LMFDB - Embedded newform 1305.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{7} -2.67513 q^{8} -1.48119 q^{10} -4.15633 q^{11} +2.96239 q^{13} +1.76845 q^{14} -4.35026 q^{16} -5.50659 q^{17} -3.19394 q^{19} -0.193937 q^{20} -6.15633 q^{22} -1.84367 q^{23} +1.00000 q^{25} +4.38787 q^{26} +0.231548 q^{28} +1.00000 q^{29} -4.80606 q^{31} -1.09332 q^{32} -8.15633 q^{34} -1.19394 q^{35} -9.50659 q^{37} -4.73084 q^{38} +2.67513 q^{40} +11.2750 q^{41} -0.0303172 q^{43} -0.806063 q^{44} -2.73084 q^{46} -4.80606 q^{47} -5.57452 q^{49} +1.48119 q^{50} +0.574515 q^{52} +1.35026 q^{53} +4.15633 q^{55} -3.19394 q^{56} +1.48119 q^{58} -13.2750 q^{59} +8.88717 q^{61} -7.11871 q^{62} +7.08110 q^{64} -2.96239 q^{65} +5.84367 q^{67} -1.06793 q^{68} -1.76845 q^{70} +1.27504 q^{71} -15.2447 q^{73} -14.0811 q^{74} -0.619421 q^{76} -4.96239 q^{77} -4.93207 q^{79} +4.35026 q^{80} +16.7005 q^{82} -4.41819 q^{83} +5.50659 q^{85} -0.0449056 q^{86} +11.1187 q^{88} +3.61213 q^{89} +3.53690 q^{91} -0.357556 q^{92} -7.11871 q^{94} +3.19394 q^{95} -1.38058 q^{97} -8.25694 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 3 q^{5} + 4 q^{7} - 3 q^{8} + q^{10} - 2 q^{11} - 2 q^{13} - 6 q^{14} - 3 q^{16} + 4 q^{17} - 10 q^{19} - q^{20} - 8 q^{22} - 16 q^{23} + 3 q^{25} + 14 q^{26} + 12 q^{28} + 3 q^{29}+ \cdots - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 3 * q^5 + 4 * q^7 - 3 * q^8 + q^10 - 2 * q^11 - 2 * q^13 - 6 * q^14 - 3 * q^16 + 4 * q^17 - 10 * q^19 - q^20 - 8 * q^22 - 16 * q^23 + 3 * q^25 + 14 * q^26 + 12 * q^28 + 3 * q^29 - 14 * q^31 + 3 * q^32 - 14 * q^34 - 4 * q^35 - 8 * q^37 + 8 * q^38 + 3 * q^40 + 2 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 5 * q^49 - q^50 - 10 * q^52 - 6 * q^53 + 2 * q^55 - 10 * q^56 - q^58 - 8 * q^59 - 6 * q^61 - 11 * q^64 + 2 * q^65 + 28 * q^67 - 12 * q^68 + 6 * q^70 - 28 * q^71 - 16 * q^73 - 10 * q^74 - 14 * q^76 - 4 * q^77 - 6 * q^79 + 3 * q^80 + 30 * q^82 - 12 * q^83 - 4 * q^85 - 24 * q^86 + 12 * q^88 + 10 * q^89 - 12 * q^91 - 4 * q^92 + 10 * q^95 + 8 * q^97 - 21 * q^98

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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	0	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.19394	0.451266	0.225633	−	0.974212i	$-0.427555\pi$
$7$	1.19394	0.451266	0.225633	+	0.974212i	$0.427555\pi$
$8$	−2.67513	−0.945802
$9$	0	0
$10$	−1.48119	−0.468395
$11$	−4.15633	−1.25318	−0.626590	−	0.779349i	$-0.715550\pi$
$11$	−4.15633	−1.25318	−0.626590	+	0.779349i	$0.715550\pi$
$12$	0	0
$13$	2.96239	0.821619	0.410809	−	0.911721i	$-0.365246\pi$
$13$	2.96239	0.821619	0.410809	+	0.911721i	$0.365246\pi$
$14$	1.76845	0.472639
$15$	0	0
$16$	−4.35026	−1.08757
$17$	−5.50659	−1.33554	−0.667772	−	0.744366i	$-0.732752\pi$
$17$	−5.50659	−1.33554	−0.667772	+	0.744366i	$0.732752\pi$
$18$	0	0
$19$	−3.19394	−0.732739	−0.366370	−	0.930469i	$-0.619399\pi$
$19$	−3.19394	−0.732739	−0.366370	+	0.930469i	$0.619399\pi$
$20$	−0.193937	−0.0433655
$21$	0	0
$22$	−6.15633	−1.31253
$23$	−1.84367	−0.384433	−0.192216	−	0.981353i	$-0.561568\pi$
$23$	−1.84367	−0.384433	−0.192216	+	0.981353i	$0.561568\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.38787	0.860533
$27$	0	0
$28$	0.231548	0.0437585
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.80606	−0.863194	−0.431597	−	0.902066i	$-0.642050\pi$
$31$	−4.80606	−0.863194	−0.431597	+	0.902066i	$0.642050\pi$
$32$	−1.09332	−0.193274
$33$	0	0
$34$	−8.15633	−1.39880
$35$	−1.19394	−0.201812
$36$	0	0
$37$	−9.50659	−1.56287	−0.781437	−	0.623985i	$-0.785513\pi$
$37$	−9.50659	−1.56287	−0.781437	+	0.623985i	$0.785513\pi$
$38$	−4.73084	−0.767444
$39$	0	0
$40$	2.67513	0.422975
$41$	11.2750	1.76087	0.880433	−	0.474171i	$-0.157252\pi$
$41$	11.2750	1.76087	0.880433	+	0.474171i	$0.157252\pi$
$42$	0	0
$43$	−0.0303172	−0.00462332	−0.00231166	−	0.999997i	$-0.500736\pi$
$43$	−0.0303172	−0.00462332	−0.00231166	+	0.999997i	$0.500736\pi$
$44$	−0.806063	−0.121519
$45$	0	0
$46$	−2.73084	−0.402640
$47$	−4.80606	−0.701036	−0.350518	−	0.936556i	$-0.613995\pi$
$47$	−4.80606	−0.701036	−0.350518	+	0.936556i	$0.613995\pi$
$48$	0	0
$49$	−5.57452	−0.796359
$50$	1.48119	0.209473
$51$	0	0
$52$	0.574515	0.0796710
$53$	1.35026	0.185473	0.0927364	−	0.995691i	$-0.470439\pi$
$53$	1.35026	0.185473	0.0927364	+	0.995691i	$0.470439\pi$
$54$	0	0
$55$	4.15633	0.560439
$56$	−3.19394	−0.426808
$57$	0	0
$58$	1.48119	0.194490
$59$	−13.2750	−1.72826	−0.864131	−	0.503266i	$-0.832132\pi$
$59$	−13.2750	−1.72826	−0.864131	+	0.503266i	$0.832132\pi$
$60$	0	0
$61$	8.88717	1.13788	0.568942	−	0.822377i	$-0.307353\pi$
$61$	8.88717	1.13788	0.568942	+	0.822377i	$0.307353\pi$
$62$	−7.11871	−0.904078
$63$	0	0
$64$	7.08110	0.885138
$65$	−2.96239	−0.367439
$66$	0	0
$67$	5.84367	0.713919	0.356959	−	0.934120i	$-0.383813\pi$
$67$	5.84367	0.713919	0.356959	+	0.934120i	$0.383813\pi$
$68$	−1.06793	−0.129505
$69$	0	0
$70$	−1.76845	−0.211370
$71$	1.27504	0.151319	0.0756596	−	0.997134i	$-0.475894\pi$
$71$	1.27504	0.151319	0.0756596	+	0.997134i	$0.475894\pi$
$72$	0	0
$73$	−15.2447	−1.78426	−0.892130	−	0.451779i	$-0.850790\pi$
