LMFDB - Embedded newform 2475.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.723686$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.723686 q^{2} -7.47628 q^{4} -1.13288 q^{7} +11.2000 q^{8} +O(q^{10})$	$q-0.723686 q^{2} -7.47628 q^{4} -1.13288 q^{7} +11.2000 q^{8} -11.0000 q^{11} +21.8566 q^{13} +0.819851 q^{14} +51.7050 q^{16} +6.18033 q^{17} -92.8393 q^{19} +7.96054 q^{22} -36.7317 q^{23} -15.8173 q^{26} +8.46975 q^{28} -71.1375 q^{29} +186.522 q^{31} -127.018 q^{32} -4.47261 q^{34} +356.581 q^{37} +67.1865 q^{38} -271.940 q^{41} -155.780 q^{43} +82.2391 q^{44} +26.5822 q^{46} +234.566 q^{47} -341.717 q^{49} -163.406 q^{52} +195.018 q^{53} -12.6882 q^{56} +51.4812 q^{58} +455.930 q^{59} -441.278 q^{61} -134.983 q^{62} -321.719 q^{64} -133.005 q^{67} -46.2058 q^{68} +1041.68 q^{71} +160.119 q^{73} -258.053 q^{74} +694.093 q^{76} +12.4617 q^{77} -761.367 q^{79} +196.799 q^{82} +51.7170 q^{83} +112.736 q^{86} -123.200 q^{88} +1075.25 q^{89} -24.7609 q^{91} +274.616 q^{92} -169.752 q^{94} +703.238 q^{97} +247.295 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * q^98

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 $

$2$	−0.723686	−0.255862	−0.127931	−	0.991783i	$-0.540834\pi$
$2$	−0.723686	−0.255862	−0.127931	+	0.991783i	$0.540834\pi$
$3$	0	0
$3$	0	0
$4$	−7.47628	−0.934535
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.13288	−0.0611699	−0.0305850	−	0.999532i	$-0.509737\pi$
$7$	−1.13288	−0.0611699	−0.0305850	+	0.999532i	$0.509737\pi$
$8$	11.2000	0.494973
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	21.8566	0.466302	0.233151	−	0.972441i	$-0.425096\pi$
$13$	21.8566	0.466302	0.233151	+	0.972441i	$0.425096\pi$
$14$	0.819851	0.0156510
$15$	0	0
$16$	51.7050	0.807890
$17$	6.18033	0.0881735	0.0440867	−	0.999028i	$-0.485962\pi$
$17$	6.18033	0.0881735	0.0440867	+	0.999028i	$0.485962\pi$
$18$	0	0
$19$	−92.8393	−1.12099	−0.560495	−	0.828158i	$-0.689389\pi$
$19$	−92.8393	−1.12099	−0.560495	+	0.828158i	$0.689389\pi$
$20$	0	0
$21$	0	0
$22$	7.96054	0.0771452
$23$	−36.7317	−0.333004	−0.166502	−	0.986041i	$-0.553247\pi$
$23$	−36.7317	−0.333004	−0.166502	+	0.986041i	$0.553247\pi$
$24$	0	0
$25$	0	0
$26$	−15.8173	−0.119309
$27$	0	0
$28$	8.46975	0.0571654
$29$	−71.1375	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$29$	−71.1375	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$30$	0	0
$31$	186.522	1.08065	0.540327	−	0.841455i	$-0.318301\pi$
$31$	186.522	1.08065	0.540327	+	0.841455i	$0.318301\pi$
$32$	−127.018	−0.701681
$33$	0	0
$34$	−4.47261	−0.0225602
$35$	0	0
$36$	0	0
$37$	356.581	1.58437	0.792184	−	0.610282i	$-0.208944\pi$
$37$	356.581	1.58437	0.792184	+	0.610282i	$0.208944\pi$
$38$	67.1865	0.286818
$39$	0	0
$40$	0	0
$41$	−271.940	−1.03585	−0.517925	−	0.855426i	$-0.673295\pi$
$41$	−271.940	−1.03585	−0.517925	+	0.855426i	$0.673295\pi$
