LMFDB - Embedded newform 4225.2.a.cb.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.76651$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76651 q^{2} -0.691389 q^{3} +1.12056 q^{4} -1.22135 q^{6} +4.36221 q^{7} -1.55354 q^{8} -2.52198 q^{9} +O(q^{10})$	\(q+1.76651 q^{2} -0.691389 q^{3} +1.12056 q^{4} -1.22135 q^{6} +4.36221 q^{7} -1.55354 q^{8} -2.52198 q^{9} +5.43722 q^{11} -0.774744 q^{12} +7.70589 q^{14} -4.98546 q^{16} +3.63696 q^{17} -4.45511 q^{18} +1.01251 q^{19} -3.01598 q^{21} +9.60491 q^{22} +3.79639 q^{23} +1.07410 q^{24} +3.81784 q^{27} +4.88812 q^{28} -1.36562 q^{29} +0.129618 q^{31} -5.69981 q^{32} -3.75923 q^{33} +6.42474 q^{34} -2.82604 q^{36} -5.24159 q^{37} +1.78862 q^{38} -4.70850 q^{41} -5.32777 q^{42} -0.0572302 q^{43} +6.09274 q^{44} +6.70637 q^{46} -7.23270 q^{47} +3.44690 q^{48} +12.0289 q^{49} -2.51456 q^{51} +10.7513 q^{53} +6.74425 q^{54} -6.77685 q^{56} -0.700040 q^{57} -2.41239 q^{58} -2.80801 q^{59} +5.30435 q^{61} +0.228972 q^{62} -11.0014 q^{63} -0.0978416 q^{64} -6.64073 q^{66} +7.94184 q^{67} +4.07545 q^{68} -2.62478 q^{69} +3.88508 q^{71} +3.91799 q^{72} -4.44994 q^{73} -9.25933 q^{74} +1.13458 q^{76} +23.7183 q^{77} -5.43195 q^{79} +4.92633 q^{81} -8.31761 q^{82} +14.1012 q^{83} -3.37960 q^{84} -0.101098 q^{86} +0.944176 q^{87} -8.44692 q^{88} +6.84215 q^{89} +4.25409 q^{92} -0.0896165 q^{93} -12.7766 q^{94} +3.94078 q^{96} +13.1040 q^{97} +21.2491 q^{98} -13.7126 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9}+O(q^{10}) $ 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9	$ 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49} - 40 q^{51} + 90 q^{54} - 20 q^{56} + 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} + 72 q^{71} + 30 q^{74} + 4 q^{76} + 16 q^{79} - 30 q^{81} + 78 q^{84} - 30 q^{86} + 94 q^{89} - 128 q^{94} - 18 q^{96} + 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * 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 $

$2$	1.76651	1.24911	0.624556	−	0.780980i	$-0.285280\pi$
$2$	1.76651	1.24911	0.624556	+	0.780980i	$0.285280\pi$
$3$	−0.691389	−0.399174	−0.199587	−	0.979880i	$-0.563960\pi$
$3$	−0.691389	−0.399174	−0.199587	+	0.979880i	$0.563960\pi$
$4$	1.12056	0.560281
$5$	0	0
$5$	0	0
$6$	−1.22135	−0.498613
$7$	4.36221	1.64876	0.824380	−	0.566037i	$-0.191524\pi$
$7$	4.36221	1.64876	0.824380	+	0.566037i	$0.191524\pi$
$8$	−1.55354	−0.549258
$9$	−2.52198	−0.840660
$10$	0	0
$11$	5.43722	1.63938	0.819692	−	0.572805i	$-0.194145\pi$
$11$	5.43722	1.63938	0.819692	+	0.572805i	$0.194145\pi$
$12$	−0.774744	−0.223649
$13$	0	0
$13$	0	0
$14$	7.70589	2.05949
$15$	0	0
$16$	−4.98546	−1.24637
$17$	3.63696	0.882094	0.441047	−	0.897484i	$-0.354607\pi$
$17$	3.63696	0.882094	0.441047	+	0.897484i	$0.354607\pi$
$18$	−4.45511	−1.05008
$19$	1.01251	0.232286	0.116143	−	0.993232i	$-0.462947\pi$
$19$	1.01251	0.232286	0.116143	+	0.993232i	$0.462947\pi$
$20$	0	0
$21$	−3.01598	−0.658141
$22$	9.60491	2.04777
$23$	3.79639	0.791602	0.395801	−	0.918336i	$-0.370467\pi$
$23$	3.79639	0.791602	0.395801	+	0.918336i	$0.370467\pi$
$24$	1.07410	0.219249
$25$	0	0
$26$	0	0
$27$	3.81784	0.734743
$28$	4.88812	0.923769
$29$	−1.36562	−0.253590	−0.126795	−	0.991929i	$-0.540469\pi$
$29$	−1.36562	−0.253590	−0.126795	+	0.991929i	$0.540469\pi$
$30$	0	0
$31$	0.129618	0.0232801	0.0116400	−	0.999932i	$-0.496295\pi$
$31$	0.129618	0.0232801	0.0116400	+	0.999932i	$0.496295\pi$
$32$	−5.69981	−1.00759
$33$	−3.75923	−0.654399
$34$	6.42474	1.10183
$35$	0	0
$36$	−2.82604	−0.471006
$37$	−5.24159	−0.861712	−0.430856	−	0.902421i	$-0.641788\pi$
$37$	−5.24159	−0.861712	−0.430856	+	0.902421i	$0.641788\pi$
$38$	1.78862	0.290152
$39$	0	0
$40$	0	0
$41$	−4.70850	−0.735343	−0.367672	−	0.929956i	$-0.619845\pi$
$41$	−4.70850	−0.735343	−0.367672	+	0.929956i	$0.619845\pi$
$42$	−5.32777	−0.822092
$43$	−0.0572302	−0.00872753	−0.00436377	−	0.999990i	$-0.501389\pi$
$43$	−0.0572302	−0.00872753	−0.00436377	+	0.999990i	$0.501389\pi$
$44$	6.09274	0.918515
$45$	0	0
$46$	6.70637	0.988800
$47$	−7.23270	−1.05500	−0.527499	−	0.849556i	$-0.676870\pi$
$47$	−7.23270	−1.05500	−0.527499	+	0.849556i	$0.676870\pi$
$48$	3.44690	0.497517
$49$	12.0289	1.71841
$50$	0	0
$51$	−2.51456	−0.352108
$52$	0	0
$53$	10.7513	1.47680	0.738402	−	0.674360i	$-0.235580\pi$
$53$	10.7513	1.47680	0.738402	+	0.674360i	$0.235580\pi$
$54$	6.74425	0.917776
$55$	0	0
$56$	−6.77685	−0.905595
$57$	−0.700040	−0.0927226
$58$	−2.41239	−0.316762
$59$	−2.80801	−0.365572	−0.182786	−	0.983153i	$-0.558512\pi$
$59$	−2.80801	−0.365572	−0.182786	+	0.983153i	$0.558512\pi$
$60$	0	0
$61$	5.30435	0.679153	0.339576	−	0.940579i	$-0.389716\pi$
$61$	5.30435	0.679153	0.339576	+	0.940579i	$0.389716\pi$
$62$	0.228972	0.0290794
$63$	−11.0014	−1.38605
$64$	−0.0978416	−0.0122302
$65$	0	0
$66$	−6.64073	−0.817417
$67$	7.94184	0.970250	0.485125	−	0.874445i	$-0.338774\pi$
$67$	7.94184	0.970250	0.485125	+	0.874445i	$0.338774\pi$
$68$	4.07545	0.494220
$69$	−2.62478	−0.315987
$70$	0	0
$71$	3.88508	0.461074	0.230537	−	0.973064i	$-0.425952\pi$
$71$	3.88508	0.461074	0.230537	+	0.973064i	$0.425952\pi$
$72$	3.91799	0.461740
$73$	−4.44994	−0.520826	−0.260413	−	0.965497i	$-0.583859\pi$
$73$	−4.44994	−0.520826	−0.260413	+	0.965497i	$0.583859\pi$
$74$	−9.25933	−1.07637
$75$	0	0
$76$	1.13458	0.130146
$77$	23.7183	2.70295
