LMFDB - Embedded newform 425.2.b.f.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("425.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.229451239931904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{7} + 64x^{6} - 30x^{5} + 2x^{4} + 136x^{3} + 324x^{2} + 180x + 50 $ x^10 - 2x^7 + 64x^6 - 30x^5 + 2x^4 + 136x^3 + 324x^2 + 180*x + 50
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$-0.328166 + 0.328166i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.f.324.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.60242i q^{2} +1.18219i q^{3} -4.77260 q^{4} +3.07656 q^{6} +3.53650i q^{7} +7.21549i q^{8} +1.60242 q^{9} +2.94609 q^{11} -5.64213i q^{12} -4.01064i q^{13} +9.20348 q^{14} +9.23255 q^{16} -1.00000i q^{17} -4.17018i q^{18} +6.97745 q^{19} -4.18083 q^{21} -7.66698i q^{22} +6.12692i q^{23} -8.53009 q^{24} -10.4374 q^{26} +5.44095i q^{27} -16.8783i q^{28} -5.30040 q^{29} +6.49485 q^{31} -9.59601i q^{32} +3.48284i q^{33} -2.60242 q^{34} -7.64773 q^{36} +3.43224i q^{37} -18.1583i q^{38} +4.74135 q^{39} +4.61307 q^{41} +10.8803i q^{42} -10.2901i q^{43} -14.0605 q^{44} +15.9448 q^{46} +3.67705i q^{47} +10.9146i q^{48} -5.50686 q^{49} +1.18219 q^{51} +19.1412i q^{52} +6.77260i q^{53} +14.1596 q^{54} -25.5176 q^{56} +8.24868i q^{57} +13.7939i q^{58} -9.92573 q^{59} -2.36438 q^{61} -16.9024i q^{62} +5.66698i q^{63} -6.50778 q^{64} +9.06383 q^{66} -9.56650i q^{67} +4.77260i q^{68} -7.24319 q^{69} +5.51248 q^{71} +11.5623i q^{72} -2.00515i q^{73} +8.93214 q^{74} -33.3006 q^{76} +10.4189i q^{77} -12.3390i q^{78} -10.5803 q^{79} -1.62497 q^{81} -12.0052i q^{82} -9.07301i q^{83} +19.9534 q^{84} -26.7792 q^{86} -6.26609i q^{87} +21.2575i q^{88} -2.63321 q^{89} +14.1837 q^{91} -29.2414i q^{92} +7.67816i q^{93} +9.56923 q^{94} +11.3443 q^{96} +5.86816i q^{97} +14.3312i q^{98} +4.72088 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) $ 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.60242i	− 1.84019i	−0.391694	−	0.920095i	$-0.628111\pi$
$2$	− 2.60242i	− 1.84019i	0.391694	−	0.920095i	$-0.371889\pi$
$3$	1.18219i	0.682539i	0.939966	+	0.341269i	$0.110857\pi$
$3$	1.18219i	0.682539i	−0.939966	+	0.341269i	$0.889143\pi$
$4$	−4.77260	−2.38630
$5$	0	0
$5$	0	0
$6$	3.07656	1.25600
$7$	3.53650i	1.33667i	0.743859	+	0.668337i	$0.232993\pi$
$7$	3.53650i	1.33667i	−0.743859	+	0.668337i	$0.767007\pi$
$8$	7.21549i	2.55106i
$9$	1.60242	0.534141
$10$	0	0
$11$	2.94609	0.888280	0.444140	−	0.895957i	$-0.353509\pi$
$11$	2.94609	0.888280	0.444140	+	0.895957i	$0.353509\pi$
$12$	− 5.64213i	− 1.62874i
$13$	− 4.01064i	− 1.11235i	−0.831064	−	0.556176i	$-0.812268\pi$
$13$	− 4.01064i	− 1.11235i	0.831064	−	0.556176i	$-0.187732\pi$
$14$	9.20348	2.45973
$15$	0	0
$16$	9.23255	2.30814
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	− 4.17018i	− 0.982921i
$19$	6.97745	1.60074	0.800368	−	0.599508i	$-0.204637\pi$
$19$	6.97745	1.60074	0.800368	+	0.599508i	$0.204637\pi$
$20$	0	0
$21$	−4.18083	−0.912331
$22$	− 7.66698i	− 1.63460i
$23$	6.12692i	1.27755i	0.769393	+	0.638775i	$0.220559\pi$
$23$	6.12692i	1.27755i	−0.769393	+	0.638775i	$0.779441\pi$
$24$	−8.53009	−1.74120
$25$	0	0
$26$	−10.4374	−2.04694
$27$	5.44095i	1.04711i
$28$	− 16.8783i	− 3.18971i
$29$	−5.30040	−0.984260	−0.492130	−	0.870522i	$-0.663782\pi$
$29$	−5.30040	−0.984260	−0.492130	+	0.870522i	$0.663782\pi$
$30$	0	0
$31$	6.49485	1.16651	0.583255	−	0.812289i	$-0.301779\pi$
$31$	6.49485	1.16651	0.583255	+	0.812289i	$0.301779\pi$
$32$	− 9.59601i	− 1.69635i
$33$	3.48284i	0.606285i
$34$	−2.60242	−0.446312
$35$	0	0
$36$	−7.64773	−1.27462
$37$	3.43224i	0.564257i	0.959377	+	0.282128i	$0.0910404\pi$
$37$	3.43224i	0.564257i	−0.959377	+	0.282128i	$0.908960\pi$
$38$	− 18.1583i	− 2.94566i
$39$	4.74135	0.759224
$40$	0	0
$41$	4.61307	0.720440	0.360220	−	0.932867i	$-0.382702\pi$
$41$	4.61307	0.720440	0.360220	+	0.932867i	$0.382702\pi$
$42$	10.8803i	1.67886i
$43$	− 10.2901i	− 1.56923i	−0.619985	−	0.784614i	$-0.712861\pi$
$43$	− 10.2901i	− 1.56923i	0.619985	−	0.784614i	$-0.287139\pi$
$44$	−14.0605	−2.11970
$45$	0	0
$46$	15.9448	2.35094
$47$	3.67705i	0.536352i	0.963370	+	0.268176i	$0.0864209\pi$
$47$	3.67705i	0.536352i	−0.963370	+	0.268176i	$0.913579\pi$
$48$	10.9146i	1.57539i
$49$	−5.50686	−0.786695
$50$	0	0
$51$	1.18219	0.165540
$52$	19.1412i	2.65441i
$53$	6.77260i	0.930289i	0.885235	+	0.465144i	$0.153998\pi$
$53$	6.77260i	0.930289i	−0.885235	+	0.465144i	$0.846002\pi$
$54$	14.1596	1.92688
$55$	0	0
$56$	−25.5176	−3.40993
$57$	8.24868i	1.09256i
$58$	13.7939i	1.81123i
$59$	−9.92573	−1.29222	−0.646110	−	0.763244i	$-0.723605\pi$
$59$	−9.92573	−1.29222	−0.646110	+	0.763244i	$0.723605\pi$
$60$	0	0
$61$	−2.36438	−0.302728	−0.151364	−	0.988478i	$-0.548367\pi$
$61$	−2.36438	−0.302728	−0.151364	+	0.988478i	$0.548367\pi$
$62$	− 16.9024i	− 2.14660i
$63$	5.66698i	0.713972i
$64$	−6.50778	−0.813473
$65$	0	0
$66$	9.06383	1.11568
$67$	− 9.56650i	− 1.16873i	−0.811490	−	0.584367i	$-0.801343\pi$
$67$	− 9.56650i	− 1.16873i	0.811490	−	0.584367i	$-0.198657\pi$
$68$	4.77260i	0.578763i
$69$	−7.24319	−0.871978
$70$	0	0
$71$	5.51248	0.654212	0.327106	−	0.944988i	$-0.393927\pi$
$71$	5.51248	0.654212	0.327106	+	0.944988i	$0.393927\pi$
$72$	11.5623i	1.36263i
$73$	− 2.00515i	− 0.234685i	−0.993091	−	0.117343i	$-0.962562\pi$
$73$	− 2.00515i	− 0.234685i	0.993091	−	0.117343i	$-0.0374376\pi$