$73$	−15.2447	−1.78426	−0.892130	+	0.451779i	$0.850790\pi$
$74$	−14.0811	−1.63689
$75$	0	0
$76$	−0.619421	−0.0710525
$77$	−4.96239	−0.565517
$78$	0	0
$79$	−4.93207	−0.554901	−0.277451	−	0.960740i	$-0.589490\pi$
$79$	−4.93207	−0.554901	−0.277451	+	0.960740i	$0.589490\pi$
$80$	4.35026	0.486374
$81$	0	0
$82$	16.7005	1.84426
$83$	−4.41819	−0.484959	−0.242480	−	0.970156i	$-0.577961\pi$
$83$	−4.41819	−0.484959	−0.242480	+	0.970156i	$0.577961\pi$
$84$	0	0
$85$	5.50659	0.597273
$86$	−0.0449056	−0.00484230
$87$	0	0
$88$	11.1187	1.18526
$89$	3.61213	0.382885	0.191442	−	0.981504i	$-0.438684\pi$
$89$	3.61213	0.382885	0.191442	+	0.981504i	$0.438684\pi$
$90$	0	0
$91$	3.53690	0.370768
$92$	−0.357556	−0.0372778
$93$	0	0
$94$	−7.11871	−0.734239
$95$	3.19394	0.327691
$96$	0	0
$97$	−1.38058	−0.140177	−0.0700883	−	0.997541i	$-0.522328\pi$
$97$	−1.38058	−0.140177	−0.0700883	+	0.997541i	$0.522328\pi$
$98$	−8.25694	−0.834077
$99$	0	0
$100$	0.193937	0.0193937
$101$	13.0132	1.29486	0.647430	−	0.762125i	$-0.275844\pi$
$101$	13.0132	1.29486	0.647430	+	0.762125i	$0.275844\pi$
$102$	0	0
$103$	5.31994	0.524190	0.262095	−	0.965042i	$-0.415587\pi$
$103$	5.31994	0.524190	0.262095	+	0.965042i	$0.415587\pi$
$104$	−7.92478	−0.777088
$105$	0	0
$106$	2.00000	0.194257
$107$	13.8192	1.33596	0.667978	−	0.744181i	$-0.267160\pi$
$107$	13.8192	1.33596	0.667978	+	0.744181i	$0.267160\pi$
$108$	0	0
$109$	−1.87399	−0.179496	−0.0897479	−	0.995965i	$-0.528606\pi$
$109$	−1.87399	−0.179496	−0.0897479	+	0.995965i	$0.528606\pi$
$110$	6.15633	0.586983
$111$	0	0
$112$	−5.19394	−0.490781
$113$	11.7685	1.10708	0.553541	−	0.832822i	$-0.313276\pi$
$113$	11.7685	1.10708	0.553541	+	0.832822i	$0.313276\pi$
$114$	0	0
$115$	1.84367	0.171924
$116$	0.193937	0.0180066
$117$	0	0
$118$	−19.6629	−1.81012
$119$	−6.57452	−0.602685
$120$	0	0
$121$	6.27504	0.570458
$122$	13.1636	1.19178
$123$	0	0
$124$	−0.932071	−0.0837025
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.2677	1.26606	0.633029	−	0.774128i	$-0.281811\pi$
$127$	14.2677	1.26606	0.633029	+	0.774128i	$0.281811\pi$
$128$	12.6751	1.12033
$129$	0	0
$130$	−4.38787	−0.384842
$131$	−5.89446	−0.515001	−0.257501	−	0.966278i	$-0.582899\pi$
$131$	−5.89446	−0.515001	−0.257501	+	0.966278i	$0.582899\pi$
$132$	0	0
$133$	−3.81336	−0.330660
$134$	8.65562	0.747731
$135$	0	0
$136$	14.7308	1.26316
$137$	−18.2823	−1.56197	−0.780983	−	0.624553i	$-0.785281\pi$
$137$	−18.2823	−1.56197	−0.780983	+	0.624553i	$0.785281\pi$
$138$	0	0
$139$	−11.5369	−0.978547	−0.489274	−	0.872130i	$-0.662738\pi$
$139$	−11.5369	−0.978547	−0.489274	+	0.872130i	$0.662738\pi$
$140$	−0.231548	−0.0195694
$141$	0	0
$142$	1.88858	0.158486
$143$	−12.3127	−1.02964
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	−22.5804	−1.86877
$147$	0	0
$148$	−1.84367	−0.151549
$149$	−2.77575	−0.227398	−0.113699	−	0.993515i	$-0.536270\pi$
$149$	−2.77575	−0.227398	−0.113699	+	0.993515i	$0.536270\pi$
$150$	0	0
$151$	1.79877	0.146382	0.0731909	−	0.997318i	$-0.476682\pi$
$151$	1.79877	0.146382	0.0731909	+	0.997318i	$0.476682\pi$
$152$	8.54420	0.693026
$153$	0	0
$154$	−7.35026	−0.592301
$155$	4.80606	0.386032
$156$	0	0
$157$	3.76845	0.300755	0.150378	−	0.988629i	$-0.451951\pi$
$157$	3.76845	0.300755	0.150378	+	0.988629i	$0.451951\pi$
$158$	−7.30536	−0.581183
$159$	0	0
$160$	1.09332	0.0864346
$161$	−2.20123	−0.173481
$162$	0	0
$163$	1.64244	0.128646	0.0643231	−	0.997929i	$-0.479511\pi$
$163$	1.64244	0.128646	0.0643231	+	0.997929i	$0.479511\pi$
$164$	2.18664	0.170748
$165$	0	0
$166$	−6.54420	−0.507928
$167$	−8.08110	−0.625334	−0.312667	−	0.949863i	$-0.601222\pi$
$167$	−8.08110	−0.625334	−0.312667	+	0.949863i	$0.601222\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	8.15633	0.625562
$171$	0	0
$172$	−0.00587961	−0.000448316 0
$173$	7.73813	0.588320	0.294160	−	0.955756i	$-0.404960\pi$
$173$	7.73813	0.588320	0.294160	+	0.955756i	$0.404960\pi$
$174$	0	0
$175$	1.19394	0.0902531
$176$	18.0811	1.36291
$177$	0	0
$178$	5.35026	0.401019
$179$	21.4010	1.59959	0.799795	−	0.600274i	$-0.204942\pi$
$179$	21.4010	1.59959	0.799795	+	0.600274i	$0.204942\pi$
$180$	0	0
$181$	15.2750	1.13538	0.567692	−	0.823241i	$-0.307836\pi$
$181$	15.2750	1.13538	0.567692	+	0.823241i	$0.307836\pi$
$182$	5.23884	0.388329
$183$	0	0
$184$	4.93207	0.363597
$185$	9.50659	0.698938
$186$	0	0
$187$	22.8872	1.67368
$188$	−0.932071	−0.0679783
$189$	0	0
$190$	4.73084	0.343211
$191$	−3.31994	−0.240223	−0.120111	−	0.992760i	$-0.538325\pi$
$191$	−3.31994	−0.240223	−0.120111	+	0.992760i	$0.538325\pi$
$192$	0	0
$193$	−4.88129	−0.351363	−0.175681	−	0.984447i	$-0.556213\pi$
$193$	−4.88129	−0.351363	−0.175681	+	0.984447i	$0.556213\pi$
$194$	−2.04491	−0.146816
$195$	0	0
$196$	−1.08110	−0.0772216
$197$	24.2374	1.72685	0.863423	−	0.504481i	$-0.168316\pi$
$197$	24.2374	1.72685	0.863423	+	0.504481i	$0.168316\pi$
$198$	0	0
$199$	16.7513	1.18747	0.593734	−	0.804661i	$-0.297653\pi$
$199$	16.7513	1.18747	0.593734	+	0.804661i	$0.297653\pi$
$200$	−2.67513	−0.189160
$201$	0	0
$202$	19.2750	1.35619
$203$	1.19394	0.0837979
$204$	0	0
