LMFDB - Embedded newform 4100.2.d.c.1149.3

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			1149.3
Root			$0.626295 - 0.626295i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.c.1149.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21551i q^{3} -5.06479i q^{7} -1.90849 q^{9} +O(q^{10})$	$q-2.21551i q^{3} -5.06479i q^{7} -1.90849 q^{9} -2.55961 q^{11} +4.93620i q^{13} +2.68821i q^{17} -4.72069 q^{19} -11.2211 q^{21} +1.49482i q^{23} -2.41826i q^{27} -2.43102 q^{29} +3.19339 q^{31} +5.67085i q^{33} +2.90849i q^{37} +10.9362 q^{39} -1.00000 q^{41} +7.62441i q^{43} +5.15171i q^{47} -18.6521 q^{49} +5.95575 q^{51} +11.1192i q^{53} +10.4587i q^{57} -1.49482 q^{59} +1.06380 q^{61} +9.66608i q^{63} +10.5596i q^{67} +3.31179 q^{69} -12.6023 q^{71} -8.28490i q^{73} +12.9639i q^{77} +4.28967 q^{79} -11.0831 q^{81} -10.5606i q^{83} +5.38595i q^{87} -9.36722 q^{89} +25.0008 q^{91} -7.07498i q^{93} -3.37641i q^{97} +4.88499 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 2.21551i	− 1.27913i	−0.768739	−	0.639563i	$-0.779115\pi$
$3$	− 2.21551i	− 1.27913i	0.768739	−	0.639563i	$-0.220885\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.06479i	− 1.91431i	−0.289574	−	0.957156i	$-0.593514\pi$
$7$	− 5.06479i	− 1.91431i	0.289574	−	0.957156i	$-0.406486\pi$
$8$	0	0
$9$	−1.90849	−0.636162
$10$	0	0
$11$	−2.55961	−0.771753	−0.385876	−	0.922551i	$-0.626101\pi$
$11$	−2.55961	−0.771753	−0.385876	+	0.922551i	$0.626101\pi$
$12$	0	0
$13$	4.93620i	1.36906i	0.728987	+	0.684528i	$0.239991\pi$
$13$	4.93620i	1.36906i	−0.728987	+	0.684528i	$0.760009\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.68821i	0.651986i	0.945372	+	0.325993i	$0.105699\pi$
$17$	2.68821i	0.651986i	−0.945372	+	0.325993i	$0.894301\pi$
$18$	0	0
$19$	−4.72069	−1.08300	−0.541500	−	0.840701i	$-0.682143\pi$
$19$	−4.72069	−1.08300	−0.541500	+	0.840701i	$0.682143\pi$
$20$	0	0
$21$	−11.2211	−2.44864
$22$	0	0
$23$	1.49482i	0.311692i	0.987781	+	0.155846i	$0.0498103\pi$
$23$	1.49482i	0.311692i	−0.987781	+	0.155846i	$0.950190\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.41826i	− 0.465395i
$28$	0	0
$29$	−2.43102	−0.451429	−0.225715	−	0.974193i	$-0.572472\pi$
$29$	−2.43102	−0.451429	−0.225715	+	0.974193i	$0.572472\pi$
$30$	0	0
$31$	3.19339	0.573549	0.286774	−	0.957998i	$-0.407417\pi$
$31$	3.19339	0.573549	0.286774	+	0.957998i	$0.407417\pi$
$32$	0	0
$33$	5.67085i	0.987168i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.90849i	0.478152i	0.971001	+	0.239076i	$0.0768445\pi$
$37$	2.90849i	0.478152i	−0.971001	+	0.239076i	$0.923155\pi$
$38$	0	0
$39$	10.9362	1.75119
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	7.62441i	1.16271i	0.813650	+	0.581356i	$0.197477\pi$
$43$	7.62441i	1.16271i	−0.813650	+	0.581356i	$0.802523\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.15171i	0.751454i	0.926730	+	0.375727i	$0.122607\pi$
$47$	5.15171i	0.751454i	−0.926730	+	0.375727i	$0.877393\pi$
$48$	0	0
$49$	−18.6521	−2.66459
$50$	0	0
$51$	5.95575	0.833972
$52$	0	0
$53$	11.1192i	1.52734i	0.645605	+	0.763672i	$0.276605\pi$
$53$	11.1192i	1.52734i	−0.645605	+	0.763672i	$0.723395\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.4587i	1.38529i
$58$	0	0
$59$	−1.49482	−0.194609	−0.0973046	−	0.995255i	$-0.531022\pi$
$59$	−1.49482	−0.194609	−0.0973046	+	0.995255i	$0.531022\pi$
$60$	0	0
$61$	1.06380	0.136206	0.0681029	−	0.997678i	$-0.478305\pi$
$61$	1.06380	0.136206	0.0681029	+	0.997678i	$0.478305\pi$
$62$	0	0
$63$	9.66608i	1.21781i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.5596i	1.29006i	0.764156	+	0.645031i	$0.223156\pi$
$67$	10.5596i	1.29006i	−0.764156	+	0.645031i	$0.776844\pi$
$68$	0	0
$69$	3.31179	0.398693
$70$	0	0
$71$	−12.6023	−1.49562	−0.747808	−	0.663915i	$-0.768894\pi$
$71$	−12.6023	−1.49562	−0.747808	+	0.663915i	$0.768894\pi$
$72$	0	0
$73$	− 8.28490i	− 0.969674i	−0.874604	−	0.484837i	$-0.838879\pi$
$73$	− 8.28490i	− 0.969674i	0.874604	−	0.484837i	$-0.161121\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.9639i	1.47737i
$78$	0	0
$79$	4.28967	0.482625	0.241313	−	0.970447i	$-0.422422\pi$
$79$	4.28967	0.482625	0.241313	+	0.970447i	$0.422422\pi$
$80$	0	0
$81$	−11.0831	−1.23146
$82$	0	0
$83$	− 10.5606i	− 1.15918i	−0.814909	−	0.579588i	$-0.803213\pi$
$83$	− 10.5606i	− 1.15918i	0.814909	−	0.579588i	$-0.196787\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.38595i	0.577435i
$88$	0	0
$89$	−9.36722	−0.992923	−0.496462	−	0.868059i	$-0.665368\pi$
$89$	−9.36722	−0.992923	−0.496462	+	0.868059i	$0.665368\pi$
$90$	0	0
$91$	25.0008	2.62080
$92$	0	0
