LMFDB - Embedded newform 3720.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.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:	$29.7043495519$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.78292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.78678$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.78678 q^{7} +1.00000 q^{9} -4.92191 q^{11} -3.39720 q^{13} +1.00000 q^{15} -2.41782 q^{17} -6.92191 q^{19} +2.78678 q^{21} -3.80740 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.02062 q^{29} +1.00000 q^{31} +4.92191 q^{33} +2.78678 q^{35} +1.39720 q^{37} +3.39720 q^{39} +4.00000 q^{41} -1.38959 q^{43} -1.00000 q^{45} -2.37657 q^{47} +0.766162 q^{49} +2.41782 q^{51} +0.233838 q^{53} +4.92191 q^{55} +6.92191 q^{57} +3.29088 q^{59} -4.31150 q^{61} -2.78678 q^{63} +3.39720 q^{65} +0.176371 q^{67} +3.80740 q^{69} +3.58979 q^{71} +2.74554 q^{73} -1.00000 q^{75} +13.7163 q^{77} +5.06947 q^{79} +1.00000 q^{81} -5.63864 q^{83} +2.41782 q^{85} -1.02062 q^{87} +9.94253 q^{89} +9.46725 q^{91} -1.00000 q^{93} +6.92191 q^{95} -12.3680 q^{97} -4.92191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.78678	−1.05331	−0.526653	−	0.850081i	$-0.676553\pi$
$7$	−2.78678	−1.05331	−0.526653	+	0.850081i	$0.676553\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.92191	−1.48401	−0.742006	−	0.670393i	$-0.766126\pi$
$11$	−4.92191	−1.48401	−0.742006	+	0.670393i	$0.766126\pi$
$12$	0	0
$13$	−3.39720	−0.942213	−0.471106	−	0.882076i	$-0.656145\pi$
$13$	−3.39720	−0.942213	−0.471106	+	0.882076i	$0.656145\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.41782	−0.586407	−0.293203	−	0.956050i	$-0.594721\pi$
$17$	−2.41782	−0.586407	−0.293203	+	0.956050i	$0.594721\pi$
$18$	0	0
$19$	−6.92191	−1.58800	−0.793998	−	0.607921i	$-0.792004\pi$
$19$	−6.92191	−1.58800	−0.793998	+	0.607921i	$0.792004\pi$
$20$	0	0
$21$	2.78678	0.608126
$22$	0	0
$23$	−3.80740	−0.793899	−0.396949	−	0.917841i	$-0.629931\pi$
$23$	−3.80740	−0.793899	−0.396949	+	0.917841i	$0.629931\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.02062	0.189525	0.0947623	−	0.995500i	$-0.469791\pi$
$29$	1.02062	0.189525	0.0947623	+	0.995500i	$0.469791\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	4.92191	0.856795
$34$	0	0
$35$	2.78678	0.471052
$36$	0	0
$37$	1.39720	0.229698	0.114849	−	0.993383i	$-0.463362\pi$
$37$	1.39720	0.229698	0.114849	+	0.993383i	$0.463362\pi$
$38$	0	0
$39$	3.39720	0.543987
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−1.38959	−0.211910	−0.105955	−	0.994371i	$-0.533790\pi$
$43$	−1.38959	−0.211910	−0.105955	+	0.994371i	$0.533790\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.37657	−0.346659	−0.173330	−	0.984864i	$-0.555453\pi$
$47$	−2.37657	−0.346659	−0.173330	+	0.984864i	$0.555453\pi$
$48$	0	0
$49$	0.766162	0.109452
$50$	0	0
$51$	2.41782	0.338562
$52$	0	0
$53$	0.233838	0.0321201	0.0160600	−	0.999871i	$-0.494888\pi$
$53$	0.233838	0.0321201	0.0160600	+	0.999871i	$0.494888\pi$
$54$	0	0
$55$	4.92191	0.663671
$56$	0	0
$57$	6.92191	0.916830
$58$	0	0
$59$	3.29088	0.428436	0.214218	−	0.976786i	$-0.431280\pi$
$59$	3.29088	0.428436	0.214218	+	0.976786i	$0.431280\pi$
$60$	0	0
$61$	−4.31150	−0.552031	−0.276015	−	0.961153i	$-0.589014\pi$
$61$	−4.31150	−0.552031	−0.276015	+	0.961153i	$0.589014\pi$
$62$	0	0
$63$	−2.78678	−0.351102
$64$	0	0
$65$	3.39720	0.421370
$66$	0	0
$67$	0.176371	0.0215472	0.0107736	−	0.999942i	$-0.496571\pi$
$67$	0.176371	0.0215472	0.0107736	+	0.999942i	$0.496571\pi$
$68$	0	0
$69$	3.80740	0.458358
$70$	0	0
$71$	3.58979	0.426030	0.213015	−	0.977049i	$-0.431672\pi$
$71$	3.58979	0.426030	0.213015	+	0.977049i	$0.431672\pi$
$72$	0	0
$73$	2.74554	0.321341	0.160671	−	0.987008i	$-0.448634\pi$
$73$	2.74554	0.321341	0.160671	+	0.987008i	$0.448634\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	13.7163	1.56312
$78$	0	0
$79$	5.06947	0.570360	0.285180	−	0.958474i	$-0.407947\pi$
$79$	5.06947	0.570360	0.285180	+	0.958474i	$0.407947\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.63864	−0.618921	−0.309461	−	0.950912i	$-0.600149\pi$
$83$	−5.63864	−0.618921	−0.309461	+	0.950912i	$0.600149\pi$
$84$	0	0
$85$	2.41782	0.262249
$86$	0	0
$87$	−1.02062	−0.109422
$88$	0	0
$89$	9.94253	1.05391	0.526953	−	0.849894i	$-0.323334\pi$
$89$	9.94253	1.05391	0.526953	+	0.849894i	$0.323334\pi$
$90$	0	0
$91$	9.46725	0.992437
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	6.92191	0.710173
