LMFDB - Embedded newform 1925.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.13935$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13935 q^{2} -2.65419 q^{3} +2.57680 q^{4} +5.67823 q^{6} -1.00000 q^{7} -1.23398 q^{8} +4.04471 q^{9} +O(q^{10})$	\(q-2.13935 q^{2} -2.65419 q^{3} +2.57680 q^{4} +5.67823 q^{6} -1.00000 q^{7} -1.23398 q^{8} +4.04471 q^{9} +1.00000 q^{11} -6.83932 q^{12} -1.44724 q^{13} +2.13935 q^{14} -2.51369 q^{16} +4.97818 q^{17} -8.65304 q^{18} +3.57680 q^{19} +2.65419 q^{21} -2.13935 q^{22} -0.0935623 q^{23} +3.27522 q^{24} +3.09614 q^{26} -2.77286 q^{27} -2.57680 q^{28} -0.0286103 q^{29} +7.58466 q^{31} +7.84562 q^{32} -2.65419 q^{33} -10.6500 q^{34} +10.4224 q^{36} -1.43746 q^{37} -7.65202 q^{38} +3.84124 q^{39} -7.91106 q^{41} -5.67823 q^{42} -4.07439 q^{43} +2.57680 q^{44} +0.200162 q^{46} +3.86750 q^{47} +6.67181 q^{48} +1.00000 q^{49} -13.2130 q^{51} -3.72924 q^{52} -1.56147 q^{53} +5.93211 q^{54} +1.23398 q^{56} -9.49351 q^{57} +0.0612073 q^{58} -5.79690 q^{59} -3.97417 q^{61} -16.2262 q^{62} -4.04471 q^{63} -11.7571 q^{64} +5.67823 q^{66} +15.2119 q^{67} +12.8278 q^{68} +0.248332 q^{69} -16.4377 q^{71} -4.99110 q^{72} -14.3530 q^{73} +3.07522 q^{74} +9.21672 q^{76} -1.00000 q^{77} -8.21773 q^{78} +7.21037 q^{79} -4.77444 q^{81} +16.9245 q^{82} +1.87568 q^{83} +6.83932 q^{84} +8.71654 q^{86} +0.0759371 q^{87} -1.23398 q^{88} -9.24303 q^{89} +1.44724 q^{91} -0.241092 q^{92} -20.1311 q^{93} -8.27391 q^{94} -20.8238 q^{96} +2.93573 q^{97} -2.13935 q^{98} +4.04471 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9}+O(q^{10}) $ 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9	$ 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9} + 7 q^{11} + 9 q^{12} + 3 q^{13} + q^{14} + 21 q^{16} + 2 q^{17} - 8 q^{18} + 20 q^{19} - q^{22} - 11 q^{23} + 18 q^{24} - 13 q^{26} + 12 q^{27} - 13 q^{28} + 4 q^{29} + 6 q^{31} - q^{32} - 7 q^{34} + 12 q^{36} - 19 q^{37} - 3 q^{38} - 10 q^{39} + 24 q^{41} - 3 q^{42} + 13 q^{44} + 33 q^{46} + q^{47} + 15 q^{48} + 7 q^{49} + 19 q^{51} + 29 q^{52} - 7 q^{53} + 9 q^{54} + 9 q^{57} - 37 q^{58} + 9 q^{59} + 18 q^{61} + 40 q^{62} - 9 q^{63} + 8 q^{64} + 3 q^{66} - 18 q^{68} - 15 q^{69} + 18 q^{71} + 64 q^{72} - 5 q^{73} - 24 q^{74} + 88 q^{76} - 7 q^{77} - 79 q^{78} + 25 q^{79} - q^{81} + 60 q^{82} + 17 q^{83} - 9 q^{84} - 41 q^{86} + 24 q^{87} - 16 q^{89} - 3 q^{91} - 28 q^{92} - 26 q^{93} - 31 q^{94} + 17 q^{96} + 4 q^{97} - q^{98} + 9 q^{99}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9 + 7 * q^11 + 9 * q^12 + 3 * q^13 + q^14 + 21 * q^16 + 2 * q^17 - 8 * q^18 + 20 * q^19 - q^22 - 11 * q^23 + 18 * q^24 - 13 * q^26 + 12 * q^27 - 13 * q^28 + 4 * q^29 + 6 * q^31 - q^32 - 7 * q^34 + 12 * q^36 - 19 * q^37 - 3 * q^38 - 10 * q^39 + 24 * q^41 - 3 * q^42 + 13 * q^44 + 33 * q^46 + q^47 + 15 * q^48 + 7 * q^49 + 19 * q^51 + 29 * q^52 - 7 * q^53 + 9 * q^54 + 9 * q^57 - 37 * q^58 + 9 * q^59 + 18 * q^61 + 40 * q^62 - 9 * q^63 + 8 * q^64 + 3 * q^66 - 18 * q^68 - 15 * q^69 + 18 * q^71 + 64 * q^72 - 5 * q^73 - 24 * q^74 + 88 * q^76 - 7 * q^77 - 79 * q^78 + 25 * q^79 - q^81 + 60 * q^82 + 17 * q^83 - 9 * q^84 - 41 * q^86 + 24 * q^87 - 16 * q^89 - 3 * q^91 - 28 * q^92 - 26 * q^93 - 31 * q^94 + 17 * q^96 + 4 * q^97 - q^98 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.13935	−1.51275	−0.756373	−	0.654140i	$-0.773031\pi$
$2$	−2.13935	−1.51275	−0.756373	+	0.654140i	$0.773031\pi$
$3$	−2.65419	−1.53240	−0.766198	−	0.642605i	$-0.777854\pi$
$3$	−2.65419	−1.53240	−0.766198	+	0.642605i	$0.777854\pi$
$4$	2.57680	1.28840
$5$	0	0
$5$	0	0
$6$	5.67823	2.31813
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.23398	−0.436278
$9$	4.04471	1.34824
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−6.83932	−1.97434
$13$	−1.44724	−0.401391	−0.200696	−	0.979654i	$-0.564320\pi$
$13$	−1.44724	−0.401391	−0.200696	+	0.979654i	$0.564320\pi$
$14$	2.13935	0.571764
$15$	0	0
$16$	−2.51369	−0.628423
$17$	4.97818	1.20739	0.603693	−	0.797217i	$-0.293695\pi$
$17$	4.97818	1.20739	0.603693	+	0.797217i	$0.293695\pi$
$18$	−8.65304	−2.03954
$19$	3.57680	0.820575	0.410287	−	0.911956i	$-0.365428\pi$
$19$	3.57680	0.820575	0.410287	+	0.911956i	$0.365428\pi$
$20$	0	0
$21$	2.65419	0.579191
$22$	−2.13935	−0.456110
$23$	−0.0935623	−0.0195091	−0.00975455	−	0.999952i	$-0.503105\pi$
$23$	−0.0935623	−0.0195091	−0.00975455	+	0.999952i	$0.503105\pi$
$24$	3.27522	0.668551
$25$	0	0
$26$	3.09614	0.607203
$27$	−2.77286	−0.533637
$28$	−2.57680	−0.486970
$29$	−0.0286103	−0.00531280	−0.00265640	−	0.999996i	$-0.500846\pi$
$29$	−0.0286103	−0.00531280	−0.00265640	+	0.999996i	$0.500846\pi$
$30$	0	0
$31$	7.58466	1.36225	0.681123	−	0.732169i	$-0.261492\pi$
$31$	7.58466	1.36225	0.681123	+	0.732169i	$0.261492\pi$
$32$	7.84562	1.38692
$33$	−2.65419	−0.462035
$34$	−10.6500	−1.82647
$35$	0	0
$36$	10.4224	1.73707
$37$	−1.43746	−0.236316	−0.118158	−	0.992995i	$-0.537699\pi$
$37$	−1.43746	−0.236316	−0.118158	+	0.992995i	$0.537699\pi$
$38$	−7.65202	−1.24132
$39$	3.84124	0.615090
$40$	0	0
$41$	−7.91106	−1.23550	−0.617750	−	0.786375i	$-0.711956\pi$
$41$	−7.91106	−1.23550	−0.617750	+	0.786375i	$0.711956\pi$
$42$	−5.67823	−0.876169
$43$	−4.07439	−0.621339	−0.310670	−	0.950518i	$-0.600553\pi$
$43$	−4.07439	−0.621339	−0.310670	+	0.950518i	$0.600553\pi$
$44$	2.57680	0.388468
$45$	0	0
$46$	0.200162	0.0295123
$47$	3.86750	0.564132	0.282066	−	0.959395i	$-0.408980\pi$
$47$	3.86750	0.564132	0.282066	+	0.959395i	$0.408980\pi$
$48$	6.67181	0.962993
$49$	1.00000	0.142857
