LMFDB - Embedded newform 5225.2.a.bb.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,0,0,32,0,-12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.790175 q^{2} +2.57534 q^{3} -1.37562 q^{4} +2.03497 q^{6} +0.0669205 q^{7} -2.66733 q^{8} +3.63236 q^{9} -1.00000 q^{11} -3.54270 q^{12} +4.14913 q^{13} +0.0528789 q^{14} +0.643589 q^{16} -6.61558 q^{17} +2.87020 q^{18} -1.00000 q^{19} +0.172343 q^{21} -0.790175 q^{22} +2.21109 q^{23} -6.86928 q^{24} +3.27854 q^{26} +1.62854 q^{27} -0.0920574 q^{28} +2.56775 q^{29} +8.77193 q^{31} +5.84321 q^{32} -2.57534 q^{33} -5.22747 q^{34} -4.99676 q^{36} +11.4277 q^{37} -0.790175 q^{38} +10.6854 q^{39} +10.9830 q^{41} +0.136181 q^{42} +3.95978 q^{43} +1.37562 q^{44} +1.74715 q^{46} -8.97459 q^{47} +1.65746 q^{48} -6.99552 q^{49} -17.0374 q^{51} -5.70765 q^{52} -3.13641 q^{53} +1.28683 q^{54} -0.178499 q^{56} -2.57534 q^{57} +2.02897 q^{58} +10.3362 q^{59} +2.76550 q^{61} +6.93135 q^{62} +0.243079 q^{63} +3.32998 q^{64} -2.03497 q^{66} +15.2686 q^{67} +9.10056 q^{68} +5.69431 q^{69} -4.11096 q^{71} -9.68871 q^{72} +7.16167 q^{73} +9.02987 q^{74} +1.37562 q^{76} -0.0669205 q^{77} +8.44334 q^{78} +8.94984 q^{79} -6.70304 q^{81} +8.67846 q^{82} +3.78408 q^{83} -0.237079 q^{84} +3.12892 q^{86} +6.61283 q^{87} +2.66733 q^{88} +0.223450 q^{89} +0.277662 q^{91} -3.04163 q^{92} +22.5907 q^{93} -7.09149 q^{94} +15.0482 q^{96} +14.1106 q^{97} -5.52768 q^{98} -3.63236 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44}+ \cdots - 34 q^{99}+O(q^{100}) $ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * q^99

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 $

$2$	0.790175	0.558738	0.279369	−	0.960184i	$-0.409875\pi$
$2$	0.790175	0.558738	0.279369	+	0.960184i	$0.409875\pi$
$3$	2.57534	1.48687	0.743436	−	0.668807i	$-0.233195\pi$
$3$	2.57534	1.48687	0.743436	+	0.668807i	$0.233195\pi$
$4$	−1.37562	−0.687812
$5$	0	0
$5$	0	0
$6$	2.03497	0.830771
$7$	0.0669205	0.0252936	0.0126468	−	0.999920i	$-0.495974\pi$
$7$	0.0669205	0.0252936	0.0126468	+	0.999920i	$0.495974\pi$
$8$	−2.66733	−0.943044
$9$	3.63236	1.21079
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.54270	−1.02269
$13$	4.14913	1.15076	0.575381	−	0.817885i	$-0.304854\pi$
$13$	4.14913	1.15076	0.575381	+	0.817885i	$0.304854\pi$
$14$	0.0528789	0.0141325
$15$	0	0
$16$	0.643589	0.160897
$17$	−6.61558	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$17$	−6.61558	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$18$	2.87020	0.676512
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.172343	0.0376083
$22$	−0.790175	−0.168466
$23$	2.21109	0.461045	0.230522	−	0.973067i	$-0.425957\pi$
$23$	2.21109	0.461045	0.230522	+	0.973067i	$0.425957\pi$
$24$	−6.86928	−1.40219
$25$	0	0
$26$	3.27854	0.642974
$27$	1.62854	0.313413
$28$	−0.0920574	−0.0173972
$29$	2.56775	0.476820	0.238410	−	0.971165i	$-0.423374\pi$
$29$	2.56775	0.476820	0.238410	+	0.971165i	$0.423374\pi$
$30$	0	0
$31$	8.77193	1.57548	0.787742	−	0.616005i	$-0.211250\pi$
$31$	8.77193	1.57548	0.787742	+	0.616005i	$0.211250\pi$
$32$	5.84321	1.03294
$33$	−2.57534	−0.448309
$34$	−5.22747	−0.896503
$35$	0	0
$36$	−4.99676	−0.832794
$37$	11.4277	1.87870	0.939350	−	0.342960i	$-0.111430\pi$
$37$	11.4277	1.87870	0.939350	+	0.342960i	$0.111430\pi$
$38$	−0.790175	−0.128183
$39$	10.6854	1.71104
$40$	0	0
$41$	10.9830	1.71525	0.857626	−	0.514274i	$-0.171939\pi$
$41$	10.9830	1.71525	0.857626	+	0.514274i	$0.171939\pi$
$42$	0.136181	0.0210132
$43$	3.95978	0.603861	0.301931	−	0.953330i	$-0.402369\pi$
$43$	3.95978	0.603861	0.301931	+	0.953330i	$0.402369\pi$
$44$	1.37562	0.207383
$45$	0	0
$46$	1.74715	0.257603
$47$	−8.97459	−1.30908	−0.654539	−	0.756028i	$-0.727137\pi$
$47$	−8.97459	−1.30908	−0.654539	+	0.756028i	$0.727137\pi$
$48$	1.65746	0.239234
$49$	−6.99552	−0.999360
$50$	0	0
$51$	−17.0374	−2.38571
$52$	−5.70765	−0.791508
$53$	−3.13641	−0.430819	−0.215410	−	0.976524i	$-0.569109\pi$
$53$	−3.13641	−0.430819	−0.215410	+	0.976524i	$0.569109\pi$
$54$	1.28683	0.175116
$55$	0	0
$56$	−0.178499	−0.0238530
$57$	−2.57534	−0.341112
$58$	2.02897	0.266417
$59$	10.3362	1.34566	0.672828	−	0.739799i	$-0.265079\pi$
$59$	10.3362	1.34566	0.672828	+	0.739799i	$0.265079\pi$
$60$	0	0
$61$	2.76550	0.354086	0.177043	−	0.984203i	$-0.443347\pi$
$61$	2.76550	0.354086	0.177043	+	0.984203i	$0.443347\pi$
$62$	6.93135	0.880283
$63$	0.243079	0.0306251
$64$	3.32998	0.416248
$65$	0	0
$66$	−2.03497	−0.250487
$67$	15.2686	1.86535	0.932677	−	0.360713i	$-0.117467\pi$
$67$	15.2686	1.86535	0.932677	+	0.360713i	$0.117467\pi$
$68$	9.10056	1.10360
$69$	5.69431	0.685514
$70$	0	0
$71$	−4.11096	−0.487881	−0.243940	−	0.969790i	$-0.578440\pi$
$71$	−4.11096	−0.487881	−0.243940	+	0.969790i	$0.578440\pi$
$72$	−9.68871	−1.14183
$73$	7.16167	0.838210	0.419105	−	0.907938i	$-0.362344\pi$
$73$	7.16167	0.838210	0.419105	+	0.907938i	$0.362344\pi$
$74$	9.02987	1.04970
$75$	0	0
$76$	1.37562	0.157795
$77$	−0.0669205	−0.00762630
$78$	8.44334	0.956020
$79$	8.94984	1.00694	0.503468	−	0.864014i	$-0.332057\pi$
$79$	8.94984	1.00694	0.503468	+	0.864014i	$0.332057\pi$
$80$	0	0
$81$	−6.70304	−0.744782
$82$	8.67846	0.958376
