LMFDB - Embedded newform 10000.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.108625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 34x^{2} + 9x + 261 $ x^4 - x^3 - 34x^2 + 9x + 261
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.33766$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +4.71963 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} +4.71963 q^{7} -0.381966 q^{9} -5.95569 q^{11} +4.91689 q^{13} -4.10159 q^{17} -8.53492 q^{19} -7.63652 q^{21} +3.48356 q^{23} +5.47214 q^{27} -0.763932 q^{29} +4.47214 q^{31} +9.63652 q^{33} +1.55525 q^{37} -7.95569 q^{39} +5.01848 q^{41} -1.37054 q^{43} +5.63652 q^{47} +15.2749 q^{49} +6.63652 q^{51} -0.472136 q^{53} +13.8098 q^{57} +0.662340 q^{59} +6.19726 q^{61} -1.80274 q^{63} -11.2545 q^{67} -5.63652 q^{69} -5.16438 q^{71} +3.40935 q^{73} -28.1087 q^{77} -4.91689 q^{79} -7.70820 q^{81} -8.00000 q^{83} +1.23607 q^{87} -14.7381 q^{89} +23.2059 q^{91} -7.23607 q^{93} +6.97418 q^{97} +2.27487 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9	$ 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9} + q^{11} + 4 q^{13} - 5 q^{17} - 14 q^{19} + q^{21} + 7 q^{23} + 4 q^{27} - 12 q^{29} + 7 q^{33} + 4 q^{37} - 7 q^{39} - 7 q^{41} + q^{43} - 9 q^{47} + 43 q^{49} - 5 q^{51} + 16 q^{53} + 17 q^{57} + 23 q^{59} + 25 q^{61} - 7 q^{63} - 9 q^{67} + 9 q^{69} - 7 q^{71} - 2 q^{73} - 63 q^{77} - 4 q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{87} - 16 q^{89} - 17 q^{91} - 20 q^{93} - 24 q^{97} - 9 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 + q^11 + 4 * q^13 - 5 * q^17 - 14 * q^19 + q^21 + 7 * q^23 + 4 * q^27 - 12 * q^29 + 7 * q^33 + 4 * q^37 - 7 * q^39 - 7 * q^41 + q^43 - 9 * q^47 + 43 * q^49 - 5 * q^51 + 16 * q^53 + 17 * q^57 + 23 * q^59 + 25 * q^61 - 7 * q^63 - 9 * q^67 + 9 * q^69 - 7 * q^71 - 2 * q^73 - 63 * q^77 - 4 * q^79 - 4 * q^81 - 32 * q^83 - 4 * q^87 - 16 * q^89 - 17 * q^91 - 20 * q^93 - 24 * q^97 - 9 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.71963	1.78385	0.891926	−	0.452182i	$-0.149354\pi$
$7$	4.71963	1.78385	0.891926	+	0.452182i	$0.149354\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−5.95569	−1.79571	−0.897855	−	0.440292i	$-0.854875\pi$
$11$	−5.95569	−1.79571	−0.897855	+	0.440292i	$0.854875\pi$
$12$	0	0
$13$	4.91689	1.36370	0.681850	−	0.731492i	$-0.261176\pi$
$13$	4.91689	1.36370	0.681850	+	0.731492i	$0.261176\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.10159	−0.994782	−0.497391	−	0.867526i	$-0.665709\pi$
$17$	−4.10159	−0.994782	−0.497391	+	0.867526i	$0.665709\pi$
$18$	0	0
$19$	−8.53492	−1.95805	−0.979023	−	0.203750i	$-0.934687\pi$
$19$	−8.53492	−1.95805	−0.979023	+	0.203750i	$0.934687\pi$
$20$	0	0
$21$	−7.63652	−1.66642
$22$	0	0
$23$	3.48356	0.726372	0.363186	−	0.931717i	$-0.381689\pi$
$23$	3.48356	0.726372	0.363186	+	0.931717i	$0.381689\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−0.763932	−0.141859	−0.0709293	−	0.997481i	$-0.522596\pi$
$29$	−0.763932	−0.141859	−0.0709293	+	0.997481i	$0.522596\pi$
$30$	0	0
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	0	0
$33$	9.63652	1.67750
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.55525	0.255681	0.127840	−	0.991795i	$-0.459195\pi$
$37$	1.55525	0.255681	0.127840	+	0.991795i	$0.459195\pi$
$38$	0	0
$39$	−7.95569	−1.27393
$40$	0	0
$41$	5.01848	0.783755	0.391878	−	0.920017i	$-0.371826\pi$
$41$	5.01848	0.783755	0.391878	+	0.920017i	$0.371826\pi$
$42$	0	0
$43$	−1.37054	−0.209006	−0.104503	−	0.994525i	$-0.533325\pi$
$43$	−1.37054	−0.209006	−0.104503	+	0.994525i	$0.533325\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.63652	0.822170	0.411085	−	0.911597i	$-0.365150\pi$
$47$	5.63652	0.822170	0.411085	+	0.911597i	$0.365150\pi$
$48$	0	0
$49$	15.2749	2.18212
$50$	0	0
$51$	6.63652	0.929298
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.8098	1.82915
$58$	0	0
$59$	0.662340	0.0862293	0.0431146	−	0.999070i	$-0.486272\pi$
$59$	0.662340	0.0862293	0.0431146	+	0.999070i	$0.486272\pi$
$60$	0	0
$61$	6.19726	0.793478	0.396739	−	0.917931i	$-0.370142\pi$
$61$	6.19726	0.793478	0.396739	+	0.917931i	$0.370142\pi$
$62$	0	0
$63$	−1.80274	−0.227123
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.2545	−1.37496	−0.687481	−	0.726202i	$-0.741284\pi$
$67$	−11.2545	−1.37496	−0.687481	+	0.726202i	$0.741284\pi$
$68$	0	0
$69$	−5.63652	−0.678557
$70$	0	0
$71$	−5.16438	−0.612899	−0.306450	−	0.951887i	$-0.599141\pi$
$71$	−5.16438	−0.612899	−0.306450	+	0.951887i	$0.599141\pi$
$72$	0	0
$73$	3.40935	0.399034	0.199517	−	0.979894i	$-0.436063\pi$