$96$	0	0
$97$	−12.3680	−1.25578	−0.627888	−	0.778304i	$-0.716080\pi$
$97$	−12.3680	−1.25578	−0.627888	+	0.778304i	$0.716080\pi$
$98$	0	0
$99$	−4.92191	−0.494671
$100$	0	0
$101$	−13.6014	−1.35339	−0.676694	−	0.736265i	$-0.736588\pi$
$101$	−13.6014	−1.35339	−0.676694	+	0.736265i	$0.736588\pi$
$102$	0	0
$103$	−0.644046	−0.0634598	−0.0317299	−	0.999496i	$-0.510102\pi$
$103$	−0.644046	−0.0634598	−0.0317299	+	0.999496i	$0.510102\pi$
$104$	0	0
$105$	−2.78678	−0.271962
$106$	0	0
$107$	−14.1189	−1.36493	−0.682463	−	0.730920i	$-0.739091\pi$
$107$	−14.1189	−1.36493	−0.682463	+	0.730920i	$0.739091\pi$
$108$	0	0
$109$	−3.11451	−0.298316	−0.149158	−	0.988813i	$-0.547656\pi$
$109$	−3.11451	−0.298316	−0.149158	+	0.988813i	$0.547656\pi$
$110$	0	0
$111$	−1.39720	−0.132616
$112$	0	0
$113$	3.81602	0.358981	0.179491	−	0.983760i	$-0.442555\pi$
$113$	3.81602	0.358981	0.179491	+	0.983760i	$0.442555\pi$
$114$	0	0
$115$	3.80740	0.355042
$116$	0	0
$117$	−3.39720	−0.314071
$118$	0	0
$119$	6.73793	0.617665
$120$	0	0
$121$	13.2252	1.20229
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.8590	−1.22979	−0.614896	−	0.788609i	$-0.710802\pi$
$127$	−13.8590	−1.22979	−0.614896	+	0.788609i	$0.710802\pi$
$128$	0	0
$129$	1.38959	0.122346
$130$	0	0
$131$	12.0440	1.05229	0.526146	−	0.850394i	$-0.323637\pi$
$131$	12.0440	1.05229	0.526146	+	0.850394i	$0.323637\pi$
$132$	0	0
$133$	19.2899	1.67264
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.7858	1.94672	0.973360	−	0.229282i	$-0.0736377\pi$
$137$	22.7858	1.94672	0.973360	+	0.229282i	$0.0736377\pi$
$138$	0	0
$139$	1.57357	0.133468	0.0667341	−	0.997771i	$-0.478742\pi$
$139$	1.57357	0.133468	0.0667341	+	0.997771i	$0.478742\pi$
$140$	0	0
$141$	2.37657	0.200144
$142$	0	0
$143$	16.7207	1.39826
$144$	0	0
$145$	−1.02062	−0.0847580
$146$	0	0
$147$	−0.766162	−0.0631920
$148$	0	0
$149$	5.07809	0.416013	0.208007	−	0.978127i	$-0.433302\pi$
$149$	5.07809	0.416013	0.208007	+	0.978127i	$0.433302\pi$
$150$	0	0
$151$	−0.729317	−0.0593510	−0.0296755	−	0.999560i	$-0.509447\pi$
$151$	−0.729317	−0.0593510	−0.0296755	+	0.999560i	$0.509447\pi$
$152$	0	0
$153$	−2.41782	−0.195469
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	4.69290	0.374534	0.187267	−	0.982309i	$-0.440037\pi$
$157$	4.69290	0.374534	0.187267	+	0.982309i	$0.440037\pi$
$158$	0	0
$159$	−0.233838	−0.0185445
$160$	0	0
$161$	10.6104	0.836218
$162$	0	0
$163$	−11.7239	−0.918288	−0.459144	−	0.888362i	$-0.651844\pi$
$163$	−11.7239	−0.918288	−0.459144	+	0.888362i	$0.651844\pi$
$164$	0	0
$165$	−4.92191	−0.383170
$166$	0	0
$167$	−8.02780	−0.621210	−0.310605	−	0.950539i	$-0.600532\pi$
$167$	−8.02780	−0.621210	−0.310605	+	0.950539i	$0.600532\pi$
$168$	0	0
$169$	−1.45906	−0.112235
$170$	0	0
$171$	−6.92191	−0.529332
$172$	0	0
$173$	−3.10589	−0.236137	−0.118068	−	0.993005i	$-0.537670\pi$
$173$	−3.10589	−0.236137	−0.118068	+	0.993005i	$0.537670\pi$
$174$	0	0
$175$	−2.78678	−0.210661
$176$	0	0
$177$	−3.29088	−0.247358
$178$	0	0
$179$	2.22902	0.166604	0.0833022	−	0.996524i	$-0.473453\pi$
$179$	2.22902	0.166604	0.0833022	+	0.996524i	$0.473453\pi$
$180$	0	0
$181$	8.16876	0.607179	0.303590	−	0.952803i	$-0.401815\pi$
$181$	8.16876	0.607179	0.303590	+	0.952803i	$0.401815\pi$
$182$	0	0
$183$	4.31150	0.318715
$184$	0	0
$185$	−1.39720	−0.102724
$186$	0	0
$187$	11.9003	0.870235
$188$	0	0
$189$	2.78678	0.202709
$190$	0	0
$191$	13.0771	0.946224	0.473112	−	0.881002i	$-0.343131\pi$
$191$	13.0771	0.946224	0.473112	+	0.881002i	$0.343131\pi$
$192$	0	0
$193$	7.53232	0.542189	0.271094	−	0.962553i	$-0.412614\pi$
$193$	7.53232	0.542189	0.271094	+	0.962553i	$0.412614\pi$
$194$	0	0
$195$	−3.39720	−0.243278
$196$	0	0
$197$	10.8855	0.775559	0.387780	−	0.921752i	$-0.373242\pi$
$197$	10.8855	0.775559	0.387780	+	0.921752i	$0.373242\pi$
$198$	0	0
$199$	5.33973	0.378523	0.189262	−	0.981927i	$-0.439391\pi$
$199$	5.33973	0.378523	0.189262	+	0.981927i	$0.439391\pi$
$200$	0	0
$201$	−0.176371	−0.0124403
$202$	0	0
$203$	−2.84425	−0.199627
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	−3.80740	−0.264633
$208$	0	0
$209$	34.0690	2.35661
$210$	0	0
$211$	11.6598	0.802697	0.401348	−	0.915925i	$-0.368542\pi$
$211$	11.6598	0.802697	0.401348	+	0.915925i	$0.368542\pi$
$212$	0	0
$213$	−3.58979	−0.245968
$214$	0	0
$215$	1.38959	0.0947691
$216$	0	0
$217$	−2.78678	−0.189179
$218$	0	0
$219$	−2.74554	−0.185526
$220$	0	0
$221$	8.21380	0.552520
