LMFDB - Embedded newform 8001.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

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:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.80317$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80317 q^{2} +5.85777 q^{4} -1.45069 q^{5} -1.00000 q^{7} -10.8140 q^{8} +O(q^{10})$	\(q-2.80317 q^{2} +5.85777 q^{4} -1.45069 q^{5} -1.00000 q^{7} -10.8140 q^{8} +4.06653 q^{10} +0.823916 q^{11} -3.71588 q^{13} +2.80317 q^{14} +18.5979 q^{16} -0.393170 q^{17} -2.77675 q^{19} -8.49779 q^{20} -2.30958 q^{22} -5.11502 q^{23} -2.89551 q^{25} +10.4162 q^{26} -5.85777 q^{28} +1.39789 q^{29} -8.61486 q^{31} -30.5052 q^{32} +1.10212 q^{34} +1.45069 q^{35} +3.26098 q^{37} +7.78371 q^{38} +15.6877 q^{40} -4.10302 q^{41} -7.13481 q^{43} +4.82631 q^{44} +14.3383 q^{46} -3.06280 q^{47} +1.00000 q^{49} +8.11660 q^{50} -21.7668 q^{52} -8.78037 q^{53} -1.19524 q^{55} +10.8140 q^{56} -3.91852 q^{58} +6.79554 q^{59} +0.528982 q^{61} +24.1489 q^{62} +48.3154 q^{64} +5.39058 q^{65} -11.2955 q^{67} -2.30310 q^{68} -4.06653 q^{70} +10.8636 q^{71} -3.59179 q^{73} -9.14109 q^{74} -16.2656 q^{76} -0.823916 q^{77} +0.905239 q^{79} -26.9798 q^{80} +11.5015 q^{82} +6.64443 q^{83} +0.570367 q^{85} +20.0001 q^{86} -8.90981 q^{88} -4.61643 q^{89} +3.71588 q^{91} -29.9626 q^{92} +8.58555 q^{94} +4.02820 q^{95} +1.82282 q^{97} -2.80317 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.80317	−1.98214	−0.991071	−	0.133338i	$-0.957430\pi$
$2$	−2.80317	−1.98214	−0.991071	+	0.133338i	$0.957430\pi$
$3$	0	0
$3$	0	0
$4$	5.85777	2.92888
$5$	−1.45069	−0.648767	−0.324384	−	0.945926i	$-0.605157\pi$
$5$	−1.45069	−0.648767	−0.324384	+	0.945926i	$0.605157\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−10.8140	−3.82332
$9$	0	0
$10$	4.06653	1.28595
$11$	0.823916	0.248420	0.124210	−	0.992256i	$-0.460360\pi$
$11$	0.823916	0.248420	0.124210	+	0.992256i	$0.460360\pi$
$12$	0	0
$13$	−3.71588	−1.03060	−0.515300	−	0.857010i	$-0.672319\pi$
$13$	−3.71588	−1.03060	−0.515300	+	0.857010i	$0.672319\pi$
$14$	2.80317	0.749179
$15$	0	0
$16$	18.5979	4.64948
$17$	−0.393170	−0.0953578	−0.0476789	−	0.998863i	$-0.515182\pi$
$17$	−0.393170	−0.0953578	−0.0476789	+	0.998863i	$0.515182\pi$
$18$	0	0
$19$	−2.77675	−0.637031	−0.318515	−	0.947918i	$-0.603184\pi$
$19$	−2.77675	−0.637031	−0.318515	+	0.947918i	$0.603184\pi$
$20$	−8.49779	−1.90016
$21$	0	0
$22$	−2.30958	−0.492403
$23$	−5.11502	−1.06656	−0.533278	−	0.845940i	$-0.679040\pi$
$23$	−5.11502	−1.06656	−0.533278	+	0.845940i	$0.679040\pi$
$24$	0	0
$25$	−2.89551	−0.579101
$26$	10.4162	2.04279
$27$	0	0
$28$	−5.85777	−1.10701
$29$	1.39789	0.259581	0.129791	−	0.991541i	$-0.458570\pi$
$29$	1.39789	0.259581	0.129791	+	0.991541i	$0.458570\pi$
$30$	0	0
$31$	−8.61486	−1.54727	−0.773637	−	0.633629i	$-0.781565\pi$
$31$	−8.61486	−1.54727	−0.773637	+	0.633629i	$0.781565\pi$
$32$	−30.5052	−5.39260
$33$	0	0
$34$	1.10212	0.189013
$35$	1.45069	0.245211
$36$	0	0
$37$	3.26098	0.536102	0.268051	−	0.963405i	$-0.413620\pi$
$37$	3.26098	0.536102	0.268051	+	0.963405i	$0.413620\pi$
$38$	7.78371	1.26269
$39$	0	0
$40$	15.6877	2.48044
$41$	−4.10302	−0.640784	−0.320392	−	0.947285i	$-0.603815\pi$
$41$	−4.10302	−0.640784	−0.320392	+	0.947285i	$0.603815\pi$
$42$	0	0
$43$	−7.13481	−1.08805	−0.544024	−	0.839070i	$-0.683100\pi$
$43$	−7.13481	−1.08805	−0.544024	+	0.839070i	$0.683100\pi$
$44$	4.82631	0.727593
$45$	0	0
$46$	14.3383	2.11407
$47$	−3.06280	−0.446755	−0.223378	−	0.974732i	$-0.571708\pi$
$47$	−3.06280	−0.446755	−0.223378	+	0.974732i	$0.571708\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.11660	1.14786
$51$	0	0
$52$	−21.7668	−3.01851
$53$	−8.78037	−1.20608	−0.603038	−	0.797712i	$-0.706043\pi$
$53$	−8.78037	−1.20608	−0.603038	+	0.797712i	$0.706043\pi$
$54$	0	0
$55$	−1.19524	−0.161167
$56$	10.8140	1.44508
$57$	0	0
$58$	−3.91852	−0.514526
$59$	6.79554	0.884704	0.442352	−	0.896842i	$-0.354144\pi$
$59$	6.79554	0.884704	0.442352	+	0.896842i	$0.354144\pi$
$60$	0	0
$61$	0.528982	0.0677292	0.0338646	−	0.999426i	$-0.489218\pi$
$61$	0.528982	0.0677292	0.0338646	+	0.999426i	$0.489218\pi$
$62$	24.1489	3.06692
$63$	0	0
$64$	48.3154	6.03942
$65$	5.39058	0.668619
$66$	0	0
$67$	−11.2955	−1.37997	−0.689983	−	0.723825i	$-0.742382\pi$
$67$	−11.2955	−1.37997	−0.689983	+	0.723825i	$0.742382\pi$
$68$	−2.30310	−0.279292
$69$	0	0
$70$	−4.06653	−0.486043
$71$	10.8636	1.28928	0.644638	−	0.764488i	$-0.277008\pi$
$71$	10.8636	1.28928	0.644638	+	0.764488i	$0.277008\pi$
$72$	0	0
$73$	−3.59179	−0.420388	−0.210194	−	0.977660i	$-0.567409\pi$
$73$	−3.59179	−0.420388	−0.210194	+	0.977660i	$0.567409\pi$
$74$	−9.14109	−1.06263
$75$	0	0
$76$	−16.2656	−1.86579
$77$	−0.823916	−0.0938939
$78$	0	0
$79$	0.905239	0.101847	0.0509237	−	0.998703i	$-0.483783\pi$