$93$	− 7.07498i	− 0.733641i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.37641i	− 0.342823i	−0.985200	−	0.171411i	$-0.945167\pi$
$97$	− 3.37641i	− 0.342823i	0.985200	−	0.171411i	$-0.0548328\pi$
$98$	0	0
$99$	4.88499	0.490960
$100$	0	0
$101$	−4.24799	−0.422691	−0.211345	−	0.977411i	$-0.567785\pi$
$101$	−4.24799	−0.422691	−0.211345	+	0.977411i	$0.567785\pi$
$102$	0	0
$103$	− 1.11923i	− 0.110281i	−0.998479	−	0.0551404i	$-0.982439\pi$
$103$	− 1.11923i	− 0.110281i	0.998479	−	0.0551404i	$-0.0175606\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.322149i	− 0.0311434i	−0.999879	−	0.0155717i	$-0.995043\pi$
$107$	− 0.322149i	− 0.0311434i	0.999879	−	0.0155717i	$-0.00495682\pi$
$108$	0	0
$109$	−3.23763	−0.310109	−0.155055	−	0.987906i	$-0.549555\pi$
$109$	−3.23763	−0.310109	−0.155055	+	0.987906i	$0.549555\pi$
$110$	0	0
$111$	6.44378	0.611616
$112$	0	0
$113$	− 2.10187i	− 0.197727i	−0.995101	−	0.0988637i	$-0.968479\pi$
$113$	− 2.10187i	− 0.197727i	0.995101	−	0.0988637i	$-0.0315208\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.42066i	− 0.870941i
$118$	0	0
$119$	13.6152	1.24810
$120$	0	0
$121$	−4.44838	−0.404398
$122$	0	0
$123$	2.21551i	0.199766i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.0658i	− 1.15940i	−0.814830	−	0.579700i	$-0.803170\pi$
$127$	− 13.0658i	− 1.15940i	0.814830	−	0.579700i	$-0.196830\pi$
$128$	0	0
$129$	16.8919	1.48725
$130$	0	0
$131$	8.43102	0.736622	0.368311	−	0.929703i	$-0.379936\pi$
$131$	8.43102	0.736622	0.368311	+	0.929703i	$0.379936\pi$
$132$	0	0
$133$	23.9093i	2.07320i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.5606i	− 1.41487i	−0.706779	−	0.707434i	$-0.749853\pi$
$137$	− 16.5606i	− 1.41487i	0.706779	−	0.707434i	$-0.250147\pi$
$138$	0	0
$139$	2.18303	0.185162	0.0925810	−	0.995705i	$-0.470488\pi$
$139$	2.18303	0.185162	0.0925810	+	0.995705i	$0.470488\pi$
$140$	0	0
$141$	11.4137	0.961204
$142$	0	0
$143$	− 12.6348i	− 1.05657i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	41.3240i	3.40834i
$148$	0	0
$149$	−2.43102	−0.199157	−0.0995785	−	0.995030i	$-0.531749\pi$
$149$	−2.43102	−0.199157	−0.0995785	+	0.995030i	$0.531749\pi$
$150$	0	0
$151$	−0.343113	−0.0279221	−0.0139611	−	0.999903i	$-0.504444\pi$
$151$	−0.343113	−0.0279221	−0.0139611	+	0.999903i	$0.504444\pi$
$152$	0	0
$153$	− 5.13040i	− 0.414769i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.6348i	− 0.848746i	−0.905487	−	0.424373i	$-0.860494\pi$
$157$	− 10.6348i	− 0.848746i	0.905487	−	0.424373i	$-0.139506\pi$
$158$	0	0
$159$	24.6348	1.95366
$160$	0	0
$161$	7.57096	0.596675
$162$	0	0
$163$	0.173834i	0.0136157i	0.999977	+	0.00680785i	$0.00216702\pi$
$163$	0.173834i	0.0136157i	−0.999977	+	0.00680785i	$0.997833\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.5596i	1.12666i	0.826233	+	0.563328i	$0.190479\pi$
$167$	14.5596i	1.12666i	−0.826233	+	0.563328i	$0.809521\pi$
$168$	0	0
$169$	−11.3661	−0.874312
$170$	0	0
$171$	9.00937	0.688963
$172$	0	0
$173$	22.2592i	1.69233i	0.532918	+	0.846167i	$0.321095\pi$
$173$	22.2592i	1.69233i	−0.532918	+	0.846167i	$0.678905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.31179i	0.248930i
$178$	0	0
$179$	17.9268	1.33991	0.669957	−	0.742400i	$-0.266312\pi$
$179$	17.9268	1.33991	0.669957	+	0.742400i	$0.266312\pi$
$180$	0	0
$181$	20.9916	1.56030	0.780148	−	0.625595i	$-0.215144\pi$
$181$	20.9916	1.56030	0.780148	+	0.625595i	$0.215144\pi$
$182$	0	0
$183$	− 2.35686i	− 0.174224i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.88077i	− 0.503172i
$188$	0	0
$189$	−12.2480	−0.890910
$190$	0	0
$191$	6.45074	0.466759	0.233380	−	0.972386i	$-0.425022\pi$
$191$	6.45074	0.466759	0.233380	+	0.972386i	$0.425022\pi$
$192$	0	0
$193$	− 6.20374i	− 0.446555i	−0.974755	−	0.223278i	$-0.928324\pi$
$193$	− 6.20374i	− 0.446555i	0.974755	−	0.223278i	$-0.0716756\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.1850i	1.01064i	0.862932	+	0.505320i	$0.168626\pi$
$197$	14.1850i	1.01064i	−0.862932	+	0.505320i	$0.831374\pi$
$198$	0	0
$199$	−18.7099	−1.32631	−0.663155	−	0.748482i	$-0.730783\pi$
$199$	−18.7099	−1.32631	−0.663155	+	0.748482i	$0.730783\pi$
$200$	0	0
$201$	23.3949	1.65015
$202$	0	0
$203$	12.3126i	0.864176i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.85285i	− 0.198286i
$208$	0	0
$209$	12.0831	0.835808
$210$	0	0
$211$	14.6984	1.01188	0.505940	−	0.862569i	$-0.331146\pi$
$211$	14.6984	1.01188	0.505940	+	0.862569i	$0.331146\pi$
$212$	0	0
$213$	27.9205i	1.91308i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 16.1738i	− 1.09795i
$218$	0	0
$219$	−18.3553	−1.24033
$220$	0	0
$221$	−13.2695	−0.892605
$222$	0	0