$222$	0	0
$223$	9.63003	0.644874	0.322437	−	0.946591i	$-0.395498\pi$
$223$	9.63003	0.644874	0.322437	+	0.946591i	$0.395498\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.20079	−0.0796990	−0.0398495	−	0.999206i	$-0.512688\pi$
$227$	−1.20079	−0.0796990	−0.0398495	+	0.999206i	$0.512688\pi$
$228$	0	0
$229$	−26.4782	−1.74973	−0.874866	−	0.484365i	$-0.839051\pi$
$229$	−26.4782	−1.74973	−0.874866	+	0.484365i	$0.839051\pi$
$230$	0	0
$231$	−13.7163	−0.902467
$232$	0	0
$233$	−25.7653	−1.68794	−0.843971	−	0.536389i	$-0.819788\pi$
$233$	−25.7653	−1.68794	−0.843971	+	0.536389i	$0.819788\pi$
$234$	0	0
$235$	2.37657	0.155031
$236$	0	0
$237$	−5.06947	−0.329298
$238$	0	0
$239$	20.3267	1.31483	0.657413	−	0.753530i	$-0.271651\pi$
$239$	20.3267	1.31483	0.657413	+	0.753530i	$0.271651\pi$
$240$	0	0
$241$	−11.9415	−0.769221	−0.384611	−	0.923079i	$-0.625664\pi$
$241$	−11.9415	−0.769221	−0.384611	+	0.923079i	$0.625664\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.766162	−0.0489483
$246$	0	0
$247$	23.5151	1.49623
$248$	0	0
$249$	5.63864	0.357334
$250$	0	0
$251$	−9.37615	−0.591817	−0.295909	−	0.955216i	$-0.595622\pi$
$251$	−9.37615	−0.591817	−0.295909	+	0.955216i	$0.595622\pi$
$252$	0	0
$253$	18.7397	1.17816
$254$	0	0
$255$	−2.41782	−0.151410
$256$	0	0
$257$	2.93053	0.182801	0.0914007	−	0.995814i	$-0.470866\pi$
$257$	2.93053	0.182801	0.0914007	+	0.995814i	$0.470866\pi$
$258$	0	0
$259$	−3.89368	−0.241942
$260$	0	0
$261$	1.02062	0.0631749
$262$	0	0
$263$	−20.8933	−1.28833	−0.644167	−	0.764885i	$-0.722796\pi$
$263$	−20.8933	−1.28833	−0.644167	+	0.764885i	$0.722796\pi$
$264$	0	0
$265$	−0.233838	−0.0143645
$266$	0	0
$267$	−9.94253	−0.608473
$268$	0	0
$269$	28.4532	1.73482	0.867412	−	0.497591i	$-0.165782\pi$
$269$	28.4532	1.73482	0.867412	+	0.497591i	$0.165782\pi$
$270$	0	0
$271$	13.8889	0.843688	0.421844	−	0.906668i	$-0.361383\pi$
$271$	13.8889	0.843688	0.421844	+	0.906668i	$0.361383\pi$
$272$	0	0
$273$	−9.46725	−0.572984
$274$	0	0
$275$	−4.92191	−0.296802
$276$	0	0
$277$	16.7321	1.00533	0.502667	−	0.864480i	$-0.332352\pi$
$277$	16.7321	1.00533	0.502667	+	0.864480i	$0.332352\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	18.0977	1.07962	0.539809	−	0.841787i	$-0.318496\pi$
$281$	18.0977	1.07962	0.539809	+	0.841787i	$0.318496\pi$
$282$	0	0
$283$	−32.3882	−1.92528	−0.962638	−	0.270792i	$-0.912715\pi$
$283$	−32.3882	−1.92528	−0.962638	+	0.270792i	$0.912715\pi$
$284$	0	0
$285$	−6.92191	−0.410019
$286$	0	0
$287$	−11.1471	−0.657995
$288$	0	0
$289$	−11.1542	−0.656127
$290$	0	0
$291$	12.3680	0.725023
$292$	0	0
$293$	−4.92673	−0.287823	−0.143911	−	0.989591i	$-0.545968\pi$
$293$	−4.92673	−0.287823	−0.143911	+	0.989591i	$0.545968\pi$
$294$	0	0
$295$	−3.29088	−0.191602
$296$	0	0
$297$	4.92191	0.285598
$298$	0	0
$299$	12.9345	0.748021
$300$	0	0
$301$	3.87248	0.223206
$302$	0	0
$303$	13.6014	0.781378
$304$	0	0
$305$	4.31150	0.246876
$306$	0	0
$307$	−15.3647	−0.876912	−0.438456	−	0.898753i	$-0.644475\pi$
$307$	−15.3647	−0.876912	−0.438456	+	0.898753i	$0.644475\pi$
$308$	0	0
$309$	0.644046	0.0366385
$310$	0	0
$311$	14.5790	0.826698	0.413349	−	0.910573i	$-0.364359\pi$
$311$	14.5790	0.826698	0.413349	+	0.910573i	$0.364359\pi$
$312$	0	0
$313$	−21.3387	−1.20614	−0.603068	−	0.797690i	$-0.706055\pi$
$313$	−21.3387	−1.20614	−0.603068	+	0.797690i	$0.706055\pi$
$314$	0	0
$315$	2.78678	0.157017
$316$	0	0
$317$	27.3111	1.53394	0.766971	−	0.641681i	$-0.221763\pi$
$317$	27.3111	1.53394	0.766971	+	0.641681i	$0.221763\pi$
$318$	0	0
$319$	−5.02341	−0.281257
$320$	0	0
$321$	14.1189	0.788040
$322$	0	0
$323$	16.7359	0.931211
$324$	0	0
$325$	−3.39720	−0.188443
$326$	0	0
$327$	3.11451	0.172233
$328$	0	0
$329$	6.62300	0.365138
$330$	0	0
$331$	5.85244	0.321679	0.160840	−	0.986981i	$-0.448580\pi$
$331$	5.85244	0.321679	0.160840	+	0.986981i	$0.448580\pi$
$332$	0	0
$333$	1.39720	0.0765659
$334$	0	0
$335$	−0.176371	−0.00963618
$336$	0	0
$337$	−6.88828	−0.375228	−0.187614	−	0.982243i	$-0.560075\pi$
$337$	−6.88828	−0.375228	−0.187614	+	0.982243i	$0.560075\pi$
$338$	0	0
$339$	−3.81602	−0.207258
$340$	0	0
$341$	−4.92191	−0.266537
$342$	0	0
$343$	17.3724	0.938019
$344$	0	0
$345$	−3.80740	−0.204984
$346$	0	0
$347$	2.58176	0.138596	0.0692980	−	0.997596i	$-0.477924\pi$
$347$	2.58176	0.138596	0.0692980	+	0.997596i	$0.477924\pi$
$348$	0	0
$349$	−5.50849	−0.294863	−0.147431	−	0.989072i	$-0.547101\pi$