$79$	0.905239	0.101847	0.0509237	+	0.998703i	$0.483783\pi$
$80$	−26.9798	−3.01643
$81$	0	0
$82$	11.5015	1.27012
$83$	6.64443	0.729321	0.364661	−	0.931140i	$-0.381185\pi$
$83$	6.64443	0.729321	0.364661	+	0.931140i	$0.381185\pi$
$84$	0	0
$85$	0.570367	0.0618650
$86$	20.0001	2.15666
$87$	0	0
$88$	−8.90981	−0.949789
$89$	−4.61643	−0.489341	−0.244671	−	0.969606i	$-0.578680\pi$
$89$	−4.61643	−0.489341	−0.244671	+	0.969606i	$0.578680\pi$
$90$	0	0
$91$	3.71588	0.389530
$92$	−29.9626	−3.12382
$93$	0	0
$94$	8.58555	0.885532
$95$	4.02820	0.413285
$96$	0	0
$97$	1.82282	0.185079	0.0925395	−	0.995709i	$-0.470502\pi$
$97$	1.82282	0.185079	0.0925395	+	0.995709i	$0.470502\pi$
$98$	−2.80317	−0.283163
$99$	0	0
$100$	−16.9612	−1.69612
$101$	−3.81514	−0.379621	−0.189810	−	0.981821i	$-0.560787\pi$
$101$	−3.81514	−0.379621	−0.189810	+	0.981821i	$0.560787\pi$
$102$	0	0
$103$	2.41268	0.237728	0.118864	−	0.992911i	$-0.462075\pi$
$103$	2.41268	0.237728	0.118864	+	0.992911i	$0.462075\pi$
$104$	40.1835	3.94031
$105$	0	0
$106$	24.6129	2.39061
$107$	−14.3153	−1.38391	−0.691957	−	0.721939i	$-0.743251\pi$
$107$	−14.3153	−1.38391	−0.691957	+	0.721939i	$0.743251\pi$
$108$	0	0
$109$	−4.80743	−0.460468	−0.230234	−	0.973135i	$-0.573949\pi$
$109$	−4.80743	−0.460468	−0.230234	+	0.973135i	$0.573949\pi$
$110$	3.35047	0.319455
$111$	0	0
$112$	−18.5979	−1.75734
$113$	−11.1881	−1.05249	−0.526244	−	0.850334i	$-0.676400\pi$
$113$	−11.1881	−1.05249	−0.526244	+	0.850334i	$0.676400\pi$
$114$	0	0
$115$	7.42030	0.691947
$116$	8.18850	0.760283
$117$	0	0
$118$	−19.0491	−1.75361
$119$	0.393170	0.0360419
$120$	0	0
$121$	−10.3212	−0.938288
$122$	−1.48283	−0.134249
$123$	0	0
$124$	−50.4638	−4.53179
$125$	11.4539	1.02447
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−74.4259	−6.57838
$129$	0	0
$130$	−15.1107	−1.32530
$131$	−12.8777	−1.12513	−0.562564	−	0.826754i	$-0.690185\pi$
$131$	−12.8777	−1.12513	−0.562564	+	0.826754i	$0.690185\pi$
$132$	0	0
$133$	2.77675	0.240775
$134$	31.6633	2.73529
$135$	0	0
$136$	4.25174	0.364584
$137$	8.25478	0.705254	0.352627	−	0.935764i	$-0.385289\pi$
$137$	8.25478	0.705254	0.352627	+	0.935764i	$0.385289\pi$
$138$	0	0
$139$	15.0499	1.27652	0.638258	−	0.769822i	$-0.279655\pi$
$139$	15.0499	1.27652	0.638258	+	0.769822i	$0.279655\pi$
$140$	8.49779	0.718194
$141$	0	0
$142$	−30.4526	−2.55553
$143$	−3.06157	−0.256021
$144$	0	0
$145$	−2.02790	−0.168408
$146$	10.0684	0.833268
$147$	0	0
$148$	19.1021	1.57018
$149$	7.30680	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$149$	7.30680	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$150$	0	0
$151$	9.49748	0.772894	0.386447	−	0.922312i	$-0.373702\pi$
$151$	9.49748	0.772894	0.386447	+	0.922312i	$0.373702\pi$
$152$	30.0278	2.43557
$153$	0	0
$154$	2.30958	0.186111
$155$	12.4975	1.00382
$156$	0	0
$157$	−17.2457	−1.37636	−0.688179	−	0.725541i	$-0.741590\pi$
$157$	−17.2457	−1.37636	−0.688179	+	0.725541i	$0.741590\pi$
$158$	−2.53754	−0.201876
$159$	0	0
$160$	44.2535	3.49854
$161$	5.11502	0.403120
$162$	0	0
$163$	−6.86945	−0.538057	−0.269029	−	0.963132i	$-0.586703\pi$
$163$	−6.86945	−0.538057	−0.269029	+	0.963132i	$0.586703\pi$
$164$	−24.0345	−1.87678
$165$	0	0
$166$	−18.6255	−1.44562
$167$	12.0240	0.930447	0.465224	−	0.885193i	$-0.345974\pi$
$167$	12.0240	0.930447	0.465224	+	0.885193i	$0.345974\pi$
$168$	0	0
$169$	0.807759	0.0621353
$170$	−1.59884	−0.122625
$171$	0	0
$172$	−41.7940	−3.18677
$173$	18.8019	1.42948	0.714742	−	0.699388i	$-0.246544\pi$
$173$	18.8019	1.42948	0.714742	+	0.699388i	$0.246544\pi$
$174$	0	0
$175$	2.89551	0.218880
$176$	15.3231	1.15502
$177$	0	0
$178$	12.9407	0.969943
$179$	−19.0555	−1.42427	−0.712137	−	0.702041i	$-0.752273\pi$
$179$	−19.0555	−1.42427	−0.712137	+	0.702041i	$0.752273\pi$
$180$	0	0
$181$	−14.6135	−1.08621	−0.543106	−	0.839664i	$-0.682752\pi$
$181$	−14.6135	−1.08621	−0.543106	+	0.839664i	$0.682752\pi$
$182$	−10.4162	−0.772103
$183$	0	0
$184$	55.3138	4.07779
$185$	−4.73066	−0.347805
$186$	0	0
$187$	−0.323939	−0.0236888
$188$	−17.9412	−1.30849
$189$	0	0
$190$	−11.2917	−0.819189
$191$	−2.40562	−0.174065	−0.0870323	−	0.996205i	$-0.527738\pi$
$191$	−2.40562	−0.174065	−0.0870323	+	0.996205i	$0.527738\pi$
$192$	0	0
$193$	26.7687	1.92685	0.963426	−	0.267976i	$-0.0863548\pi$
$193$	26.7687	1.92685	0.963426	+	0.267976i	$0.0863548\pi$
$194$	−5.10967	−0.366853
$195$	0	0
$196$	5.85777	0.418412
$197$	−16.0401	−1.14281	−0.571404	−	0.820669i	$-0.693601\pi$
$197$	−16.0401	−1.14281	−0.571404	+	0.820669i	$0.693601\pi$
$198$	0	0
$199$	−21.3502	−1.51347	−0.756737	−	0.653719i	$-0.773208\pi$
$199$	−21.3502	−1.51347	−0.756737	+	0.653719i	$0.773208\pi$
$200$	31.3120	2.21409
$201$	0	0
$202$	10.6945	0.752462
$203$	−1.39789	−0.0981124
$204$	0	0
$205$	5.95220	0.415720
$206$	−6.76315	−0.471211
$207$	0	0