$50$	0	0
$51$	−13.2130	−1.85019
$52$	−3.72924	−0.517153
$53$	−1.56147	−0.214485	−0.107242	−	0.994233i	$-0.534202\pi$
$53$	−1.56147	−0.214485	−0.107242	+	0.994233i	$0.534202\pi$
$54$	5.93211	0.807258
$55$	0	0
$56$	1.23398	0.164898
$57$	−9.49351	−1.25745
$58$	0.0612073	0.00803691
$59$	−5.79690	−0.754692	−0.377346	−	0.926072i	$-0.623163\pi$
$59$	−5.79690	−0.754692	−0.377346	+	0.926072i	$0.623163\pi$
$60$	0	0
$61$	−3.97417	−0.508840	−0.254420	−	0.967094i	$-0.581885\pi$
$61$	−3.97417	−0.508840	−0.254420	+	0.967094i	$0.581885\pi$
$62$	−16.2262	−2.06073
$63$	−4.04471	−0.509586
$64$	−11.7571	−1.46964
$65$	0	0
$66$	5.67823	0.698941
$67$	15.2119	1.85844	0.929218	−	0.369533i	$-0.120482\pi$
$67$	15.2119	1.85844	0.929218	+	0.369533i	$0.120482\pi$
$68$	12.8278	1.55560
$69$	0.248332	0.0298957
$70$	0	0
$71$	−16.4377	−1.95079	−0.975395	−	0.220463i	$-0.929243\pi$
$71$	−16.4377	−1.95079	−0.975395	+	0.220463i	$0.929243\pi$
$72$	−4.99110	−0.588206
$73$	−14.3530	−1.67989	−0.839946	−	0.542669i	$-0.817414\pi$
$73$	−14.3530	−1.67989	−0.839946	+	0.542669i	$0.817414\pi$
$74$	3.07522	0.357487
$75$	0	0
$76$	9.21672	1.05723
$77$	−1.00000	−0.113961
$78$	−8.21773	−0.930475
$79$	7.21037	0.811231	0.405615	−	0.914044i	$-0.367057\pi$
$79$	7.21037	0.811231	0.405615	+	0.914044i	$0.367057\pi$
$80$	0	0
$81$	−4.77444	−0.530494
$82$	16.9245	1.86900
$83$	1.87568	0.205883	0.102941	−	0.994687i	$-0.467175\pi$
$83$	1.87568	0.205883	0.102941	+	0.994687i	$0.467175\pi$
$84$	6.83932	0.746231
$85$	0	0
$86$	8.71654	0.939929
$87$	0.0759371	0.00814131
$88$	−1.23398	−0.131543
$89$	−9.24303	−0.979759	−0.489880	−	0.871790i	$-0.662959\pi$
$89$	−9.24303	−0.979759	−0.489880	+	0.871790i	$0.662959\pi$
$90$	0	0
$91$	1.44724	0.151712
$92$	−0.241092	−0.0251355
$93$	−20.1311	−2.08750
$94$	−8.27391	−0.853389
$95$	0	0
$96$	−20.8238	−2.12532
$97$	2.93573	0.298078	0.149039	−	0.988831i	$-0.452382\pi$
$97$	2.93573	0.298078	0.149039	+	0.988831i	$0.452382\pi$
$98$	−2.13935	−0.216107
$99$	4.04471	0.406509
$100$	0	0
$101$	−1.04693	−0.104174	−0.0520868	−	0.998643i	$-0.516587\pi$
$101$	−1.04693	−0.104174	−0.0520868	+	0.998643i	$0.516587\pi$
$102$	28.2672	2.79887
$103$	11.1215	1.09583	0.547917	−	0.836532i	$-0.315421\pi$
$103$	11.1215	1.09583	0.547917	+	0.836532i	$0.315421\pi$
$104$	1.78586	0.175118
$105$	0	0
$106$	3.34053	0.324461
$107$	3.91977	0.378938	0.189469	−	0.981887i	$-0.439323\pi$
$107$	3.91977	0.378938	0.189469	+	0.981887i	$0.439323\pi$
$108$	−7.14512	−0.687539
$109$	14.2332	1.36329	0.681647	−	0.731681i	$-0.261264\pi$
$109$	14.2332	1.36329	0.681647	+	0.731681i	$0.261264\pi$
$110$	0	0
$111$	3.81528	0.362130
$112$	2.51369	0.237522
$113$	−17.5713	−1.65297	−0.826483	−	0.562962i	$-0.809662\pi$
$113$	−17.5713	−1.65297	−0.826483	+	0.562962i	$0.809662\pi$
$114$	20.3099	1.90220
$115$	0	0
$116$	−0.0737231	−0.00684502
$117$	−5.85365	−0.541170
$118$	12.4016	1.14166
$119$	−4.97818	−0.456349
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.50212	0.769746
$123$	20.9974	1.89327
$124$	19.5442	1.75512
$125$	0	0
$126$	8.65304	0.770874
$127$	0.0853402	0.00757272	0.00378636	−	0.999993i	$-0.498795\pi$
$127$	0.0853402	0.00757272	0.00378636	+	0.999993i	$0.498795\pi$
$128$	9.46130	0.836269
$129$	10.8142	0.952138
$130$	0	0
$131$	16.2223	1.41735	0.708675	−	0.705535i	$-0.249293\pi$
$131$	16.2223	1.41735	0.708675	+	0.705535i	$0.249293\pi$
$132$	−6.83932	−0.595286
$133$	−3.57680	−0.310148
$134$	−32.5436	−2.81134
$135$	0	0
$136$	−6.14298	−0.526756
$137$	1.63402	0.139604	0.0698020	−	0.997561i	$-0.477763\pi$
$137$	1.63402	0.139604	0.0698020	+	0.997561i	$0.477763\pi$
$138$	−0.531268	−0.0452245
$139$	2.74737	0.233029	0.116515	−	0.993189i	$-0.462828\pi$
$139$	2.74737	0.233029	0.116515	+	0.993189i	$0.462828\pi$
$140$	0	0
$141$	−10.2651	−0.864474
$142$	35.1658	2.95105
$143$	−1.44724	−0.121024
$144$	−10.1672	−0.847264
$145$	0	0
$146$	30.7061	2.54125
$147$	−2.65419	−0.218914
$148$	−3.70404	−0.304470
$149$	−3.13184	−0.256570	−0.128285	−	0.991737i	$-0.540947\pi$
$149$	−3.13184	−0.256570	−0.128285	+	0.991737i	$0.540947\pi$
$150$	0	0
$151$	7.52290	0.612205	0.306102	−	0.951999i	$-0.400975\pi$
$151$	7.52290	0.612205	0.306102	+	0.951999i	$0.400975\pi$
$152$	−4.41371	−0.357999
$153$	20.1353	1.62784
$154$	2.13935	0.172393
$155$	0	0
$156$	9.89811	0.792483
$157$	23.0371	1.83856	0.919280	−	0.393605i	$-0.128772\pi$
$157$	23.0371	1.83856	0.919280	+	0.393605i	$0.128772\pi$
$158$	−15.4255	−1.22719
$159$	4.14444	0.328675
$160$	0	0
$161$	0.0935623	0.00737374
$162$	10.2142	0.802502
$163$	−19.9728	−1.56439	−0.782196	−	0.623033i	$-0.785900\pi$
$163$	−19.9728	−1.56439	−0.782196	+	0.623033i	$0.785900\pi$
$164$	−20.3852	−1.59182
$165$	0	0
$166$	−4.01274	−0.311449
$167$	0.512923	0.0396912	0.0198456	−	0.999803i	$-0.493683\pi$
$167$	0.512923	0.0396912	0.0198456	+	0.999803i	$0.493683\pi$
$168$	−3.27522	−0.252688
$169$	−10.9055	−0.838885
$170$	0	0
$171$	14.4671	1.10633
$172$	−10.4989	−0.800534
$173$	21.5953	1.64186	0.820930	−	0.571028i	$-0.193455\pi$
$173$	21.5953	1.64186	0.820930	+	0.571028i	$0.193455\pi$
$174$	−0.162456	−0.0123157
$175$	0	0
$176$	−2.51369	−0.189477
$177$	15.3861	1.15649
$178$	19.7740	1.48213
$179$	12.2476	0.915430	0.457715	−	0.889099i	$-0.348668\pi$
$179$	12.2476	0.915430	0.457715	+	0.889099i	$0.348668\pi$
$180$	0	0
$181$	20.6384	1.53404	0.767019	−	0.641625i	$-0.221739\pi$
$181$	20.6384	1.53404	0.767019	+	0.641625i	$0.221739\pi$
$182$	−3.09614	−0.229501
$183$	10.5482	0.779745
$184$	0.115454	0.00851139
$185$	0	0
$186$	43.0674	3.15786
$187$	4.97818	0.364040
$188$	9.96577	0.726829