$74$	8.93214	1.03834
$75$	0	0
$76$	−33.3006	−3.81984
$77$	10.4189i	1.18734i
$78$	− 12.3390i	− 1.39712i
$79$	−10.5803	−1.19038	−0.595191	−	0.803584i	$-0.702924\pi$
$79$	−10.5803	−1.19038	−0.595191	+	0.803584i	$0.702924\pi$
$80$	0	0
$81$	−1.62497	−0.180552
$82$	− 12.0052i	− 1.32575i
$83$	− 9.07301i	− 0.995892i	−0.867208	−	0.497946i	$-0.834088\pi$
$83$	− 9.07301i	− 0.995892i	0.867208	−	0.497946i	$-0.165912\pi$
$84$	19.9534	2.17710
$85$	0	0
$86$	−26.7792	−2.88768
$87$	− 6.26609i	− 0.671796i
$88$	21.2575i	2.26606i
$89$	−2.63321	−0.279119	−0.139560	−	0.990214i	$-0.544569\pi$
$89$	−2.63321	−0.279119	−0.139560	+	0.990214i	$0.544569\pi$
$90$	0	0
$91$	14.1837	1.48685
$92$	− 29.2414i	− 3.04862i
$93$	7.67816i	0.796188i
$94$	9.56923	0.986991
$95$	0	0
$96$	11.3443	1.15783
$97$	5.86816i	0.595822i	0.954594	+	0.297911i	$0.0962898\pi$
$97$	5.86816i	0.595822i	−0.954594	+	0.297911i	$0.903710\pi$
$98$	14.3312i	1.44767i
$99$	4.72088	0.474467
$100$	0	0
$101$	−7.90283	−0.786361	−0.393180	−	0.919461i	$-0.628625\pi$
$101$	−7.90283	−0.786361	−0.393180	+	0.919461i	$0.628625\pi$
$102$	− 3.07656i	− 0.304625i
$103$	6.36826i	0.627483i	0.949508	+	0.313742i	$0.101583\pi$
$103$	6.36826i	0.627483i	−0.949508	+	0.313742i	$0.898417\pi$
$104$	28.9388	2.83768
$105$	0	0
$106$	17.6252	1.71191
$107$	6.85432i	0.662632i	0.943520	+	0.331316i	$0.107493\pi$
$107$	6.85432i	0.662632i	−0.943520	+	0.331316i	$0.892507\pi$
$108$	− 25.9675i	− 2.49872i
$109$	14.6758	1.40569	0.702843	−	0.711345i	$-0.251914\pi$
$109$	14.6758	1.40569	0.702843	+	0.711345i	$0.251914\pi$
$110$	0	0
$111$	−4.05757	−0.385127
$112$	32.6509i	3.08522i
$113$	− 13.3994i	− 1.26051i	−0.776388	−	0.630255i	$-0.782950\pi$
$113$	− 13.3994i	− 1.26051i	0.776388	−	0.630255i	$-0.217050\pi$
$114$	21.4666	2.01053
$115$	0	0
$116$	25.2967	2.34874
$117$	− 6.42675i	− 0.594153i
$118$	25.8309i	2.37793i
$119$	3.53650	0.324191
$120$	0	0
$121$	−2.32055	−0.210959
$122$	6.15313i	0.557078i
$123$	5.45353i	0.491728i
$124$	−30.9974	−2.78365
$125$	0	0
$126$	14.7479	1.31384
$127$	4.63321i	0.411131i	0.978643	+	0.205565i	$0.0659033\pi$
$127$	4.63321i	0.411131i	−0.978643	+	0.205565i	$0.934097\pi$
$128$	− 2.25602i	− 0.199405i
$129$	12.1649	1.07106
$130$	0	0
$131$	−12.1496	−1.06151	−0.530757	−	0.847524i	$-0.678092\pi$
$131$	−12.1496	−1.06151	−0.530757	+	0.847524i	$0.678092\pi$
$132$	− 16.6222i	− 1.44678i
$133$	24.6758i	2.13966i
$134$	−24.8961	−2.15069
$135$	0	0
$136$	7.21549	0.618723
$137$	− 8.86852i	− 0.757689i	−0.925460	−	0.378844i	$-0.876322\pi$
$137$	− 8.86852i	− 0.757689i	0.925460	−	0.378844i	$-0.123678\pi$
$138$	18.8498i	1.60461i
$139$	−7.32306	−0.621134	−0.310567	−	0.950552i	$-0.600519\pi$
$139$	−7.32306	−0.621134	−0.310567	+	0.950552i	$0.600519\pi$
$140$	0	0
$141$	−4.34697	−0.366081
$142$	− 14.3458i	− 1.20387i
$143$	− 11.8157i	− 0.988081i
$144$	14.7944	1.23287
$145$	0	0
$146$	−5.21825	−0.431866
$147$	− 6.51017i	− 0.536950i
$148$	− 16.3807i	− 1.34649i
$149$	−13.9059	−1.13922	−0.569608	−	0.821916i	$-0.692905\pi$
$149$	−13.9059	−1.13922	−0.569608	+	0.821916i	$0.692905\pi$
$150$	0	0
$151$	14.3884	1.17091	0.585456	−	0.810704i	$-0.300916\pi$
$151$	14.3884	1.17091	0.585456	+	0.810704i	$0.300916\pi$
$152$	50.3457i	4.08358i
$153$	− 1.60242i	− 0.129548i
$154$	27.1143	2.18493
$155$	0	0
$156$	−22.6286	−1.81174
$157$	− 8.68608i	− 0.693224i	−0.938009	−	0.346612i	$-0.887332\pi$
$157$	− 8.68608i	− 0.693224i	0.938009	−	0.346612i	$-0.112668\pi$
$158$	27.5345i	2.19053i
$159$	−8.00652	−0.634958
$160$	0	0
$161$	−21.6679	−1.70767
$162$	4.22887i	0.332251i
$163$	− 8.95868i	− 0.701698i	−0.936432	−	0.350849i	$-0.885893\pi$
$163$	− 8.95868i	− 0.701698i	0.936432	−	0.350849i	$-0.114107\pi$
$164$	−22.0163	−1.71919
$165$	0	0
$166$	−23.6118	−1.83263
$167$	− 4.37318i	− 0.338407i	−0.985581	−	0.169203i	$-0.945881\pi$
$167$	− 4.37318i	− 0.338407i	0.985581	−	0.169203i	$-0.0541195\pi$
$168$	− 30.1667i	− 2.32741i
$169$	−3.08527	−0.237328
$170$	0	0
$171$	11.1808	0.855019
$172$	49.1106i	3.74465i
$173$	8.82433i	0.670901i	0.942058	+	0.335451i	$0.108889\pi$
$173$	8.82433i	0.670901i	−0.942058	+	0.335451i	$0.891111\pi$
$174$	−16.3070	−1.23623
$175$	0	0
$176$	27.1999	2.05027
$177$	− 11.7341i	− 0.881990i
$178$	6.85272i	0.513633i
$179$	−9.42951	−0.704795	−0.352397	−	0.935850i	$-0.614633\pi$
$179$	−9.42951	−0.704795	−0.352397	+	0.935850i	$0.614633\pi$
$180$	0	0
$181$	−11.7939	−0.876633	−0.438317	−	0.898821i	$-0.644425\pi$
$181$	−11.7939	−0.876633	−0.438317	+	0.898821i	$0.644425\pi$
$182$	− 36.9119i	− 2.73609i
$183$	− 2.79515i	− 0.206624i
$184$	−44.2087	−3.25911
$185$	0	0
$186$	19.9818	1.46514
$187$	− 2.94609i	− 0.215440i
$188$	− 17.5491i	− 1.27990i
$189$	−19.2419	−1.39964
$190$	0	0
$191$	5.19969	0.376237	0.188118	−	0.982146i	$-0.439761\pi$
$191$	5.19969	0.376237	0.188118	+	0.982146i	$0.439761\pi$
$192$	− 7.69345i	− 0.555227i
$193$	− 14.3936i	− 1.03607i	−0.855359	−	0.518035i	$-0.826664\pi$
$193$	− 14.3936i	− 1.03607i	0.855359	−	0.518035i	$-0.173336\pi$
$194$	15.2714	1.09643
$195$	0	0
$196$	26.2821	1.87729
$197$	− 16.0840i	− 1.14594i	−0.819577	−	0.572969i	$-0.805792\pi$
$197$	− 16.0840i	− 1.14594i	0.819577	−	0.572969i	$-0.194208\pi$
$198$	− 12.2857i	− 0.873109i
$199$	−18.1750	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$199$	−18.1750	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$200$	0	0
$201$	11.3094	0.797706
$202$	20.5665i	1.44705i
