LMFDB - Embedded newform 1225.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.70156 q^{2} -5.70156 q^{3} +5.70156 q^{4} +21.1047 q^{6} +8.50781 q^{8} +5.50781 q^{9} -60.0156 q^{11} -32.5078 q^{12} +0.387503 q^{13} -77.1047 q^{16} -35.4922 q^{17} -20.3875 q^{18} +6.08907 q^{19} +222.152 q^{22} +31.5078 q^{23} -48.5078 q^{24} -1.43437 q^{26} +122.539 q^{27} -292.942 q^{29} -130.303 q^{31} +217.345 q^{32} +342.183 q^{33} +131.377 q^{34} +31.4031 q^{36} -219.989 q^{37} -22.5391 q^{38} -2.20937 q^{39} +447.795 q^{41} -210.020 q^{43} -342.183 q^{44} -116.628 q^{46} +457.769 q^{47} +439.617 q^{48} +202.361 q^{51} +2.20937 q^{52} -144.334 q^{53} -453.586 q^{54} -34.7172 q^{57} +1084.34 q^{58} -767.328 q^{59} -667.884 q^{61} +482.325 q^{62} -187.680 q^{64} -1266.61 q^{66} +77.4593 q^{67} -202.361 q^{68} -179.644 q^{69} -906.573 q^{71} +46.8594 q^{72} -1029.72 q^{73} +814.303 q^{74} +34.7172 q^{76} +8.17813 q^{78} -690.764 q^{79} -847.375 q^{81} -1657.54 q^{82} -979.408 q^{83} +777.403 q^{86} +1670.23 q^{87} -510.602 q^{88} +910.927 q^{89} +179.644 q^{92} +742.931 q^{93} -1694.46 q^{94} -1239.21 q^{96} +11.1751 q^{97} -330.555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 5 q^{3} + 5 q^{4} + 23 q^{6} - 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} - 135 q^{16} - 103 q^{17} - 92 q^{18} + 57 q^{19} + 233 q^{22} + 31 q^{23} - 65 q^{24} + 138 q^{26}+ \cdots - 437 q^{99}+O(q^{100}) $ 2 * q - q^2 - 5 * q^3 + 5 * q^4 + 23 * q^6 - 15 * q^8 - 21 * q^9 - 56 * q^11 - 33 * q^12 + 52 * q^13 - 135 * q^16 - 103 * q^17 - 92 * q^18 + 57 * q^19 + 233 * q^22 + 31 * q^23 - 65 * q^24 + 138 * q^26 + 85 * q^27 - 413 * q^29 + 162 * q^31 + 249 * q^32 + 345 * q^33 - 51 * q^34 + 50 * q^36 - 75 * q^37 + 115 * q^38 + 34 * q^39 + 505 * q^41 + 73 * q^43 - 345 * q^44 - 118 * q^46 + 224 * q^47 + 399 * q^48 + 155 * q^51 - 34 * q^52 + 262 * q^53 - 555 * q^54 + q^57 + 760 * q^58 - 190 * q^59 - 990 * q^61 + 1272 * q^62 + 361 * q^64 - 1259 * q^66 - 908 * q^67 - 155 * q^68 - 180 * q^69 + 127 * q^71 + 670 * q^72 - 337 * q^73 + 1206 * q^74 - q^76 + 106 * q^78 - 1119 * q^79 - 158 * q^81 - 1503 * q^82 - 1517 * q^83 + 1542 * q^86 + 1586 * q^87 - 605 * q^88 + 1713 * q^89 + 180 * q^92 + 948 * q^93 - 2326 * q^94 - 1217 * q^96 + 1764 * q^97 - 437 * 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 $

$2$	−3.70156	−1.30870	−0.654350	−	0.756192i	$-0.727058\pi$
$2$	−3.70156	−1.30870	−0.654350	+	0.756192i	$0.727058\pi$
$3$	−5.70156	−1.09727	−0.548633	−	0.836063i	$-0.684852\pi$
$3$	−5.70156	−1.09727	−0.548633	+	0.836063i	$0.684852\pi$
$4$	5.70156	0.712695
$5$	0	0
$5$	0	0
$6$	21.1047	1.43599
$7$	0	0
$7$	0	0
$8$	8.50781	0.375996
$9$	5.50781	0.203993
$10$	0	0
$11$	−60.0156	−1.64504	−0.822518	−	0.568739i	$-0.807431\pi$
$11$	−60.0156	−1.64504	−0.822518	+	0.568739i	$0.807431\pi$
$12$	−32.5078	−0.782016
$13$	0.387503	0.00826723	0.00413362	−	0.999991i	$-0.498684\pi$
$13$	0.387503	0.00826723	0.00413362	+	0.999991i	$0.498684\pi$
$14$	0	0
$15$	0	0
$16$	−77.1047	−1.20476
$17$	−35.4922	−0.506360	−0.253180	−	0.967419i	$-0.581476\pi$
$17$	−35.4922	−0.506360	−0.253180	+	0.967419i	$0.581476\pi$
$18$	−20.3875	−0.266966
$19$	6.08907	0.0735225	0.0367612	−	0.999324i	$-0.488296\pi$
$19$	6.08907	0.0735225	0.0367612	+	0.999324i	$0.488296\pi$
$20$	0	0
$21$	0	0
$22$	222.152	2.15286
$23$	31.5078	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$23$	31.5078	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$24$	−48.5078	−0.412567
$25$	0	0
$26$	−1.43437	−0.0108193
$27$	122.539	0.873432
$28$	0	0
$29$	−292.942	−1.87579	−0.937896	−	0.346915i	$-0.887229\pi$
$29$	−292.942	−1.87579	−0.937896	+	0.346915i	$0.887229\pi$
$30$	0	0
$31$	−130.303	−0.754940	−0.377470	−	0.926022i	$-0.623206\pi$
$31$	−130.303	−0.754940	−0.377470	+	0.926022i	$0.623206\pi$
$32$	217.345	1.20067
$33$	342.183	1.80504
$34$	131.377	0.662673
$35$	0	0
$36$	31.4031	0.145385
$37$	−219.989	−0.977459	−0.488729	−	0.872435i	$-0.662539\pi$
$37$	−219.989	−0.977459	−0.488729	+	0.872435i	$0.662539\pi$
$38$	−22.5391	−0.0962189
$39$	−2.20937	−0.00907135
$40$	0	0
$41$	447.795	1.70570	0.852852	−	0.522153i	$-0.174871\pi$
$41$	447.795	1.70570	0.852852	+	0.522153i	$0.174871\pi$
$42$	0	0
$43$	−210.020	−0.744832	−0.372416	−	0.928066i	$-0.621471\pi$
$43$	−210.020	−0.744832	−0.372416	+	0.928066i	$0.621471\pi$
$44$	−342.183	−1.17241
$45$	0	0
$46$	−116.628	−0.373823
$47$	457.769	1.42069	0.710345	−	0.703854i	$-0.248539\pi$
$47$	457.769	1.42069	0.710345	+	0.703854i	$0.248539\pi$
$48$	439.617	1.32194
$49$	0	0
$50$	0	0
$51$	202.361	0.555612
$52$	2.20937	0.00589202
$53$	−144.334	−0.374073	−0.187036	−	0.982353i	$-0.559888\pi$
$53$	−144.334	−0.374073	−0.187036	+	0.982353i	$0.559888\pi$
$54$	−453.586	−1.14306
$55$	0	0
$56$	0	0
$57$	−34.7172	−0.0806737
$58$	1084.34	2.45485
$59$	−767.328	−1.69318	−0.846590	−	0.532246i	$-0.821348\pi$
$59$	−767.328	−1.69318	−0.846590	+	0.532246i	$0.821348\pi$
$60$	0	0
$61$	−667.884	−1.40187	−0.700933	−	0.713227i	$-0.747233\pi$
$61$	−667.884	−1.40187	−0.700933	+	0.713227i	$0.747233\pi$
$62$	482.325	0.987989
$63$	0	0
$64$	−187.680	−0.366562
$65$	0	0
$66$	−1266.61	−2.36226