$42$	0	0
$43$	−155.780	−0.552470	−0.276235	−	0.961090i	$-0.589087\pi$
$43$	−155.780	−0.552470	−0.276235	+	0.961090i	$0.589087\pi$
$44$	82.2391	0.281773
$45$	0	0
$46$	26.5822	0.0852028
$47$	234.566	0.727978	0.363989	−	0.931403i	$-0.381415\pi$
$47$	234.566	0.727978	0.363989	+	0.931403i	$0.381415\pi$
$48$	0	0
$49$	−341.717	−0.996258
$50$	0	0
$51$	0	0
$52$	−163.406	−0.435775
$53$	195.018	0.505429	0.252715	−	0.967541i	$-0.418677\pi$
$53$	195.018	0.505429	0.252715	+	0.967541i	$0.418677\pi$
$54$	0	0
$55$	0	0
$56$	−12.6882	−0.0302775
$57$	0	0
$58$	51.4812	0.116548
$59$	455.930	1.00605	0.503025	−	0.864272i	$-0.332220\pi$
$59$	455.930	1.00605	0.503025	+	0.864272i	$0.332220\pi$
$60$	0	0
$61$	−441.278	−0.926227	−0.463114	−	0.886299i	$-0.653268\pi$
$61$	−441.278	−0.926227	−0.463114	+	0.886299i	$0.653268\pi$
$62$	−134.983	−0.276498
$63$	0	0
$64$	−321.719	−0.628357
$65$	0	0
$66$	0	0
$67$	−133.005	−0.242524	−0.121262	−	0.992621i	$-0.538694\pi$
$67$	−133.005	−0.242524	−0.121262	+	0.992621i	$0.538694\pi$
$68$	−46.2058	−0.0824012
$69$	0	0
$70$	0	0
$71$	1041.68	1.74119	0.870596	−	0.491999i	$-0.163734\pi$
$71$	1041.68	1.74119	0.870596	+	0.491999i	$0.163734\pi$
$72$	0	0
$73$	160.119	0.256719	0.128360	−	0.991728i	$-0.459029\pi$
$73$	160.119	0.256719	0.128360	+	0.991728i	$0.459029\pi$
$74$	−258.053	−0.405379
$75$	0	0
$76$	694.093	1.04760
$77$	12.4617	0.0184434
$78$	0	0
$79$	−761.367	−1.08431	−0.542155	−	0.840278i	$-0.682392\pi$
$79$	−761.367	−1.08431	−0.542155	+	0.840278i	$0.682392\pi$
$80$	0	0
$81$	0	0
$82$	196.799	0.265034
$83$	51.7170	0.0683938	0.0341969	−	0.999415i	$-0.489113\pi$
$83$	51.7170	0.0683938	0.0341969	+	0.999415i	$0.489113\pi$
$84$	0	0
$85$	0	0
$86$	112.736	0.141356
$87$	0	0
$88$	−123.200	−0.149240
$89$	1075.25	1.28064	0.640318	−	0.768110i	$-0.278802\pi$
$89$	1075.25	1.28064	0.640318	+	0.768110i	$0.278802\pi$
$90$	0	0
$91$	−24.7609	−0.0285236
$92$	274.616	0.311203
$93$	0	0
$94$	−169.752	−0.186262
$95$	0	0
$96$	0	0
$97$	703.238	0.736114	0.368057	−	0.929803i	$-0.380023\pi$
$97$	703.238	0.736114	0.368057	+	0.929803i	$0.380023\pi$
$98$	247.295	0.254904
$99$	0	0
$100$	0	0
$101$	856.886	0.844192	0.422096	−	0.906551i	$-0.361295\pi$
$101$	856.886	0.844192	0.422096	+	0.906551i	$0.361295\pi$
$102$	0	0
$103$	805.989	0.771034	0.385517	−	0.922701i	$-0.374023\pi$
$103$	805.989	0.771034	0.385517	+	0.922701i	$0.374023\pi$
$104$	244.793	0.230807
$105$	0	0
$106$	−141.132	−0.129320
$107$	−1608.55	−1.45331	−0.726654	−	0.687004i	$-0.758926\pi$
$107$	−1608.55	−1.45331	−0.726654	+	0.687004i	$0.758926\pi$
$108$	0	0
$109$	−925.724	−0.813471	−0.406735	−	0.913546i	$-0.633333\pi$
$109$	−925.724	−0.813471	−0.406735	+	0.913546i	$0.633333\pi$
$110$	0	0
$111$	0	0
$112$	−58.5757	−0.0494186
$113$	1214.81	1.01133	0.505663	−	0.862731i	$-0.331248\pi$
$113$	1214.81	1.01133	0.505663	+	0.862731i	$0.331248\pi$