$203$	− 18.7449i	− 1.31563i
$204$	−5.64213	−0.395028
$205$	0	0
$206$	16.5729	1.15469
$207$	9.81791i	0.682392i
$208$	− 37.0285i	− 2.56746i
$209$	20.5562	1.42190
$210$	0	0
$211$	−8.40614	−0.578702	−0.289351	−	0.957223i	$-0.593440\pi$
$211$	−8.40614	−0.578702	−0.289351	+	0.957223i	$0.593440\pi$
$212$	− 32.3230i	− 2.21995i
$213$	6.51681i	0.446525i
$214$	17.8378	1.21937
$215$	0	0
$216$	−39.2591	−2.67124
$217$	22.9691i	1.55924i
$218$	− 38.1926i	− 2.58673i
$219$	2.37047	0.160182
$220$	0	0
$221$	−4.01064	−0.269785
$222$	10.5595i	0.708708i
$223$	2.90591i	0.194594i	0.995255	+	0.0972971i	$0.0310197\pi$
$223$	2.90591i	0.194594i	−0.995255	+	0.0972971i	$0.968980\pi$
$224$	33.9363	2.26747
$225$	0	0
$226$	−34.8709	−2.31958
$227$	− 15.8127i	− 1.04952i	−0.851249	−	0.524762i	$-0.824154\pi$
$227$	− 15.8127i	− 1.04952i	0.851249	−	0.524762i	$-0.175846\pi$
$228$	− 39.3677i	− 2.60719i
$229$	23.1302	1.52849	0.764244	−	0.644927i	$-0.223112\pi$
$229$	23.1302	1.52849	0.764244	+	0.644927i	$0.223112\pi$
$230$	0	0
$231$	−12.3171	−0.810405
$232$	− 38.2450i	− 2.51091i
$233$	14.5265i	0.951665i	0.879536	+	0.475833i	$0.157853\pi$
$233$	14.5265i	0.951665i	−0.879536	+	0.475833i	$0.842147\pi$
$234$	−16.7251	−1.09336
$235$	0	0
$236$	47.3716	3.08363
$237$	− 12.5080i	− 0.812481i
$238$	− 9.20348i	− 0.596573i
$239$	−3.56923	−0.230874	−0.115437	−	0.993315i	$-0.536827\pi$
$239$	−3.56923	−0.230874	−0.115437	+	0.993315i	$0.536827\pi$
$240$	0	0
$241$	−17.7990	−1.14654	−0.573269	−	0.819367i	$-0.694325\pi$
$241$	−17.7990	−1.14654	−0.573269	+	0.819367i	$0.694325\pi$
$242$	6.03904i	0.388204i
$243$	14.4018i	0.923876i
$244$	11.2843	0.722401
$245$	0	0
$246$	14.1924	0.904874
$247$	− 27.9841i	− 1.78058i
$248$	46.8636i	2.97584i
$249$	10.7260	0.679735
$250$	0	0
$251$	7.45480	0.470543	0.235271	−	0.971930i	$-0.424402\pi$
$251$	7.45480	0.470543	0.235271	+	0.971930i	$0.424402\pi$
$252$	− 27.0462i	− 1.70375i
$253$	18.0505i	1.13482i
$254$	12.0576	0.756559
$255$	0	0
$256$	−18.8867	−1.18042
$257$	26.4740i	1.65140i	0.564106	+	0.825702i	$0.309221\pi$
$257$	26.4740i	1.65140i	−0.564106	+	0.825702i	$0.690779\pi$
$258$	− 31.6582i	− 1.97095i
$259$	−12.1381	−0.754227
$260$	0	0
$261$	−8.49349	−0.525734
$262$	31.6183i	1.95339i
$263$	− 12.4974i	− 0.770621i	−0.922787	−	0.385310i	$-0.874094\pi$
$263$	− 12.4974i	− 0.770621i	0.922787	−	0.385310i	$-0.125906\pi$
$264$	−25.1304	−1.54667
$265$	0	0
$266$	64.2168	3.93739
$267$	− 3.11296i	− 0.190510i
$268$	45.6571i	2.78895i
$269$	−2.83773	−0.173020	−0.0865098	−	0.996251i	$-0.527571\pi$
$269$	−2.83773	−0.173020	−0.0865098	+	0.996251i	$0.527571\pi$
$270$	0	0
$271$	−7.60005	−0.461670	−0.230835	−	0.972993i	$-0.574146\pi$
$271$	−7.60005	−0.461670	−0.230835	+	0.972993i	$0.574146\pi$
$272$	− 9.23255i	− 0.559805i
$273$	16.7678i	1.01483i
$274$	−23.0796	−1.39429
$275$	0	0
$276$	34.5689	2.08080
$277$	− 30.4187i	− 1.82768i	−0.406070	−	0.913842i	$-0.633101\pi$
$277$	− 30.4187i	− 1.82768i	0.406070	−	0.913842i	$-0.366899\pi$
$278$	19.0577i	1.14300i
$279$	10.4075	0.623081
$280$	0	0
$281$	−20.5944	−1.22856	−0.614279	−	0.789089i	$-0.710553\pi$
$281$	−20.5944	−1.22856	−0.614279	+	0.789089i	$0.710553\pi$
$282$	11.3127i	0.673659i
$283$	4.14433i	0.246355i	0.992385	+	0.123177i	$0.0393084\pi$
$283$	4.14433i	0.246355i	−0.992385	+	0.123177i	$0.960692\pi$
$284$	−26.3089	−1.56115
$285$	0	0
$286$	−30.7495	−1.81826
$287$	16.3141i	0.962993i
$288$	− 15.3769i	− 0.906091i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.93729	−0.406671
$292$	9.56980i	0.560030i
$293$	7.85031i	0.458620i	0.973353	+	0.229310i	$0.0736470\pi$
$293$	7.85031i	0.458620i	−0.973353	+	0.229310i	$0.926353\pi$
$294$	−16.9422	−0.988090
$295$	0	0
$296$	−24.7653	−1.43945
$297$	16.0295i	0.930127i
$298$	36.1891i	2.09638i
$299$	24.5729	1.42109
$300$	0	0
$301$	36.3910	2.09754
$302$	− 37.4447i	− 2.15470i
$303$	− 9.34266i	− 0.536722i
$304$	64.4196	3.69472
$305$	0	0
$306$	−4.17018	−0.238393
$307$	0.473348i	0.0270154i	0.999909	+	0.0135077i	$0.00429977\pi$
$307$	0.473348i	0.0270154i	−0.999909	+	0.0135077i	$0.995700\pi$
$308$	− 49.7251i	− 2.83335i
$309$	−7.52851	−0.428282
$310$	0	0
$311$	18.3062	1.03805	0.519023	−	0.854760i	$-0.326296\pi$
$311$	18.3062	1.03805	0.519023	+	0.854760i	$0.326296\pi$
$312$	34.2112i	1.93683i
$313$	3.84193i	0.217159i	0.994088	+	0.108579i	$0.0346302\pi$
$313$	3.84193i	0.217159i	−0.994088	+	0.108579i	$0.965370\pi$
$314$	−22.6048	−1.27567
$315$	0	0
$316$	50.4958	2.84061
$317$	4.52266i	0.254018i	0.991902	+	0.127009i	$0.0405377\pi$
$317$	4.52266i	0.254018i	−0.991902	+	0.127009i	$0.959462\pi$
$318$	20.8363i	1.16844i
$319$	−15.6155	−0.874299
$320$	0	0
$321$	−8.10312	−0.452272
$322$	56.3890i	3.14243i
$323$	− 6.97745i	− 0.388236i
$324$	7.75535	0.430853
$325$	0	0
$326$	−23.3143	−1.29126
$327$	17.3496i	0.959435i
$328$	33.2855i	1.83789i
$329$	−13.0039	−0.716928
$330$	0	0
$331$	17.4347	0.958296	0.479148	−	0.877734i	$-0.340946\pi$
$331$	17.4347	0.958296	0.479148	+	0.877734i	$0.340946\pi$
$332$	43.3019i	2.37650i
$333$	5.49990i	0.301393i
$334$	−11.3809	−0.622733
$335$	0	0
$336$	−38.5997	−2.10578
$337$	0.943903i	0.0514177i	0.999669	+	0.0257088i	$0.00818428\pi$
$337$	0.943903i	0.0514177i	−0.999669	+	0.0257088i	$0.991816\pi$
$338$	8.02917i	0.436729i
$339$	15.8407	0.860347
$340$	0	0
$341$	19.1344	1.03619
$342$	− 29.0972i	− 1.57340i
$343$	5.28048i	0.285119i
$344$	74.2482	4.00320
$345$	0	0