$223$	− 20.1109i	− 1.34672i	−0.739314	−	0.673361i	$-0.764850\pi$
$223$	− 20.1109i	− 1.34672i	0.739314	−	0.673361i	$-0.235150\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.71149i	− 0.445457i	−0.974880	−	0.222729i	$-0.928504\pi$
$227$	− 6.71149i	− 0.445457i	0.974880	−	0.222729i	$-0.0714964\pi$
$228$	0	0
$229$	−12.7624	−0.843361	−0.421680	−	0.906745i	$-0.638560\pi$
$229$	−12.7624	−0.843361	−0.421680	+	0.906745i	$0.638560\pi$
$230$	0	0
$231$	28.7217	1.88975
$232$	0	0
$233$	29.0750i	1.90477i	0.304906	+	0.952383i	$0.401375\pi$
$233$	29.0750i	1.90477i	−0.304906	+	0.952383i	$0.598625\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.50380i	− 0.617338i
$238$	0	0
$239$	−26.0971	−1.68808	−0.844041	−	0.536279i	$-0.819829\pi$
$239$	−26.0971	−1.68808	−0.844041	+	0.536279i	$0.819829\pi$
$240$	0	0
$241$	1.06380	0.0685255	0.0342627	−	0.999413i	$-0.489092\pi$
$241$	1.06380	0.0685255	0.0342627	+	0.999413i	$0.489092\pi$
$242$	0	0
$243$	17.3000i	1.10980i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 23.3023i	− 1.48269i
$248$	0	0
$249$	−23.3971	−1.48273
$250$	0	0
$251$	−7.82815	−0.494108	−0.247054	−	0.969002i	$-0.579463\pi$
$251$	−7.82815	−0.494108	−0.247054	+	0.969002i	$0.579463\pi$
$252$	0	0
$253$	− 3.82617i	− 0.240549i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.49482i	0.467514i	0.972295	+	0.233757i	$0.0751020\pi$
$257$	7.49482i	0.467514i	−0.972295	+	0.233757i	$0.924898\pi$
$258$	0	0
$259$	14.7309	0.915332
$260$	0	0
$261$	4.63957	0.287182
$262$	0	0
$263$	− 15.5919i	− 0.961439i	−0.876874	−	0.480720i	$-0.840375\pi$
$263$	− 15.5919i	− 0.961439i	0.876874	−	0.480720i	$-0.159625\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.7532i	1.27007i
$268$	0	0
$269$	−8.80860	−0.537070	−0.268535	−	0.963270i	$-0.586539\pi$
$269$	−8.80860	−0.537070	−0.268535	+	0.963270i	$0.586539\pi$
$270$	0	0
$271$	−19.8816	−1.20772	−0.603860	−	0.797090i	$-0.706372\pi$
$271$	−19.8816	−1.20772	−0.603860	+	0.797090i	$0.706372\pi$
$272$	0	0
$273$	− 55.3896i	− 3.35233i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.08232i	− 0.425535i	−0.977103	−	0.212768i	$-0.931752\pi$
$277$	− 7.08232i	− 0.425535i	0.977103	−	0.212768i	$-0.0682477\pi$
$278$	0	0
$279$	−6.09453	−0.364870
$280$	0	0
$281$	−31.2695	−1.86538	−0.932692	−	0.360675i	$-0.882547\pi$
$281$	−31.2695	−1.86538	−0.932692	+	0.360675i	$0.882547\pi$
$282$	0	0
$283$	2.18303i	0.129768i	0.997893	+	0.0648838i	$0.0206677\pi$
$283$	2.18303i	0.129768i	−0.997893	+	0.0648838i	$0.979332\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.06479i	0.298965i
$288$	0	0
$289$	9.77354	0.574914
$290$	0	0
$291$	−7.48048	−0.438514
$292$	0	0
$293$	33.6798i	1.96760i	0.179278	+	0.983798i	$0.442624\pi$
$293$	33.6798i	1.96760i	−0.179278	+	0.983798i	$0.557376\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.18982i	0.359170i
$298$	0	0
$299$	−7.37874	−0.426723
$300$	0	0
$301$	38.6160	2.22579
$302$	0	0
$303$	9.41147i	0.540675i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.7532i	− 0.727862i	−0.931426	−	0.363931i	$-0.881434\pi$
$307$	− 12.7532i	− 0.727862i	0.931426	−	0.363931i	$-0.118566\pi$
$308$	0	0
$309$	−2.47966	−0.141063
$310$	0	0
$311$	2.18204	0.123732	0.0618660	−	0.998084i	$-0.480295\pi$
$311$	2.18204	0.123732	0.0618660	+	0.998084i	$0.480295\pi$
$312$	0	0
$313$	29.3042i	1.65637i	0.560452	+	0.828187i	$0.310627\pi$
$313$	29.3042i	1.65637i	−0.560452	+	0.828187i	$0.689373\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.5511i	− 1.60359i	−0.597601	−	0.801794i	$-0.703879\pi$
$317$	− 28.5511i	− 1.60359i	0.597601	−	0.801794i	$-0.296121\pi$
$318$	0	0
$319$	6.22247	0.348392
$320$	0	0
$321$	−0.713725	−0.0398363
$322$	0	0
$323$	− 12.6902i	− 0.706101i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.17301i	0.396669i
$328$	0	0
$329$	26.0923	1.43852
$330$	0	0
$331$	−4.80761	−0.264250	−0.132125	−	0.991233i	$-0.542180\pi$
$331$	−4.80761	−0.264250	−0.132125	+	0.991233i	$0.542180\pi$
$332$	0	0
$333$	− 5.55080i	− 0.304182i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.3148i	0.888724i	0.895847	+	0.444362i	$0.146570\pi$
$337$	16.3148i	0.888724i	−0.895847	+	0.444362i	$0.853430\pi$
$338$	0	0
$339$	−4.65672	−0.252918
$340$	0	0
$341$	−8.17383	−0.442638
$342$	0	0
$343$	59.0156i	3.18654i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.6034i	1.69656i	0.529547	+	0.848281i	$0.322362\pi$
$347$	31.6034i	1.69656i	−0.529547	+	0.848281i	$0.677638\pi$
$348$	0	0
$349$	−4.97311	−0.266204	−0.133102	−	0.991102i	$-0.542494\pi$
$349$	−4.97311	−0.266204	−0.133102	+	0.991102i	$0.542494\pi$
$350$	0	0
$351$	11.9370	0.637151
$352$	0	0