$205$	−11.2750	−0.787483
$206$	7.87987	0.549017
$207$	0	0
$208$	−12.8872	−0.893564
$209$	13.2750	0.918254
$210$	0	0
$211$	−25.3054	−1.74209	−0.871046	−	0.491201i	$-0.836558\pi$
$211$	−25.3054	−1.74209	−0.871046	+	0.491201i	$0.836558\pi$
$212$	0.261865	0.0179850
$213$	0	0
$214$	20.4690	1.39923
$215$	0.0303172	0.00206761
$216$	0	0
$217$	−5.73813	−0.389530
$218$	−2.77575	−0.187997
$219$	0	0
$220$	0.806063	0.0543448
$221$	−16.3127	−1.09731
$222$	0	0
$223$	17.6932	1.18483	0.592413	−	0.805634i	$-0.298175\pi$
$223$	17.6932	1.18483	0.592413	+	0.805634i	$0.298175\pi$
$224$	−1.30536	−0.0872178
$225$	0	0
$226$	17.4314	1.15952
$227$	−26.8423	−1.78158	−0.890792	−	0.454412i	$-0.849849\pi$
$227$	−26.8423	−1.78158	−0.890792	+	0.454412i	$0.849849\pi$
$228$	0	0
$229$	−17.2243	−1.13821	−0.569105	−	0.822265i	$-0.692710\pi$
$229$	−17.2243	−1.13821	−0.569105	+	0.822265i	$0.692710\pi$
$230$	2.73084	0.180066
$231$	0	0
$232$	−2.67513	−0.175631
$233$	9.07381	0.594445	0.297222	−	0.954808i	$-0.403940\pi$
$233$	9.07381	0.594445	0.297222	+	0.954808i	$0.403940\pi$
$234$	0	0
$235$	4.80606	0.313513
$236$	−2.57452	−0.167587
$237$	0	0
$238$	−9.73813	−0.631230
$239$	−20.4993	−1.32599	−0.662995	−	0.748624i	$-0.730715\pi$
$239$	−20.4993	−1.32599	−0.662995	+	0.748624i	$0.730715\pi$
$240$	0	0
$241$	5.47627	0.352758	0.176379	−	0.984322i	$-0.443562\pi$
$241$	5.47627	0.352758	0.176379	+	0.984322i	$0.443562\pi$
$242$	9.29455	0.597476
$243$	0	0
$244$	1.72355	0.110339
$245$	5.57452	0.356143
$246$	0	0
$247$	−9.46168	−0.602032
$248$	12.8568	0.816411
$249$	0	0
$250$	−1.48119	−0.0936790
$251$	29.6180	1.86947	0.934736	−	0.355343i	$-0.115636\pi$
$251$	29.6180	1.86947	0.934736	+	0.355343i	$0.115636\pi$
$252$	0	0
$253$	7.66291	0.481763
$254$	21.1333	1.32602
$255$	0	0
$256$	4.61213	0.288258
$257$	−17.6629	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$257$	−17.6629	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$258$	0	0
$259$	−11.3503	−0.705271
$260$	−0.574515	−0.0356299
$261$	0	0
$262$	−8.73084	−0.539393
$263$	−27.3561	−1.68685	−0.843426	−	0.537245i	$-0.819465\pi$
$263$	−27.3561	−1.68685	−0.843426	+	0.537245i	$0.819465\pi$
$264$	0	0
$265$	−1.35026	−0.0829459
$266$	−5.64832	−0.346321
$267$	0	0
$268$	1.13330	0.0692275
$269$	−10.4993	−0.640153	−0.320077	−	0.947392i	$-0.603709\pi$
$269$	−10.4993	−0.640153	−0.320077	+	0.947392i	$0.603709\pi$
$270$	0	0
$271$	−9.61801	−0.584252	−0.292126	−	0.956380i	$-0.594363\pi$
$271$	−9.61801	−0.584252	−0.292126	+	0.956380i	$0.594363\pi$
$272$	23.9551	1.45249
$273$	0	0
$274$	−27.0797	−1.63594
$275$	−4.15633	−0.250636
$276$	0	0
$277$	13.3503	0.802139	0.401070	−	0.916048i	$-0.368638\pi$
$277$	13.3503	0.802139	0.401070	+	0.916048i	$0.368638\pi$
$278$	−17.0884	−1.02489
$279$	0	0
$280$	3.19394	0.190874
$281$	−20.4241	−1.21840	−0.609199	−	0.793017i	$-0.708509\pi$
$281$	−20.4241	−1.21840	−0.609199	+	0.793017i	$0.708509\pi$
$282$	0	0
$283$	−8.02047	−0.476767	−0.238384	−	0.971171i	$-0.576618\pi$
$283$	−8.02047	−0.476767	−0.238384	+	0.971171i	$0.576618\pi$
$284$	0.247277	0.0146732
$285$	0	0
$286$	−18.2374	−1.07840
$287$	13.4617	0.794618
$288$	0	0
$289$	13.3225	0.783676
$290$	−1.48119	−0.0869787
$291$	0	0
$292$	−2.95651	−0.173017
$293$	23.3054	1.36151	0.680757	−	0.732510i	$-0.261651\pi$
$293$	23.3054	1.36151	0.680757	+	0.732510i	$0.261651\pi$
$294$	0	0
$295$	13.2750	0.772903
$296$	25.4314	1.47817
$297$	0	0
$298$	−4.11142	−0.238168
$299$	−5.46168	−0.315857
$300$	0	0
$301$	−0.0361968	−0.00208635
$302$	2.66433	0.153315
$303$	0	0
$304$	13.8945	0.796902
$305$	−8.88717	−0.508878
$306$	0	0
$307$	−6.73084	−0.384149	−0.192075	−	0.981380i	$-0.561522\pi$
$307$	−6.73084	−0.384149	−0.192075	+	0.981380i	$0.561522\pi$
$308$	−0.962389	−0.0548372
$309$	0	0
$310$	7.11871	0.404316
$311$	22.0567	1.25072	0.625359	−	0.780337i	$-0.284952\pi$
$311$	22.0567	1.25072	0.625359	+	0.780337i	$0.284952\pi$
$312$	0	0
$313$	5.03761	0.284743	0.142371	−	0.989813i	$-0.454527\pi$
$313$	5.03761	0.284743	0.142371	+	0.989813i	$0.454527\pi$
$314$	5.58181	0.315000
$315$	0	0
$316$	−0.956509	−0.0538078
$317$	−34.2941	−1.92615	−0.963074	−	0.269237i	$-0.913229\pi$
$317$	−34.2941	−1.92615	−0.963074	+	0.269237i	$0.913229\pi$
$318$	0	0
$319$	−4.15633	−0.232710
$320$	−7.08110	−0.395846
$321$	0	0
$322$	−3.26045	−0.181698
$323$	17.5877	0.978605
$324$	0	0
$325$	2.96239	0.164324
$326$	2.43278	0.134739
$327$	0	0
$328$	−30.1622	−1.66543
$329$	−5.73813	−0.316354
$330$	0	0
$331$	−34.8324	−1.91456	−0.957281	−	0.289159i	$-0.906625\pi$
$331$	−34.8324	−1.91456	−0.957281	+	0.289159i	$0.906625\pi$
$332$	−0.856849	−0.0470257
$333$	0	0
$334$	−11.9697	−0.654952
$335$	−5.84367	−0.319274
$336$	0	0
$337$	−17.6326	−0.960509	−0.480254	−	0.877129i	$-0.659456\pi$
$337$	−17.6326	−0.960509	−0.480254	+	0.877129i	$0.659456\pi$
$338$	−6.25694	−0.340333
$339$	0	0
$340$	1.06793	0.0579166
$341$	19.9756	1.08174
$342$	0	0
$343$	−15.0132	−0.810635
$344$	0.0811024	0.00437275
$345$	0	0
$346$	11.4617	0.616184
$347$	3.11871	0.167421	0.0837107	−	0.996490i	$-0.473323\pi$
$347$	3.11871	0.167421	0.0837107	+	0.996490i	$0.473323\pi$
$348$	0	0