$346$	22.9646	1.23459
$347$	− 19.2108i	− 1.03129i	−0.856802	−	0.515645i	$-0.827552\pi$
$347$	− 19.2108i	− 1.03129i	0.856802	−	0.515645i	$-0.172448\pi$
$348$	29.9056i	1.60311i
$349$	31.7831	1.70131	0.850655	−	0.525724i	$-0.176205\pi$
$349$	31.7831	1.70131	0.850655	+	0.525724i	$0.176205\pi$
$350$	0	0
$351$	21.8217	1.16476
$352$	− 28.2707i	− 1.50683i
$353$	− 15.4511i	− 0.822380i	−0.911550	−	0.411190i	$-0.865113\pi$
$353$	− 15.4511i	− 0.822380i	0.911550	−	0.411190i	$-0.134887\pi$
$354$	−30.5371	−1.62303
$355$	0	0
$356$	12.5673	0.666064
$357$	4.18083i	0.221273i
$358$	24.5396i	1.29696i
$359$	27.9639	1.47588	0.737940	−	0.674866i	$-0.235799\pi$
$359$	27.9639	1.47588	0.737940	+	0.674866i	$0.235799\pi$
$360$	0	0
$361$	29.6848	1.56236
$362$	30.6927i	1.61317i
$363$	− 2.74333i	− 0.143987i
$364$	−67.6930	−3.54808
$365$	0	0
$366$	−7.27417	−0.380227
$367$	− 22.7225i	− 1.18610i	−0.805165	−	0.593051i	$-0.797923\pi$
$367$	− 22.7225i	− 1.18610i	0.805165	−	0.593051i	$-0.202077\pi$
$368$	56.5671i	2.94876i
$369$	7.39208	0.384817
$370$	0	0
$371$	−23.9513	−1.24349
$372$	− 36.6448i	− 1.89995i
$373$	− 35.5230i	− 1.83931i	−0.392725	−	0.919656i	$-0.628468\pi$
$373$	− 35.5230i	− 1.83931i	0.392725	−	0.919656i	$-0.371532\pi$
$374$	−7.66698	−0.396450
$375$	0	0
$376$	−26.5317	−1.36827
$377$	21.2580i	1.09484i
$378$	50.0756i	2.57561i
$379$	30.4727	1.56528	0.782640	−	0.622475i	$-0.213873\pi$
$379$	30.4727	1.56528	0.782640	+	0.622475i	$0.213873\pi$
$380$	0	0
$381$	−5.47734	−0.280613
$382$	− 13.5318i	− 0.692347i
$383$	− 25.9667i	− 1.32683i	−0.748250	−	0.663417i	$-0.769105\pi$
$383$	− 25.9667i	− 1.32683i	0.748250	−	0.663417i	$-0.230895\pi$
$384$	2.66704	0.136102
$385$	0	0
$386$	−37.4581	−1.90657
$387$	− 16.4891i	− 0.838189i
$388$	− 28.0064i	− 1.42181i
$389$	−6.37729	−0.323342	−0.161671	−	0.986845i	$-0.551688\pi$
$389$	−6.37729	−0.323342	−0.161671	+	0.986845i	$0.551688\pi$
$390$	0	0
$391$	6.12692	0.309852
$392$	− 39.7347i	− 2.00691i
$393$	− 14.3631i	− 0.724524i
$394$	−41.8574	−2.10874
$395$	0	0
$396$	−22.5309	−1.13222
$397$	13.1874i	0.661858i	0.943656	+	0.330929i	$0.107362\pi$
$397$	13.1874i	0.661858i	−0.943656	+	0.330929i	$0.892638\pi$
$398$	47.2989i	2.37088i
$399$	−29.1715	−1.46040
$400$	0	0
$401$	−28.2411	−1.41030	−0.705148	−	0.709061i	$-0.749119\pi$
$401$	−28.2411	−1.41030	−0.705148	+	0.709061i	$0.749119\pi$
$402$	− 29.4319i	− 1.46793i
$403$	− 26.0486i	− 1.29757i
$404$	37.7171	1.87649
$405$	0	0
$406$	−48.7822	−2.42102
$407$	10.1117i	0.501218i
$408$	8.53009i	0.422302i
$409$	21.0374	1.04023	0.520117	−	0.854095i	$-0.325888\pi$
$409$	21.0374	1.04023	0.520117	+	0.854095i	$0.325888\pi$
$410$	0	0
$411$	10.4843	0.517152
$412$	− 30.3932i	− 1.49737i
$413$	− 35.1024i	− 1.72728i
$414$	25.5504	1.25573
$415$	0	0
$416$	−38.4862	−1.88694
$417$	− 8.65726i	− 0.423948i
$418$	− 53.4959i	− 2.61657i
$419$	28.1482	1.37513	0.687565	−	0.726123i	$-0.258680\pi$
$419$	28.1482	1.37513	0.687565	+	0.726123i	$0.258680\pi$
$420$	0	0
$421$	−16.6639	−0.812147	−0.406074	−	0.913840i	$-0.633102\pi$
$421$	−16.6639	−0.812147	−0.406074	+	0.913840i	$0.633102\pi$
$422$	21.8763i	1.06492i
$423$	5.89218i	0.286488i
$424$	−48.8677	−2.37322
$425$	0	0
$426$	16.9595	0.821691
$427$	− 8.36165i	− 0.404649i
$428$	− 32.7130i	− 1.58124i
$429$	13.9685	0.674403
$430$	0	0
$431$	−11.1833	−0.538682	−0.269341	−	0.963045i	$-0.586806\pi$
$431$	−11.1833	−0.538682	−0.269341	+	0.963045i	$0.586806\pi$
$432$	50.2338i	2.41687i
$433$	11.0440i	0.530743i	0.964146	+	0.265372i	$0.0854946\pi$
$433$	11.0440i	0.530743i	−0.964146	+	0.265372i	$0.914505\pi$
$434$	59.7753	2.86930
$435$	0	0
$436$	−70.0417	−3.35439
$437$	42.7503i	2.04502i
$438$	− 6.16897i	− 0.294765i
$439$	5.34654	0.255176	0.127588	−	0.991827i	$-0.459276\pi$
$439$	5.34654	0.255176	0.127588	+	0.991827i	$0.459276\pi$
$440$	0	0
$441$	−8.82433	−0.420206
$442$	10.4374i	0.496456i
$443$	16.5863i	0.788040i	0.919102	+	0.394020i	$0.128916\pi$
$443$	16.5863i	0.788040i	−0.919102	+	0.394020i	$0.871084\pi$
$444$	19.3652	0.919030
$445$	0	0
$446$	7.56241	0.358091
$447$	− 16.4395i	− 0.777559i
$448$	− 23.0148i	− 1.08735i
$449$	−35.2901	−1.66544	−0.832722	−	0.553691i	$-0.813219\pi$
$449$	−35.2901	−1.66544	−0.832722	+	0.553691i	$0.813219\pi$
$450$	0	0
$451$	13.5905	0.639952
$452$	63.9500i	3.00796i
$453$	17.0098i	0.799192i
$454$	−41.1513	−1.93132
$455$	0	0
$456$	−59.5183	−2.78720
$457$	− 9.73794i	− 0.455522i	−0.973717	−	0.227761i	$-0.926860\pi$
$457$	− 9.73794i	− 0.455522i	0.973717	−	0.227761i	$-0.0731404\pi$
$458$	− 60.1947i	− 2.81271i
$459$	5.44095	0.253962
$460$	0	0
$461$	−16.4097	−0.764276	−0.382138	−	0.924105i	$-0.624812\pi$
$461$	−16.4097	−0.764276	−0.382138	+	0.924105i	$0.624812\pi$
$462$	32.0543i	1.49130i
$463$	16.9720i	0.788755i	0.918948	+	0.394378i	$0.129040\pi$
$463$	16.9720i	0.788755i	−0.918948	+	0.394378i	$0.870960\pi$
$464$	−48.9362	−2.27181
$465$	0	0
$466$	37.8042	1.75125
$467$	23.2884i	1.07766i	0.842416	+	0.538828i	$0.181133\pi$
$467$	23.2884i	1.07766i	−0.842416	+	0.538828i	$0.818867\pi$
$468$	30.6723i	1.41783i
$469$	33.8320	1.56221
$470$	0	0
$471$	10.2686	0.473152
$472$	− 71.6190i	− 3.29653i
$473$	− 30.3156i	− 1.39391i
$474$	−32.5511	−1.49512
$475$	0	0
$476$	−16.8783	−0.773617
$477$	10.8526i	0.496905i
$478$	9.28864i	0.424853i
$479$	−2.30406	−0.105275	−0.0526376	−	0.998614i	$-0.516763\pi$
$479$	−2.30406	−0.105275	−0.0526376	+	0.998614i	$0.516763\pi$
