LMFDB - Embedded newform 7605.2.a.cx.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 18 x^{10} + 16 x^{9} + 118 x^{8} - 90 x^{7} - 339 x^{6} + 212 x^{5} + 388 x^{4} + \cdots + 7 $ x^12 - x^11 - 18x^10 + 16x^9 + 118x^8 - 90x^7 - 339x^6 + 212x^5 + 388x^4 - 183x^3 - 91x^2 + 21x + 7
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			1.12
Root			$2.56140$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56140 q^{2} +4.56078 q^{4} -1.00000 q^{5} -1.44606 q^{7} +6.55920 q^{8} +O(q^{10})$	$q+2.56140 q^{2} +4.56078 q^{4} -1.00000 q^{5} -1.44606 q^{7} +6.55920 q^{8} -2.56140 q^{10} -5.29280 q^{11} -3.70395 q^{14} +7.67918 q^{16} +2.51555 q^{17} +7.66639 q^{19} -4.56078 q^{20} -13.5570 q^{22} +0.944022 q^{23} +1.00000 q^{25} -6.59518 q^{28} +7.41501 q^{29} +6.76445 q^{31} +6.55107 q^{32} +6.44334 q^{34} +1.44606 q^{35} -0.0748598 q^{37} +19.6367 q^{38} -6.55920 q^{40} +10.5018 q^{41} -5.69766 q^{43} -24.1393 q^{44} +2.41802 q^{46} -2.33083 q^{47} -4.90890 q^{49} +2.56140 q^{50} +12.2474 q^{53} +5.29280 q^{55} -9.48501 q^{56} +18.9928 q^{58} -1.85841 q^{59} -2.05945 q^{61} +17.3265 q^{62} +1.42157 q^{64} -13.1493 q^{67} +11.4729 q^{68} +3.70395 q^{70} +5.93919 q^{71} -7.22147 q^{73} -0.191746 q^{74} +34.9648 q^{76} +7.65372 q^{77} +11.5410 q^{79} -7.67918 q^{80} +26.8994 q^{82} +7.84386 q^{83} -2.51555 q^{85} -14.5940 q^{86} -34.7165 q^{88} +10.0293 q^{89} +4.30548 q^{92} -5.97020 q^{94} -7.66639 q^{95} +4.27112 q^{97} -12.5737 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8}+O(q^{10}) $ 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8	$ 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8} - q^{10} - 6 q^{11} + 18 q^{14} + 11 q^{16} + 14 q^{17} - 13 q^{20} - 12 q^{22} + 28 q^{23} + 12 q^{25} - 26 q^{28} + 24 q^{29} - 2 q^{31} + 7 q^{32} - 6 q^{34} + 12 q^{35} - 8 q^{37} + 8 q^{38} - 3 q^{40} - 4 q^{41} + 4 q^{43} - 30 q^{44} + 9 q^{46} + 22 q^{47} + 8 q^{49} + q^{50} + 36 q^{53} + 6 q^{55} + 32 q^{56} + 8 q^{58} - 28 q^{59} - 38 q^{61} + 26 q^{62} - 31 q^{64} - 22 q^{67} + 46 q^{68} - 18 q^{70} + 24 q^{71} - 12 q^{73} + 26 q^{74} + 17 q^{76} + 28 q^{77} + 20 q^{79} - 11 q^{80} - 36 q^{82} - 26 q^{83} - 14 q^{85} + 74 q^{86} - 34 q^{88} - 2 q^{89} + 86 q^{92} - 16 q^{94} - 26 q^{97} - 49 q^{98}+O(q^{100}) $ 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8 - q^10 - 6 * q^11 + 18 * q^14 + 11 * q^16 + 14 * q^17 - 13 * q^20 - 12 * q^22 + 28 * q^23 + 12 * q^25 - 26 * q^28 + 24 * q^29 - 2 * q^31 + 7 * q^32 - 6 * q^34 + 12 * q^35 - 8 * q^37 + 8 * q^38 - 3 * q^40 - 4 * q^41 + 4 * q^43 - 30 * q^44 + 9 * q^46 + 22 * q^47 + 8 * q^49 + q^50 + 36 * q^53 + 6 * q^55 + 32 * q^56 + 8 * q^58 - 28 * q^59 - 38 * q^61 + 26 * q^62 - 31 * q^64 - 22 * q^67 + 46 * q^68 - 18 * q^70 + 24 * q^71 - 12 * q^73 + 26 * q^74 + 17 * q^76 + 28 * q^77 + 20 * q^79 - 11 * q^80 - 36 * q^82 - 26 * q^83 - 14 * q^85 + 74 * q^86 - 34 * q^88 - 2 * q^89 + 86 * q^92 - 16 * q^94 - 26 * q^97 - 49 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.56140	1.81119	0.905593	−	0.424149i	$-0.139427\pi$
$2$	2.56140	1.81119	0.905593	+	0.424149i	$0.139427\pi$
$3$	0	0
$3$	0	0
$4$	4.56078	2.28039
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.44606	−0.546560	−0.273280	−	0.961934i	$-0.588109\pi$
$7$	−1.44606	−0.546560	−0.273280	+	0.961934i	$0.588109\pi$
$8$	6.55920	2.31903
$9$	0	0
$10$	−2.56140	−0.809987
$11$	−5.29280	−1.59584	−0.797920	−	0.602763i	$-0.794066\pi$
$11$	−5.29280	−1.59584	−0.797920	+	0.602763i	$0.794066\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.70395	−0.989922
$15$	0	0
$16$	7.67918	1.91979
$17$	2.51555	0.610111	0.305055	−	0.952335i	$-0.401325\pi$
$17$	2.51555	0.610111	0.305055	+	0.952335i	$0.401325\pi$
$18$	0	0
$19$	7.66639	1.75879	0.879396	−	0.476092i	$-0.157947\pi$
$19$	7.66639	1.75879	0.879396	+	0.476092i	$0.157947\pi$
$20$	−4.56078	−1.01982
$21$	0	0
$22$	−13.5570	−2.89036
$23$	0.944022	0.196842	0.0984211	−	0.995145i	$-0.468621\pi$
$23$	0.944022	0.196842	0.0984211	+	0.995145i	$0.468621\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−6.59518	−1.24637
$29$	7.41501	1.37693	0.688467	−	0.725268i	$-0.258284\pi$
$29$	7.41501	1.37693	0.688467	+	0.725268i	$0.258284\pi$
$30$	0	0
$31$	6.76445	1.21493	0.607465	−	0.794346i	$-0.292186\pi$
$31$	6.76445	1.21493	0.607465	+	0.794346i	$0.292186\pi$
$32$	6.55107	1.15808
$33$	0	0
$34$	6.44334	1.10502
$35$	1.44606	0.244429
$36$	0	0
$37$	−0.0748598	−0.0123069	−0.00615344	−	0.999981i	$-0.501959\pi$
$37$	−0.0748598	−0.0123069	−0.00615344	+	0.999981i	$0.501959\pi$
$38$	19.6367	3.18550
$39$	0	0
$40$	−6.55920	−1.03710
$41$	10.5018	1.64011	0.820054	−	0.572286i	$-0.193943\pi$
$41$	10.5018	1.64011	0.820054	+	0.572286i	$0.193943\pi$
$42$	0	0
$43$	−5.69766	−0.868886	−0.434443	−	0.900699i	$-0.643055\pi$
$43$	−5.69766	−0.868886	−0.434443	+	0.900699i	$0.643055\pi$
$44$	−24.1393	−3.63914
$45$	0	0
$46$	2.41802	0.356518
$47$	−2.33083	−0.339987	−0.169993	−	0.985445i	$-0.554375\pi$
$47$	−2.33083	−0.339987	−0.169993	+	0.985445i	$0.554375\pi$
$48$	0	0
$49$	−4.90890	−0.701272
$50$	2.56140	0.362237
$51$	0	0
$52$	0	0