$78$	0	0
$79$	−5.43195	−0.611142	−0.305571	−	0.952169i	$-0.598847\pi$
$79$	−5.43195	−0.611142	−0.305571	+	0.952169i	$0.598847\pi$
$80$	0	0
$81$	4.92633	0.547370
$82$	−8.31761	−0.918526
$83$	14.1012	1.54781	0.773906	−	0.633301i	$-0.218300\pi$
$83$	14.1012	1.54781	0.773906	+	0.633301i	$0.218300\pi$
$84$	−3.37960	−0.368744
$85$	0	0
$86$	−0.101098	−0.0109017
$87$	0.944176	0.101226
$88$	−8.44692	−0.900445
$89$	6.84215	0.725267	0.362633	−	0.931932i	$-0.381878\pi$
$89$	6.84215	0.725267	0.362633	+	0.931932i	$0.381878\pi$
$90$	0	0
$91$	0	0
$92$	4.25409	0.443520
$93$	−0.0896165	−0.00929280
$94$	−12.7766	−1.31781
$95$	0	0
$96$	3.94078	0.402204
$97$	13.1040	1.33051	0.665256	−	0.746616i	$-0.268322\pi$
$97$	13.1040	1.33051	0.665256	+	0.746616i	$0.268322\pi$
$98$	21.2491	2.14648
$99$	−13.7126	−1.37816
$100$	0	0
$101$	1.59542	0.158750	0.0793750	−	0.996845i	$-0.474708\pi$
$101$	1.59542	0.158750	0.0793750	+	0.996845i	$0.474708\pi$
$102$	−4.44199	−0.439823
$103$	2.66136	0.262232	0.131116	−	0.991367i	$-0.458144\pi$
$103$	2.66136	0.262232	0.131116	+	0.991367i	$0.458144\pi$
$104$	0	0
$105$	0	0
$106$	18.9923	1.84469
$107$	−2.01632	−0.194925	−0.0974626	−	0.995239i	$-0.531073\pi$
$107$	−2.01632	−0.194925	−0.0974626	+	0.995239i	$0.531073\pi$
$108$	4.27812	0.411663
$109$	14.6022	1.39864	0.699321	−	0.714808i	$-0.253486\pi$
$109$	14.6022	1.39864	0.699321	+	0.714808i	$0.253486\pi$
$110$	0	0
$111$	3.62398	0.343973
$112$	−21.7476	−2.05496
$113$	−4.29215	−0.403772	−0.201886	−	0.979409i	$-0.564707\pi$
$113$	−4.29215	−0.403772	−0.201886	+	0.979409i	$0.564707\pi$
$114$	−1.23663	−0.115821
$115$	0	0
$116$	−1.53026	−0.142081
$117$	0	0
$118$	−4.96038	−0.456640
$119$	15.8652	1.45436
$120$	0	0
$121$	18.5633	1.68758
$122$	9.37020	0.848338
$123$	3.25540	0.293530
$124$	0.145245	0.0130434
$125$	0	0
$126$	−19.4341	−1.73133
$127$	−3.27565	−0.290667	−0.145333	−	0.989383i	$-0.546425\pi$
$127$	−3.27565	−0.290667	−0.145333	+	0.989383i	$0.546425\pi$
$128$	11.2268	0.992316
$129$	0.0395684	0.00348380
$130$	0	0
$131$	14.5462	1.27091	0.635455	−	0.772138i	$-0.280812\pi$
$131$	14.5462	1.27091	0.635455	+	0.772138i	$0.280812\pi$
$132$	−4.21245	−0.366647
$133$	4.41679	0.382985
$134$	14.0293	1.21195
$135$	0	0
$136$	−5.65016	−0.484497
$137$	−1.88763	−0.161272	−0.0806358	−	0.996744i	$-0.525695\pi$
$137$	−1.88763	−0.161272	−0.0806358	+	0.996744i	$0.525695\pi$
$138$	−4.63671	−0.394703
$139$	−10.0693	−0.854069	−0.427034	−	0.904235i	$-0.640442\pi$
$139$	−10.0693	−0.854069	−0.427034	+	0.904235i	$0.640442\pi$
$140$	0	0
$141$	5.00061	0.421127
$142$	6.86304	0.575934
$143$	0	0
$144$	12.5732	1.04777
$145$	0	0
$146$	−7.86086	−0.650570
$147$	−8.31662	−0.685943
$148$	−5.87353	−0.482801
$149$	−15.6539	−1.28242	−0.641210	−	0.767365i	$-0.721567\pi$
$149$	−15.6539	−1.28242	−0.641210	+	0.767365i	$0.721567\pi$
$150$	0	0
$151$	7.41670	0.603563	0.301781	−	0.953377i	$-0.402419\pi$
$151$	7.41670	0.603563	0.301781	+	0.953377i	$0.402419\pi$
$152$	−1.57298	−0.127585
$153$	−9.17236	−0.741541
$154$	41.8986	3.37629
$155$	0	0
$156$	0	0
$157$	12.2365	0.976582	0.488291	−	0.872681i	$-0.337620\pi$
$157$	12.2365	0.976582	0.488291	+	0.872681i	$0.337620\pi$
$158$	−9.59559	−0.763384
$159$	−7.43333	−0.589502
$160$	0	0
$161$	16.5606	1.30516
$162$	8.70242	0.683727
$163$	−19.4961	−1.52705	−0.763525	−	0.645778i	$-0.776533\pi$
$163$	−19.4961	−1.52705	−0.763525	+	0.645778i	$0.776533\pi$
$164$	−5.27616	−0.411999
$165$	0	0
$166$	24.9100	1.93339
$167$	−19.6198	−1.51822	−0.759111	−	0.650961i	$-0.774366\pi$
$167$	−19.6198	−1.51822	−0.759111	+	0.650961i	$0.774366\pi$
$168$	4.68544	0.361490
$169$	0	0
$170$	0	0
$171$	−2.55354	−0.195274
$172$	−0.0641300	−0.00488987
$173$	−22.9374	−1.74390	−0.871949	−	0.489597i	$-0.837144\pi$
$173$	−22.9374	−1.74390	−0.871949	+	0.489597i	$0.837144\pi$
$174$	1.66790	0.126443
$175$	0	0
$176$	−27.1071	−2.04327
$177$	1.94143	0.145927
$178$	12.0867	0.905940
$179$	−14.3131	−1.06981	−0.534905	−	0.844912i	$-0.679653\pi$
$179$	−14.3131	−1.06981	−0.534905	+	0.844912i	$0.679653\pi$
$180$	0	0
$181$	14.1550	1.05214	0.526068	−	0.850442i	$-0.323666\pi$
$181$	14.1550	1.05214	0.526068	+	0.850442i	$0.323666\pi$
$182$	0	0
$183$	−3.66737	−0.271100
$184$	−5.89783	−0.434794
$185$	0	0
$186$	−0.158309	−0.0116077
$187$	19.7750	1.44609
$188$	−8.10469	−0.591095
$189$	16.6542	1.21141
$190$	0	0
$191$	−13.0345	−0.943141	−0.471571	−	0.881828i	$-0.656313\pi$
$191$	−13.0345	−0.943141	−0.471571	+	0.881828i	$0.656313\pi$
$192$	0.0676466	0.00488198
$193$	−3.81266	−0.274441	−0.137221	−	0.990541i	$-0.543817\pi$
$193$	−3.81266	−0.274441	−0.137221	+	0.990541i	$0.543817\pi$
$194$	23.1484	1.66196
$195$	0	0
$196$	13.4791	0.962792
$197$	−6.91512	−0.492682	−0.246341	−	0.969183i	$-0.579228\pi$
$197$	−6.91512	−0.492682	−0.246341	+	0.969183i	$0.579228\pi$
$198$	−24.2234	−1.72148
$199$	23.3248	1.65345	0.826726	−	0.562604i	$-0.190200\pi$
$199$	23.3248	1.65345	0.826726	+	0.562604i	$0.190200\pi$
$200$	0	0
$201$	−5.49090	−0.387298
$202$	2.81832	0.198296
$203$	−5.95713	−0.418108
$204$	−2.81772	−0.197280
$205$	0	0
$206$	4.70133	0.327557
$207$	−9.57442	−0.665468
$208$	0	0
$209$	5.50526	0.380806
$210$	0	0
$211$	−1.02673	−0.0706831	−0.0353415	−	0.999375i	$-0.511252\pi$
$211$	−1.02673	−0.0706831	−0.0353415	+	0.999375i	$0.511252\pi$
$212$	12.0475	0.827426
$213$	−2.68610	−0.184049