$480$	0	0
$481$	13.7655	0.627653
$482$	46.3206i	2.10985i
$483$	− 25.6156i	− 1.16555i
$484$	11.0750	0.503411
$485$	0	0
$486$	37.4796	1.70011
$487$	14.0889i	0.638430i	0.947682	+	0.319215i	$0.103419\pi$
$487$	14.0889i	0.638430i	−0.947682	+	0.319215i	$0.896581\pi$
$488$	− 17.0602i	− 0.772278i
$489$	10.5909	0.478936
$490$	0	0
$491$	21.0485	0.949905	0.474953	−	0.880011i	$-0.342465\pi$
$491$	21.0485	0.949905	0.474953	+	0.880011i	$0.342465\pi$
$492$	− 26.0275i	− 1.17341i
$493$	5.30040i	0.238718i
$494$	−72.8264	−3.27661
$495$	0	0
$496$	59.9640	2.69247
$497$	19.4949i	0.874467i
$498$	− 27.9137i	− 1.25084i
$499$	−27.6747	−1.23889	−0.619446	−	0.785039i	$-0.712643\pi$
$499$	−27.6747	−1.23889	−0.619446	+	0.785039i	$0.712643\pi$
$500$	0	0
$501$	5.16994	0.230976
$502$	− 19.4005i	− 0.865889i
$503$	31.0855i	1.38603i	0.720921	+	0.693017i	$0.243719\pi$
$503$	31.0855i	1.38603i	−0.720921	+	0.693017i	$0.756281\pi$
$504$	−40.8900	−1.82139
$505$	0	0
$506$	46.9749	2.08829
$507$	− 3.64738i	− 0.161986i
$508$	− 22.1125i	− 0.981082i
$509$	20.5481	0.910781	0.455390	−	0.890292i	$-0.349500\pi$
$509$	20.5481	0.910781	0.455390	+	0.890292i	$0.349500\pi$
$510$	0	0
$511$	7.09123	0.313697
$512$	44.6391i	1.97279i
$513$	37.9639i	1.67615i
$514$	68.8966	3.03890
$515$	0	0
$516$	−58.0582	−2.55587
$517$	10.8329i	0.476431i
$518$	31.5886i	1.38792i
$519$	−10.4320	−0.457916
$520$	0	0
$521$	−27.9505	−1.22453	−0.612267	−	0.790651i	$-0.709742\pi$
$521$	−27.9505	−1.22453	−0.612267	+	0.790651i	$0.709742\pi$
$522$	22.1037i	0.967450i
$523$	− 0.0826499i	− 0.00361403i	−0.999998	−	0.00180701i	$-0.999425\pi$
$523$	− 0.0826499i	− 0.00361403i	0.999998	−	0.00180701i	$-0.000575191\pi$
$524$	57.9851	2.53309
$525$	0	0
$526$	−32.5234	−1.41809
$527$	− 6.49485i	− 0.282920i
$528$	32.1555i	1.39939i
$529$	−14.5391	−0.632136
$530$	0	0
$531$	−15.9052	−0.690228
$532$	− 117.768i	− 5.10588i
$533$	− 18.5014i	− 0.801383i
$534$	−8.10123	−0.350574
$535$	0	0
$536$	69.0270	2.98151
$537$	− 11.1475i	− 0.481050i
$538$	7.38498i	0.318389i
$539$	−16.2237	−0.698805
$540$	0	0
$541$	−10.7378	−0.461654	−0.230827	−	0.972995i	$-0.574143\pi$
$541$	−10.7378	−0.461654	−0.230827	+	0.972995i	$0.574143\pi$
$542$	19.7785i	0.849561i
$543$	− 13.9426i	− 0.598336i
$544$	−9.59601	−0.411426
$545$	0	0
$546$	43.6369	1.86749
$547$	− 36.8538i	− 1.57576i	−0.615832	−	0.787878i	$-0.711180\pi$
$547$	− 36.8538i	− 1.57576i	0.615832	−	0.787878i	$-0.288820\pi$
$548$	42.3259i	1.80807i
$549$	−3.78874	−0.161700
$550$	0	0
$551$	−36.9833	−1.57554
$552$	− 52.2632i	− 2.22447i
$553$	− 37.4174i	− 1.59115i
$554$	−79.1624	−3.36329
$555$	0	0
$556$	34.9501	1.48221
$557$	2.41625i	0.102380i	0.998689	+	0.0511899i	$0.0163014\pi$
$557$	2.41625i	0.102380i	−0.998689	+	0.0511899i	$0.983699\pi$
$558$	− 27.0847i	− 1.14659i
$559$	−41.2700	−1.74553
$560$	0	0
$561$	3.48284	0.147046
$562$	53.5953i	2.26078i
$563$	− 4.74857i	− 0.200129i	−0.994981	−	0.100064i	$-0.968095\pi$
$563$	− 4.74857i	− 0.200129i	0.994981	−	0.100064i	$-0.0319048\pi$
$564$	20.7464	0.873580
$565$	0	0
$566$	10.7853	0.453340
$567$	− 5.74672i	− 0.241340i
$568$	39.7753i	1.66893i
$569$	−46.0041	−1.92859	−0.964296	−	0.264827i	$-0.914685\pi$
$569$	−46.0041	−1.92859	−0.964296	+	0.264827i	$0.914685\pi$
$570$	0	0
$571$	9.95720	0.416696	0.208348	−	0.978055i	$-0.433191\pi$
$571$	9.95720	0.416696	0.208348	+	0.978055i	$0.433191\pi$
$572$	56.3918i	2.35786i
$573$	6.14704i	0.256796i
$574$	42.4563	1.77209
$575$	0	0
$576$	−10.4282	−0.434509
$577$	18.8669i	0.785439i	0.919658	+	0.392720i	$0.128466\pi$
$577$	18.8669i	0.785439i	−0.919658	+	0.392720i	$0.871534\pi$
$578$	2.60242i	0.108247i
$579$	17.0159	0.707158
$580$	0	0
$581$	32.0867	1.33118
$582$	18.0538i	0.748353i
$583$	19.9527i	0.826357i
$584$	14.4682	0.598696
$585$	0	0
$586$	20.4298	0.843949
$587$	− 23.8851i	− 0.985843i	−0.870074	−	0.492921i	$-0.835929\pi$
$587$	− 23.8851i	− 0.985843i	0.870074	−	0.492921i	$-0.164071\pi$
$588$	31.0705i	1.28132i
$589$	45.3175	1.86728
$590$	0	0
$591$	19.0144	0.782147
$592$	31.6883i	1.30238i
$593$	37.9019i	1.55644i	0.627989	+	0.778222i	$0.283878\pi$
$593$	37.9019i	1.55644i	−0.627989	+	0.778222i	$0.716122\pi$
$594$	41.7156	1.71161
$595$	0	0
$596$	66.3674	2.71852
$597$	− 21.4863i	− 0.879375i
$598$	− 63.9490i	− 2.61507i
$599$	−17.8049	−0.727488	−0.363744	−	0.931499i	$-0.618502\pi$
$599$	−17.8049	−0.727488	−0.363744	+	0.931499i	$0.618502\pi$
$600$	0	0
$601$	33.6532	1.37274	0.686372	−	0.727251i	$-0.259202\pi$
$601$	33.6532	1.37274	0.686372	+	0.727251i	$0.259202\pi$
$602$	− 94.7049i	− 3.85988i
$603$	− 15.3296i	− 0.624269i
$604$	−68.6702	−2.79415
$605$	0	0
$606$	−24.3135	−0.987670
$607$	5.89699i	0.239351i	0.992813	+	0.119676i	$0.0381855\pi$
$607$	5.89699i	0.239351i	−0.992813	+	0.119676i	$0.961815\pi$
$608$	− 66.9557i	− 2.71541i
$609$	22.1601	0.897971
$610$	0	0
$611$	14.7473	0.596613
$612$	7.64773i	0.309141i
$613$	− 7.04143i	− 0.284401i	−0.989838	−	0.142200i	$-0.954582\pi$
$613$	− 7.04143i	− 0.284401i	0.989838	−	0.142200i	$-0.0454177\pi$
$614$	1.23185	0.0497135
$615$	0	0
$616$	−75.1772	−3.02898
$617$	6.04823i	0.243492i	0.992561	+	0.121746i	$0.0388494\pi$
$617$	6.04823i	0.243492i	−0.992561	+	0.121746i	$0.961151\pi$
$618$	19.5924i	0.788120i
$619$	34.0992	1.37056	0.685282	−	0.728278i	$-0.259679\pi$
$619$	34.0992	1.37056	0.685282	+	0.728278i	$0.259679\pi$
$620$	0	0
$621$	−33.3362	−1.33774