$53$	12.2474	1.68231	0.841153	−	0.540797i	$-0.181877\pi$
$53$	12.2474	1.68231	0.841153	+	0.540797i	$0.181877\pi$
$54$	0	0
$55$	5.29280	0.713681
$56$	−9.48501	−1.26749
$57$	0	0
$58$	18.9928	2.49388
$59$	−1.85841	−0.241945	−0.120972	−	0.992656i	$-0.538601\pi$
$59$	−1.85841	−0.241945	−0.120972	+	0.992656i	$0.538601\pi$
$60$	0	0
$61$	−2.05945	−0.263686	−0.131843	−	0.991271i	$-0.542089\pi$
$61$	−2.05945	−0.263686	−0.131843	+	0.991271i	$0.542089\pi$
$62$	17.3265	2.20046
$63$	0	0
$64$	1.42157	0.177697
$65$	0	0
$66$	0	0
$67$	−13.1493	−1.60644	−0.803222	−	0.595679i	$-0.796883\pi$
$67$	−13.1493	−1.60644	−0.803222	+	0.595679i	$0.796883\pi$
$68$	11.4729	1.39129
$69$	0	0
$70$	3.70395	0.442707
$71$	5.93919	0.704852	0.352426	−	0.935840i	$-0.385357\pi$
$71$	5.93919	0.704852	0.352426	+	0.935840i	$0.385357\pi$
$72$	0	0
$73$	−7.22147	−0.845210	−0.422605	−	0.906314i	$-0.638884\pi$
$73$	−7.22147	−0.845210	−0.422605	+	0.906314i	$0.638884\pi$
$74$	−0.191746	−0.0222900
$75$	0	0
$76$	34.9648	4.01073
$77$	7.65372	0.872223
$78$	0	0
$79$	11.5410	1.29847	0.649234	−	0.760589i	$-0.275090\pi$
$79$	11.5410	1.29847	0.649234	+	0.760589i	$0.275090\pi$
$80$	−7.67918	−0.858558
$81$	0	0
$82$	26.8994	2.97054
$83$	7.84386	0.860976	0.430488	−	0.902596i	$-0.358342\pi$
$83$	7.84386	0.860976	0.430488	+	0.902596i	$0.358342\pi$
$84$	0	0
$85$	−2.51555	−0.272850
$86$	−14.5940	−1.57371
$87$	0	0
$88$	−34.7165	−3.70079
$89$	10.0293	1.06311	0.531554	−	0.847025i	$-0.321608\pi$
$89$	10.0293	1.06311	0.531554	+	0.847025i	$0.321608\pi$
$90$	0	0
$91$	0	0
$92$	4.30548	0.448877
$93$	0	0
$94$	−5.97020	−0.615779
$95$	−7.66639	−0.786555
$96$	0	0
$97$	4.27112	0.433667	0.216833	−	0.976209i	$-0.430427\pi$
$97$	4.27112	0.433667	0.216833	+	0.976209i	$0.430427\pi$
$98$	−12.5737	−1.27013
$99$	0	0
$100$	4.56078	0.456078
$101$	−0.0213198	−0.00212140	−0.00106070	−	0.999999i	$-0.500338\pi$
$101$	−0.0213198	−0.00212140	−0.00106070	+	0.999999i	$0.500338\pi$
$102$	0	0
$103$	17.0672	1.68168	0.840841	−	0.541283i	$-0.182061\pi$
$103$	17.0672	1.68168	0.840841	+	0.541283i	$0.182061\pi$
$104$	0	0
$105$	0	0
$106$	31.3705	3.04697
$107$	12.8123	1.23861	0.619307	−	0.785149i	$-0.287414\pi$
$107$	12.8123	1.23861	0.619307	+	0.785149i	$0.287414\pi$
$108$	0	0
$109$	−13.3953	−1.28304	−0.641518	−	0.767108i	$-0.721695\pi$
$109$	−13.3953	−1.28304	−0.641518	+	0.767108i	$0.721695\pi$
$110$	13.5570	1.29261
$111$	0	0
$112$	−11.1046	−1.04928
$113$	−8.90096	−0.837332	−0.418666	−	0.908140i	$-0.637502\pi$
$113$	−8.90096	−0.837332	−0.418666	+	0.908140i	$0.637502\pi$
$114$	0	0
$115$	−0.944022	−0.0880305
$116$	33.8183	3.13995
$117$	0	0
$118$	−4.76014	−0.438207
$119$	−3.63764	−0.333462
$120$	0	0
$121$	17.0138	1.54671
$122$	−5.27509	−0.477584
$123$	0	0
$124$	30.8512	2.77052
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.6609	0.946002	0.473001	−	0.881062i	$-0.343171\pi$
$127$	10.6609	0.946002	0.473001	+	0.881062i	$0.343171\pi$
$128$	−9.46092	−0.836235
$129$	0	0
$130$	0	0
$131$	16.9436	1.48037	0.740185	−	0.672403i	$-0.234738\pi$
$131$	16.9436	1.48037	0.740185	+	0.672403i	$0.234738\pi$
$132$	0	0
$133$	−11.0861	−0.961285
$134$	−33.6807	−2.90957
$135$	0	0
$136$	16.5000	1.41486
$137$	−9.72098	−0.830519	−0.415260	−	0.909703i	$-0.636309\pi$
$137$	−9.72098	−0.830519	−0.415260	+	0.909703i	$0.636309\pi$
$138$	0	0
$139$	−1.25051	−0.106067	−0.0530336	−	0.998593i	$-0.516889\pi$
$139$	−1.25051	−0.106067	−0.0530336	+	0.998593i	$0.516889\pi$
$140$	6.59518	0.557394
$141$	0	0
$142$	15.2127	1.27662
$143$	0	0
$144$	0	0
$145$	−7.41501	−0.615783
$146$	−18.4971	−1.53083
$147$	0	0
$148$	−0.341419	−0.0280645
$149$	13.0614	1.07003	0.535016	−	0.844842i	$-0.320305\pi$
$149$	13.0614	1.07003	0.535016	+	0.844842i	$0.320305\pi$
$150$	0	0
$151$	2.60928	0.212341	0.106170	−	0.994348i	$-0.466141\pi$
$151$	2.60928	0.212341	0.106170	+	0.994348i	$0.466141\pi$
$152$	50.2854	4.07868
$153$	0	0
$154$	19.6043	1.57976
$155$	−6.76445	−0.543334
$156$	0	0
$157$	−23.5169	−1.87686	−0.938428	−	0.345475i	$-0.887718\pi$
$157$	−23.5169	−1.87686	−0.938428	+	0.345475i	$0.887718\pi$
$158$	29.5612	2.35176
$159$	0	0
$160$	−6.55107	−0.517908
$161$	−1.36512	−0.107586
$162$	0	0
$163$	21.8047	1.70787	0.853937	−	0.520377i	$-0.174209\pi$
$163$	21.8047	1.70787	0.853937	+	0.520377i	$0.174209\pi$
$164$	47.8965	3.74009
$165$	0	0
$166$	20.0913	1.55939
$167$	−11.4569	−0.886565	−0.443283	−	0.896382i	$-0.646186\pi$
$167$	−11.4569	−0.886565	−0.443283	+	0.896382i	$0.646186\pi$
$168$	0	0
$169$	0	0
$170$	−6.44334	−0.494182
$171$	0	0
$172$	−25.9858	−1.98140
$173$	22.5545	1.71479	0.857395	−	0.514660i	$-0.172082\pi$
$173$	22.5545	1.71479	0.857395	+	0.514660i	$0.172082\pi$
$174$	0	0
$175$	−1.44606	−0.109312
$176$	−40.6444	−3.06368
$177$	0	0
$178$	25.6892	1.92548
$179$	18.5005	1.38279	0.691397	−	0.722475i	$-0.256996\pi$
$179$	18.5005	1.38279	0.691397	+	0.722475i	$0.256996\pi$
$180$	0	0
$181$	5.02873	0.373782	0.186891	−	0.982381i	$-0.440159\pi$
$181$	5.02873	0.373782	0.186891	+	0.982381i	$0.440159\pi$
$182$	0	0
$183$	0	0
$184$	6.19203	0.456482
$185$	0.0748598	0.00550380
$186$	0	0
$187$	−13.3143	−0.973639