$73$	3.40935	0.399034	0.199517	+	0.979894i	$0.436063\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−28.1087	−3.20328
$78$	0	0
$79$	−4.91689	−0.553193	−0.276597	−	0.960986i	$-0.589207\pi$
$79$	−4.91689	−0.553193	−0.276597	+	0.960986i	$0.589207\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.23607	0.132520
$88$	0	0
$89$	−14.7381	−1.56224	−0.781118	−	0.624383i	$-0.785350\pi$
$89$	−14.7381	−1.56224	−0.781118	+	0.624383i	$0.785350\pi$
$90$	0	0
$91$	23.2059	2.43264
$92$	0	0
$93$	−7.23607	−0.750345
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.97418	0.708120	0.354060	−	0.935223i	$-0.384801\pi$
$97$	6.97418	0.708120	0.354060	+	0.935223i	$0.384801\pi$
$98$	0	0
$99$	2.27487	0.228633
$100$	0	0
$101$	12.1530	1.20926	0.604632	−	0.796505i	$-0.293320\pi$
$101$	12.1530	1.20926	0.604632	+	0.796505i	$0.293320\pi$
$102$	0	0
$103$	−4.96120	−0.488841	−0.244421	−	0.969669i	$-0.578598\pi$
$103$	−4.96120	−0.488841	−0.244421	+	0.969669i	$0.578598\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.5294	−1.79131	−0.895653	−	0.444753i	$-0.853292\pi$
$107$	−18.5294	−1.79131	−0.895653	+	0.444753i	$0.853292\pi$
$108$	0	0
$109$	−9.68674	−0.927822	−0.463911	−	0.885882i	$-0.653554\pi$
$109$	−9.68674	−0.927822	−0.463911	+	0.885882i	$0.653554\pi$
$110$	0	0
$111$	−2.51644	−0.238850
$112$	0	0
$113$	−6.22717	−0.585803	−0.292901	−	0.956143i	$-0.594621\pi$
$113$	−6.22717	−0.585803	−0.292901	+	0.956143i	$0.594621\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.87808	−0.173629
$118$	0	0
$119$	−19.3580	−1.77454
$120$	0	0
$121$	24.4703	2.22457
$122$	0	0
$123$	−8.12007	−0.732162
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.07169	0.361304	0.180652	−	0.983547i	$-0.442179\pi$
$127$	4.07169	0.361304	0.180652	+	0.983547i	$0.442179\pi$
$128$	0	0
$129$	2.21759	0.195248
$130$	0	0
$131$	4.60661	0.402482	0.201241	−	0.979542i	$-0.435503\pi$
$131$	4.60661	0.402482	0.201241	+	0.979542i	$0.435503\pi$
$132$	0	0
$133$	−40.2816	−3.49286
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5146	0.983759	0.491879	−	0.870663i	$-0.336310\pi$
$137$	11.5146	0.983759	0.491879	+	0.870663i	$0.336310\pi$
$138$	0	0
$139$	−7.13448	−0.605138	−0.302569	−	0.953127i	$-0.597844\pi$
$139$	−7.13448	−0.605138	−0.302569	+	0.953127i	$0.597844\pi$
$140$	0	0
$141$	−9.12007	−0.768049
$142$	0	0
$143$	−29.2835	−2.44881
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−24.7153	−2.03848
$148$	0	0
$149$	−1.24199	−0.101748	−0.0508739	−	0.998705i	$-0.516201\pi$
$149$	−1.24199	−0.101748	−0.0508739	+	0.998705i	$0.516201\pi$
$150$	0	0
$151$	−16.3561	−1.33104	−0.665522	−	0.746378i	$-0.731791\pi$
$151$	−16.3561	−1.33104	−0.665522	+	0.746378i	$0.731791\pi$
$152$	0	0
$153$	1.56667	0.126658
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.18584	0.733110	0.366555	−	0.930396i	$-0.380537\pi$
$157$	9.18584	0.733110	0.366555	+	0.930396i	$0.380537\pi$
$158$	0	0
$159$	0.763932	0.0605838
$160$	0	0
$161$	16.4411	1.29574
$162$	0	0
$163$	1.61253	0.126303	0.0631517	−	0.998004i	$-0.479885\pi$
$163$	1.61253	0.126303	0.0631517	+	0.998004i	$0.479885\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.62509	0.667430	0.333715	−	0.942674i	$-0.391698\pi$
$167$	8.62509	0.667430	0.333715	+	0.942674i	$0.391698\pi$
$168$	0	0
$169$	11.1758	0.859677
$170$	0	0
$171$	3.26005	0.249302
$172$	0	0
$173$	14.2173	1.08092	0.540461	−	0.841369i	$-0.318250\pi$
$173$	14.2173	1.08092	0.540461	+	0.841369i	$0.318250\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.07169	−0.0805530
$178$	0	0
$179$	−1.21209	−0.0905955	−0.0452978	−	0.998974i	$-0.514424\pi$
$179$	−1.21209	−0.0905955	−0.0452978	+	0.998974i	$0.514424\pi$
$180$	0	0
$181$	−6.27487	−0.466408	−0.233204	−	0.972428i	$-0.574921\pi$
$181$	−6.27487	−0.466408	−0.233204	+	0.972428i	$0.574921\pi$
$182$	0	0
$183$	−10.0274	−0.741245
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.4278	1.78634
$188$	0	0
$189$	25.8264	1.87860
$190$	0	0
$191$	−0.570329	−0.0412675	−0.0206338	−	0.999787i	$-0.506568\pi$
$191$	−0.570329	−0.0412675	−0.0206338	+	0.999787i	$0.506568\pi$
$192$	0	0
$193$	−9.37054	−0.674506	−0.337253	−	0.941414i	$-0.609498\pi$
$193$	−9.37054	−0.674506	−0.337253	+	0.941414i	$0.609498\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.73105	−0.123332	−0.0616661	−	0.998097i	$-0.519641\pi$
$197$	−1.73105	−0.123332	−0.0616661	+	0.998097i	$0.519641\pi$
$198$	0	0
$199$	−17.3676	−1.23115	−0.615577	−	0.788077i	$-0.711077\pi$
$199$	−17.3676	−1.23115	−0.615577	+	0.788077i	$0.711077\pi$
$200$	0	0
$201$	18.2102	1.28445
$202$	0	0
$203$	−3.60547	−0.253055