$349$	−5.50849	−0.294863	−0.147431	+	0.989072i	$0.547101\pi$
$350$	0	0
$351$	3.39720	0.181329
$352$	0	0
$353$	−4.26164	−0.226824	−0.113412	−	0.993548i	$-0.536178\pi$
$353$	−4.26164	−0.226824	−0.113412	+	0.993548i	$0.536178\pi$
$354$	0	0
$355$	−3.58979	−0.190526
$356$	0	0
$357$	−6.73793	−0.356609
$358$	0	0
$359$	−27.6714	−1.46044	−0.730221	−	0.683211i	$-0.760583\pi$
$359$	−27.6714	−1.46044	−0.730221	+	0.683211i	$0.760583\pi$
$360$	0	0
$361$	28.9129	1.52173
$362$	0	0
$363$	−13.2252	−0.694144
$364$	0	0
$365$	−2.74554	−0.143708
$366$	0	0
$367$	22.4668	1.17276	0.586379	−	0.810037i	$-0.300553\pi$
$367$	22.4668	1.17276	0.586379	+	0.810037i	$0.300553\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	−0.651655	−0.0338322
$372$	0	0
$373$	−17.9453	−0.929173	−0.464587	−	0.885528i	$-0.653797\pi$
$373$	−17.9453	−0.929173	−0.464587	+	0.885528i	$0.653797\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.46725	−0.178572
$378$	0	0
$379$	−35.3724	−1.81696	−0.908478	−	0.417933i	$-0.862755\pi$
$379$	−35.3724	−1.81696	−0.908478	+	0.417933i	$0.862755\pi$
$380$	0	0
$381$	13.8590	0.710020
$382$	0	0
$383$	11.1798	0.571259	0.285630	−	0.958340i	$-0.407797\pi$
$383$	11.1798	0.571259	0.285630	+	0.958340i	$0.407797\pi$
$384$	0	0
$385$	−13.7163	−0.699048
$386$	0	0
$387$	−1.38959	−0.0706367
$388$	0	0
$389$	17.2496	0.874591	0.437296	−	0.899318i	$-0.355936\pi$
$389$	17.2496	0.874591	0.437296	+	0.899318i	$0.355936\pi$
$390$	0	0
$391$	9.20561	0.465548
$392$	0	0
$393$	−12.0440	−0.607541
$394$	0	0
$395$	−5.06947	−0.255073
$396$	0	0
$397$	23.6014	1.18452	0.592259	−	0.805747i	$-0.298236\pi$
$397$	23.6014	1.18452	0.592259	+	0.805747i	$0.298236\pi$
$398$	0	0
$399$	−19.2899	−0.965701
$400$	0	0
$401$	−11.1634	−0.557472	−0.278736	−	0.960368i	$-0.589915\pi$
$401$	−11.1634	−0.557472	−0.278736	+	0.960368i	$0.589915\pi$
$402$	0	0
$403$	−3.39720	−0.169226
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−6.87688	−0.340874
$408$	0	0
$409$	4.13894	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$409$	4.13894	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$410$	0	0
$411$	−22.7858	−1.12394
$412$	0	0
$413$	−9.17097	−0.451274
$414$	0	0
$415$	5.63864	0.276790
$416$	0	0
$417$	−1.57357	−0.0770579
$418$	0	0
$419$	−10.8969	−0.532348	−0.266174	−	0.963925i	$-0.585760\pi$
$419$	−10.8969	−0.532348	−0.266174	+	0.963925i	$0.585760\pi$
$420$	0	0
$421$	−2.37657	−0.115827	−0.0579136	−	0.998322i	$-0.518445\pi$
$421$	−2.37657	−0.115827	−0.0579136	+	0.998322i	$0.518445\pi$
$422$	0	0
$423$	−2.37657	−0.115553
$424$	0	0
$425$	−2.41782	−0.117281
$426$	0	0
$427$	12.0152	0.581457
$428$	0	0
$429$	−16.7207	−0.807283
$430$	0	0
$431$	29.7261	1.43186	0.715928	−	0.698174i	$-0.246004\pi$
$431$	29.7261	1.43186	0.715928	+	0.698174i	$0.246004\pi$
$432$	0	0
$433$	−15.0158	−0.721613	−0.360807	−	0.932641i	$-0.617499\pi$
$433$	−15.0158	−0.721613	−0.360807	+	0.932641i	$0.617499\pi$
$434$	0	0
$435$	1.02062	0.0489350
$436$	0	0
$437$	26.3545	1.26071
$438$	0	0
$439$	−13.3033	−0.634932	−0.317466	−	0.948270i	$-0.602832\pi$
$439$	−13.3033	−0.634932	−0.317466	+	0.948270i	$0.602832\pi$
$440$	0	0
$441$	0.766162	0.0364839
$442$	0	0
$443$	−34.4869	−1.63852	−0.819260	−	0.573422i	$-0.805616\pi$
$443$	−34.4869	−1.63852	−0.819260	+	0.573422i	$0.805616\pi$
$444$	0	0
$445$	−9.94253	−0.471321
$446$	0	0
$447$	−5.07809	−0.240185
$448$	0	0
$449$	35.2736	1.66467	0.832333	−	0.554276i	$-0.187005\pi$
$449$	35.2736	1.66467	0.832333	+	0.554276i	$0.187005\pi$
$450$	0	0
$451$	−19.6876	−0.927055
$452$	0	0
$453$	0.729317	0.0342663
$454$	0	0
$455$	−9.46725	−0.443832
$456$	0	0
$457$	−3.96374	−0.185416	−0.0927079	−	0.995693i	$-0.529552\pi$
$457$	−3.96374	−0.185416	−0.0927079	+	0.995693i	$0.529552\pi$
$458$	0	0
$459$	2.41782	0.112854
$460$	0	0
$461$	40.4208	1.88258	0.941292	−	0.337594i	$-0.109613\pi$
$461$	40.4208	1.88258	0.941292	+	0.337594i	$0.109613\pi$
$462$	0	0
$463$	5.23721	0.243394	0.121697	−	0.992567i	$-0.461166\pi$
$463$	5.23721	0.243394	0.121697	+	0.992567i	$0.461166\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−2.76177	−0.127799	−0.0638996	−	0.997956i	$-0.520354\pi$
$467$	−2.76177	−0.127799	−0.0638996	+	0.997956i	$0.520354\pi$
$468$	0	0
$469$	−0.491508	−0.0226957
$470$	0	0
$471$	−4.69290	−0.216237
$472$	0	0
$473$	6.83943	0.314477
$474$	0	0
$475$	−6.92191	−0.317599
$476$	0	0
$477$	0.233838	0.0107067
$478$	0	0
$479$	−32.8510	−1.50100	−0.750500	−	0.660870i	$-0.770187\pi$