$188$	−10.6304	−0.775303
$189$	0	0
$190$	−19.6367	−1.42460
$191$	11.0095	0.796622	0.398311	−	0.917250i	$-0.369596\pi$
$191$	11.0095	0.796622	0.398311	+	0.917250i	$0.369596\pi$
$192$	0	0
$193$	−25.1511	−1.81041	−0.905207	−	0.424972i	$-0.860284\pi$
$193$	−25.1511	−1.81041	−0.905207	+	0.424972i	$0.860284\pi$
$194$	10.9401	0.785450
$195$	0	0
$196$	−22.3884	−1.59917
$197$	−14.3338	−1.02124	−0.510621	−	0.859806i	$-0.670585\pi$
$197$	−14.3338	−1.02124	−0.510621	+	0.859806i	$0.670585\pi$
$198$	0	0
$199$	17.7358	1.25726	0.628628	−	0.777706i	$-0.283617\pi$
$199$	17.7358	1.25726	0.628628	+	0.777706i	$0.283617\pi$
$200$	6.55920	0.463805
$201$	0	0
$202$	−0.0546086	−0.00384225
$203$	−10.7226	−0.752577
$204$	0	0
$205$	−10.5018	−0.733479
$206$	43.7160	3.04584
$207$	0	0
$208$	0	0
$209$	−40.5767	−2.80675
$210$	0	0
$211$	−15.2154	−1.04747	−0.523736	−	0.851881i	$-0.675462\pi$
$211$	−15.2154	−1.04747	−0.523736	+	0.851881i	$0.675462\pi$
$212$	55.8576	3.83632
$213$	0	0
$214$	32.8175	2.24336
$215$	5.69766	0.388577
$216$	0	0
$217$	−9.78182	−0.664033
$218$	−34.3107	−2.32382
$219$	0	0
$220$	24.1393	1.62747
$221$	0	0
$222$	0	0
$223$	−24.9535	−1.67101	−0.835506	−	0.549481i	$-0.814826\pi$
$223$	−24.9535	−1.67101	−0.835506	+	0.549481i	$0.814826\pi$
$224$	−9.47326	−0.632959
$225$	0	0
$226$	−22.7989	−1.51656
$227$	−7.69341	−0.510630	−0.255315	−	0.966858i	$-0.582179\pi$
$227$	−7.69341	−0.510630	−0.255315	+	0.966858i	$0.582179\pi$
$228$	0	0
$229$	−25.2551	−1.66890	−0.834452	−	0.551081i	$-0.814216\pi$
$229$	−25.2551	−1.66890	−0.834452	+	0.551081i	$0.814216\pi$
$230$	−2.41802	−0.159440
$231$	0	0
$232$	48.6365	3.19314
$233$	−8.09830	−0.530537	−0.265269	−	0.964175i	$-0.585461\pi$
$233$	−8.09830	−0.530537	−0.265269	+	0.964175i	$0.585461\pi$
$234$	0	0
$235$	2.33083	0.152047
$236$	−8.47582	−0.551729
$237$	0	0
$238$	−9.31747	−0.603962
$239$	−19.2339	−1.24414	−0.622068	−	0.782964i	$-0.713707\pi$
$239$	−19.2339	−1.24414	−0.622068	+	0.782964i	$0.713707\pi$
$240$	0	0
$241$	3.20809	0.206651	0.103326	−	0.994648i	$-0.467052\pi$
$241$	3.20809	0.206651	0.103326	+	0.994648i	$0.467052\pi$
$242$	43.5791	2.80137
$243$	0	0
$244$	−9.39271	−0.601307
$245$	4.90890	0.313618
$246$	0	0
$247$	0	0
$248$	44.3693	2.81746
$249$	0	0
$250$	−2.56140	−0.161997
$251$	−9.31638	−0.588044	−0.294022	−	0.955799i	$-0.594994\pi$
$251$	−9.31638	−0.588044	−0.294022	+	0.955799i	$0.594994\pi$
$252$	0	0
$253$	−4.99652	−0.314129
$254$	27.3068	1.71338
$255$	0	0
$256$	−27.0764	−1.69227
$257$	−6.47095	−0.403647	−0.201823	−	0.979422i	$-0.564687\pi$
$257$	−6.47095	−0.403647	−0.201823	+	0.979422i	$0.564687\pi$
$258$	0	0
$259$	0.108252	0.00672645
$260$	0	0
$261$	0	0
$262$	43.3994	2.68122
$263$	−1.91873	−0.118314	−0.0591571	−	0.998249i	$-0.518841\pi$
$263$	−1.91873	−0.118314	−0.0591571	+	0.998249i	$0.518841\pi$
$264$	0	0
$265$	−12.2474	−0.752350
$266$	−28.3959	−1.74107
$267$	0	0
$268$	−59.9712	−3.66332
$269$	4.99555	0.304584	0.152292	−	0.988336i	$-0.451335\pi$
$269$	4.99555	0.304584	0.152292	+	0.988336i	$0.451335\pi$
$270$	0	0
$271$	−7.45094	−0.452613	−0.226306	−	0.974056i	$-0.572665\pi$
$271$	−7.45094	−0.452613	−0.226306	+	0.974056i	$0.572665\pi$
$272$	19.3174	1.17129
$273$	0	0
$274$	−24.8993	−1.50422
$275$	−5.29280	−0.319168
$276$	0	0
$277$	9.88513	0.593940	0.296970	−	0.954887i	$-0.404024\pi$
$277$	9.88513	0.593940	0.296970	+	0.954887i	$0.404024\pi$
$278$	−3.20307	−0.192107
$279$	0	0
$280$	9.48501	0.566838
$281$	−13.4442	−0.802013	−0.401006	−	0.916075i	$-0.631340\pi$
$281$	−13.4442	−0.802013	−0.401006	+	0.916075i	$0.631340\pi$
$282$	0	0
$283$	−14.0702	−0.836389	−0.418194	−	0.908358i	$-0.637337\pi$
$283$	−14.0702	−0.836389	−0.418194	+	0.908358i	$0.637337\pi$
$284$	27.0874	1.60734
$285$	0	0
$286$	0	0
$287$	−15.1863	−0.896418
$288$	0	0
$289$	−10.6720	−0.627765
$290$	−18.9928	−1.11530
$291$	0	0
$292$	−32.9356	−1.92741
$293$	15.1769	0.886644	0.443322	−	0.896362i	$-0.353800\pi$
$293$	15.1769	0.886644	0.443322	+	0.896362i	$0.353800\pi$
$294$	0	0
$295$	1.85841	0.108201
$296$	−0.491020	−0.0285400
$297$	0	0
$298$	33.4555	1.93803
$299$	0	0
$300$	0	0
$301$	8.23918	0.474898
$302$	6.68343	0.384588
$303$	0	0
$304$	58.8716	3.37652
$305$	2.05945	0.117924
$306$	0	0
$307$	15.9924	0.912736	0.456368	−	0.889791i	$-0.349150\pi$
$307$	15.9924	0.912736	0.456368	+	0.889791i	$0.349150\pi$
$308$	34.9070	1.98901
$309$	0	0
$310$	−17.3265	−0.984078
$311$	−15.1164	−0.857172	−0.428586	−	0.903501i	$-0.640988\pi$
$311$	−15.1164	−0.857172	−0.428586	+	0.903501i	$0.640988\pi$
$312$	0	0
$313$	10.9994	0.621723	0.310861	−	0.950455i	$-0.399382\pi$
$313$	10.9994	0.621723	0.310861	+	0.950455i	$0.399382\pi$
$314$	−60.2363	−3.39933
$315$	0	0
$316$	52.6361	2.96101
$317$	7.81860	0.439136	0.219568	−	0.975597i	$-0.429535\pi$
$317$	7.81860	0.439136	0.219568	+	0.975597i	$0.429535\pi$
$318$	0	0
$319$	−39.2462	−2.19737
$320$	−1.42157	−0.0794683
$321$	0	0
$322$	−3.49661	−0.194858
$323$	19.2852	1.07306
$324$	0	0
$325$	0	0
$326$	55.8505	3.09327
$327$	0	0
$328$	68.8835	3.80345
$329$	3.37053	0.185823
$330$	0	0
$331$	28.2963	1.55530	0.777652	−	0.628695i	$-0.216410\pi$