$67$	77.4593	0.141241	0.0706206	−	0.997503i	$-0.477502\pi$
$67$	77.4593	0.141241	0.0706206	+	0.997503i	$0.477502\pi$
$68$	−202.361	−0.360880
$69$	−179.644	−0.313428
$70$	0	0
$71$	−906.573	−1.51536	−0.757679	−	0.652627i	$-0.773667\pi$
$71$	−906.573	−1.51536	−0.757679	+	0.652627i	$0.773667\pi$
$72$	46.8594	0.0767005
$73$	−1029.72	−1.65095	−0.825477	−	0.564436i	$-0.809094\pi$
$73$	−1029.72	−1.65095	−0.825477	+	0.564436i	$0.809094\pi$
$74$	814.303	1.27920
$75$	0	0
$76$	34.7172	0.0523991
$77$	0	0
$78$	8.17813	0.0118717
$79$	−690.764	−0.983760	−0.491880	−	0.870663i	$-0.663690\pi$
$79$	−690.764	−0.983760	−0.491880	+	0.870663i	$0.663690\pi$
$80$	0	0
$81$	−847.375	−1.16238
$82$	−1657.54	−2.23225
$83$	−979.408	−1.29523	−0.647614	−	0.761968i	$-0.724233\pi$
$83$	−979.408	−1.29523	−0.647614	+	0.761968i	$0.724233\pi$
$84$	0	0
$85$	0	0
$86$	777.403	0.974762
$87$	1670.23	2.05824
$88$	−510.602	−0.618526
$89$	910.927	1.08492	0.542461	−	0.840081i	$-0.317493\pi$
$89$	910.927	1.08492	0.542461	+	0.840081i	$0.317493\pi$
$90$	0	0
$91$	0	0
$92$	179.644	0.203578
$93$	742.931	0.828370
$94$	−1694.46	−1.85926
$95$	0	0
$96$	−1239.21	−1.31746
$97$	11.1751	0.0116975	0.00584876	−	0.999983i	$-0.498138\pi$
$97$	11.1751	0.0116975	0.00584876	+	0.999983i	$0.498138\pi$
$98$	0	0
$99$	−330.555	−0.335576
$100$	0	0
$101$	−675.850	−0.665837	−0.332919	−	0.942956i	$-0.608033\pi$
$101$	−675.850	−0.665837	−0.332919	+	0.942956i	$0.608033\pi$
$102$	−749.052	−0.727129
$103$	−1528.88	−1.46257	−0.731286	−	0.682071i	$-0.761079\pi$
$103$	−1528.88	−1.46257	−0.731286	+	0.682071i	$0.761079\pi$
$104$	3.29680	0.00310844
$105$	0	0
$106$	534.263	0.489549
$107$	−701.595	−0.633886	−0.316943	−	0.948445i	$-0.602656\pi$
$107$	−701.595	−0.633886	−0.316943	+	0.948445i	$0.602656\pi$
$108$	698.664	0.622491
$109$	−597.130	−0.524722	−0.262361	−	0.964970i	$-0.584501\pi$
$109$	−597.130	−0.524722	−0.262361	+	0.964970i	$0.584501\pi$
$110$	0	0
$111$	1254.28	1.07253
$112$	0	0
$113$	−88.1187	−0.0733585	−0.0366792	−	0.999327i	$-0.511678\pi$
$113$	−88.1187	−0.0733585	−0.0366792	+	0.999327i	$0.511678\pi$
$114$	128.508	0.105578
$115$	0	0
$116$	−1670.23	−1.33687
$117$	2.13429	0.00168646
$118$	2840.31	2.21586
$119$	0	0
$120$	0	0
$121$	2270.87	1.70614
$122$	2472.22	1.83462
$123$	−2553.13	−1.87161
$124$	−742.931	−0.538042
$125$	0	0
$126$	0	0
$127$	2119.35	1.48080	0.740401	−	0.672166i	$-0.234636\pi$
$127$	2119.35	1.48080	0.740401	+	0.672166i	$0.234636\pi$
$128$	−1044.05	−0.720955
$129$	1197.44	0.817279
$130$	0	0
$131$	264.294	0.176271	0.0881353	−	0.996109i	$-0.471909\pi$
$131$	264.294	0.176271	0.0881353	+	0.996109i	$0.471909\pi$
$132$	1950.98	1.28644
$133$	0	0
$134$	−286.720	−0.184842
$135$	0	0
$136$	−301.961	−0.190389
$137$	−468.227	−0.291995	−0.145997	−	0.989285i	$-0.546639\pi$
$137$	−468.227	−0.291995	−0.145997	+	0.989285i	$0.546639\pi$
$138$	664.962	0.410184
$139$	−956.908	−0.583913	−0.291956	−	0.956432i	$-0.594306\pi$
$139$	−956.908	−0.583913	−0.291956	+	0.956432i	$0.594306\pi$
$140$	0	0
$141$	−2610.00	−1.55888
$142$	3355.74	1.98315
$143$	−23.2562	−0.0135999
$144$	−424.678	−0.245763
$145$	0	0
$146$	3811.57	2.16060
$147$	0	0
$148$	−1254.28	−0.696630
$149$	−382.967	−0.210563	−0.105282	−	0.994442i	$-0.533574\pi$
$149$	−382.967	−0.210563	−0.105282	+	0.994442i	$0.533574\pi$
$150$	0	0
$151$	1302.98	0.702218	0.351109	−	0.936335i	$-0.385805\pi$
$151$	1302.98	0.702218	0.351109	+	0.936335i	$0.385805\pi$
$152$	51.8046	0.0276441
$153$	−195.484	−0.103294
$154$	0	0
$155$	0	0
$156$	−12.5969	−0.00646511
$157$	−3203.32	−1.62836	−0.814182	−	0.580610i	$-0.802814\pi$
$157$	−3203.32	−1.62836	−0.814182	+	0.580610i	$0.802814\pi$
$158$	2556.91	1.28745
$159$	822.931	0.410457
$160$	0	0
$161$	0	0
$162$	3136.61	1.52121
$163$	−1368.81	−0.657752	−0.328876	−	0.944373i	$-0.606670\pi$
$163$	−1368.81	−0.657752	−0.328876	+	0.944373i	$0.606670\pi$
$164$	2553.13	1.21565
$165$	0	0
$166$	3625.34	1.69507
$167$	3073.34	1.42409	0.712043	−	0.702136i	$-0.247770\pi$
$167$	3073.34	1.42409	0.712043	+	0.702136i	$0.247770\pi$
$168$	0	0
$169$	−2196.85	−0.999932
$170$	0	0
$171$	33.5374	0.0149981
$172$	−1197.44	−0.530839
$173$	−2402.16	−1.05568	−0.527841	−	0.849343i	$-0.676998\pi$
$173$	−2402.16	−1.05568	−0.527841	+	0.849343i	$0.676998\pi$
$174$	−6182.45	−2.69362
$175$	0	0
$176$	4627.49	1.98187
$177$	4374.97	1.85787
$178$	−3371.85	−1.41984
$179$	−1146.25	−0.478628	−0.239314	−	0.970942i	$-0.576923\pi$
$179$	−1146.25	−0.478628	−0.239314	+	0.970942i	$0.576923\pi$
$180$	0	0
$181$	475.847	0.195411	0.0977056	−	0.995215i	$-0.468850\pi$
$181$	475.847	0.195411	0.0977056	+	0.995215i	$0.468850\pi$
$182$	0	0
$183$	3807.98	1.53822
$184$	268.062	0.107401
$185$	0	0
$186$	−2750.01	−1.08409
$187$	2130.09	0.832980
$188$	2610.00	1.01252
$189$	0	0
$190$	0	0
$191$	990.003	0.375048	0.187524	−	0.982260i	$-0.439954\pi$
$191$	990.003	0.375048	0.187524	+	0.982260i	$0.439954\pi$
$192$	1070.07	0.402216
$193$	−1392.88	−0.519492	−0.259746	−	0.965677i	$-0.583639\pi$
$193$	−1392.88	−0.519492	−0.259746	+	0.965677i	$0.583639\pi$
$194$	−41.3653	−0.0153085
$195$	0	0
$196$	0	0
$197$	3583.39	1.29597	0.647985	−	0.761653i	$-0.275612\pi$
$197$	3583.39	1.29597	0.647985	+	0.761653i	$0.275612\pi$