$479$	−32.8510	−1.50100	−0.750500	+	0.660870i	$0.770187\pi$
$480$	0	0
$481$	−4.74655	−0.216424
$482$	0	0
$483$	−10.6104	−0.482790
$484$	0	0
$485$	12.3680	0.561600
$486$	0	0
$487$	−9.72889	−0.440858	−0.220429	−	0.975403i	$-0.570746\pi$
$487$	−9.72889	−0.440858	−0.220429	+	0.975403i	$0.570746\pi$
$488$	0	0
$489$	11.7239	0.530174
$490$	0	0
$491$	37.6166	1.69761	0.848806	−	0.528704i	$-0.177322\pi$
$491$	37.6166	1.69761	0.848806	+	0.528704i	$0.177322\pi$
$492$	0	0
$493$	−2.46768	−0.111138
$494$	0	0
$495$	4.92191	0.221224
$496$	0	0
$497$	−10.0040	−0.448739
$498$	0	0
$499$	−15.1167	−0.676716	−0.338358	−	0.941017i	$-0.609872\pi$
$499$	−15.1167	−0.676716	−0.338358	+	0.941017i	$0.609872\pi$
$500$	0	0
$501$	8.02780	0.358656
$502$	0	0
$503$	−41.8917	−1.86786	−0.933929	−	0.357460i	$-0.883643\pi$
$503$	−41.8917	−1.86786	−0.933929	+	0.357460i	$0.883643\pi$
$504$	0	0
$505$	13.6014	0.605253
$506$	0	0
$507$	1.45906	0.0647991
$508$	0	0
$509$	−24.8472	−1.10133	−0.550667	−	0.834725i	$-0.685626\pi$
$509$	−24.8472	−1.10133	−0.550667	+	0.834725i	$0.685626\pi$
$510$	0	0
$511$	−7.65123	−0.338470
$512$	0	0
$513$	6.92191	0.305610
$514$	0	0
$515$	0.644046	0.0283801
$516$	0	0
$517$	11.6973	0.514446
$518$	0	0
$519$	3.10589	0.136334
$520$	0	0
$521$	20.9910	0.919631	0.459815	−	0.888015i	$-0.347916\pi$
$521$	20.9910	0.919631	0.459815	+	0.888015i	$0.347916\pi$
$522$	0	0
$523$	−34.1267	−1.49226	−0.746128	−	0.665803i	$-0.768089\pi$
$523$	−34.1267	−1.49226	−0.746128	+	0.665803i	$0.768089\pi$
$524$	0	0
$525$	2.78678	0.121625
$526$	0	0
$527$	−2.41782	−0.105322
$528$	0	0
$529$	−8.50367	−0.369725
$530$	0	0
$531$	3.29088	0.142812
$532$	0	0
$533$	−13.5888	−0.588596
$534$	0	0
$535$	14.1189	0.610413
$536$	0	0
$537$	−2.22902	−0.0961891
$538$	0	0
$539$	−3.77098	−0.162428
$540$	0	0
$541$	−22.9171	−0.985283	−0.492641	−	0.870232i	$-0.663969\pi$
$541$	−22.9171	−0.985283	−0.492641	+	0.870232i	$0.663969\pi$
$542$	0	0
$543$	−8.16876	−0.350555
$544$	0	0
$545$	3.11451	0.133411
$546$	0	0
$547$	34.2580	1.46477	0.732383	−	0.680893i	$-0.238408\pi$
$547$	34.2580	1.46477	0.732383	+	0.680893i	$0.238408\pi$
$548$	0	0
$549$	−4.31150	−0.184010
$550$	0	0
$551$	−7.06465	−0.300964
$552$	0	0
$553$	−14.1275	−0.600763
$554$	0	0
$555$	1.39720	0.0593077
$556$	0	0
$557$	1.22681	0.0519817	0.0259908	−	0.999662i	$-0.491726\pi$
$557$	1.22681	0.0519817	0.0259908	+	0.999662i	$0.491726\pi$
$558$	0	0
$559$	4.72070	0.199664
$560$	0	0
$561$	−11.9003	−0.502430
$562$	0	0
$563$	33.0964	1.39485	0.697424	−	0.716659i	$-0.254329\pi$
$563$	33.0964	1.39485	0.697424	+	0.716659i	$0.254329\pi$
$564$	0	0
$565$	−3.81602	−0.160541
$566$	0	0
$567$	−2.78678	−0.117034
$568$	0	0
$569$	18.4819	0.774801	0.387400	−	0.921912i	$-0.373373\pi$
$569$	18.4819	0.774801	0.387400	+	0.921912i	$0.373373\pi$
$570$	0	0
$571$	13.5736	0.568036	0.284018	−	0.958819i	$-0.408332\pi$
$571$	13.5736	0.568036	0.284018	+	0.958819i	$0.408332\pi$
$572$	0	0
$573$	−13.0771	−0.546303
$574$	0	0
$575$	−3.80740	−0.158780
$576$	0	0
$577$	−33.8590	−1.40957	−0.704785	−	0.709421i	$-0.748957\pi$
$577$	−33.8590	−1.40957	−0.704785	+	0.709421i	$0.748957\pi$
$578$	0	0
$579$	−7.53232	−0.313033
$580$	0	0
$581$	15.7137	0.651913
$582$	0	0
$583$	−1.15093	−0.0476666
$584$	0	0
$585$	3.39720	0.140457
$586$	0	0
$587$	−4.27026	−0.176252	−0.0881262	−	0.996109i	$-0.528088\pi$
$587$	−4.27026	−0.176252	−0.0881262	+	0.996109i	$0.528088\pi$
$588$	0	0
$589$	−6.92191	−0.285212
$590$	0	0
$591$	−10.8855	−0.447769
$592$	0	0
$593$	25.4478	1.04502	0.522508	−	0.852634i	$-0.324996\pi$
$593$	25.4478	1.04502	0.522508	+	0.852634i	$0.324996\pi$
$594$	0	0
$595$	−6.73793	−0.276228
$596$	0	0
$597$	−5.33973	−0.218541
$598$	0	0
$599$	−20.3257	−0.830486	−0.415243	−	0.909711i	$-0.636303\pi$
$599$	−20.3257	−0.830486	−0.415243	+	0.909711i	$0.636303\pi$
$600$	0	0
$601$	−18.8933	−0.770672	−0.385336	−	0.922776i	$-0.625914\pi$
$601$	−18.8933	−0.770672	−0.385336	+	0.922776i	$0.625914\pi$
$602$	0	0
$603$	0.176371	0.00718238
$604$	0	0
$605$	−13.2252	−0.537682
$606$	0	0
$607$	20.2042	0.820062	0.410031	−	0.912072i	$-0.365518\pi$
$607$	20.2042	0.820062	0.410031	+	0.912072i	$0.365518\pi$
$608$	0	0
$609$	2.84425	0.115255
$610$	0	0
$611$	8.07369	0.326627
$612$	0	0
$613$	−5.45567	−0.220352	−0.110176	−	0.993912i	$-0.535142\pi$
$613$	−5.45567	−0.220352	−0.110176	+	0.993912i	$0.535142\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−6.68325	−0.269058	−0.134529	−	0.990910i	$-0.542952\pi$