$331$	28.2963	1.55530	0.777652	+	0.628695i	$0.216410\pi$
$332$	35.7741	1.96336
$333$	0	0
$334$	−29.3459	−1.60573
$335$	13.1493	0.718424
$336$	0	0
$337$	−5.90691	−0.321770	−0.160885	−	0.986973i	$-0.551435\pi$
$337$	−5.90691	−0.321770	−0.160885	+	0.986973i	$0.551435\pi$
$338$	0	0
$339$	0	0
$340$	−11.4729	−0.622205
$341$	−35.8029	−1.93884
$342$	0	0
$343$	17.2210	0.929848
$344$	−37.3721	−2.01497
$345$	0	0
$346$	57.7712	3.10580
$347$	11.0703	0.594287	0.297143	−	0.954833i	$-0.403966\pi$
$347$	11.0703	0.594287	0.297143	+	0.954833i	$0.403966\pi$
$348$	0	0
$349$	0.498429	0.0266803	0.0133401	−	0.999911i	$-0.495754\pi$
$349$	0.498429	0.0266803	0.0133401	+	0.999911i	$0.495754\pi$
$350$	−3.70395	−0.197984
$351$	0	0
$352$	−34.6735	−1.84810
$353$	−19.4630	−1.03591	−0.517955	−	0.855408i	$-0.673307\pi$
$353$	−19.4630	−1.03591	−0.517955	+	0.855408i	$0.673307\pi$
$354$	0	0
$355$	−5.93919	−0.315219
$356$	45.7416	2.42430
$357$	0	0
$358$	47.3873	2.50450
$359$	0.128577	0.00678605	0.00339302	−	0.999994i	$-0.498920\pi$
$359$	0.128577	0.00678605	0.00339302	+	0.999994i	$0.498920\pi$
$360$	0	0
$361$	39.7736	2.09335
$362$	12.8806	0.676989
$363$	0	0
$364$	0	0
$365$	7.22147	0.377989
$366$	0	0
$367$	−8.12443	−0.424092	−0.212046	−	0.977260i	$-0.568013\pi$
$367$	−8.12443	−0.424092	−0.212046	+	0.977260i	$0.568013\pi$
$368$	7.24931	0.377897
$369$	0	0
$370$	0.191746	0.00996841
$371$	−17.7105	−0.919482
$372$	0	0
$373$	−26.3918	−1.36651	−0.683257	−	0.730178i	$-0.739437\pi$
$373$	−26.3918	−1.36651	−0.683257	+	0.730178i	$0.739437\pi$
$374$	−34.1033	−1.76344
$375$	0	0
$376$	−15.2884	−0.788438
$377$	0	0
$378$	0	0
$379$	−2.79108	−0.143368	−0.0716842	−	0.997427i	$-0.522837\pi$
$379$	−2.79108	−0.143368	−0.0716842	+	0.997427i	$0.522837\pi$
$380$	−34.9648	−1.79365
$381$	0	0
$382$	28.1999	1.44283
$383$	−29.7140	−1.51831	−0.759157	−	0.650908i	$-0.774388\pi$
$383$	−29.7140	−1.51831	−0.759157	+	0.650908i	$0.774388\pi$
$384$	0	0
$385$	−7.65372	−0.390070
$386$	−64.4220	−3.27899
$387$	0	0
$388$	19.4797	0.988930
$389$	−11.0412	−0.559810	−0.279905	−	0.960028i	$-0.590303\pi$
$389$	−11.0412	−0.559810	−0.279905	+	0.960028i	$0.590303\pi$
$390$	0	0
$391$	2.37474	0.120096
$392$	−32.1985	−1.62627
$393$	0	0
$394$	−36.7147	−1.84966
$395$	−11.5410	−0.580692
$396$	0	0
$397$	13.0766	0.656295	0.328147	−	0.944627i	$-0.393576\pi$
$397$	13.0766	0.656295	0.328147	+	0.944627i	$0.393576\pi$
$398$	45.4285	2.27712
$399$	0	0
$400$	7.67918	0.383959
$401$	2.69801	0.134732	0.0673660	−	0.997728i	$-0.478540\pi$
$401$	2.69801	0.134732	0.0673660	+	0.997728i	$0.478540\pi$
$402$	0	0
$403$	0	0
$404$	−0.0972349	−0.00483762
$405$	0	0
$406$	−27.4648	−1.36306
$407$	0.396218	0.0196398
$408$	0	0
$409$	−24.2630	−1.19973	−0.599863	−	0.800103i	$-0.704778\pi$
$409$	−24.2630	−1.19973	−0.599863	+	0.800103i	$0.704778\pi$
$410$	−26.8994	−1.32847
$411$	0	0
$412$	77.8398	3.83489
$413$	2.68738	0.132237
$414$	0	0
$415$	−7.84386	−0.385040
$416$	0	0
$417$	0	0
$418$	−103.933	−5.08354
$419$	−9.15399	−0.447202	−0.223601	−	0.974681i	$-0.571781\pi$
$419$	−9.15399	−0.447202	−0.223601	+	0.974681i	$0.571781\pi$
$420$	0	0
$421$	−13.9302	−0.678917	−0.339458	−	0.940621i	$-0.610244\pi$
$421$	−13.9302	−0.678917	−0.339458	+	0.940621i	$0.610244\pi$
$422$	−38.9728	−1.89716
$423$	0	0
$424$	80.3330	3.90131
$425$	2.51555	0.122022
$426$	0	0
$427$	2.97810	0.144120
$428$	58.4342	2.82452
$429$	0	0
$430$	14.5940	0.703786
$431$	−3.61627	−0.174190	−0.0870949	−	0.996200i	$-0.527758\pi$
$431$	−3.61627	−0.174190	−0.0870949	+	0.996200i	$0.527758\pi$
$432$	0	0
$433$	−31.1502	−1.49698	−0.748491	−	0.663145i	$-0.769221\pi$
$433$	−31.1502	−1.49698	−0.748491	+	0.663145i	$0.769221\pi$
$434$	−25.0552	−1.20269
$435$	0	0
$436$	−61.0930	−2.92582
$437$	7.23725	0.346204
$438$	0	0
$439$	−13.2279	−0.631335	−0.315668	−	0.948870i	$-0.602228\pi$
$439$	−13.2279	−0.631335	−0.315668	+	0.948870i	$0.602228\pi$
$440$	34.7165	1.65505
$441$	0	0
$442$	0	0
$443$	14.2964	0.679243	0.339621	−	0.940562i	$-0.389701\pi$
$443$	14.2964	0.679243	0.339621	+	0.940562i	$0.389701\pi$
$444$	0	0
$445$	−10.0293	−0.475436
$446$	−63.9161	−3.02651
$447$	0	0
$448$	−2.05568	−0.0971219
$449$	−2.55742	−0.120692	−0.0603462	−	0.998178i	$-0.519220\pi$
$449$	−2.55742	−0.120692	−0.0603462	+	0.998178i	$0.519220\pi$
$450$	0	0
$451$	−55.5840	−2.61735
$452$	−40.5953	−1.90944
$453$	0	0
$454$	−19.7059	−0.924845
$455$	0	0
$456$	0	0
$457$	16.4774	0.770779	0.385390	−	0.922754i	$-0.374067\pi$
$457$	16.4774	0.770779	0.385390	+	0.922754i	$0.374067\pi$
$458$	−64.6885	−3.02269
$459$	0	0
$460$	−4.30548	−0.200744
$461$	27.0006	1.25754	0.628772	−	0.777590i	$-0.283558\pi$
$461$	27.0006	1.25754	0.628772	+	0.777590i	$0.283558\pi$
$462$	0	0
$463$	5.00924	0.232799	0.116400	−	0.993202i	$-0.462865\pi$
$463$	5.00924	0.232799	0.116400	+	0.993202i	$0.462865\pi$
$464$	56.9412	2.64343
$465$	0	0
$466$	−20.7430	−0.960901
$467$	−40.1904	−1.85979	−0.929894	−	0.367828i	$-0.880102\pi$
$467$	−40.1904	−1.85979	−0.929894	+	0.367828i	$0.880102\pi$
$468$	0	0
$469$	19.0147	0.878019
$470$	5.97020	0.275385
$471$	0	0
$472$	−12.1897	−0.561076
$473$	30.1566	1.38660