$198$	1223.57	0.439168
$199$	−623.947	−0.222263	−0.111132	−	0.993806i	$-0.535448\pi$
$199$	−623.947	−0.222263	−0.111132	+	0.993806i	$0.535448\pi$
$200$	0	0
$201$	−441.639	−0.154979
$202$	2501.70	0.871381
$203$	0	0
$204$	1153.77	0.395982
$205$	0	0
$206$	5659.24	1.91407
$207$	173.539	0.0582696
$208$	−29.8783	−0.00996004
$209$	−365.439	−0.120947
$210$	0	0
$211$	2266.48	0.739482	0.369741	−	0.929135i	$-0.379446\pi$
$211$	2266.48	0.739482	0.369741	+	0.929135i	$0.379446\pi$
$212$	−822.931	−0.266600
$213$	5168.88	1.66275
$214$	2597.00	0.829566
$215$	0	0
$216$	1042.54	0.328406
$217$	0	0
$218$	2210.31	0.686703
$219$	5871.01	1.81154
$220$	0	0
$221$	−13.7533	−0.00418620
$222$	−4642.80	−1.40362
$223$	2118.67	0.636217	0.318108	−	0.948054i	$-0.396952\pi$
$223$	2118.67	0.636217	0.318108	+	0.948054i	$0.396952\pi$
$224$	0	0
$225$	0	0
$226$	326.177	0.0960042
$227$	2790.62	0.815948	0.407974	−	0.912994i	$-0.366235\pi$
$227$	2790.62	0.815948	0.407974	+	0.912994i	$0.366235\pi$
$228$	−197.942	−0.0574958
$229$	5813.77	1.67766	0.838832	−	0.544390i	$-0.183239\pi$
$229$	5813.77	1.67766	0.838832	+	0.544390i	$0.183239\pi$
$230$	0	0
$231$	0	0
$232$	−2492.30	−0.705290
$233$	−1936.18	−0.544392	−0.272196	−	0.962242i	$-0.587750\pi$
$233$	−1936.18	−0.544392	−0.272196	+	0.962242i	$0.587750\pi$
$234$	−7.90022	−0.00220707
$235$	0	0
$236$	−4374.97	−1.20672
$237$	3938.43	1.07945
$238$	0	0
$239$	−2755.79	−0.745845	−0.372923	−	0.927862i	$-0.621644\pi$
$239$	−2755.79	−0.745845	−0.372923	+	0.927862i	$0.621644\pi$
$240$	0	0
$241$	4025.23	1.07588	0.537942	−	0.842982i	$-0.319202\pi$
$241$	4025.23	1.07588	0.537942	+	0.842982i	$0.319202\pi$
$242$	−8405.78	−2.23283
$243$	1522.81	0.402009
$244$	−3807.98	−0.999103
$245$	0	0
$246$	9450.58	2.44938
$247$	2.35953	0.000607827 0
$248$	−1108.59	−0.283854
$249$	5584.15	1.42121
$250$	0	0
$251$	−7166.64	−1.80221	−0.901104	−	0.433602i	$-0.857242\pi$
$251$	−7166.64	−1.80221	−0.901104	+	0.433602i	$0.857242\pi$
$252$	0	0
$253$	−1890.96	−0.469896
$254$	−7844.90	−1.93792
$255$	0	0
$256$	5366.07	1.31008
$257$	647.756	0.157221	0.0786107	−	0.996905i	$-0.474952\pi$
$257$	647.756	0.157221	0.0786107	+	0.996905i	$0.474952\pi$
$258$	−4432.41	−1.06957
$259$	0	0
$260$	0	0
$261$	−1613.47	−0.382649
$262$	−978.300	−0.230685
$263$	6131.33	1.43754	0.718771	−	0.695246i	$-0.244705\pi$
$263$	6131.33	1.43754	0.718771	+	0.695246i	$0.244705\pi$
$264$	2911.23	0.678688
$265$	0	0
$266$	0	0
$267$	−5193.70	−1.19045
$268$	441.639	0.100662
$269$	975.631	0.221135	0.110567	−	0.993869i	$-0.464733\pi$
$269$	975.631	0.221135	0.110567	+	0.993869i	$0.464733\pi$
$270$	0	0
$271$	−4672.83	−1.04743	−0.523716	−	0.851893i	$-0.675455\pi$
$271$	−4672.83	−1.04743	−0.523716	+	0.851893i	$0.675455\pi$
$272$	2736.61	0.610043
$273$	0	0
$274$	1733.17	0.382134
$275$	0	0
$276$	−1024.25	−0.223379
$277$	−2390.16	−0.518451	−0.259226	−	0.965817i	$-0.583467\pi$
$277$	−2390.16	−0.518451	−0.259226	+	0.965817i	$0.583467\pi$
$278$	3542.05	0.764166
$279$	−717.685	−0.154002
$280$	0	0
$281$	−3321.28	−0.705092	−0.352546	−	0.935794i	$-0.614684\pi$
$281$	−3321.28	−0.705092	−0.352546	+	0.935794i	$0.614684\pi$
$282$	9661.07	2.04010
$283$	8260.15	1.73504	0.867518	−	0.497406i	$-0.165714\pi$
$283$	8260.15	1.73504	0.867518	+	0.497406i	$0.165714\pi$
$284$	−5168.88	−1.07999
$285$	0	0
$286$	86.0844	0.0177982
$287$	0	0
$288$	1197.10	0.244929
$289$	−3653.30	−0.743600
$290$	0	0
$291$	−63.7155	−0.0128353
$292$	−5871.01	−1.17663
$293$	−628.249	−0.125265	−0.0626326	−	0.998037i	$-0.519950\pi$
$293$	−628.249	−0.125265	−0.0626326	+	0.998037i	$0.519950\pi$
$294$	0	0
$295$	0	0
$296$	−1871.63	−0.367520
$297$	−7354.26	−1.43683
$298$	1417.58	0.275564
$299$	12.2094	0.00236149
$300$	0	0
$301$	0	0
$302$	−4823.06	−0.918993
$303$	3853.40	0.730601
$304$	−469.495	−0.0885770
$305$	0	0
$306$	723.597	0.135181
$307$	25.2045	0.00468565	0.00234283	−	0.999997i	$-0.499254\pi$
$307$	25.2045	0.00468565	0.00234283	+	0.999997i	$0.499254\pi$
$308$	0	0
$309$	8716.99	1.60483
$310$	0	0
$311$	9436.18	1.72050	0.860252	−	0.509870i	$-0.170306\pi$
$311$	9436.18	1.72050	0.860252	+	0.509870i	$0.170306\pi$
$312$	−18.7969	−0.00341079
$313$	−2778.60	−0.501776	−0.250888	−	0.968016i	$-0.580723\pi$
$313$	−2778.60	−0.501776	−0.250888	+	0.968016i	$0.580723\pi$
$314$	11857.3	2.13104
$315$	0	0
$316$	−3938.43	−0.701121
$317$	2980.21	0.528030	0.264015	−	0.964519i	$-0.414953\pi$
$317$	2980.21	0.528030	0.264015	+	0.964519i	$0.414953\pi$
$318$	−3046.13	−0.537165
$319$	17581.1	3.08575
$320$	0	0
$321$	4000.19	0.695541
$322$	0	0
$323$	−216.114	−0.0372289
$324$	−4831.36	−0.828423
$325$	0	0
$326$	5066.74	0.860800
$327$	3404.57	0.575759
$328$	3809.76	0.641337
$329$	0	0
$330$	0	0
$331$	2588.16	0.429782	0.214891	−	0.976638i	$-0.431060\pi$
$331$	2588.16	0.429782	0.214891	+	0.976638i	$0.431060\pi$
$332$	−5584.15	−0.923103
$333$	−1211.66	−0.199395
$334$	−11376.2	−1.86370
$335$	0	0
$336$	0	0
$337$	−5284.64	−0.854221	−0.427110	−	0.904199i	$-0.640468\pi$
$337$	−5284.64	−0.854221	−0.427110	+	0.904199i	$0.640468\pi$
$338$	8131.78	1.30861
$339$	502.414	0.0804938
$340$	0	0
$341$	7820.22	1.24190
$342$	−124.141	−0.0196280