$617$	−6.68325	−0.269058	−0.134529	+	0.990910i	$0.542952\pi$
$618$	0	0
$619$	−36.7445	−1.47689	−0.738444	−	0.674315i	$-0.764439\pi$
$619$	−36.7445	−1.47689	−0.738444	+	0.674315i	$0.764439\pi$
$620$	0	0
$621$	3.80740	0.152786
$622$	0	0
$623$	−27.7077	−1.11009
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−34.0690	−1.36059
$628$	0	0
$629$	−3.37816	−0.134696
$630$	0	0
$631$	−13.2162	−0.526128	−0.263064	−	0.964778i	$-0.584733\pi$
$631$	−13.2162	−0.526128	−0.263064	+	0.964778i	$0.584733\pi$
$632$	0	0
$633$	−11.6598	−0.463437
$634$	0	0
$635$	13.8590	0.549979
$636$	0	0
$637$	−2.60280	−0.103127
$638$	0	0
$639$	3.58979	0.142010
$640$	0	0
$641$	11.9051	0.470223	0.235111	−	0.971968i	$-0.424455\pi$
$641$	11.9051	0.470223	0.235111	+	0.971968i	$0.424455\pi$
$642$	0	0
$643$	−13.2486	−0.522475	−0.261237	−	0.965275i	$-0.584131\pi$
$643$	−13.2486	−0.522475	−0.261237	+	0.965275i	$0.584131\pi$
$644$	0	0
$645$	−1.38959	−0.0547150
$646$	0	0
$647$	24.2981	0.955255	0.477628	−	0.878562i	$-0.341497\pi$
$647$	24.2981	0.955255	0.477628	+	0.878562i	$0.341497\pi$
$648$	0	0
$649$	−16.1974	−0.635804
$650$	0	0
$651$	2.78678	0.109223
$652$	0	0
$653$	−3.15575	−0.123494	−0.0617470	−	0.998092i	$-0.519667\pi$
$653$	−3.15575	−0.123494	−0.0617470	+	0.998092i	$0.519667\pi$
$654$	0	0
$655$	−12.0440	−0.470599
$656$	0	0
$657$	2.74554	0.107114
$658$	0	0
$659$	−15.4202	−0.600685	−0.300342	−	0.953831i	$-0.597101\pi$
$659$	−15.4202	−0.600685	−0.300342	+	0.953831i	$0.597101\pi$
$660$	0	0
$661$	35.6768	1.38767	0.693834	−	0.720135i	$-0.255920\pi$
$661$	35.6768	1.38767	0.693834	+	0.720135i	$0.255920\pi$
$662$	0	0
$663$	−8.21380	−0.318998
$664$	0	0
$665$	−19.2899	−0.748029
$666$	0	0
$667$	−3.88592	−0.150463
$668$	0	0
$669$	−9.63003	−0.372318
$670$	0	0
$671$	21.2208	0.819221
$672$	0	0
$673$	−15.2975	−0.589675	−0.294837	−	0.955547i	$-0.595265\pi$
$673$	−15.2975	−0.589675	−0.294837	+	0.955547i	$0.595265\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−9.43743	−0.362710	−0.181355	−	0.983418i	$-0.558048\pi$
$677$	−9.43743	−0.362710	−0.181355	+	0.983418i	$0.558048\pi$
$678$	0	0
$679$	34.4668	1.32272
$680$	0	0
$681$	1.20079	0.0460142
$682$	0	0
$683$	6.27423	0.240077	0.120038	−	0.992769i	$-0.461698\pi$
$683$	6.27423	0.240077	0.120038	+	0.992769i	$0.461698\pi$
$684$	0	0
$685$	−22.7858	−0.870600
$686$	0	0
$687$	26.4782	1.01021
$688$	0	0
$689$	−0.794392	−0.0302639
$690$	0	0
$691$	20.7081	0.787774	0.393887	−	0.919159i	$-0.371130\pi$
$691$	20.7081	0.787774	0.393887	+	0.919159i	$0.371130\pi$
$692$	0	0
$693$	13.7163	0.521039
$694$	0	0
$695$	−1.57357	−0.0596888
$696$	0	0
$697$	−9.67127	−0.366325
$698$	0	0
$699$	25.7653	0.974534
$700$	0	0
$701$	−25.9713	−0.980924	−0.490462	−	0.871463i	$-0.663172\pi$
$701$	−25.9713	−0.980924	−0.490462	+	0.871463i	$0.663172\pi$
$702$	0	0
$703$	−9.67127	−0.364759
$704$	0	0
$705$	−2.37657	−0.0895070
$706$	0	0
$707$	37.9041	1.42553
$708$	0	0
$709$	−5.93975	−0.223072	−0.111536	−	0.993760i	$-0.535577\pi$
$709$	−5.93975	−0.223072	−0.111536	+	0.993760i	$0.535577\pi$
$710$	0	0
$711$	5.06947	0.190120
$712$	0	0
$713$	−3.80740	−0.142588
$714$	0	0
$715$	−16.7207	−0.625319
$716$	0	0
$717$	−20.3267	−0.759115
$718$	0	0
$719$	26.6534	0.994005	0.497003	−	0.867749i	$-0.334434\pi$
$719$	26.6534	0.994005	0.497003	+	0.867749i	$0.334434\pi$
$720$	0	0
$721$	1.79482	0.0668425
$722$	0	0
$723$	11.9415	0.444110
$724$	0	0
$725$	1.02062	0.0379049
$726$	0	0
$727$	−11.0006	−0.407989	−0.203995	−	0.978972i	$-0.565392\pi$
$727$	−11.0006	−0.407989	−0.203995	+	0.978972i	$0.565392\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.35977	0.124266
$732$	0	0
$733$	23.7865	0.878575	0.439287	−	0.898347i	$-0.355231\pi$
$733$	23.7865	0.878575	0.439287	+	0.898347i	$0.355231\pi$
$734$	0	0
$735$	0.766162	0.0282603
$736$	0	0
$737$	−0.868083	−0.0319762
$738$	0	0
$739$	−33.4652	−1.23104	−0.615519	−	0.788122i	$-0.711054\pi$
$739$	−33.4652	−1.23104	−0.615519	+	0.788122i	$0.711054\pi$
$740$	0	0
$741$	−23.5151	−0.863849
$742$	0	0
$743$	11.3051	0.414743	0.207372	−	0.978262i	$-0.433509\pi$
$743$	11.3051	0.414743	0.207372	+	0.978262i	$0.433509\pi$
$744$	0	0
$745$	−5.07809	−0.186047
$746$	0	0
$747$	−5.63864	−0.206307
$748$	0	0
$749$	39.3463	1.43768
$750$	0	0
$751$	43.5555	1.58936	0.794681	−	0.607027i	$-0.207638\pi$
$751$	43.5555	1.58936	0.794681	+	0.607027i	$0.207638\pi$
$752$	0	0
$753$	9.37615	0.341686
$754$	0	0