$474$	0	0
$475$	7.66639	0.351758
$476$	−16.5905	−0.760425
$477$	0	0
$478$	−49.2657	−2.25336
$479$	−11.3250	−0.517451	−0.258726	−	0.965951i	$-0.583303\pi$
$479$	−11.3250	−0.517451	−0.258726	+	0.965951i	$0.583303\pi$
$480$	0	0
$481$	0	0
$482$	8.21721	0.374284
$483$	0	0
$484$	77.5960	3.52709
$485$	−4.27112	−0.193942
$486$	0	0
$487$	35.4507	1.60643	0.803213	−	0.595692i	$-0.203122\pi$
$487$	35.4507	1.60643	0.803213	+	0.595692i	$0.203122\pi$
$488$	−13.5083	−0.611494
$489$	0	0
$490$	12.5737	0.568021
$491$	−27.0086	−1.21888	−0.609440	−	0.792832i	$-0.708606\pi$
$491$	−27.0086	−1.21888	−0.609440	+	0.792832i	$0.708606\pi$
$492$	0	0
$493$	18.6528	0.840082
$494$	0	0
$495$	0	0
$496$	51.9454	2.33242
$497$	−8.58844	−0.385244
$498$	0	0
$499$	−18.2083	−0.815117	−0.407559	−	0.913179i	$-0.633620\pi$
$499$	−18.2083	−0.815117	−0.407559	+	0.913179i	$0.633620\pi$
$500$	−4.56078	−0.203964
$501$	0	0
$502$	−23.8630	−1.06506
$503$	21.2111	0.945756	0.472878	−	0.881128i	$-0.343215\pi$
$503$	21.2111	0.945756	0.472878	+	0.881128i	$0.343215\pi$
$504$	0	0
$505$	0.0213198	0.000948718 0
$506$	−12.7981	−0.568945
$507$	0	0
$508$	48.6220	2.15725
$509$	7.79526	0.345519	0.172759	−	0.984964i	$-0.444732\pi$
$509$	7.79526	0.345519	0.172759	+	0.984964i	$0.444732\pi$
$510$	0	0
$511$	10.4427	0.461958
$512$	−50.4316	−2.22878
$513$	0	0
$514$	−16.5747	−0.731079
$515$	−17.0672	−0.752071
$516$	0	0
$517$	12.3366	0.542564
$518$	0.277277	0.0121828
$519$	0	0
$520$	0	0
$521$	−37.9763	−1.66377	−0.831886	−	0.554947i	$-0.812739\pi$
$521$	−37.9763	−1.66377	−0.831886	+	0.554947i	$0.812739\pi$
$522$	0	0
$523$	21.3080	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$523$	21.3080	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$524$	77.2761	3.37582
$525$	0	0
$526$	−4.91465	−0.214289
$527$	17.0163	0.741242
$528$	0	0
$529$	−22.1088	−0.961253
$530$	−31.3705	−1.36265
$531$	0	0
$532$	−50.5612	−2.19211
$533$	0	0
$534$	0	0
$535$	−12.8123	−0.553925
$536$	−86.2490	−3.72539
$537$	0	0
$538$	12.7956	0.551658
$539$	25.9819	1.11912
$540$	0	0
$541$	40.2176	1.72909	0.864544	−	0.502557i	$-0.167607\pi$
$541$	40.2176	1.72909	0.864544	+	0.502557i	$0.167607\pi$
$542$	−19.0849	−0.819765
$543$	0	0
$544$	16.4796	0.706555
$545$	13.3953	0.573791
$546$	0	0
$547$	45.0283	1.92527	0.962636	−	0.270799i	$-0.0872879\pi$
$547$	45.0283	1.92527	0.962636	+	0.270799i	$0.0872879\pi$
$548$	−44.3353	−1.89391
$549$	0	0
$550$	−13.5570	−0.578072
$551$	56.8464	2.42174
$552$	0	0
$553$	−16.6891	−0.709691
$554$	25.3198	1.07573
$555$	0	0
$556$	−5.70332	−0.241875
$557$	−30.8296	−1.30629	−0.653145	−	0.757233i	$-0.726551\pi$
$557$	−30.8296	−1.30629	−0.653145	+	0.757233i	$0.726551\pi$
$558$	0	0
$559$	0	0
$560$	11.1046	0.469254
$561$	0	0
$562$	−34.4360	−1.45259
$563$	13.9330	0.587206	0.293603	−	0.955927i	$-0.405146\pi$
$563$	13.9330	0.587206	0.293603	+	0.955927i	$0.405146\pi$
$564$	0	0
$565$	8.90096	0.374466
$566$	−36.0395	−1.51485
$567$	0	0
$568$	38.9563	1.63457
$569$	0.120707	0.00506032	0.00253016	−	0.999997i	$-0.499195\pi$
$569$	0.120707	0.00506032	0.00253016	+	0.999997i	$0.499195\pi$
$570$	0	0
$571$	−18.2919	−0.765491	−0.382746	−	0.923854i	$-0.625021\pi$
$571$	−18.2919	−0.765491	−0.382746	+	0.923854i	$0.625021\pi$
$572$	0	0
$573$	0	0
$574$	−38.8982	−1.62358
$575$	0.944022	0.0393685
$576$	0	0
$577$	−22.7796	−0.948326	−0.474163	−	0.880437i	$-0.657249\pi$
$577$	−22.7796	−0.948326	−0.474163	+	0.880437i	$0.657249\pi$
$578$	−27.3353	−1.13700
$579$	0	0
$580$	−33.8183	−1.40423
$581$	−11.3427	−0.470575
$582$	0	0
$583$	−64.8229	−2.68469
$584$	−47.3671	−1.96006
$585$	0	0
$586$	38.8742	1.60588
$587$	−0.926647	−0.0382468	−0.0191234	−	0.999817i	$-0.506088\pi$
$587$	−0.926647	−0.0382468	−0.0191234	+	0.999817i	$0.506088\pi$
$588$	0	0
$589$	51.8589	2.13681
$590$	4.76014	0.195972
$591$	0	0
$592$	−0.574862	−0.0236267
$593$	2.82265	0.115912	0.0579561	−	0.998319i	$-0.481542\pi$
$593$	2.82265	0.115912	0.0579561	+	0.998319i	$0.481542\pi$
$594$	0	0
$595$	3.63764	0.149129
$596$	59.5703	2.44009
$597$	0	0
$598$	0	0
$599$	46.1094	1.88398	0.941989	−	0.335643i	$-0.108954\pi$
$599$	46.1094	1.88398	0.941989	+	0.335643i	$0.108954\pi$
$600$	0	0
$601$	−25.4122	−1.03658	−0.518292	−	0.855204i	$-0.673432\pi$
$601$	−25.4122	−1.03658	−0.518292	+	0.855204i	$0.673432\pi$
$602$	21.1039	0.860129
$603$	0	0
$604$	11.9004	0.484220
$605$	−17.0138	−0.691708
$606$	0	0
$607$	−24.7675	−1.00528	−0.502642	−	0.864495i	$-0.667639\pi$
$607$	−24.7675	−1.00528	−0.502642	+	0.864495i	$0.667639\pi$
$608$	50.2231	2.03681
$609$	0	0
$610$	5.27509	0.213582
$611$	0	0
$612$	0	0
$613$	−5.35628	−0.216338	−0.108169	−	0.994133i	$-0.534499\pi$
$613$	−5.35628	−0.216338	−0.108169	+	0.994133i	$0.534499\pi$
$614$	40.9630	1.65313
$615$	0	0
$616$	50.2023	2.02271
$617$	−22.8124	−0.918393	−0.459197	−	0.888335i	$-0.651863\pi$
$617$	−22.8124	−0.918393	−0.459197	+	0.888335i	$0.651863\pi$
$618$	0	0
$619$	24.2953	0.976509	0.488255	−	0.872701i	$-0.337634\pi$
$619$	24.2953	0.976509	0.488255	+	0.872701i	$0.337634\pi$
$620$	−30.8512	−1.23901
$621$	0	0
$622$	−38.7191	−1.55250
$623$	−14.5030	−0.581052