$214$	−3.56186	−0.243484
$215$	0	0
$216$	−5.93115	−0.403564
$217$	0.565421	0.0383833
$218$	25.7950	1.74706
$219$	3.07664	0.207900
$220$	0	0
$221$	0	0
$222$	6.40180	0.429660
$223$	5.61855	0.376246	0.188123	−	0.982146i	$-0.439760\pi$
$223$	5.61855	0.376246	0.188123	+	0.982146i	$0.439760\pi$
$224$	−24.8637	−1.66128
$225$	0	0
$226$	−7.58214	−0.504356
$227$	−12.3749	−0.821348	−0.410674	−	0.911782i	$-0.634707\pi$
$227$	−12.3749	−0.821348	−0.410674	+	0.911782i	$0.634707\pi$
$228$	−0.784439	−0.0519507
$229$	19.7494	1.30508	0.652538	−	0.757756i	$-0.273704\pi$
$229$	19.7494	1.30508	0.652538	+	0.757756i	$0.273704\pi$
$230$	0	0
$231$	−16.3986	−1.07895
$232$	2.12154	0.139286
$233$	16.2048	1.06161	0.530804	−	0.847494i	$-0.321890\pi$
$233$	16.2048	1.06161	0.530804	+	0.847494i	$0.321890\pi$
$234$	0	0
$235$	0	0
$236$	−3.14655	−0.204823
$237$	3.75559	0.243952
$238$	28.0261	1.81666
$239$	21.8233	1.41163	0.705816	−	0.708395i	$-0.250581\pi$
$239$	21.8233	1.41163	0.705816	+	0.708395i	$0.250581\pi$
$240$	0	0
$241$	−0.301660	−0.0194317	−0.00971583	−	0.999953i	$-0.503093\pi$
$241$	−0.301660	−0.0194317	−0.00971583	+	0.999953i	$0.503093\pi$
$242$	32.7924	2.10797
$243$	−14.8595	−0.953239
$244$	5.94385	0.380516
$245$	0	0
$246$	5.75070	0.366652
$247$	0	0
$248$	−0.201366	−0.0127868
$249$	−9.74944	−0.617845
$250$	0	0
$251$	2.05366	0.129626	0.0648129	−	0.997897i	$-0.479355\pi$
$251$	2.05366	0.129626	0.0648129	+	0.997897i	$0.479355\pi$
$252$	−12.3278	−0.776576
$253$	20.6418	1.29774
$254$	−5.78646	−0.363075
$255$	0	0
$256$	20.0279	1.25174
$257$	−3.60458	−0.224847	−0.112424	−	0.993660i	$-0.535861\pi$
$257$	−3.60458	−0.224847	−0.112424	+	0.993660i	$0.535861\pi$
$258$	0.0698980	0.00435166
$259$	−22.8649	−1.42076
$260$	0	0
$261$	3.44407	0.213183
$262$	25.6961	1.58751
$263$	2.36961	0.146117	0.0730583	−	0.997328i	$-0.476724\pi$
$263$	2.36961	0.146117	0.0730583	+	0.997328i	$0.476724\pi$
$264$	5.84011	0.359434
$265$	0	0
$266$	7.80231	0.478391
$267$	−4.73059	−0.289507
$268$	8.89932	0.543613
$269$	6.16524	0.375901	0.187951	−	0.982178i	$-0.439816\pi$
$269$	6.16524	0.375901	0.187951	+	0.982178i	$0.439816\pi$
$270$	0	0
$271$	−30.6859	−1.86403	−0.932017	−	0.362414i	$-0.881952\pi$
$271$	−30.6859	−1.86403	−0.932017	+	0.362414i	$0.881952\pi$
$272$	−18.1320	−1.09941
$273$	0	0
$274$	−3.33453	−0.201446
$275$	0	0
$276$	−2.94123	−0.177041
$277$	21.4700	1.29001	0.645004	−	0.764180i	$-0.276856\pi$
$277$	21.4700	1.29001	0.645004	+	0.764180i	$0.276856\pi$
$278$	−17.7876	−1.06683
$279$	−0.326894	−0.0195706
$280$	0	0
$281$	30.1639	1.79943	0.899714	−	0.436480i	$-0.143775\pi$
$281$	30.1639	1.79943	0.899714	+	0.436480i	$0.143775\pi$
$282$	8.83363	0.526035
$283$	−11.3752	−0.676185	−0.338092	−	0.941113i	$-0.609782\pi$
$283$	−11.3752	−0.676185	−0.338092	+	0.941113i	$0.609782\pi$
$284$	4.35348	0.258331
$285$	0	0
$286$	0	0
$287$	−20.5394	−1.21240
$288$	14.3748	0.847043
$289$	−3.77249	−0.221911
$290$	0	0
$291$	−9.05997	−0.531105
$292$	−4.98643	−0.291809
$293$	16.6684	0.973778	0.486889	−	0.873464i	$-0.338132\pi$
$293$	16.6684	0.973778	0.486889	+	0.873464i	$0.338132\pi$
$294$	−14.6914	−0.856820
$295$	0	0
$296$	8.14300	0.473302
$297$	20.7584	1.20453
$298$	−27.6528	−1.60189
$299$	0	0
$300$	0	0
$301$	−0.249650	−0.0143896
$302$	13.1017	0.753918
$303$	−1.10305	−0.0633688
$304$	−5.04785	−0.289514
$305$	0	0
$306$	−16.2031	−0.926268
$307$	8.83409	0.504188	0.252094	−	0.967703i	$-0.418881\pi$
$307$	8.83409	0.504188	0.252094	+	0.967703i	$0.418881\pi$
$308$	26.5778	1.51441
$309$	−1.84004	−0.104676
$310$	0	0
$311$	−15.9186	−0.902661	−0.451331	−	0.892357i	$-0.649051\pi$
$311$	−15.9186	−0.902661	−0.451331	+	0.892357i	$0.649051\pi$
$312$	0	0
$313$	−27.6188	−1.56111	−0.780554	−	0.625089i	$-0.785063\pi$
$313$	−27.6188	−1.56111	−0.780554	+	0.625089i	$0.785063\pi$
$314$	21.6160	1.21986
$315$	0	0
$316$	−6.08683	−0.342411
$317$	−30.9950	−1.74086	−0.870428	−	0.492296i	$-0.836158\pi$
$317$	−30.9950	−1.74086	−0.870428	+	0.492296i	$0.836158\pi$
$318$	−13.1311	−0.736354
$319$	−7.42518	−0.415731
$320$	0	0
$321$	1.39406	0.0778090
$322$	29.2546	1.63029
$323$	3.68247	0.204898
$324$	5.52026	0.306681
$325$	0	0
$326$	−34.4400	−1.90746
$327$	−10.0958	−0.558301
$328$	7.31482	0.403893
$329$	−31.5505	−1.73944
$330$	0	0
$331$	−19.1915	−1.05486	−0.527431	−	0.849598i	$-0.676845\pi$
$331$	−19.1915	−1.05486	−0.527431	+	0.849598i	$0.676845\pi$
$332$	15.8013	0.867209
$333$	13.2192	0.724407
$334$	−34.6585	−1.89643
$335$	0	0
$336$	15.0361	0.820285
$337$	−15.6473	−0.852362	−0.426181	−	0.904638i	$-0.640141\pi$
$337$	−15.6473	−0.852362	−0.426181	+	0.904638i	$0.640141\pi$
$338$	0	0
$339$	2.96755	0.161175
$340$	0	0
$341$	0.704762	0.0381650
$342$	−4.51086	−0.243919
$343$	21.9369	1.18448
$344$	0.0889093	0.00479367
$345$	0	0
$346$	−40.5192	−2.17832
$347$	−9.61628	−0.516229	−0.258115	−	0.966114i	$-0.583101\pi$
$347$	−9.61628	−0.516229	−0.258115	+	0.966114i	$0.583101\pi$
$348$	1.05801	0.0567152
$349$	−13.3991	−0.717240	−0.358620	−	0.933484i	$-0.616753\pi$
$349$	−13.3991	−0.717240	−0.358620	+	0.933484i	$0.616753\pi$
$350$	0	0
$351$	0	0
$352$	−30.9911	−1.65183
$353$	−24.9985	−1.33054	−0.665269	−	0.746604i	$-0.731683\pi$
$353$	−24.9985	−1.33054	−0.665269	+	0.746604i	$0.731683\pi$
$354$	3.42955	0.182279
$355$	0	0
$356$	7.66706	0.406353
$357$	−10.9690	−0.580542