$83$	3.78408	0.415357	0.207679	−	0.978197i	$-0.433409\pi$
$83$	3.78408	0.415357	0.207679	+	0.978197i	$0.433409\pi$
$84$	−0.237079	−0.0258674
$85$	0	0
$86$	3.12892	0.337400
$87$	6.61283	0.708970
$88$	2.66733	0.284339
$89$	0.223450	0.0236856	0.0118428	−	0.999930i	$-0.496230\pi$
$89$	0.223450	0.0236856	0.0118428	+	0.999930i	$0.496230\pi$
$90$	0	0
$91$	0.277662	0.0291069
$92$	−3.04163	−0.317112
$93$	22.5907	2.34254
$94$	−7.09149	−0.731431
$95$	0	0
$96$	15.0482	1.53585
$97$	14.1106	1.43272	0.716359	−	0.697732i	$-0.245807\pi$
$97$	14.1106	1.43272	0.716359	+	0.697732i	$0.245807\pi$
$98$	−5.52768	−0.558380
$99$	−3.63236	−0.365066
$100$	0	0
$101$	−11.6761	−1.16182	−0.580909	−	0.813968i	$-0.697303\pi$
$101$	−11.6761	−1.16182	−0.580909	+	0.813968i	$0.697303\pi$
$102$	−13.4625	−1.33298
$103$	−11.6529	−1.14820	−0.574098	−	0.818786i	$-0.694647\pi$
$103$	−11.6529	−1.14820	−0.574098	+	0.818786i	$0.694647\pi$
$104$	−11.0671	−1.08522
$105$	0	0
$106$	−2.47831	−0.240715
$107$	2.72445	0.263382	0.131691	−	0.991291i	$-0.457959\pi$
$107$	2.72445	0.263382	0.131691	+	0.991291i	$0.457959\pi$
$108$	−2.24026	−0.215569
$109$	−3.96882	−0.380144	−0.190072	−	0.981770i	$-0.560872\pi$
$109$	−3.96882	−0.380144	−0.190072	+	0.981770i	$0.560872\pi$
$110$	0	0
$111$	29.4301	2.79339
$112$	0.0430693	0.00406967
$113$	2.81732	0.265031	0.132516	−	0.991181i	$-0.457695\pi$
$113$	2.81732	0.265031	0.132516	+	0.991181i	$0.457695\pi$
$114$	−2.03497	−0.190592
$115$	0	0
$116$	−3.53226	−0.327962
$117$	15.0711	1.39333
$118$	8.16739	0.751869
$119$	−0.442718	−0.0405839
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.18523	0.197841
$123$	28.2848	2.55036
$124$	−12.0669	−1.08364
$125$	0	0
$126$	0.192075	0.0171114
$127$	−1.05561	−0.0936707	−0.0468353	−	0.998903i	$-0.514914\pi$
$127$	−1.05561	−0.0936707	−0.0468353	+	0.998903i	$0.514914\pi$
$128$	−9.05516	−0.800371
$129$	10.1978	0.897864
$130$	0	0
$131$	−18.2156	−1.59151	−0.795753	−	0.605621i	$-0.792925\pi$
$131$	−18.2156	−1.59151	−0.795753	+	0.605621i	$0.792925\pi$
$132$	3.54270	0.308352
$133$	−0.0669205	−0.00580274
$134$	12.0648	1.04224
$135$	0	0
$136$	17.6460	1.51313
$137$	7.14378	0.610334	0.305167	−	0.952299i	$-0.401288\pi$
$137$	7.14378	0.610334	0.305167	+	0.952299i	$0.401288\pi$
$138$	4.49950	0.383023
$139$	2.25667	0.191408	0.0957041	−	0.995410i	$-0.469490\pi$
$139$	2.25667	0.191408	0.0957041	+	0.995410i	$0.469490\pi$
$140$	0	0
$141$	−23.1126	−1.94643
$142$	−3.24837	−0.272598
$143$	−4.14913	−0.346968
$144$	2.33775	0.194812
$145$	0	0
$146$	5.65897	0.468340
$147$	−18.0158	−1.48592
$148$	−15.7202	−1.29219
$149$	5.60288	0.459006	0.229503	−	0.973308i	$-0.426290\pi$
$149$	5.60288	0.459006	0.229503	+	0.973308i	$0.426290\pi$
$150$	0	0
$151$	−20.8631	−1.69782	−0.848908	−	0.528541i	$-0.822739\pi$
$151$	−20.8631	−1.69782	−0.848908	+	0.528541i	$0.822739\pi$
$152$	2.66733	0.216349
$153$	−24.0302	−1.94273
$154$	−0.0528789	−0.00426110
$155$	0	0
$156$	−14.6991	−1.17687
$157$	−0.664796	−0.0530565	−0.0265282	−	0.999648i	$-0.508445\pi$
$157$	−0.664796	−0.0530565	−0.0265282	+	0.999648i	$0.508445\pi$
$158$	7.07194	0.562613
$159$	−8.07732	−0.640573
$160$	0	0
$161$	0.147967	0.0116615
$162$	−5.29657	−0.416138
$163$	0.357030	0.0279647	0.0139824	−	0.999902i	$-0.495549\pi$
$163$	0.357030	0.0279647	0.0139824	+	0.999902i	$0.495549\pi$
$164$	−15.1084	−1.17977
$165$	0	0
$166$	2.99009	0.232076
$167$	−12.2396	−0.947127	−0.473563	−	0.880760i	$-0.657033\pi$
$167$	−12.2396	−0.947127	−0.473563	+	0.880760i	$0.657033\pi$
$168$	−0.459696	−0.0354663
$169$	4.21530	0.324253
$170$	0	0
$171$	−3.63236	−0.277774
$172$	−5.44717	−0.415343
$173$	−14.4260	−1.09679	−0.548394	−	0.836220i	$-0.684761\pi$
$173$	−14.4260	−1.09679	−0.548394	+	0.836220i	$0.684761\pi$
$174$	5.22529	0.396128
$175$	0	0
$176$	−0.643589	−0.0485124
$177$	26.6191	2.00082
$178$	0.176564	0.0132341
$179$	8.72275	0.651969	0.325984	−	0.945375i	$-0.394304\pi$
$179$	8.72275	0.651969	0.325984	+	0.945375i	$0.394304\pi$
$180$	0	0
$181$	−13.9631	−1.03787	−0.518936	−	0.854813i	$-0.673672\pi$
$181$	−13.9631	−1.03787	−0.518936	+	0.854813i	$0.673672\pi$
$182$	0.219401	0.0162631
$183$	7.12209	0.526480
$184$	−5.89772	−0.434786
$185$	0	0
$186$	17.8506	1.30887
$187$	6.61558	0.483779
$188$	12.3457	0.900400
$189$	0.108983	0.00792733
$190$	0	0
$191$	4.76262	0.344611	0.172306	−	0.985044i	$-0.444878\pi$
$191$	4.76262	0.344611	0.172306	+	0.985044i	$0.444878\pi$
$192$	8.57582	0.618907
$193$	−5.69191	−0.409713	−0.204856	−	0.978792i	$-0.565673\pi$
$193$	−5.69191	−0.409713	−0.204856	+	0.978792i	$0.565673\pi$
$194$	11.1499	0.800514
$195$	0	0
$196$	9.62321	0.687372
$197$	19.1425	1.36385	0.681923	−	0.731424i	$-0.261144\pi$
$197$	19.1425	1.36385	0.681923	+	0.731424i	$0.261144\pi$
$198$	−2.87020	−0.203976
$199$	21.1279	1.49772	0.748859	−	0.662730i	$-0.230602\pi$
$199$	21.1279	1.49772	0.748859	+	0.662730i	$0.230602\pi$
$200$	0	0
$201$	39.3217	2.77354
$202$	−9.22619	−0.649152
$203$	0.171835	0.0120605
$204$	23.4370	1.64092
$205$	0	0
$206$	−9.20784	−0.641541
$207$	8.03149	0.558227
$208$	2.67034	0.185155
$209$	1.00000	0.0691714
$210$	0	0
$211$	21.9394	1.51037	0.755187	−	0.655510i	$-0.227546\pi$