$208$	−69.1076	−4.79175
$209$	−2.28781	−0.158251
$210$	0	0
$211$	25.7720	1.77422	0.887109	−	0.461560i	$-0.152710\pi$
$211$	25.7720	1.77422	0.887109	+	0.461560i	$0.152710\pi$
$212$	−51.4334	−3.53246
$213$	0	0
$214$	40.1283	2.74311
$215$	10.3504	0.705890
$216$	0	0
$217$	8.61486	0.584815
$218$	13.4760	0.912713
$219$	0	0
$220$	−7.00146	−0.472039
$221$	1.46097	0.0982757
$222$	0	0
$223$	3.09046	0.206953	0.103476	−	0.994632i	$-0.467003\pi$
$223$	3.09046	0.206953	0.103476	+	0.994632i	$0.467003\pi$
$224$	30.5052	2.03821
$225$	0	0
$226$	31.3621	2.08618
$227$	12.4458	0.826058	0.413029	−	0.910718i	$-0.364471\pi$
$227$	12.4458	0.826058	0.413029	+	0.910718i	$0.364471\pi$
$228$	0	0
$229$	26.8922	1.77708	0.888542	−	0.458796i	$-0.151719\pi$
$229$	26.8922	1.77708	0.888542	+	0.458796i	$0.151719\pi$
$230$	−20.8004	−1.37154
$231$	0	0
$232$	−15.1167	−0.992462
$233$	1.31306	0.0860213	0.0430107	−	0.999075i	$-0.486305\pi$
$233$	1.31306	0.0860213	0.0430107	+	0.999075i	$0.486305\pi$
$234$	0	0
$235$	4.44317	0.289840
$236$	39.8067	2.59120
$237$	0	0
$238$	−1.10212	−0.0714401
$239$	−4.53296	−0.293213	−0.146606	−	0.989195i	$-0.546835\pi$
$239$	−4.53296	−0.293213	−0.146606	+	0.989195i	$0.546835\pi$
$240$	0	0
$241$	9.13924	0.588710	0.294355	−	0.955696i	$-0.404895\pi$
$241$	9.13924	0.588710	0.294355	+	0.955696i	$0.404895\pi$
$242$	28.9320	1.85982
$243$	0	0
$244$	3.09865	0.198371
$245$	−1.45069	−0.0926810
$246$	0	0
$247$	10.3181	0.656524
$248$	93.1609	5.91572
$249$	0	0
$250$	−32.1073	−2.03064
$251$	−11.9503	−0.754297	−0.377149	−	0.926153i	$-0.623095\pi$
$251$	−11.9503	−0.754297	−0.377149	+	0.926153i	$0.623095\pi$
$252$	0	0
$253$	−4.21435	−0.264954
$254$	2.80317	0.175887
$255$	0	0
$256$	111.998	6.99986
$257$	−0.737972	−0.0460334	−0.0230167	−	0.999735i	$-0.507327\pi$
$257$	−0.737972	−0.0460334	−0.0230167	+	0.999735i	$0.507327\pi$
$258$	0	0
$259$	−3.26098	−0.202627
$260$	31.5768	1.95831
$261$	0	0
$262$	36.0983	2.23016
$263$	13.6894	0.844125	0.422063	−	0.906567i	$-0.361306\pi$
$263$	13.6894	0.844125	0.422063	+	0.906567i	$0.361306\pi$
$264$	0	0
$265$	12.7376	0.782463
$266$	−7.78371	−0.477250
$267$	0	0
$268$	−66.1665	−4.04176
$269$	−26.3805	−1.60845	−0.804224	−	0.594326i	$-0.797419\pi$
$269$	−26.3805	−1.60845	−0.804224	+	0.594326i	$0.797419\pi$
$270$	0	0
$271$	16.1864	0.983256	0.491628	−	0.870805i	$-0.336402\pi$
$271$	16.1864	0.983256	0.491628	+	0.870805i	$0.336402\pi$
$272$	−7.31215	−0.443364
$273$	0	0
$274$	−23.1396	−1.39791
$275$	−2.38565	−0.143860
$276$	0	0
$277$	−16.3981	−0.985267	−0.492634	−	0.870237i	$-0.663966\pi$
$277$	−16.3981	−0.985267	−0.492634	+	0.870237i	$0.663966\pi$
$278$	−42.1875	−2.53024
$279$	0	0
$280$	−15.6877	−0.937520
$281$	−17.0406	−1.01655	−0.508277	−	0.861194i	$-0.669717\pi$
$281$	−17.0406	−1.01655	−0.508277	+	0.861194i	$0.669717\pi$
$282$	0	0
$283$	−13.7749	−0.818835	−0.409417	−	0.912347i	$-0.634268\pi$
$283$	−13.7749	−0.818835	−0.409417	+	0.912347i	$0.634268\pi$
$284$	63.6367	3.77614
$285$	0	0
$286$	8.58211	0.507471
$287$	4.10302	0.242194
$288$	0	0
$289$	−16.8454	−0.990907
$290$	5.68454	0.333808
$291$	0	0
$292$	−21.0399	−1.23127
$293$	18.6537	1.08976	0.544880	−	0.838514i	$-0.316575\pi$
$293$	18.6537	1.08976	0.544880	+	0.838514i	$0.316575\pi$
$294$	0	0
$295$	−9.85820	−0.573967
$296$	−35.2642	−2.04969
$297$	0	0
$298$	−20.4822	−1.18650
$299$	19.0068	1.09919
$300$	0	0
$301$	7.13481	0.411243
$302$	−26.6231	−1.53199
$303$	0	0
$304$	−51.6418	−2.96186
$305$	−0.767388	−0.0439405
$306$	0	0
$307$	28.1144	1.60457	0.802287	−	0.596938i	$-0.203616\pi$
$307$	28.1144	1.60457	0.802287	+	0.596938i	$0.203616\pi$
$308$	−4.82631	−0.275004
$309$	0	0
$310$	−35.0325	−1.98971
$311$	20.6845	1.17291	0.586456	−	0.809981i	$-0.300523\pi$
$311$	20.6845	1.17291	0.586456	+	0.809981i	$0.300523\pi$
$312$	0	0
$313$	6.99001	0.395099	0.197549	−	0.980293i	$-0.436702\pi$
$313$	6.99001	0.395099	0.197549	+	0.980293i	$0.436702\pi$
$314$	48.3427	2.72814
$315$	0	0
$316$	5.30268	0.298299
$317$	−23.4240	−1.31562	−0.657810	−	0.753184i	$-0.728517\pi$
$317$	−23.4240	−1.31562	−0.657810	+	0.753184i	$0.728517\pi$
$318$	0	0
$319$	1.15174	0.0644851
$320$	−70.0905	−3.91818
$321$	0	0
$322$	−14.3383	−0.799042
$323$	1.09174	0.0607459
$324$	0	0
$325$	10.7594	0.596821
$326$	19.2563	1.06651
$327$	0	0
$328$	44.3700	2.44992
$329$	3.06280	0.168858
$330$	0	0
$331$	−11.8270	−0.650071	−0.325035	−	0.945702i	$-0.605376\pi$
$331$	−11.8270	−0.650071	−0.325035	+	0.945702i	$0.605376\pi$
$332$	38.9215	2.13610
$333$	0	0
$334$	−33.7054	−1.84428
$335$	16.3863	0.895277
$336$	0	0
$337$	5.78630	0.315200	0.157600	−	0.987503i	$-0.449624\pi$
$337$	5.78630	0.315200	0.157600	+	0.987503i	$0.449624\pi$
$338$	−2.26429	−0.123161
$339$	0	0
$340$	3.34108	0.181195
$341$	−7.09792	−0.384374
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	77.1557	4.15996