$358$	−25.2842	−1.33631
$359$	12.4028	0.654596	0.327298	−	0.944921i	$-0.393862\pi$
$359$	12.4028	0.654596	0.327298	+	0.944921i	$0.393862\pi$
$360$	0	0
$361$	−17.9748	−0.946043
$362$	25.0051	1.31424
$363$	−12.8345	−0.673636
$364$	0	0
$365$	0	0
$366$	−6.47845	−0.338634
$367$	−11.1223	−0.580581	−0.290290	−	0.956939i	$-0.593752\pi$
$367$	−11.1223	−0.580581	−0.290290	+	0.956939i	$0.593752\pi$
$368$	−18.9268	−0.986626
$369$	11.8747	0.618174
$370$	0	0
$371$	46.8994	2.43490
$372$	−0.100421	−0.00520658
$373$	−18.6801	−0.967218	−0.483609	−	0.875284i	$-0.660674\pi$
$373$	−18.6801	−0.967218	−0.483609	+	0.875284i	$0.660674\pi$
$374$	34.9327	1.80633
$375$	0	0
$376$	11.2363	0.579466
$377$	0	0
$378$	29.4198	1.51319
$379$	26.5107	1.36176	0.680881	−	0.732394i	$-0.261597\pi$
$379$	26.5107	1.36176	0.680881	+	0.732394i	$0.261597\pi$
$380$	0	0
$381$	2.26475	0.116026
$382$	−23.0255	−1.17809
$383$	3.72734	0.190458	0.0952291	−	0.995455i	$-0.469642\pi$
$383$	3.72734	0.190458	0.0952291	+	0.995455i	$0.469642\pi$
$384$	−7.76207	−0.396106
$385$	0	0
$386$	−6.73510	−0.342808
$387$	0.144334	0.00733689
$388$	14.6839	0.745460
$389$	11.5239	0.584287	0.292144	−	0.956374i	$-0.405631\pi$
$389$	11.5239	0.584287	0.292144	+	0.956374i	$0.405631\pi$
$390$	0	0
$391$	13.8073	0.698267
$392$	−18.6873	−0.943850
$393$	−10.0571	−0.507314
$394$	−12.2156	−0.615415
$395$	0	0
$396$	−15.3658	−0.772159
$397$	−31.2706	−1.56943	−0.784714	−	0.619858i	$-0.787190\pi$
$397$	−31.2706	−1.56943	−0.784714	+	0.619858i	$0.787190\pi$
$398$	41.2036	2.06535
$399$	−3.05372	−0.152877
$400$	0	0
$401$	22.2171	1.10947	0.554734	−	0.832028i	$-0.312820\pi$
$401$	22.2171	1.10947	0.554734	+	0.832028i	$0.312820\pi$
$402$	−9.69974	−0.483779
$403$	0	0
$404$	1.78776	0.0889446
$405$	0	0
$406$	−10.5233	−0.522264
$407$	−28.4997	−1.41268
$408$	3.90646	0.193398
$409$	−22.4733	−1.11123	−0.555617	−	0.831439i	$-0.687518\pi$
$409$	−22.4733	−1.11123	−0.555617	+	0.831439i	$0.687518\pi$
$410$	0	0
$411$	1.30509	0.0643753
$412$	2.98222	0.146924
$413$	−12.2491	−0.602740
$414$	−16.9133	−0.831245
$415$	0	0
$416$	0	0
$417$	6.96182	0.340922
$418$	9.72510	0.475670
$419$	7.08755	0.346249	0.173125	−	0.984900i	$-0.444614\pi$
$419$	7.08755	0.346249	0.173125	+	0.984900i	$0.444614\pi$
$420$	0	0
$421$	30.8537	1.50372	0.751858	−	0.659325i	$-0.229158\pi$
$421$	30.8537	1.50372	0.751858	+	0.659325i	$0.229158\pi$
$422$	−1.81373	−0.0882911
$423$	18.2407	0.886895
$424$	−16.7025	−0.811147
$425$	0	0
$426$	−4.74503	−0.229898
$427$	23.1387	1.11976
$428$	−2.25941	−0.109213
$429$	0	0
$430$	0	0
$431$	25.2922	1.21828	0.609142	−	0.793062i	$-0.291514\pi$
$431$	25.2922	1.21828	0.609142	+	0.793062i	$0.291514\pi$
$432$	−19.0337	−0.915759
$433$	37.5017	1.80222	0.901109	−	0.433592i	$-0.142754\pi$
$433$	37.5017	1.80222	0.901109	+	0.433592i	$0.142754\pi$
$434$	0.998822	0.0479450
$435$	0	0
$436$	16.3627	0.783632
$437$	3.84389	0.183878
$438$	5.43491	0.259690
$439$	7.55770	0.360709	0.180355	−	0.983602i	$-0.442275\pi$
$439$	7.55770	0.360709	0.180355	+	0.983602i	$0.442275\pi$
$440$	0	0
$441$	−30.3366	−1.44460
$442$	0	0
$443$	−9.24630	−0.439305	−0.219652	−	0.975578i	$-0.570492\pi$
$443$	−9.24630	−0.439305	−0.219652	+	0.975578i	$0.570492\pi$
$444$	4.06089	0.192721
$445$	0	0
$446$	9.92522	0.469973
$447$	10.8230	0.511908
$448$	−0.426806	−0.0201647
$449$	2.01940	0.0953012	0.0476506	−	0.998864i	$-0.484827\pi$
$449$	2.01940	0.0953012	0.0476506	+	0.998864i	$0.484827\pi$
$450$	0	0
$451$	−25.6011	−1.20551
$452$	−4.80963	−0.226226
$453$	−5.12783	−0.240926
$454$	−21.8603	−1.02596
$455$	0	0
$456$	1.08754	0.0509287
$457$	−32.6478	−1.52720	−0.763600	−	0.645690i	$-0.776570\pi$
$457$	−32.6478	−1.52720	−0.763600	+	0.645690i	$0.776570\pi$
$458$	34.8875	1.63019
$459$	13.8853	0.648112
$460$	0	0
$461$	27.4978	1.28070	0.640350	−	0.768083i	$-0.278789\pi$
$461$	27.4978	1.28070	0.640350	+	0.768083i	$0.278789\pi$
$462$	−28.9682	−1.34772
$463$	−26.0866	−1.21235	−0.606174	−	0.795332i	$-0.707297\pi$
$463$	−26.0866	−1.21235	−0.606174	+	0.795332i	$0.707297\pi$
$464$	6.80826	0.316066
$465$	0	0
$466$	28.6259	1.32607
$467$	−9.04138	−0.418385	−0.209193	−	0.977874i	$-0.567084\pi$
$467$	−9.04138	−0.418385	−0.209193	+	0.977874i	$0.567084\pi$
$468$	0	0
$469$	34.6440	1.59971
$470$	0	0
$471$	−8.46021	−0.389826
$472$	4.36235	0.200793
$473$	−0.311173	−0.0143078
$474$	6.63429	0.304723
$475$	0	0
$476$	17.7779	0.814850
$477$	−27.1146	−1.24149
$478$	38.5511	1.76329
$479$	14.6976	0.671550	0.335775	−	0.941942i	$-0.391002\pi$
$479$	14.6976	0.671550	0.335775	+	0.941942i	$0.391002\pi$
$480$	0	0
$481$	0	0
$482$	−0.532887	−0.0242723
$483$	−11.4498	−0.520986
$484$	20.8014	0.945517
$485$	0	0
$486$	−26.2495	−1.19070
$487$	−1.08011	−0.0489443	−0.0244722	−	0.999701i	$-0.507791\pi$
$487$	−1.08011	−0.0489443	−0.0244722	+	0.999701i	$0.507791\pi$
$488$	−8.24051	−0.373030
$489$	13.4794	0.609558
$490$	0	0
$491$	−2.17260	−0.0980479	−0.0490239	−	0.998798i	$-0.515611\pi$
$491$	−2.17260	−0.0980479	−0.0490239	+	0.998798i	$0.515611\pi$
$492$	3.64788	0.164459
$493$	−4.96672	−0.223690
$494$	0	0
$495$	0	0
$496$	−0.646206	−0.0290155
$497$	16.9475	0.760201
$498$	−17.2225	−0.771758
$499$	−28.5639	−1.27869	−0.639347	−	0.768918i	$-0.720795\pi$
$499$	−28.5639	−1.27869	−0.639347	+	0.768918i	$0.720795\pi$
$500$	0	0
$501$	13.5649	0.606034
$502$	3.62781	0.161917