$211$	21.9394	1.51037	0.755187	+	0.655510i	$0.227546\pi$
$212$	4.31452	0.296323
$213$	−10.5871	−0.725416
$214$	2.15279	0.147162
$215$	0	0
$216$	−4.34386	−0.295562
$217$	0.587022	0.0398496
$218$	−3.13606	−0.212401
$219$	18.4437	1.24631
$220$	0	0
$221$	−27.4489	−1.84641
$222$	23.2549	1.56077
$223$	23.0514	1.54363	0.771817	−	0.635844i	$-0.219348\pi$
$223$	23.0514	1.54363	0.771817	+	0.635844i	$0.219348\pi$
$224$	0.391031	0.0261268
$225$	0	0
$226$	2.22618	0.148083
$227$	−12.1150	−0.804101	−0.402050	−	0.915618i	$-0.631702\pi$
$227$	−12.1150	−0.804101	−0.402050	+	0.915618i	$0.631702\pi$
$228$	3.54270	0.234621
$229$	−17.0184	−1.12461	−0.562303	−	0.826931i	$-0.690085\pi$
$229$	−17.0184	−1.12461	−0.562303	+	0.826931i	$0.690085\pi$
$230$	0	0
$231$	−0.172343	−0.0113393
$232$	−6.84905	−0.449662
$233$	−2.08834	−0.136812	−0.0684060	−	0.997658i	$-0.521791\pi$
$233$	−2.08834	−0.136812	−0.0684060	+	0.997658i	$0.521791\pi$
$234$	11.9088	0.778505
$235$	0	0
$236$	−14.2187	−0.925558
$237$	23.0488	1.49718
$238$	−0.349825	−0.0226758
$239$	−16.8870	−1.09233	−0.546165	−	0.837677i	$-0.683913\pi$
$239$	−16.8870	−1.09233	−0.546165	+	0.837677i	$0.683913\pi$
$240$	0	0
$241$	19.6461	1.26551	0.632757	−	0.774351i	$-0.281923\pi$
$241$	19.6461	1.26551	0.632757	+	0.774351i	$0.281923\pi$
$242$	0.790175	0.0507944
$243$	−22.1482	−1.42081
$244$	−3.80429	−0.243545
$245$	0	0
$246$	22.3500	1.42498
$247$	−4.14913	−0.264003
$248$	−23.3976	−1.48575
$249$	9.74529	0.617583
$250$	0	0
$251$	21.9810	1.38743	0.693714	−	0.720251i	$-0.255973\pi$
$251$	21.9810	1.38743	0.693714	+	0.720251i	$0.255973\pi$
$252$	−0.334386	−0.0210643
$253$	−2.21109	−0.139010
$254$	−0.834120	−0.0523374
$255$	0	0
$256$	−13.8151	−0.863445
$257$	−27.5365	−1.71768	−0.858839	−	0.512246i	$-0.828814\pi$
$257$	−27.5365	−1.71768	−0.858839	+	0.512246i	$0.828814\pi$
$258$	8.05802	0.501671
$259$	0.764746	0.0475190
$260$	0	0
$261$	9.32701	0.577327
$262$	−14.3935	−0.889235
$263$	−21.4214	−1.32090	−0.660449	−	0.750871i	$-0.729634\pi$
$263$	−21.4214	−1.32090	−0.660449	+	0.750871i	$0.729634\pi$
$264$	6.86928	0.422775
$265$	0	0
$266$	−0.0528789	−0.00324221
$267$	0.575459	0.0352175
$268$	−21.0038	−1.28301
$269$	8.06510	0.491738	0.245869	−	0.969303i	$-0.420927\pi$
$269$	8.06510	0.491738	0.245869	+	0.969303i	$0.420927\pi$
$270$	0	0
$271$	−14.7370	−0.895211	−0.447606	−	0.894231i	$-0.647723\pi$
$271$	−14.7370	−0.895211	−0.447606	+	0.894231i	$0.647723\pi$
$272$	−4.25772	−0.258162
$273$	0.715073	0.0432782
$274$	5.64483	0.341017
$275$	0	0
$276$	−7.83323	−0.471505
$277$	32.5834	1.95775	0.978873	−	0.204470i	$-0.0655472\pi$
$277$	32.5834	1.95775	0.978873	+	0.204470i	$0.0655472\pi$
$278$	1.78316	0.106947
$279$	31.8628	1.90758
$280$	0	0
$281$	−16.8550	−1.00548	−0.502741	−	0.864437i	$-0.667675\pi$
$281$	−16.8550	−1.00548	−0.502741	+	0.864437i	$0.667675\pi$
$282$	−18.2630	−1.08754
$283$	6.26244	0.372263	0.186132	−	0.982525i	$-0.440405\pi$
$283$	6.26244	0.372263	0.186132	+	0.982525i	$0.440405\pi$
$284$	5.65513	0.335570
$285$	0	0
$286$	−3.27854	−0.193864
$287$	0.734986	0.0433848
$288$	21.2247	1.25067
$289$	26.7659	1.57447
$290$	0	0
$291$	36.3396	2.13027
$292$	−9.85177	−0.576531
$293$	−7.67080	−0.448133	−0.224067	−	0.974574i	$-0.571933\pi$
$293$	−7.67080	−0.448133	−0.224067	+	0.974574i	$0.571933\pi$
$294$	−14.2356	−0.830240
$295$	0	0
$296$	−30.4814	−1.77170
$297$	−1.62854	−0.0944976
$298$	4.42725	0.256464
$299$	9.17411	0.530553
$300$	0	0
$301$	0.264991	0.0152738
$302$	−16.4855	−0.948634
$303$	−30.0700	−1.72748
$304$	−0.643589	−0.0369124
$305$	0	0
$306$	−18.9880	−1.08547
$307$	−0.00440478	−0.000251394 0	−0.000125697	−	1.00000i	$-0.500040\pi$
$307$	−0.00440478	−0.000251394 0	−0.000125697		1.00000i	$0.500040\pi$
$308$	0.0920574	0.00524546
$309$	−30.0102	−1.70722
$310$	0	0
$311$	12.0604	0.683883	0.341942	−	0.939721i	$-0.388915\pi$
$311$	12.0604	0.683883	0.341942	+	0.939721i	$0.388915\pi$
$312$	−28.5015	−1.61358
$313$	16.4882	0.931967	0.465983	−	0.884793i	$-0.345701\pi$
$313$	16.4882	0.931967	0.465983	+	0.884793i	$0.345701\pi$
$314$	−0.525305	−0.0296447
$315$	0	0
$316$	−12.3116	−0.692582
$317$	25.9591	1.45801	0.729005	−	0.684508i	$-0.239983\pi$
$317$	25.9591	1.45801	0.729005	+	0.684508i	$0.239983\pi$
$318$	−6.38249	−0.357912
$319$	−2.56775	−0.143767
$320$	0	0
$321$	7.01637	0.391616
$322$	0.116920	0.00651570
$323$	6.61558	0.368101
$324$	9.22086	0.512270
$325$	0	0
$326$	0.282116	0.0156249
$327$	−10.2210	−0.565225
$328$	−29.2952	−1.61756
$329$	−0.600584	−0.0331113
$330$	0	0
$331$	25.4456	1.39862	0.699310	−	0.714819i	$-0.253491\pi$
$331$	25.4456	1.39862	0.699310	+	0.714819i	$0.253491\pi$
$332$	−5.20548	−0.285688
$333$	41.5095	2.27470
$334$	−9.67140	−0.529196
$335$	0	0
$336$	0.110918	0.00605107
$337$	−28.8344	−1.57071	−0.785355	−	0.619046i	$-0.787520\pi$
$337$	−28.8344	−1.57071	−0.785355	+	0.619046i	$0.787520\pi$
$338$	3.33082	0.181173
$339$	7.25555	0.394067
$340$	0	0
$341$	−8.77193	−0.475026
$342$	−2.87020	−0.155203
$343$	−0.936587	−0.0505710
$344$	−10.5621	−0.569468
$345$	0	0
$346$	−11.3991	−0.612817
$347$	−20.2741	−1.08837	−0.544185	−	0.838965i	$-0.683161\pi$