$345$	0	0
$346$	−52.7050	−2.83344
$347$	23.8178	1.27861	0.639304	−	0.768954i	$-0.279223\pi$
$347$	23.8178	1.27861	0.639304	+	0.768954i	$0.279223\pi$
$348$	0	0
$349$	29.5571	1.58215	0.791077	−	0.611717i	$-0.209521\pi$
$349$	29.5571	1.58215	0.791077	+	0.611717i	$0.209521\pi$
$350$	−8.11660	−0.433850
$351$	0	0
$352$	−25.1337	−1.33963
$353$	18.5255	0.986011	0.493005	−	0.870026i	$-0.335898\pi$
$353$	18.5255	0.986011	0.493005	+	0.870026i	$0.335898\pi$
$354$	0	0
$355$	−15.7597	−0.836440
$356$	−27.0420	−1.43322
$357$	0	0
$358$	53.4158	2.82311
$359$	−1.27264	−0.0671672	−0.0335836	−	0.999436i	$-0.510692\pi$
$359$	−1.27264	−0.0671672	−0.0335836	+	0.999436i	$0.510692\pi$
$360$	0	0
$361$	−11.2896	−0.594192
$362$	40.9641	2.15302
$363$	0	0
$364$	21.7668	1.14089
$365$	5.21057	0.272734
$366$	0	0
$367$	−22.5288	−1.17599	−0.587996	−	0.808864i	$-0.700083\pi$
$367$	−22.5288	−1.17599	−0.587996	+	0.808864i	$0.700083\pi$
$368$	−95.1288	−4.95893
$369$	0	0
$370$	13.2609	0.689399
$371$	8.78037	0.455854
$372$	0	0
$373$	−7.94023	−0.411130	−0.205565	−	0.978643i	$-0.565903\pi$
$373$	−7.94023	−0.411130	−0.205565	+	0.978643i	$0.565903\pi$
$374$	0.908057	0.0469545
$375$	0	0
$376$	33.1211	1.70809
$377$	−5.19438	−0.267524
$378$	0	0
$379$	−16.1684	−0.830515	−0.415258	−	0.909704i	$-0.636309\pi$
$379$	−16.1684	−0.830515	−0.415258	+	0.909704i	$0.636309\pi$
$380$	23.5963	1.21046
$381$	0	0
$382$	6.74337	0.345021
$383$	−19.3267	−0.987550	−0.493775	−	0.869590i	$-0.664383\pi$
$383$	−19.3267	−0.987550	−0.493775	+	0.869590i	$0.664383\pi$
$384$	0	0
$385$	1.19524	0.0609153
$386$	−75.0372	−3.81929
$387$	0	0
$388$	10.6776	0.542075
$389$	−12.8065	−0.649316	−0.324658	−	0.945832i	$-0.605249\pi$
$389$	−12.8065	−0.649316	−0.324658	+	0.945832i	$0.605249\pi$
$390$	0	0
$391$	2.01108	0.101704
$392$	−10.8140	−0.546189
$393$	0	0
$394$	44.9631	2.26521
$395$	−1.31322	−0.0660752
$396$	0	0
$397$	6.39314	0.320862	0.160431	−	0.987047i	$-0.448712\pi$
$397$	6.39314	0.320862	0.160431	+	0.987047i	$0.448712\pi$
$398$	59.8482	2.99992
$399$	0	0
$400$	−53.8504	−2.69252
$401$	−35.9471	−1.79511	−0.897555	−	0.440902i	$-0.854659\pi$
$401$	−35.9471	−1.79511	−0.897555	+	0.440902i	$0.854659\pi$
$402$	0	0
$403$	32.0118	1.59462
$404$	−22.3482	−1.11186
$405$	0	0
$406$	3.91852	0.194473
$407$	2.68677	0.133178
$408$	0	0
$409$	−19.7091	−0.974552	−0.487276	−	0.873248i	$-0.662009\pi$
$409$	−19.7091	−0.974552	−0.487276	+	0.873248i	$0.662009\pi$
$410$	−16.6850	−0.824015
$411$	0	0
$412$	14.1329	0.696278
$413$	−6.79554	−0.334387
$414$	0	0
$415$	−9.63900	−0.473160
$416$	113.353	5.55761
$417$	0	0
$418$	6.41312	0.313676
$419$	4.70863	0.230031	0.115016	−	0.993364i	$-0.463308\pi$
$419$	4.70863	0.230031	0.115016	+	0.993364i	$0.463308\pi$
$420$	0	0
$421$	−4.49103	−0.218880	−0.109440	−	0.993993i	$-0.534906\pi$
$421$	−4.49103	−0.218880	−0.109440	+	0.993993i	$0.534906\pi$
$422$	−72.2433	−3.51675
$423$	0	0
$424$	94.9507	4.61122
$425$	1.13843	0.0552218
$426$	0	0
$427$	−0.528982	−0.0255992
$428$	−83.8558	−4.05332
$429$	0	0
$430$	−29.0139	−1.39917
$431$	−29.8665	−1.43862	−0.719309	−	0.694691i	$-0.755541\pi$
$431$	−29.8665	−1.43862	−0.719309	+	0.694691i	$0.755541\pi$
$432$	0	0
$433$	−23.1220	−1.11117	−0.555586	−	0.831459i	$-0.687506\pi$
$433$	−23.1220	−1.11117	−0.555586	+	0.831459i	$0.687506\pi$
$434$	−24.1489	−1.15918
$435$	0	0
$436$	−28.1608	−1.34866
$437$	14.2032	0.679429
$438$	0	0
$439$	9.40503	0.448877	0.224439	−	0.974488i	$-0.427945\pi$
$439$	9.40503	0.448877	0.224439	+	0.974488i	$0.427945\pi$
$440$	12.9254	0.616192
$441$	0	0
$442$	−4.09536	−0.194796
$443$	0.276021	0.0131142	0.00655708	−	0.999979i	$-0.497913\pi$
$443$	0.276021	0.0131142	0.00655708	+	0.999979i	$0.497913\pi$
$444$	0	0
$445$	6.69700	0.317468
$446$	−8.66309	−0.410209
$447$	0	0
$448$	−48.3154	−2.28269
$449$	22.9639	1.08373	0.541867	−	0.840465i	$-0.317718\pi$
$449$	22.9639	1.08373	0.541867	+	0.840465i	$0.317718\pi$
$450$	0	0
$451$	−3.38054	−0.159184
$452$	−65.5373	−3.08261
$453$	0	0
$454$	−34.8878	−1.63736
$455$	−5.39058	−0.252714
$456$	0	0
$457$	−23.3729	−1.09334	−0.546669	−	0.837349i	$-0.684104\pi$
$457$	−23.3729	−1.09334	−0.546669	+	0.837349i	$0.684104\pi$
$458$	−75.3833	−3.52243
$459$	0	0
$460$	43.4664	2.02663
$461$	−17.3216	−0.806748	−0.403374	−	0.915035i	$-0.632163\pi$
$461$	−17.3216	−0.806748	−0.403374	+	0.915035i	$0.632163\pi$
$462$	0	0
$463$	16.1833	0.752100	0.376050	−	0.926599i	$-0.377282\pi$
$463$	16.1833	0.752100	0.376050	+	0.926599i	$0.377282\pi$
$464$	25.9978	1.20692
$465$	0	0
$466$	−3.68073	−0.170506
$467$	−30.9019	−1.42997	−0.714985	−	0.699140i	$-0.753566\pi$
$467$	−30.9019	−1.42997	−0.714985	+	0.699140i	$0.753566\pi$
$468$	0	0
$469$	11.2955	0.521578
$470$	−12.4550	−0.574504
$471$	0	0
$472$	−73.4869	−3.38251
$473$	−5.87848	−0.270293
$474$	0	0