$343$	0	0
$344$	−1786.81	−0.280054
$345$	0	0
$346$	8891.75	1.38157
$347$	11787.7	1.82363	0.911813	−	0.410606i	$-0.134683\pi$
$347$	11787.7	1.82363	0.911813	+	0.410606i	$0.134683\pi$
$348$	9522.91	1.46690
$349$	−8765.61	−1.34445	−0.672224	−	0.740347i	$-0.734661\pi$
$349$	−8765.61	−1.34445	−0.672224	+	0.740347i	$0.734661\pi$
$350$	0	0
$351$	47.4843	0.00722086
$352$	−13044.1	−1.97515
$353$	8578.29	1.29342	0.646709	−	0.762737i	$-0.276145\pi$
$353$	8578.29	1.29342	0.646709	+	0.762737i	$0.276145\pi$
$354$	−16194.2	−2.43139
$355$	0	0
$356$	5193.70	0.773218
$357$	0	0
$358$	4242.90	0.626380
$359$	3730.38	0.548417	0.274209	−	0.961670i	$-0.411584\pi$
$359$	3730.38	0.548417	0.274209	+	0.961670i	$0.411584\pi$
$360$	0	0
$361$	−6821.92	−0.994594
$362$	−1761.38	−0.255735
$363$	−12947.5	−1.87209
$364$	0	0
$365$	0	0
$366$	−14095.5	−2.01307
$367$	−515.769	−0.0733594	−0.0366797	−	0.999327i	$-0.511678\pi$
$367$	−515.769	−0.0733594	−0.0366797	+	0.999327i	$0.511678\pi$
$368$	−2429.40	−0.344134
$369$	2466.37	0.347952
$370$	0	0
$371$	0	0
$372$	4235.87	0.590375
$373$	−10922.0	−1.51614	−0.758071	−	0.652173i	$-0.773858\pi$
$373$	−10922.0	−1.51614	−0.758071	+	0.652173i	$0.773858\pi$
$374$	−7884.64	−1.09012
$375$	0	0
$376$	3894.61	0.534173
$377$	−113.516	−0.0155076
$378$	0	0
$379$	−8403.14	−1.13889	−0.569446	−	0.822029i	$-0.692842\pi$
$379$	−8403.14	−1.13889	−0.569446	+	0.822029i	$0.692842\pi$
$380$	0	0
$381$	−12083.6	−1.62483
$382$	−3664.56	−0.490825
$383$	−3030.68	−0.404336	−0.202168	−	0.979351i	$-0.564799\pi$
$383$	−3030.68	−0.404336	−0.202168	+	0.979351i	$0.564799\pi$
$384$	5952.74	0.791080
$385$	0	0
$386$	5155.85	0.679859
$387$	−1156.75	−0.151941
$388$	63.7155	0.00833677
$389$	1403.25	0.182899	0.0914497	−	0.995810i	$-0.470850\pi$
$389$	1403.25	0.182899	0.0914497	+	0.995810i	$0.470850\pi$
$390$	0	0
$391$	−1118.28	−0.144639
$392$	0	0
$393$	−1506.89	−0.193416
$394$	−13264.1	−1.69604
$395$	0	0
$396$	−1884.68	−0.239163
$397$	55.9118	0.00706835	0.00353417	−	0.999994i	$-0.498875\pi$
$397$	55.9118	0.00706835	0.00353417	+	0.999994i	$0.498875\pi$
$398$	2309.58	0.290876
$399$	0	0
$400$	0	0
$401$	−3730.18	−0.464529	−0.232264	−	0.972653i	$-0.574613\pi$
$401$	−3730.18	−0.464529	−0.232264	+	0.972653i	$0.574613\pi$
$402$	1634.75	0.202821
$403$	−50.4928	−0.00624126
$404$	−3853.40	−0.474539
$405$	0	0
$406$	0	0
$407$	13202.8	1.60795
$408$	1721.65	0.208908
$409$	1968.50	0.237985	0.118993	−	0.992895i	$-0.462033\pi$
$409$	1968.50	0.237985	0.118993	+	0.992895i	$0.462033\pi$
$410$	0	0
$411$	2669.62	0.320396
$412$	−8716.99	−1.04237
$413$	0	0
$414$	−642.366	−0.0762574
$415$	0	0
$416$	84.2220	0.00992625
$417$	5455.87	0.640708
$418$	1352.70	0.158283
$419$	13208.4	1.54003	0.770015	−	0.638026i	$-0.220249\pi$
$419$	13208.4	1.54003	0.770015	+	0.638026i	$0.220249\pi$
$420$	0	0
$421$	−7485.74	−0.866586	−0.433293	−	0.901253i	$-0.642648\pi$
$421$	−7485.74	−0.866586	−0.433293	+	0.901253i	$0.642648\pi$
$422$	−8389.50	−0.967760
$423$	2521.30	0.289811
$424$	−1227.97	−0.140650
$425$	0	0
$426$	−19132.9	−2.17604
$427$	0	0
$428$	−4000.19	−0.451767
$429$	132.597	0.0149227
$430$	0	0
$431$	10623.4	1.18726	0.593631	−	0.804737i	$-0.297694\pi$
$431$	10623.4	1.18726	0.593631	+	0.804737i	$0.297694\pi$
$432$	−9448.34	−1.05228
$433$	7268.73	0.806727	0.403364	−	0.915040i	$-0.367841\pi$
$433$	7268.73	0.806727	0.403364	+	0.915040i	$0.367841\pi$
$434$	0	0
$435$	0	0
$436$	−3404.57	−0.373967
$437$	191.853	0.0210013
$438$	−21731.9	−2.37076
$439$	743.352	0.0808161	0.0404080	−	0.999183i	$-0.487134\pi$
$439$	743.352	0.0808161	0.0404080	+	0.999183i	$0.487134\pi$
$440$	0	0
$441$	0	0
$442$	50.9088	0.00547847
$443$	−10857.8	−1.16449	−0.582246	−	0.813013i	$-0.697826\pi$
$443$	−10857.8	−1.16449	−0.582246	+	0.813013i	$0.697826\pi$
$444$	7151.36	0.764389
$445$	0	0
$446$	−7842.37	−0.832617
$447$	2183.51	0.231044
$448$	0	0
$449$	−11162.9	−1.17329	−0.586645	−	0.809844i	$-0.699552\pi$
$449$	−11162.9	−1.17329	−0.586645	+	0.809844i	$0.699552\pi$
$450$	0	0
$451$	−26874.7	−2.80594
$452$	−502.414	−0.0522822
$453$	−7429.02	−0.770520
$454$	−10329.7	−1.06783
$455$	0	0
$456$	−295.367	−0.0303330
$457$	−13451.0	−1.37683	−0.688414	−	0.725318i	$-0.741693\pi$
$457$	−13451.0	−1.37683	−0.688414	+	0.725318i	$0.741693\pi$
$458$	−21520.0	−2.19556
$459$	−4349.18	−0.442271
$460$	0	0
$461$	9139.46	0.923356	0.461678	−	0.887048i	$-0.347248\pi$
$461$	9139.46	0.923356	0.461678	+	0.887048i	$0.347248\pi$
$462$	0	0
$463$	13122.5	1.31718	0.658588	−	0.752504i	$-0.271154\pi$
$463$	13122.5	1.31718	0.658588	+	0.752504i	$0.271154\pi$
$464$	22587.2	2.25988
$465$	0	0
$466$	7166.89	0.712446
$467$	11921.2	1.18126	0.590629	−	0.806943i	$-0.298880\pi$
$467$	11921.2	1.18126	0.590629	+	0.806943i	$0.298880\pi$
$468$	12.1688	0.00120193
$469$	0	0
$470$	0	0
$471$	18264.0	1.78675
$472$	−6528.28	−0.636628
$473$	12604.5	1.22528
$474$	−14578.4	−1.41267
$475$	0	0
$476$	0	0
$477$	−794.966	−0.0763082
$478$	10200.7	0.976088
$479$	−825.281	−0.0787224	−0.0393612	−	0.999225i	$-0.512532\pi$
$479$	−825.281	−0.0787224	−0.0393612	+	0.999225i	$0.512532\pi$
$480$	0	0
$481$	−85.2464	−0.00808088
$482$	−14899.6	−1.40801
$483$	0	0
$484$	12947.5	1.21596