$353$	− 1.72430i	− 0.0917750i	−0.998947	−	0.0458875i	$-0.985388\pi$
$353$	− 1.72430i	− 0.0917750i	0.998947	−	0.0458875i	$-0.0146116\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 30.1646i	− 1.59648i
$358$	0	0
$359$	7.13796	0.376727	0.188364	−	0.982099i	$-0.439682\pi$
$359$	7.13796	0.376727	0.188364	+	0.982099i	$0.439682\pi$
$360$	0	0
$361$	3.28490	0.172889
$362$	0	0
$363$	9.85542i	0.517276i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.38595i	− 0.489943i	−0.969530	−	0.244971i	$-0.921221\pi$
$367$	− 9.38595i	− 0.489943i	0.969530	−	0.244971i	$-0.0787785\pi$
$368$	0	0
$369$	1.90849	0.0993518
$370$	0	0
$371$	56.3166	2.92381
$372$	0	0
$373$	− 23.1212i	− 1.19717i	−0.801059	−	0.598585i	$-0.795730\pi$
$373$	− 23.1212i	− 1.19717i	0.801059	−	0.598585i	$-0.204270\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−5.99080	−0.307727	−0.153863	−	0.988092i	$-0.549172\pi$
$379$	−5.99080	−0.307727	−0.153863	+	0.988092i	$0.549172\pi$
$380$	0	0
$381$	−28.9474	−1.48302
$382$	0	0
$383$	29.5811i	1.51153i	0.654845	+	0.755763i	$0.272734\pi$
$383$	29.5811i	1.51153i	−0.654845	+	0.755763i	$0.727266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 14.5511i	− 0.739672i
$388$	0	0
$389$	−22.8620	−1.15915	−0.579576	−	0.814918i	$-0.696782\pi$
$389$	−22.8620	−1.15915	−0.579576	+	0.814918i	$0.696782\pi$
$390$	0	0
$391$	−4.01839	−0.203219
$392$	0	0
$393$	− 18.6790i	− 0.942232i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.588531i	− 0.0295375i	−0.999891	−	0.0147688i	$-0.995299\pi$
$397$	− 0.588531i	− 0.0295375i	0.999891	−	0.0147688i	$-0.00470122\pi$
$398$	0	0
$399$	52.9713	2.65188
$400$	0	0
$401$	−0.0926758	−0.00462801	−0.00231400	−	0.999997i	$-0.500737\pi$
$401$	−0.0926758	−0.00462801	−0.00231400	+	0.999997i	$0.500737\pi$
$402$	0	0
$403$	15.7632i	0.785220i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.44460i	− 0.369015i
$408$	0	0
$409$	1.29526	0.0640463	0.0320232	−	0.999487i	$-0.489805\pi$
$409$	1.29526	0.0640463	0.0320232	+	0.999487i	$0.489805\pi$
$410$	0	0
$411$	−36.6902	−1.80979
$412$	0	0
$413$	7.57096i	0.372543i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.83652i	− 0.236846i
$418$	0	0
$419$	−1.75201	−0.0855912	−0.0427956	−	0.999084i	$-0.513626\pi$
$419$	−1.75201	−0.0855912	−0.0427956	+	0.999084i	$0.513626\pi$
$420$	0	0
$421$	36.7364	1.79042	0.895212	−	0.445641i	$-0.147024\pi$
$421$	36.7364	1.79042	0.895212	+	0.445641i	$0.147024\pi$
$422$	0	0
$423$	− 9.83196i	− 0.478046i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5.38793i	− 0.260740i
$428$	0	0
$429$	−27.9924	−1.35149
$430$	0	0
$431$	12.3126	0.593078	0.296539	−	0.955021i	$-0.404168\pi$
$431$	12.3126	0.593078	0.296539	+	0.955021i	$0.404168\pi$
$432$	0	0
$433$	17.7632i	0.853644i	0.904336	+	0.426822i	$0.140367\pi$
$433$	17.7632i	0.853644i	−0.904336	+	0.426822i	$0.859633\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.05659i	− 0.337562i
$438$	0	0
$439$	−3.22230	−0.153792	−0.0768961	−	0.997039i	$-0.524501\pi$
$439$	−3.22230	−0.153792	−0.0768961	+	0.997039i	$0.524501\pi$
$440$	0	0
$441$	35.5973	1.69511
$442$	0	0
$443$	35.8537i	1.70346i	0.523982	+	0.851730i	$0.324446\pi$
$443$	35.8537i	1.70346i	−0.523982	+	0.851730i	$0.675554\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.38595i	0.254747i
$448$	0	0
$449$	15.3437	0.724113	0.362057	−	0.932156i	$-0.382075\pi$
$449$	15.3437	0.724113	0.362057	+	0.932156i	$0.382075\pi$
$450$	0	0
$451$	2.55961	0.120528
$452$	0	0
$453$	0.760170i	0.0357159i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.5917i	1.29068i	0.763894	+	0.645342i	$0.223285\pi$
$457$	27.5917i	1.29068i	−0.763894	+	0.645342i	$0.776715\pi$
$458$	0	0
$459$	6.50079	0.303431
$460$	0	0
$461$	−8.08116	−0.376377	−0.188189	−	0.982133i	$-0.560262\pi$
$461$	−8.08116	−0.376377	−0.188189	+	0.982133i	$0.560262\pi$
$462$	0	0
$463$	8.23622i	0.382770i	0.981515	+	0.191385i	$0.0612979\pi$
$463$	8.23622i	0.382770i	−0.981515	+	0.191385i	$0.938702\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 23.7333i	− 1.09825i	−0.835742	−	0.549123i	$-0.814962\pi$
$467$	− 23.7333i	− 1.09825i	0.835742	−	0.549123i	$-0.185038\pi$
$468$	0	0
$469$	53.4822	2.46958
$470$	0	0
$471$	−23.5614	−1.08565
$472$	0	0
$473$	− 19.5155i	− 0.897325i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 21.2209i	− 0.971638i
$478$	0	0
$479$	16.6987	0.762985	0.381492	−	0.924372i	$-0.375410\pi$
$479$	16.6987	0.762985	0.381492	+	0.924372i	$0.375410\pi$
$480$	0	0
$481$	−14.3569	−0.654617
$482$	0	0
$483$	− 16.7735i	− 0.763223i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.927003i	0.0420065i	0.999779	+	0.0210033i	$0.00668604\pi$
