	−20.8188	−0.718745	−0.359373	+	0.933194i	$0.617009\pi$
$840$	0	0
$841$	−25.7246	−0.887054
$842$	0	0
$843$	− 1.54256i	− 0.0531287i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25.7641i	− 0.885265i
$848$	0	0
$849$	1.13941	0.0391044
$850$	0	0
$851$	17.5617	0.602006
$852$	0	0
$853$	− 48.6666i	− 1.66631i	−0.553037	−	0.833157i	$-0.686531\pi$
$853$	− 48.6666i	− 1.66631i	0.553037	−	0.833157i	$-0.313469\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.8515i	− 0.439001i	−0.975612	−	0.219500i	$-0.929557\pi$
$857$	− 12.8515i	− 0.439001i	0.975612	−	0.219500i	$-0.0704427\pi$
$858$	0	0
$859$	1.60904	0.0548998	0.0274499	−	0.999623i	$-0.491261\pi$
$859$	1.60904	0.0548998	0.0274499	+	0.999623i	$0.491261\pi$
$860$	0	0
$861$	−0.299072	−0.0101923
$862$	0	0
$863$	46.0494i	1.56754i	0.621052	+	0.783770i	$0.286706\pi$
$863$	46.0494i	1.56754i	−0.621052	+	0.783770i	$0.713294\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.916518i	0.0311266i
$868$	0	0
$869$	1.53461	0.0520581
$870$	0	0
$871$	64.1549	2.17381
$872$	0	0
$873$	− 24.8920i	− 0.842465i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.63392i	0.122709i	0.998116	+	0.0613543i	$0.0195420\pi$
$877$	3.63392i	0.122709i	−0.998116	+	0.0613543i	$0.980458\pi$
$878$	0	0
$879$	1.06995	0.0360887
$880$	0	0
$881$	28.2426	0.951519	0.475759	−	0.879575i	$-0.342173\pi$
$881$	28.2426	0.951519	0.475759	+	0.879575i	$0.342173\pi$
$882$	0	0
$883$	4.23962i	0.142674i	0.997452	+	0.0713372i	$0.0227266\pi$
$883$	4.23962i	0.142674i	−0.997452	+	0.0713372i	$0.977273\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 46.4003i	− 1.55797i	−0.627043	−	0.778984i	$-0.715735\pi$
$887$	− 46.4003i	− 1.55797i	0.627043	−	0.778984i	$-0.284265\pi$
$888$	0	0
$889$	−46.9254	−1.57383
$890$	0	0
$891$	−14.9450	−0.500677
$892$	0	0
$893$	− 40.3352i	− 1.34976i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.56166i	− 0.185698i
$898$	0	0
$899$	−3.00339	−0.100169
$900$	0	0
$901$	−48.2644	−1.60792
$902$	0	0
$903$	− 0.439432i	− 0.0146234i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27.0358i	− 0.897708i	−0.893605	−	0.448854i	$-0.851832\pi$
$907$	− 27.0358i	− 0.897708i	0.893605	−	0.448854i	$-0.148168\pi$
$908$	0	0
$909$	29.2866	0.971376
$910$	0	0
$911$	−42.3500	−1.40312	−0.701559	−	0.712612i	$-0.747512\pi$
$911$	−42.3500	−1.40312	−0.701559	+	0.712612i	$0.747512\pi$
$912$	0	0
$913$	− 16.9243i	− 0.560111i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.1782i	0.435183i
$918$	0	0
$919$	−8.17400	−0.269635	−0.134818	−	0.990870i	$-0.543045\pi$
$919$	−8.17400	−0.269635	−0.134818	+	0.990870i	$0.543045\pi$
$920$	0	0
$921$	0.818991	0.0269867
$922$	0	0
$923$	5.14582i	0.169377i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.93996i	0.0637166i
$928$	0	0
$929$	−42.2596	−1.38649	−0.693247	−	0.720700i	$-0.743820\pi$
$929$	−42.2596	−1.38649	−0.693247	+	0.720700i	$0.743820\pi$
$930$	0	0
$931$	−13.6623	−0.447764
$932$	0	0
$933$	− 3.01994i	− 0.0988684i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.16600i	0.0380917i	0.999819	+	0.0190458i	$0.00606284\pi$
$937$	1.16600i	0.0380917i	−0.999819	+	0.0190458i	$0.993937\pi$
$938$	0	0
$939$	−0.210896	−0.00688234
$940$	0	0
$941$	4.93864	0.160995	0.0804975	−	0.996755i	$-0.474349\pi$
$941$	4.93864	0.160995	0.0804975	+	0.996755i	$0.474349\pi$
$942$	0	0
$943$	− 8.82071i	− 0.287242i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.3210i	0.822822i	0.911450	+	0.411411i	$0.134964\pi$