$485$	0	0
$486$	−5636.76	−0.526108
$487$	6678.32	0.621404	0.310702	−	0.950507i	$-0.399436\pi$
$487$	6678.32	0.621404	0.310702	+	0.950507i	$0.399436\pi$
$488$	−5682.23	−0.527096
$489$	7804.36	0.721729
$490$	0	0
$491$	−3098.65	−0.284807	−0.142403	−	0.989809i	$-0.545483\pi$
$491$	−3098.65	−0.284807	−0.142403	+	0.989809i	$0.545483\pi$
$492$	−14556.8	−1.33389
$493$	10397.2	0.949827
$494$	−8.73395	−0.000795464 0
$495$	0	0
$496$	10047.0	0.909522
$497$	0	0
$498$	−20670.1	−1.85994
$499$	−2265.10	−0.203206	−0.101603	−	0.994825i	$-0.532397\pi$
$499$	−2265.10	−0.203206	−0.101603	+	0.994825i	$0.532397\pi$
$500$	0	0
$501$	−17522.8	−1.56260
$502$	26527.8	2.35855
$503$	−9980.43	−0.884703	−0.442351	−	0.896842i	$-0.645856\pi$
$503$	−9980.43	−0.884703	−0.442351	+	0.896842i	$0.645856\pi$
$504$	0	0
$505$	0	0
$506$	6999.51	0.614953
$507$	12525.5	1.09719
$508$	12083.6	1.05536
$509$	4956.16	0.431588	0.215794	−	0.976439i	$-0.430766\pi$
$509$	4956.16	0.431588	0.215794	+	0.976439i	$0.430766\pi$
$510$	0	0
$511$	0	0
$512$	−11510.4	−0.993541
$513$	746.148	0.0642169
$514$	−2397.71	−0.205756
$515$	0	0
$516$	6827.30	0.582471
$517$	−27473.3	−2.33709
$518$	0	0
$519$	13696.1	1.15836
$520$	0	0
$521$	−3442.97	−0.289519	−0.144759	−	0.989467i	$-0.546241\pi$
$521$	−3442.97	−0.289519	−0.144759	+	0.989467i	$0.546241\pi$
$522$	5972.36	0.500772
$523$	−11109.9	−0.928878	−0.464439	−	0.885605i	$-0.653744\pi$
$523$	−11109.9	−0.928878	−0.464439	+	0.885605i	$0.653744\pi$
$524$	1506.89	0.125627
$525$	0	0
$526$	−22695.5	−1.88131
$527$	4624.74	0.382271
$528$	−26383.9	−2.17464
$529$	−11174.3	−0.918407
$530$	0	0
$531$	−4226.30	−0.345397
$532$	0	0
$533$	173.522	0.0141015
$534$	19224.8	1.55794
$535$	0	0
$536$	659.009	0.0531061
$537$	6535.39	0.525182
$538$	−3611.36	−0.289399
$539$	0	0
$540$	0	0
$541$	3680.65	0.292502	0.146251	−	0.989248i	$-0.453279\pi$
$541$	3680.65	0.292502	0.146251	+	0.989248i	$0.453279\pi$
$542$	17296.8	1.37078
$543$	−2713.07	−0.214418
$544$	−7714.06	−0.607974
$545$	0	0
$546$	0	0
$547$	−14657.4	−1.14572	−0.572858	−	0.819655i	$-0.694165\pi$
$547$	−14657.4	−1.14572	−0.572858	+	0.819655i	$0.694165\pi$
$548$	−2669.62	−0.208103
$549$	−3678.58	−0.285971
$550$	0	0
$551$	−1783.74	−0.137913
$552$	−1528.37	−0.117848
$553$	0	0
$554$	8847.33	0.678497
$555$	0	0
$556$	−5455.87	−0.416152
$557$	6547.61	0.498081	0.249040	−	0.968493i	$-0.419885\pi$
$557$	6547.61	0.498081	0.249040	+	0.968493i	$0.419885\pi$
$558$	2656.55	0.201543
$559$	−81.3835	−0.00615770
$560$	0	0
$561$	−12144.8	−0.914001
$562$	12293.9	0.922754
$563$	−11983.3	−0.897047	−0.448523	−	0.893771i	$-0.648050\pi$
$563$	−11983.3	−0.897047	−0.448523	+	0.893771i	$0.648050\pi$
$564$	−14881.1	−1.11100
$565$	0	0
$566$	−30575.5	−2.27064
$567$	0	0
$568$	−7712.95	−0.569768
$569$	19164.8	1.41200	0.706002	−	0.708210i	$-0.250497\pi$
$569$	19164.8	1.41200	0.706002	+	0.708210i	$0.250497\pi$
$570$	0	0
$571$	14618.8	1.07142	0.535708	−	0.844403i	$-0.320045\pi$
$571$	14618.8	1.07142	0.535708	+	0.844403i	$0.320045\pi$
$572$	−132.597	−0.00969258
$573$	−5644.57	−0.411527
$574$	0	0
$575$	0	0
$576$	−1033.70	−0.0747760
$577$	12446.8	0.898033	0.449017	−	0.893523i	$-0.351774\pi$
$577$	12446.8	0.898033	0.449017	+	0.893523i	$0.351774\pi$
$578$	13522.9	0.973149
$579$	7941.62	0.570021
$580$	0	0
$581$	0	0
$582$	235.847	0.0167976
$583$	8662.32	0.615363
$584$	−8760.66	−0.620752
$585$	0	0
$586$	2325.50	0.163935
$587$	−7467.53	−0.525073	−0.262537	−	0.964922i	$-0.584559\pi$
$587$	−7467.53	−0.525073	−0.262537	+	0.964922i	$0.584559\pi$
$588$	0	0
$589$	−793.424	−0.0555050
$590$	0	0
$591$	−20430.9	−1.42202
$592$	16962.2	1.17760
$593$	26703.9	1.84924	0.924619	−	0.380892i	$-0.124383\pi$
$593$	26703.9	1.84924	0.924619	+	0.380892i	$0.124383\pi$
$594$	27222.2	1.88037
$595$	0	0
$596$	−2183.51	−0.150067
$597$	3557.47	0.243882
$598$	−45.1938	−0.00309048
$599$	6896.60	0.470430	0.235215	−	0.971943i	$-0.424421\pi$
$599$	6896.60	0.470430	0.235215	+	0.971943i	$0.424421\pi$
$600$	0	0
$601$	−18156.5	−1.23231	−0.616154	−	0.787626i	$-0.711310\pi$
$601$	−18156.5	−1.23231	−0.616154	+	0.787626i	$0.711310\pi$
$602$	0	0
$603$	426.631	0.0288122
$604$	7429.02	0.500468
$605$	0	0
$606$	−14263.6	−0.956137
$607$	−2074.50	−0.138717	−0.0693587	−	0.997592i	$-0.522095\pi$
$607$	−2074.50	−0.138717	−0.0693587	+	0.997592i	$0.522095\pi$
$608$	1323.43	0.0882766
$609$	0	0
$610$	0	0
$611$	177.387	0.0117452
$612$	−1114.57	−0.0736171
$613$	−12159.4	−0.801162	−0.400581	−	0.916261i	$-0.631192\pi$
$613$	−12159.4	−0.801162	−0.400581	+	0.916261i	$0.631192\pi$
$614$	−93.2959	−0.00613211
$615$	0	0
$616$	0	0
$617$	4359.84	0.284474	0.142237	−	0.989833i	$-0.454570\pi$
$617$	4359.84	0.284474	0.142237	+	0.989833i	$0.454570\pi$
$618$	−32266.5	−2.10024
$619$	29046.6	1.88608	0.943038	−	0.332684i	$-0.107954\pi$
$619$	29046.6	1.88608	0.943038	+	0.332684i	$0.107954\pi$
$620$	0	0
$621$	3860.94	0.249491
$622$	−34928.6	−2.25162
$623$	0	0
$624$	170.353	0.0109288
$625$	0	0
$626$	10285.2	0.656674
$627$	2083.57	0.132711
$628$	−18264.0	−1.16053
$629$	7807.89	0.494946
$630$	0	0
$631$	−14708.4	−0.927946	−0.463973	−	0.885849i	$-0.653577\pi$
$631$	−14708.4	−0.927946	−0.463973	+	0.885849i	$0.653577\pi$