$114$	0	0
$115$	0	0
$116$	531.844	0.425693
$117$	0	0
$118$	−329.950	−0.257410
$119$	−7.00159	−0.00539357
$120$	0	0
$121$	121.000	0.0909091
$122$	319.347	0.236986
$123$	0	0
$124$	−1394.49	−1.00991
$125$	0	0
$126$	0	0
$127$	−1128.96	−0.788810	−0.394405	−	0.918937i	$-0.629049\pi$
$127$	−1128.96	−0.788810	−0.394405	+	0.918937i	$0.629049\pi$
$128$	1248.97	0.862454
$129$	0	0
$130$	0	0
$131$	1562.32	1.04199	0.520993	−	0.853561i	$-0.325562\pi$
$131$	1562.32	1.04199	0.520993	+	0.853561i	$0.325562\pi$
$132$	0	0
$133$	105.176	0.0685708
$134$	96.2535	0.0620525
$135$	0	0
$136$	69.2194	0.0436435
$137$	44.9323	0.0280206	0.0140103	−	0.999902i	$-0.495540\pi$
$137$	44.9323	0.0280206	0.0140103	+	0.999902i	$0.495540\pi$
$138$	0	0
$139$	−415.574	−0.253587	−0.126793	−	0.991929i	$-0.540468\pi$
$139$	−415.574	−0.253587	−0.126793	+	0.991929i	$0.540468\pi$
$140$	0	0
$141$	0	0
$142$	−753.849	−0.445504
$143$	−240.422	−0.140595
$144$	0	0
$145$	0	0
$146$	−115.876	−0.0656846
$147$	0	0
$148$	−2665.90	−1.48065
$149$	−678.402	−0.372999	−0.186499	−	0.982455i	$-0.559714\pi$
$149$	−678.402	−0.372999	−0.186499	+	0.982455i	$0.559714\pi$
$150$	0	0
$151$	616.083	0.332027	0.166014	−	0.986123i	$-0.446910\pi$
$151$	616.083	0.332027	0.166014	+	0.986123i	$0.446910\pi$
$152$	−1039.80	−0.554860
$153$	0	0
$154$	−9.01837	−0.00471896
$155$	0	0
$156$	0	0
$157$	616.599	0.313439	0.156720	−	0.987643i	$-0.449908\pi$
$157$	616.599	0.313439	0.156720	+	0.987643i	$0.449908\pi$
$158$	550.991	0.277433
$159$	0	0
$160$	0	0
$161$	41.6127	0.0203698
$162$	0	0
$163$	−2031.15	−0.976026	−0.488013	−	0.872836i	$-0.662278\pi$
$163$	−2031.15	−0.976026	−0.488013	+	0.872836i	$0.662278\pi$
$164$	2033.10	0.968038
$165$	0	0
$166$	−37.4269	−0.0174993
$167$	−2114.90	−0.979973	−0.489987	−	0.871730i	$-0.662998\pi$
$167$	−2114.90	−0.979973	−0.489987	+	0.871730i	$0.662998\pi$
$168$	0	0
$169$	−1719.29	−0.782563
$170$	0	0
$171$	0	0
$172$	1164.65	0.516302
$173$	−15.5096	−0.00681602	−0.00340801	−	0.999994i	$-0.501085\pi$
$173$	−15.5096	−0.00681602	−0.00340801	+	0.999994i	$0.501085\pi$
$174$	0	0
$175$	0	0
$176$	−568.755	−0.243588
$177$	0	0
$178$	−778.146	−0.327666
$179$	−487.991	−0.203766	−0.101883	−	0.994796i	$-0.532487\pi$
$179$	−487.991	−0.203766	−0.101883	+	0.994796i	$0.532487\pi$
$180$	0	0
$181$	1248.71	0.512795	0.256397	−	0.966571i	$-0.417464\pi$
$181$	1248.71	0.512795	0.256397	+	0.966571i	$0.417464\pi$
$182$	17.9191	0.00729810
$183$	0	0
$184$	−411.393	−0.164828
$185$	0	0
$186$	0	0
$187$	−67.9836	−0.0265853
$188$	−1753.68	−0.680321
$189$	0	0
$190$	0	0
$191$	2571.60	0.974213	0.487107	−	0.873342i	$-0.338052\pi$
$191$	2571.60	0.974213	0.487107	+	0.873342i	$0.338052\pi$
$192$	0	0
$193$	−1221.80	−0.455686	−0.227843	−	0.973698i	$-0.573167\pi$
$193$	−1221.80	−0.455686	−0.227843	+	0.973698i	$0.573167\pi$
$194$	−508.924	−0.188343
$195$	0	0
$196$	2554.77	0.931038