$349$	−13.0738	−0.699825	−0.349912	−	0.936782i	$-0.613789\pi$
$349$	−13.0738	−0.699825	−0.349912	+	0.936782i	$0.613789\pi$
$350$	1.76845	0.0945277
$351$	0	0
$352$	4.54420	0.242207
$353$	−5.19982	−0.276758	−0.138379	−	0.990379i	$-0.544189\pi$
$353$	−5.19982	−0.276758	−0.138379	+	0.990379i	$0.544189\pi$
$354$	0	0
$355$	−1.27504	−0.0676720
$356$	0.700523	0.0371277
$357$	0	0
$358$	31.6991	1.67535
$359$	−30.4182	−1.60541	−0.802705	−	0.596376i	$-0.796607\pi$
$359$	−30.4182	−1.60541	−0.802705	+	0.596376i	$0.796607\pi$
$360$	0	0
$361$	−8.79877	−0.463093
$362$	22.6253	1.18916
$363$	0	0
$364$	0.685935	0.0359528
$365$	15.2447	0.797945
$366$	0	0
$367$	20.6556	1.07821	0.539107	−	0.842237i	$-0.318762\pi$
$367$	20.6556	1.07821	0.539107	+	0.842237i	$0.318762\pi$
$368$	8.02047	0.418096
$369$	0	0
$370$	14.0811	0.732042
$371$	1.61213	0.0836975
$372$	0	0
$373$	11.0884	0.574135	0.287068	−	0.957910i	$-0.407320\pi$
$373$	11.0884	0.574135	0.287068	+	0.957910i	$0.407320\pi$
$374$	33.9003	1.75294
$375$	0	0
$376$	12.8568	0.663041
$377$	2.96239	0.152571
$378$	0	0
$379$	10.0811	0.517831	0.258916	−	0.965900i	$-0.416635\pi$
$379$	10.0811	0.517831	0.258916	+	0.965900i	$0.416635\pi$
$380$	0.619421	0.0317756
$381$	0	0
$382$	−4.91748	−0.251600
$383$	−16.3576	−0.835832	−0.417916	−	0.908486i	$-0.637239\pi$
$383$	−16.3576	−0.835832	−0.417916	+	0.908486i	$0.637239\pi$
$384$	0	0
$385$	4.96239	0.252907
$386$	−7.23013	−0.368004
$387$	0	0
$388$	−0.267745	−0.0135927
$389$	−31.9003	−1.61741	−0.808706	−	0.588213i	$-0.799831\pi$
$389$	−31.9003	−1.61741	−0.808706	+	0.588213i	$0.799831\pi$
$390$	0	0
$391$	10.1524	0.513427
$392$	14.9126	0.753198
$393$	0	0
$394$	35.9003	1.80863
$395$	4.93207	0.248159
$396$	0	0
$397$	2.98683	0.149905	0.0749523	−	0.997187i	$-0.476120\pi$
$397$	2.98683	0.149905	0.0749523	+	0.997187i	$0.476120\pi$
$398$	24.8119	1.24371
$399$	0	0
$400$	−4.35026	−0.217513
$401$	21.9756	1.09741	0.548704	−	0.836017i	$-0.315122\pi$
$401$	21.9756	1.09741	0.548704	+	0.836017i	$0.315122\pi$
$402$	0	0
$403$	−14.2374	−0.709217
$404$	2.52373	0.125560
$405$	0	0
$406$	1.76845	0.0877668
$407$	39.5125	1.95856
$408$	0	0
$409$	−22.4387	−1.10952	−0.554760	−	0.832010i	$-0.687190\pi$
$409$	−22.4387	−1.10952	−0.554760	+	0.832010i	$0.687190\pi$
$410$	−16.7005	−0.824780
$411$	0	0
$412$	1.03173	0.0508298
$413$	−15.8496	−0.779906
$414$	0	0
$415$	4.41819	0.216880
$416$	−3.23884	−0.158797
$417$	0	0
$418$	19.6629	0.961744
$419$	−10.3634	−0.506287	−0.253143	−	0.967429i	$-0.581464\pi$
$419$	−10.3634	−0.506287	−0.253143	+	0.967429i	$0.581464\pi$
$420$	0	0
$421$	34.0362	1.65882	0.829411	−	0.558638i	$-0.188676\pi$
$421$	34.0362	1.65882	0.829411	+	0.558638i	$0.188676\pi$
$422$	−37.4821	−1.82460
$423$	0	0
$424$	−3.61213	−0.175420
$425$	−5.50659	−0.267109
$426$	0	0
$427$	10.6107	0.513488
$428$	2.68006	0.129545
$429$	0	0
$430$	0.0449056	0.00216554
$431$	−25.7743	−1.24151	−0.620753	−	0.784006i	$-0.713173\pi$
$431$	−25.7743	−1.24151	−0.620753	+	0.784006i	$0.713173\pi$
$432$	0	0
$433$	2.18076	0.104801	0.0524004	−	0.998626i	$-0.483313\pi$
$433$	2.18076	0.104801	0.0524004	+	0.998626i	$0.483313\pi$
$434$	−8.49929	−0.407979
$435$	0	0
$436$	−0.363436	−0.0174054
$437$	5.88858	0.281689
$438$	0	0
$439$	−35.5125	−1.69492	−0.847459	−	0.530861i	$-0.821869\pi$
$439$	−35.5125	−1.69492	−0.847459	+	0.530861i	$0.821869\pi$
$440$	−11.1187	−0.530064
$441$	0	0
$442$	−24.1622	−1.14928
$443$	4.34297	0.206341	0.103170	−	0.994664i	$-0.467101\pi$
$443$	4.34297	0.206341	0.103170	+	0.994664i	$0.467101\pi$
$444$	0	0
$445$	−3.61213	−0.171231
$446$	26.2071	1.24094
$447$	0	0
$448$	8.45439	0.399432
$449$	−31.3357	−1.47882	−0.739411	−	0.673254i	$-0.764896\pi$
$449$	−31.3357	−1.47882	−0.739411	+	0.673254i	$0.764896\pi$
$450$	0	0
$451$	−46.8627	−2.20668
$452$	2.28233	0.107352
$453$	0	0
$454$	−39.7586	−1.86596
$455$	−3.53690	−0.165813
$456$	0	0
$457$	34.3488	1.60677	0.803386	−	0.595459i	$-0.203030\pi$
$457$	34.3488	1.60677	0.803386	+	0.595459i	$0.203030\pi$
$458$	−25.5125	−1.19212
$459$	0	0
$460$	0.357556	0.0166711
$461$	−11.8641	−0.552568	−0.276284	−	0.961076i	$-0.589103\pi$
$461$	−11.8641	−0.552568	−0.276284	+	0.961076i	$0.589103\pi$
$462$	0	0
$463$	40.4953	1.88198	0.940989	−	0.338438i	$-0.109899\pi$
$463$	40.4953	1.88198	0.940989	+	0.338438i	$0.109899\pi$
$464$	−4.35026	−0.201956
$465$	0	0
$466$	13.4401	0.622599
$467$	30.2071	1.39782	0.698909	−	0.715210i	$-0.253669\pi$
$467$	30.2071	1.39782	0.698909	+	0.715210i	$0.253669\pi$
$468$	0	0
$469$	6.97698	0.322167
$470$	7.11871	0.328362
$471$	0	0
$472$	35.5125	1.63459
$473$	0.126008	0.00579385
$474$	0	0
$475$	−3.19394	−0.146548
$476$	−1.27504	−0.0584413
$477$	0	0
$478$	−30.3634	−1.38879
$479$	−0.0547547	−0.00250181	−0.00125090	−	0.999999i	$-0.500398\pi$
$479$	−0.0547547	−0.00250181	−0.00125090	+	0.999999i	$0.500398\pi$
$480$	0	0
$481$	−28.1622	−1.28409
$482$	8.11142	0.369465
$483$	0	0
$484$	1.21696	0.0553163
$485$	1.38058	0.0626889
$486$	0	0
$487$	−0.881286	−0.0399349	−0.0199674	−	0.999801i	$-0.506356\pi$
$487$	−0.881286	−0.0399349	−0.0199674	+	0.999801i	$0.506356\pi$
$488$	−23.7743	−1.07621