$503$	−37.4525	−1.66992	−0.834962	−	0.550307i	$-0.814511\pi$
$503$	−37.4525	−1.66992	−0.834962	+	0.550307i	$0.814511\pi$
$504$	17.0911	0.761298
$505$	0	0
$506$	36.4640	1.62102
$507$	0	0
$508$	−3.67056	−0.162855
$509$	−21.3881	−0.948011	−0.474005	−	0.880522i	$-0.657192\pi$
$509$	−21.3881	−0.948011	−0.474005	+	0.880522i	$0.657192\pi$
$510$	0	0
$511$	−19.4116	−0.858716
$512$	12.9260	0.571253
$513$	3.86561	0.170671
$514$	−6.36753	−0.280860
$515$	0	0
$516$	0.0443388	0.00195191
$517$	−39.3258	−1.72955
$518$	−40.3911	−1.77468
$519$	15.8587	0.696118
$520$	0	0
$521$	−29.6689	−1.29982	−0.649910	−	0.760011i	$-0.725193\pi$
$521$	−29.6689	−1.29982	−0.649910	+	0.760011i	$0.725193\pi$
$522$	6.08399	0.266289
$523$	17.7702	0.777036	0.388518	−	0.921441i	$-0.372987\pi$
$523$	17.7702	0.777036	0.388518	+	0.921441i	$0.372987\pi$
$524$	16.3000	0.712067
$525$	0	0
$526$	4.18595	0.182516
$527$	0.471416	0.0205352
$528$	18.7415	0.815620
$529$	−8.58743	−0.373366
$530$	0	0
$531$	7.08175	0.307322
$532$	4.94929	0.214579
$533$	0	0
$534$	−8.35664	−0.361627
$535$	0	0
$536$	−12.3379	−0.532918
$537$	9.89591	0.427040
$538$	10.8910	0.469543
$539$	65.4035	2.81713
$540$	0	0
$541$	−27.9280	−1.20072	−0.600358	−	0.799731i	$-0.704975\pi$
$541$	−27.9280	−1.20072	−0.600358	+	0.799731i	$0.704975\pi$
$542$	−54.2069	−2.32839
$543$	−9.78664	−0.419985
$544$	−20.7300	−0.888791
$545$	0	0
$546$	0	0
$547$	−41.7436	−1.78483	−0.892414	−	0.451218i	$-0.850990\pi$
$547$	−41.7436	−1.78483	−0.892414	+	0.451218i	$0.850990\pi$
$548$	−2.11521	−0.0903574
$549$	−13.3775	−0.570937
$550$	0	0
$551$	−1.38271	−0.0589054
$552$	4.07770	0.173558
$553$	−23.6953	−1.00763
$554$	37.9270	1.61136
$555$	0	0
$556$	−11.2833	−0.478518
$557$	10.3200	0.437273	0.218636	−	0.975806i	$-0.429839\pi$
$557$	10.3200	0.437273	0.218636	+	0.975806i	$0.429839\pi$
$558$	−0.577462	−0.0244459
$559$	0	0
$560$	0	0
$561$	−13.6722	−0.577241
$562$	53.2849	2.24769
$563$	−20.3028	−0.855660	−0.427830	−	0.903859i	$-0.640722\pi$
$563$	−20.3028	−0.855660	−0.427830	+	0.903859i	$0.640722\pi$
$564$	5.60349	0.235950
$565$	0	0
$566$	−20.0944	−0.844630
$567$	21.4897	0.902482
$568$	−6.03562	−0.253249
$569$	−17.2259	−0.722148	−0.361074	−	0.932537i	$-0.617590\pi$
$569$	−17.2259	−0.722148	−0.361074	+	0.932537i	$0.617590\pi$
$570$	0	0
$571$	−11.0786	−0.463626	−0.231813	−	0.972760i	$-0.574466\pi$
$571$	−11.0786	−0.463626	−0.231813	+	0.972760i	$0.574466\pi$
$572$	0	0
$573$	9.01189	0.376477
$574$	−36.2831	−1.51443
$575$	0	0
$576$	0.246755	0.0102814
$577$	16.1346	0.671691	0.335845	−	0.941917i	$-0.390978\pi$
$577$	16.1346	0.671691	0.335845	+	0.941917i	$0.390978\pi$
$578$	−6.66414	−0.277192
$579$	2.63603	0.109550
$580$	0	0
$581$	61.5125	2.55197
$582$	−16.0045	−0.663410
$583$	58.4572	2.42105
$584$	6.91314	0.286068
$585$	0	0
$586$	29.4449	1.21636
$587$	3.29336	0.135932	0.0679658	−	0.997688i	$-0.478349\pi$
$587$	3.29336	0.135932	0.0679658	+	0.997688i	$0.478349\pi$
$588$	−9.31929	−0.384321
$589$	0.131240	0.00540765
$590$	0	0
$591$	4.78104	0.196666
$592$	26.1318	1.07401
$593$	−4.66535	−0.191583	−0.0957915	−	0.995401i	$-0.530538\pi$
$593$	−4.66535	−0.191583	−0.0957915	+	0.995401i	$0.530538\pi$
$594$	36.6700	1.50459
$595$	0	0
$596$	−17.5412	−0.718516
$597$	−16.1265	−0.660015
$598$	0	0
$599$	−13.3301	−0.544651	−0.272326	−	0.962205i	$-0.587793\pi$
$599$	−13.3301	−0.544651	−0.272326	+	0.962205i	$0.587793\pi$
$600$	0	0
$601$	32.4692	1.32445	0.662223	−	0.749307i	$-0.269613\pi$
$601$	32.4692	1.32445	0.662223	+	0.749307i	$0.269613\pi$
$602$	−0.441010	−0.0179742
$603$	−20.0292	−0.815651
$604$	8.31088	0.338165
$605$	0	0
$606$	−1.94856	−0.0791547
$607$	−35.5116	−1.44137	−0.720686	−	0.693262i	$-0.756173\pi$
$607$	−35.5116	−1.44137	−0.720686	+	0.693262i	$0.756173\pi$
$608$	−5.77113	−0.234050
$609$	4.11869	0.166898
$610$	0	0
$611$	0	0
$612$	−10.2782	−0.415471
$613$	−13.2854	−0.536593	−0.268296	−	0.963336i	$-0.586461\pi$
$613$	−13.2854	−0.536593	−0.268296	+	0.963336i	$0.586461\pi$
$614$	15.6055	0.629788
$615$	0	0
$616$	−36.8472	−1.48462
$617$	32.4545	1.30657	0.653284	−	0.757113i	$-0.273391\pi$
$617$	32.4545	1.30657	0.653284	+	0.757113i	$0.273391\pi$
$618$	−3.25045	−0.130752
$619$	−27.8149	−1.11797	−0.558987	−	0.829176i	$-0.688810\pi$
$619$	−27.8149	−1.11797	−0.558987	+	0.829176i	$0.688810\pi$
$620$	0	0
$621$	14.4940	0.581624
$622$	−28.1204	−1.12753
$623$	29.8469	1.19579
$624$	0	0
$625$	0	0
$626$	−48.7889	−1.95000
$627$	−3.80627	−0.152008
$628$	13.7118	0.547160
$629$	−19.0635	−0.760110
$630$	0	0
$631$	−12.6301	−0.502796	−0.251398	−	0.967884i	$-0.580890\pi$
$631$	−12.6301	−0.502796	−0.251398	+	0.967884i	$0.580890\pi$
$632$	8.43873	0.335675
$633$	0.709870	0.0282148
$634$	−54.7531	−2.17452
$635$	0	0
$636$	−8.32951	−0.330287
$637$	0	0
$638$	−13.1167	−0.519294
$639$	−9.79811	−0.387607
$640$	0	0
$641$	7.25209	0.286440	0.143220	−	0.989691i	$-0.454254\pi$
$641$	7.25209	0.286440	0.143220	+	0.989691i	$0.454254\pi$
$642$	2.46263	0.0971922
$643$	3.94228	0.155468	0.0777341	−	0.996974i	$-0.475231\pi$
$643$	3.94228	0.155468	0.0777341	+	0.996974i	$0.475231\pi$
$644$	18.5572	0.731257
$645$	0	0
$646$	6.50513	0.255941
$647$	4.09732	0.161082	0.0805412	−	0.996751i	$-0.474335\pi$
$647$	4.09732	0.161082	0.0805412	+	0.996751i	$0.474335\pi$
$648$	−7.65324	−0.300648
$649$	−15.2678	−0.599312
$650$	0	0
$651$	−0.390926	−0.0153216