$475$	8.04010	0.368905
$476$	2.30310	0.105562
$477$	0	0
$478$	12.7067	0.581190
$479$	−30.5265	−1.39479	−0.697395	−	0.716687i	$-0.745658\pi$
$479$	−30.5265	−1.39479	−0.697395	+	0.716687i	$0.745658\pi$
$480$	0	0
$481$	−12.1174	−0.552506
$482$	−25.6189	−1.16691
$483$	0	0
$484$	−60.4590	−2.74814
$485$	−2.64434	−0.120073
$486$	0	0
$487$	0.219254	0.00993536	0.00496768	−	0.999988i	$-0.498419\pi$
$487$	0.219254	0.00993536	0.00496768	+	0.999988i	$0.498419\pi$
$488$	−5.72040	−0.258951
$489$	0	0
$490$	4.06653	0.183707
$491$	17.1673	0.774750	0.387375	−	0.921922i	$-0.373382\pi$
$491$	17.1673	0.774750	0.387375	+	0.921922i	$0.373382\pi$
$492$	0	0
$493$	−0.549608	−0.0247531
$494$	−28.9233	−1.30132
$495$	0	0
$496$	−160.218	−7.19401
$497$	−10.8636	−0.487301
$498$	0	0
$499$	−20.0425	−0.897224	−0.448612	−	0.893727i	$-0.648081\pi$
$499$	−20.0425	−0.897224	−0.448612	+	0.893727i	$0.648081\pi$
$500$	67.0944	3.00055
$501$	0	0
$502$	33.4988	1.49512
$503$	40.6923	1.81438	0.907190	−	0.420722i	$-0.138223\pi$
$503$	40.6923	1.81438	0.907190	+	0.420722i	$0.138223\pi$
$504$	0	0
$505$	5.53457	0.246285
$506$	11.8135	0.525176
$507$	0	0
$508$	−5.85777	−0.259896
$509$	20.1488	0.893079	0.446540	−	0.894764i	$-0.352656\pi$
$509$	20.1488	0.893079	0.446540	+	0.894764i	$0.352656\pi$
$510$	0	0
$511$	3.59179	0.158892
$512$	−165.097	−7.29633
$513$	0	0
$514$	2.06866	0.0912448
$515$	−3.50004	−0.154230
$516$	0	0
$517$	−2.52349	−0.110983
$518$	9.14109	0.401636
$519$	0	0
$520$	−58.2936	−2.55635
$521$	43.4612	1.90407	0.952034	−	0.305992i	$-0.0989881\pi$
$521$	43.4612	1.90407	0.952034	+	0.305992i	$0.0989881\pi$
$522$	0	0
$523$	−4.23106	−0.185011	−0.0925056	−	0.995712i	$-0.529488\pi$
$523$	−4.23106	−0.185011	−0.0925056	+	0.995712i	$0.529488\pi$
$524$	−75.4345	−3.29537
$525$	0	0
$526$	−38.3738	−1.67318
$527$	3.38711	0.147545
$528$	0	0
$529$	3.16347	0.137542
$530$	−35.7056	−1.55095
$531$	0	0
$532$	16.2656	0.705202
$533$	15.2463	0.660392
$534$	0	0
$535$	20.7671	0.897838
$536$	122.150	5.27606
$537$	0	0
$538$	73.9491	3.18817
$539$	0.823916	0.0354886
$540$	0	0
$541$	46.1063	1.98227	0.991133	−	0.132875i	$-0.0424210\pi$
$541$	46.1063	1.98227	0.991133	+	0.132875i	$0.0424210\pi$
$542$	−45.3734	−1.94895
$543$	0	0
$544$	11.9937	0.514227
$545$	6.97408	0.298737
$546$	0	0
$547$	−31.8560	−1.36207	−0.681033	−	0.732253i	$-0.738469\pi$
$547$	−31.8560	−1.36207	−0.681033	+	0.732253i	$0.738469\pi$
$548$	48.3546	2.06561
$549$	0	0
$550$	6.68739	0.285151
$551$	−3.88159	−0.165361
$552$	0	0
$553$	−0.905239	−0.0384947
$554$	45.9667	1.95294
$555$	0	0
$556$	88.1589	3.73877
$557$	−41.5869	−1.76209	−0.881046	−	0.473031i	$-0.843160\pi$
$557$	−41.5869	−1.76209	−0.881046	+	0.473031i	$0.843160\pi$
$558$	0	0
$559$	26.5121	1.12134
$560$	26.9798	1.14010
$561$	0	0
$562$	47.7676	2.01495
$563$	−23.1484	−0.975588	−0.487794	−	0.872959i	$-0.662198\pi$
$563$	−23.1484	−0.975588	−0.487794	+	0.872959i	$0.662198\pi$
$564$	0	0
$565$	16.2304	0.682819
$566$	38.6135	1.62305
$567$	0	0
$568$	−117.479	−4.92932
$569$	22.4983	0.943176	0.471588	−	0.881819i	$-0.343681\pi$
$569$	22.4983	0.943176	0.471588	+	0.881819i	$0.343681\pi$
$570$	0	0
$571$	10.5998	0.443587	0.221794	−	0.975094i	$-0.428809\pi$
$571$	10.5998	0.443587	0.221794	+	0.975094i	$0.428809\pi$
$572$	−17.9340	−0.749857
$573$	0	0
$574$	−11.5015	−0.480062
$575$	14.8106	0.617644
$576$	0	0
$577$	41.1738	1.71409	0.857044	−	0.515243i	$-0.172298\pi$
$577$	41.1738	1.71409	0.857044	+	0.515243i	$0.172298\pi$
$578$	47.2206	1.96412
$579$	0	0
$580$	−11.8790	−0.493247
$581$	−6.64443	−0.275658
$582$	0	0
$583$	−7.23428	−0.299613
$584$	38.8416	1.60728
$585$	0	0
$586$	−52.2895	−2.16006
$587$	9.01585	0.372124	0.186062	−	0.982538i	$-0.440427\pi$
$587$	9.01585	0.372124	0.186062	+	0.982538i	$0.440427\pi$
$588$	0	0
$589$	23.9213	0.985661
$590$	27.6342	1.13768
$591$	0	0
$592$	60.6474	2.49259
$593$	−24.6544	−1.01244	−0.506218	−	0.862406i	$-0.668957\pi$
$593$	−24.6544	−1.01244	−0.506218	+	0.862406i	$0.668957\pi$
$594$	0	0
$595$	−0.570367	−0.0233828
$596$	42.8016	1.75322
$597$	0	0
$598$	−53.2793	−2.17875
$599$	−22.2761	−0.910175	−0.455088	−	0.890447i	$-0.650392\pi$
$599$	−22.2761	−0.910175	−0.455088	+	0.890447i	$0.650392\pi$
$600$	0	0
$601$	31.3730	1.27973	0.639866	−	0.768486i	$-0.278990\pi$
$601$	31.3730	1.27973	0.639866	+	0.768486i	$0.278990\pi$
$602$	−20.0001	−0.815142
$603$	0	0
$604$	55.6341	2.26372
$605$	14.9728	0.608730
$606$	0	0
$607$	−38.6356	−1.56817	−0.784086	−	0.620652i	$-0.786868\pi$
$607$	−38.6356	−1.56817	−0.784086	+	0.620652i	$0.786868\pi$
$608$	84.7053	3.43525
$609$	0	0
$610$	2.15112	0.0870963
$611$	11.3810	0.460426
$612$	0	0
$613$	35.5111	1.43428	0.717141	−	0.696929i	$-0.245451\pi$
$613$	35.5111	1.43428	0.717141	+	0.696929i	$0.245451\pi$
$614$	−78.8095	−3.18049
$615$	0	0
$616$	8.90981	0.358987