$489$	0	0
$490$	8.25694	0.373011
$491$	−41.0698	−1.85346	−0.926728	−	0.375733i	$-0.877391\pi$
$491$	−41.0698	−1.85346	−0.926728	+	0.375733i	$0.877391\pi$
$492$	0	0
$493$	−5.50659	−0.248004
$494$	−14.0146	−0.630546
$495$	0	0
$496$	20.9076	0.938780
$497$	1.52232	0.0682852
$498$	0	0
$499$	12.3733	0.553904	0.276952	−	0.960884i	$-0.410676\pi$
$499$	12.3733	0.553904	0.276952	+	0.960884i	$0.410676\pi$
$500$	−0.193937	−0.00867311
$501$	0	0
$502$	43.8700	1.95801
$503$	2.26774	0.101114	0.0505569	−	0.998721i	$-0.483900\pi$
$503$	2.26774	0.101114	0.0505569	+	0.998721i	$0.483900\pi$
$504$	0	0
$505$	−13.0132	−0.579079
$506$	11.3503	0.504581
$507$	0	0
$508$	2.76704	0.122767
$509$	10.9018	0.483212	0.241606	−	0.970374i	$-0.422326\pi$
$509$	10.9018	0.483212	0.241606	+	0.970374i	$0.422326\pi$
$510$	0	0
$511$	−18.2012	−0.805175
$512$	−18.5188	−0.818423
$513$	0	0
$514$	−26.1622	−1.15397
$515$	−5.31994	−0.234425
$516$	0	0
$517$	19.9756	0.878524
$518$	−16.8119	−0.738674
$519$	0	0
$520$	7.92478	0.347524
$521$	4.72496	0.207004	0.103502	−	0.994629i	$-0.466995\pi$
$521$	4.72496	0.207004	0.103502	+	0.994629i	$0.466995\pi$
$522$	0	0
$523$	−1.06793	−0.0466973	−0.0233486	−	0.999727i	$-0.507433\pi$
$523$	−1.06793	−0.0466973	−0.0233486	+	0.999727i	$0.507433\pi$
$524$	−1.14315	−0.0499388
$525$	0	0
$526$	−40.5198	−1.76675
$527$	26.4650	1.15283
$528$	0	0
$529$	−19.6009	−0.852211
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−0.739549	−0.0320635
$533$	33.4010	1.44676
$534$	0	0
$535$	−13.8192	−0.597458
$536$	−15.6326	−0.675225
$537$	0	0
$538$	−15.5515	−0.670472
$539$	23.1695	0.997981
$540$	0	0
$541$	−7.46168	−0.320803	−0.160401	−	0.987052i	$-0.551279\pi$
$541$	−7.46168	−0.320803	−0.160401	+	0.987052i	$0.551279\pi$
$542$	−14.2461	−0.611924
$543$	0	0
$544$	6.02047	0.258125
$545$	1.87399	0.0802730
$546$	0	0
$547$	−38.9683	−1.66616	−0.833081	−	0.553150i	$-0.813425\pi$
$547$	−38.9683	−1.66616	−0.833081	+	0.553150i	$0.813425\pi$
$548$	−3.54561	−0.151461
$549$	0	0
$550$	−6.15633	−0.262507
$551$	−3.19394	−0.136066
$552$	0	0
$553$	−5.88858	−0.250408
$554$	19.7743	0.840131
$555$	0	0
$556$	−2.23743	−0.0948881
$557$	22.9986	0.974481	0.487241	−	0.873268i	$-0.338003\pi$
$557$	22.9986	0.974481	0.487241	+	0.873268i	$0.338003\pi$
$558$	0	0
$559$	−0.0898112	−0.00379861
$560$	5.19394	0.219484
$561$	0	0
$562$	−30.2520	−1.27610
$563$	−11.6688	−0.491781	−0.245890	−	0.969298i	$-0.579080\pi$
$563$	−11.6688	−0.491781	−0.245890	+	0.969298i	$0.579080\pi$
$564$	0	0
$565$	−11.7685	−0.495102
$566$	−11.8799	−0.499348
$567$	0	0
$568$	−3.41090	−0.143118
$569$	−11.3357	−0.475216	−0.237608	−	0.971361i	$-0.576363\pi$
$569$	−11.3357	−0.475216	−0.237608	+	0.971361i	$0.576363\pi$
$570$	0	0
$571$	27.1754	1.13725	0.568627	−	0.822595i	$-0.307475\pi$
$571$	27.1754	1.13725	0.568627	+	0.822595i	$0.307475\pi$
$572$	−2.38787	−0.0998420
$573$	0	0
$574$	19.9394	0.832253
$575$	−1.84367	−0.0768866
$576$	0	0
$577$	22.5950	0.940641	0.470321	−	0.882496i	$-0.344138\pi$
$577$	22.5950	0.940641	0.470321	+	0.882496i	$0.344138\pi$
$578$	19.7332	0.820793
$579$	0	0
$580$	−0.193937	−0.00805278
$581$	−5.27504	−0.218845
$582$	0	0
$583$	−5.61213	−0.232431
$584$	40.7816	1.68756
$585$	0	0
$586$	34.5198	1.42600
$587$	−9.31994	−0.384675	−0.192338	−	0.981329i	$-0.561607\pi$
$587$	−9.31994	−0.384675	−0.192338	+	0.981329i	$0.561607\pi$
$588$	0	0
$589$	15.3503	0.632497
$590$	19.6629	0.809509
$591$	0	0
$592$	41.3561	1.69973
$593$	15.1246	0.621093	0.310546	−	0.950558i	$-0.399488\pi$
$593$	15.1246	0.621093	0.310546	+	0.950558i	$0.399488\pi$
$594$	0	0
$595$	6.57452	0.269529
$596$	−0.538319	−0.0220504
$597$	0	0
$598$	−8.08981	−0.330817
$599$	−4.09569	−0.167345	−0.0836727	−	0.996493i	$-0.526665\pi$
$599$	−4.09569	−0.167345	−0.0836727	+	0.996493i	$0.526665\pi$
$600$	0	0
$601$	22.2276	0.906682	0.453341	−	0.891337i	$-0.350232\pi$
$601$	22.2276	0.906682	0.453341	+	0.891337i	$0.350232\pi$
$602$	−0.0536145	−0.00218516
$603$	0	0
$604$	0.348847	0.0141944
$605$	−6.27504	−0.255117
$606$	0	0
$607$	−48.2941	−1.96020	−0.980098	−	0.198512i	$-0.936389\pi$
$607$	−48.2941	−1.96020	−0.980098	+	0.198512i	$0.936389\pi$
$608$	3.49200	0.141619
$609$	0	0
$610$	−13.1636	−0.532979
$611$	−14.2374	−0.575985
$612$	0	0
$613$	−9.74798	−0.393717	−0.196859	−	0.980432i	$-0.563074\pi$
$613$	−9.74798	−0.393717	−0.196859	+	0.980432i	$0.563074\pi$
$614$	−9.96968	−0.402344
$615$	0	0
$616$	13.2750	0.534867
$617$	−18.2170	−0.733387	−0.366694	−	0.930342i	$-0.619510\pi$
$617$	−18.2170	−0.733387	−0.366694	+	0.930342i	$0.619510\pi$
$618$	0	0
$619$	25.0943	1.00862	0.504312	−	0.863521i	$-0.331746\pi$
$619$	25.0943	1.00862	0.504312	+	0.863521i	$0.331746\pi$
$620$	0.932071	0.0374329
$621$	0	0
$622$	32.6702	1.30996
$623$	4.31265	0.172783
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.46168	0.298229
$627$	0	0
$628$	0.730841	0.0291637
$629$	52.3488	2.08729
$630$	0	0
$631$	21.4617	0.854376	0.427188	−	0.904163i	$-0.359504\pi$
$631$	21.4617	0.854376	0.427188	+	0.904163i	$0.359504\pi$
$632$	13.1939	0.524827
$633$	0	0
$634$	−50.7962	−2.01738
$635$	−14.2677	−0.566198