$652$	−21.8466	−0.855577
$653$	−37.1255	−1.45283	−0.726416	−	0.687255i	$-0.758815\pi$
$653$	−37.1255	−1.45283	−0.726416	+	0.687255i	$0.758815\pi$
$654$	−17.8344	−0.697380
$655$	0	0
$656$	23.4740	0.916507
$657$	11.2227	0.437838
$658$	−55.7344	−2.17275
$659$	42.8526	1.66930	0.834651	−	0.550779i	$-0.185669\pi$
$659$	42.8526	1.66930	0.834651	+	0.550779i	$0.185669\pi$
$660$	0	0
$661$	−36.1806	−1.40726	−0.703632	−	0.710565i	$-0.748440\pi$
$661$	−36.1806	−1.40726	−0.703632	+	0.710565i	$0.748440\pi$
$662$	−33.9021	−1.31764
$663$	0	0
$664$	−21.9068	−0.850148
$665$	0	0
$666$	23.3518	0.904866
$667$	−5.18443	−0.200742
$668$	−21.9852	−0.850631
$669$	−3.88460	−0.150187
$670$	0	0
$671$	28.8409	1.11339
$672$	17.1905	0.663139
$673$	−32.9716	−1.27096	−0.635481	−	0.772116i	$-0.719198\pi$
$673$	−32.9716	−1.27096	−0.635481	+	0.772116i	$0.719198\pi$
$674$	−27.6411	−1.06470
$675$	0	0
$676$	0	0
$677$	17.9308	0.689137	0.344568	−	0.938761i	$-0.388025\pi$
$677$	17.9308	0.689137	0.344568	+	0.938761i	$0.388025\pi$
$678$	5.24221	0.201326
$679$	57.1625	2.19369
$680$	0	0
$681$	8.55584	0.327860
$682$	1.24497	0.0476723
$683$	−10.5968	−0.405476	−0.202738	−	0.979233i	$-0.564984\pi$
$683$	−10.5968	−0.405476	−0.202738	+	0.979233i	$0.564984\pi$
$684$	−2.86140	−0.109408
$685$	0	0
$686$	38.7518	1.47955
$687$	−13.6545	−0.520952
$688$	0.285319	0.0108777
$689$	0	0
$690$	0	0
$691$	4.11847	0.156674	0.0783369	−	0.996927i	$-0.475039\pi$
$691$	4.11847	0.156674	0.0783369	+	0.996927i	$0.475039\pi$
$692$	−25.7028	−0.977073
$693$	−59.8171	−2.27226
$694$	−16.9873	−0.644828
$695$	0	0
$696$	−1.46681	−0.0555994
$697$	−17.1246	−0.648642
$698$	−23.6697	−0.895913
$699$	−11.2038	−0.423766
$700$	0	0
$701$	−21.0511	−0.795089	−0.397545	−	0.917583i	$-0.630138\pi$
$701$	−21.0511	−0.795089	−0.397545	+	0.917583i	$0.630138\pi$
$702$	0	0
$703$	−5.30718	−0.200164
$704$	−0.531986	−0.0200500
$705$	0	0
$706$	−44.1602	−1.66199
$707$	6.95954	0.261741
$708$	2.17549	0.0817599
$709$	−21.9555	−0.824558	−0.412279	−	0.911058i	$-0.635267\pi$
$709$	−21.9555	−0.824558	−0.412279	+	0.911058i	$0.635267\pi$
$710$	0	0
$711$	13.6993	0.513763
$712$	−10.6295	−0.398359
$713$	0.492080	0.0184286
$714$	−19.3769	−0.725162
$715$	0	0
$716$	−16.0387	−0.599394
$717$	−15.0884	−0.563486
$718$	21.9097	0.817663
$719$	15.1917	0.566553	0.283276	−	0.959038i	$-0.408579\pi$
$719$	15.1917	0.566553	0.283276	+	0.959038i	$0.408579\pi$
$720$	0	0
$721$	11.6094	0.432357
$722$	−31.7527	−1.18171
$723$	0.208565	0.00775661
$724$	15.8616	0.589492
$725$	0	0
$726$	−22.6723	−0.841447
$727$	−0.952345	−0.0353205	−0.0176603	−	0.999844i	$-0.505622\pi$
$727$	−0.952345	−0.0353205	−0.0176603	+	0.999844i	$0.505622\pi$
$728$	0	0
$729$	−4.50529	−0.166863
$730$	0	0
$731$	−0.208144	−0.00769850
$732$	−4.10952	−0.151892
$733$	−16.5074	−0.609716	−0.304858	−	0.952398i	$-0.598609\pi$
$733$	−16.5074	−0.609716	−0.304858	+	0.952398i	$0.598609\pi$
$734$	−19.6477	−0.725210
$735$	0	0
$736$	−21.6387	−0.797612
$737$	43.1815	1.59061
$738$	20.9769	0.772169
$739$	−5.66334	−0.208329	−0.104165	−	0.994560i	$-0.533217\pi$
$739$	−5.66334	−0.208329	−0.104165	+	0.994560i	$0.533217\pi$
$740$	0	0
$741$	0	0
$742$	82.8484	3.04146
$743$	32.3940	1.18842	0.594211	−	0.804309i	$-0.297465\pi$
$743$	32.3940	1.18842	0.594211	+	0.804309i	$0.297465\pi$
$744$	0.139223	0.00510414
$745$	0	0
$746$	−32.9986	−1.20816
$747$	−35.5630	−1.30118
$748$	22.1591	0.810216
$749$	−8.79562	−0.321385
$750$	0	0
$751$	−27.7912	−1.01412	−0.507058	−	0.861912i	$-0.669267\pi$
$751$	−27.7912	−1.01412	−0.507058	+	0.861912i	$0.669267\pi$
$752$	36.0584	1.31491
$753$	−1.41988	−0.0517432
$754$	0	0
$755$	0	0
$756$	18.6621	0.678733
$757$	−1.65955	−0.0603173	−0.0301586	−	0.999545i	$-0.509601\pi$
$757$	−1.65955	−0.0603173	−0.0301586	+	0.999545i	$0.509601\pi$
$758$	46.8314	1.70099
$759$	−14.2715	−0.518023
$760$	0	0
$761$	−46.8781	−1.69933	−0.849665	−	0.527323i	$-0.823196\pi$
$761$	−46.8781	−1.69933	−0.849665	+	0.527323i	$0.823196\pi$
$762$	4.00070	0.144930
$763$	63.6980	2.30602
$764$	−14.6059	−0.528424
$765$	0	0
$766$	6.58439	0.237904
$767$	0	0
$768$	−13.8471	−0.499663
$769$	−33.8126	−1.21931	−0.609657	−	0.792665i	$-0.708693\pi$
$769$	−33.8126	−1.21931	−0.609657	+	0.792665i	$0.708693\pi$
$770$	0	0
$771$	2.49217	0.0897531
$772$	−4.27232	−0.153764
$773$	−35.9618	−1.29346	−0.646728	−	0.762721i	$-0.723863\pi$
$773$	−35.9618	−1.29346	−0.646728	+	0.762721i	$0.723863\pi$
$774$	0.254967	0.00916460
$775$	0	0
$776$	−20.3576	−0.730794
$777$	15.8085	0.567128
$778$	20.3572	0.729840
$779$	−4.76741	−0.170810
$780$	0	0
$781$	21.1240	0.755878
$782$	24.3908	0.872214
$783$	−5.21372	−0.186323
$784$	−59.9694	−2.14177
$785$	0	0
$786$	−17.7660	−0.633692
$787$	−11.8483	−0.422346	−0.211173	−	0.977449i	$-0.567728\pi$
$787$	−11.8483	−0.422346	−0.211173	+	0.977449i	$0.567728\pi$
$788$	−7.74883	−0.276040
$789$	−1.63832	−0.0583259
$790$	0	0
$791$	−18.7233	−0.665723
$792$	21.3030	0.756968
$793$	0	0
$794$	−55.2399	−1.96039
$795$	0	0
$796$	26.1369	0.926398
$797$	−18.5060	−0.655517	−0.327759	−	0.944761i	$-0.606293\pi$
$797$	−18.5060	−0.655517	−0.327759	+	0.944761i	$0.606293\pi$
$798$	−5.39443	−0.190961
$799$	−26.3051	−0.930607
$800$	0	0
$801$	−17.2558	−0.609703
$802$	39.2467	1.38585
$803$	−24.1953	−0.853833
$804$	−6.15289	−0.216996
$805$	0	0
$806$	0	0
$807$	−4.26258	−0.150050