$347$	−20.2741	−1.08837	−0.544185	+	0.838965i	$0.683161\pi$
$348$	−9.09677	−0.487638
$349$	23.5879	1.26263	0.631316	−	0.775525i	$-0.282515\pi$
$349$	23.5879	1.26263	0.631316	+	0.775525i	$0.282515\pi$
$350$	0	0
$351$	6.75703	0.360664
$352$	−5.84321	−0.311444
$353$	−26.1715	−1.39297	−0.696484	−	0.717573i	$-0.745253\pi$
$353$	−26.1715	−1.39297	−0.696484	+	0.717573i	$0.745253\pi$
$354$	21.0338	1.11793
$355$	0	0
$356$	−0.307383	−0.0162913
$357$	−1.14015	−0.0603431
$358$	6.89249	0.364280
$359$	32.5322	1.71698	0.858492	−	0.512828i	$-0.171402\pi$
$359$	32.5322	1.71698	0.858492	+	0.512828i	$0.171402\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−11.0333	−0.579898
$363$	2.57534	0.135170
$364$	−0.381958	−0.0200201
$365$	0	0
$366$	5.62770	0.294164
$367$	−9.12225	−0.476178	−0.238089	−	0.971243i	$-0.576521\pi$
$367$	−9.12225	−0.476178	−0.238089	+	0.971243i	$0.576521\pi$
$368$	1.42304	0.0741808
$369$	39.8941	2.07680
$370$	0	0
$371$	−0.209890	−0.0108970
$372$	−31.0763	−1.61123
$373$	−32.3890	−1.67704	−0.838519	−	0.544872i	$-0.816578\pi$
$373$	−32.3890	−1.67704	−0.838519	+	0.544872i	$0.816578\pi$
$374$	5.22747	0.270306
$375$	0	0
$376$	23.9382	1.23452
$377$	10.6539	0.548706
$378$	0.0861155	0.00442930
$379$	−20.8497	−1.07098	−0.535489	−	0.844542i	$-0.679873\pi$
$379$	−20.8497	−1.07098	−0.535489	+	0.844542i	$0.679873\pi$
$380$	0	0
$381$	−2.71856	−0.139276
$382$	3.76330	0.192547
$383$	−14.8606	−0.759342	−0.379671	−	0.925122i	$-0.623963\pi$
$383$	−14.8606	−0.759342	−0.379671	+	0.925122i	$0.623963\pi$
$384$	−23.3201	−1.19005
$385$	0	0
$386$	−4.49760	−0.228922
$387$	14.3834	0.731147
$388$	−19.4109	−0.985441
$389$	−2.79047	−0.141482	−0.0707412	−	0.997495i	$-0.522536\pi$
$389$	−2.79047	−0.141482	−0.0707412	+	0.997495i	$0.522536\pi$
$390$	0	0
$391$	−14.6277	−0.739753
$392$	18.6594	0.942441
$393$	−46.9114	−2.36637
$394$	15.1259	0.762032
$395$	0	0
$396$	4.99676	0.251097
$397$	−9.55809	−0.479707	−0.239853	−	0.970809i	$-0.577099\pi$
$397$	−9.55809	−0.479707	−0.239853	+	0.970809i	$0.577099\pi$
$398$	16.6947	0.836832
$399$	−0.172343	−0.00862793
$400$	0	0
$401$	−9.84081	−0.491427	−0.245713	−	0.969343i	$-0.579022\pi$
$401$	−9.84081	−0.491427	−0.245713	+	0.969343i	$0.579022\pi$
$402$	31.0710	1.54968
$403$	36.3959	1.81301
$404$	16.0620	0.799113
$405$	0	0
$406$	0.135780	0.00673864
$407$	−11.4277	−0.566449
$408$	45.4443	2.24983
$409$	5.23929	0.259066	0.129533	−	0.991575i	$-0.458652\pi$
$409$	5.23929	0.259066	0.129533	+	0.991575i	$0.458652\pi$
$410$	0	0
$411$	18.3976	0.907488
$412$	16.0300	0.789743
$413$	0.691702	0.0340364
$414$	6.34628	0.311902
$415$	0	0
$416$	24.2443	1.18867
$417$	5.81168	0.284599
$418$	0.790175	0.0386487
$419$	−11.2351	−0.548870	−0.274435	−	0.961606i	$-0.588491\pi$
$419$	−11.2351	−0.548870	−0.274435	+	0.961606i	$0.588491\pi$
$420$	0	0
$421$	34.2914	1.67126	0.835630	−	0.549292i	$-0.185103\pi$
$421$	34.2914	1.67126	0.835630	+	0.549292i	$0.185103\pi$
$422$	17.3360	0.843903
$423$	−32.5989	−1.58501
$424$	8.36586	0.406282
$425$	0	0
$426$	−8.36566	−0.405318
$427$	0.185069	0.00895610
$428$	−3.74782	−0.181158
$429$	−10.6854	−0.515897
$430$	0	0
$431$	24.2325	1.16724	0.583620	−	0.812027i	$-0.301636\pi$
$431$	24.2325	1.16724	0.583620	+	0.812027i	$0.301636\pi$
$432$	1.04811	0.0504273
$433$	−11.5535	−0.555228	−0.277614	−	0.960693i	$-0.589544\pi$
$433$	−11.5535	−0.555228	−0.277614	+	0.960693i	$0.589544\pi$
$434$	0.463850	0.0222655
$435$	0	0
$436$	5.45960	0.261467
$437$	−2.21109	−0.105771
$438$	14.5738	0.696361
$439$	−38.1975	−1.82307	−0.911534	−	0.411225i	$-0.865101\pi$
$439$	−38.1975	−1.82307	−0.911534	+	0.411225i	$0.865101\pi$
$440$	0	0
$441$	−25.4103	−1.21001
$442$	−21.6894	−1.03166
$443$	19.8322	0.942256	0.471128	−	0.882065i	$-0.343847\pi$
$443$	19.8322	0.942256	0.471128	+	0.882065i	$0.343847\pi$
$444$	−40.4848	−1.92132
$445$	0	0
$446$	18.2146	0.862487
$447$	14.4293	0.682483
$448$	0.222844	0.0105284
$449$	19.6541	0.927532	0.463766	−	0.885958i	$-0.346498\pi$
$449$	19.6541	0.927532	0.463766	+	0.885958i	$0.346498\pi$
$450$	0	0
$451$	−10.9830	−0.517168
$452$	−3.87557	−0.182292
$453$	−53.7295	−2.52443
$454$	−9.57296	−0.449282
$455$	0	0
$456$	6.86928	0.321684
$457$	12.6434	0.591432	0.295716	−	0.955276i	$-0.404442\pi$
$457$	12.6434	0.591432	0.295716	+	0.955276i	$0.404442\pi$
$458$	−13.4475	−0.628360
$459$	−10.7738	−0.502876
$460$	0	0
$461$	−2.24816	−0.104707	−0.0523536	−	0.998629i	$-0.516672\pi$
$461$	−2.24816	−0.104707	−0.0523536	+	0.998629i	$0.516672\pi$
$462$	−0.136181	−0.00633571
$463$	16.2839	0.756778	0.378389	−	0.925647i	$-0.376478\pi$
$463$	16.2839	0.756778	0.378389	+	0.925647i	$0.376478\pi$
$464$	1.65258	0.0767190
$465$	0	0
$466$	−1.65016	−0.0764420
$467$	−2.08404	−0.0964378	−0.0482189	−	0.998837i	$-0.515355\pi$
$467$	−2.08404	−0.0964378	−0.0482189	+	0.998837i	$0.515355\pi$
$468$	−20.7322	−0.958347
$469$	1.02178	0.0471815
$470$	0	0
$471$	−1.71207	−0.0788881
$472$	−27.5700	−1.26901
$473$	−3.95978	−0.182071
$474$	18.2126	0.836533
$475$	0	0
$476$	0.609014	0.0279141
$477$	−11.3926	−0.521630
$478$	−13.3437	−0.610326
$479$	16.3963	0.749164	0.374582	−	0.927194i	$-0.377786\pi$
$479$	16.3963	0.749164	0.374582	+	0.927194i	$0.377786\pi$