$636$	0	0
$637$	−16.5139	−0.654304
$638$	−6.15633	−0.243731
$639$	0	0
$640$	−12.6751	−0.501029
$641$	−3.17347	−0.125344	−0.0626722	−	0.998034i	$-0.519962\pi$
$641$	−3.17347	−0.125344	−0.0626722	+	0.998034i	$0.519962\pi$
$642$	0	0
$643$	−2.74069	−0.108082	−0.0540411	−	0.998539i	$-0.517210\pi$
$643$	−2.74069	−0.108082	−0.0540411	+	0.998539i	$0.517210\pi$
$644$	−0.426899	−0.0168222
$645$	0	0
$646$	26.0508	1.02495
$647$	−6.34297	−0.249368	−0.124684	−	0.992197i	$-0.539792\pi$
$647$	−6.34297	−0.249368	−0.124684	+	0.992197i	$0.539792\pi$
$648$	0	0
$649$	55.1754	2.16582
$650$	4.38787	0.172107
$651$	0	0
$652$	0.318530	0.0124746
$653$	−4.08110	−0.159706	−0.0798529	−	0.996807i	$-0.525445\pi$
$653$	−4.08110	−0.159706	−0.0798529	+	0.996807i	$0.525445\pi$
$654$	0	0
$655$	5.89446	0.230316
$656$	−49.0494	−1.91506
$657$	0	0
$658$	−8.49929	−0.331337
$659$	−9.58181	−0.373254	−0.186627	−	0.982431i	$-0.559756\pi$
$659$	−9.58181	−0.373254	−0.186627	+	0.982431i	$0.559756\pi$
$660$	0	0
$661$	−27.5271	−1.07068	−0.535339	−	0.844637i	$-0.679816\pi$
$661$	−27.5271	−1.07068	−0.535339	+	0.844637i	$0.679816\pi$
$662$	−51.5936	−2.00524
$663$	0	0
$664$	11.8192	0.458675
$665$	3.81336	0.147876
$666$	0	0
$667$	−1.84367	−0.0713874
$668$	−1.56722	−0.0606376
$669$	0	0
$670$	−8.65562	−0.334396
$671$	−36.9380	−1.42597
$672$	0	0
$673$	3.13727	0.120933	0.0604665	−	0.998170i	$-0.480741\pi$
$673$	3.13727	0.120933	0.0604665	+	0.998170i	$0.480741\pi$
$674$	−26.1173	−1.00600
$675$	0	0
$676$	−0.819237	−0.0315091
$677$	46.2579	1.77784	0.888918	−	0.458067i	$-0.151458\pi$
$677$	46.2579	1.77784	0.888918	+	0.458067i	$0.151458\pi$
$678$	0	0
$679$	−1.64832	−0.0632569
$680$	−14.7308	−0.564902
$681$	0	0
$682$	29.5877	1.13297
$683$	9.01905	0.345104	0.172552	−	0.985000i	$-0.444799\pi$
$683$	9.01905	0.345104	0.172552	+	0.985000i	$0.444799\pi$
$684$	0	0
$685$	18.2823	0.698532
$686$	−22.2374	−0.849029
$687$	0	0
$688$	0.131888	0.00502817
$689$	4.00000	0.152388
$690$	0	0
$691$	−50.0625	−1.90447	−0.952234	−	0.305368i	$-0.901221\pi$
$691$	−50.0625	−1.90447	−0.952234	+	0.305368i	$0.901221\pi$
$692$	1.50071	0.0570483
$693$	0	0
$694$	4.61942	0.175351
$695$	11.5369	0.437620
$696$	0	0
$697$	−62.0870	−2.35171
$698$	−19.3649	−0.732970
$699$	0	0
$700$	0.231548	0.00875169
$701$	−45.3014	−1.71101	−0.855505	−	0.517795i	$-0.826753\pi$
$701$	−45.3014	−1.71101	−0.855505	+	0.517795i	$0.826753\pi$
$702$	0	0
$703$	30.3634	1.14518
$704$	−29.4314	−1.10924
$705$	0	0
$706$	−7.70194	−0.289866
$707$	15.5369	0.584325
$708$	0	0
$709$	−3.27504	−0.122997	−0.0614983	−	0.998107i	$-0.519588\pi$
$709$	−3.27504	−0.122997	−0.0614983	+	0.998107i	$0.519588\pi$
$710$	−1.88858	−0.0708772
$711$	0	0
$712$	−9.66291	−0.362133
$713$	8.86082	0.331840
$714$	0	0
$715$	12.3127	0.460467
$716$	4.15045	0.155109
$717$	0	0
$718$	−45.0553	−1.68145
$719$	−27.7235	−1.03391	−0.516957	−	0.856011i	$-0.672935\pi$
$719$	−27.7235	−1.03391	−0.516957	+	0.856011i	$0.672935\pi$
$720$	0	0
$721$	6.35168	0.236549
$722$	−13.0327	−0.485026
$723$	0	0
$724$	2.96239	0.110096
$725$	1.00000	0.0371391
$726$	0	0
$727$	−26.8930	−0.997408	−0.498704	−	0.866772i	$-0.666190\pi$
$727$	−26.8930	−0.997408	−0.498704	+	0.866772i	$0.666190\pi$
$728$	−9.46168	−0.350673
$729$	0	0
$730$	22.5804	0.835738
$731$	0.166944	0.00617465
$732$	0	0
$733$	3.17935	0.117432	0.0587160	−	0.998275i	$-0.481299\pi$
$733$	3.17935	0.117432	0.0587160	+	0.998275i	$0.481299\pi$
$734$	30.5950	1.12928
$735$	0	0
$736$	2.01573	0.0743007
$737$	−24.2882	−0.894668
$738$	0	0
$739$	−29.7440	−1.09415	−0.547076	−	0.837083i	$-0.684259\pi$
$739$	−29.7440	−1.09415	−0.547076	+	0.837083i	$0.684259\pi$
$740$	1.84367	0.0677748
$741$	0	0
$742$	2.38787	0.0876616
$743$	4.34297	0.159328	0.0796640	−	0.996822i	$-0.474615\pi$
$743$	4.34297	0.159328	0.0796640	+	0.996822i	$0.474615\pi$
$744$	0	0
$745$	2.77575	0.101695
$746$	16.4241	0.601328
$747$	0	0
$748$	4.43866	0.162293
$749$	16.4993	0.602871
$750$	0	0
$751$	22.5804	0.823970	0.411985	−	0.911191i	$-0.364836\pi$
$751$	22.5804	0.823970	0.411985	+	0.911191i	$0.364836\pi$
$752$	20.9076	0.762423
$753$	0	0
$754$	4.38787	0.159797
$755$	−1.79877	−0.0654639
$756$	0	0
$757$	9.88461	0.359262	0.179631	−	0.983734i	$-0.442510\pi$
$757$	9.88461	0.359262	0.179631	+	0.983734i	$0.442510\pi$
$758$	14.9321	0.542357
$759$	0	0
$760$	−8.54420	−0.309931
$761$	−13.6991	−0.496592	−0.248296	−	0.968684i	$-0.579871\pi$
$761$	−13.6991	−0.496592	−0.248296	+	0.968684i	$0.579871\pi$
$762$	0	0
$763$	−2.23743	−0.0810003
$764$	−0.643859	−0.0232940
$765$	0	0
$766$	−24.2287	−0.875419
$767$	−39.3258	−1.41997
$768$	0	0
$769$	25.0132	0.901998	0.450999	−	0.892524i	$-0.351068\pi$
$769$	25.0132	0.901998	0.450999	+	0.892524i	$0.351068\pi$
$770$	7.35026	0.264885
$771$	0	0
$772$	−0.946660	−0.0340710
$773$	35.9062	1.29146	0.645728	−	0.763567i	$-0.276554\pi$
$773$	35.9062	1.29146	0.645728	+	0.763567i	$0.276554\pi$
$774$	0	0
$775$	−4.80606	−0.172639
$776$	3.69323	0.132579
$777$	0	0
$778$	−47.2506	−1.69402
$779$	−36.0118	−1.29026
$780$	0	0
$781$	−5.29948	−0.189630
$782$	15.0376	0.537744
$783$	0	0
$784$	24.2506	0.866093