$487$	0.927003i	0.0420065i	−0.999779	+	0.0210033i	$0.993314\pi$
$488$	0	0
$489$	0.385130	0.0174162
$490$	0	0
$491$	−26.8752	−1.21286	−0.606430	−	0.795137i	$-0.707399\pi$
$491$	−26.8752	−1.21286	−0.606430	+	0.795137i	$0.707399\pi$
$492$	0	0
$493$	− 6.53509i	− 0.294326i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	63.8279i	2.86307i
$498$	0	0
$499$	−18.2155	−0.815438	−0.407719	−	0.913107i	$-0.633676\pi$
$499$	−18.2155	−0.815438	−0.407719	+	0.913107i	$0.633676\pi$
$500$	0	0
$501$	32.2570	1.44114
$502$	0	0
$503$	− 18.9591i	− 0.845346i	−0.906282	−	0.422673i	$-0.861092\pi$
$503$	− 18.9591i	− 0.845346i	0.906282	−	0.422673i	$-0.138908\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	25.1816i	1.11835i
$508$	0	0
$509$	−12.4218	−0.550588	−0.275294	−	0.961360i	$-0.588775\pi$
$509$	−12.4218	−0.550588	−0.275294	+	0.961360i	$0.588775\pi$
$510$	0	0
$511$	−41.9613	−1.85626
$512$	0	0
$513$	11.4159i	0.504022i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.1864i	− 0.579937i
$518$	0	0
$519$	49.3154	2.16471
$520$	0	0
$521$	−7.17267	−0.314240	−0.157120	−	0.987579i	$-0.550221\pi$
$521$	−7.17267	−0.314240	−0.157120	+	0.987579i	$0.550221\pi$
$522$	0	0
$523$	− 35.6563i	− 1.55914i	−0.626315	−	0.779570i	$-0.715437\pi$
$523$	− 35.6563i	− 1.55914i	0.626315	−	0.779570i	$-0.284563\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.58448i	0.373946i
$528$	0	0
$529$	20.7655	0.902848
$530$	0	0
$531$	2.85285	0.123803
$532$	0	0
$533$	− 4.93620i	− 0.213811i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 39.7171i	− 1.71392i
$538$	0	0
$539$	47.7422	2.05640
$540$	0	0
$541$	28.9731	1.24565	0.622826	−	0.782361i	$-0.285985\pi$
$541$	28.9731	1.24565	0.622826	+	0.782361i	$0.285985\pi$
$542$	0	0
$543$	− 46.5072i	− 1.99581i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.1178i	0.689148i	0.938759	+	0.344574i	$0.111977\pi$
$547$	16.1178i	0.689148i	−0.938759	+	0.344574i	$0.888023\pi$
$548$	0	0
$549$	−2.03025	−0.0866489
$550$	0	0
$551$	11.4761	0.488898
$552$	0	0
$553$	− 21.7263i	− 0.923895i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.2675i	− 1.15536i	−0.816262	−	0.577681i	$-0.803958\pi$
$557$	− 27.2675i	− 1.15536i	0.816262	−	0.577681i	$-0.196042\pi$
$558$	0	0
$559$	−37.6356	−1.59182
$560$	0	0
$561$	−15.2444	−0.643620
$562$	0	0
$563$	− 41.9173i	− 1.76660i	−0.468805	−	0.883302i	$-0.655315\pi$
$563$	− 41.9173i	− 1.76660i	0.468805	−	0.883302i	$-0.344685\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	56.1338i	2.35740i
$568$	0	0
$569$	−37.3762	−1.56689	−0.783446	−	0.621460i	$-0.786540\pi$
$569$	−37.3762	−1.56689	−0.783446	+	0.621460i	$0.786540\pi$
$570$	0	0
$571$	−26.3228	−1.10157	−0.550787	−	0.834646i	$-0.685673\pi$
$571$	−26.3228	−1.10157	−0.550787	+	0.834646i	$0.685673\pi$
$572$	0	0
$573$	− 14.2917i	− 0.597044i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 13.8262i	− 0.575591i	−0.957692	−	0.287795i	$-0.907078\pi$
$577$	− 13.8262i	− 0.575591i	0.957692	−	0.287795i	$-0.0929223\pi$
$578$	0	0
$579$	−13.7445	−0.571200
$580$	0	0
$581$	−53.4873	−2.21903
$582$	0	0
$583$	− 28.4609i	− 1.17873i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.7004i	0.936945i	0.883478	+	0.468472i	$0.155195\pi$
$587$	22.7004i	0.936945i	−0.883478	+	0.468472i	$0.844805\pi$
$588$	0	0
$589$	−15.0750	−0.621154
$590$	0	0
$591$	31.4270	1.29274
$592$	0	0
$593$	− 16.4167i	− 0.674152i	−0.941477	−	0.337076i	$-0.890562\pi$
$593$	− 16.4167i	− 0.674152i	0.941477	−	0.337076i	$-0.109438\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.4520i	1.69652i
$598$	0	0
$599$	6.54909	0.267588	0.133794	−	0.991009i	$-0.457284\pi$
$599$	6.54909	0.267588	0.133794	+	0.991009i	$0.457284\pi$
$600$	0	0
$601$	−24.7783	−1.01073	−0.505365	−	0.862906i	$-0.668642\pi$
$601$	−24.7783	−1.01073	−0.505365	+	0.862906i	$0.668642\pi$
$602$	0	0
$603$	− 20.1529i	− 0.820688i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 15.7982i	− 0.641231i	−0.947209	−	0.320615i	$-0.896110\pi$
$607$	− 15.7982i	− 0.641231i	0.947209	−	0.320615i	$-0.103890\pi$
$608$	0	0
$609$	27.2787	1.10539
$610$	0	0
$611$	−25.4299	−1.02878
$612$	0	0
$613$	− 39.3415i	− 1.58899i	−0.607272	−	0.794494i	$-0.707736\pi$
$613$	− 39.3415i	− 1.58899i	0.607272	−	0.794494i	$-0.292264\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.4075i	− 1.06313i	−0.847019	−	0.531563i	$-0.821605\pi$
$617$	− 26.4075i	− 1.06313i	0.847019	−	0.531563i	$-0.178395\pi$
$618$	0	0
$619$	−33.9278	−1.36367	−0.681837	−	0.731504i	$-0.738819\pi$
$619$	−33.9278	−1.36367	−0.681837	+	0.731504i	$0.738819\pi$
$620$	0	0
$621$	3.61487	0.145060
$622$	0	0