$617$	−44.0995	−1.77538	−0.887689	−	0.460443i	$-0.847690\pi$
$617$	−44.0995	−1.77538	−0.887689	+	0.460443i	$0.847690\pi$
$618$	0	0
$619$	30.2260	1.21488	0.607442	−	0.794364i	$-0.292196\pi$
$619$	30.2260	1.21488	0.607442	+	0.794364i	$0.292196\pi$
$620$	73.2072	2.94007
$621$	0	0
$622$	−57.9823	−2.32488
$623$	4.61643	0.184954
$624$	0	0
$625$	−2.13852	−0.0855406
$626$	−19.5942	−0.783142
$627$	0	0
$628$	−101.021	−4.03119
$629$	−1.28212	−0.0511215
$630$	0	0
$631$	−26.7462	−1.06475	−0.532375	−	0.846509i	$-0.678700\pi$
$631$	−26.7462	−1.06475	−0.532375	+	0.846509i	$0.678700\pi$
$632$	−9.78924	−0.389395
$633$	0	0
$634$	65.6613	2.60774
$635$	1.45069	0.0575688
$636$	0	0
$637$	−3.71588	−0.147229
$638$	−3.22853	−0.127819
$639$	0	0
$640$	107.969	4.26784
$641$	39.6400	1.56568	0.782842	−	0.622220i	$-0.213769\pi$
$641$	39.6400	1.56568	0.782842	+	0.622220i	$0.213769\pi$
$642$	0	0
$643$	−10.2482	−0.404149	−0.202074	−	0.979370i	$-0.564768\pi$
$643$	−10.2482	−0.404149	−0.202074	+	0.979370i	$0.564768\pi$
$644$	29.9626	1.18069
$645$	0	0
$646$	−3.06033	−0.120407
$647$	−4.12550	−0.162190	−0.0810951	−	0.996706i	$-0.525842\pi$
$647$	−4.12550	−0.162190	−0.0810951	+	0.996706i	$0.525842\pi$
$648$	0	0
$649$	5.59895	0.219778
$650$	−30.1603	−1.18298
$651$	0	0
$652$	−40.2397	−1.57591
$653$	31.8892	1.24792	0.623961	−	0.781456i	$-0.285523\pi$
$653$	31.8892	1.24792	0.623961	+	0.781456i	$0.285523\pi$
$654$	0	0
$655$	18.6815	0.729946
$656$	−76.3076	−2.97931
$657$	0	0
$658$	−8.58555	−0.334700
$659$	46.5189	1.81212	0.906060	−	0.423150i	$-0.139076\pi$
$659$	46.5189	1.81212	0.906060	+	0.423150i	$0.139076\pi$
$660$	0	0
$661$	−20.4562	−0.795655	−0.397828	−	0.917460i	$-0.630236\pi$
$661$	−20.4562	−0.795655	−0.397828	+	0.917460i	$0.630236\pi$
$662$	33.1531	1.28853
$663$	0	0
$664$	−71.8528	−2.78843
$665$	−4.02820	−0.156207
$666$	0	0
$667$	−7.15023	−0.276858
$668$	70.4340	2.72517
$669$	0	0
$670$	−45.9335	−1.77457
$671$	0.435837	0.0168253
$672$	0	0
$673$	−24.7924	−0.955675	−0.477837	−	0.878448i	$-0.658579\pi$
$673$	−24.7924	−0.955675	−0.477837	+	0.878448i	$0.658579\pi$
$674$	−16.2200	−0.624770
$675$	0	0
$676$	4.73167	0.181987
$677$	−12.2689	−0.471531	−0.235766	−	0.971810i	$-0.575760\pi$
$677$	−12.2689	−0.471531	−0.235766	+	0.971810i	$0.575760\pi$
$678$	0	0
$679$	−1.82282	−0.0699533
$680$	−6.16794	−0.236530
$681$	0	0
$682$	19.8967	0.761883
$683$	37.7152	1.44313	0.721566	−	0.692346i	$-0.243423\pi$
$683$	37.7152	1.44313	0.721566	+	0.692346i	$0.243423\pi$
$684$	0	0
$685$	−11.9751	−0.457545
$686$	2.80317	0.107026
$687$	0	0
$688$	−132.692	−5.05885
$689$	32.6268	1.24298
$690$	0	0
$691$	−3.68159	−0.140054	−0.0700272	−	0.997545i	$-0.522309\pi$
$691$	−3.68159	−0.140054	−0.0700272	+	0.997545i	$0.522309\pi$
$692$	110.137	4.18679
$693$	0	0
$694$	−66.7654	−2.53438
$695$	−21.8327	−0.828162
$696$	0	0
$697$	1.61319	0.0611038
$698$	−82.8535	−3.13605
$699$	0	0
$700$	16.9612	0.641073
$701$	−16.6339	−0.628254	−0.314127	−	0.949381i	$-0.601712\pi$
$701$	−16.6339	−0.628254	−0.314127	+	0.949381i	$0.601712\pi$
$702$	0	0
$703$	−9.05494	−0.341513
$704$	39.8078	1.50031
$705$	0	0
$706$	−51.9300	−1.95441
$707$	3.81514	0.143483
$708$	0	0
$709$	49.5433	1.86064	0.930319	−	0.366752i	$-0.119530\pi$
$709$	49.5433	1.86064	0.930319	+	0.366752i	$0.119530\pi$
$710$	44.1773	1.65794
$711$	0	0
$712$	49.9220	1.87091
$713$	44.0652	1.65025
$714$	0	0
$715$	4.44138	0.166098
$716$	−111.623	−4.17153
$717$	0	0
$718$	3.56742	0.133135
$719$	20.5779	0.767428	0.383714	−	0.923452i	$-0.374645\pi$
$719$	20.5779	0.767428	0.383714	+	0.923452i	$0.374645\pi$
$720$	0	0
$721$	−2.41268	−0.0898528
$722$	31.6468	1.17777
$723$	0	0
$724$	−85.6024	−3.18139
$725$	−4.04759	−0.150324
$726$	0	0
$727$	16.0003	0.593419	0.296709	−	0.954968i	$-0.404111\pi$
$727$	16.0003	0.593419	0.296709	+	0.954968i	$0.404111\pi$
$728$	−40.1835	−1.48930
$729$	0	0
$730$	−14.6061	−0.540597
$731$	2.80519	0.103754
$732$	0	0
$733$	52.0610	1.92292	0.961458	−	0.274953i	$-0.0886621\pi$
$733$	52.0610	1.92292	0.961458	+	0.274953i	$0.0886621\pi$
$734$	63.1520	2.33098
$735$	0	0
$736$	156.035	5.75151
$737$	−9.30655	−0.342811
$738$	0	0
$739$	25.7539	0.947373	0.473687	−	0.880694i	$-0.342923\pi$
$739$	25.7539	0.947373	0.473687	+	0.880694i	$0.342923\pi$
$740$	−27.7111	−1.01868
$741$	0	0
$742$	−24.6129	−0.903567
$743$	11.3361	0.415881	0.207940	−	0.978141i	$-0.433324\pi$
$743$	11.3361	0.415881	0.207940	+	0.978141i	$0.433324\pi$
$744$	0	0
$745$	−10.5999	−0.388350
$746$	22.2578	0.814917
$747$	0	0
$748$	−1.89756	−0.0693817
$749$	14.3153	0.523070
$750$	0	0
$751$	5.23002	0.190846	0.0954230	−	0.995437i	$-0.469580\pi$
$751$	5.23002	0.190846	0.0954230	+	0.995437i	$0.469580\pi$
$752$	−56.9617	−2.07718
$753$	0	0
$754$	14.5607	0.530271
$755$	−13.7779	−0.501428
$756$	0	0
$757$	−22.3482	−0.812259	−0.406129	−	0.913816i	$-0.633122\pi$