$189$	2.77286	0.201696
$190$	0	0
$191$	−0.644390	−0.0466264	−0.0233132	−	0.999728i	$-0.507421\pi$
$191$	−0.644390	−0.0466264	−0.0233132	+	0.999728i	$0.507421\pi$
$192$	31.2056	2.25207
$193$	12.4834	0.898574	0.449287	−	0.893388i	$-0.351678\pi$
$193$	12.4834	0.898574	0.449287	+	0.893388i	$0.351678\pi$
$194$	−6.28053	−0.450916
$195$	0	0
$196$	2.57680	0.184057
$197$	−12.7791	−0.910470	−0.455235	−	0.890371i	$-0.650445\pi$
$197$	−12.7791	−0.910470	−0.455235	+	0.890371i	$0.650445\pi$
$198$	−8.65304	−0.614945
$199$	8.77764	0.622230	0.311115	−	0.950372i	$-0.399297\pi$
$199$	8.77764	0.622230	0.311115	+	0.950372i	$0.399297\pi$
$200$	0	0
$201$	−40.3754	−2.84786
$202$	2.23975	0.157588
$203$	0.0286103	0.00200805
$204$	−34.0473	−2.38379
$205$	0	0
$206$	−23.7928	−1.65772
$207$	−0.378433	−0.0263029
$208$	3.63791	0.252243
$209$	3.57680	0.247413
$210$	0	0
$211$	−16.4338	−1.13135	−0.565675	−	0.824628i	$-0.691384\pi$
$211$	−16.4338	−1.13135	−0.565675	+	0.824628i	$0.691384\pi$
$212$	−4.02360	−0.276342
$213$	43.6286	2.98938
$214$	−8.38574	−0.573238
$215$	0	0
$216$	3.42166	0.232814
$217$	−7.58466	−0.514881
$218$	−30.4498	−2.06232
$219$	38.0956	2.57426
$220$	0	0
$221$	−7.20460	−0.484634
$222$	−8.16220	−0.547811
$223$	9.47714	0.634636	0.317318	−	0.948319i	$-0.397218\pi$
$223$	9.47714	0.634636	0.317318	+	0.948319i	$0.397218\pi$
$224$	−7.84562	−0.524208
$225$	0	0
$226$	37.5910	2.50052
$227$	18.1313	1.20342	0.601709	−	0.798716i	$-0.294487\pi$
$227$	18.1313	1.20342	0.601709	+	0.798716i	$0.294487\pi$
$228$	−24.4629	−1.62009
$229$	19.3600	1.27935	0.639674	−	0.768646i	$-0.279069\pi$
$229$	19.3600	1.27935	0.639674	+	0.768646i	$0.279069\pi$
$230$	0	0
$231$	2.65419	0.174633
$232$	0.0353046	0.00231786
$233$	−21.6903	−1.42098	−0.710489	−	0.703709i	$-0.751526\pi$
$233$	−21.6903	−1.42098	−0.710489	+	0.703709i	$0.751526\pi$
$234$	12.5230	0.818654
$235$	0	0
$236$	−14.9375	−0.972346
$237$	−19.1377	−1.24313
$238$	10.6500	0.690340
$239$	17.1784	1.11118	0.555590	−	0.831456i	$-0.312492\pi$
$239$	17.1784	1.11118	0.555590	+	0.831456i	$0.312492\pi$
$240$	0	0
$241$	17.3725	1.11906	0.559531	−	0.828809i	$-0.310981\pi$
$241$	17.3725	1.11906	0.559531	+	0.828809i	$0.310981\pi$
$242$	−2.13935	−0.137522
$243$	20.9908	1.34656
$244$	−10.2406	−0.655590
$245$	0	0
$246$	−44.9208	−2.86404
$247$	−5.17648	−0.329371
$248$	−9.35933	−0.594318
$249$	−4.97841	−0.315494
$250$	0	0
$251$	4.81052	0.303637	0.151819	−	0.988408i	$-0.451487\pi$
$251$	4.81052	0.303637	0.151819	+	0.988408i	$0.451487\pi$
$252$	−10.4224	−0.656551
$253$	−0.0935623	−0.00588221
$254$	−0.182572	−0.0114556
$255$	0	0
$256$	3.27323	0.204577
$257$	−26.0449	−1.62464	−0.812318	−	0.583214i	$-0.801795\pi$
$257$	−26.0449	−1.62464	−0.812318	+	0.583214i	$0.801795\pi$
$258$	−23.1353	−1.44034
$259$	1.43746	0.0893192
$260$	0	0
$261$	−0.115720	−0.00716291
$262$	−34.7052	−2.14409
$263$	24.8179	1.53034	0.765168	−	0.643830i	$-0.222656\pi$
$263$	24.8179	1.53034	0.765168	+	0.643830i	$0.222656\pi$
$264$	3.27522	0.201576
$265$	0	0
$266$	7.65202	0.469175
$267$	24.5327	1.50138
$268$	39.1982	2.39441
$269$	−28.7038	−1.75010	−0.875050	−	0.484032i	$-0.839172\pi$
$269$	−28.7038	−1.75010	−0.875050	+	0.484032i	$0.839172\pi$
$270$	0	0
$271$	−16.7160	−1.01543	−0.507714	−	0.861526i	$-0.669509\pi$
$271$	−16.7160	−1.01543	−0.507714	+	0.861526i	$0.669509\pi$
$272$	−12.5136	−0.758749
$273$	−3.84124	−0.232482
$274$	−3.49574	−0.211185
$275$	0	0
$276$	0.639903	0.0385176
$277$	−10.3098	−0.619458	−0.309729	−	0.950825i	$-0.600238\pi$
$277$	−10.3098	−0.619458	−0.309729	+	0.950825i	$0.600238\pi$
$278$	−5.87759	−0.352514
$279$	30.6778	1.83663
$280$	0	0
$281$	26.7582	1.59626	0.798131	−	0.602483i	$-0.205822\pi$
$281$	26.7582	1.59626	0.798131	+	0.602483i	$0.205822\pi$
$282$	21.9605	1.30773
$283$	−7.46951	−0.444016	−0.222008	−	0.975045i	$-0.571261\pi$
$283$	−7.46951	−0.444016	−0.222008	+	0.975045i	$0.571261\pi$
$284$	−42.3566	−2.51340
$285$	0	0
$286$	3.09614	0.183079
$287$	7.91106	0.466975
$288$	31.7333	1.86990
$289$	7.78227	0.457780
$290$	0	0
$291$	−7.79197	−0.456773
$292$	−36.9849	−2.16438
$293$	18.3741	1.07342	0.536712	−	0.843766i	$-0.319666\pi$
$293$	18.3741	1.07342	0.536712	+	0.843766i	$0.319666\pi$
$294$	5.67823	0.331161
$295$	0	0
$296$	1.77379	0.103100
$297$	−2.77286	−0.160898
$298$	6.70009	0.388126
$299$	0.135407	0.00783078
$300$	0	0
$301$	4.07439	0.234844
$302$	−16.0941	−0.926110
$303$	2.77875	0.159635
$304$	−8.99098	−0.515668
$305$	0	0
$306$	−43.0764	−2.46251
$307$	20.9302	1.19455	0.597274	−	0.802037i	$-0.296250\pi$
$307$	20.9302	1.19455	0.597274	+	0.802037i	$0.296250\pi$
$308$	−2.57680	−0.146827
$309$	−29.5186	−1.67925
$310$	0	0
$311$	−1.00226	−0.0568329	−0.0284165	−	0.999596i	$-0.509046\pi$
$311$	−1.00226	−0.0568329	−0.0284165	+	0.999596i	$0.509046\pi$
$312$	−4.74001	−0.268350
$313$	22.6903	1.28253	0.641265	−	0.767319i	$-0.278410\pi$
$313$	22.6903	1.28253	0.641265	+	0.767319i	$0.278410\pi$
$314$	−49.2843	−2.78127
$315$	0	0
$316$	18.5797	1.04519
$317$	−30.1883	−1.69555	−0.847773	−	0.530359i	$-0.822057\pi$
$317$	−30.1883	−1.69555	−0.847773	+	0.530359i	$0.822057\pi$
$318$	−8.86639	−0.497202
$319$	−0.0286103	−0.00160187
$320$	0	0
$321$	−10.4038	−0.580684
$322$	−0.200162	−0.0111546
$323$	17.8060	0.990750
$324$	−12.3028	−0.683489
$325$	0	0
$326$	42.7288	2.36653
$327$	−37.7776	−2.08911
$328$	9.76210	0.539022
$329$	−3.86750	−0.213222
$330$	0	0
$331$	12.3262	0.677511	0.338755	−	0.940875i	$-0.389994\pi$
$331$	12.3262	0.677511	0.338755	+	0.940875i	$0.389994\pi$
$332$	4.83326	0.265260