$623$	47.4430i	1.90076i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 26.7703i	− 1.06910i
$628$	0	0
$629$	−7.81861	−0.311748
$630$	0	0
$631$	−5.68017	−0.226124	−0.113062	−	0.993588i	$-0.536066\pi$
$631$	−5.68017	−0.226124	−0.113062	+	0.993588i	$0.536066\pi$
$632$	0	0
$633$	− 32.5644i	− 1.29432i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 92.0706i	− 3.64797i
$638$	0	0
$639$	24.0513	0.951454
$640$	0	0
$641$	15.5802	0.615379	0.307690	−	0.951487i	$-0.400444\pi$
$641$	15.5802	0.615379	0.307690	+	0.951487i	$0.400444\pi$
$642$	0	0
$643$	− 22.9372i	− 0.904554i	−0.891877	−	0.452277i	$-0.850612\pi$
$643$	− 22.9372i	− 0.904554i	0.891877	−	0.452277i	$-0.149388\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.0546i	0.591858i	0.955210	+	0.295929i	$0.0956291\pi$
$647$	15.0546i	0.591858i	−0.955210	+	0.295929i	$0.904371\pi$
$648$	0	0
$649$	3.82617	0.150190
$650$	0	0
$651$	−35.8333	−1.40442
$652$	0	0
$653$	45.8652i	1.79484i	0.441174	+	0.897422i	$0.354562\pi$
$653$	45.8652i	1.79484i	−0.441174	+	0.897422i	$0.645438\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.8116i	0.616870i
$658$	0	0
$659$	4.37302	0.170349	0.0851744	−	0.996366i	$-0.472855\pi$
$659$	4.37302	0.170349	0.0851744	+	0.996366i	$0.472855\pi$
$660$	0	0
$661$	−25.4946	−0.991625	−0.495813	−	0.868430i	$-0.665130\pi$
$661$	−25.4946	−0.991625	−0.495813	+	0.868430i	$0.665130\pi$
$662$	0	0
$663$	29.3988i	1.14175i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.63394i	− 0.140707i
$668$	0	0
$669$	−44.5558	−1.72263
$670$	0	0
$671$	−2.72292	−0.105117
$672$	0	0
$673$	− 31.7540i	− 1.22403i	−0.790848	−	0.612013i	$-0.790360\pi$
$673$	− 31.7540i	− 1.22403i	0.790848	−	0.612013i	$-0.209640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.8998i	− 1.07228i	−0.844131	−	0.536138i	$-0.819883\pi$
$677$	− 27.8998i	− 1.07228i	0.844131	−	0.536138i	$-0.180117\pi$
$678$	0	0
$679$	−17.1008	−0.656270
$680$	0	0
$681$	−14.8694	−0.569796
$682$	0	0
$683$	− 8.11782i	− 0.310620i	−0.987866	−	0.155310i	$-0.950362\pi$
$683$	− 8.11782i	− 0.310620i	0.987866	−	0.155310i	$-0.0496376\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.2752i	1.07876i
$688$	0	0
$689$	−54.8867	−2.09102
$690$	0	0
$691$	−26.6390	−1.01339	−0.506697	−	0.862124i	$-0.669134\pi$
$691$	−26.6390	−1.01339	−0.506697	+	0.862124i	$0.669134\pi$
$692$	0	0
$693$	− 24.7414i	− 0.939850i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.68821i	− 0.101823i
$698$	0	0
$699$	64.4159	2.43643
$700$	0	0
$701$	43.1302	1.62900	0.814502	−	0.580160i	$-0.197010\pi$
$701$	43.1302	1.62900	0.814502	+	0.580160i	$0.197010\pi$
$702$	0	0
$703$	− 13.7301i	− 0.517839i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.5152i	0.809162i
$708$	0	0
$709$	−49.5175	−1.85967	−0.929835	−	0.367978i	$-0.880050\pi$
$709$	−49.5175	−1.85967	−0.929835	+	0.367978i	$0.880050\pi$
$710$	0	0
$711$	−8.18677	−0.307028
$712$	0	0
$713$	4.77354i	0.178771i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	57.8184i	2.15927i
$718$	0	0
$719$	16.4839	0.614745	0.307372	−	0.951589i	$-0.400550\pi$
$719$	16.4839	0.614745	0.307372	+	0.951589i	$0.400550\pi$
$720$	0	0
$721$	−5.66866	−0.211112
$722$	0	0
$723$	− 2.35686i	− 0.0876527i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.5947i	1.80228i	0.433530	+	0.901139i	$0.357268\pi$
$727$	48.5947i	1.80228i	−0.433530	+	0.901139i	$0.642732\pi$
$728$	0	0
$729$	5.07896	0.188110
$730$	0	0
$731$	−20.4960	−0.758071
$732$	0	0
$733$	− 11.8166i	− 0.436457i	−0.975898	−	0.218229i	$-0.929972\pi$
$733$	− 11.8166i	− 0.436457i	0.975898	−	0.218229i	$-0.0700278\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 27.0285i	− 0.995609i
$738$	0	0
$739$	5.68017	0.208949	0.104474	−	0.994528i	$-0.466684\pi$
$739$	5.68017	0.208949	0.104474	+	0.994528i	$0.466684\pi$
$740$	0	0
$741$	−51.6264	−1.89654
$742$	0	0
$743$	24.4980i	0.898743i	0.893345	+	0.449372i	$0.148352\pi$
$743$	24.4980i	0.898743i	−0.893345	+	0.449372i	$0.851648\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.1548i	0.737424i
$748$	0	0
$749$	−1.63162	−0.0596181
$750$	0	0
$751$	−35.7897	−1.30598	−0.652992	−	0.757365i	$-0.726487\pi$
$751$	−35.7897	−1.30598	−0.652992	+	0.757365i	$0.726487\pi$
$752$	0	0
$753$	17.3433i	0.632027i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	37.7169i	1.37084i	0.728147	+	0.685421i	$0.240382\pi$
$757$	37.7169i	1.37084i	−0.728147	+	0.685421i	$0.759618\pi$
$758$	0	0
$759$	−8.47691	−0.307692
$760$	0	0
$761$	−22.6533	−0.821181	−0.410590	−	0.911820i	$-0.634677\pi$
$761$	−22.6533	−0.821181	−0.410590	+	0.911820i	$0.634677\pi$
$762$	0	0
$763$	16.3979i	0.593646i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.37874i	− 0.266431i