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.1739	1.12606
$627$	0	0
$628$	−107.256	−4.27997
$629$	−0.188314	−0.00750856
$630$	0	0
$631$	−16.5843	−0.660212	−0.330106	−	0.943944i	$-0.607085\pi$
$631$	−16.5843	−0.660212	−0.330106	+	0.943944i	$0.607085\pi$
$632$	75.6999	3.01118
$633$	0	0
$634$	20.0266	0.795357
$635$	−10.6609	−0.423065
$636$	0	0
$637$	0	0
$638$	−100.525	−3.97984
$639$	0	0
$640$	9.46092	0.373976
$641$	18.7185	0.739336	0.369668	−	0.929164i	$-0.379471\pi$
$641$	18.7185	0.739336	0.369668	+	0.929164i	$0.379471\pi$
$642$	0	0
$643$	−5.16560	−0.203712	−0.101856	−	0.994799i	$-0.532478\pi$
$643$	−5.16560	−0.203712	−0.101856	+	0.994799i	$0.532478\pi$
$644$	−6.22600	−0.245339
$645$	0	0
$646$	49.3972	1.94351
$647$	19.6675	0.773207	0.386604	−	0.922246i	$-0.373648\pi$
$647$	19.6675	0.773207	0.386604	+	0.922246i	$0.373648\pi$
$648$	0	0
$649$	9.83621	0.386105
$650$	0	0
$651$	0	0
$652$	99.4464	3.89462
$653$	14.4003	0.563527	0.281764	−	0.959484i	$-0.409081\pi$
$653$	14.4003	0.563527	0.281764	+	0.959484i	$0.409081\pi$
$654$	0	0
$655$	−16.9436	−0.662041
$656$	80.6453	3.14867
$657$	0	0
$658$	8.63328	0.336560
$659$	13.8017	0.537637	0.268819	−	0.963191i	$-0.413367\pi$
$659$	13.8017	0.537637	0.268819	+	0.963191i	$0.413367\pi$
$660$	0	0
$661$	5.57183	0.216719	0.108360	−	0.994112i	$-0.465440\pi$
$661$	5.57183	0.216719	0.108360	+	0.994112i	$0.465440\pi$
$662$	72.4782	2.81694
$663$	0	0
$664$	51.4494	1.99662
$665$	11.0861	0.429900
$666$	0	0
$667$	6.99994	0.271039
$668$	−52.2526	−2.02172
$669$	0	0
$670$	33.6807	1.30120
$671$	10.9003	0.420800
$672$	0	0
$673$	33.8447	1.30462	0.652308	−	0.757954i	$-0.273801\pi$
$673$	33.8447	1.30462	0.652308	+	0.757954i	$0.273801\pi$
$674$	−15.1300	−0.582784
$675$	0	0
$676$	0	0
$677$	24.2412	0.931667	0.465833	−	0.884872i	$-0.345755\pi$
$677$	24.2412	0.931667	0.465833	+	0.884872i	$0.345755\pi$
$678$	0	0
$679$	−6.17631	−0.237025
$680$	−16.5000	−0.632746
$681$	0	0
$682$	−91.7056	−3.51159
$683$	24.0669	0.920895	0.460448	−	0.887687i	$-0.347689\pi$
$683$	24.0669	0.920895	0.460448	+	0.887687i	$0.347689\pi$
$684$	0	0
$685$	9.72098	0.371419
$686$	44.1100	1.68413
$687$	0	0
$688$	−43.7534	−1.66808
$689$	0	0
$690$	0	0
$691$	2.08135	0.0791784	0.0395892	−	0.999216i	$-0.487395\pi$
$691$	2.08135	0.0791784	0.0395892	+	0.999216i	$0.487395\pi$
$692$	102.866	3.91039
$693$	0	0
$694$	28.3556	1.07636
$695$	1.25051	0.0474347
$696$	0	0
$697$	26.4179	1.00065
$698$	1.27668	0.0483230
$699$	0	0
$700$	−6.59518	−0.249274
$701$	9.47793	0.357977	0.178988	−	0.983851i	$-0.442718\pi$
$701$	9.47793	0.357977	0.178988	+	0.983851i	$0.442718\pi$
$702$	0	0
$703$	−0.573905	−0.0216452
$704$	−7.52410	−0.283575
$705$	0	0
$706$	−49.8525	−1.87622
$707$	0.0308298	0.00115947
$708$	0	0
$709$	9.47784	0.355948	0.177974	−	0.984035i	$-0.443046\pi$
$709$	9.47784	0.355948	0.177974	+	0.984035i	$0.443046\pi$
$710$	−15.2127	−0.570921
$711$	0	0
$712$	65.7844	2.46537
$713$	6.38579	0.239150
$714$	0	0
$715$	0	0
$716$	84.3769	3.15331
$717$	0	0
$718$	0.329338	0.0122908
$719$	35.0508	1.30717	0.653587	−	0.756851i	$-0.273263\pi$
$719$	35.0508	1.30717	0.653587	+	0.756851i	$0.273263\pi$
$720$	0	0
$721$	−24.6802	−0.919140
$722$	101.876	3.79144
$723$	0	0
$724$	22.9349	0.852370
$725$	7.41501	0.275387
$726$	0	0
$727$	−15.0005	−0.556338	−0.278169	−	0.960532i	$-0.589728\pi$
$727$	−15.0005	−0.556338	−0.278169	+	0.960532i	$0.589728\pi$
$728$	0	0
$729$	0	0
$730$	18.4971	0.684608
$731$	−14.3328	−0.530117
$732$	0	0
$733$	−30.2340	−1.11672	−0.558359	−	0.829599i	$-0.688569\pi$
$733$	−30.2340	−1.11672	−0.558359	+	0.829599i	$0.688569\pi$
$734$	−20.8099	−0.768108
$735$	0	0
$736$	6.18436	0.227958
$737$	69.5967	2.56363
$738$	0	0
$739$	−30.1770	−1.11008	−0.555040	−	0.831824i	$-0.687297\pi$
$739$	−30.1770	−1.11008	−0.555040	+	0.831824i	$0.687297\pi$
$740$	0.341419	0.0125508
$741$	0	0
$742$	−45.3637	−1.66535
$743$	−4.15655	−0.152489	−0.0762444	−	0.997089i	$-0.524293\pi$
$743$	−4.15655	−0.152489	−0.0762444	+	0.997089i	$0.524293\pi$
$744$	0	0
$745$	−13.0614	−0.478533
$746$	−67.6000	−2.47501
$747$	0	0
$748$	−60.7237	−2.22028
$749$	−18.5274	−0.676977
$750$	0	0
$751$	2.45495	0.0895825	0.0447913	−	0.998996i	$-0.485738\pi$
$751$	2.45495	0.0895825	0.0447913	+	0.998996i	$0.485738\pi$
$752$	−17.8989	−0.652705
$753$	0	0
$754$	0	0
$755$	−2.60928	−0.0949616
$756$	0	0
$757$	−42.8047	−1.55576	−0.777882	−	0.628410i	$-0.783706\pi$
$757$	−42.8047	−1.55576	−0.777882	+	0.628410i	$0.783706\pi$
$758$	−7.14909	−0.259667
$759$	0	0
$760$	−50.2854	−1.82404
$761$	−44.4319	−1.61066	−0.805328	−	0.592829i	$-0.798011\pi$
$761$	−44.4319	−1.61066	−0.805328	+	0.592829i	$0.798011\pi$
$762$	0	0
$763$	19.3704	0.701256
$764$	50.2121	1.81661
$765$	0	0
$766$	−76.1094	−2.74995
$767$	0	0
$768$	0	0
$769$	−8.11378	−0.292591	−0.146295	−	0.989241i	$-0.546735\pi$
$769$	−8.11378	−0.292591	−0.146295	+	0.989241i	$0.546735\pi$
$770$	−19.6043	−0.706489
$771$	0	0
$772$	−114.709	−4.12845
$773$	41.3542	1.48741	0.743703	−	0.668510i	$-0.233068\pi$
$773$	41.3542	1.48741	0.743703	+	0.668510i	$0.233068\pi$
$774$	0	0
$775$	6.76445	0.242986
$776$	28.0151	1.00568
$777$	0	0