$632$	−5876.89	−0.369889
$633$	−12922.5	−0.811408
$634$	−11031.4	−0.691033
$635$	0	0
$636$	4691.99	0.292531
$637$	0	0
$638$	−65077.6	−4.03832
$639$	−4993.23	−0.309123
$640$	0	0
$641$	−24048.7	−1.48185	−0.740925	−	0.671588i	$-0.765612\pi$
$641$	−24048.7	−1.48185	−0.740925	+	0.671588i	$0.765612\pi$
$642$	−14806.9	−0.910255
$643$	19196.1	1.17732	0.588661	−	0.808380i	$-0.299655\pi$
$643$	19196.1	1.17732	0.588661	+	0.808380i	$0.299655\pi$
$644$	0	0
$645$	0	0
$646$	799.960	0.0487214
$647$	7185.82	0.436636	0.218318	−	0.975878i	$-0.429943\pi$
$647$	7185.82	0.436636	0.218318	+	0.975878i	$0.429943\pi$
$648$	−7209.31	−0.437050
$649$	46051.7	2.78534
$650$	0	0
$651$	0	0
$652$	−7804.36	−0.468777
$653$	−403.329	−0.0241707	−0.0120854	−	0.999927i	$-0.503847\pi$
$653$	−403.329	−0.0241707	−0.0120854	+	0.999927i	$0.503847\pi$
$654$	−12602.2	−0.753496
$655$	0	0
$656$	−34527.1	−2.05497
$657$	−5671.50	−0.336783
$658$	0	0
$659$	20527.9	1.21343	0.606716	−	0.794919i	$-0.292487\pi$
$659$	20527.9	1.21343	0.606716	+	0.794919i	$0.292487\pi$
$660$	0	0
$661$	4372.75	0.257307	0.128654	−	0.991690i	$-0.458934\pi$
$661$	4372.75	0.257307	0.128654	+	0.991690i	$0.458934\pi$
$662$	−9580.22	−0.562456
$663$	78.4155	0.00459337
$664$	−8332.62	−0.487000
$665$	0	0
$666$	4485.03	0.260948
$667$	−9229.97	−0.535811
$668$	17522.8	1.01494
$669$	−12079.7	−0.698099
$670$	0	0
$671$	40083.5	2.30612
$672$	0	0
$673$	19902.7	1.13996	0.569980	−	0.821659i	$-0.306951\pi$
$673$	19902.7	1.13996	0.569980	+	0.821659i	$0.306951\pi$
$674$	19561.4	1.11792
$675$	0	0
$676$	−12525.5	−0.712647
$677$	9714.40	0.551484	0.275742	−	0.961232i	$-0.411076\pi$
$677$	9714.40	0.551484	0.275742	+	0.961232i	$0.411076\pi$
$678$	−1859.72	−0.105342
$679$	0	0
$680$	0	0
$681$	−15910.9	−0.895312
$682$	−28947.0	−1.62528
$683$	88.0227	0.00493132	0.00246566	−	0.999997i	$-0.499215\pi$
$683$	88.0227	0.00493132	0.00246566	+	0.999997i	$0.499215\pi$
$684$	191.216	0.0106891
$685$	0	0
$686$	0	0
$687$	−33147.6	−1.84084
$688$	16193.5	0.897345
$689$	−55.9300	−0.00309254
$690$	0	0
$691$	−3722.48	−0.204934	−0.102467	−	0.994736i	$-0.532674\pi$
$691$	−3722.48	−0.204934	−0.102467	+	0.994736i	$0.532674\pi$
$692$	−13696.1	−0.752380
$693$	0	0
$694$	−43633.0	−2.38658
$695$	0	0
$696$	14210.0	0.773891
$697$	−15893.2	−0.863700
$698$	32446.5	1.75948
$699$	11039.2	0.597343
$700$	0	0
$701$	13167.2	0.709443	0.354722	−	0.934972i	$-0.384576\pi$
$701$	13167.2	0.709443	0.354722	+	0.934972i	$0.384576\pi$
$702$	−175.766	−0.00944994
$703$	−1339.53	−0.0718652
$704$	11263.7	0.603007
$705$	0	0
$706$	−31753.1	−1.69270
$707$	0	0
$708$	24944.2	1.32409
$709$	−14539.6	−0.770166	−0.385083	−	0.922882i	$-0.625827\pi$
$709$	−14539.6	−0.770166	−0.385083	+	0.922882i	$0.625827\pi$
$710$	0	0
$711$	−3804.60	−0.200680
$712$	7749.99	0.407926
$713$	−4105.57	−0.215645
$714$	0	0
$715$	0	0
$716$	−6535.39	−0.341116
$717$	15712.3	0.818391
$718$	−13808.2	−0.717713
$719$	−26509.9	−1.37504	−0.687519	−	0.726166i	$-0.741300\pi$
$719$	−26509.9	−1.37504	−0.687519	+	0.726166i	$0.741300\pi$
$720$	0	0
$721$	0	0
$722$	25251.8	1.30163
$723$	−22950.1	−1.18053
$724$	2713.07	0.139269
$725$	0	0
$726$	47926.1	2.45001
$727$	11761.8	0.600030	0.300015	−	0.953934i	$-0.403008\pi$
$727$	11761.8	0.600030	0.300015	+	0.953934i	$0.403008\pi$
$728$	0	0
$729$	14196.7	0.721270
$730$	0	0
$731$	7454.08	0.377153
$732$	21711.5	1.09628
$733$	−6458.47	−0.325442	−0.162721	−	0.986672i	$-0.552027\pi$
$733$	−6458.47	−0.325442	−0.162721	+	0.986672i	$0.552027\pi$
$734$	1909.15	0.0960055
$735$	0	0
$736$	6848.07	0.342967
$737$	−4648.77	−0.232347
$738$	−9129.43	−0.455364
$739$	33663.0	1.67566	0.837830	−	0.545932i	$-0.183824\pi$
$739$	33663.0	1.67566	0.837830	+	0.545932i	$0.183824\pi$
$740$	0	0
$741$	−13.4530	−0.000666948 0
$742$	0	0
$743$	19979.4	0.986506	0.493253	−	0.869886i	$-0.335808\pi$
$743$	19979.4	0.986506	0.493253	+	0.869886i	$0.335808\pi$
$744$	6320.72	0.311463
$745$	0	0
$746$	40428.5	1.98417
$747$	−5394.39	−0.264218
$748$	12144.8	0.593661
$749$	0	0
$750$	0	0
$751$	−31850.0	−1.54757	−0.773785	−	0.633449i	$-0.781639\pi$
$751$	−31850.0	−1.54757	−0.773785	+	0.633449i	$0.781639\pi$
$752$	−35296.1	−1.71159
$753$	40861.1	1.97750
$754$	420.186	0.0202948
$755$	0	0
$756$	0	0
$757$	39396.6	1.89154	0.945769	−	0.324840i	$-0.105310\pi$
$757$	39396.6	1.89154	0.945769	+	0.324840i	$0.105310\pi$
$758$	31104.7	1.49047
$759$	10781.4	0.515601
$760$	0	0
$761$	7170.87	0.341582	0.170791	−	0.985307i	$-0.445368\pi$
$761$	7170.87	0.341582	0.170791	+	0.985307i	$0.445368\pi$
$762$	44728.2	2.12642
$763$	0	0
$764$	5644.57	0.267295
$765$	0	0
$766$	11218.3	0.529154
$767$	−297.342	−0.0139979
$768$	−30595.0	−1.43750
$769$	−27056.7	−1.26878	−0.634388	−	0.773015i	$-0.718748\pi$
$769$	−27056.7	−1.26878	−0.634388	+	0.773015i	$0.718748\pi$
$770$	0	0
$771$	−3693.22	−0.172514
$772$	−7941.62	−0.370240
$773$	40087.8	1.86527	0.932637	−	0.360815i	$-0.117501\pi$
$773$	40087.8	1.86527	0.932637	+	0.360815i	$0.117501\pi$
$774$	4281.79	0.198845
$775$	0	0
$776$	95.0757	0.00439822
$777$	0	0
$778$	−5194.23	−0.239360
$779$	2726.65	0.125408
$780$	0	0
$781$	54408.6	2.49282
$782$	4139.39	0.189289