$622$	− 47.6404i	− 1.91020i
$623$	− 9.31235i	− 0.373092i
$624$	43.7747	1.75239
$625$	0	0
$626$	9.99833	0.399614
$627$	24.3014i	0.970504i
$628$	41.4552i	1.65424i
$629$	3.43224	0.136852
$630$	0	0
$631$	18.9841	0.755743	0.377872	−	0.925858i	$-0.376656\pi$
$631$	18.9841	0.755743	0.377872	+	0.925858i	$0.376656\pi$
$632$	− 76.3424i	− 3.03674i
$633$	− 9.93767i	− 0.394987i
$634$	11.7699	0.467441
$635$	0	0
$636$	38.2119	1.51520
$637$	22.0861i	0.875082i
$638$	40.6381i	1.60888i
$639$	8.83333	0.349441
$640$	0	0
$641$	−16.3869	−0.647245	−0.323623	−	0.946186i	$-0.604901\pi$
$641$	−16.3869	−0.647245	−0.323623	+	0.946186i	$0.604901\pi$
$642$	21.0877i	0.832267i
$643$	30.8332i	1.21594i	0.793958	+	0.607972i	$0.208017\pi$
$643$	30.8332i	1.21594i	−0.793958	+	0.607972i	$0.791983\pi$
$644$	103.412	4.07501
$645$	0	0
$646$	−18.1583	−0.714428
$647$	15.8642i	0.623684i	0.950134	+	0.311842i	$0.100946\pi$
$647$	15.8642i	0.623684i	−0.950134	+	0.311842i	$0.899054\pi$
$648$	− 11.7250i	− 0.460600i
$649$	−29.2421	−1.14785
$650$	0	0
$651$	−27.1539	−1.06424
$652$	42.7562i	1.67446i
$653$	− 7.51669i	− 0.294151i	−0.989125	−	0.147075i	$-0.953014\pi$
$653$	− 7.51669i	− 0.294151i	0.989125	−	0.147075i	$-0.0469860\pi$
$654$	45.1510	1.76554
$655$	0	0
$656$	42.5904	1.66287
$657$	− 3.21310i	− 0.125355i
$658$	33.8416i	1.31928i
$659$	3.89783	0.151838	0.0759190	−	0.997114i	$-0.475811\pi$
$659$	3.89783	0.151838	0.0759190	+	0.997114i	$0.475811\pi$
$660$	0	0
$661$	−16.6761	−0.648625	−0.324313	−	0.945950i	$-0.605133\pi$
$661$	−16.6761	−0.648625	−0.324313	+	0.945950i	$0.605133\pi$
$662$	− 45.3724i	− 1.76345i
$663$	− 4.74135i	− 0.184139i
$664$	65.4662	2.54058
$665$	0	0
$666$	14.3131	0.554620
$667$	− 32.4751i	− 1.25744i
$668$	20.8715i	0.807541i
$669$	−3.43534	−0.132818
$670$	0	0
$671$	−6.96569	−0.268907
$672$	40.1193i	1.54763i
$673$	− 39.2465i	− 1.51284i	−0.654086	−	0.756420i	$-0.726946\pi$
$673$	− 39.2465i	− 1.51284i	0.654086	−	0.756420i	$-0.273054\pi$
$674$	2.45644	0.0946184
$675$	0	0
$676$	14.7248	0.566337
$677$	15.3340i	0.589332i	0.955600	+	0.294666i	$0.0952083\pi$
$677$	15.3340i	0.589332i	−0.955600	+	0.294666i	$0.904792\pi$
$678$	− 41.2241i	− 1.58320i
$679$	−20.7528	−0.796419
$680$	0	0
$681$	18.6936	0.716341
$682$	− 49.7959i	− 1.90678i
$683$	15.0386i	0.575435i	0.957715	+	0.287717i	$0.0928964\pi$
$683$	15.0386i	0.575435i	−0.957715	+	0.287717i	$0.907104\pi$
$684$	−53.3617	−2.04033
$685$	0	0
$686$	13.7420	0.524674
$687$	27.3444i	1.04325i
$688$	− 95.0040i	− 3.62199i
$689$	27.1625	1.03481
$690$	0	0
$691$	0.735679	0.0279866	0.0139933	−	0.999902i	$-0.495546\pi$
$691$	0.735679	0.0279866	0.0139933	+	0.999902i	$0.495546\pi$
$692$	− 42.1150i	− 1.60097i
$693$	16.6954i	0.634207i
$694$	−49.9947	−1.89777
$695$	0	0
$696$	45.2129	1.71379
$697$	− 4.61307i	− 0.174732i
$698$	− 82.7131i	− 3.13074i
$699$	−17.1732	−0.649548
$700$	0	0
$701$	−16.8115	−0.634962	−0.317481	−	0.948265i	$-0.602837\pi$
$701$	−16.8115	−0.634962	−0.317481	+	0.948265i	$0.602837\pi$
$702$	− 56.7893i	− 2.14337i
$703$	23.9483i	0.903227i
$704$	−19.1725	−0.722592
$705$	0	0
$706$	−40.2104	−1.51334
$707$	− 27.9484i	− 1.05111i
$708$	56.0023i	2.10470i
$709$	−15.0110	−0.563750	−0.281875	−	0.959451i	$-0.590956\pi$
$709$	−15.0110	−0.563750	−0.281875	+	0.959451i	$0.590956\pi$
$710$	0	0
$711$	−16.9542	−0.635832
$712$	− 18.9999i	− 0.712051i
$713$	39.7934i	1.49028i
$714$	10.8803	0.407184
$715$	0	0
$716$	45.0033	1.68185
$717$	− 4.21951i	− 0.157581i
$718$	− 72.7740i	− 2.71590i
$719$	−37.4098	−1.39515	−0.697575	−	0.716512i	$-0.745738\pi$
$719$	−37.4098	−1.39515	−0.697575	+	0.716512i	$0.745738\pi$
$720$	0	0
$721$	−22.5214	−0.838740
$722$	− 77.2524i	− 2.87504i
$723$	− 21.0419i	− 0.782556i
$724$	56.2876	2.09191
$725$	0	0
$726$	−7.13930	−0.264964
$727$	− 6.76798i	− 0.251011i	−0.992093	−	0.125505i	$-0.959945\pi$
$727$	− 6.76798i	− 0.251011i	0.992093	−	0.125505i	$-0.0400552\pi$
$728$	102.342i	3.79305i
$729$	−21.9006	−0.811134
$730$	0	0
$731$	−10.2901	−0.380594
$732$	13.3402i	0.493067i
$733$	18.3230i	0.676774i	0.941007	+	0.338387i	$0.109881\pi$
$733$	18.3230i	0.676774i	−0.941007	+	0.338387i	$0.890119\pi$
$734$	−59.1334	−2.18266
$735$	0	0
$736$	58.7940	2.16717
$737$	− 28.1838i	− 1.03816i
$738$	− 19.2373i	− 0.708136i
$739$	0.240801	0.00885802	0.00442901	−	0.999990i	$-0.498590\pi$
$739$	0.240801	0.00885802	0.00442901	+	0.999990i	$0.498590\pi$
$740$	0	0
$741$	33.0825	1.21532
$742$	62.3315i	2.28826i
$743$	38.0128i	1.39455i	0.716801	+	0.697277i	$0.245605\pi$
$743$	38.0128i	1.39455i	−0.716801	+	0.697277i	$0.754395\pi$
$744$	−55.4017	−2.03113
$745$	0	0
$746$	−92.4459	−3.38468
$747$	− 14.5388i	− 0.531947i
$748$	14.0605i	0.514104i
$749$	−24.2403	−0.885722
$750$	0	0
$751$	51.5263	1.88022	0.940111	−	0.340868i	$-0.110721\pi$
$751$	51.5263	1.88022	0.940111	+	0.340868i	$0.110721\pi$
$752$	33.9485i	1.23797i
$753$	8.81301i	0.321164i
$754$	55.3224	2.01472
$755$	0	0
$756$	91.8341	3.33997
$757$	− 0.984558i	− 0.0357844i	−0.999840	−	0.0178922i	$-0.994304\pi$
$757$	− 0.984558i	− 0.0357844i	0.999840	−	0.0178922i	$-0.00569556\pi$
$758$	− 79.3029i	− 2.88041i
$759$	−21.3391	−0.774560
$760$	0	0
$761$	−25.8638	−0.937563	−0.468781	−	0.883314i	$-0.655307\pi$
$761$	−25.8638	−0.937563	−0.468781	+	0.883314i	$0.655307\pi$
$762$	14.2544i	0.516381i
$763$	51.9010i	1.87894i
$764$	−24.8161	−0.897814
$765$	0	0
$766$	−67.5762	−2.44163
$767$	39.8086i	1.43740i