$480$	0	0
$481$	47.4150	2.16194
$482$	15.5238	0.707090
$483$	0.381066	0.0173391
$484$	−1.37562	−0.0625284
$485$	0	0
$486$	−17.5010	−0.793859
$487$	−6.55183	−0.296892	−0.148446	−	0.988921i	$-0.547427\pi$
$487$	−6.55183	−0.296892	−0.148446	+	0.988921i	$0.547427\pi$
$488$	−7.37651	−0.333919
$489$	0.919471	0.0415799
$490$	0	0
$491$	29.7599	1.34305	0.671524	−	0.740983i	$-0.265640\pi$
$491$	29.7599	1.34305	0.671524	+	0.740983i	$0.265640\pi$
$492$	−38.9093	−1.75417
$493$	−16.9872	−0.765064
$494$	−3.27854	−0.147508
$495$	0	0
$496$	5.64552	0.253491
$497$	−0.275107	−0.0123403
$498$	7.70048	0.345067
$499$	−25.0420	−1.12104	−0.560518	−	0.828142i	$-0.689398\pi$
$499$	−25.0420	−1.12104	−0.560518	+	0.828142i	$0.689398\pi$
$500$	0	0
$501$	−31.5210	−1.40826
$502$	17.3688	0.775208
$503$	−14.9942	−0.668560	−0.334280	−	0.942474i	$-0.608493\pi$
$503$	−14.9942	−0.668560	−0.334280	+	0.942474i	$0.608493\pi$
$504$	−0.648374	−0.0288809
$505$	0	0
$506$	−1.74715	−0.0776703
$507$	10.8558	0.482123
$508$	1.45213	0.0644278
$509$	−21.5572	−0.955506	−0.477753	−	0.878494i	$-0.658549\pi$
$509$	−21.5572	−0.955506	−0.477753	+	0.878494i	$0.658549\pi$
$510$	0	0
$511$	0.479263	0.0212013
$512$	7.19396	0.317931
$513$	−1.62854	−0.0719019
$514$	−21.7586	−0.959731
$515$	0	0
$516$	−14.0283	−0.617562
$517$	8.97459	0.394702
$518$	0.604283	0.0265507
$519$	−37.1518	−1.63078
$520$	0	0
$521$	−34.7658	−1.52312	−0.761558	−	0.648097i	$-0.775565\pi$
$521$	−34.7658	−1.52312	−0.761558	+	0.648097i	$0.775565\pi$
$522$	7.36996	0.322575
$523$	−26.7047	−1.16771	−0.583857	−	0.811856i	$-0.698457\pi$
$523$	−26.7047	−1.16771	−0.583857	+	0.811856i	$0.698457\pi$
$524$	25.0579	1.09466
$525$	0	0
$526$	−16.9266	−0.738036
$527$	−58.0314	−2.52789
$528$	−1.65746	−0.0721316
$529$	−18.1111	−0.787438
$530$	0	0
$531$	37.5447	1.62930
$532$	0.0920574	0.00399120
$533$	45.5698	1.97385
$534$	0.454713	0.0196773
$535$	0	0
$536$	−40.7264	−1.75911
$537$	22.4640	0.969394
$538$	6.37284	0.274753
$539$	6.99552	0.301318
$540$	0	0
$541$	9.54398	0.410328	0.205164	−	0.978728i	$-0.434227\pi$
$541$	9.54398	0.410328	0.205164	+	0.978728i	$0.434227\pi$
$542$	−11.6448	−0.500188
$543$	−35.9598	−1.54318
$544$	−38.6563	−1.65737
$545$	0	0
$546$	0.565033	0.0241812
$547$	−27.6080	−1.18043	−0.590217	−	0.807245i	$-0.700958\pi$
$547$	−27.6080	−1.18043	−0.590217	+	0.807245i	$0.700958\pi$
$548$	−9.82715	−0.419795
$549$	10.0453	0.428723
$550$	0	0
$551$	−2.56775	−0.109390
$552$	−15.1886	−0.646470
$553$	0.598928	0.0254690
$554$	25.7466	1.09387
$555$	0	0
$556$	−3.10433	−0.131653
$557$	−14.9783	−0.634651	−0.317325	−	0.948317i	$-0.602785\pi$
$557$	−14.9783	−0.634651	−0.317325	+	0.948317i	$0.602785\pi$
$558$	25.1772	1.06583
$559$	16.4297	0.694900
$560$	0	0
$561$	17.0374	0.719318
$562$	−13.3184	−0.561801
$563$	−0.475683	−0.0200476	−0.0100238	−	0.999950i	$-0.503191\pi$
$563$	−0.475683	−0.0200476	−0.0100238	+	0.999950i	$0.503191\pi$
$564$	31.7942	1.33878
$565$	0	0
$566$	4.94842	0.207998
$567$	−0.448571	−0.0188382
$568$	10.9653	0.460093
$569$	−9.06417	−0.379990	−0.189995	−	0.981785i	$-0.560847\pi$
$569$	−9.06417	−0.379990	−0.189995	+	0.981785i	$0.560847\pi$
$570$	0	0
$571$	4.75692	0.199071	0.0995355	−	0.995034i	$-0.468264\pi$
$571$	4.75692	0.199071	0.0995355	+	0.995034i	$0.468264\pi$
$572$	5.70765	0.238649
$573$	12.2654	0.512393
$574$	0.580767	0.0242408
$575$	0	0
$576$	12.0957	0.503987
$577$	−24.9499	−1.03868	−0.519338	−	0.854569i	$-0.673822\pi$
$577$	−24.9499	−1.03868	−0.519338	+	0.854569i	$0.673822\pi$
$578$	21.1498	0.879715
$579$	−14.6586	−0.609190
$580$	0	0
$581$	0.253233	0.0105059
$582$	28.7147	1.19026
$583$	3.13641	0.129897
$584$	−19.1026	−0.790470
$585$	0	0
$586$	−6.06127	−0.250389
$587$	−19.3052	−0.796810	−0.398405	−	0.917210i	$-0.630436\pi$
$587$	−19.3052	−0.796810	−0.398405	+	0.917210i	$0.630436\pi$
$588$	24.7830	1.02203
$589$	−8.77193	−0.361441
$590$	0	0
$591$	49.2983	2.02786
$592$	7.35473	0.302278
$593$	6.49883	0.266875	0.133437	−	0.991057i	$-0.457398\pi$
$593$	6.49883	0.266875	0.133437	+	0.991057i	$0.457398\pi$
$594$	−1.28683	−0.0527994
$595$	0	0
$596$	−7.70746	−0.315710
$597$	54.4115	2.22691
$598$	7.24915	0.296440
$599$	8.10308	0.331083	0.165541	−	0.986203i	$-0.447063\pi$
$599$	8.10308	0.331083	0.165541	+	0.986203i	$0.447063\pi$
$600$	0	0
$601$	−34.2650	−1.39770	−0.698848	−	0.715270i	$-0.746304\pi$
$601$	−34.2650	−1.39770	−0.698848	+	0.715270i	$0.746304\pi$
$602$	0.209389	0.00853405
$603$	55.4610	2.25855
$604$	28.6998	1.16778
$605$	0	0
$606$	−23.7605	−0.965206
$607$	16.9455	0.687797	0.343898	−	0.939007i	$-0.388252\pi$
$607$	16.9455	0.687797	0.343898	+	0.939007i	$0.388252\pi$
$608$	−5.84321	−0.236974
$609$	0.442534	0.0179324
$610$	0	0
$611$	−37.2367	−1.50644
$612$	33.0565	1.33623
$613$	1.47243	0.0594709	0.0297354	−	0.999558i	$-0.490534\pi$
$613$	1.47243	0.0594709	0.0297354	+	0.999558i	$0.490534\pi$
$614$	−0.00348055	−0.000140464 0
$615$	0	0
$616$	0.178499	0.00719194
$617$	19.4675	0.783731	0.391866	−	0.920022i	$-0.371830\pi$
$617$	19.4675	0.783731	0.391866	+	0.920022i	$0.371830\pi$
$618$	−23.7133	−0.953889
$619$	21.8377	0.877732	0.438866	−	0.898552i	$-0.355380\pi$