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.33060	−0.0924832
$208$	0	0
$209$	50.8314	3.51608
$210$	0	0
$211$	7.74953	0.533500	0.266750	−	0.963766i	$-0.414050\pi$
$211$	7.74953	0.533500	0.266750	+	0.963766i	$0.414050\pi$
$212$	0	0
$213$	8.35614	0.572553
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.1068	1.43282
$218$	0	0
$219$	−5.51644	−0.372767
$220$	0	0
$221$	−20.1671	−1.35658
$222$	0	0
$223$	−18.4278	−1.23402	−0.617009	−	0.786956i	$-0.711656\pi$
$223$	−18.4278	−1.23402	−0.617009	+	0.786956i	$0.711656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.33766	−0.553390	−0.276695	−	0.960958i	$-0.589239\pi$
$227$	−8.33766	−0.553390	−0.276695	+	0.960958i	$0.589239\pi$
$228$	0	0
$229$	17.3004	1.14324	0.571622	−	0.820517i	$-0.306314\pi$
$229$	17.3004	1.14324	0.571622	+	0.820517i	$0.306314\pi$
$230$	0	0
$231$	45.4808	2.99241
$232$	0	0
$233$	17.9139	1.17358	0.586790	−	0.809739i	$-0.300392\pi$
$233$	17.9139	1.17358	0.586790	+	0.809739i	$0.300392\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.95569	0.516778
$238$	0	0
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	−7.41485	−0.477632	−0.238816	−	0.971065i	$-0.576759\pi$
$241$	−7.41485	−0.477632	−0.238816	+	0.971065i	$0.576759\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−41.9653	−2.67019
$248$	0	0
$249$	12.9443	0.820310
$250$	0	0
$251$	−13.5463	−0.855038	−0.427519	−	0.904006i	$-0.640612\pi$
$251$	−13.5463	−0.855038	−0.427519	+	0.904006i	$0.640612\pi$
$252$	0	0
$253$	−20.7470	−1.30435
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.44815	0.464603	0.232302	−	0.972644i	$-0.425374\pi$
$257$	7.44815	0.464603	0.232302	+	0.972644i	$0.425374\pi$
$258$	0	0
$259$	7.34018	0.456097
$260$	0	0
$261$	0.291796	0.0180617
$262$	0	0
$263$	18.9273	1.16711	0.583555	−	0.812074i	$-0.301661\pi$
$263$	18.9273	1.16711	0.583555	+	0.812074i	$0.301661\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	23.8468	1.45940
$268$	0	0
$269$	−14.3466	−0.874725	−0.437363	−	0.899285i	$-0.644087\pi$
$269$	−14.3466	−0.874725	−0.437363	+	0.899285i	$0.644087\pi$
$270$	0	0
$271$	23.0315	1.39906	0.699531	−	0.714602i	$-0.253392\pi$
$271$	23.0315	1.39906	0.699531	+	0.714602i	$0.253392\pi$
$272$	0	0
$273$	−37.5479	−2.27250
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.42967	−0.446406	−0.223203	−	0.974772i	$-0.571651\pi$
$277$	−7.42967	−0.446406	−0.223203	+	0.974772i	$0.571651\pi$
$278$	0	0
$279$	−1.70820	−0.102267
$280$	0	0
$281$	−5.73219	−0.341954	−0.170977	−	0.985275i	$-0.554692\pi$
$281$	−5.73219	−0.341954	−0.170977	+	0.985275i	$0.554692\pi$
$282$	0	0
$283$	−3.92125	−0.233094	−0.116547	−	0.993185i	$-0.537183\pi$
$283$	−3.92125	−0.233094	−0.116547	+	0.993185i	$0.537183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.6854	1.39810
$288$	0	0
$289$	−0.176940	−0.0104082
$290$	0	0
$291$	−11.2845	−0.661506
$292$	0	0
$293$	−17.8671	−1.04381	−0.521903	−	0.853005i	$-0.674778\pi$
$293$	−17.8671	−1.04381	−0.521903	+	0.853005i	$0.674778\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−32.5904	−1.89109
$298$	0	0
$299$	17.1283	0.990554
$300$	0	0
$301$	−6.46845	−0.372835
$302$	0	0
$303$	−19.6639	−1.12966
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.4965	0.998580	0.499290	−	0.866435i	$-0.333594\pi$
$307$	17.4965	0.998580	0.499290	+	0.866435i	$0.333594\pi$
$308$	0	0
$309$	8.02738	0.456662
$310$	0	0
$311$	−30.6178	−1.73617	−0.868087	−	0.496413i	$-0.834650\pi$
$311$	−30.6178	−1.73617	−0.868087	+	0.496413i	$0.834650\pi$
$312$	0	0
$313$	−17.0299	−0.962587	−0.481294	−	0.876559i	$-0.659833\pi$
$313$	−17.0299	−0.962587	−0.481294	+	0.876559i	$0.659833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.4278	−1.48433	−0.742167	−	0.670215i	$-0.766202\pi$
$317$	−26.4278	−1.48433	−0.742167	+	0.670215i	$0.766202\pi$
$318$	0	0
$319$	4.54975	0.254737
$320$	0	0
$321$	29.9812	1.67339
$322$	0	0
$323$	35.0068	1.94783
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.6735	0.866745
$328$	0	0
$329$	26.6022	1.46663
$330$	0	0
$331$	−8.37054	−0.460087	−0.230043	−	0.973180i	$-0.573887\pi$
$331$	−8.37054	−0.460087	−0.230043	+	0.973180i	$0.573887\pi$
$332$	0	0
$333$	−0.594051	−0.0325538
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.4359	−0.622951	−0.311475	−	0.950254i	$-0.600823\pi$
$337$	−11.4359	−0.622951	−0.311475	+	0.950254i	$0.600823\pi$
$338$	0	0
$339$	10.0758	0.547241
$340$	0	0
$341$	−26.6347	−1.44235
$342$	0	0
$343$	39.0543	2.10873
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.2395	−0.979146	−0.489573	−	0.871962i	$-0.662847\pi$
$347$	−18.2395	−0.979146	−0.489573	+	0.871962i	$0.662847\pi$