$197$	−1428.32	−0.516568	−0.258284	−	0.966069i	$-0.583157\pi$
$197$	−1428.32	−0.516568	−0.258284	+	0.966069i	$0.583157\pi$
$198$	0	0
$199$	−816.166	−0.290736	−0.145368	−	0.989378i	$-0.546437\pi$
$199$	−816.166	−0.290736	−0.145368	+	0.989378i	$0.546437\pi$
$200$	0	0
$201$	0	0
$202$	−620.116	−0.215996
$203$	80.5904	0.0278637
$204$	0	0
$205$	0	0
$206$	−583.283	−0.197278
$207$	0	0
$208$	1130.09	0.376721
$209$	1021.23	0.337991
$210$	0	0
$211$	−903.360	−0.294739	−0.147369	−	0.989082i	$-0.547081\pi$
$211$	−903.360	−0.294739	−0.147369	+	0.989082i	$0.547081\pi$
$212$	−1458.01	−0.472341
$213$	0	0
$214$	1164.08	0.371846
$215$	0	0
$216$	0	0
$217$	−211.307	−0.0661035
$218$	669.934	0.208136
$219$	0	0
$220$	0	0
$221$	135.081	0.0411154
$222$	0	0
$223$	−533.966	−0.160345	−0.0801727	−	0.996781i	$-0.525547\pi$
$223$	−533.966	−0.160345	−0.0801727	+	0.996781i	$0.525547\pi$
$224$	143.896	0.0429218
$225$	0	0
$226$	−879.142	−0.258760
$227$	696.531	0.203658	0.101829	−	0.994802i	$-0.467531\pi$
$227$	696.531	0.203658	0.101829	+	0.994802i	$0.467531\pi$
$228$	0	0
$229$	−796.794	−0.229929	−0.114964	−	0.993370i	$-0.536675\pi$
$229$	−796.794	−0.229929	−0.114964	+	0.993370i	$0.536675\pi$
$230$	0	0
$231$	0	0
$232$	−796.737	−0.225467
$233$	−5657.08	−1.59059	−0.795295	−	0.606223i	$-0.792684\pi$
$233$	−5657.08	−1.59059	−0.795295	+	0.606223i	$0.792684\pi$
$234$	0	0
$235$	0	0
$236$	−3408.66	−0.940190
$237$	0	0
$238$	5.06695	0.00138001
$239$	2023.06	0.547534	0.273767	−	0.961796i	$-0.411730\pi$
$239$	2023.06	0.547534	0.273767	+	0.961796i	$0.411730\pi$
$240$	0	0
$241$	−3513.86	−0.939203	−0.469601	−	0.882879i	$-0.655602\pi$
$241$	−3513.86	−0.939203	−0.469601	+	0.882879i	$0.655602\pi$
$242$	−87.5660	−0.0232601
$243$	0	0
$244$	3299.12	0.865592
$245$	0	0
$246$	0	0
$247$	−2029.15	−0.522719
$248$	2089.03	0.534895
$249$	0	0
$250$	0	0
$251$	−1814.32	−0.456250	−0.228125	−	0.973632i	$-0.573260\pi$
$251$	−1814.32	−0.456250	−0.228125	+	0.973632i	$0.573260\pi$
$252$	0	0
$253$	404.048	0.100404
$254$	817.011	0.201826
$255$	0	0
$256$	1669.89	0.407688
$257$	1445.46	0.350839	0.175419	−	0.984494i	$-0.443872\pi$
$257$	1445.46	0.350839	0.175419	+	0.984494i	$0.443872\pi$
$258$	0	0
$259$	−403.965	−0.0969157
$260$	0	0
$261$	0	0
$262$	−1130.63	−0.266604
$263$	−4411.79	−1.03438	−0.517191	−	0.855870i	$-0.673022\pi$
$263$	−4411.79	−1.03438	−0.517191	+	0.855870i	$0.673022\pi$
$264$	0	0
$265$	0	0
$266$	−76.1144	−0.0175446
$267$	0	0
$268$	994.379	0.226647
$269$	−3955.14	−0.896465	−0.448232	−	0.893917i	$-0.647946\pi$
$269$	−3955.14	−0.896465	−0.448232	+	0.893917i	$0.647946\pi$
$270$	0	0
$271$	6656.96	1.49218	0.746091	−	0.665844i	$-0.231928\pi$
$271$	6656.96	1.49218	0.746091	+	0.665844i	$0.231928\pi$
$272$	319.554	0.0712345
$273$	0	0
$274$	−32.5169	−0.00716941
$275$	0	0
$276$	0	0
$277$	−8523.79	−1.84890	−0.924450	−	0.381304i	$-0.875475\pi$