$619$	21.8377	0.877732	0.438866	+	0.898552i	$0.355380\pi$
$620$	0	0
$621$	3.60086	0.144497
$622$	9.52983	0.382112
$623$	0.0149534	0.000599094 0
$624$	6.87702	0.275301
$625$	0	0
$626$	13.0285	0.520725
$627$	2.57534	0.102849
$628$	0.914509	0.0364929
$629$	−75.6008	−3.01440
$630$	0	0
$631$	0.393330	0.0156582	0.00782912	−	0.999969i	$-0.497508\pi$
$631$	0.393330	0.0156582	0.00782912	+	0.999969i	$0.497508\pi$
$632$	−23.8722	−0.949585
$633$	56.5015	2.24573
$634$	20.5123	0.814646
$635$	0	0
$636$	11.1114	0.440594
$637$	−29.0253	−1.15003
$638$	−2.02897	−0.0803278
$639$	−14.9325	−0.590720
$640$	0	0
$641$	2.08000	0.0821549	0.0410774	−	0.999156i	$-0.486921\pi$
$641$	2.08000	0.0821549	0.0410774	+	0.999156i	$0.486921\pi$
$642$	5.54416	0.218811
$643$	−15.7787	−0.622250	−0.311125	−	0.950369i	$-0.600706\pi$
$643$	−15.7787	−0.622250	−0.311125	+	0.950369i	$0.600706\pi$
$644$	−0.203548	−0.00802090
$645$	0	0
$646$	5.22747	0.205672
$647$	−6.67760	−0.262524	−0.131262	−	0.991348i	$-0.541903\pi$
$647$	−6.67760	−0.262524	−0.131262	+	0.991348i	$0.541903\pi$
$648$	17.8792	0.702363
$649$	−10.3362	−0.405731
$650$	0	0
$651$	1.51178	0.0592513
$652$	−0.491138	−0.0192345
$653$	−6.00890	−0.235147	−0.117573	−	0.993064i	$-0.537512\pi$
$653$	−6.00890	−0.235147	−0.117573	+	0.993064i	$0.537512\pi$
$654$	−8.07641	−0.315813
$655$	0	0
$656$	7.06852	0.275979
$657$	26.0138	1.01489
$658$	−0.474566	−0.0185005
$659$	14.3389	0.558566	0.279283	−	0.960209i	$-0.409903\pi$
$659$	14.3389	0.558566	0.279283	+	0.960209i	$0.409903\pi$
$660$	0	0
$661$	−4.17451	−0.162370	−0.0811849	−	0.996699i	$-0.525870\pi$
$661$	−4.17451	−0.162370	−0.0811849	+	0.996699i	$0.525870\pi$
$662$	20.1065	0.781462
$663$	−70.6902	−2.74538
$664$	−10.0934	−0.391700
$665$	0	0
$666$	32.7997	1.27096
$667$	5.67754	0.219835
$668$	16.8371	0.651445
$669$	59.3651	2.29519
$670$	0	0
$671$	−2.76550	−0.106761
$672$	1.00704	0.0388473
$673$	1.71412	0.0660745	0.0330373	−	0.999454i	$-0.489482\pi$
$673$	1.71412	0.0660745	0.0330373	+	0.999454i	$0.489482\pi$
$674$	−22.7842	−0.877615
$675$	0	0
$676$	−5.79866	−0.223025
$677$	−0.206434	−0.00793392	−0.00396696	−	0.999992i	$-0.501263\pi$
$677$	−0.206434	−0.00793392	−0.00396696	+	0.999992i	$0.501263\pi$
$678$	5.73315	0.220180
$679$	0.944291	0.0362386
$680$	0	0
$681$	−31.2002	−1.19559
$682$	−6.93135	−0.265415
$683$	29.3173	1.12179	0.560897	−	0.827886i	$-0.310456\pi$
$683$	29.3173	1.12179	0.560897	+	0.827886i	$0.310456\pi$
$684$	4.99676	0.191056
$685$	0	0
$686$	−0.740068	−0.0282559
$687$	−43.8280	−1.67214
$688$	2.54847	0.0971596
$689$	−13.0134	−0.495771
$690$	0	0
$691$	−44.8795	−1.70730	−0.853649	−	0.520849i	$-0.825615\pi$
$691$	−44.8795	−1.70730	−0.853649	+	0.520849i	$0.825615\pi$
$692$	19.8448	0.754384
$693$	−0.243079	−0.00923382
$694$	−16.0201	−0.608114
$695$	0	0
$696$	−17.6386	−0.668590
$697$	−72.6587	−2.75215
$698$	18.6386	0.705481
$699$	−5.37819	−0.203422
$700$	0	0
$701$	7.13370	0.269436	0.134718	−	0.990884i	$-0.456987\pi$
$701$	7.13370	0.269436	0.134718	+	0.990884i	$0.456987\pi$
$702$	5.33924	0.201517
$703$	−11.4277	−0.431003
$704$	−3.32998	−0.125503
$705$	0	0
$706$	−20.6800	−0.778304
$707$	−0.781373	−0.0293865
$708$	−36.6179	−1.37619
$709$	−37.5143	−1.40888	−0.704440	−	0.709763i	$-0.748802\pi$
$709$	−37.5143	−1.40888	−0.704440	+	0.709763i	$0.748802\pi$
$710$	0	0
$711$	32.5090	1.21918
$712$	−0.596015	−0.0223366
$713$	19.3955	0.726369
$714$	−0.900916	−0.0337160
$715$	0	0
$716$	−11.9992	−0.448432
$717$	−43.4898	−1.62415
$718$	25.7061	0.959343
$719$	−9.92990	−0.370323	−0.185161	−	0.982708i	$-0.559281\pi$
$719$	−9.92990	−0.370323	−0.185161	+	0.982708i	$0.559281\pi$
$720$	0	0
$721$	−0.779819	−0.0290420
$722$	0.790175	0.0294073
$723$	50.5952	1.88166
$724$	19.2080	0.713860
$725$	0	0
$726$	2.03497	0.0755247
$727$	−16.4750	−0.611025	−0.305513	−	0.952188i	$-0.598828\pi$
$727$	−16.4750	−0.611025	−0.305513	+	0.952188i	$0.598828\pi$
$728$	−0.740617	−0.0274491
$729$	−36.9300	−1.36778
$730$	0	0
$731$	−26.1963	−0.968904
$732$	−9.79732	−0.362119
$733$	35.6615	1.31719	0.658594	−	0.752498i	$-0.271151\pi$
$733$	35.6615	1.31719	0.658594	+	0.752498i	$0.271151\pi$
$734$	−7.20817	−0.266059
$735$	0	0
$736$	12.9199	0.476233
$737$	−15.2686	−0.562425
$738$	31.5233	1.16039
$739$	12.0150	0.441977	0.220989	−	0.975276i	$-0.429072\pi$
$739$	12.0150	0.441977	0.220989	+	0.975276i	$0.429072\pi$
$740$	0	0
$741$	−10.6854	−0.392538
$742$	−0.165850	−0.00608854
$743$	6.68005	0.245067	0.122534	−	0.992464i	$-0.460898\pi$
$743$	6.68005	0.245067	0.122534	+	0.992464i	$0.460898\pi$
$744$	−60.2568	−2.20912
$745$	0	0
$746$	−25.5930	−0.937025
$747$	13.7452	0.502909
$748$	−9.10056	−0.332749
$749$	0.182321	0.00666188
$750$	0	0
$751$	−6.22015	−0.226976	−0.113488	−	0.993539i	$-0.536202\pi$
$751$	−6.22015	−0.226976	−0.113488	+	0.993539i	$0.536202\pi$
$752$	−5.77595	−0.210627
$753$	56.6084	2.06293
$754$	8.41848	0.306583
$755$	0	0
$756$	−0.149919	−0.00545251
$757$	−14.2550	−0.518105	−0.259053	−	0.965863i	$-0.583410\pi$
$757$	−14.2550	−0.518105	−0.259053	+	0.965863i	$0.583410\pi$
$758$	−16.4749	−0.598396
$759$	−5.69431	−0.206690
$760$	0	0
$761$	−3.46765	−0.125702	−0.0628512	−	0.998023i	$-0.520019\pi$