$783$	−35896.9	−1.63838
$784$	0	0
$785$	0	0
$786$	5577.84	0.253123
$787$	−23501.4	−1.06447	−0.532233	−	0.846598i	$-0.678647\pi$
$787$	−23501.4	−1.06447	−0.532233	+	0.846598i	$0.678647\pi$
$788$	20430.9	0.923632
$789$	−34958.1	−1.57737
$790$	0	0
$791$	0	0
$792$	−2812.30	−0.126175
$793$	−258.807	−0.0115896
$794$	−206.961	−0.00925035
$795$	0	0
$796$	−3557.47	−0.158406
$797$	−1635.92	−0.0727066	−0.0363533	−	0.999339i	$-0.511574\pi$
$797$	−1635.92	−0.0727066	−0.0363533	+	0.999339i	$0.511574\pi$
$798$	0	0
$799$	−16247.2	−0.719381
$800$	0	0
$801$	5017.21	0.221316
$802$	13807.5	0.607929
$803$	61799.3	2.71588
$804$	−2518.03	−0.110453
$805$	0	0
$806$	186.902	0.00816794
$807$	−5562.62	−0.242644
$808$	−5750.00	−0.250352
$809$	−33824.6	−1.46998	−0.734988	−	0.678080i	$-0.762812\pi$
$809$	−33824.6	−1.46998	−0.734988	+	0.678080i	$0.762812\pi$
$810$	0	0
$811$	3636.30	0.157445	0.0787224	−	0.996897i	$-0.474916\pi$
$811$	3636.30	0.157445	0.0787224	+	0.996897i	$0.474916\pi$
$812$	0	0
$813$	26642.4	1.14931
$814$	−48870.9	−2.10433
$815$	0	0
$816$	−15603.0	−0.669379
$817$	−1278.83	−0.0547619
$818$	−7286.52	−0.311451
$819$	0	0
$820$	0	0
$821$	2417.39	0.102762	0.0513808	−	0.998679i	$-0.483638\pi$
$821$	2417.39	0.102762	0.0513808	+	0.998679i	$0.483638\pi$
$822$	−9881.78	−0.419302
$823$	−15752.2	−0.667178	−0.333589	−	0.942719i	$-0.608260\pi$
$823$	−15752.2	−0.667178	−0.333589	+	0.942719i	$0.608260\pi$
$824$	−13007.4	−0.549920
$825$	0	0
$826$	0	0
$827$	−4850.49	−0.203952	−0.101976	−	0.994787i	$-0.532516\pi$
$827$	−4850.49	−0.203952	−0.101976	+	0.994787i	$0.532516\pi$
$828$	989.444	0.0415284
$829$	−33160.3	−1.38927	−0.694636	−	0.719362i	$-0.744434\pi$
$829$	−33160.3	−1.38927	−0.694636	+	0.719362i	$0.744434\pi$
$830$	0	0
$831$	13627.7	0.568879
$832$	−72.7264	−0.00303045
$833$	0	0
$834$	−20195.2	−0.838494
$835$	0	0
$836$	−2083.57	−0.0861984
$837$	−15967.2	−0.659388
$838$	−48891.7	−2.01544
$839$	−14966.6	−0.615857	−0.307929	−	0.951409i	$-0.599636\pi$
$839$	−14966.6	−0.615857	−0.307929	+	0.951409i	$0.599636\pi$
$840$	0	0
$841$	61426.1	2.51860
$842$	27708.9	1.13410
$843$	18936.5	0.773674
$844$	12922.5	0.527025
$845$	0	0
$846$	−9332.76	−0.379275
$847$	0	0
$848$	11128.9	0.450668
$849$	−47095.8	−1.90380
$850$	0	0
$851$	−6931.37	−0.279206
$852$	29470.7	1.18504
$853$	−5806.35	−0.233066	−0.116533	−	0.993187i	$-0.537178\pi$
$853$	−5806.35	−0.233066	−0.116533	+	0.993187i	$0.537178\pi$
$854$	0	0
$855$	0	0
$856$	−5969.04	−0.238338
$857$	−9191.52	−0.366367	−0.183183	−	0.983079i	$-0.558640\pi$
$857$	−9191.52	−0.366367	−0.183183	+	0.983079i	$0.558640\pi$
$858$	−490.816	−0.0195293
$859$	−15029.2	−0.596962	−0.298481	−	0.954416i	$-0.596480\pi$
$859$	−15029.2	−0.596962	−0.298481	+	0.954416i	$0.596480\pi$
$860$	0	0
$861$	0	0
$862$	−39323.1	−1.55377
$863$	−17147.5	−0.676369	−0.338185	−	0.941080i	$-0.609813\pi$
$863$	−17147.5	−0.676369	−0.338185	+	0.941080i	$0.609813\pi$
$864$	26633.3	1.04871
$865$	0	0
$866$	−26905.7	−1.05576
$867$	20829.5	0.815927
$868$	0	0
$869$	41456.6	1.61832
$870$	0	0
$871$	30.0157	0.00116767
$872$	−5080.27	−0.197293
$873$	61.5504	0.00238621
$874$	−710.156	−0.0274844
$875$	0	0
$876$	33473.9	1.29107
$877$	−33483.6	−1.28924	−0.644618	−	0.764505i	$-0.722984\pi$
$877$	−33483.6	−1.28924	−0.644618	+	0.764505i	$0.722984\pi$
$878$	−2751.56	−0.105764
$879$	3582.00	0.137449
$880$	0	0
$881$	10695.6	0.409019	0.204509	−	0.978865i	$-0.434440\pi$
$881$	10695.6	0.409019	0.204509	+	0.978865i	$0.434440\pi$
$882$	0	0
$883$	31934.0	1.21706	0.608531	−	0.793530i	$-0.291759\pi$
$883$	31934.0	1.21706	0.608531	+	0.793530i	$0.291759\pi$
$884$	−78.4155	−0.00298348
$885$	0	0
$886$	40190.9	1.52397
$887$	−10655.2	−0.403345	−0.201672	−	0.979453i	$-0.564638\pi$
$887$	−10655.2	−0.403345	−0.201672	+	0.979453i	$0.564638\pi$
$888$	10671.2	0.403268
$889$	0	0
$890$	0	0
$891$	50855.7	1.91216
$892$	12079.7	0.453429
$893$	2787.38	0.104453
$894$	−8082.41	−0.302367
$895$	0	0
$896$	0	0
$897$	−69.6125	−0.00259119
$898$	41320.0	1.53549
$899$	38171.3	1.41611
$900$	0	0
$901$	5122.74	0.189415
$902$	99478.4	3.67214
$903$	0	0
$904$	−749.697	−0.0275825
$905$	0	0
$906$	27499.0	1.00838
$907$	−20737.2	−0.759171	−0.379585	−	0.925157i	$-0.623933\pi$
$907$	−20737.2	−0.759171	−0.379585	+	0.925157i	$0.623933\pi$
$908$	15910.9	0.581522
$909$	−3722.45	−0.135826
$910$	0	0
$911$	11734.8	0.426774	0.213387	−	0.976968i	$-0.431550\pi$
$911$	11734.8	0.426774	0.213387	+	0.976968i	$0.431550\pi$
$912$	2676.86	0.0971926
$913$	58779.8	2.13070
$914$	49789.7	1.80185
$915$	0	0
$916$	33147.6	1.19566
$917$	0	0
$918$	16098.8	0.578800
$919$	−21922.8	−0.786906	−0.393453	−	0.919345i	$-0.628720\pi$
$919$	−21922.8	−0.786906	−0.393453	+	0.919345i	$0.628720\pi$
$920$	0	0
$921$	−143.705	−0.00514141
$922$	−33830.3	−1.20840
$923$	−351.300	−0.0125278
$924$	0	0
$925$	0	0
$926$	−48573.6	−1.72379
$927$	−8420.77	−0.298354
$928$	−63669.6	−2.25222
$929$	36647.7	1.29427	0.647133	−	0.762377i	$-0.275968\pi$
$929$	36647.7	1.29427	0.647133	+	0.762377i	$0.275968\pi$
$930$	0	0
$931$	0	0
$932$	−11039.2	−0.387986
$933$	−53801.0	−1.88785
$934$	−44127.1	−1.54591
$935$	0	0
$936$	18.1582	0.000634101 0
$937$	−24959.7	−0.870222	−0.435111	−	0.900377i	$-0.643291\pi$