$785$	−3.76845	−0.134502
$786$	0	0
$787$	50.3839	1.79599	0.897996	−	0.440003i	$-0.145023\pi$
$787$	50.3839	1.79599	0.897996	+	0.440003i	$0.145023\pi$
$788$	4.70052	0.167449
$789$	0	0
$790$	7.30536	0.259913
$791$	14.0508	0.499588
$792$	0	0
$793$	26.3272	0.934908
$794$	4.42407	0.157004
$795$	0	0
$796$	3.24869	0.115147
$797$	5.69323	0.201665	0.100832	−	0.994903i	$-0.467849\pi$
$797$	5.69323	0.201665	0.100832	+	0.994903i	$0.467849\pi$
$798$	0	0
$799$	26.4650	0.936265
$800$	−1.09332	−0.0386547
$801$	0	0
$802$	32.5501	1.14938
$803$	63.3620	2.23600
$804$	0	0
$805$	2.20123	0.0775832
$806$	−21.0884	−0.742807
$807$	0	0
$808$	−34.8119	−1.22468
$809$	7.76257	0.272918	0.136459	−	0.990646i	$-0.456428\pi$
$809$	7.76257	0.272918	0.136459	+	0.990646i	$0.456428\pi$
$810$	0	0
$811$	−26.4894	−0.930170	−0.465085	−	0.885266i	$-0.653976\pi$
$811$	−26.4894	−0.930170	−0.465085	+	0.885266i	$0.653976\pi$
$812$	0.231548	0.00812574
$813$	0	0
$814$	58.5256	2.05132
$815$	−1.64244	−0.0575323
$816$	0	0
$817$	0.0968311	0.00338769
$818$	−33.2360	−1.16207
$819$	0	0
$820$	−2.18664	−0.0763609
$821$	−25.4763	−0.889128	−0.444564	−	0.895747i	$-0.646641\pi$
$821$	−25.4763	−0.889128	−0.444564	+	0.895747i	$0.646641\pi$
$822$	0	0
$823$	−9.22028	−0.321399	−0.160699	−	0.987003i	$-0.551375\pi$
$823$	−9.22028	−0.321399	−0.160699	+	0.987003i	$0.551375\pi$
$824$	−14.2315	−0.495779
$825$	0	0
$826$	−23.4763	−0.816844
$827$	−24.5343	−0.853143	−0.426571	−	0.904454i	$-0.640279\pi$
$827$	−24.5343	−0.853143	−0.426571	+	0.904454i	$0.640279\pi$
$828$	0	0
$829$	0.201231	0.00698903	0.00349452	−	0.999994i	$-0.498888\pi$
$829$	0.201231	0.00698903	0.00349452	+	0.999994i	$0.498888\pi$
$830$	6.54420	0.227152
$831$	0	0
$832$	20.9770	0.727246
$833$	30.6966	1.06357
$834$	0	0
$835$	8.08110	0.279658
$836$	2.57452	0.0890415
$837$	0	0
$838$	−15.3503	−0.530266
$839$	−1.45580	−0.0502599	−0.0251299	−	0.999684i	$-0.508000\pi$
$839$	−1.45580	−0.0502599	−0.0251299	+	0.999684i	$0.508000\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	50.4142	1.73739
$843$	0	0
$844$	−4.90763	−0.168928
$845$	4.22425	0.145319
$846$	0	0
$847$	7.49200	0.257428
$848$	−5.87399	−0.201714
$849$	0	0
$850$	−8.15633	−0.279760
$851$	17.5271	0.600820
$852$	0	0
$853$	43.1793	1.47843	0.739216	−	0.673468i	$-0.235196\pi$
$853$	43.1793	1.47843	0.739216	+	0.673468i	$0.235196\pi$
$854$	15.7165	0.537808
$855$	0	0
$856$	−36.9683	−1.26355
$857$	20.9887	0.716962	0.358481	−	0.933537i	$-0.383295\pi$
$857$	20.9887	0.716962	0.358481	+	0.933537i	$0.383295\pi$
$858$	0	0
$859$	−49.4069	−1.68574	−0.842871	−	0.538115i	$-0.819137\pi$
$859$	−49.4069	−1.68574	−0.842871	+	0.538115i	$0.819137\pi$
$860$	0.00587961	0.000200493 0
$861$	0	0
$862$	−38.1768	−1.30031
$863$	−56.6820	−1.92948	−0.964738	−	0.263211i	$-0.915218\pi$
$863$	−56.6820	−1.92948	−0.964738	+	0.263211i	$0.915218\pi$
$864$	0	0
$865$	−7.73813	−0.263104
$866$	3.23013	0.109764
$867$	0	0
$868$	−1.11283	−0.0377721
$869$	20.4993	0.695391
$870$	0	0
$871$	17.3112	0.586569
$872$	5.01317	0.169767
$873$	0	0
$874$	8.72213	0.295031
$875$	−1.19394	−0.0403624
$876$	0	0
$877$	−13.1998	−0.445726	−0.222863	−	0.974850i	$-0.571540\pi$
$877$	−13.1998	−0.445726	−0.222863	+	0.974850i	$0.571540\pi$
$878$	−52.6009	−1.77519
$879$	0	0
$880$	−18.0811	−0.609514
$881$	−6.37802	−0.214881	−0.107441	−	0.994212i	$-0.534266\pi$
$881$	−6.37802	−0.214881	−0.107441	+	0.994212i	$0.534266\pi$
$882$	0	0
$883$	48.6213	1.63624	0.818119	−	0.575049i	$-0.195017\pi$
$883$	48.6213	1.63624	0.818119	+	0.575049i	$0.195017\pi$
$884$	−3.16362	−0.106404
$885$	0	0
$886$	6.43278	0.216113
$887$	15.0317	0.504716	0.252358	−	0.967634i	$-0.418794\pi$
$887$	15.0317	0.504716	0.252358	+	0.967634i	$0.418794\pi$
$888$	0	0
$889$	17.0348	0.571328
$890$	−5.35026	−0.179341
$891$	0	0
$892$	3.43136	0.114891
$893$	15.3503	0.513677
$894$	0	0
$895$	−21.4010	−0.715358
$896$	15.1333	0.505568
$897$	0	0
$898$	−46.4142	−1.54886
$899$	−4.80606	−0.160291
$900$	0	0
$901$	−7.43533	−0.247707
$902$	−69.4128	−2.31119
$903$	0	0
$904$	−31.4821	−1.04708
$905$	−15.2750	−0.507759
$906$	0	0
$907$	−0.342968	−0.0113880	−0.00569402	−	0.999984i	$-0.501812\pi$
$907$	−0.342968	−0.0113880	−0.00569402	+	0.999984i	$0.501812\pi$
$908$	−5.20570	−0.172757
$909$	0	0
$910$	−5.23884	−0.173666
$911$	−20.9076	−0.692701	−0.346350	−	0.938105i	$-0.612579\pi$
$911$	−20.9076	−0.692701	−0.346350	+	0.938105i	$0.612579\pi$
$912$	0	0
$913$	18.3634	0.607741
$914$	50.8773	1.68287
$915$	0	0
$916$	−3.34041	−0.110370
$917$	−7.03761	−0.232402
$918$	0	0
$919$	−1.90034	−0.0626864	−0.0313432	−	0.999509i	$-0.509978\pi$
$919$	−1.90034	−0.0626864	−0.0313432	+	0.999509i	$0.509978\pi$
$920$	−4.93207	−0.162606
$921$	0	0
$922$	−17.5731	−0.578739
$923$	3.77716	0.124327
$924$	0	0
$925$	−9.50659	−0.312575
$926$	59.9814	1.97111
$927$	0	0
$928$	−1.09332	−0.0358900
$929$	−39.3522	−1.29110	−0.645551	−	0.763717i	$-0.723372\pi$
$929$	−39.3522	−1.29110	−0.645551	+	0.763717i	$0.723372\pi$
$930$	0	0
$931$	17.8046	0.583524
$932$	1.75974	0.0576423
$933$	0	0
$934$	44.7426	1.46402
$935$	−22.8872	−0.748490
$936$	0	0