$761$	−3.46765	−0.125702	−0.0628512	+	0.998023i	$0.520019\pi$
$762$	−2.14814	−0.0778189
$763$	−0.265595	−0.00961519
$764$	−6.55158	−0.237028
$765$	0	0
$766$	−11.7425	−0.424273
$767$	42.8862	1.54853
$768$	−35.5786	−1.28383
$769$	−38.6283	−1.39297	−0.696485	−	0.717571i	$-0.745254\pi$
$769$	−38.6283	−1.39297	−0.696485	+	0.717571i	$0.745254\pi$
$770$	0	0
$771$	−70.9157	−2.55397
$772$	7.82993	0.281805
$773$	27.8564	1.00193	0.500963	−	0.865469i	$-0.332979\pi$
$773$	27.8564	1.00193	0.500963	+	0.865469i	$0.332979\pi$
$774$	11.3654	0.408520
$775$	0	0
$776$	−37.6378	−1.35112
$777$	1.96948	0.0706547
$778$	−2.20496	−0.0790515
$779$	−10.9830	−0.393506
$780$	0	0
$781$	4.11096	0.147102
$782$	−11.5584	−0.413328
$783$	4.18169	0.149441
$784$	−4.50224	−0.160794
$785$	0	0
$786$	−37.0682	−1.32218
$787$	−9.77533	−0.348453	−0.174226	−	0.984706i	$-0.555742\pi$
$787$	−9.77533	−0.348453	−0.174226	+	0.984706i	$0.555742\pi$
$788$	−26.3329	−0.938069
$789$	−55.1672	−1.96401
$790$	0	0
$791$	0.188537	0.00670359
$792$	9.68871	0.344273
$793$	11.4744	0.407469
$794$	−7.55256	−0.268030
$795$	0	0
$796$	−29.0641	−1.03015
$797$	2.96470	0.105015	0.0525075	−	0.998621i	$-0.483279\pi$
$797$	2.96470	0.105015	0.0525075	+	0.998621i	$0.483279\pi$
$798$	−0.136181	−0.00482075
$799$	59.3721	2.10043
$800$	0	0
$801$	0.811650	0.0286782
$802$	−7.77596	−0.274579
$803$	−7.16167	−0.252730
$804$	−54.0919	−1.90767
$805$	0	0
$806$	28.7591	1.01300
$807$	20.7704	0.731151
$808$	31.1441	1.09565
$809$	36.2770	1.27543	0.637715	−	0.770272i	$-0.279880\pi$
$809$	36.2770	1.27543	0.637715	+	0.770272i	$0.279880\pi$
$810$	0	0
$811$	28.4399	0.998661	0.499330	−	0.866412i	$-0.333579\pi$
$811$	28.4399	0.998661	0.499330	+	0.866412i	$0.333579\pi$
$812$	−0.236381	−0.00829534
$813$	−37.9528	−1.33106
$814$	−9.02987	−0.316497
$815$	0	0
$816$	−10.9651	−0.383854
$817$	−3.95978	−0.138535
$818$	4.13995	0.144750
$819$	1.00857	0.0352422
$820$	0	0
$821$	34.1352	1.19133	0.595663	−	0.803235i	$-0.296890\pi$
$821$	34.1352	1.19133	0.595663	+	0.803235i	$0.296890\pi$
$822$	14.5373	0.507048
$823$	−21.6135	−0.753400	−0.376700	−	0.926335i	$-0.622941\pi$
$823$	−21.6135	−0.753400	−0.376700	+	0.926335i	$0.622941\pi$
$824$	31.0822	1.08280
$825$	0	0
$826$	0.546566	0.0190175
$827$	3.22960	0.112304	0.0561522	−	0.998422i	$-0.482117\pi$
$827$	3.22960	0.112304	0.0561522	+	0.998422i	$0.482117\pi$
$828$	−11.0483	−0.383955
$829$	54.2752	1.88506	0.942528	−	0.334126i	$-0.108441\pi$
$829$	54.2752	1.88506	0.942528	+	0.334126i	$0.108441\pi$
$830$	0	0
$831$	83.9132	2.91092
$832$	13.8165	0.479002
$833$	46.2795	1.60349
$834$	4.59224	0.159016
$835$	0	0
$836$	−1.37562	−0.0475769
$837$	14.2854	0.493777
$838$	−8.87769	−0.306675
$839$	18.4633	0.637422	0.318711	−	0.947852i	$-0.396750\pi$
$839$	18.4633	0.637422	0.318711	+	0.947852i	$0.396750\pi$
$840$	0	0
$841$	−22.4066	−0.772643
$842$	27.0962	0.933797
$843$	−43.4072	−1.49502
$844$	−30.1804	−1.03885
$845$	0	0
$846$	−25.7589	−0.885608
$847$	0.0669205	0.00229942
$848$	−2.01856	−0.0693177
$849$	16.1279	0.553508
$850$	0	0
$851$	25.2677	0.866164
$852$	14.5639	0.498950
$853$	−39.7042	−1.35945	−0.679723	−	0.733469i	$-0.737900\pi$
$853$	−39.7042	−1.35945	−0.679723	+	0.733469i	$0.737900\pi$
$854$	0.146237	0.00500411
$855$	0	0
$856$	−7.26701	−0.248381
$857$	−52.0593	−1.77831	−0.889155	−	0.457606i	$-0.848707\pi$
$857$	−52.0593	−1.77831	−0.889155	+	0.457606i	$0.848707\pi$
$858$	−8.44334	−0.288251
$859$	14.9753	0.510952	0.255476	−	0.966815i	$-0.417768\pi$
$859$	14.9753	0.510952	0.255476	+	0.966815i	$0.417768\pi$
$860$	0	0
$861$	1.89284	0.0645077
$862$	19.1479	0.652182
$863$	17.1586	0.584086	0.292043	−	0.956405i	$-0.405665\pi$
$863$	17.1586	0.584086	0.292043	+	0.956405i	$0.405665\pi$
$864$	9.51591	0.323738
$865$	0	0
$866$	−9.12932	−0.310227
$867$	68.9313	2.34103
$868$	−0.807521	−0.0274091
$869$	−8.94984	−0.303602
$870$	0	0
$871$	63.3513	2.14658
$872$	10.5862	0.358493
$873$	51.2549	1.73472
$874$	−1.74715	−0.0590982
$875$	0	0
$876$	−25.3716	−0.857228
$877$	7.93413	0.267917	0.133958	−	0.990987i	$-0.457231\pi$
$877$	7.93413	0.267917	0.133958	+	0.990987i	$0.457231\pi$
$878$	−30.1827	−1.01862
$879$	−19.7549	−0.666316
$880$	0	0
$881$	−15.2979	−0.515399	−0.257700	−	0.966225i	$-0.582964\pi$
$881$	−15.2979	−0.515399	−0.257700	+	0.966225i	$0.582964\pi$
$882$	−20.0785	−0.676080
$883$	0.534788	0.0179971	0.00899853	−	0.999960i	$-0.497136\pi$
$883$	0.534788	0.0179971	0.00899853	+	0.999960i	$0.497136\pi$
$884$	37.7594	1.26999
$885$	0	0
$886$	15.6709	0.526474
$887$	9.87389	0.331533	0.165766	−	0.986165i	$-0.446990\pi$
$887$	9.87389	0.331533	0.165766	+	0.986165i	$0.446990\pi$
$888$	−78.5000	−2.63429
$889$	−0.0706423	−0.00236927
$890$	0	0
$891$	6.70304	0.224560
$892$	−31.7100	−1.06173
$893$	8.97459	0.300323
$894$	11.4017	0.381329
$895$	0	0
$896$	−0.605976	−0.0202442
$897$	23.6264	0.788864
$898$	15.5301	0.518247
$899$	22.5241	0.751222
$900$	0	0
$901$	20.7492	0.691256
$902$	−8.67846	−0.288961
$903$	0.682440	0.0227102
$904$	−7.51473	−0.249936
$905$	0	0
$906$	−42.4557	−1.41050
$907$	−32.3531	−1.07427	−0.537133	−	0.843497i	$-0.680493\pi$
$907$	−32.3531	−1.07427	−0.537133	+	0.843497i	$0.680493\pi$