$808$	−2.47854	−0.0871947
$809$	−17.9832	−0.632255	−0.316127	−	0.948717i	$-0.602383\pi$
$809$	−17.9832	−0.632255	−0.316127	+	0.948717i	$0.602383\pi$
$810$	0	0
$811$	49.4247	1.73554	0.867768	−	0.496969i	$-0.165554\pi$
$811$	49.4247	1.73554	0.867768	+	0.496969i	$0.165554\pi$
$812$	−6.67533	−0.234258
$813$	21.2159	0.744073
$814$	−50.3450	−1.76459
$815$	0	0
$816$	12.5362	0.438856
$817$	−0.0579464	−0.00202729
$818$	−39.6993	−1.38805
$819$	0	0
$820$	0	0
$821$	23.0664	0.805022	0.402511	−	0.915415i	$-0.368138\pi$
$821$	23.0664	0.805022	0.402511	+	0.915415i	$0.368138\pi$
$822$	2.30546	0.0804120
$823$	23.8292	0.830635	0.415317	−	0.909677i	$-0.363671\pi$
$823$	23.8292	0.830635	0.415317	+	0.909677i	$0.363671\pi$
$824$	−4.13453	−0.144033
$825$	0	0
$826$	−21.6382	−0.752890
$827$	−31.4303	−1.09294	−0.546468	−	0.837480i	$-0.684028\pi$
$827$	−31.4303	−1.09294	−0.546468	+	0.837480i	$0.684028\pi$
$828$	−10.7287	−0.372849
$829$	−9.13940	−0.317425	−0.158712	−	0.987325i	$-0.550734\pi$
$829$	−9.13940	−0.317425	−0.158712	+	0.987325i	$0.550734\pi$
$830$	0	0
$831$	−14.8441	−0.514937
$832$	0	0
$833$	43.7485	1.51580
$834$	12.2981	0.425849
$835$	0	0
$836$	6.16898	0.213359
$837$	0.494860	0.0171049
$838$	12.5202	0.432504
$839$	38.0484	1.31358	0.656788	−	0.754075i	$-0.271915\pi$
$839$	38.0484	1.31358	0.656788	+	0.754075i	$0.271915\pi$
$840$	0	0
$841$	−27.1351	−0.935692
$842$	54.5034	1.87831
$843$	−20.8550	−0.718284
$844$	−1.15052	−0.0396024
$845$	0	0
$846$	32.2225	1.10783
$847$	80.9772	2.78241
$848$	−53.6002	−1.84064
$849$	7.86468	0.269915
$850$	0	0
$851$	−19.8991	−0.682133
$852$	−3.00995	−0.103119
$853$	9.60563	0.328891	0.164445	−	0.986386i	$-0.447417\pi$
$853$	9.60563	0.328891	0.164445	+	0.986386i	$0.447417\pi$
$854$	40.8747	1.39870
$855$	0	0
$856$	3.13243	0.107064
$857$	30.6098	1.04561	0.522805	−	0.852452i	$-0.324886\pi$
$857$	30.6098	1.04561	0.522805	+	0.852452i	$0.324886\pi$
$858$	0	0
$859$	−33.5778	−1.14566	−0.572829	−	0.819675i	$-0.694154\pi$
$859$	−33.5778	−1.14566	−0.572829	+	0.819675i	$0.694154\pi$
$860$	0	0
$861$	14.2007	0.483960
$862$	44.6790	1.52177
$863$	34.9343	1.18918	0.594589	−	0.804030i	$-0.297315\pi$
$863$	34.9343	1.18918	0.594589	+	0.804030i	$0.297315\pi$
$864$	−21.7609	−0.740322
$865$	0	0
$866$	66.2473	2.25117
$867$	2.60826	0.0885810
$868$	0.633589	0.0215054
$869$	−29.5347	−1.00190
$870$	0	0
$871$	0	0
$872$	−22.6851	−0.768215
$873$	−33.0481	−1.11851
$874$	6.79028	0.229685
$875$	0	0
$876$	3.44756	0.116482
$877$	22.1218	0.747001	0.373500	−	0.927630i	$-0.378157\pi$
$877$	22.1218	0.747001	0.373500	+	0.927630i	$0.378157\pi$
$878$	13.3508	0.450566
$879$	−11.5243	−0.388707
$880$	0	0
$881$	−3.01725	−0.101654	−0.0508269	−	0.998707i	$-0.516186\pi$
$881$	−3.01725	−0.101654	−0.0508269	+	0.998707i	$0.516186\pi$
$882$	−53.5899	−1.80446
$883$	−21.4931	−0.723300	−0.361650	−	0.932314i	$-0.617786\pi$
$883$	−21.4931	−0.723300	−0.361650	+	0.932314i	$0.617786\pi$
$884$	0	0
$885$	0	0
$886$	−16.3337	−0.548741
$887$	−0.273357	−0.00917842	−0.00458921	−	0.999989i	$-0.501461\pi$
$887$	−0.273357	−0.00917842	−0.00458921	+	0.999989i	$0.501461\pi$
$888$	−5.62998	−0.188930
$889$	−14.2890	−0.479239
$890$	0	0
$891$	26.7856	0.897350
$892$	6.29593	0.210803
$893$	−7.32320	−0.245062
$894$	19.1189	0.639431
$895$	0	0
$896$	48.9735	1.63609
$897$	0	0
$898$	3.56729	0.119042
$899$	−0.177009	−0.00590359
$900$	0	0
$901$	39.1021	1.30268
$902$	−45.2247	−1.50582
$903$	0.172605	0.00574395
$904$	6.66802	0.221775
$905$	0	0
$906$	−9.05837	−0.300944
$907$	36.0688	1.19765	0.598823	−	0.800881i	$-0.295635\pi$
$907$	36.0688	1.19765	0.598823	+	0.800881i	$0.295635\pi$
$908$	−13.8668	−0.460186
$909$	−4.02361	−0.133455
$910$	0	0
$911$	−22.0257	−0.729744	−0.364872	−	0.931058i	$-0.618887\pi$
$911$	−22.0257	−0.729744	−0.364872	+	0.931058i	$0.618887\pi$
$912$	3.49003	0.115566
$913$	76.6715	2.53746
$914$	−57.6727	−1.90764
$915$	0	0
$916$	22.1304	0.731209
$917$	63.4537	2.09543
$918$	24.5286	0.809565
$919$	−8.72185	−0.287707	−0.143854	−	0.989599i	$-0.545949\pi$
$919$	−8.72185	−0.287707	−0.143854	+	0.989599i	$0.545949\pi$
$920$	0	0
$921$	−6.10780	−0.201259
$922$	48.5751	1.59974
$923$	0	0
$924$	−18.3756	−0.604513
$925$	0	0
$926$	−46.0823	−1.51436
$927$	−6.71191	−0.220448
$928$	7.78378	0.255515
$929$	30.2187	0.991442	0.495721	−	0.868482i	$-0.334904\pi$
$929$	30.2187	0.991442	0.495721	+	0.868482i	$0.334904\pi$
$930$	0	0
$931$	12.1794	0.399163
$932$	18.1584	0.594799
$933$	11.0060	0.360319
$934$	−15.9717	−0.522610
$935$	0	0
$936$	0	0
$937$	−27.7430	−0.906323	−0.453161	−	0.891428i	$-0.649704\pi$
$937$	−27.7430	−0.906323	−0.453161	+	0.891428i	$0.649704\pi$
$938$	61.1989	1.99822
$939$	19.0953	0.623153
$940$	0	0
$941$	52.5726	1.71382	0.856909	−	0.515468i	$-0.172382\pi$
$941$	52.5726	1.71382	0.856909	+	0.515468i	$0.172382\pi$
$942$	−14.9451	−0.486936
$943$	−17.8753	−0.582099
$944$	13.9992	0.455636
$945$	0	0
$946$	−0.549691	−0.0178720
$947$	−48.2034	−1.56640	−0.783200	−	0.621770i	$-0.786414\pi$
$947$	−48.2034	−1.56640	−0.783200	+	0.621770i	$0.786414\pi$
$948$	4.20837	0.136681
$949$	0	0
$950$	0	0
$951$	21.4296	0.694903
$952$	−24.6472	−0.798819
$953$	39.0286	1.26426	0.632130	−	0.774862i	$-0.282181\pi$
$953$	39.0286	1.26426	0.632130	+	0.774862i	$0.282181\pi$
$954$	−47.8982	−1.55076
$955$	0	0
$956$	24.4544	0.790911