$333$	−5.81410	−0.318611
$334$	−1.09732	−0.0600427
$335$	0	0
$336$	−6.67181	−0.363977
$337$	−9.15370	−0.498634	−0.249317	−	0.968422i	$-0.580206\pi$
$337$	−9.15370	−0.498634	−0.249317	+	0.968422i	$0.580206\pi$
$338$	23.3307	1.26902
$339$	46.6374	2.53300
$340$	0	0
$341$	7.58466	0.410733
$342$	−30.9502	−1.67360
$343$	−1.00000	−0.0539949
$344$	5.02772	0.271077
$345$	0	0
$346$	−46.1998	−2.48372
$347$	−17.8726	−0.959451	−0.479725	−	0.877419i	$-0.659264\pi$
$347$	−17.8726	−0.959451	−0.479725	+	0.877419i	$0.659264\pi$
$348$	0.195675	0.0104893
$349$	27.2960	1.46112	0.730562	−	0.682847i	$-0.239258\pi$
$349$	27.2960	1.46112	0.730562	+	0.682847i	$0.239258\pi$
$350$	0	0
$351$	4.01298	0.214197
$352$	7.84562	0.418173
$353$	10.1063	0.537904	0.268952	−	0.963154i	$-0.413323\pi$
$353$	10.1063	0.537904	0.268952	+	0.963154i	$0.413323\pi$
$354$	−32.9161	−1.74947
$355$	0	0
$356$	−23.8175	−1.26232
$357$	13.2130	0.699307
$358$	−26.2019	−1.38481
$359$	−16.4626	−0.868865	−0.434433	−	0.900704i	$-0.643051\pi$
$359$	−16.4626	−0.868865	−0.434433	+	0.900704i	$0.643051\pi$
$360$	0	0
$361$	−6.20648	−0.326657
$362$	−44.1526	−2.32061
$363$	−2.65419	−0.139309
$364$	3.72924	0.195465
$365$	0	0
$366$	−22.5662	−1.17956
$367$	−8.01567	−0.418414	−0.209207	−	0.977871i	$-0.567088\pi$
$367$	−8.01567	−0.418414	−0.209207	+	0.977871i	$0.567088\pi$
$368$	0.235187	0.0122600
$369$	−31.9980	−1.66575
$370$	0	0
$371$	1.56147	0.0810676
$372$	−51.8739	−2.68954
$373$	19.5753	1.01357	0.506785	−	0.862072i	$-0.330834\pi$
$373$	19.5753	1.01357	0.506785	+	0.862072i	$0.330834\pi$
$374$	−10.6500	−0.550701
$375$	0	0
$376$	−4.77242	−0.246119
$377$	0.0414058	0.00213251
$378$	−5.93211	−0.305115
$379$	8.43035	0.433038	0.216519	−	0.976278i	$-0.430530\pi$
$379$	8.43035	0.433038	0.216519	+	0.976278i	$0.430530\pi$
$380$	0	0
$381$	−0.226509	−0.0116044
$382$	1.37857	0.0705339
$383$	−10.0136	−0.511671	−0.255835	−	0.966720i	$-0.582351\pi$
$383$	−10.0136	−0.511671	−0.255835	+	0.966720i	$0.582351\pi$
$384$	−25.1121	−1.28149
$385$	0	0
$386$	−26.7063	−1.35931
$387$	−16.4798	−0.837713
$388$	7.56479	0.384044
$389$	−10.3628	−0.525416	−0.262708	−	0.964875i	$-0.584616\pi$
$389$	−10.3628	−0.525416	−0.262708	+	0.964875i	$0.584616\pi$
$390$	0	0
$391$	−0.465770	−0.0235550
$392$	−1.23398	−0.0623254
$393$	−43.0571	−2.17194
$394$	27.3388	1.37731
$395$	0	0
$396$	10.4224	0.523747
$397$	16.4804	0.827127	0.413563	−	0.910475i	$-0.364284\pi$
$397$	16.4804	0.827127	0.413563	+	0.910475i	$0.364284\pi$
$398$	−18.7784	−0.941277
$399$	9.49351	0.475270
$400$	0	0
$401$	33.5768	1.67674	0.838372	−	0.545099i	$-0.183508\pi$
$401$	33.5768	1.67674	0.838372	+	0.545099i	$0.183508\pi$
$402$	86.3769	4.30809
$403$	−10.9768	−0.546793
$404$	−2.69774	−0.134217
$405$	0	0
$406$	−0.0612073	−0.00303767
$407$	−1.43746	−0.0712521
$408$	16.3046	0.807199
$409$	21.9354	1.08464	0.542318	−	0.840173i	$-0.317547\pi$
$409$	21.9354	1.08464	0.542318	+	0.840173i	$0.317547\pi$
$410$	0	0
$411$	−4.33700	−0.213929
$412$	28.6579	1.41188
$413$	5.79690	0.285247
$414$	0.809599	0.0397896
$415$	0	0
$416$	−11.3545	−0.556699
$417$	−7.29205	−0.357093
$418$	−7.65202	−0.374273
$419$	−17.9475	−0.876794	−0.438397	−	0.898781i	$-0.644454\pi$
$419$	−17.9475	−0.876794	−0.438397	+	0.898781i	$0.644454\pi$
$420$	0	0
$421$	−7.45515	−0.363342	−0.181671	−	0.983359i	$-0.558151\pi$
$421$	−7.45515	−0.363342	−0.181671	+	0.983359i	$0.558151\pi$
$422$	35.1576	1.71145
$423$	15.6429	0.760584
$424$	1.92683	0.0935749
$425$	0	0
$426$	−93.3367	−4.52218
$427$	3.97417	0.192324
$428$	10.1005	0.488225
$429$	3.84124	0.185457
$430$	0	0
$431$	9.54762	0.459893	0.229946	−	0.973203i	$-0.426145\pi$
$431$	9.54762	0.459893	0.229946	+	0.973203i	$0.426145\pi$
$432$	6.97012	0.335350
$433$	−28.0455	−1.34778	−0.673890	−	0.738832i	$-0.735378\pi$
$433$	−28.0455	−1.34778	−0.673890	+	0.738832i	$0.735378\pi$
$434$	16.2262	0.778884
$435$	0	0
$436$	36.6762	1.75647
$437$	−0.334654	−0.0160087
$438$	−81.4996	−3.89420
$439$	−5.54490	−0.264643	−0.132322	−	0.991207i	$-0.542243\pi$
$439$	−5.54490	−0.264643	−0.132322	+	0.991207i	$0.542243\pi$
$440$	0	0
$441$	4.04471	0.192605
$442$	15.4131	0.733128
$443$	−20.1025	−0.955097	−0.477548	−	0.878606i	$-0.658474\pi$
$443$	−20.1025	−0.955097	−0.477548	+	0.878606i	$0.658474\pi$
$444$	9.83122	0.466569
$445$	0	0
$446$	−20.2749	−0.960044
$447$	8.31249	0.393168
$448$	11.7571	0.555472
$449$	3.41450	0.161140	0.0805700	−	0.996749i	$-0.474326\pi$
$449$	3.41450	0.161140	0.0805700	+	0.996749i	$0.474326\pi$
$450$	0	0
$451$	−7.91106	−0.372517
$452$	−45.2777	−2.12968
$453$	−19.9672	−0.938140
$454$	−38.7892	−1.82047
$455$	0	0
$456$	11.7148	0.548596
$457$	−38.2390	−1.78875	−0.894373	−	0.447323i	$-0.852378\pi$
$457$	−38.2390	−1.78875	−0.894373	+	0.447323i	$0.852378\pi$
$458$	−41.4178	−1.93533
$459$	−13.8038	−0.644306
$460$	0	0
$461$	−4.87777	−0.227180	−0.113590	−	0.993528i	$-0.536235\pi$
$461$	−4.87777	−0.227180	−0.113590	+	0.993528i	$0.536235\pi$
$462$	−5.67823	−0.264175
$463$	−27.2273	−1.26536	−0.632679	−	0.774414i	$-0.718045\pi$
$463$	−27.2273	−1.26536	−0.632679	+	0.774414i	$0.718045\pi$
$464$	0.0719175	0.00333869
$465$	0	0
$466$	46.4030	2.14958
$467$	19.2412	0.890377	0.445189	−	0.895437i	$-0.353137\pi$
$467$	19.2412	0.890377	0.445189	+	0.895437i	$0.353137\pi$
$468$	−15.0837	−0.697245
$469$	−15.2119	−0.702423
$470$	0	0
$471$	−61.1447	−2.81740
$472$	7.15326	0.329256
$473$	−4.07439	−0.187341
$474$	40.9421	1.88053
$475$	0	0
$476$	−12.8278	−0.587961
$477$	−6.31570	−0.289176
$478$	−36.7506	−1.68093
$479$	11.7101	0.535048	0.267524	−	0.963551i	$-0.413795\pi$