$768$	− 22.3277i	− 0.805680i
$769$	−47.2099	−1.70243	−0.851216	−	0.524816i	$-0.824134\pi$
$769$	−47.2099	−1.70243	−0.851216	+	0.524816i	$0.824134\pi$
$770$	0	0
$771$	−31.2974	−1.12715
$772$	68.6947i	2.47238i
$773$	1.18874i	0.0427560i	0.999771	+	0.0213780i	$0.00680535\pi$
$773$	1.18874i	0.0427560i	−0.999771	+	0.0213780i	$0.993195\pi$
$774$	−42.9116	−1.54243
$775$	0	0
$776$	−42.3417	−1.51998
$777$	− 14.3496i	− 0.514789i
$778$	16.5964i	0.595010i
$779$	32.1874	1.15323
$780$	0	0
$781$	16.2403	0.581123
$782$	− 15.9448i	− 0.570186i
$783$	− 28.8392i	− 1.03063i
$784$	−50.8424	−1.81580
$785$	0	0
$786$	−37.3789	−1.33326
$787$	38.6087i	1.37625i	0.725592	+	0.688125i	$0.241566\pi$
$787$	38.6087i	1.37625i	−0.725592	+	0.688125i	$0.758434\pi$
$788$	76.7626i	2.73455i
$789$	14.7743	0.525978
$790$	0	0
$791$	47.3870	1.68489
$792$	34.0635i	1.21039i
$793$	9.48270i	0.336741i
$794$	34.3193	1.21795
$795$	0	0
$796$	86.7419	3.07448
$797$	7.77134i	0.275275i	0.990483	+	0.137638i	$0.0439509\pi$
$797$	7.77134i	0.275275i	−0.990483	+	0.137638i	$0.956049\pi$
$798$	75.9166i	2.68742i
$799$	3.67705	0.130085
$800$	0	0
$801$	−4.21951	−0.149089
$802$	73.4954i	2.59521i
$803$	− 5.90736i	− 0.208466i
$804$	−53.9755	−1.90357
$805$	0	0
$806$	−67.7893	−2.38778
$807$	− 3.35474i	− 0.118093i
$808$	− 57.0228i	− 2.00605i
$809$	52.6740	1.85192	0.925960	−	0.377621i	$-0.123258\pi$
$809$	52.6740	1.85192	0.925960	+	0.377621i	$0.123258\pi$
$810$	0	0
$811$	−38.0502	−1.33612	−0.668062	−	0.744105i	$-0.732876\pi$
$811$	−38.0502	−1.33612	−0.668062	+	0.744105i	$0.732876\pi$
$812$	89.4620i	3.13950i
$813$	− 8.98471i	− 0.315108i
$814$	26.3149	0.922337
$815$	0	0
$816$	10.9146	0.382089
$817$	− 71.7988i	− 2.51192i
$818$	− 54.7483i	− 1.91423i
$819$	22.7282	0.794188
$820$	0	0
$821$	22.8093	0.796051	0.398026	−	0.917374i	$-0.369695\pi$
$821$	22.8093	0.796051	0.398026	+	0.917374i	$0.369695\pi$
$822$	− 27.2845i	− 0.951658i
$823$	− 26.6089i	− 0.927530i	−0.885958	−	0.463765i	$-0.846498\pi$
$823$	− 26.6089i	− 0.927530i	0.885958	−	0.463765i	$-0.153502\pi$
$824$	−45.9501	−1.60075
$825$	0	0
$826$	−91.3513	−3.17852
$827$	− 0.707585i	− 0.0246052i	−0.999924	−	0.0123026i	$-0.996084\pi$
$827$	− 0.707585i	− 0.0246052i	0.999924	−	0.0123026i	$-0.00391613\pi$
$828$	− 46.8570i	− 1.62839i
$829$	−39.7559	−1.38078	−0.690390	−	0.723437i	$-0.742561\pi$
$829$	−39.7559	−1.38078	−0.690390	+	0.723437i	$0.742561\pi$
$830$	0	0
$831$	35.9608	1.24746
$832$	26.1004i	0.904869i
$833$	5.50686i	0.190802i
$834$	−22.5298	−0.780145
$835$	0	0
$836$	−98.1067	−3.39309
$837$	35.3382i	1.22147i
$838$	− 73.2535i	− 2.53050i
$839$	−15.4254	−0.532542	−0.266271	−	0.963898i	$-0.585792\pi$
$839$	−15.4254	−0.532542	−0.266271	+	0.963898i	$0.585792\pi$
$840$	0	0
$841$	−0.905714	−0.0312315
$842$	43.3664i	1.49451i
$843$	− 24.3465i	− 0.838539i
$844$	40.1192	1.38096
$845$	0	0
$846$	15.3340	0.527192
$847$	− 8.20662i	− 0.281983i
$848$	62.5284i	2.14723i
$849$	−4.89939	−0.168147
$850$	0	0
$851$	−21.0291	−0.720867
$852$	− 31.1022i	− 1.06554i
$853$	50.0605i	1.71404i	0.515284	+	0.857020i	$0.327686\pi$
$853$	50.0605i	1.71404i	−0.515284	+	0.857020i	$0.672314\pi$
$854$	−21.7606	−0.744631
$855$	0	0
$856$	−49.4573	−1.69041
$857$	− 46.0441i	− 1.57284i	−0.617695	−	0.786418i	$-0.711933\pi$
$857$	− 46.0441i	− 1.57284i	0.617695	−	0.786418i	$-0.288067\pi$
$858$	− 36.3518i	− 1.24103i
$859$	37.1872	1.26881	0.634406	−	0.773000i	$-0.281245\pi$
$859$	37.1872	1.26881	0.634406	+	0.773000i	$0.281245\pi$
$860$	0	0
$861$	−19.2864	−0.657280
$862$	29.1038i	0.991279i
$863$	− 48.2016i	− 1.64080i	−0.571788	−	0.820401i	$-0.693750\pi$
$863$	− 48.2016i	− 1.64080i	0.571788	−	0.820401i	$-0.306250\pi$
$864$	52.2114	1.77627
$865$	0	0
$866$	28.7413	0.976668
$867$	− 1.18219i	− 0.0401493i
$868$	− 109.622i	− 3.72083i
$869$	−31.1707	−1.05739
$870$	0	0
$871$	−38.3678	−1.30004
$872$	105.893i	3.58599i
$873$	9.40328i	0.318253i
$874$	111.254	3.76323
$875$	0	0
$876$	−11.3133	−0.382242
$877$	7.97840i	0.269411i	0.990886	+	0.134706i	$0.0430089\pi$
$877$	7.97840i	0.269411i	−0.990886	+	0.134706i	$0.956991\pi$
$878$	− 13.9139i	− 0.469573i
$879$	−9.28057	−0.313026
$880$	0	0
$881$	48.5755	1.63655	0.818276	−	0.574826i	$-0.194930\pi$
$881$	48.5755	1.63655	0.818276	+	0.574826i	$0.194930\pi$
$882$	22.9646i	0.773259i
$883$	− 42.6792i	− 1.43627i	−0.695905	−	0.718134i	$-0.744996\pi$
$883$	− 42.6792i	− 1.43627i	0.695905	−	0.718134i	$-0.255004\pi$
$884$	19.1412	0.643789
$885$	0	0
$886$	43.1646	1.45014
$887$	− 53.1721i	− 1.78535i	−0.450706	−	0.892673i	$-0.648828\pi$
$887$	− 53.1721i	− 1.78535i	0.450706	−	0.892673i	$-0.351172\pi$
$888$	− 29.2773i	− 0.982483i
$889$	−16.3854	−0.549547
$890$	0	0
$891$	−4.78732	−0.160381
$892$	− 13.8688i	− 0.464361i
$893$	25.6564i	0.858559i
$894$	−42.7824	−1.43086
$895$	0	0
$896$	7.97841	0.266540
$897$	29.0499i	0.969947i
$898$	91.8398i	3.06473i
$899$	−34.4254	−1.14815
$900$	0	0
$901$	6.77260	0.225628
$902$	− 35.3683i	− 1.17763i
$903$	43.0212i	1.43166i
$904$	96.6832	3.21564
$905$	0	0
$906$	44.2668	1.47067
$907$	42.0949i	1.39774i	0.715250	+	0.698868i	$0.246313\pi$
$907$	42.0949i	1.39774i	−0.715250	+	0.698868i	$0.753687\pi$
$908$	75.4676i	2.50448i
$909$	−12.6637	−0.420027
$910$	0	0
$911$	−32.3227	−1.07090	−0.535450	−	0.844567i	$-0.679858\pi$
$911$	−32.3227	−1.07090	−0.535450	+	0.844567i	$0.679858\pi$
$912$	76.1564i	2.52179i
$913$	− 26.7299i	− 0.884631i