$348$	0	0
$349$	32.3931	1.73396	0.866982	−	0.498340i	$-0.166057\pi$
$349$	32.3931	1.73396	0.866982	+	0.498340i	$0.166057\pi$
$350$	0	0
$351$	26.9059	1.43613
$352$	0	0
$353$	−24.0404	−1.27954	−0.639770	−	0.768567i	$-0.720970\pi$
$353$	−24.0404	−1.27954	−0.639770	+	0.768567i	$0.720970\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	31.3219	1.65773
$358$	0	0
$359$	−15.0196	−0.792705	−0.396353	−	0.918098i	$-0.629724\pi$
$359$	−15.0196	−0.792705	−0.396353	+	0.918098i	$0.629724\pi$
$360$	0	0
$361$	53.8449	2.83394
$362$	0	0
$363$	−39.5938	−2.07813
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.6753	1.18364	0.591821	−	0.806069i	$-0.298409\pi$
$367$	22.6753	1.18364	0.591821	+	0.806069i	$0.298409\pi$
$368$	0	0
$369$	−1.91689	−0.0997893
$370$	0	0
$371$	−2.22831	−0.115688
$372$	0	0
$373$	−15.3676	−0.795702	−0.397851	−	0.917450i	$-0.630244\pi$
$373$	−15.3676	−0.795702	−0.397851	+	0.917450i	$0.630244\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.75617	−0.193453
$378$	0	0
$379$	20.1235	1.03367	0.516837	−	0.856084i	$-0.327109\pi$
$379$	20.1235	1.03367	0.516837	+	0.856084i	$0.327109\pi$
$380$	0	0
$381$	−6.58813	−0.337520
$382$	0	0
$383$	16.4685	0.841500	0.420750	−	0.907177i	$-0.361767\pi$
$383$	16.4685	0.841500	0.420750	+	0.907177i	$0.361767\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.523501	0.0266111
$388$	0	0
$389$	−28.5306	−1.44656	−0.723278	−	0.690557i	$-0.757366\pi$
$389$	−28.5306	−1.44656	−0.723278	+	0.690557i	$0.757366\pi$
$390$	0	0
$391$	−14.2881	−0.722582
$392$	0	0
$393$	−7.45365	−0.375987
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4721	0.525581	0.262791	−	0.964853i	$-0.415357\pi$
$397$	10.4721	0.525581	0.262791	+	0.964853i	$0.415357\pi$
$398$	0	0
$399$	65.1771	3.26294
$400$	0	0
$401$	−32.9961	−1.64774	−0.823872	−	0.566776i	$-0.808191\pi$
$401$	−32.9961	−1.64774	−0.823872	+	0.566776i	$0.808191\pi$
$402$	0	0
$403$	21.9890	1.09535
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.26257	−0.459129
$408$	0	0
$409$	5.19910	0.257079	0.128540	−	0.991704i	$-0.458971\pi$
$409$	5.19910	0.257079	0.128540	+	0.991704i	$0.458971\pi$
$410$	0	0
$411$	−18.6310	−0.919000
$412$	0	0
$413$	3.12600	0.153820
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.5438	0.565303
$418$	0	0
$419$	−3.48104	−0.170060	−0.0850299	−	0.996378i	$-0.527099\pi$
$419$	−3.48104	−0.170060	−0.0850299	+	0.996378i	$0.527099\pi$
$420$	0	0
$421$	−16.0415	−0.781815	−0.390907	−	0.920430i	$-0.627839\pi$
$421$	−16.0415	−0.781815	−0.390907	+	0.920430i	$0.627839\pi$
$422$	0	0
$423$	−2.15296	−0.104680
$424$	0	0
$425$	0	0
$426$	0	0
$427$	29.2488	1.41545
$428$	0	0
$429$	47.3817	2.28761
$430$	0	0
$431$	−33.6018	−1.61854	−0.809271	−	0.587436i	$-0.800137\pi$
$431$	−33.6018	−1.61854	−0.809271	+	0.587436i	$0.800137\pi$
$432$	0	0
$433$	−22.1929	−1.06652	−0.533261	−	0.845951i	$-0.679034\pi$
$433$	−22.1929	−1.06652	−0.533261	+	0.845951i	$0.679034\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.7319	−1.42227
$438$	0	0
$439$	3.21461	0.153425	0.0767124	−	0.997053i	$-0.475558\pi$
$439$	3.21461	0.153425	0.0767124	+	0.997053i	$0.475558\pi$
$440$	0	0
$441$	−5.83448	−0.277832
$442$	0	0
$443$	−31.2358	−1.48406	−0.742028	−	0.670368i	$-0.766136\pi$
$443$	−31.2358	−1.48406	−0.742028	+	0.670368i	$0.766136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.00958	0.0950499
$448$	0	0
$449$	−9.99564	−0.471723	−0.235862	−	0.971787i	$-0.575791\pi$
$449$	−9.99564	−0.471723	−0.235862	+	0.971787i	$0.575791\pi$
$450$	0	0
$451$	−29.8885	−1.40740
$452$	0	0
$453$	26.4648	1.24342
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.54635	−0.446559	−0.223280	−	0.974754i	$-0.571676\pi$
$457$	−9.54635	−0.446559	−0.223280	+	0.974754i	$0.571676\pi$
$458$	0	0
$459$	−22.4445	−1.04762
$460$	0	0
$461$	−21.2835	−0.991271	−0.495635	−	0.868531i	$-0.665065\pi$
$461$	−21.2835	−0.991271	−0.495635	+	0.868531i	$0.665065\pi$
$462$	0	0
$463$	−9.46071	−0.439677	−0.219838	−	0.975536i	$-0.570553\pi$
$463$	−9.46071	−0.439677	−0.219838	+	0.975536i	$0.570553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.7938	−1.37869	−0.689347	−	0.724431i	$-0.742103\pi$
$467$	−29.7938	−1.37869	−0.689347	+	0.724431i	$0.742103\pi$
$468$	0	0
$469$	−53.1173	−2.45273
$470$	0	0
$471$	−14.8630	−0.684851
$472$	0	0
$473$	8.16254	0.375314
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.180340	0.00825720
$478$	0	0
$479$	30.2173	1.38066	0.690332	−	0.723493i	$-0.257465\pi$
$479$	30.2173	1.38066	0.690332	+	0.723493i	$0.257465\pi$
$480$	0	0
$481$	7.64697	0.348672
$482$	0	0
$483$	−26.6022	−1.21044
$484$	0	0