$757$	−22.3482	−0.812259	−0.406129	+	0.913816i	$0.633122\pi$
$758$	45.3228	1.64620
$759$	0	0
$760$	−43.5609	−1.58012
$761$	0.525467	0.0190482	0.00952409	−	0.999955i	$-0.496968\pi$
$761$	0.525467	0.0190482	0.00952409	+	0.999955i	$0.496968\pi$
$762$	0	0
$763$	4.80743	0.174041
$764$	−14.0916	−0.509815
$765$	0	0
$766$	54.1761	1.95746
$767$	−25.2514	−0.911775
$768$	0	0
$769$	18.9555	0.683552	0.341776	−	0.939781i	$-0.388972\pi$
$769$	18.9555	0.683552	0.341776	+	0.939781i	$0.388972\pi$
$770$	−3.35047	−0.120743
$771$	0	0
$772$	156.805	5.64352
$773$	−33.3510	−1.19955	−0.599775	−	0.800168i	$-0.704743\pi$
$773$	−33.3510	−1.19955	−0.599775	+	0.800168i	$0.704743\pi$
$774$	0	0
$775$	24.9444	0.896028
$776$	−19.7119	−0.707617
$777$	0	0
$778$	35.8988	1.28704
$779$	11.3931	0.408199
$780$	0	0
$781$	8.95072	0.320282
$782$	−5.63739	−0.201593
$783$	0	0
$784$	18.5979	0.664211
$785$	25.0181	0.892936
$786$	0	0
$787$	−11.7190	−0.417736	−0.208868	−	0.977944i	$-0.566978\pi$
$787$	−11.7190	−0.417736	−0.208868	+	0.977944i	$0.566978\pi$
$788$	−93.9590	−3.34715
$789$	0	0
$790$	3.68118	0.130970
$791$	11.1881	0.397803
$792$	0	0
$793$	−1.96563	−0.0698017
$794$	−17.9211	−0.635994
$795$	0	0
$796$	−125.064	−4.43279
$797$	42.9928	1.52288	0.761441	−	0.648234i	$-0.224492\pi$
$797$	42.9928	1.52288	0.761441	+	0.648234i	$0.224492\pi$
$798$	0	0
$799$	1.20420	0.0426016
$800$	88.3279	3.12286
$801$	0	0
$802$	100.766	3.55816
$803$	−2.95933	−0.104433
$804$	0	0
$805$	−7.42030	−0.261531
$806$	−89.7345	−3.16076
$807$	0	0
$808$	41.2569	1.45141
$809$	−25.6804	−0.902876	−0.451438	−	0.892302i	$-0.649089\pi$
$809$	−25.6804	−0.902876	−0.451438	+	0.892302i	$0.649089\pi$
$810$	0	0
$811$	−50.5316	−1.77440	−0.887202	−	0.461382i	$-0.847354\pi$
$811$	−50.5316	−1.77440	−0.887202	+	0.461382i	$0.847354\pi$
$812$	−8.18850	−0.287360
$813$	0	0
$814$	−7.53148	−0.263978
$815$	9.96543	0.349074
$816$	0	0
$817$	19.8116	0.693120
$818$	55.2480	1.93170
$819$	0	0
$820$	34.8666	1.21759
$821$	20.1671	0.703838	0.351919	−	0.936030i	$-0.385529\pi$
$821$	20.1671	0.703838	0.351919	+	0.936030i	$0.385529\pi$
$822$	0	0
$823$	−31.3478	−1.09272	−0.546358	−	0.837552i	$-0.683986\pi$
$823$	−31.3478	−1.09272	−0.546358	+	0.837552i	$0.683986\pi$
$824$	−26.0907	−0.908911
$825$	0	0
$826$	19.0491	0.662802
$827$	−22.3659	−0.777738	−0.388869	−	0.921293i	$-0.627134\pi$
$827$	−22.3659	−0.777738	−0.388869	+	0.921293i	$0.627134\pi$
$828$	0	0
$829$	54.5745	1.89545	0.947726	−	0.319085i	$-0.103375\pi$
$829$	54.5745	1.89545	0.947726	+	0.319085i	$0.103375\pi$
$830$	27.0198	0.937869
$831$	0	0
$832$	−179.534	−6.22422
$833$	−0.393170	−0.0136225
$834$	0	0
$835$	−17.4431	−0.603644
$836$	−13.4015	−0.463499
$837$	0	0
$838$	−13.1991	−0.455955
$839$	−2.65487	−0.0916564	−0.0458282	−	0.998949i	$-0.514593\pi$
$839$	−2.65487	−0.0916564	−0.0458282	+	0.998949i	$0.514593\pi$
$840$	0	0
$841$	−27.0459	−0.932618
$842$	12.5891	0.433850
$843$	0	0
$844$	150.966	5.19648
$845$	−1.17181	−0.0403114
$846$	0	0
$847$	10.3212	0.354639
$848$	−163.296	−5.60762
$849$	0	0
$850$	−3.19121	−0.109457
$851$	−16.6800	−0.571783
$852$	0	0
$853$	40.0863	1.37253	0.686264	−	0.727353i	$-0.259249\pi$
$853$	40.0863	1.37253	0.686264	+	0.727353i	$0.259249\pi$
$854$	1.48283	0.0507413
$855$	0	0
$856$	154.806	5.29115
$857$	−9.98258	−0.340998	−0.170499	−	0.985358i	$-0.554538\pi$
$857$	−9.98258	−0.340998	−0.170499	+	0.985358i	$0.554538\pi$
$858$	0	0
$859$	18.0371	0.615419	0.307709	−	0.951480i	$-0.400438\pi$
$859$	18.0371	0.615419	0.307709	+	0.951480i	$0.400438\pi$
$860$	60.6301	2.06747
$861$	0	0
$862$	83.7208	2.85154
$863$	11.6703	0.397262	0.198631	−	0.980074i	$-0.436351\pi$
$863$	11.6703	0.397262	0.198631	+	0.980074i	$0.436351\pi$
$864$	0	0
$865$	−27.2757	−0.927402
$866$	64.8149	2.20250
$867$	0	0
$868$	50.4638	1.71285
$869$	0.745841	0.0253009
$870$	0	0
$871$	41.9728	1.42219
$872$	51.9875	1.76052
$873$	0	0
$874$	−39.8139	−1.34672
$875$	−11.4539	−0.387213
$876$	0	0
$877$	9.02118	0.304623	0.152312	−	0.988333i	$-0.451328\pi$
$877$	9.02118	0.304623	0.152312	+	0.988333i	$0.451328\pi$
$878$	−26.3639	−0.889739
$879$	0	0
$880$	−22.2290	−0.749341
$881$	−1.15493	−0.0389106	−0.0194553	−	0.999811i	$-0.506193\pi$
$881$	−1.15493	−0.0389106	−0.0194553	+	0.999811i	$0.506193\pi$
$882$	0	0
$883$	34.8778	1.17373	0.586866	−	0.809684i	$-0.300362\pi$
$883$	34.8778	1.17373	0.586866	+	0.809684i	$0.300362\pi$
$884$	8.55805	0.287838
$885$	0	0
$886$	−0.773734	−0.0259941
$887$	4.75159	0.159543	0.0797714	−	0.996813i	$-0.474581\pi$
$887$	4.75159	0.159543	0.0797714	+	0.996813i	$0.474581\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−18.7728	−0.629267
$891$	0	0
$892$	18.1032	0.606140
$893$	8.50464	0.284597
$894$	0	0
$895$	27.6436	0.924022
$896$	74.4259	2.48639
$897$	0	0
$898$	−64.3717	−2.14811