$768$	0	0
$769$	42.6292	1.53725	0.768624	−	0.639701i	$-0.220942\pi$
$769$	42.6292	1.53725	0.768624	+	0.639701i	$0.220942\pi$
$770$	0	0
$771$	16.6049	0.598009
$772$	0	0
$773$	− 11.7635i	− 0.423105i	−0.977367	−	0.211552i	$-0.932148\pi$
$773$	− 11.7635i	− 0.423105i	0.977367	−	0.211552i	$-0.0678519\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 32.6364i	− 1.17082i
$778$	0	0
$779$	4.72069	0.169136
$780$	0	0
$781$	32.2570	1.15425
$782$	0	0
$783$	5.87884i	0.210093i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.4517i	1.22807i	0.789278	+	0.614036i	$0.210455\pi$
$787$	34.4517i	1.22807i	−0.789278	+	0.614036i	$0.789545\pi$
$788$	0	0
$789$	−34.5441	−1.22980
$790$	0	0
$791$	−10.6455	−0.378512
$792$	0	0
$793$	5.25113i	0.186473i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.2161i	1.45995i	0.683475	+	0.729974i	$0.260468\pi$
$797$	41.2161i	1.45995i	−0.683475	+	0.729974i	$0.739532\pi$
$798$	0	0
$799$	−13.8489	−0.489937
$800$	0	0
$801$	17.8772	0.631660
$802$	0	0
$803$	21.2061i	0.748349i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.5155i	0.686979i
$808$	0	0
$809$	−12.4123	−0.436393	−0.218196	−	0.975905i	$-0.570017\pi$
$809$	−12.4123	−0.436393	−0.218196	+	0.975905i	$0.570017\pi$
$810$	0	0
$811$	3.91665	0.137532	0.0687660	−	0.997633i	$-0.478094\pi$
$811$	3.91665	0.137532	0.0687660	+	0.997633i	$0.478094\pi$
$812$	0	0
$813$	44.0479i	1.54483i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 35.9924i	− 1.25922i
$818$	0	0
$819$	−47.7137	−1.66725
$820$	0	0
$821$	28.6326	0.999283	0.499642	−	0.866232i	$-0.333465\pi$
$821$	28.6326	0.999283	0.499642	+	0.866232i	$0.333465\pi$
$822$	0	0
$823$	− 43.8212i	− 1.52751i	−0.645506	−	0.763755i	$-0.723353\pi$
$823$	− 43.8212i	− 1.52751i	0.645506	−	0.763755i	$-0.276647\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.1924i	0.389198i	0.980883	+	0.194599i	$0.0623405\pi$
$827$	11.1924i	0.389198i	−0.980883	+	0.194599i	$0.937659\pi$
$828$	0	0
$829$	−26.6278	−0.924820	−0.462410	−	0.886666i	$-0.653015\pi$
$829$	−26.6278	−0.924820	−0.462410	+	0.886666i	$0.653015\pi$
$830$	0	0
$831$	−15.6910	−0.544313
$832$	0	0
$833$	− 50.1408i	− 1.73727i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7.72244i	− 0.266927i
$838$	0	0
$839$	−9.43085	−0.325589	−0.162795	−	0.986660i	$-0.552051\pi$
$839$	−9.43085	−0.325589	−0.162795	+	0.986660i	$0.552051\pi$
$840$	0	0
$841$	−23.0901	−0.796212
$842$	0	0
$843$	69.2780i	2.38606i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	22.5301i	0.774144i
$848$	0	0
$849$	4.83652	0.165989
$850$	0	0
$851$	−4.34767	−0.149036
$852$	0	0
$853$	− 32.6045i	− 1.11636i	−0.829721	−	0.558179i	$-0.811500\pi$
$853$	− 32.6045i	− 1.11636i	0.829721	−	0.558179i	$-0.188500\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.25499i	− 0.145348i	−0.997356	−	0.0726739i	$-0.976847\pi$
$857$	− 4.25499i	− 0.145348i	0.997356	−	0.0726739i	$-0.0231532\pi$
$858$	0	0
$859$	−24.5164	−0.836487	−0.418244	−	0.908335i	$-0.637354\pi$
$859$	−24.5164	−0.836487	−0.418244	+	0.908335i	$0.637354\pi$
$860$	0	0
$861$	11.2211	0.382414
$862$	0	0
$863$	4.11643i	0.140125i	0.997543	+	0.0700624i	$0.0223198\pi$
$863$	4.11643i	0.140125i	−0.997543	+	0.0700624i	$0.977680\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 21.6534i	− 0.735387i
$868$	0	0
$869$	−10.9799	−0.372467
$870$	0	0
$871$	−52.1244	−1.76617
$872$	0	0
$873$	6.44384i	0.218091i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.17301i	− 0.107145i	−0.998564	−	0.0535725i	$-0.982939\pi$
$877$	− 3.17301i	− 0.107145i	0.998564	−	0.0535725i	$-0.0170608\pi$
$878$	0	0
$879$	74.6180	2.51680
$880$	0	0
$881$	16.0391	0.540371	0.270186	−	0.962808i	$-0.412915\pi$
$881$	16.0391	0.540371	0.270186	+	0.962808i	$0.412915\pi$
$882$	0	0
$883$	36.5648i	1.23050i	0.788330	+	0.615252i	$0.210946\pi$
$883$	36.5648i	1.23050i	−0.788330	+	0.615252i	$0.789054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 44.6498i	− 1.49919i	−0.661896	−	0.749596i	$-0.730248\pi$
$887$	− 44.6498i	− 1.49919i	0.661896	−	0.749596i	$-0.269752\pi$
$888$	0	0
$889$	−66.1755	−2.21945
$890$	0	0
$891$	28.3686	0.950382
$892$	0	0
$893$	− 24.3196i	− 0.813825i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.3477i	0.545833i
$898$	0	0
$899$	−7.76319	−0.258917
$900$	0	0
$901$	−29.8908	−0.995807
$902$	0	0
$903$	− 85.5542i	− 2.84707i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 43.2692i	− 1.43673i	−0.695667	−	0.718365i	$-0.744891\pi$
$907$	− 43.2692i	− 1.43673i	0.695667	−	0.718365i	$-0.255109\pi$
$908$	0	0
$909$	8.10723	0.268900
$910$	0	0