$937$	−6.37802	−0.208361	−0.104180	−	0.994558i	$-0.533222\pi$
$937$	−6.37802	−0.208361	−0.104180	+	0.994558i	$0.533222\pi$
$938$	10.3343	0.337426
$939$	0	0
$940$	0.932071	0.0304008
$941$	26.6253	0.867960	0.433980	−	0.900923i	$-0.357109\pi$
$941$	26.6253	0.867960	0.433980	+	0.900923i	$0.357109\pi$
$942$	0	0
$943$	−20.7875	−0.676934
$944$	57.7499	1.87960
$945$	0	0
$946$	0.186642	0.00606827
$947$	12.2823	0.399122	0.199561	−	0.979885i	$-0.436048\pi$
$947$	12.2823	0.399122	0.199561	+	0.979885i	$0.436048\pi$
$948$	0	0
$949$	−45.1608	−1.46598
$950$	−4.73084	−0.153489
$951$	0	0
$952$	17.5877	0.570020
$953$	−0.821792	−0.0266205	−0.0133102	−	0.999911i	$-0.504237\pi$
$953$	−0.821792	−0.0266205	−0.0133102	+	0.999911i	$0.504237\pi$
$954$	0	0
$955$	3.31994	0.107431
$956$	−3.97556	−0.128579
$957$	0	0
$958$	−0.0811024	−0.00262030
$959$	−21.8279	−0.704861
$960$	0	0
$961$	−7.90175	−0.254895
$962$	−41.7137	−1.34490
$963$	0	0
$964$	1.06205	0.0342063
$965$	4.88129	0.157134
$966$	0	0
$967$	−37.4314	−1.20371	−0.601856	−	0.798605i	$-0.705572\pi$
$967$	−37.4314	−1.20371	−0.601856	+	0.798605i	$0.705572\pi$
$968$	−16.7866	−0.539540
$969$	0	0
$970$	2.04491	0.0656580
$971$	8.71625	0.279718	0.139859	−	0.990171i	$-0.455335\pi$
$971$	8.71625	0.279718	0.139859	+	0.990171i	$0.455335\pi$
$972$	0	0
$973$	−13.7743	−0.441585
$974$	−1.30536	−0.0418263
$975$	0	0
$976$	−38.6615	−1.23752
$977$	33.7645	1.08022	0.540111	−	0.841594i	$-0.318382\pi$
$977$	33.7645	1.08022	0.540111	+	0.841594i	$0.318382\pi$
$978$	0	0
$979$	−15.0132	−0.479823
$980$	1.08110	0.0345345
$981$	0	0
$982$	−60.8324	−1.94124
$983$	−43.6082	−1.39088	−0.695442	−	0.718582i	$-0.744791\pi$
$983$	−43.6082	−1.39088	−0.695442	+	0.718582i	$0.744791\pi$
$984$	0	0
$985$	−24.2374	−0.772269
$986$	−8.15633	−0.259750
$987$	0	0
$988$	−1.83497	−0.0583780
$989$	0.0558950	0.00177736
$990$	0	0
$991$	−52.9741	−1.68278	−0.841390	−	0.540429i	$-0.818262\pi$
$991$	−52.9741	−1.68278	−0.841390	+	0.540429i	$0.818262\pi$
$992$	5.25457	0.166833
$993$	0	0
$994$	2.25485	0.0715193
$995$	−16.7513	−0.531052
$996$	0	0
$997$	13.6326	0.431749	0.215874	−	0.976421i	$-0.430740\pi$
$997$	13.6326	0.431749	0.215874	+	0.976421i	$0.430740\pi$
$998$	18.3272	0.580139
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.p.1.3		3
3.2	odd	2		145.2.a.c.1.1	✓	3
5.4	even	2		6525.2.a.be.1.1		3
12.11	even	2		2320.2.a.n.1.2		3
15.2	even	4		725.2.b.e.349.2		6
15.8	even	4		725.2.b.e.349.5		6
15.14	odd	2		725.2.a.e.1.3		3
21.20	even	2		7105.2.a.o.1.1		3
24.5	odd	2		9280.2.a.bj.1.2		3
24.11	even	2		9280.2.a.br.1.2		3
87.86	odd	2		4205.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.c.1.1	✓	3	3.2	odd	2
725.2.a.e.1.3		3	15.14	odd	2
725.2.b.e.349.2		6	15.2	even	4
725.2.b.e.349.5		6	15.8	even	4
1305.2.a.p.1.3		3	1.1	even	1	trivial
2320.2.a.n.1.2		3	12.11	even	2
4205.2.a.f.1.3		3	87.86	odd	2
6525.2.a.be.1.1		3	5.4	even	2
7105.2.a.o.1.1		3	21.20	even	2
9280.2.a.bj.1.2		3	24.5	odd	2
9280.2.a.br.1.2		3	24.11	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{7} -2.67513 q^{8} -1.48119 q^{10} -4.15633 q^{11} +2.96239 q^{13} +1.76845 q^{14} -4.35026 q^{16} -5.50659 q^{17} -3.19394 q^{19} -0.193937 q^{20} -6.15633 q^{22} -1.84367 q^{23} +1.00000 q^{25} +4.38787 q^{26} +0.231548 q^{28} +1.00000 q^{29} -4.80606 q^{31} -1.09332 q^{32} -8.15633 q^{34} -1.19394 q^{35} -9.50659 q^{37} -4.73084 q^{38} +2.67513 q^{40} +11.2750 q^{41} -0.0303172 q^{43} -0.806063 q^{44} -2.73084 q^{46} -4.80606 q^{47} -5.57452 q^{49} +1.48119 q^{50} +0.574515 q^{52} +1.35026 q^{53} +4.15633 q^{55} -3.19394 q^{56} +1.48119 q^{58} -13.2750 q^{59} +8.88717 q^{61} -7.11871 q^{62} +7.08110 q^{64} -2.96239 q^{65} +5.84367 q^{67} -1.06793 q^{68} -1.76845 q^{70} +1.27504 q^{71} -15.2447 q^{73} -14.0811 q^{74} -0.619421 q^{76} -4.96239 q^{77} -4.93207 q^{79} +4.35026 q^{80} +16.7005 q^{82} -4.41819 q^{83} +5.50659 q^{85} -0.0449056 q^{86} +11.1187 q^{88} +3.61213 q^{89} +3.53690 q^{91} -0.357556 q^{92} -7.11871 q^{94} +3.19394 q^{95} -1.38058 q^{97} -8.25694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 3 q^{5} + 4 q^{7} - 3 q^{8} + q^{10} - 2 q^{11} - 2 q^{13} - 6 q^{14} - 3 q^{16} + 4 q^{17} - 10 q^{19} - q^{20} - 8 q^{22} - 16 q^{23} + 3 q^{25} + 14 q^{26} + 12 q^{28} + 3 q^{29}+ \cdots - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 3 * q^5 + 4 * q^7 - 3 * q^8 + q^10 - 2 * q^11 - 2 * q^13 - 6 * q^14 - 3 * q^16 + 4 * q^17 - 10 * q^19 - q^20 - 8 * q^22 - 16 * q^23 + 3 * q^25 + 14 * q^26 + 12 * q^28 + 3 * q^29 - 14 * q^31 + 3 * q^32 - 14 * q^34 - 4 * q^35 - 8 * q^37 + 8 * q^38 + 3 * q^40 + 2 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 5 * q^49 - q^50 - 10 * q^52 - 6 * q^53 + 2 * q^55 - 10 * q^56 - q^58 - 8 * q^59 - 6 * q^61 - 11 * q^64 + 2 * q^65 + 28 * q^67 - 12 * q^68 + 6 * q^70 - 28 * q^71 - 16 * q^73 - 10 * q^74 - 14 * q^76 - 4 * q^77 - 6 * q^79 + 3 * q^80 + 30 * q^82 - 12 * q^83 - 4 * q^85 - 24 * q^86 + 12 * q^88 + 10 * q^89 - 12 * q^91 - 4 * q^92 + 10 * q^95 + 8 * q^97 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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