$485$	0	0
$486$	0	0
$487$	43.5584	1.97382	0.986909	−	0.161279i	$-0.0515617\pi$
$487$	43.5584	1.97382	0.986909	+	0.161279i	$0.0515617\pi$
$488$	0	0
$489$	−2.60913	−0.117989
$490$	0	0
$491$	24.2669	1.09515	0.547574	−	0.836757i	$-0.315552\pi$
$491$	24.2669	1.09515	0.547574	+	0.836757i	$0.315552\pi$
$492$	0	0
$493$	3.13334	0.141118
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.3739	−1.09332
$498$	0	0
$499$	−16.4130	−0.734747	−0.367374	−	0.930073i	$-0.619743\pi$
$499$	−16.4130	−0.734747	−0.367374	+	0.930073i	$0.619743\pi$
$500$	0	0
$501$	−13.9557	−0.623494
$502$	0	0
$503$	−9.46664	−0.422096	−0.211048	−	0.977476i	$-0.567688\pi$
$503$	−9.46664	−0.422096	−0.211048	+	0.977476i	$0.567688\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−18.0828	−0.803087
$508$	0	0
$509$	−37.5753	−1.66550	−0.832748	−	0.553653i	$-0.813234\pi$
$509$	−37.5753	−1.66550	−0.832748	+	0.553653i	$0.813234\pi$
$510$	0	0
$511$	16.0909	0.711817
$512$	0	0
$513$	−46.7043	−2.06204
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−33.5694	−1.47638
$518$	0	0
$519$	−23.0041	−1.00977
$520$	0	0
$521$	−11.3801	−0.498572	−0.249286	−	0.968430i	$-0.580196\pi$
$521$	−11.3801	−0.498572	−0.249286	+	0.968430i	$0.580196\pi$
$522$	0	0
$523$	−3.45025	−0.150869	−0.0754345	−	0.997151i	$-0.524034\pi$
$523$	−3.45025	−0.150869	−0.0754345	+	0.997151i	$0.524034\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.3429	−0.799028
$528$	0	0
$529$	−10.8648	−0.472384
$530$	0	0
$531$	−0.252991	−0.0109789
$532$	0	0
$533$	24.6753	1.06881
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.96120	0.0846318
$538$	0	0
$539$	−90.9725	−3.91846
$540$	0	0
$541$	43.5789	1.87361	0.936803	−	0.349858i	$-0.113770\pi$
$541$	43.5789	1.87361	0.936803	+	0.349858i	$0.113770\pi$
$542$	0	0
$543$	10.1530	0.435705
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17.0196	−0.727706	−0.363853	−	0.931456i	$-0.618539\pi$
$547$	−17.0196	−0.727706	−0.363853	+	0.931456i	$0.618539\pi$
$548$	0	0
$549$	−2.36714	−0.101027
$550$	0	0
$551$	6.52010	0.277766
$552$	0	0
$553$	−23.2059	−0.986814
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.76985	0.117362	0.0586812	−	0.998277i	$-0.481310\pi$
$557$	2.76985	0.117362	0.0586812	+	0.998277i	$0.481310\pi$
$558$	0	0
$559$	−6.73881	−0.285021
$560$	0	0
$561$	−39.5251	−1.66875
$562$	0	0
$563$	−8.35642	−0.352181	−0.176091	−	0.984374i	$-0.556345\pi$
$563$	−8.35642	−0.352181	−0.176091	+	0.984374i	$0.556345\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−36.3798	−1.52781
$568$	0	0
$569$	20.1157	0.843294	0.421647	−	0.906760i	$-0.361452\pi$
$569$	20.1157	0.843294	0.421647	+	0.906760i	$0.361452\pi$
$570$	0	0
$571$	12.1674	0.509188	0.254594	−	0.967048i	$-0.418058\pi$
$571$	12.1674	0.509188	0.254594	+	0.967048i	$0.418058\pi$
$572$	0	0
$573$	0.922811	0.0385510
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.56937	0.315117	0.157558	−	0.987510i	$-0.449638\pi$
$577$	7.56937	0.315117	0.157558	+	0.987510i	$0.449638\pi$
$578$	0	0
$579$	15.1619	0.630105
$580$	0	0
$581$	−37.7570	−1.56642
$582$	0	0
$583$	2.81190	0.116457
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.6365	−0.562839	−0.281420	−	0.959585i	$-0.590805\pi$
$587$	−13.6365	−0.562839	−0.281420	+	0.959585i	$0.590805\pi$
$588$	0	0
$589$	−38.1693	−1.57274
$590$	0	0
$591$	2.80090	0.115213
$592$	0	0
$593$	20.6568	0.848275	0.424137	−	0.905598i	$-0.360577\pi$
$593$	20.6568	0.848275	0.424137	+	0.905598i	$0.360577\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.1013	1.15011
$598$	0	0
$599$	−41.6707	−1.70262	−0.851309	−	0.524665i	$-0.824191\pi$
$599$	−41.6707	−1.70262	−0.851309	+	0.524665i	$0.824191\pi$
$600$	0	0
$601$	25.3559	1.03429	0.517144	−	0.855899i	$-0.326995\pi$
$601$	25.3559	1.03429	0.517144	+	0.855899i	$0.326995\pi$
$602$	0	0
$603$	4.29886	0.175063
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.9735	1.54130	0.770648	−	0.637261i	$-0.219933\pi$
$607$	37.9735	1.54130	0.770648	+	0.637261i	$0.219933\pi$
$608$	0	0
$609$	5.83378	0.236397
$610$	0	0
$611$	27.7141	1.12119
$612$	0	0
$613$	3.50502	0.141566	0.0707832	−	0.997492i	$-0.477450\pi$
$613$	3.50502	0.141566	0.0707832	+	0.997492i	$0.477450\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.0025	−0.442945	−0.221472	−	0.975167i	$-0.571086\pi$
$617$	−11.0025	−0.442945	−0.221472	+	0.975167i	$0.571086\pi$
$618$	0	0
$619$	−8.93763	−0.359234	−0.179617	−	0.983737i	$-0.557486\pi$
$619$	−8.93763	−0.359234	−0.179617	+	0.983737i	$0.557486\pi$
$620$	0	0
$621$	19.0625	0.764952
$622$	0	0
$623$	−69.5584	−2.78680
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−82.2469	−3.28463