$479$	11.7101	0.535048	0.267524	+	0.963551i	$0.413795\pi$
$480$	0	0
$481$	2.08034	0.0948553
$482$	−37.1659	−1.69286
$483$	−0.248332	−0.0112995
$484$	2.57680	0.117127
$485$	0	0
$486$	−44.9067	−2.03701
$487$	−5.85079	−0.265125	−0.132562	−	0.991175i	$-0.542320\pi$
$487$	−5.85079	−0.265125	−0.132562	+	0.991175i	$0.542320\pi$
$488$	4.90405	0.221996
$489$	53.0116	2.39727
$490$	0	0
$491$	38.8495	1.75325	0.876627	−	0.481171i	$-0.159788\pi$
$491$	38.8495	1.75325	0.876627	+	0.481171i	$0.159788\pi$
$492$	54.1062	2.43930
$493$	−0.142427	−0.00641460
$494$	11.0743	0.498255
$495$	0	0
$496$	−19.0655	−0.856067
$497$	16.4377	0.737330
$498$	10.6506	0.477263
$499$	13.3622	0.598174	0.299087	−	0.954226i	$-0.403318\pi$
$499$	13.3622	0.598174	0.299087	+	0.954226i	$0.403318\pi$
$500$	0	0
$501$	−1.36139	−0.0608226
$502$	−10.2914	−0.459326
$503$	−7.11973	−0.317453	−0.158727	−	0.987323i	$-0.550739\pi$
$503$	−7.11973	−0.317453	−0.158727	+	0.987323i	$0.550739\pi$
$504$	4.99110	0.222321
$505$	0	0
$506$	0.200162	0.00889830
$507$	28.9453	1.28550
$508$	0.219905	0.00975670
$509$	−40.4144	−1.79134	−0.895668	−	0.444723i	$-0.853302\pi$
$509$	−40.4144	−1.79134	−0.895668	+	0.444723i	$0.853302\pi$
$510$	0	0
$511$	14.3530	0.634940
$512$	−25.9252	−1.14574
$513$	−9.91798	−0.437889
$514$	55.7191	2.45766
$515$	0	0
$516$	27.8661	1.22674
$517$	3.86750	0.170092
$518$	−3.07522	−0.135117
$519$	−57.3180	−2.51598
$520$	0	0
$521$	19.0269	0.833582	0.416791	−	0.909002i	$-0.363155\pi$
$521$	19.0269	0.833582	0.416791	+	0.909002i	$0.363155\pi$
$522$	0.247566	0.0108357
$523$	33.7332	1.47505	0.737525	−	0.675320i	$-0.235994\pi$
$523$	33.7332	1.47505	0.737525	+	0.675320i	$0.235994\pi$
$524$	41.8017	1.82612
$525$	0	0
$526$	−53.0941	−2.31501
$527$	37.7578	1.64476
$528$	6.67181	0.290353
$529$	−22.9912	−0.999619
$530$	0	0
$531$	−23.4468	−1.01750
$532$	−9.21672	−0.399595
$533$	11.4492	0.495919
$534$	−52.4840	−2.27121
$535$	0	0
$536$	−18.7713	−0.810795
$537$	−32.5075	−1.40280
$538$	61.4073	2.64746
$539$	1.00000	0.0430730
$540$	0	0
$541$	1.08651	0.0467127	0.0233563	−	0.999727i	$-0.492565\pi$
$541$	1.08651	0.0467127	0.0233563	+	0.999727i	$0.492565\pi$
$542$	35.7614	1.53608
$543$	−54.7781	−2.35075
$544$	39.0569	1.67455
$545$	0	0
$546$	8.21773	0.351687
$547$	29.7693	1.27284	0.636422	−	0.771341i	$-0.280414\pi$
$547$	29.7693	1.27284	0.636422	+	0.771341i	$0.280414\pi$
$548$	4.21056	0.179866
$549$	−16.0744	−0.686037
$550$	0	0
$551$	−0.102333	−0.00435955
$552$	−0.306437	−0.0130428
$553$	−7.21037	−0.306616
$554$	22.0563	0.937083
$555$	0	0
$556$	7.07944	0.300235
$557$	−31.4593	−1.33297	−0.666486	−	0.745517i	$-0.732202\pi$
$557$	−31.4593	−1.33297	−0.666486	+	0.745517i	$0.732202\pi$
$558$	−65.6304	−2.77836
$559$	5.89661	0.249400
$560$	0	0
$561$	−13.2130	−0.557854
$562$	−57.2451	−2.41474
$563$	−10.0869	−0.425114	−0.212557	−	0.977149i	$-0.568179\pi$
$563$	−10.0869	−0.425114	−0.212557	+	0.977149i	$0.568179\pi$
$564$	−26.4510	−1.11379
$565$	0	0
$566$	15.9799	0.671684
$567$	4.77444	0.200508
$568$	20.2838	0.851087
$569$	32.8331	1.37643	0.688217	−	0.725505i	$-0.258394\pi$
$569$	32.8331	1.37643	0.688217	+	0.725505i	$0.258394\pi$
$570$	0	0
$571$	33.5869	1.40557	0.702784	−	0.711403i	$-0.251940\pi$
$571$	33.5869	1.40557	0.702784	+	0.711403i	$0.251940\pi$
$572$	−3.72924	−0.155927
$573$	1.71033	0.0714501
$574$	−16.9245	−0.706415
$575$	0	0
$576$	−47.5541	−1.98142
$577$	44.4427	1.85017	0.925087	−	0.379754i	$-0.123991\pi$
$577$	44.4427	1.85017	0.925087	+	0.379754i	$0.123991\pi$
$578$	−16.6490	−0.692505
$579$	−33.1332	−1.37697
$580$	0	0
$581$	−1.87568	−0.0778164
$582$	16.6697	0.690982
$583$	−1.56147	−0.0646695
$584$	17.7113	0.732901
$585$	0	0
$586$	−39.3085	−1.62382
$587$	20.1796	0.832899	0.416450	−	0.909159i	$-0.363274\pi$
$587$	20.1796	0.832899	0.416450	+	0.909159i	$0.363274\pi$
$588$	−6.83932	−0.282049
$589$	27.1288	1.11782
$590$	0	0
$591$	33.9180	1.39520
$592$	3.61332	0.148507
$593$	43.3635	1.78072	0.890362	−	0.455254i	$-0.150451\pi$
$593$	43.3635	1.78072	0.890362	+	0.455254i	$0.150451\pi$
$594$	5.93211	0.243397
$595$	0	0
$596$	−8.07014	−0.330566
$597$	−23.2975	−0.953503
$598$	−0.289682	−0.0118460
$599$	31.5041	1.28722	0.643612	−	0.765352i	$-0.277435\pi$
$599$	31.5041	1.28722	0.643612	+	0.765352i	$0.277435\pi$
$600$	0	0
$601$	26.9776	1.10044	0.550221	−	0.835019i	$-0.314544\pi$
$601$	26.9776	1.10044	0.550221	+	0.835019i	$0.314544\pi$
$602$	−8.71654	−0.355260
$603$	61.5279	2.50561
$604$	19.3850	0.788765
$605$	0	0
$606$	−5.94472	−0.241488
$607$	46.7606	1.89796	0.948978	−	0.315343i	$-0.102120\pi$
$607$	46.7606	1.89796	0.948978	+	0.315343i	$0.102120\pi$
$608$	28.0622	1.13807
$609$	−0.0759371	−0.00307713
$610$	0	0
$611$	−5.59718	−0.226438
$612$	51.8847	2.09731
$613$	−16.0111	−0.646684	−0.323342	−	0.946282i	$-0.604806\pi$
$613$	−16.0111	−0.646684	−0.323342	+	0.946282i	$0.604806\pi$
$614$	−44.7769	−1.80705
$615$	0	0
$616$	1.23398	0.0497185
$617$	6.41357	0.258201	0.129100	−	0.991632i	$-0.458791\pi$
$617$	6.41357	0.258201	0.129100	+	0.991632i	$0.458791\pi$
$618$	63.1505	2.54028
$619$	23.2515	0.934556	0.467278	−	0.884111i	$-0.345235\pi$
$619$	23.2515	0.934556	0.467278	+	0.884111i	$0.345235\pi$
$620$	0	0
$621$	0.259435	0.0104108
$622$	2.14418	0.0859738
$623$	9.24303	0.370314
$624$	−9.65569	−0.386537
$625$	0	0
$626$	−48.5424	−1.94014
$627$	−9.49351	−0.379134
$628$	59.3620	2.36880
$629$	−7.15592	−0.285325
$630$	0	0
$631$	−38.8561	−1.54684	−0.773419	−	0.633895i	$-0.781455\pi$
$631$	−38.8561	−1.54684	−0.773419	+	0.633895i	$0.781455\pi$
$632$	−8.89746	−0.353922