$277$	−8523.79	−1.84890	−0.924450	+	0.381304i	$0.875475\pi$
$278$	300.745	0.0648831
$279$	0	0
$280$	0	0
$281$	4210.99	0.893973	0.446986	−	0.894541i	$-0.352497\pi$
$281$	4210.99	0.893973	0.446986	+	0.894541i	$0.352497\pi$
$282$	0	0
$283$	−3529.66	−0.741400	−0.370700	−	0.928753i	$-0.620882\pi$
$283$	−3529.66	−0.741400	−0.370700	+	0.928753i	$0.620882\pi$
$284$	−7787.88	−1.62720
$285$	0	0
$286$	173.990	0.0359729
$287$	308.076	0.0633629
$288$	0	0
$289$	−4874.80	−0.992225
$290$	0	0
$291$	0	0
$292$	−1197.09	−0.239913
$293$	−7110.96	−1.41784	−0.708919	−	0.705290i	$-0.750817\pi$
$293$	−7110.96	−1.41784	−0.708919	+	0.705290i	$0.750817\pi$
$294$	0	0
$295$	0	0
$296$	3993.70	0.784220
$297$	0	0
$298$	490.950	0.0954361
$299$	−802.828	−0.155280
$300$	0	0
$301$	176.480	0.0337945
$302$	−445.851	−0.0849531
$303$	0	0
$304$	−4800.25	−0.905636
$305$	0	0
$306$	0	0
$307$	−9474.72	−1.76140	−0.880702	−	0.473671i	$-0.842929\pi$
$307$	−9474.72	−1.76140	−0.880702	+	0.473671i	$0.842929\pi$
$308$	−93.1672	−0.0172360
$309$	0	0
$310$	0	0
$311$	5210.73	0.950075	0.475038	−	0.879965i	$-0.342434\pi$
$311$	5210.73	0.950075	0.475038	+	0.879965i	$0.342434\pi$
$312$	0	0
$313$	−1944.27	−0.351108	−0.175554	−	0.984470i	$-0.556172\pi$
$313$	−1944.27	−0.351108	−0.175554	+	0.984470i	$0.556172\pi$
$314$	−446.224	−0.0801970
$315$	0	0
$316$	5692.20	1.01333
$317$	−6852.73	−1.21416	−0.607079	−	0.794642i	$-0.707659\pi$
$317$	−6852.73	−1.21416	−0.607079	+	0.794642i	$0.707659\pi$
$318$	0	0
$319$	782.512	0.137343
$320$	0	0
$321$	0	0
$322$	−30.1145	−0.00521185
$323$	−573.777	−0.0988415
$324$	0	0
$325$	0	0
$326$	1469.92	0.249727
$327$	0	0
$328$	−3045.72	−0.512718
$329$	−265.736	−0.0445304
$330$	0	0
$331$	−2019.85	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$331$	−2019.85	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$332$	−386.651	−0.0639163
$333$	0	0
$334$	1530.52	0.250737
$335$	0	0
$336$	0	0
$337$	−11738.7	−1.89748	−0.948739	−	0.316060i	$-0.897640\pi$
$337$	−11738.7	−1.89748	−0.948739	+	0.316060i	$0.897640\pi$
$338$	1244.23	0.200228
$339$	0	0
$340$	0	0
$341$	−2051.74	−0.325829
$342$	0	0
$343$	775.704	0.122111
$344$	−1744.73	−0.273458
$345$	0	0
$346$	11.2241	0.00174396
$347$	12692.2	1.96355	0.981777	−	0.190036i	$-0.0608603\pi$
$347$	12692.2	1.96355	0.981777	+	0.190036i	$0.0608603\pi$
$348$	0	0
$349$	12026.8	1.84465	0.922324	−	0.386417i	$-0.126287\pi$
$349$	12026.8	1.84465	0.922324	+	0.386417i	$0.126287\pi$
$350$	0	0
$351$	0	0
$352$	1397.20	0.211565
$353$	−8948.39	−1.34922	−0.674610	−	0.738174i	$-0.735688\pi$
$353$	−8948.39	−1.34922	−0.674610	+	0.738174i	$0.735688\pi$
$354$	0	0
$355$	0	0
$356$	−8038.90	−1.19680
$357$	0	0
$358$	353.152	0.0521360
$359$	−3906.52	−0.574313	−0.287156	−	0.957884i	$-0.592710\pi$
$359$	−3906.52	−0.574313	−0.287156	+	0.957884i	$0.592710\pi$
$360$	0	0
$361$	1760.14	0.256617