$908$	16.6657	0.553070
$909$	−42.4119	−1.40671
$910$	0	0
$911$	−58.8040	−1.94826	−0.974132	−	0.225978i	$-0.927442\pi$
$911$	−58.8040	−1.94826	−0.974132	+	0.225978i	$0.927442\pi$
$912$	−1.65746	−0.0548840
$913$	−3.78408	−0.125235
$914$	9.99047	0.330455
$915$	0	0
$916$	23.4109	0.773517
$917$	−1.21900	−0.0402549
$918$	−8.51315	−0.280976
$919$	−22.2112	−0.732679	−0.366340	−	0.930481i	$-0.619389\pi$
$919$	−22.2112	−0.732679	−0.366340	+	0.930481i	$0.619389\pi$
$920$	0	0
$921$	−0.0113438	−0.000373791 0
$922$	−1.77644	−0.0585038
$923$	−17.0569	−0.561435
$924$	0.237079	0.00779932
$925$	0	0
$926$	12.8672	0.422841
$927$	−42.3276	−1.39022
$928$	15.0039	0.492528
$929$	8.60843	0.282433	0.141217	−	0.989979i	$-0.454899\pi$
$929$	8.60843	0.282433	0.141217	+	0.989979i	$0.454899\pi$
$930$	0	0
$931$	6.99552	0.229269
$932$	2.87278	0.0941009
$933$	31.0596	1.01685
$934$	−1.64675	−0.0538835
$935$	0	0
$936$	−40.1998	−1.31397
$937$	−25.1791	−0.822566	−0.411283	−	0.911508i	$-0.634919\pi$
$937$	−25.1791	−0.822566	−0.411283	+	0.911508i	$0.634919\pi$
$938$	0.807385	0.0263621
$939$	42.4626	1.38571
$940$	0	0
$941$	25.5413	0.832622	0.416311	−	0.909222i	$-0.363323\pi$
$941$	25.5413	0.832622	0.416311	+	0.909222i	$0.363323\pi$
$942$	−1.35284	−0.0440778
$943$	24.2844	0.790808
$944$	6.65225	0.216512
$945$	0	0
$946$	−3.12892	−0.101730
$947$	21.1214	0.686353	0.343176	−	0.939271i	$-0.388497\pi$
$947$	21.1214	0.686353	0.343176	+	0.939271i	$0.388497\pi$
$948$	−31.7065	−1.02978
$949$	29.7147	0.964581
$950$	0	0
$951$	66.8535	2.16787
$952$	1.18088	0.0382724
$953$	−14.3865	−0.466025	−0.233012	−	0.972474i	$-0.574858\pi$
$953$	−14.3865	−0.466025	−0.233012	+	0.972474i	$0.574858\pi$
$954$	−9.00213	−0.291455
$955$	0	0
$956$	23.2302	0.751318
$957$	−6.61283	−0.213762
$958$	12.9559	0.418587
$959$	0.478065	0.0154375
$960$	0	0
$961$	45.9467	1.48215
$962$	37.4661	1.20796
$963$	9.89618	0.318900
$964$	−27.0256	−0.870436
$965$	0	0
$966$	0.301109	0.00968801
$967$	−41.6857	−1.34052	−0.670261	−	0.742125i	$-0.733818\pi$
$967$	−41.6857	−1.34052	−0.670261	+	0.742125i	$0.733818\pi$
$968$	−2.66733	−0.0857313
$969$	17.0374	0.547319
$970$	0	0
$971$	45.0663	1.44625	0.723123	−	0.690719i	$-0.242706\pi$
$971$	45.0663	1.44625	0.723123	+	0.690719i	$0.242706\pi$
$972$	30.4676	0.977249
$973$	0.151017	0.00484139
$974$	−5.17709	−0.165885
$975$	0	0
$976$	1.77985	0.0569715
$977$	9.51039	0.304264	0.152132	−	0.988360i	$-0.451386\pi$
$977$	9.51039	0.304264	0.152132	+	0.988360i	$0.451386\pi$
$978$	0.726543	0.0232323
$979$	−0.223450	−0.00714149
$980$	0	0
$981$	−14.4162	−0.460273
$982$	23.5156	0.750412
$983$	38.9961	1.24378	0.621891	−	0.783104i	$-0.286365\pi$
$983$	38.9961	1.24378	0.621891	+	0.783104i	$0.286365\pi$
$984$	−75.4451	−2.40510
$985$	0	0
$986$	−13.4228	−0.427470
$987$	−1.54671	−0.0492322
$988$	5.70765	0.181584
$989$	8.75544	0.278407
$990$	0	0
$991$	61.4220	1.95113	0.975566	−	0.219705i	$-0.0705095\pi$
$991$	61.4220	1.95113	0.975566	+	0.219705i	$0.0705095\pi$
$992$	51.2562	1.62739
$993$	65.5311	2.07957
$994$	−0.217383	−0.00689497
$995$	0	0
$996$	−13.4059	−0.424781
$997$	8.89988	0.281862	0.140931	−	0.990019i	$-0.454990\pi$
$997$	8.89988	0.281862	0.140931	+	0.990019i	$0.454990\pi$
$998$	−19.7876	−0.626365
$999$	18.6105	0.588809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.bb.1.13		22
5.2	odd	4		1045.2.b.d.419.13	yes	22
5.3	odd	4		1045.2.b.d.419.10	✓	22
5.4	even	2	inner	5225.2.a.bb.1.10		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.d.419.10	✓	22	5.3	odd	4
1045.2.b.d.419.13	yes	22	5.2	odd	4
5225.2.a.bb.1.10		22	5.4	even	2	inner
5225.2.a.bb.1.13		22	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.790175 q^{2} +2.57534 q^{3} -1.37562 q^{4} +2.03497 q^{6} +0.0669205 q^{7} -2.66733 q^{8} +3.63236 q^{9} -1.00000 q^{11} -3.54270 q^{12} +4.14913 q^{13} +0.0528789 q^{14} +0.643589 q^{16} -6.61558 q^{17} +2.87020 q^{18} -1.00000 q^{19} +0.172343 q^{21} -0.790175 q^{22} +2.21109 q^{23} -6.86928 q^{24} +3.27854 q^{26} +1.62854 q^{27} -0.0920574 q^{28} +2.56775 q^{29} +8.77193 q^{31} +5.84321 q^{32} -2.57534 q^{33} -5.22747 q^{34} -4.99676 q^{36} +11.4277 q^{37} -0.790175 q^{38} +10.6854 q^{39} +10.9830 q^{41} +0.136181 q^{42} +3.95978 q^{43} +1.37562 q^{44} +1.74715 q^{46} -8.97459 q^{47} +1.65746 q^{48} -6.99552 q^{49} -17.0374 q^{51} -5.70765 q^{52} -3.13641 q^{53} +1.28683 q^{54} -0.178499 q^{56} -2.57534 q^{57} +2.02897 q^{58} +10.3362 q^{59} +2.76550 q^{61} +6.93135 q^{62} +0.243079 q^{63} +3.32998 q^{64} -2.03497 q^{66} +15.2686 q^{67} +9.10056 q^{68} +5.69431 q^{69} -4.11096 q^{71} -9.68871 q^{72} +7.16167 q^{73} +9.02987 q^{74} +1.37562 q^{76} -0.0669205 q^{77} +8.44334 q^{78} +8.94984 q^{79} -6.70304 q^{81} +8.67846 q^{82} +3.78408 q^{83} -0.237079 q^{84} +3.12892 q^{86} +6.61283 q^{87} +2.66733 q^{88} +0.223450 q^{89} +0.277662 q^{91} -3.04163 q^{92} +22.5907 q^{93} -7.09149 q^{94} +15.0482 q^{96} +14.1106 q^{97} -5.52768 q^{98} -3.63236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44}+ \cdots - 34 q^{99}+O(q^{100}) \) 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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