$633$	43.6184	1.73368
$634$	64.5833	2.56493
$635$	0	0
$636$	10.6794	0.423466
$637$	−1.44724	−0.0573416
$638$	0.0612073	0.00242322
$639$	−66.4856	−2.63013
$640$	0	0
$641$	2.82122	0.111432	0.0557158	−	0.998447i	$-0.482256\pi$
$641$	2.82122	0.111432	0.0557158	+	0.998447i	$0.482256\pi$
$642$	22.2573	0.878427
$643$	−8.39312	−0.330992	−0.165496	−	0.986210i	$-0.552923\pi$
$643$	−8.39312	−0.330992	−0.165496	+	0.986210i	$0.552923\pi$
$644$	0.241092	0.00950034
$645$	0	0
$646$	−38.0931	−1.49875
$647$	−2.37335	−0.0933060	−0.0466530	−	0.998911i	$-0.514856\pi$
$647$	−2.37335	−0.0933060	−0.0466530	+	0.998911i	$0.514856\pi$
$648$	5.89157	0.231443
$649$	−5.79690	−0.227548
$650$	0	0
$651$	20.1311	0.789001
$652$	−51.4660	−2.01556
$653$	23.7869	0.930855	0.465427	−	0.885086i	$-0.345901\pi$
$653$	23.7869	0.930855	0.465427	+	0.885086i	$0.345901\pi$
$654$	80.8194	3.16029
$655$	0	0
$656$	19.8860	0.776417
$657$	−58.0538	−2.26489
$658$	8.27391	0.322551
$659$	2.32216	0.0904585	0.0452293	−	0.998977i	$-0.485598\pi$
$659$	2.32216	0.0904585	0.0452293	+	0.998977i	$0.485598\pi$
$660$	0	0
$661$	−3.08598	−0.120031	−0.0600153	−	0.998197i	$-0.519115\pi$
$661$	−3.08598	−0.120031	−0.0600153	+	0.998197i	$0.519115\pi$
$662$	−26.3701	−1.02490
$663$	19.1224	0.742651
$664$	−2.31456	−0.0898222
$665$	0	0
$666$	12.4384	0.481977
$667$	0.00267685	0.000103648 0
$668$	1.32170	0.0511382
$669$	−25.1541	−0.972514
$670$	0	0
$671$	−3.97417	−0.153421
$672$	20.8238	0.803294
$673$	29.3935	1.13304	0.566518	−	0.824050i	$-0.308290\pi$
$673$	29.3935	1.13304	0.566518	+	0.824050i	$0.308290\pi$
$674$	19.5829	0.754306
$675$	0	0
$676$	−28.1013	−1.08082
$677$	25.6569	0.986077	0.493038	−	0.870008i	$-0.335886\pi$
$677$	25.6569	0.986077	0.493038	+	0.870008i	$0.335886\pi$
$678$	−99.7736	−3.83178
$679$	−2.93573	−0.112663
$680$	0	0
$681$	−48.1239	−1.84411
$682$	−16.2262	−0.621334
$683$	29.3691	1.12378	0.561888	−	0.827213i	$-0.310075\pi$
$683$	29.3691	1.12378	0.561888	+	0.827213i	$0.310075\pi$
$684$	37.2790	1.42540
$685$	0	0
$686$	2.13935	0.0816806
$687$	−51.3852	−1.96047
$688$	10.2418	0.390464
$689$	2.25982	0.0860922
$690$	0	0
$691$	16.9292	0.644018	0.322009	−	0.946737i	$-0.395642\pi$
$691$	16.9292	0.644018	0.322009	+	0.946737i	$0.395642\pi$
$692$	55.6468	2.11538
$693$	−4.04471	−0.153646
$694$	38.2357	1.45141
$695$	0	0
$696$	−0.0937049	−0.00355188
$697$	−39.3827	−1.49172
$698$	−58.3957	−2.21031
$699$	57.5701	2.17750
$700$	0	0
$701$	37.4212	1.41338	0.706690	−	0.707524i	$-0.250188\pi$
$701$	37.4212	1.41338	0.706690	+	0.707524i	$0.250188\pi$
$702$	−8.58516	−0.324026
$703$	−5.14150	−0.193915
$704$	−11.7571	−0.443113
$705$	0	0
$706$	−21.6209	−0.813712
$707$	1.04693	0.0393739
$708$	39.6468	1.49002
$709$	−48.1585	−1.80863	−0.904315	−	0.426866i	$-0.859618\pi$
$709$	−48.1585	−1.80863	−0.904315	+	0.426866i	$0.859618\pi$
$710$	0	0
$711$	29.1639	1.09373
$712$	11.4057	0.427447
$713$	−0.709639	−0.0265762
$714$	−28.2672	−1.05787
$715$	0	0
$716$	31.5597	1.17944
$717$	−45.5948	−1.70277
$718$	35.2193	1.31437
$719$	−37.3298	−1.39217	−0.696083	−	0.717961i	$-0.745075\pi$
$719$	−37.3298	−1.39217	−0.696083	+	0.717961i	$0.745075\pi$
$720$	0	0
$721$	−11.1215	−0.414187
$722$	13.2778	0.494149
$723$	−46.1100	−1.71485
$724$	53.1810	1.97646
$725$	0	0
$726$	5.67823	0.210739
$727$	−29.2226	−1.08380	−0.541902	−	0.840441i	$-0.682296\pi$
$727$	−29.2226	−1.08380	−0.541902	+	0.840441i	$0.682296\pi$
$728$	−1.78586	−0.0661884
$729$	−41.3903	−1.53297
$730$	0	0
$731$	−20.2831	−0.750196
$732$	27.1806	1.00462
$733$	2.64099	0.0975471	0.0487735	−	0.998810i	$-0.484469\pi$
$733$	2.64099	0.0975471	0.0487735	+	0.998810i	$0.484469\pi$
$734$	17.1483	0.632955
$735$	0	0
$736$	−0.734055	−0.0270576
$737$	15.2119	0.560339
$738$	68.4547	2.51985
$739$	18.9936	0.698690	0.349345	−	0.936994i	$-0.386404\pi$
$739$	18.9936	0.698690	0.349345	+	0.936994i	$0.386404\pi$
$740$	0	0
$741$	13.7393	0.504727
$742$	−3.34053	−0.122635
$743$	−38.6308	−1.41723	−0.708614	−	0.705596i	$-0.750679\pi$
$743$	−38.6308	−1.41723	−0.708614	+	0.705596i	$0.750679\pi$
$744$	24.8414	0.910731
$745$	0	0
$746$	−41.8784	−1.53328
$747$	7.58660	0.277579
$748$	12.8278	0.469030
$749$	−3.91977	−0.143225
$750$	0	0
$751$	−38.7209	−1.41294	−0.706472	−	0.707741i	$-0.749714\pi$
$751$	−38.7209	−1.41294	−0.706472	+	0.707741i	$0.749714\pi$
$752$	−9.72170	−0.354514
$753$	−12.7680	−0.465293
$754$	−0.0885814	−0.00322595
$755$	0	0
$756$	7.14512	0.259865
$757$	1.11213	0.0404212	0.0202106	−	0.999796i	$-0.493566\pi$
$757$	1.11213	0.0404212	0.0202106	+	0.999796i	$0.493566\pi$
$758$	−18.0354	−0.655076
$759$	0.248332	0.00901388
$760$	0	0
$761$	9.88898	0.358475	0.179238	−	0.983806i	$-0.442637\pi$
$761$	9.88898	0.358475	0.179238	+	0.983806i	$0.442637\pi$
$762$	0.484581	0.0175545
$763$	−14.2332	−0.515277
$764$	−1.66047	−0.0600735
$765$	0	0
$766$	21.4225	0.774028
$767$	8.38948	0.302927
$768$	−8.68778	−0.313493
$769$	17.2952	0.623679	0.311840	−	0.950135i	$-0.399055\pi$
$769$	17.2952	0.623679	0.311840	+	0.950135i	$0.399055\pi$
$770$	0	0
$771$	69.1281	2.48959
$772$	32.1672	1.15772
$773$	0.542505	0.0195126	0.00975628	−	0.999952i	$-0.496894\pi$
$773$	0.542505	0.0195126	0.00975628	+	0.999952i	$0.496894\pi$
$774$	35.2559	1.26725
$775$	0	0
$776$	−3.62263	−0.130045
$777$	−3.81528	−0.136872
$778$	22.1697	0.794821
$779$	−28.2963	−1.01382
$780$	0	0
$781$	−16.4377	−0.588186
$782$	0.996443	0.0356327
$783$	0.0793324	0.00283511
$784$	−2.51369	−0.0897748
$785$	0	0
$786$	92.1140	3.28560
$787$	−39.1688	−1.39622	−0.698108	−	0.715992i	$-0.745975\pi$