$362$	−903.673	−0.131204
$363$	0	0
$364$	185.120	0.0266563
$365$	0	0
$366$	0	0
$367$	−4225.04	−0.600941	−0.300471	−	0.953791i	$-0.597144\pi$
$367$	−4225.04	−0.600941	−0.300471	+	0.953791i	$0.597144\pi$
$368$	−1899.21	−0.269030
$369$	0	0
$370$	0	0
$371$	−220.932	−0.0309171
$372$	0	0
$373$	−4223.50	−0.586285	−0.293143	−	0.956069i	$-0.594701\pi$
$373$	−4223.50	−0.586285	−0.293143	+	0.956069i	$0.594701\pi$
$374$	49.1988	0.00680216
$375$	0	0
$376$	2627.13	0.360330
$377$	−1554.82	−0.212407
$378$	0	0
$379$	−748.009	−0.101379	−0.0506895	−	0.998714i	$-0.516142\pi$
$379$	−748.009	−0.101379	−0.0506895	+	0.998714i	$0.516142\pi$
$380$	0	0
$381$	0	0
$382$	−1861.03	−0.249264
$383$	4668.07	0.622786	0.311393	−	0.950281i	$-0.399204\pi$
$383$	4668.07	0.622786	0.311393	+	0.950281i	$0.399204\pi$
$384$	0	0
$385$	0	0
$386$	884.203	0.116593
$387$	0	0
$388$	−5257.61	−0.687924
$389$	−6544.12	−0.852957	−0.426478	−	0.904498i	$-0.640246\pi$
$389$	−6544.12	−0.852957	−0.426478	+	0.904498i	$0.640246\pi$
$390$	0	0
$391$	−227.014	−0.0293621
$392$	−3827.21	−0.493121
$393$	0	0
$394$	1033.66	0.132170
$395$	0	0
$396$	0	0
$397$	−6439.84	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$397$	−6439.84	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$398$	590.648	0.0743882
$399$	0	0
$400$	0	0
$401$	−7727.03	−0.962268	−0.481134	−	0.876647i	$-0.659775\pi$
$401$	−7727.03	−0.962268	−0.481134	+	0.876647i	$0.659775\pi$
$402$	0	0
$403$	4076.72	0.503911
$404$	−6406.32	−0.788927
$405$	0	0
$406$	−58.3222	−0.00712926
$407$	−3922.40	−0.477705
$408$	0	0
$409$	−2539.97	−0.307074	−0.153537	−	0.988143i	$-0.549066\pi$
$409$	−2539.97	−0.307074	−0.153537	+	0.988143i	$0.549066\pi$
$410$	0	0
$411$	0	0
$412$	−6025.80	−0.720558
$413$	−516.515	−0.0615401
$414$	0	0
$415$	0	0
$416$	−2776.17	−0.327195
$417$	0	0
$418$	−739.052	−0.0864789
$419$	−1170.17	−0.136436	−0.0682181	−	0.997670i	$-0.521731\pi$
$419$	−1170.17	−0.136436	−0.0682181	+	0.997670i	$0.521731\pi$
$420$	0	0
$421$	4009.01	0.464103	0.232052	−	0.972703i	$-0.425456\pi$
$421$	4009.01	0.464103	0.232052	+	0.972703i	$0.425456\pi$
$422$	653.749	0.0754123
$423$	0	0
$424$	2184.19	0.250174
$425$	0	0
$426$	0	0
$427$	499.916	0.0566573
$428$	12025.9	1.35817
$429$	0	0
$430$	0	0
$431$	2676.70	0.299146	0.149573	−	0.988751i	$-0.452210\pi$
$431$	2676.70	0.299146	0.149573	+	0.988751i	$0.452210\pi$
$432$	0	0
$433$	−5655.83	−0.627718	−0.313859	−	0.949470i	$-0.601622\pi$
$433$	−5655.83	−0.627718	−0.313859	+	0.949470i	$0.601622\pi$
$434$	152.920	0.0169133
$435$	0	0
$436$	6920.97	0.760217
$437$	3410.14	0.373294
$438$	0	0
$439$	−6305.94	−0.685572	−0.342786	−	0.939414i	$-0.611371\pi$
$439$	−6305.94	−0.685572	−0.342786	+	0.939414i	$0.611371\pi$
$440$	0	0
$441$	0	0
$442$	−97.7560	−0.0105199
$443$	−6772.22	−0.726315	−0.363158	−	0.931728i	$-0.618301\pi$
$443$	−6772.22	−0.726315	−0.363158	+	0.931728i	$0.618301\pi$