$755$	0.729317	0.0265426
$756$	0	0
$757$	−9.70870	−0.352869	−0.176434	−	0.984312i	$-0.556456\pi$
$757$	−9.70870	−0.352869	−0.176434	+	0.984312i	$0.556456\pi$
$758$	0	0
$759$	−18.7397	−0.680208
$760$	0	0
$761$	−12.9083	−0.467926	−0.233963	−	0.972245i	$-0.575169\pi$
$761$	−12.9083	−0.467926	−0.233963	+	0.972245i	$0.575169\pi$
$762$	0	0
$763$	8.67946	0.314218
$764$	0	0
$765$	2.41782	0.0874164
$766$	0	0
$767$	−11.1798	−0.403678
$768$	0	0
$769$	−18.1189	−0.653384	−0.326692	−	0.945131i	$-0.605934\pi$
$769$	−18.1189	−0.653384	−0.326692	+	0.945131i	$0.605934\pi$
$770$	0	0
$771$	−2.93053	−0.105540
$772$	0	0
$773$	−15.9808	−0.574790	−0.287395	−	0.957812i	$-0.592789\pi$
$773$	−15.9808	−0.574790	−0.287395	+	0.957812i	$0.592789\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	3.89368	0.139685
$778$	0	0
$779$	−27.6876	−0.992013
$780$	0	0
$781$	−17.6686	−0.632234
$782$	0	0
$783$	−1.02062	−0.0364740
$784$	0	0
$785$	−4.69290	−0.167497
$786$	0	0
$787$	38.8634	1.38533	0.692666	−	0.721259i	$-0.256436\pi$
$787$	38.8634	1.38533	0.692666	+	0.721259i	$0.256436\pi$
$788$	0	0
$789$	20.8933	0.743820
$790$	0	0
$791$	−10.6344	−0.378117
$792$	0	0
$793$	14.6470	0.520131
$794$	0	0
$795$	0.233838	0.00829336
$796$	0	0
$797$	−41.6378	−1.47489	−0.737443	−	0.675409i	$-0.763967\pi$
$797$	−41.6378	−1.47489	−0.737443	+	0.675409i	$0.763967\pi$
$798$	0	0
$799$	5.74612	0.203283
$800$	0	0
$801$	9.94253	0.351302
$802$	0	0
$803$	−13.5133	−0.476874
$804$	0	0
$805$	−10.6104	−0.373968
$806$	0	0
$807$	−28.4532	−1.00160
$808$	0	0
$809$	−40.6524	−1.42926	−0.714632	−	0.699501i	$-0.753406\pi$
$809$	−40.6524	−1.42926	−0.714632	+	0.699501i	$0.753406\pi$
$810$	0	0
$811$	−28.7810	−1.01064	−0.505318	−	0.862933i	$-0.668625\pi$
$811$	−28.7810	−1.01064	−0.505318	+	0.862933i	$0.668625\pi$
$812$	0	0
$813$	−13.8889	−0.487104
$814$	0	0
$815$	11.7239	0.410671
$816$	0	0
$817$	9.61860	0.336512
$818$	0	0
$819$	9.46725	0.330812
$820$	0	0
$821$	−44.9873	−1.57007	−0.785034	−	0.619452i	$-0.787355\pi$
$821$	−44.9873	−1.57007	−0.785034	+	0.619452i	$0.787355\pi$
$822$	0	0
$823$	22.5405	0.785713	0.392856	−	0.919600i	$-0.371487\pi$
$823$	22.5405	0.785713	0.392856	+	0.919600i	$0.371487\pi$
$824$	0	0
$825$	4.92191	0.171359
$826$	0	0
$827$	−25.2438	−0.877813	−0.438907	−	0.898533i	$-0.644634\pi$
$827$	−25.2438	−0.877813	−0.438907	+	0.898533i	$0.644634\pi$
$828$	0	0
$829$	−48.4946	−1.68429	−0.842144	−	0.539253i	$-0.818707\pi$
$829$	−48.4946	−1.68429	−0.842144	+	0.539253i	$0.818707\pi$
$830$	0	0
$831$	−16.7321	−0.580430
$832$	0	0
$833$	−1.85244	−0.0641833
$834$	0	0
$835$	8.02780	0.277814
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	24.2444	0.837009	0.418505	−	0.908215i	$-0.362554\pi$
$839$	24.2444	0.837009	0.418505	+	0.908215i	$0.362554\pi$
$840$	0	0
$841$	−27.9583	−0.964080
$842$	0	0
$843$	−18.0977	−0.623318
$844$	0	0
$845$	1.45906	0.0501932
$846$	0	0
$847$	−36.8558	−1.26638
$848$	0	0
$849$	32.3882	1.11156
$850$	0	0
$851$	−5.31969	−0.182357
$852$	0	0
$853$	−36.7906	−1.25969	−0.629843	−	0.776722i	$-0.716881\pi$
$853$	−36.7906	−1.25969	−0.629843	+	0.776722i	$0.716881\pi$
$854$	0	0
$855$	6.92191	0.236724
$856$	0	0
$857$	−19.8114	−0.676744	−0.338372	−	0.941012i	$-0.609876\pi$
$857$	−19.8114	−0.676744	−0.338372	+	0.941012i	$0.609876\pi$
$858$	0	0
$859$	1.39338	0.0475416	0.0237708	−	0.999717i	$-0.492433\pi$
$859$	1.39338	0.0475416	0.0237708	+	0.999717i	$0.492433\pi$
$860$	0	0
$861$	11.1471	0.379893
$862$	0	0
$863$	−0.995603	−0.0338907	−0.0169454	−	0.999856i	$-0.505394\pi$
$863$	−0.995603	−0.0338907	−0.0169454	+	0.999856i	$0.505394\pi$
$864$	0	0
$865$	3.10589	0.105603
$866$	0	0
$867$	11.1542	0.378815
$868$	0	0
$869$	−24.9515	−0.846422
$870$	0	0
$871$	−0.599167	−0.0203020
$872$	0	0
$873$	−12.3680	−0.418592
$874$	0	0
$875$	2.78678	0.0942105
$876$	0	0
$877$	28.8857	0.975400	0.487700	−	0.873011i	$-0.337836\pi$
$877$	28.8857	0.975400	0.487700	+	0.873011i	$0.337836\pi$
$878$	0	0
$879$	4.92673	0.166175
$880$	0	0
$881$	−10.7504	−0.362189	−0.181094	−	0.983466i	$-0.557964\pi$
$881$	−10.7504	−0.362189	−0.181094	+	0.983466i	$0.557964\pi$
$882$	0	0
$883$	26.1831	0.881132	0.440566	−	0.897720i	$-0.354778\pi$
$883$	26.1831	0.881132	0.440566	+	0.897720i	$0.354778\pi$
$884$	0	0
$885$	3.29088	0.110622
$886$	0	0
$887$	14.0642	0.472230	0.236115	−	0.971725i	$-0.424126\pi$
$887$	14.0642	0.472230	0.236115	+	0.971725i	$0.424126\pi$
$888$	0	0
$889$	38.6221	1.29535