$899$	−12.0426	−0.401643
$900$	0	0
$901$	3.45218	0.115009
$902$	9.47624	0.315524
$903$	0	0
$904$	120.988	4.02400
$905$	21.1996	0.704698
$906$	0	0
$907$	−33.4720	−1.11142	−0.555709	−	0.831377i	$-0.687553\pi$
$907$	−33.4720	−1.11142	−0.555709	+	0.831377i	$0.687553\pi$
$908$	72.9048	2.41943
$909$	0	0
$910$	15.1107	0.500915
$911$	−0.237931	−0.00788301	−0.00394151	−	0.999992i	$-0.501255\pi$
$911$	−0.237931	−0.00788301	−0.00394151	+	0.999992i	$0.501255\pi$
$912$	0	0
$913$	5.47445	0.181178
$914$	65.5182	2.16715
$915$	0	0
$916$	157.528	5.20487
$917$	12.8777	0.425258
$918$	0	0
$919$	−22.9760	−0.757910	−0.378955	−	0.925415i	$-0.623716\pi$
$919$	−22.9760	−0.757910	−0.378955	+	0.925415i	$0.623716\pi$
$920$	−80.2430	−2.64553
$921$	0	0
$922$	48.5554	1.59909
$923$	−40.3680	−1.32873
$924$	0	0
$925$	−9.44219	−0.310457
$926$	−45.3645	−1.49077
$927$	0	0
$928$	−42.6428	−1.39982
$929$	−50.0413	−1.64180	−0.820901	−	0.571071i	$-0.806528\pi$
$929$	−50.0413	−1.64180	−0.820901	+	0.571071i	$0.806528\pi$
$930$	0	0
$931$	−2.77675	−0.0910044
$932$	7.69159	0.251947
$933$	0	0
$934$	86.6233	2.83440
$935$	0.469935	0.0153685
$936$	0	0
$937$	−2.29608	−0.0750096	−0.0375048	−	0.999296i	$-0.511941\pi$
$937$	−2.29608	−0.0750096	−0.0375048	+	0.999296i	$0.511941\pi$
$938$	−31.6633	−1.03384
$939$	0	0
$940$	26.0270	0.848908
$941$	8.21890	0.267929	0.133964	−	0.990986i	$-0.457229\pi$
$941$	8.21890	0.267929	0.133964	+	0.990986i	$0.457229\pi$
$942$	0	0
$943$	20.9870	0.683432
$944$	126.383	4.11341
$945$	0	0
$946$	16.4784	0.535758
$947$	58.4845	1.90049	0.950246	−	0.311501i	$-0.100832\pi$
$947$	58.4845	1.90049	0.950246	+	0.311501i	$0.100832\pi$
$948$	0	0
$949$	13.3467	0.433251
$950$	−22.5378	−0.731222
$951$	0	0
$952$	−4.25174	−0.137800
$953$	−18.3103	−0.593128	−0.296564	−	0.955013i	$-0.595841\pi$
$953$	−18.3103	−0.593128	−0.296564	+	0.955013i	$0.595841\pi$
$954$	0	0
$955$	3.48980	0.112927
$956$	−26.5530	−0.858787
$957$	0	0
$958$	85.5710	2.76467
$959$	−8.25478	−0.266561
$960$	0	0
$961$	43.2157	1.39406
$962$	33.9672	1.09515
$963$	0	0
$964$	53.5355	1.72426
$965$	−38.8330	−1.25008
$966$	0	0
$967$	58.4921	1.88098	0.940490	−	0.339822i	$-0.110367\pi$
$967$	58.4921	1.88098	0.940490	+	0.339822i	$0.110367\pi$
$968$	111.613	3.58737
$969$	0	0
$970$	7.41253	0.238002
$971$	36.6287	1.17547	0.587736	−	0.809053i	$-0.300019\pi$
$971$	36.6287	1.17547	0.587736	+	0.809053i	$0.300019\pi$
$972$	0	0
$973$	−15.0499	−0.482478
$974$	−0.614607	−0.0196933
$975$	0	0
$976$	9.83796	0.314906
$977$	49.0117	1.56802	0.784011	−	0.620747i	$-0.213171\pi$
$977$	49.0117	1.56802	0.784011	+	0.620747i	$0.213171\pi$
$978$	0	0
$979$	−3.80355	−0.121562
$980$	−8.49779	−0.271452
$981$	0	0
$982$	−48.1229	−1.53566
$983$	−1.33794	−0.0426738	−0.0213369	−	0.999772i	$-0.506792\pi$
$983$	−1.33794	−0.0426738	−0.0213369	+	0.999772i	$0.506792\pi$
$984$	0	0
$985$	23.2691	0.741416
$986$	1.54064	0.0490641
$987$	0	0
$988$	60.4409	1.92288
$989$	36.4947	1.16046
$990$	0	0
$991$	0.0709378	0.00225341	0.00112671	−	0.999999i	$-0.499641\pi$
$991$	0.0709378	0.00225341	0.00112671	+	0.999999i	$0.499641\pi$
$992$	262.798	8.34383
$993$	0	0
$994$	30.4526	0.965899
$995$	30.9724	0.981893
$996$	0	0
$997$	16.2009	0.513087	0.256544	−	0.966533i	$-0.417416\pi$
$997$	16.2009	0.513087	0.256544	+	0.966533i	$0.417416\pi$
$998$	56.1825	1.77842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.1		16
3.2	odd	2		2667.2.a.n.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.16	✓	16	3.2	odd	2
8001.2.a.s.1.1		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80317 q^{2} +5.85777 q^{4} -1.45069 q^{5} -1.00000 q^{7} -10.8140 q^{8} +O(q^{10})\)	\(q-2.80317 q^{2} +5.85777 q^{4} -1.45069 q^{5} -1.00000 q^{7} -10.8140 q^{8} +4.06653 q^{10} +0.823916 q^{11} -3.71588 q^{13} +2.80317 q^{14} +18.5979 q^{16} -0.393170 q^{17} -2.77675 q^{19} -8.49779 q^{20} -2.30958 q^{22} -5.11502 q^{23} -2.89551 q^{25} +10.4162 q^{26} -5.85777 q^{28} +1.39789 q^{29} -8.61486 q^{31} -30.5052 q^{32} +1.10212 q^{34} +1.45069 q^{35} +3.26098 q^{37} +7.78371 q^{38} +15.6877 q^{40} -4.10302 q^{41} -7.13481 q^{43} +4.82631 q^{44} +14.3383 q^{46} -3.06280 q^{47} +1.00000 q^{49} +8.11660 q^{50} -21.7668 q^{52} -8.78037 q^{53} -1.19524 q^{55} +10.8140 q^{56} -3.91852 q^{58} +6.79554 q^{59} +0.528982 q^{61} +24.1489 q^{62} +48.3154 q^{64} +5.39058 q^{65} -11.2955 q^{67} -2.30310 q^{68} -4.06653 q^{70} +10.8636 q^{71} -3.59179 q^{73} -9.14109 q^{74} -16.2656 q^{76} -0.823916 q^{77} +0.905239 q^{79} -26.9798 q^{80} +11.5015 q^{82} +6.64443 q^{83} +0.570367 q^{85} +20.0001 q^{86} -8.90981 q^{88} -4.61643 q^{89} +3.71588 q^{91} -29.9626 q^{92} +8.58555 q^{94} +4.02820 q^{95} +1.82282 q^{97} -2.80317 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

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