$947$	25.3210i	0.822822i	−0.911450	+	0.411411i	$0.865036\pi$
$948$	0	0
$949$	−55.2419	−1.79323
$950$	0	0
$951$	−1.74922	−0.0567224
$952$	0	0
$953$	47.6143i	1.54238i	0.636606	+	0.771189i	$0.280338\pi$
$953$	47.6143i	1.54238i	−0.636606	+	0.771189i	$0.719662\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.288391i	0.00932235i
$958$	0	0
$959$	−12.8941	−0.416371
$960$	0	0
$961$	−28.2460	−0.911163
$962$	0	0
$963$	50.2559i	1.61947i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 25.9763i	− 0.835342i	−0.908598	−	0.417671i	$-0.862847\pi$
$967$	− 25.9763i	− 0.835342i	0.908598	−	0.417671i	$-0.137153\pi$
$968$	0	0
$969$	2.31933	0.0745076
$970$	0	0
$971$	−55.8687	−1.79291	−0.896457	−	0.443132i	$-0.853867\pi$
$971$	−55.8687	−1.79291	−0.896457	+	0.443132i	$0.853867\pi$
$972$	0	0
$973$	37.6833i	1.20807i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.5241i	− 1.36047i	−0.732996	−	0.680233i	$-0.761879\pi$
$977$	− 42.5241i	− 1.36047i	0.732996	−	0.680233i	$-0.238121\pi$
$978$	0	0
$979$	−10.7920	−0.344915
$980$	0	0
$981$	−11.5143	−0.367622
$982$	0	0
$983$	− 50.8977i	− 1.62338i	−0.584086	−	0.811692i	$-0.698547\pi$
$983$	− 50.8977i	− 1.62338i	0.584086	−	0.811692i	$-0.301453\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.55271i	− 0.0812535i
$988$	0	0
$989$	12.9604	0.412118
$990$	0	0
$991$	−27.7147	−0.880387	−0.440194	−	0.897903i	$-0.645090\pi$
$991$	−27.7147	−0.880387	−0.440194	+	0.897903i	$0.645090\pi$
$992$	0	0
$993$	− 0.961973i	− 0.0305273i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.3480i	− 0.707768i	−0.935289	−	0.353884i	$-0.884861\pi$
$997$	− 22.3480i	− 0.707768i	0.935289	−	0.353884i	$-0.115139\pi$
$998$	0	0
$999$	−1.13426	−0.0358863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.c.1149.4		8
5.2	odd	4		4100.2.a.c.1.3		4
5.3	odd	4		164.2.a.a.1.2	✓	4
5.4	even	2	inner	4100.2.d.c.1149.5		8
15.8	even	4		1476.2.a.g.1.3		4
20.3	even	4		656.2.a.i.1.3		4
35.13	even	4		8036.2.a.i.1.3		4
40.3	even	4		2624.2.a.y.1.2		4
40.13	odd	4		2624.2.a.v.1.3		4
60.23	odd	4		5904.2.a.bp.1.3		4
205.163	odd	4		6724.2.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.2	✓	4	5.3	odd	4
656.2.a.i.1.3		4	20.3	even	4
1476.2.a.g.1.3		4	15.8	even	4
2624.2.a.v.1.3		4	40.13	odd	4
2624.2.a.y.1.2		4	40.3	even	4
4100.2.a.c.1.3		4	5.2	odd	4
4100.2.d.c.1149.4		8	1.1	even	1	trivial
4100.2.d.c.1149.5		8	5.4	even	2	inner
5904.2.a.bp.1.3		4	60.23	odd	4
6724.2.a.c.1.3		4	205.163	odd	4
8036.2.a.i.1.3		4	35.13	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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0950939i q^{3} +3.14501i q^{7} +2.99096 q^{9} +O(q^{10})\)	\(q-0.0950939i q^{3} +3.14501i q^{7} +2.99096 q^{9} -1.67570 q^{11} -6.63052i q^{13} +5.16120i q^{17} +4.72562 q^{19} +0.299072 q^{21} +8.82071i q^{23} -0.569704i q^{27} +1.80981 q^{29} -1.65951 q^{31} +0.159349i q^{33} -1.99096i q^{37} -0.630522 q^{39} -1.00000 q^{41} -1.46932i q^{43} -8.53543i q^{47} -2.89112 q^{49} +0.490799 q^{51} +9.35139i q^{53} -0.449377i q^{57} -8.82071 q^{59} +12.6305 q^{61} +9.40660i q^{63} +9.67570i q^{67} +0.838796 q^{69} -0.776081 q^{71} -8.33145i q^{73} -5.27009i q^{77} -0.915804 q^{79} +8.91870 q^{81} +10.0998i q^{83} -0.172102i q^{87} +6.44033 q^{89} +20.8531 q^{91} +0.157809i q^{93} -8.32241i q^{97} -5.01193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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