$778$	−28.2809	−1.01392
$779$	80.5110	2.88461
$780$	0	0
$781$	−31.4350	−1.12483
$782$	6.08266	0.217515
$783$	0	0
$784$	−37.6963	−1.34630
$785$	23.5169	0.839355
$786$	0	0
$787$	−0.720519	−0.0256837	−0.0128419	−	0.999918i	$-0.504088\pi$
$787$	−0.720519	−0.0256837	−0.0128419	+	0.999918i	$0.504088\pi$
$788$	−65.3735	−2.32883
$789$	0	0
$790$	−29.5612	−1.05174
$791$	12.8713	0.457652
$792$	0	0
$793$	0	0
$794$	33.4944	1.18867
$795$	0	0
$796$	80.8890	2.86704
$797$	31.1479	1.10331	0.551657	−	0.834071i	$-0.313996\pi$
$797$	31.1479	1.10331	0.551657	+	0.834071i	$0.313996\pi$
$798$	0	0
$799$	−5.86333	−0.207430
$800$	6.55107	0.231615
$801$	0	0
$802$	6.91068	0.244025
$803$	38.2218	1.34882
$804$	0	0
$805$	1.36512	0.0481140
$806$	0	0
$807$	0	0
$808$	−0.139841	−0.00491958
$809$	43.4578	1.52789	0.763947	−	0.645278i	$-0.223259\pi$
$809$	43.4578	1.52789	0.763947	+	0.645278i	$0.223259\pi$
$810$	0	0
$811$	−24.3734	−0.855867	−0.427933	−	0.903810i	$-0.640758\pi$
$811$	−24.3734	−0.855867	−0.427933	+	0.903810i	$0.640758\pi$
$812$	−48.9033	−1.71617
$813$	0	0
$814$	1.01487	0.0355713
$815$	−21.8047	−0.763784
$816$	0	0
$817$	−43.6805	−1.52819
$818$	−62.1472	−2.17293
$819$	0	0
$820$	−47.8965	−1.67262
$821$	−48.3721	−1.68820	−0.844098	−	0.536188i	$-0.819864\pi$
$821$	−48.3721	−1.68820	−0.844098	+	0.536188i	$0.819864\pi$
$822$	0	0
$823$	29.1395	1.01574	0.507870	−	0.861434i	$-0.330433\pi$
$823$	29.1395	1.01574	0.507870	+	0.861434i	$0.330433\pi$
$824$	111.947	3.89986
$825$	0	0
$826$	6.88346	0.239506
$827$	36.6295	1.27373	0.636866	−	0.770974i	$-0.280230\pi$
$827$	36.6295	1.27373	0.636866	+	0.770974i	$0.280230\pi$
$828$	0	0
$829$	36.2504	1.25903	0.629515	−	0.776989i	$-0.283254\pi$
$829$	36.2504	1.25903	0.629515	+	0.776989i	$0.283254\pi$
$830$	−20.0913	−0.697379
$831$	0	0
$832$	0	0
$833$	−12.3486	−0.427854
$834$	0	0
$835$	11.4569	0.396484
$836$	−185.062	−6.40049
$837$	0	0
$838$	−23.4471	−0.809965
$839$	−1.48384	−0.0512278	−0.0256139	−	0.999672i	$-0.508154\pi$
$839$	−1.48384	−0.0512278	−0.0256139	+	0.999672i	$0.508154\pi$
$840$	0	0
$841$	25.9824	0.895945
$842$	−35.6809	−1.22964
$843$	0	0
$844$	−69.3941	−2.38864
$845$	0	0
$846$	0	0
$847$	−24.6030	−0.845368
$848$	94.0498	3.22968
$849$	0	0
$850$	6.44334	0.221005
$851$	−0.0706693	−0.00242251
$852$	0	0
$853$	−56.0890	−1.92045	−0.960225	−	0.279228i	$-0.909921\pi$
$853$	−56.0890	−1.92045	−0.960225	+	0.279228i	$0.909921\pi$
$854$	7.62810	0.261028
$855$	0	0
$856$	84.0385	2.87238
$857$	6.12447	0.209208	0.104604	−	0.994514i	$-0.466642\pi$
$857$	6.12447	0.209208	0.104604	+	0.994514i	$0.466642\pi$
$858$	0	0
$859$	11.6674	0.398088	0.199044	−	0.979991i	$-0.436216\pi$
$859$	11.6674	0.398088	0.199044	+	0.979991i	$0.436216\pi$
$860$	25.9858	0.886109
$861$	0	0
$862$	−9.26273	−0.315490
$863$	−19.4269	−0.661301	−0.330651	−	0.943753i	$-0.607268\pi$
$863$	−19.4269	−0.661301	−0.330651	+	0.943753i	$0.607268\pi$
$864$	0	0
$865$	−22.5545	−0.766877
$866$	−79.7881	−2.71131
$867$	0	0
$868$	−44.6127	−1.51426
$869$	−61.0844	−2.07215
$870$	0	0
$871$	0	0
$872$	−87.8623	−2.97539
$873$	0	0
$874$	18.5375	0.627040
$875$	1.44606	0.0488858
$876$	0	0
$877$	−10.2311	−0.345481	−0.172740	−	0.984967i	$-0.555262\pi$
$877$	−10.2311	−0.345481	−0.172740	+	0.984967i	$0.555262\pi$
$878$	−33.8821	−1.14347
$879$	0	0
$880$	40.6444	1.37012
$881$	−13.4850	−0.454320	−0.227160	−	0.973857i	$-0.572944\pi$
$881$	−13.4850	−0.454320	−0.227160	+	0.973857i	$0.572944\pi$
$882$	0	0
$883$	−29.8622	−1.00494	−0.502472	−	0.864594i	$-0.667576\pi$
$883$	−29.8622	−1.00494	−0.502472	+	0.864594i	$0.667576\pi$
$884$	0	0
$885$	0	0
$886$	36.6188	1.23023
$887$	8.16826	0.274263	0.137132	−	0.990553i	$-0.456212\pi$
$887$	8.16826	0.274263	0.137132	+	0.990553i	$0.456212\pi$
$888$	0	0
$889$	−15.4163	−0.517047
$890$	−25.6892	−0.861103
$891$	0	0
$892$	−113.808	−3.81056
$893$	−17.8691	−0.597966
$894$	0	0
$895$	−18.5005	−0.618404
$896$	13.6811	0.457053
$897$	0	0
$898$	−6.55059	−0.218596
$899$	50.1585	1.67288
$900$	0	0
$901$	30.8089	1.02639
$902$	−142.373	−4.74051
$903$	0	0
$904$	−58.3831	−1.94179
$905$	−5.02873	−0.167161
$906$	0	0
$907$	14.0702	0.467194	0.233597	−	0.972333i	$-0.424950\pi$
$907$	14.0702	0.467194	0.233597	+	0.972333i	$0.424950\pi$
$908$	−35.0880	−1.16444
$909$	0	0
$910$	0	0
$911$	−6.03489	−0.199945	−0.0999725	−	0.994990i	$-0.531875\pi$
$911$	−6.03489	−0.199945	−0.0999725	+	0.994990i	$0.531875\pi$
$912$	0	0
$913$	−41.5160	−1.37398
$914$	42.2052	1.39602
$915$	0	0
$916$	−115.183	−3.80575
$917$	−24.5015	−0.809111
$918$	0	0
$919$	14.2407	0.469757	0.234879	−	0.972025i	$-0.424531\pi$
$919$	14.2407	0.469757	0.234879	+	0.972025i	$0.424531\pi$
$920$	−6.19203	−0.204145
$921$	0	0
$922$	69.1595	2.27765
$923$	0	0
$924$	0	0
$925$	−0.0748598	−0.00246138
$926$	12.8307	0.421642
$927$	0	0
$928$	48.5763	1.59459
$929$	23.2586	0.763091	0.381545	−	0.924350i	$-0.375392\pi$
$929$	23.2586	0.763091	0.381545	+	0.924350i	$0.375392\pi$
$930$	0	0
$931$	−37.6336	−1.23339
$932$	−36.9346	−1.20983
$933$	0	0
$934$	−102.944	−3.36842
$935$	13.3143	0.435425
$936$	0	0
$937$	4.97074	0.162387	0.0811935	−	0.996698i	$-0.474127\pi$