$628$	0	0
$629$	−6.37899	−0.254347
$630$	0	0
$631$	4.30138	0.171235	0.0856176	−	0.996328i	$-0.472714\pi$
$631$	4.30138	0.171235	0.0856176	+	0.996328i	$0.472714\pi$
$632$	0	0
$633$	−12.5390	−0.498381
$634$	0	0
$635$	0	0
$636$	0	0
$637$	75.1049	2.97576
$638$	0	0
$639$	1.97262	0.0780355
$640$	0	0
$641$	−41.9221	−1.65582	−0.827912	−	0.560858i	$-0.810471\pi$
$641$	−41.9221	−1.65582	−0.827912	+	0.560858i	$0.810471\pi$
$642$	0	0
$643$	18.8534	0.743505	0.371753	−	0.928332i	$-0.378757\pi$
$643$	18.8534	0.743505	0.371753	+	0.928332i	$0.378757\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.00958	−0.157633	−0.0788165	−	0.996889i	$-0.525114\pi$
$647$	−4.00958	−0.157633	−0.0788165	+	0.996889i	$0.525114\pi$
$648$	0	0
$649$	−3.94469	−0.154843
$650$	0	0
$651$	−34.1515	−1.33850
$652$	0	0
$653$	11.5886	0.453495	0.226748	−	0.973954i	$-0.427191\pi$
$653$	11.5886	0.453495	0.226748	+	0.973954i	$0.427191\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.30226	−0.0508058
$658$	0	0
$659$	22.4057	0.872801	0.436400	−	0.899753i	$-0.356253\pi$
$659$	22.4057	0.872801	0.436400	+	0.899753i	$0.356253\pi$
$660$	0	0
$661$	−15.4872	−0.602383	−0.301191	−	0.953564i	$-0.597384\pi$
$661$	−15.4872	−0.602383	−0.301191	+	0.953564i	$0.597384\pi$
$662$	0	0
$663$	32.6310	1.26728
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.66120	−0.103042
$668$	0	0
$669$	29.8169	1.15279
$670$	0	0
$671$	−36.9090	−1.42486
$672$	0	0
$673$	15.4984	0.597419	0.298709	−	0.954344i	$-0.403444\pi$
$673$	15.4984	0.597419	0.298709	+	0.954344i	$0.403444\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.49864	0.0960306	0.0480153	−	0.998847i	$-0.484710\pi$
$677$	2.49864	0.0960306	0.0480153	+	0.998847i	$0.484710\pi$
$678$	0	0
$679$	32.9155	1.26318
$680$	0	0
$681$	13.4906	0.516962
$682$	0	0
$683$	31.7248	1.21392	0.606959	−	0.794733i	$-0.292389\pi$
$683$	31.7248	1.21392	0.606959	+	0.794733i	$0.292389\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−27.9927	−1.06799
$688$	0	0
$689$	−2.32144	−0.0884398
$690$	0	0
$691$	−7.82420	−0.297647	−0.148823	−	0.988864i	$-0.547549\pi$
$691$	−7.82420	−0.297647	−0.148823	+	0.988864i	$0.547549\pi$
$692$	0	0
$693$	10.7365	0.407848
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.5838	−0.779666
$698$	0	0
$699$	−28.9853	−1.09633
$700$	0	0
$701$	44.6104	1.68491	0.842456	−	0.538765i	$-0.181109\pi$
$701$	44.6104	1.68491	0.842456	+	0.538765i	$0.181109\pi$
$702$	0	0
$703$	−13.2739	−0.500635
$704$	0	0
$705$	0	0
$706$	0	0
$707$	57.3574	2.15715
$708$	0	0
$709$	2.19865	0.0825719	0.0412860	−	0.999147i	$-0.486855\pi$
$709$	2.19865	0.0825719	0.0412860	+	0.999147i	$0.486855\pi$
$710$	0	0
$711$	1.87808	0.0704337
$712$	0	0
$713$	15.5789	0.583436
$714$	0	0
$715$	0	0
$716$	0	0
$717$	35.4164	1.32265
$718$	0	0
$719$	30.3008	1.13003	0.565015	−	0.825080i	$-0.308870\pi$
$719$	30.3008	1.13003	0.565015	+	0.825080i	$0.308870\pi$
$720$	0	0
$721$	−23.4150	−0.872020
$722$	0	0
$723$	11.9975	0.446191
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.96670	0.221293	0.110646	−	0.993860i	$-0.464708\pi$
$727$	5.96670	0.221293	0.110646	+	0.993860i	$0.464708\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	5.62141	0.207915
$732$	0	0
$733$	−4.40779	−0.162805	−0.0814027	−	0.996681i	$-0.525940\pi$
$733$	−4.40779	−0.162805	−0.0814027	+	0.996681i	$0.525940\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	67.0287	2.46903
$738$	0	0
$739$	35.0138	1.28800	0.644002	−	0.765023i	$-0.277273\pi$
$739$	35.0138	1.28800	0.644002	+	0.765023i	$0.277273\pi$
$740$	0	0
$741$	67.9012	2.49441
$742$	0	0
$743$	−46.2483	−1.69669	−0.848344	−	0.529446i	$-0.822400\pi$
$743$	−46.2483	−1.69669	−0.848344	+	0.529446i	$0.822400\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.05573	0.111803
$748$	0	0
$749$	−87.4520	−3.19542
$750$	0	0
$751$	49.8812	1.82019	0.910095	−	0.414399i	$-0.136008\pi$
$751$	49.8812	1.82019	0.910095	+	0.414399i	$0.136008\pi$
$752$	0	0
$753$	21.9184	0.798753
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.6839	0.933499	0.466749	−	0.884390i	$-0.345425\pi$
$757$	25.6839	0.933499	0.466749	+	0.884390i	$0.345425\pi$
$758$	0	0
$759$	33.5694	1.21849
$760$	0	0
$761$	−24.5618	−0.890366	−0.445183	−	0.895440i	$-0.646861\pi$
$761$	−24.5618	−0.890366	−0.445183	+	0.895440i	$0.646861\pi$
$762$	0	0
$763$	−45.7178	−1.65510
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.25665	0.117591
$768$	0	0
$769$	−15.7747	−0.568848	−0.284424	−	0.958699i	$-0.591802\pi$
$769$	−15.7747	−0.568848	−0.284424	+	0.958699i	$0.591802\pi$
$770$	0	0
$771$	−12.0514	−0.434019