$444$	0	0
$445$	0	0
$446$	386.424	0.0410262
$447$	0	0
$448$	364.470	0.0384365
$449$	5049.16	0.530701	0.265351	−	0.964152i	$-0.414512\pi$
$449$	5049.16	0.530701	0.265351	+	0.964152i	$0.414512\pi$
$450$	0	0
$451$	2991.34	0.312321
$452$	−9082.27	−0.945120
$453$	0	0
$454$	−504.070	−0.0521083
$455$	0	0
$456$	0	0
$457$	−3866.14	−0.395733	−0.197867	−	0.980229i	$-0.563401\pi$
$457$	−3866.14	−0.395733	−0.197867	+	0.980229i	$0.563401\pi$
$458$	576.629	0.0588299
$459$	0	0
$460$	0	0
$461$	9108.85	0.920263	0.460132	−	0.887851i	$-0.347802\pi$
$461$	9108.85	0.920263	0.460132	+	0.887851i	$0.347802\pi$
$462$	0	0
$463$	17421.8	1.74873	0.874363	−	0.485273i	$-0.161280\pi$
$463$	17421.8	1.74873	0.874363	+	0.485273i	$0.161280\pi$
$464$	−3678.16	−0.368005
$465$	0	0
$466$	4093.95	0.406971
$467$	2602.36	0.257864	0.128932	−	0.991653i	$-0.458845\pi$
$467$	2602.36	0.257864	0.128932	+	0.991653i	$0.458845\pi$
$468$	0	0
$469$	150.679	0.0148352
$470$	0	0
$471$	0	0
$472$	5106.40	0.497968
$473$	1713.58	0.166576
$474$	0	0
$475$	0	0
$476$	52.3458	0.00504047
$477$	0	0
$478$	−1464.06	−0.140093
$479$	−10751.8	−1.02560	−0.512801	−	0.858508i	$-0.671392\pi$
$479$	−10751.8	−1.02560	−0.512801	+	0.858508i	$0.671392\pi$
$480$	0	0
$481$	7793.65	0.738794
$482$	2542.93	0.240306
$483$	0	0
$484$	−904.630	−0.0849577
$485$	0	0
$486$	0	0
$487$	2115.37	0.196831	0.0984154	−	0.995145i	$-0.468623\pi$
$487$	2115.37	0.196831	0.0984154	+	0.995145i	$0.468623\pi$
$488$	−4942.30	−0.458458
$489$	0	0
$490$	0	0
$491$	−4634.30	−0.425953	−0.212977	−	0.977057i	$-0.568316\pi$
$491$	−4634.30	−0.425953	−0.212977	+	0.977057i	$0.568316\pi$
$492$	0	0
$493$	−439.653	−0.0401642
$494$	1468.47	0.133744
$495$	0	0
$496$	9644.09	0.873049
$497$	−1180.10	−0.106509
$498$	0	0
$499$	2495.43	0.223870	0.111935	−	0.993716i	$-0.464295\pi$
$499$	2495.43	0.223870	0.111935	+	0.993716i	$0.464295\pi$
$500$	0	0
$501$	0	0
$502$	1313.00	0.116737
$503$	18521.0	1.64177	0.820886	−	0.571092i	$-0.193480\pi$
$503$	18521.0	1.64177	0.820886	+	0.571092i	$0.193480\pi$
$504$	0	0
$505$	0	0
$506$	−292.404	−0.0256896
$507$	0	0
$508$	8440.41	0.737170
$509$	6102.40	0.531403	0.265702	−	0.964055i	$-0.414396\pi$
$509$	6102.40	0.531403	0.265702	+	0.964055i	$0.414396\pi$
$510$	0	0
$511$	−181.396	−0.0157035
$512$	−11200.2	−0.966765
$513$	0	0
$514$	−1046.06	−0.0897661
$515$	0	0
$516$	0	0
$517$	−2580.23	−0.219494
$518$	292.344	0.0247970
$519$	0	0
$520$	0	0
$521$	2813.12	0.236555	0.118277	−	0.992981i	$-0.462263\pi$
$521$	2813.12	0.236555	0.118277	+	0.992981i	$0.462263\pi$
$522$	0	0
$523$	−1563.27	−0.130702	−0.0653509	−	0.997862i	$-0.520817\pi$
$523$	−1563.27	−0.130702	−0.0653509	+	0.997862i	$0.520817\pi$
$524$	−11680.3	−0.973772
$525$	0	0
$526$	3192.75	0.264659
$527$	1152.76	0.0952850
$528$	0	0
$529$	−10817.8	−0.889109
$530$	0	0
$531$	0	0
$532$	−786.326	−0.0640818