$787$	−39.1688	−1.39622	−0.698108	+	0.715992i	$0.745975\pi$
$788$	−32.9291	−1.17305
$789$	−65.8713	−2.34508
$790$	0	0
$791$	17.5713	0.624762
$792$	−4.99110	−0.177351
$793$	5.75156	0.204244
$794$	−35.2572	−1.25123
$795$	0	0
$796$	22.6182	0.801683
$797$	−33.2935	−1.17932	−0.589658	−	0.807653i	$-0.700738\pi$
$797$	−33.2935	−1.17932	−0.589658	+	0.807653i	$0.700738\pi$
$798$	−20.3099	−0.718963
$799$	19.2531	0.681125
$800$	0	0
$801$	−37.3854	−1.32095
$802$	−71.8323	−2.53649
$803$	−14.3530	−0.506507
$804$	−104.039	−3.66919
$805$	0	0
$806$	23.4832	0.827160
$807$	76.1852	2.68185
$808$	1.29189	0.0454487
$809$	49.5167	1.74091	0.870457	−	0.492244i	$-0.163823\pi$
$809$	49.5167	1.74091	0.870457	+	0.492244i	$0.163823\pi$
$810$	0	0
$811$	−8.53630	−0.299750	−0.149875	−	0.988705i	$-0.547887\pi$
$811$	−8.53630	−0.299750	−0.149875	+	0.988705i	$0.547887\pi$
$812$	0.0737231	0.00258717
$813$	44.3675	1.55604
$814$	3.07522	0.107786
$815$	0	0
$816$	33.2135	1.16270
$817$	−14.5733	−0.509855
$818$	−46.9274	−1.64078
$819$	5.85365	0.204543
$820$	0	0
$821$	4.19863	0.146533	0.0732666	−	0.997312i	$-0.476658\pi$
$821$	4.19863	0.146533	0.0732666	+	0.997312i	$0.476658\pi$
$822$	9.27835	0.323620
$823$	55.8410	1.94650	0.973248	−	0.229759i	$-0.0737937\pi$
$823$	55.8410	1.94650	0.973248	+	0.229759i	$0.0737937\pi$
$824$	−13.7237	−0.478089
$825$	0	0
$826$	−12.4016	−0.431506
$827$	14.6093	0.508014	0.254007	−	0.967202i	$-0.418251\pi$
$827$	14.6093	0.508014	0.254007	+	0.967202i	$0.418251\pi$
$828$	−0.975146	−0.0338887
$829$	−43.3917	−1.50705	−0.753527	−	0.657417i	$-0.771649\pi$
$829$	−43.3917	−1.50705	−0.753527	+	0.657417i	$0.771649\pi$
$830$	0	0
$831$	27.3643	0.949255
$832$	17.0153	0.589900
$833$	4.97818	0.172484
$834$	15.6002	0.540191
$835$	0	0
$836$	9.21672	0.318767
$837$	−21.0312	−0.726945
$838$	38.3960	1.32637
$839$	17.4846	0.603635	0.301818	−	0.953366i	$-0.402407\pi$
$839$	17.4846	0.603635	0.301818	+	0.953366i	$0.402407\pi$
$840$	0	0
$841$	−28.9992	−0.999972
$842$	15.9491	0.549644
$843$	−71.0214	−2.44611
$844$	−42.3467	−1.45763
$845$	0	0
$846$	−33.4656	−1.15057
$847$	−1.00000	−0.0343604
$848$	3.92506	0.134787
$849$	19.8255	0.680408
$850$	0	0
$851$	0.134492	0.00461032
$852$	112.422	3.85153
$853$	−22.0429	−0.754734	−0.377367	−	0.926064i	$-0.623170\pi$
$853$	−22.0429	−0.754734	−0.377367	+	0.926064i	$0.623170\pi$
$854$	−8.50212	−0.290937
$855$	0	0
$856$	−4.83692	−0.165323
$857$	−39.6431	−1.35418	−0.677092	−	0.735899i	$-0.736760\pi$
$857$	−39.6431	−1.35418	−0.677092	+	0.735899i	$0.736760\pi$
$858$	−8.21773	−0.280549
$859$	22.5293	0.768690	0.384345	−	0.923190i	$-0.374427\pi$
$859$	22.5293	0.768690	0.384345	+	0.923190i	$0.374427\pi$
$860$	0	0
$861$	−20.9974	−0.715591
$862$	−20.4257	−0.695701
$863$	−22.9053	−0.779706	−0.389853	−	0.920877i	$-0.627474\pi$
$863$	−22.9053	−0.779706	−0.389853	+	0.920877i	$0.627474\pi$
$864$	−21.7548	−0.740114
$865$	0	0
$866$	59.9990	2.03885
$867$	−20.6556	−0.701501
$868$	−19.5442	−0.663373
$869$	7.21037	0.244595
$870$	0	0
$871$	−22.0153	−0.745959
$872$	−17.5635	−0.594775
$873$	11.8742	0.401880
$874$	0.715941	0.0242171
$875$	0	0
$876$	98.1648	3.31668
$877$	40.0339	1.35185	0.675925	−	0.736970i	$-0.263744\pi$
$877$	40.0339	1.35185	0.675925	+	0.736970i	$0.263744\pi$
$878$	11.8625	0.400338
$879$	−48.7682	−1.64491
$880$	0	0
$881$	−7.67069	−0.258432	−0.129216	−	0.991616i	$-0.541246\pi$
$881$	−7.67069	−0.258432	−0.129216	+	0.991616i	$0.541246\pi$
$882$	−8.65304	−0.291363
$883$	1.57160	0.0528886	0.0264443	−	0.999650i	$-0.491582\pi$
$883$	1.57160	0.0528886	0.0264443	+	0.999650i	$0.491582\pi$
$884$	−18.5648	−0.624403
$885$	0	0
$886$	43.0061	1.44482
$887$	14.5540	0.488677	0.244338	−	0.969690i	$-0.421429\pi$
$887$	14.5540	0.488677	0.244338	+	0.969690i	$0.421429\pi$
$888$	−4.70798	−0.157990
$889$	−0.0853402	−0.00286222
$890$	0	0
$891$	−4.77444	−0.159950
$892$	24.4207	0.817666
$893$	13.8333	0.462913
$894$	−17.7833	−0.594763
$895$	0	0
$896$	−9.46130	−0.316080
$897$	−0.359395	−0.0119999
$898$	−7.30479	−0.243764
$899$	−0.216999	−0.00723734
$900$	0	0
$901$	−7.77328	−0.258966
$902$	16.9245	0.563524
$903$	−10.8142	−0.359874
$904$	21.6826	0.721153
$905$	0	0
$906$	42.7167	1.41917
$907$	7.26471	0.241221	0.120610	−	0.992700i	$-0.461515\pi$
$907$	7.26471	0.241221	0.120610	+	0.992700i	$0.461515\pi$
$908$	46.7208	1.55048
$909$	−4.23454	−0.140451
$910$	0	0
$911$	33.3354	1.10445	0.552225	−	0.833695i	$-0.313779\pi$
$911$	33.3354	1.10445	0.552225	+	0.833695i	$0.313779\pi$
$912$	23.8638	0.790208
$913$	1.87568	0.0620760
$914$	81.8065	2.70592
$915$	0	0
$916$	49.8870	1.64831
$917$	−16.2223	−0.535708
$918$	29.5311	0.974672
$919$	−2.78198	−0.0917690	−0.0458845	−	0.998947i	$-0.514611\pi$
$919$	−2.78198	−0.0917690	−0.0458845	+	0.998947i	$0.514611\pi$
$920$	0	0
$921$	−55.5526	−1.83052
$922$	10.4352	0.343666
$923$	23.7892	0.783030
$924$	6.83932	0.224997
$925$	0	0
$926$	58.2486	1.91417
$927$	44.9833	1.47745
$928$	−0.224466	−0.00736844
$929$	−11.2001	−0.367464	−0.183732	−	0.982976i	$-0.558818\pi$
$929$	−11.2001	−0.367464	−0.183732	+	0.982976i	$0.558818\pi$
$930$	0	0
$931$	3.57680	0.117225
$932$	−55.8916	−1.83079
$933$	2.66018	0.0870905
$934$	−41.1636	−1.34692
$935$	0	0
$936$	7.22330	0.236101
$937$	12.8469	0.419691	0.209845	−	0.977735i	$-0.432704\pi$
$937$	12.8469	0.419691	0.209845	+	0.977735i	$0.432704\pi$
$938$	32.5436	1.06259
$939$	−60.2243	−1.96534
$940$	0	0
$941$	9.37647	0.305664	0.152832	−	0.988252i	$-0.451161\pi$
$941$	9.37647	0.305664	0.152832	+	0.988252i	$0.451161\pi$
$942$	130.810	4.26201
$943$	0.740177	0.0241035