$772$	0	0
$773$	−41.9639	−1.50934	−0.754668	−	0.656107i	$-0.772202\pi$
$773$	−41.9639	−1.50934	−0.754668	+	0.656107i	$0.772202\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.8767	−0.426073
$778$	0	0
$779$	−42.8324	−1.53463
$780$	0	0
$781$	30.7575	1.10059
$782$	0	0
$783$	−4.18034	−0.149393
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.6251	−0.450036	−0.225018	−	0.974355i	$-0.572244\pi$
$787$	−12.6251	−0.450036	−0.225018	+	0.974355i	$0.572244\pi$
$788$	0	0
$789$	−30.6251	−1.09028
$790$	0	0
$791$	−29.3899	−1.04498
$792$	0	0
$793$	30.4713	1.08207
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.0365	−0.532622	−0.266311	−	0.963887i	$-0.585805\pi$
$797$	−15.0365	−0.532622	−0.266311	+	0.963887i	$0.585805\pi$
$798$	0	0
$799$	−23.1187	−0.817880
$800$	0	0
$801$	5.62946	0.198907
$802$	0	0
$803$	−20.3050	−0.716549
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.2132	0.817144
$808$	0	0
$809$	−48.8148	−1.71624	−0.858118	−	0.513453i	$-0.828366\pi$
$809$	−48.8148	−1.71624	−0.858118	+	0.513453i	$0.828366\pi$
$810$	0	0
$811$	−31.4257	−1.10351	−0.551753	−	0.834008i	$-0.686041\pi$
$811$	−31.4257	−1.10351	−0.551753	+	0.834008i	$0.686041\pi$
$812$	0	0
$813$	−37.2657	−1.30696
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.6975	0.409243
$818$	0	0
$819$	−8.86386	−0.309728
$820$	0	0
$821$	−41.0995	−1.43438	−0.717191	−	0.696877i	$-0.754573\pi$
$821$	−41.0995	−1.43438	−0.717191	+	0.696877i	$0.754573\pi$
$822$	0	0
$823$	−43.2776	−1.50856	−0.754281	−	0.656552i	$-0.772014\pi$
$823$	−43.2776	−1.50856	−0.754281	+	0.656552i	$0.772014\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.8929	−1.00470	−0.502352	−	0.864663i	$-0.667532\pi$
$827$	−28.8929	−1.00470	−0.502352	+	0.864663i	$0.667532\pi$
$828$	0	0
$829$	−36.7442	−1.27618	−0.638089	−	0.769962i	$-0.720275\pi$
$829$	−36.7442	−1.27618	−0.638089	+	0.769962i	$0.720275\pi$
$830$	0	0
$831$	12.0215	0.417020
$832$	0	0
$833$	−62.6513	−2.17074
$834$	0	0
$835$	0	0
$836$	0	0
$837$	24.4721	0.845881
$838$	0	0
$839$	37.0114	1.27778	0.638888	−	0.769300i	$-0.279395\pi$
$839$	37.0114	1.27778	0.638888	+	0.769300i	$0.279395\pi$
$840$	0	0
$841$	−28.4164	−0.979876
$842$	0	0
$843$	9.27487	0.319444
$844$	0	0
$845$	0	0
$846$	0	0
$847$	115.491	3.96831
$848$	0	0
$849$	6.34472	0.217750
$850$	0	0
$851$	5.41779	0.185720
$852$	0	0
$853$	−2.99816	−0.102655	−0.0513275	−	0.998682i	$-0.516345\pi$
$853$	−2.99816	−0.102655	−0.0513275	+	0.998682i	$0.516345\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.14224	−0.243974	−0.121987	−	0.992532i	$-0.538927\pi$
$857$	−7.14224	−0.243974	−0.121987	+	0.992532i	$0.538927\pi$
$858$	0	0
$859$	−35.1678	−1.19991	−0.599954	−	0.800034i	$-0.704815\pi$
$859$	−35.1678	−1.19991	−0.599954	+	0.800034i	$0.704815\pi$
$860$	0	0
$861$	−38.3237	−1.30607
$862$	0	0
$863$	−4.55933	−0.155201	−0.0776006	−	0.996985i	$-0.524726\pi$
$863$	−4.55933	−0.155201	−0.0776006	+	0.996985i	$0.524726\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.286295	0.00972310
$868$	0	0
$869$	29.2835	0.993374
$870$	0	0
$871$	−55.3374	−1.87504
$872$	0	0
$873$	−2.66390	−0.0901593
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−51.8679	−1.75146	−0.875728	−	0.482805i	$-0.839618\pi$
$877$	−51.8679	−1.75146	−0.875728	+	0.482805i	$0.839618\pi$
$878$	0	0
$879$	28.9095	0.975095
$880$	0	0
$881$	40.3650	1.35993	0.679966	−	0.733244i	$-0.261995\pi$
$881$	40.3650	1.35993	0.679966	+	0.733244i	$0.261995\pi$
$882$	0	0
$883$	−38.1243	−1.28299	−0.641493	−	0.767129i	$-0.721685\pi$
$883$	−38.1243	−1.28299	−0.641493	+	0.767129i	$0.721685\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.5611	1.32833	0.664167	−	0.747585i	$-0.268787\pi$
$887$	39.5611	1.32833	0.664167	+	0.747585i	$0.268787\pi$
$888$	0	0
$889$	19.2168	0.644512
$890$	0	0
$891$	45.9077	1.53797
$892$	0	0
$893$	−48.1072	−1.60985
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−27.7141	−0.925348
$898$	0	0
$899$	−3.41641	−0.113944
$900$	0	0
$901$	1.93651	0.0645145
$902$	0	0
$903$	10.4662	0.348293
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−46.1202	−1.53140	−0.765698	−	0.643200i	$-0.777606\pi$
$907$	−46.1202	−1.53140	−0.765698	+	0.643200i	$0.777606\pi$
$908$	0	0
$909$	−4.64202	−0.153966
$910$	0	0
$911$	48.5771	1.60943	0.804716	−	0.593660i	$-0.202318\pi$
$911$	48.5771	1.60943	0.804716	+	0.593660i	$0.202318\pi$
$912$	0	0
$913$	47.6456	1.57684
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.7415	0.717967
$918$	0	0
$919$	−17.5036	−0.577390	−0.288695	−	0.957421i	$-0.593221\pi$
$919$	−17.5036	−0.577390	−0.288695	+	0.957421i	$0.593221\pi$