$937$	−24959.7	−0.870222	−0.435111	+	0.900377i	$0.643291\pi$
$938$	0	0
$939$	15842.4	0.550581
$940$	0	0
$941$	−17086.7	−0.591935	−0.295968	−	0.955198i	$-0.595642\pi$
$941$	−17086.7	−0.591935	−0.295968	+	0.955198i	$0.595642\pi$
$942$	−67605.2	−2.33832
$943$	14109.0	0.487226
$944$	59164.6	2.03988
$945$	0	0
$946$	−46656.3	−1.60352
$947$	54034.9	1.85417	0.927085	−	0.374852i	$-0.122307\pi$
$947$	54034.9	1.85417	0.927085	+	0.374852i	$0.122307\pi$
$948$	22455.2	0.769316
$949$	−399.020	−0.0136488
$950$	0	0
$951$	−16991.9	−0.579389
$952$	0	0
$953$	−24377.9	−0.828622	−0.414311	−	0.910135i	$-0.635977\pi$
$953$	−24377.9	−0.828622	−0.414311	+	0.910135i	$0.635977\pi$
$954$	2942.62	0.0998645
$955$	0	0
$956$	−15712.3	−0.531561
$957$	−100240.	−3.38588
$958$	3054.83	0.103024
$959$	0	0
$960$	0	0
$961$	−12812.1	−0.430066
$962$	315.545	0.0105754
$963$	−3864.25	−0.129308
$964$	22950.1	0.766777
$965$	0	0
$966$	0	0
$967$	−32668.1	−1.08639	−0.543194	−	0.839607i	$-0.682785\pi$
$967$	−32668.1	−1.08639	−0.543194	+	0.839607i	$0.682785\pi$
$968$	19320.2	0.641502
$969$	1232.19	0.0408500
$970$	0	0
$971$	−24623.2	−0.813795	−0.406898	−	0.913474i	$-0.633389\pi$
$971$	−24623.2	−0.813795	−0.406898	+	0.913474i	$0.633389\pi$
$972$	8682.37	0.286510
$973$	0	0
$974$	−24720.2	−0.813231
$975$	0	0
$976$	51497.0	1.68891
$977$	−18320.6	−0.599926	−0.299963	−	0.953951i	$-0.596974\pi$
$977$	−18320.6	−0.599926	−0.299963	+	0.953951i	$0.596974\pi$
$978$	−28888.3	−0.944526
$979$	−54669.8	−1.78473
$980$	0	0
$981$	−3288.88	−0.107040
$982$	11469.8	0.372727
$983$	−2176.33	−0.0706145	−0.0353072	−	0.999377i	$-0.511241\pi$
$983$	−2176.33	−0.0706145	−0.0353072	+	0.999377i	$0.511241\pi$
$984$	−21721.6	−0.703718
$985$	0	0
$986$	−38485.7	−1.24304
$987$	0	0
$988$	13.4530	0.000433196 0
$989$	−6617.28	−0.212758
$990$	0	0
$991$	−21487.2	−0.688761	−0.344381	−	0.938830i	$-0.611911\pi$
$991$	−21487.2	−0.688761	−0.344381	+	0.938830i	$0.611911\pi$
$992$	−28320.8	−0.906437
$993$	−14756.5	−0.471586
$994$	0	0
$995$	0	0
$996$	31838.4	1.01289
$997$	19083.0	0.606182	0.303091	−	0.952962i	$-0.401981\pi$
$997$	19083.0	0.606182	0.303091	+	0.952962i	$0.401981\pi$
$998$	8384.41	0.265936
$999$	−26957.2	−0.853743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.r.1.1		2
5.4	even	2		1225.4.a.t.1.2		2
7.6	odd	2		175.4.a.d.1.1	✓	2
21.20	even	2		1575.4.a.v.1.2		2
35.13	even	4		175.4.b.d.99.4		4
35.27	even	4		175.4.b.d.99.1		4
35.34	odd	2		175.4.a.e.1.2	yes	2
105.104	even	2		1575.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.1	✓	2	7.6	odd	2
175.4.a.e.1.2	yes	2	35.34	odd	2
175.4.b.d.99.1		4	35.27	even	4
175.4.b.d.99.4		4	35.13	even	4
1225.4.a.r.1.1		2	1.1	even	1	trivial
1225.4.a.t.1.2		2	5.4	even	2
1575.4.a.s.1.1		2	105.104	even	2
1575.4.a.v.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-3.70156 q^{2} -5.70156 q^{3} +5.70156 q^{4} +21.1047 q^{6} +8.50781 q^{8} +5.50781 q^{9} -60.0156 q^{11} -32.5078 q^{12} +0.387503 q^{13} -77.1047 q^{16} -35.4922 q^{17} -20.3875 q^{18} +6.08907 q^{19} +222.152 q^{22} +31.5078 q^{23} -48.5078 q^{24} -1.43437 q^{26} +122.539 q^{27} -292.942 q^{29} -130.303 q^{31} +217.345 q^{32} +342.183 q^{33} +131.377 q^{34} +31.4031 q^{36} -219.989 q^{37} -22.5391 q^{38} -2.20937 q^{39} +447.795 q^{41} -210.020 q^{43} -342.183 q^{44} -116.628 q^{46} +457.769 q^{47} +439.617 q^{48} +202.361 q^{51} +2.20937 q^{52} -144.334 q^{53} -453.586 q^{54} -34.7172 q^{57} +1084.34 q^{58} -767.328 q^{59} -667.884 q^{61} +482.325 q^{62} -187.680 q^{64} -1266.61 q^{66} +77.4593 q^{67} -202.361 q^{68} -179.644 q^{69} -906.573 q^{71} +46.8594 q^{72} -1029.72 q^{73} +814.303 q^{74} +34.7172 q^{76} +8.17813 q^{78} -690.764 q^{79} -847.375 q^{81} -1657.54 q^{82} -979.408 q^{83} +777.403 q^{86} +1670.23 q^{87} -510.602 q^{88} +910.927 q^{89} +179.644 q^{92} +742.931 q^{93} -1694.46 q^{94} -1239.21 q^{96} +11.1751 q^{97} -330.555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 5 q^{3} + 5 q^{4} + 23 q^{6} - 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} - 135 q^{16} - 103 q^{17} - 92 q^{18} + 57 q^{19} + 233 q^{22} + 31 q^{23} - 65 q^{24} + 138 q^{26}+ \cdots - 437 q^{99}+O(q^{100}) \) 2 * q - q^2 - 5 * q^3 + 5 * q^4 + 23 * q^6 - 15 * q^8 - 21 * q^9 - 56 * q^11 - 33 * q^12 + 52 * q^13 - 135 * q^16 - 103 * q^17 - 92 * q^18 + 57 * q^19 + 233 * q^22 + 31 * q^23 - 65 * q^24 + 138 * q^26 + 85 * q^27 - 413 * q^29 + 162 * q^31 + 249 * q^32 + 345 * q^33 - 51 * q^34 + 50 * q^36 - 75 * q^37 + 115 * q^38 + 34 * q^39 + 505 * q^41 + 73 * q^43 - 345 * q^44 - 118 * q^46 + 224 * q^47 + 399 * q^48 + 155 * q^51 - 34 * q^52 + 262 * q^53 - 555 * q^54 + q^57 + 760 * q^58 - 190 * q^59 - 990 * q^61 + 1272 * q^62 + 361 * q^64 - 1259 * q^66 - 908 * q^67 - 155 * q^68 - 180 * q^69 + 127 * q^71 + 670 * q^72 - 337 * q^73 + 1206 * q^74 - q^76 + 106 * q^78 - 1119 * q^79 - 158 * q^81 - 1503 * q^82 - 1517 * q^83 + 1542 * q^86 + 1586 * q^87 - 605 * q^88 + 1713 * q^89 + 180 * q^92 + 948 * q^93 - 2326 * q^94 - 1217 * q^96 + 1764 * q^97 - 437 * q^99

Introduction

L-functions

Modular forms

Varieties

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