$944$	14.5716	0.474266
$945$	0	0
$946$	8.71654	0.283399
$947$	17.4137	0.565868	0.282934	−	0.959139i	$-0.408692\pi$
$947$	17.4137	0.565868	0.282934	+	0.959139i	$0.408692\pi$
$948$	−49.3140	−1.60165
$949$	20.7722	0.674294
$950$	0	0
$951$	80.1255	2.59825
$952$	6.14298	0.199095
$953$	−41.9815	−1.35991	−0.679957	−	0.733252i	$-0.738002\pi$
$953$	−41.9815	−1.35991	−0.679957	+	0.733252i	$0.738002\pi$
$954$	13.5115	0.437450
$955$	0	0
$956$	44.2654	1.43165
$957$	0.0759371	0.00245470
$958$	−25.0519	−0.809391
$959$	−1.63402	−0.0527654
$960$	0	0
$961$	26.5271	0.855714
$962$	−4.45057	−0.143492
$963$	15.8543	0.510899
$964$	44.7656	1.44180
$965$	0	0
$966$	0.531268	0.0170933
$967$	52.0296	1.67316	0.836580	−	0.547845i	$-0.184552\pi$
$967$	52.0296	1.67316	0.836580	+	0.547845i	$0.184552\pi$
$968$	−1.23398	−0.0396616
$969$	−47.2604	−1.51822
$970$	0	0
$971$	−24.8363	−0.797036	−0.398518	−	0.917161i	$-0.630475\pi$
$971$	−24.8363	−0.797036	−0.398518	+	0.917161i	$0.630475\pi$
$972$	54.0893	1.73491
$973$	−2.74737	−0.0880768
$974$	12.5169	0.401067
$975$	0	0
$976$	9.98984	0.319767
$977$	27.3773	0.875877	0.437938	−	0.899005i	$-0.355709\pi$
$977$	27.3773	0.875877	0.437938	+	0.899005i	$0.355709\pi$
$978$	−113.410	−3.62646
$979$	−9.24303	−0.295408
$980$	0	0
$981$	57.5692	1.83804
$982$	−83.1126	−2.65223
$983$	−4.24625	−0.135434	−0.0677172	−	0.997705i	$-0.521572\pi$
$983$	−4.24625	−0.135434	−0.0677172	+	0.997705i	$0.521572\pi$
$984$	−25.9104	−0.825994
$985$	0	0
$986$	0.304701	0.00970366
$987$	10.2651	0.326740
$988$	−13.3388	−0.424363
$989$	0.381210	0.0121218
$990$	0	0
$991$	−25.5549	−0.811777	−0.405889	−	0.913923i	$-0.633038\pi$
$991$	−25.5549	−0.811777	−0.405889	+	0.913923i	$0.633038\pi$
$992$	59.5064	1.88933
$993$	−32.7161	−1.03821
$994$	−35.1658	−1.11539
$995$	0	0
$996$	−12.8284	−0.406483
$997$	−26.5563	−0.841047	−0.420524	−	0.907282i	$-0.638154\pi$
$997$	−26.5563	−0.841047	−0.420524	+	0.907282i	$0.638154\pi$
$998$	−28.5864	−0.904886
$999$	3.98587	0.126107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.bb.1.2	✓	7
5.2	odd	4		1925.2.b.r.1849.4		14
5.3	odd	4		1925.2.b.r.1849.11		14
5.4	even	2		1925.2.a.bd.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.bb.1.2	✓	7	1.1	even	1	trivial
1925.2.a.bd.1.6	yes	7	5.4	even	2
1925.2.b.r.1849.4		14	5.2	odd	4
1925.2.b.r.1849.11		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.13935 q^{2} -2.65419 q^{3} +2.57680 q^{4} +5.67823 q^{6} -1.00000 q^{7} -1.23398 q^{8} +4.04471 q^{9} +O(q^{10})\)	\(q-2.13935 q^{2} -2.65419 q^{3} +2.57680 q^{4} +5.67823 q^{6} -1.00000 q^{7} -1.23398 q^{8} +4.04471 q^{9} +1.00000 q^{11} -6.83932 q^{12} -1.44724 q^{13} +2.13935 q^{14} -2.51369 q^{16} +4.97818 q^{17} -8.65304 q^{18} +3.57680 q^{19} +2.65419 q^{21} -2.13935 q^{22} -0.0935623 q^{23} +3.27522 q^{24} +3.09614 q^{26} -2.77286 q^{27} -2.57680 q^{28} -0.0286103 q^{29} +7.58466 q^{31} +7.84562 q^{32} -2.65419 q^{33} -10.6500 q^{34} +10.4224 q^{36} -1.43746 q^{37} -7.65202 q^{38} +3.84124 q^{39} -7.91106 q^{41} -5.67823 q^{42} -4.07439 q^{43} +2.57680 q^{44} +0.200162 q^{46} +3.86750 q^{47} +6.67181 q^{48} +1.00000 q^{49} -13.2130 q^{51} -3.72924 q^{52} -1.56147 q^{53} +5.93211 q^{54} +1.23398 q^{56} -9.49351 q^{57} +0.0612073 q^{58} -5.79690 q^{59} -3.97417 q^{61} -16.2262 q^{62} -4.04471 q^{63} -11.7571 q^{64} +5.67823 q^{66} +15.2119 q^{67} +12.8278 q^{68} +0.248332 q^{69} -16.4377 q^{71} -4.99110 q^{72} -14.3530 q^{73} +3.07522 q^{74} +9.21672 q^{76} -1.00000 q^{77} -8.21773 q^{78} +7.21037 q^{79} -4.77444 q^{81} +16.9245 q^{82} +1.87568 q^{83} +6.83932 q^{84} +8.71654 q^{86} +0.0759371 q^{87} -1.23398 q^{88} -9.24303 q^{89} +1.44724 q^{91} -0.241092 q^{92} -20.1311 q^{93} -8.27391 q^{94} -20.8238 q^{96} +2.93573 q^{97} -2.13935 q^{98} +4.04471 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9	\( 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9} + 7 q^{11} + 9 q^{12} + 3 q^{13} + q^{14} + 21 q^{16} + 2 q^{17} - 8 q^{18} + 20 q^{19} - q^{22} - 11 q^{23} + 18 q^{24} - 13 q^{26} + 12 q^{27} - 13 q^{28} + 4 q^{29} + 6 q^{31} - q^{32} - 7 q^{34} + 12 q^{36} - 19 q^{37} - 3 q^{38} - 10 q^{39} + 24 q^{41} - 3 q^{42} + 13 q^{44} + 33 q^{46} + q^{47} + 15 q^{48} + 7 q^{49} + 19 q^{51} + 29 q^{52} - 7 q^{53} + 9 q^{54} + 9 q^{57} - 37 q^{58} + 9 q^{59} + 18 q^{61} + 40 q^{62} - 9 q^{63} + 8 q^{64} + 3 q^{66} - 18 q^{68} - 15 q^{69} + 18 q^{71} + 64 q^{72} - 5 q^{73} - 24 q^{74} + 88 q^{76} - 7 q^{77} - 79 q^{78} + 25 q^{79} - q^{81} + 60 q^{82} + 17 q^{83} - 9 q^{84} - 41 q^{86} + 24 q^{87} - 16 q^{89} - 3 q^{91} - 28 q^{92} - 26 q^{93} - 31 q^{94} + 17 q^{96} + 4 q^{97} - q^{98} + 9 q^{99}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9 + 7 * q^11 + 9 * q^12 + 3 * q^13 + q^14 + 21 * q^16 + 2 * q^17 - 8 * q^18 + 20 * q^19 - q^22 - 11 * q^23 + 18 * q^24 - 13 * q^26 + 12 * q^27 - 13 * q^28 + 4 * q^29 + 6 * q^31 - q^32 - 7 * q^34 + 12 * q^36 - 19 * q^37 - 3 * q^38 - 10 * q^39 + 24 * q^41 - 3 * q^42 + 13 * q^44 + 33 * q^46 + q^47 + 15 * q^48 + 7 * q^49 + 19 * q^51 + 29 * q^52 - 7 * q^53 + 9 * q^54 + 9 * q^57 - 37 * q^58 + 9 * q^59 + 18 * q^61 + 40 * q^62 - 9 * q^63 + 8 * q^64 + 3 * q^66 - 18 * q^68 - 15 * q^69 + 18 * q^71 + 64 * q^72 - 5 * q^73 - 24 * q^74 + 88 * q^76 - 7 * q^77 - 79 * q^78 + 25 * q^79 - q^81 + 60 * q^82 + 17 * q^83 - 9 * q^84 - 41 * q^86 + 24 * q^87 - 16 * q^89 - 3 * q^91 - 28 * q^92 - 26 * q^93 - 31 * q^94 + 17 * q^96 + 4 * q^97 - q^98 + 9 * q^99

Introduction

L-functions

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