$920$	0	0
$921$	−28.3100	−0.932846
$922$	0	0
$923$	−25.3927	−0.835810
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.89501	0.0622402
$928$	0	0
$929$	26.3842	0.865638	0.432819	−	0.901481i	$-0.357519\pi$
$929$	26.3842	0.865638	0.432819	+	0.901481i	$0.357519\pi$
$930$	0	0
$931$	−130.370	−4.27270
$932$	0	0
$933$	49.5406	1.62189
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−45.1623	−1.47539	−0.737693	−	0.675136i	$-0.764085\pi$
$937$	−45.1623	−1.47539	−0.737693	+	0.675136i	$0.764085\pi$
$938$	0	0
$939$	27.5550	0.899222
$940$	0	0
$941$	19.6063	0.639148	0.319574	−	0.947561i	$-0.396460\pi$
$941$	19.6063	0.639148	0.319574	+	0.947561i	$0.396460\pi$
$942$	0	0
$943$	17.4822	0.569298
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.1998	0.493927	0.246963	−	0.969025i	$-0.420567\pi$
$947$	15.1998	0.493927	0.246963	+	0.969025i	$0.420567\pi$
$948$	0	0
$949$	16.7634	0.544163
$950$	0	0
$951$	42.7611	1.38662
$952$	0	0
$953$	−20.9546	−0.678785	−0.339392	−	0.940645i	$-0.610221\pi$
$953$	−20.9546	−0.678785	−0.339392	+	0.940645i	$0.610221\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.36164	−0.237968
$958$	0	0
$959$	54.3446	1.75488
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	7.07761	0.228073
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.1100	0.646695	0.323348	−	0.946280i	$-0.395192\pi$
$967$	20.1100	0.646695	0.323348	+	0.946280i	$0.395192\pi$
$968$	0	0
$969$	−56.6422	−1.81961
$970$	0	0
$971$	58.3728	1.87327	0.936636	−	0.350304i	$-0.113922\pi$
$971$	58.3728	1.87327	0.936636	+	0.350304i	$0.113922\pi$
$972$	0	0
$973$	−33.6721	−1.07948
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.4477	−0.590194	−0.295097	−	0.955467i	$-0.595352\pi$
$977$	−18.4477	−0.590194	−0.295097	+	0.955467i	$0.595352\pi$
$978$	0	0
$979$	87.7757	2.80532
$980$	0	0
$981$	3.70001	0.118132
$982$	0	0
$983$	4.52010	0.144169	0.0720844	−	0.997399i	$-0.477035\pi$
$983$	4.52010	0.144169	0.0720844	+	0.997399i	$0.477035\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−43.0433	−1.37008
$988$	0	0
$989$	−4.77437	−0.151816
$990$	0	0
$991$	−60.3022	−1.91556	−0.957782	−	0.287496i	$-0.907177\pi$
$991$	−60.3022	−1.91556	−0.957782	+	0.287496i	$0.907177\pi$
$992$	0	0
$993$	13.5438	0.429800
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.1447	−0.923020	−0.461510	−	0.887135i	$-0.652692\pi$
$997$	−29.1447	−0.923020	−0.461510	+	0.887135i	$0.652692\pi$
$998$	0	0
$999$	8.51052	0.269261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.r.1.2		4
4.3	odd	2		5000.2.a.h.1.3	yes	4
5.4	even	2		10000.2.a.y.1.3		4
20.19	odd	2		5000.2.a.g.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.g.1.2	✓	4	20.19	odd	2
5000.2.a.h.1.3	yes	4	4.3	odd	2
10000.2.a.r.1.2		4	1.1	even	1	trivial
10000.2.a.y.1.3		4	5.4	even	2

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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +4.71963 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} +4.71963 q^{7} -0.381966 q^{9} -5.95569 q^{11} +4.91689 q^{13} -4.10159 q^{17} -8.53492 q^{19} -7.63652 q^{21} +3.48356 q^{23} +5.47214 q^{27} -0.763932 q^{29} +4.47214 q^{31} +9.63652 q^{33} +1.55525 q^{37} -7.95569 q^{39} +5.01848 q^{41} -1.37054 q^{43} +5.63652 q^{47} +15.2749 q^{49} +6.63652 q^{51} -0.472136 q^{53} +13.8098 q^{57} +0.662340 q^{59} +6.19726 q^{61} -1.80274 q^{63} -11.2545 q^{67} -5.63652 q^{69} -5.16438 q^{71} +3.40935 q^{73} -28.1087 q^{77} -4.91689 q^{79} -7.70820 q^{81} -8.00000 q^{83} +1.23607 q^{87} -14.7381 q^{89} +23.2059 q^{91} -7.23607 q^{93} +6.97418 q^{97} +2.27487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9	\( 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9} + q^{11} + 4 q^{13} - 5 q^{17} - 14 q^{19} + q^{21} + 7 q^{23} + 4 q^{27} - 12 q^{29} + 7 q^{33} + 4 q^{37} - 7 q^{39} - 7 q^{41} + q^{43} - 9 q^{47} + 43 q^{49} - 5 q^{51} + 16 q^{53} + 17 q^{57} + 23 q^{59} + 25 q^{61} - 7 q^{63} - 9 q^{67} + 9 q^{69} - 7 q^{71} - 2 q^{73} - 63 q^{77} - 4 q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{87} - 16 q^{89} - 17 q^{91} - 20 q^{93} - 24 q^{97} - 9 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 + q^11 + 4 * q^13 - 5 * q^17 - 14 * q^19 + q^21 + 7 * q^23 + 4 * q^27 - 12 * q^29 + 7 * q^33 + 4 * q^37 - 7 * q^39 - 7 * q^41 + q^43 - 9 * q^47 + 43 * q^49 - 5 * q^51 + 16 * q^53 + 17 * q^57 + 23 * q^59 + 25 * q^61 - 7 * q^63 - 9 * q^67 + 9 * q^69 - 7 * q^71 - 2 * q^73 - 63 * q^77 - 4 * q^79 - 4 * q^81 - 32 * q^83 - 4 * q^87 - 16 * q^89 - 17 * q^91 - 20 * q^93 - 24 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more