$911$	−1.64116	−0.0543739	−0.0271870	−	0.999630i	$-0.508655\pi$
$911$	−1.64116	−0.0543739	−0.0271870	+	0.999630i	$0.508655\pi$
$912$	0	0
$913$	27.0311i	0.894598i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 42.7014i	− 1.41012i
$918$	0	0
$919$	−6.67802	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$919$	−6.67802	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$920$	0	0
$921$	−28.2548	−0.931027
$922$	0	0
$923$	− 62.2074i	− 2.04758i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.13603i	0.0701564i
$928$	0	0
$929$	23.3505	0.766104	0.383052	−	0.923727i	$-0.374873\pi$
$929$	23.3505	0.766104	0.383052	+	0.923727i	$0.374873\pi$
$930$	0	0
$931$	88.0508	2.88575
$932$	0	0
$933$	− 4.83433i	− 0.158269i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.9295i	1.50045i	0.661182	+	0.750225i	$0.270055\pi$
$937$	45.9295i	1.50045i	−0.661182	+	0.750225i	$0.729945\pi$
$938$	0	0
$939$	64.9238	2.11871
$940$	0	0
$941$	−13.2488	−0.431899	−0.215949	−	0.976405i	$-0.569285\pi$
$941$	−13.2488	−0.431899	−0.215949	+	0.976405i	$0.569285\pi$
$942$	0	0
$943$	− 1.49482i	− 0.0486781i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.1017i	− 0.718207i	−0.933298	−	0.359104i	$-0.883082\pi$
$947$	− 22.1017i	− 0.718207i	0.933298	−	0.359104i	$-0.116918\pi$
$948$	0	0
$949$	40.8959	1.32754
$950$	0	0
$951$	−63.2552	−2.05119
$952$	0	0
$953$	18.2642i	0.591635i	0.955244	+	0.295818i	$0.0955920\pi$
$953$	18.2642i	0.591635i	−0.955244	+	0.295818i	$0.904408\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 13.7860i	− 0.445637i
$958$	0	0
$959$	−83.8760	−2.70850
$960$	0	0
$961$	−20.8023	−0.671042
$962$	0	0
$963$	0.614817i	0.0198122i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 28.0843i	− 0.903132i	−0.892238	−	0.451566i	$-0.850866\pi$
$967$	− 28.0843i	− 0.903132i	0.892238	−	0.451566i	$-0.149134\pi$
$968$	0	0
$969$	−28.1152	−0.903192
$970$	0	0
$971$	−34.1505	−1.09594	−0.547972	−	0.836497i	$-0.684600\pi$
$971$	−34.1505	−1.09594	−0.547972	+	0.836497i	$0.684600\pi$
$972$	0	0
$973$	− 11.0566i	− 0.354458i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.4597i	1.32641i	0.748437	+	0.663206i	$0.230805\pi$
$977$	41.4597i	1.32641i	−0.748437	+	0.663206i	$0.769195\pi$
$978$	0	0
$979$	23.9765	0.766291
$980$	0	0
$981$	6.17898	0.197280
$982$	0	0
$983$	34.1240i	1.08839i	0.838960	+	0.544193i	$0.183164\pi$
$983$	34.1240i	1.08839i	−0.838960	+	0.544193i	$0.816836\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 57.8078i	− 1.84004i
$988$	0	0
$989$	−11.3971	−0.362408
$990$	0	0
$991$	−21.3535	−0.678315	−0.339158	−	0.940730i	$-0.610142\pi$
$991$	−21.3535	−0.678315	−0.339158	+	0.940730i	$0.610142\pi$
$992$	0	0
$993$	10.6513i	0.338009i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 19.3560i	− 0.613012i	−0.951869	−	0.306506i	$-0.900840\pi$
$997$	− 19.3560i	− 0.613012i	0.951869	−	0.306506i	$-0.0991600\pi$
$998$	0	0
$999$	7.03348	0.222529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.c.1149.3		8
5.2	odd	4		4100.2.a.c.1.2		4
5.3	odd	4		164.2.a.a.1.3	✓	4
5.4	even	2	inner	4100.2.d.c.1149.6		8
15.8	even	4		1476.2.a.g.1.1		4
20.3	even	4		656.2.a.i.1.2		4
35.13	even	4		8036.2.a.i.1.2		4
40.3	even	4		2624.2.a.y.1.3		4
40.13	odd	4		2624.2.a.v.1.2		4
60.23	odd	4		5904.2.a.bp.1.1		4
205.163	odd	4		6724.2.a.c.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.3	✓	4	5.3	odd	4
656.2.a.i.1.2		4	20.3	even	4
1476.2.a.g.1.1		4	15.8	even	4
2624.2.a.v.1.2		4	40.13	odd	4
2624.2.a.y.1.3		4	40.3	even	4
4100.2.a.c.1.2		4	5.2	odd	4
4100.2.d.c.1149.3		8	1.1	even	1	trivial
4100.2.d.c.1149.6		8	5.4	even	2	inner
5904.2.a.bp.1.1		4	60.23	odd	4
6724.2.a.c.1.2		4	205.163	odd	4
8036.2.a.i.1.2		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)

\(f(q)\)	\(=\)	\(q-2.21551i q^{3} -5.06479i q^{7} -1.90849 q^{9} +O(q^{10})\)	\(q-2.21551i q^{3} -5.06479i q^{7} -1.90849 q^{9} -2.55961 q^{11} +4.93620i q^{13} +2.68821i q^{17} -4.72069 q^{19} -11.2211 q^{21} +1.49482i q^{23} -2.41826i q^{27} -2.43102 q^{29} +3.19339 q^{31} +5.67085i q^{33} +2.90849i q^{37} +10.9362 q^{39} -1.00000 q^{41} +7.62441i q^{43} +5.15171i q^{47} -18.6521 q^{49} +5.95575 q^{51} +11.1192i q^{53} +10.4587i q^{57} -1.49482 q^{59} +1.06380 q^{61} +9.66608i q^{63} +10.5596i q^{67} +3.31179 q^{69} -12.6023 q^{71} -8.28490i q^{73} +12.9639i q^{77} +4.28967 q^{79} -11.0831 q^{81} -10.5606i q^{83} +5.38595i q^{87} -9.36722 q^{89} +25.0008 q^{91} -7.07498i q^{93} -3.37641i q^{97} +4.88499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 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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