$914$	−25.3422	−0.838247
$915$	0	0
$916$	−110.391	−3.64744
$917$	− 42.9670i	− 1.41890i
$918$	− 14.1596i	− 0.467338i
$919$	−20.8713	−0.688480	−0.344240	−	0.938882i	$-0.611863\pi$
$919$	−20.8713	−0.688480	−0.344240	+	0.938882i	$0.611863\pi$
$920$	0	0
$921$	−0.559588	−0.0184391
$922$	42.7050i	1.40641i
$923$	− 22.1086i	− 0.727714i
$924$	58.7846	1.93387
$925$	0	0
$926$	44.1683	1.45146
$927$	10.2046i	0.335165i
$928$	50.8627i	1.66965i
$929$	−41.3293	−1.35597	−0.677985	−	0.735076i	$-0.737146\pi$
$929$	−41.3293	−1.35597	−0.677985	+	0.735076i	$0.737146\pi$
$930$	0	0
$931$	−38.4239	−1.25929
$932$	− 69.3294i	− 2.27096i
$933$	21.6414i	0.708507i
$934$	60.6061	1.98309
$935$	0	0
$936$	46.3721	1.51572
$937$	− 24.3395i	− 0.795138i	−0.917572	−	0.397569i	$-0.869854\pi$
$937$	− 24.3395i	− 0.795138i	0.917572	−	0.397569i	$-0.130146\pi$
$938$	− 88.0451i	− 2.87477i
$939$	−4.54190	−0.148219
$940$	0	0
$941$	−33.4007	−1.08883	−0.544415	−	0.838816i	$-0.683248\pi$
$941$	−33.4007	−1.08883	−0.544415	+	0.838816i	$0.683248\pi$
$942$	− 26.7233i	− 0.870691i
$943$	28.2639i	0.920399i
$944$	−91.6398	−2.98262
$945$	0	0
$946$	−78.8940	−2.56507
$947$	− 14.7087i	− 0.477969i	−0.971023	−	0.238985i	$-0.923185\pi$
$947$	− 14.7087i	− 0.477969i	0.971023	−	0.238985i	$-0.0768146\pi$
$948$	59.6957i	1.93883i
$949$	−8.04195	−0.261053
$950$	0	0
$951$	−5.34665	−0.173377
$952$	25.5176i	0.827031i
$953$	20.9652i	0.679128i	0.940583	+	0.339564i	$0.110280\pi$
$953$	20.9652i	0.679128i	−0.940583	+	0.339564i	$0.889720\pi$
$954$	28.2430	0.914401
$955$	0	0
$956$	17.0345	0.550936
$957$	− 18.4605i	− 0.596743i
$958$	5.99614i	0.193727i
$959$	31.3635	1.01278
$960$	0	0
$961$	11.1831	0.360746
$962$	− 35.8236i	− 1.15500i
$963$	10.9835i	0.353939i
$964$	84.9478	2.73598
$965$	0	0
$966$	−66.6626	−2.14483
$967$	31.1916i	1.00306i	0.865141	+	0.501528i	$0.167229\pi$
$967$	31.1916i	1.00306i	−0.865141	+	0.501528i	$0.832771\pi$
$968$	− 16.7439i	− 0.538168i
$969$	8.24868	0.264986
$970$	0	0
$971$	3.55989	0.114242	0.0571211	−	0.998367i	$-0.481808\pi$
$971$	3.55989	0.114242	0.0571211	+	0.998367i	$0.481808\pi$
$972$	− 68.7341i	− 2.20465i
$973$	− 25.8980i	− 0.830253i
$974$	36.6653	1.17483
$975$	0	0
$976$	−21.8293	−0.698738
$977$	− 51.0404i	− 1.63293i	−0.577397	−	0.816463i	$-0.695932\pi$
$977$	− 51.0404i	− 1.63293i	0.577397	−	0.816463i	$-0.304068\pi$
$978$	− 27.5619i	− 0.881333i
$979$	−7.75767	−0.247936
$980$	0	0
$981$	23.5168	0.750834
$982$	− 54.7771i	− 1.74801i
$983$	43.4903i	1.38712i	0.720397	+	0.693562i	$0.243960\pi$
$983$	43.4903i	1.38712i	−0.720397	+	0.693562i	$0.756040\pi$
$984$	−39.3499	−1.25443
$985$	0	0
$986$	13.7939	0.439287
$987$	− 15.3731i	− 0.489331i
$988$	133.557i	4.24901i
$989$	63.0467	2.00477
$990$	0	0
$991$	−20.5455	−0.652650	−0.326325	−	0.945258i	$-0.605810\pi$
$991$	−20.5455	−0.652650	−0.326325	+	0.945258i	$0.605810\pi$
$992$	− 62.3247i	− 1.97881i
$993$	20.6111i	0.654074i
$994$	50.7340	1.60919
$995$	0	0
$996$	−51.1911	−1.62205
$997$	− 48.6107i	− 1.53952i	−0.638336	−	0.769758i	$-0.720377\pi$
$997$	− 48.6107i	− 1.53952i	0.638336	−	0.769758i	$-0.279623\pi$
$998$	72.0214i	2.27980i
$999$	−18.6746	−0.590839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.f.324.2		10
5.2	odd	4		425.2.a.i.1.5	✓	5
5.3	odd	4		425.2.a.j.1.1	yes	5
5.4	even	2	inner	425.2.b.f.324.9		10
15.2	even	4		3825.2.a.bq.1.1		5
15.8	even	4		3825.2.a.bl.1.5		5
20.3	even	4		6800.2.a.cd.1.3		5
20.7	even	4		6800.2.a.bz.1.3		5
85.33	odd	4		7225.2.a.y.1.1		5
85.67	odd	4		7225.2.a.x.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.5	✓	5	5.2	odd	4
425.2.a.j.1.1	yes	5	5.3	odd	4
425.2.b.f.324.2		10	1.1	even	1	trivial
425.2.b.f.324.9		10	5.4	even	2	inner
3825.2.a.bl.1.5		5	15.8	even	4
3825.2.a.bq.1.1		5	15.2	even	4
6800.2.a.bz.1.3		5	20.7	even	4
6800.2.a.cd.1.3		5	20.3	even	4
7225.2.a.x.1.5		5	85.67	odd	4
7225.2.a.y.1.1		5	85.33	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.60242i q^{2} +1.18219i q^{3} -4.77260 q^{4} +3.07656 q^{6} +3.53650i q^{7} +7.21549i q^{8} +1.60242 q^{9} +2.94609 q^{11} -5.64213i q^{12} -4.01064i q^{13} +9.20348 q^{14} +9.23255 q^{16} -1.00000i q^{17} -4.17018i q^{18} +6.97745 q^{19} -4.18083 q^{21} -7.66698i q^{22} +6.12692i q^{23} -8.53009 q^{24} -10.4374 q^{26} +5.44095i q^{27} -16.8783i q^{28} -5.30040 q^{29} +6.49485 q^{31} -9.59601i q^{32} +3.48284i q^{33} -2.60242 q^{34} -7.64773 q^{36} +3.43224i q^{37} -18.1583i q^{38} +4.74135 q^{39} +4.61307 q^{41} +10.8803i q^{42} -10.2901i q^{43} -14.0605 q^{44} +15.9448 q^{46} +3.67705i q^{47} +10.9146i q^{48} -5.50686 q^{49} +1.18219 q^{51} +19.1412i q^{52} +6.77260i q^{53} +14.1596 q^{54} -25.5176 q^{56} +8.24868i q^{57} +13.7939i q^{58} -9.92573 q^{59} -2.36438 q^{61} -16.9024i q^{62} +5.66698i q^{63} -6.50778 q^{64} +9.06383 q^{66} -9.56650i q^{67} +4.77260i q^{68} -7.24319 q^{69} +5.51248 q^{71} +11.5623i q^{72} -2.00515i q^{73} +8.93214 q^{74} -33.3006 q^{76} +10.4189i q^{77} -12.3390i q^{78} -10.5803 q^{79} -1.62497 q^{81} -12.0052i q^{82} -9.07301i q^{83} +19.9534 q^{84} -26.7792 q^{86} -6.26609i q^{87} +21.2575i q^{88} -2.63321 q^{89} +14.1837 q^{91} -29.2414i q^{92} +7.67816i q^{93} +9.56923 q^{94} +11.3443 q^{96} +5.86816i q^{97} +14.3312i q^{98} +4.72088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) \) 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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