$957$	5.13369	0.165949
$958$	25.9635	0.838841
$959$	−8.23426	−0.265898
$960$	0	0
$961$	−30.9832	−0.999458
$962$	0	0
$963$	5.08513	0.163866
$964$	−0.338029	−0.0108872
$965$	0	0
$966$	−20.2263	−0.650770
$967$	−43.4644	−1.39772	−0.698861	−	0.715258i	$-0.746309\pi$
$967$	−43.4644	−1.39772	−0.698861	+	0.715258i	$0.746309\pi$
$968$	−28.8388	−0.926916
$969$	−2.54602	−0.0817900
$970$	0	0
$971$	12.6094	0.404655	0.202328	−	0.979318i	$-0.435149\pi$
$971$	12.6094	0.404655	0.202328	+	0.979318i	$0.435149\pi$
$972$	−16.6510	−0.534082
$973$	−43.9245	−1.40815
$974$	−1.90802	−0.0611370
$975$	0	0
$976$	−26.4447	−0.846473
$977$	36.8798	1.17989	0.589944	−	0.807444i	$-0.299150\pi$
$977$	36.8798	1.17989	0.589944	+	0.807444i	$0.299150\pi$
$978$	23.8115	0.761407
$979$	37.2023	1.18899
$980$	0	0
$981$	−36.8266	−1.17578
$982$	−3.83791	−0.122473
$983$	15.6468	0.499056	0.249528	−	0.968368i	$-0.419725\pi$
$983$	15.6468	0.499056	0.249528	+	0.968368i	$0.419725\pi$
$984$	−5.05739	−0.161224
$985$	0	0
$986$	−8.77376	−0.279414
$987$	21.8137	0.694338
$988$	0	0
$989$	−0.217268	−0.00690873
$990$	0	0
$991$	21.2433	0.674816	0.337408	−	0.941359i	$-0.390450\pi$
$991$	21.2433	0.674816	0.337408	+	0.941359i	$0.390450\pi$
$992$	−0.738798	−0.0234568
$993$	13.2688	0.421073
$994$	29.9380	0.949576
$995$	0	0
$996$	−10.9248	−0.346167
$997$	17.7179	0.561132	0.280566	−	0.959835i	$-0.409478\pi$
$997$	17.7179	0.561132	0.280566	+	0.959835i	$0.409478\pi$
$998$	−50.4584	−1.59723
$999$	−20.0115	−0.633137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.cb.1.14		18
5.2	odd	4		845.2.b.h.339.14	yes	18
5.3	odd	4		845.2.b.h.339.5	yes	18
5.4	even	2	inner	4225.2.a.cb.1.5		18
13.12	even	2		4225.2.a.ca.1.5		18
65.2	even	12		845.2.l.g.654.9		72
65.3	odd	12		845.2.n.h.529.14		36
65.7	even	12		845.2.l.g.699.28		72
65.8	even	4		845.2.d.e.844.27		36
65.12	odd	4		845.2.b.g.339.5	✓	18
65.17	odd	12		845.2.n.i.484.5		36
65.18	even	4		845.2.d.e.844.9		36
65.22	odd	12		845.2.n.h.484.14		36
65.23	odd	12		845.2.n.i.529.5		36
65.28	even	12		845.2.l.g.654.28		72
65.32	even	12		845.2.l.g.699.10		72
65.33	even	12		845.2.l.g.699.9		72
65.37	even	12		845.2.l.g.654.27		72
65.38	odd	4		845.2.b.g.339.14	yes	18
65.42	odd	12		845.2.n.h.529.5		36
65.43	odd	12		845.2.n.i.484.14		36
65.47	even	4		845.2.d.e.844.10		36
65.48	odd	12		845.2.n.h.484.5		36
65.57	even	4		845.2.d.e.844.28		36
65.58	even	12		845.2.l.g.699.27		72
65.62	odd	12		845.2.n.i.529.14		36
65.63	even	12		845.2.l.g.654.10		72
65.64	even	2		4225.2.a.ca.1.14		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.5	✓	18	65.12	odd	4
845.2.b.g.339.14	yes	18	65.38	odd	4
845.2.b.h.339.5	yes	18	5.3	odd	4
845.2.b.h.339.14	yes	18	5.2	odd	4
845.2.d.e.844.9		36	65.18	even	4
845.2.d.e.844.10		36	65.47	even	4
845.2.d.e.844.27		36	65.8	even	4
845.2.d.e.844.28		36	65.57	even	4
845.2.l.g.654.9		72	65.2	even	12
845.2.l.g.654.10		72	65.63	even	12
845.2.l.g.654.27		72	65.37	even	12
845.2.l.g.654.28		72	65.28	even	12
845.2.l.g.699.9		72	65.33	even	12
845.2.l.g.699.10		72	65.32	even	12
845.2.l.g.699.27		72	65.58	even	12
845.2.l.g.699.28		72	65.7	even	12
845.2.n.h.484.5		36	65.48	odd	12
845.2.n.h.484.14		36	65.22	odd	12
845.2.n.h.529.5		36	65.42	odd	12
845.2.n.h.529.14		36	65.3	odd	12
845.2.n.i.484.5		36	65.17	odd	12
845.2.n.i.484.14		36	65.43	odd	12
845.2.n.i.529.5		36	65.23	odd	12
845.2.n.i.529.14		36	65.62	odd	12
4225.2.a.ca.1.5		18	13.12	even	2
4225.2.a.ca.1.14		18	65.64	even	2
4225.2.a.cb.1.5		18	5.4	even	2	inner
4225.2.a.cb.1.14		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76651 q^{2} -0.691389 q^{3} +1.12056 q^{4} -1.22135 q^{6} +4.36221 q^{7} -1.55354 q^{8} -2.52198 q^{9} +O(q^{10})\)	\(q+1.76651 q^{2} -0.691389 q^{3} +1.12056 q^{4} -1.22135 q^{6} +4.36221 q^{7} -1.55354 q^{8} -2.52198 q^{9} +5.43722 q^{11} -0.774744 q^{12} +7.70589 q^{14} -4.98546 q^{16} +3.63696 q^{17} -4.45511 q^{18} +1.01251 q^{19} -3.01598 q^{21} +9.60491 q^{22} +3.79639 q^{23} +1.07410 q^{24} +3.81784 q^{27} +4.88812 q^{28} -1.36562 q^{29} +0.129618 q^{31} -5.69981 q^{32} -3.75923 q^{33} +6.42474 q^{34} -2.82604 q^{36} -5.24159 q^{37} +1.78862 q^{38} -4.70850 q^{41} -5.32777 q^{42} -0.0572302 q^{43} +6.09274 q^{44} +6.70637 q^{46} -7.23270 q^{47} +3.44690 q^{48} +12.0289 q^{49} -2.51456 q^{51} +10.7513 q^{53} +6.74425 q^{54} -6.77685 q^{56} -0.700040 q^{57} -2.41239 q^{58} -2.80801 q^{59} +5.30435 q^{61} +0.228972 q^{62} -11.0014 q^{63} -0.0978416 q^{64} -6.64073 q^{66} +7.94184 q^{67} +4.07545 q^{68} -2.62478 q^{69} +3.88508 q^{71} +3.91799 q^{72} -4.44994 q^{73} -9.25933 q^{74} +1.13458 q^{76} +23.7183 q^{77} -5.43195 q^{79} +4.92633 q^{81} -8.31761 q^{82} +14.1012 q^{83} -3.37960 q^{84} -0.101098 q^{86} +0.944176 q^{87} -8.44692 q^{88} +6.84215 q^{89} +4.25409 q^{92} -0.0896165 q^{93} -12.7766 q^{94} +3.94078 q^{96} +13.1040 q^{97} +21.2491 q^{98} -13.7126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9}+O(q^{10}) \) 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9	\( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49} - 40 q^{51} + 90 q^{54} - 20 q^{56} + 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} + 72 q^{71} + 30 q^{74} + 4 q^{76} + 16 q^{79} - 30 q^{81} + 78 q^{84} - 30 q^{86} + 94 q^{89} - 128 q^{94} - 18 q^{96} + 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more