$890$	0	0
$891$	−4.92191	−0.164890
$892$	0	0
$893$	16.4504	0.550493
$894$	0	0
$895$	−2.22902	−0.0745078
$896$	0	0
$897$	−12.9345	−0.431870
$898$	0	0
$899$	1.02062	0.0340396
$900$	0	0
$901$	−0.565376	−0.0188354
$902$	0	0
$903$	−3.87248	−0.128868
$904$	0	0
$905$	−8.16876	−0.271539
$906$	0	0
$907$	11.5698	0.384167	0.192084	−	0.981379i	$-0.438476\pi$
$907$	11.5698	0.384167	0.192084	+	0.981379i	$0.438476\pi$
$908$	0	0
$909$	−13.6014	−0.451129
$910$	0	0
$911$	27.8438	0.922507	0.461254	−	0.887268i	$-0.347400\pi$
$911$	27.8438	0.922507	0.461254	+	0.887268i	$0.347400\pi$
$912$	0	0
$913$	27.7529	0.918487
$914$	0	0
$915$	−4.31150	−0.142534
$916$	0	0
$917$	−33.5641	−1.10838
$918$	0	0
$919$	35.4154	1.16825	0.584123	−	0.811665i	$-0.301439\pi$
$919$	35.4154	1.16825	0.584123	+	0.811665i	$0.301439\pi$
$920$	0	0
$921$	15.3647	0.506286
$922$	0	0
$923$	−12.1952	−0.401411
$924$	0	0
$925$	1.39720	0.0459395
$926$	0	0
$927$	−0.644046	−0.0211533
$928$	0	0
$929$	20.0250	0.657000	0.328500	−	0.944504i	$-0.393457\pi$
$929$	20.0250	0.657000	0.328500	+	0.944504i	$0.393457\pi$
$930$	0	0
$931$	−5.30331	−0.173809
$932$	0	0
$933$	−14.5790	−0.477294
$934$	0	0
$935$	−11.9003	−0.389181
$936$	0	0
$937$	−32.0825	−1.04809	−0.524045	−	0.851691i	$-0.675577\pi$
$937$	−32.0825	−1.04809	−0.524045	+	0.851691i	$0.675577\pi$
$938$	0	0
$939$	21.3387	0.696363
$940$	0	0
$941$	−6.05282	−0.197316	−0.0986582	−	0.995121i	$-0.531455\pi$
$941$	−6.05282	−0.197316	−0.0986582	+	0.995121i	$0.531455\pi$
$942$	0	0
$943$	−15.2296	−0.495945
$944$	0	0
$945$	−2.78678	−0.0906541
$946$	0	0
$947$	−31.1873	−1.01345	−0.506726	−	0.862107i	$-0.669145\pi$
$947$	−31.1873	−1.01345	−0.506726	+	0.862107i	$0.669145\pi$
$948$	0	0
$949$	−9.32714	−0.302772
$950$	0	0
$951$	−27.3111	−0.885622
$952$	0	0
$953$	14.7129	0.476599	0.238299	−	0.971192i	$-0.423410\pi$
$953$	14.7129	0.476599	0.238299	+	0.971192i	$0.423410\pi$
$954$	0	0
$955$	−13.0771	−0.423164
$956$	0	0
$957$	5.02341	0.162384
$958$	0	0
$959$	−63.4990	−2.05049
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−14.1189	−0.454975
$964$	0	0
$965$	−7.53232	−0.242474
$966$	0	0
$967$	−9.83302	−0.316208	−0.158104	−	0.987422i	$-0.550538\pi$
$967$	−9.83302	−0.316208	−0.158104	+	0.987422i	$0.550538\pi$
$968$	0	0
$969$	−16.7359	−0.537635
$970$	0	0
$971$	−46.9048	−1.50525	−0.752624	−	0.658451i	$-0.771212\pi$
$971$	−46.9048	−1.50525	−0.752624	+	0.658451i	$0.771212\pi$
$972$	0	0
$973$	−4.38519	−0.140583
$974$	0	0
$975$	3.39720	0.108797
$976$	0	0
$977$	15.6713	0.501368	0.250684	−	0.968069i	$-0.419344\pi$
$977$	15.6713	0.501368	0.250684	+	0.968069i	$0.419344\pi$
$978$	0	0
$979$	−48.9363	−1.56401
$980$	0	0
$981$	−3.11451	−0.0994386
$982$	0	0
$983$	0.212634	0.00678198	0.00339099	−	0.999994i	$-0.498921\pi$
$983$	0.212634	0.00678198	0.00339099	+	0.999994i	$0.498921\pi$
$984$	0	0
$985$	−10.8855	−0.346841
$986$	0	0
$987$	−6.62300	−0.210812
$988$	0	0
$989$	5.29072	0.168235
$990$	0	0
$991$	21.9638	0.697702	0.348851	−	0.937178i	$-0.386572\pi$
$991$	21.9638	0.697702	0.348851	+	0.937178i	$0.386572\pi$
$992$	0	0
$993$	−5.85244	−0.185722
$994$	0	0
$995$	−5.33973	−0.169281
$996$	0	0
$997$	−9.31768	−0.295094	−0.147547	−	0.989055i	$-0.547138\pi$
$997$	−9.31768	−0.295094	−0.147547	+	0.989055i	$0.547138\pi$
$998$	0	0
$999$	−1.39720	−0.0442053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.r.1.1	✓	4
4.3	odd	2		7440.2.a.bx.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.r.1.1	✓	4	1.1	even	1	trivial
7440.2.a.bx.1.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.78678 q^{7} +1.00000 q^{9} -4.92191 q^{11} -3.39720 q^{13} +1.00000 q^{15} -2.41782 q^{17} -6.92191 q^{19} +2.78678 q^{21} -3.80740 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.02062 q^{29} +1.00000 q^{31} +4.92191 q^{33} +2.78678 q^{35} +1.39720 q^{37} +3.39720 q^{39} +4.00000 q^{41} -1.38959 q^{43} -1.00000 q^{45} -2.37657 q^{47} +0.766162 q^{49} +2.41782 q^{51} +0.233838 q^{53} +4.92191 q^{55} +6.92191 q^{57} +3.29088 q^{59} -4.31150 q^{61} -2.78678 q^{63} +3.39720 q^{65} +0.176371 q^{67} +3.80740 q^{69} +3.58979 q^{71} +2.74554 q^{73} -1.00000 q^{75} +13.7163 q^{77} +5.06947 q^{79} +1.00000 q^{81} -5.63864 q^{83} +2.41782 q^{85} -1.02062 q^{87} +9.94253 q^{89} +9.46725 q^{91} -1.00000 q^{93} +6.92191 q^{95} -12.3680 q^{97} -4.92191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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