$937$	4.97074	0.162387	0.0811935	+	0.996698i	$0.474127\pi$
$938$	48.7044	1.59025
$939$	0	0
$940$	10.6304	0.346726
$941$	19.0015	0.619432	0.309716	−	0.950829i	$-0.399766\pi$
$941$	19.0015	0.619432	0.309716	+	0.950829i	$0.399766\pi$
$942$	0	0
$943$	9.91395	0.322843
$944$	−14.2711	−0.464484
$945$	0	0
$946$	77.2432	2.51139
$947$	−4.97145	−0.161550	−0.0807752	−	0.996732i	$-0.525740\pi$
$947$	−4.97145	−0.161550	−0.0807752	+	0.996732i	$0.525740\pi$
$948$	0	0
$949$	0	0
$950$	19.6367	0.637099
$951$	0	0
$952$	−23.8600	−0.773308
$953$	−38.4337	−1.24499	−0.622495	−	0.782624i	$-0.713881\pi$
$953$	−38.4337	−1.24499	−0.622495	+	0.782624i	$0.713881\pi$
$954$	0	0
$955$	−11.0095	−0.356260
$956$	−87.7215	−2.83712
$957$	0	0
$958$	−29.0078	−0.937200
$959$	14.0571	0.453929
$960$	0	0
$961$	14.7578	0.476057
$962$	0	0
$963$	0	0
$964$	14.6314	0.471246
$965$	25.1511	0.809641
$966$	0	0
$967$	11.8497	0.381061	0.190530	−	0.981681i	$-0.438979\pi$
$967$	11.8497	0.381061	0.190530	+	0.981681i	$0.438979\pi$
$968$	111.597	3.58685
$969$	0	0
$970$	−10.9401	−0.351264
$971$	−25.8783	−0.830475	−0.415237	−	0.909713i	$-0.636301\pi$
$971$	−25.8783	−0.830475	−0.415237	+	0.909713i	$0.636301\pi$
$972$	0	0
$973$	1.80832	0.0579721
$974$	90.8036	2.90954
$975$	0	0
$976$	−15.8149	−0.506222
$977$	43.1220	1.37959	0.689797	−	0.724003i	$-0.257700\pi$
$977$	43.1220	1.37959	0.689797	+	0.724003i	$0.257700\pi$
$978$	0	0
$979$	−53.0833	−1.69655
$980$	22.3884	0.715173
$981$	0	0
$982$	−69.1798	−2.20762
$983$	−26.2863	−0.838404	−0.419202	−	0.907893i	$-0.637690\pi$
$983$	−26.2863	−0.838404	−0.419202	+	0.907893i	$0.637690\pi$
$984$	0	0
$985$	14.3338	0.456714
$986$	47.7774	1.52154
$987$	0	0
$988$	0	0
$989$	−5.37872	−0.171033
$990$	0	0
$991$	−17.3240	−0.550317	−0.275158	−	0.961399i	$-0.588730\pi$
$991$	−17.3240	−0.550317	−0.275158	+	0.961399i	$0.588730\pi$
$992$	44.3144	1.40698
$993$	0	0
$994$	−21.9985	−0.697749
$995$	−17.7358	−0.562262
$996$	0	0
$997$	−13.0489	−0.413262	−0.206631	−	0.978419i	$-0.566250\pi$
$997$	−13.0489	−0.413262	−0.206631	+	0.978419i	$0.566250\pi$
$998$	−46.6389	−1.47633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cx.1.12	yes	12
3.2	odd	2		7605.2.a.cv.1.1	✓	12
13.12	even	2		7605.2.a.cw.1.1	yes	12
39.38	odd	2		7605.2.a.cy.1.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7605.2.a.cv.1.1	✓	12	3.2	odd	2
7605.2.a.cw.1.1	yes	12	13.12	even	2
7605.2.a.cx.1.12	yes	12	1.1	even	1	trivial
7605.2.a.cy.1.12	yes	12	39.38	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56140 q^{2} +4.56078 q^{4} -1.00000 q^{5} -1.44606 q^{7} +6.55920 q^{8} +O(q^{10})\)	\(q+2.56140 q^{2} +4.56078 q^{4} -1.00000 q^{5} -1.44606 q^{7} +6.55920 q^{8} -2.56140 q^{10} -5.29280 q^{11} -3.70395 q^{14} +7.67918 q^{16} +2.51555 q^{17} +7.66639 q^{19} -4.56078 q^{20} -13.5570 q^{22} +0.944022 q^{23} +1.00000 q^{25} -6.59518 q^{28} +7.41501 q^{29} +6.76445 q^{31} +6.55107 q^{32} +6.44334 q^{34} +1.44606 q^{35} -0.0748598 q^{37} +19.6367 q^{38} -6.55920 q^{40} +10.5018 q^{41} -5.69766 q^{43} -24.1393 q^{44} +2.41802 q^{46} -2.33083 q^{47} -4.90890 q^{49} +2.56140 q^{50} +12.2474 q^{53} +5.29280 q^{55} -9.48501 q^{56} +18.9928 q^{58} -1.85841 q^{59} -2.05945 q^{61} +17.3265 q^{62} +1.42157 q^{64} -13.1493 q^{67} +11.4729 q^{68} +3.70395 q^{70} +5.93919 q^{71} -7.22147 q^{73} -0.191746 q^{74} +34.9648 q^{76} +7.65372 q^{77} +11.5410 q^{79} -7.67918 q^{80} +26.8994 q^{82} +7.84386 q^{83} -2.51555 q^{85} -14.5940 q^{86} -34.7165 q^{88} +10.0293 q^{89} +4.30548 q^{92} -5.97020 q^{94} -7.66639 q^{95} +4.27112 q^{97} -12.5737 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8	\( 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8} - q^{10} - 6 q^{11} + 18 q^{14} + 11 q^{16} + 14 q^{17} - 13 q^{20} - 12 q^{22} + 28 q^{23} + 12 q^{25} - 26 q^{28} + 24 q^{29} - 2 q^{31} + 7 q^{32} - 6 q^{34} + 12 q^{35} - 8 q^{37} + 8 q^{38} - 3 q^{40} - 4 q^{41} + 4 q^{43} - 30 q^{44} + 9 q^{46} + 22 q^{47} + 8 q^{49} + q^{50} + 36 q^{53} + 6 q^{55} + 32 q^{56} + 8 q^{58} - 28 q^{59} - 38 q^{61} + 26 q^{62} - 31 q^{64} - 22 q^{67} + 46 q^{68} - 18 q^{70} + 24 q^{71} - 12 q^{73} + 26 q^{74} + 17 q^{76} + 28 q^{77} + 20 q^{79} - 11 q^{80} - 36 q^{82} - 26 q^{83} - 14 q^{85} + 74 q^{86} - 34 q^{88} - 2 q^{89} + 86 q^{92} - 16 q^{94} - 26 q^{97} - 49 q^{98}+O(q^{100}) \) 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8 - q^10 - 6 * q^11 + 18 * q^14 + 11 * q^16 + 14 * q^17 - 13 * q^20 - 12 * q^22 + 28 * q^23 + 12 * q^25 - 26 * q^28 + 24 * q^29 - 2 * q^31 + 7 * q^32 - 6 * q^34 + 12 * q^35 - 8 * q^37 + 8 * q^38 - 3 * q^40 - 4 * q^41 + 4 * q^43 - 30 * q^44 + 9 * q^46 + 22 * q^47 + 8 * q^49 + q^50 + 36 * q^53 + 6 * q^55 + 32 * q^56 + 8 * q^58 - 28 * q^59 - 38 * q^61 + 26 * q^62 - 31 * q^64 - 22 * q^67 + 46 * q^68 - 18 * q^70 + 24 * q^71 - 12 * q^73 + 26 * q^74 + 17 * q^76 + 28 * q^77 + 20 * q^79 - 11 * q^80 - 36 * q^82 - 26 * q^83 - 14 * q^85 + 74 * q^86 - 34 * q^88 - 2 * q^89 + 86 * q^92 - 16 * q^94 - 26 * q^97 - 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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