$533$	−5943.67	−0.483019
$534$	0	0
$535$	0	0
$536$	−1489.65	−0.120043
$537$	0	0
$538$	2862.28	0.229371
$539$	3758.88	0.300383
$540$	0	0
$541$	−5959.61	−0.473611	−0.236806	−	0.971557i	$-0.576100\pi$
$541$	−5959.61	−0.473611	−0.236806	+	0.971557i	$0.576100\pi$
$542$	−4817.55	−0.381792
$543$	0	0
$544$	−785.012	−0.0618697
$545$	0	0
$546$	0	0
$547$	−18084.2	−1.41357	−0.706787	−	0.707426i	$-0.749856\pi$
$547$	−18084.2	−1.41357	−0.706787	+	0.707426i	$0.749856\pi$
$548$	−335.927	−0.0261863
$549$	0	0
$550$	0	0
$551$	6604.35	0.510626
$552$	0	0
$553$	862.540	0.0663272
$554$	6168.55	0.473062
$555$	0	0
$556$	3106.95	0.236985
$557$	18129.5	1.37912	0.689561	−	0.724227i	$-0.257803\pi$
$557$	18129.5	1.37912	0.689561	+	0.724227i	$0.257803\pi$
$558$	0	0
$559$	−3404.81	−0.257618
$560$	0	0
$561$	0	0
$562$	−3047.43	−0.228733
$563$	−17910.2	−1.34072	−0.670361	−	0.742035i	$-0.733861\pi$
$563$	−17910.2	−1.34072	−0.670361	+	0.742035i	$0.733861\pi$
$564$	0	0
$565$	0	0
$566$	2554.36	0.189696
$567$	0	0
$568$	11666.8	0.861843
$569$	−24176.6	−1.78126	−0.890629	−	0.454731i	$-0.849735\pi$
$569$	−24176.6	−1.78126	−0.890629	+	0.454731i	$0.849735\pi$
$570$	0	0
$571$	−13167.8	−0.965073	−0.482537	−	0.875876i	$-0.660285\pi$
$571$	−13167.8	−0.965073	−0.482537	+	0.875876i	$0.660285\pi$
$572$	1797.46	0.131391
$573$	0	0
$574$	−222.950	−0.0162121
$575$	0	0
$576$	0	0
$577$	722.513	0.0521293	0.0260646	−	0.999660i	$-0.491702\pi$
$577$	722.513	0.0521293	0.0260646	+	0.999660i	$0.491702\pi$
$578$	3527.83	0.253872
$579$	0	0
$580$	0	0
$581$	−58.5893	−0.00418364
$582$	0	0
$583$	−2145.19	−0.152393
$584$	1793.33	0.127069
$585$	0	0
$586$	5146.10	0.362770
$587$	−17038.8	−1.19807	−0.599034	−	0.800723i	$-0.704449\pi$
$587$	−17038.8	−1.19807	−0.599034	+	0.800723i	$0.704449\pi$
$588$	0	0
$589$	−17316.5	−1.21140
$590$	0	0
$591$	0	0
$592$	18437.0	1.28000
$593$	6206.74	0.429815	0.214908	−	0.976634i	$-0.431055\pi$
$593$	6206.74	0.429815	0.214908	+	0.976634i	$0.431055\pi$
$594$	0	0
$595$	0	0
$596$	5071.92	0.348580
$597$	0	0
$598$	580.996	0.0397302
$599$	23050.7	1.57233	0.786164	−	0.618018i	$-0.212064\pi$
$599$	23050.7	1.57233	0.786164	+	0.618018i	$0.212064\pi$
$600$	0	0
$601$	−6323.69	−0.429199	−0.214600	−	0.976702i	$-0.568845\pi$
$601$	−6323.69	−0.429199	−0.214600	+	0.976702i	$0.568845\pi$
$602$	−127.716	−0.00864673
$603$	0	0
$604$	−4606.01	−0.310291
$605$	0	0
$606$	0	0
$607$	−4625.14	−0.309273	−0.154636	−	0.987971i	$-0.549421\pi$
$607$	−4625.14	−0.309273	−0.154636	+	0.987971i	$0.549421\pi$
$608$	11792.3	0.786577
$609$	0	0
$610$	0	0
$611$	5126.81	0.339457
$612$	0	0
$613$	28075.3	1.84984	0.924919	−	0.380164i	$-0.124132\pi$
$613$	28075.3	1.84984	0.924919	+	0.380164i	$0.124132\pi$
$614$	6856.72	0.450676
$615$	0	0
$616$	139.571	0.00912900
$617$	−7264.85	−0.474023	−0.237011	−	0.971507i	$-0.576168\pi$
$617$	−7264.85	−0.474023	−0.237011	+	0.971507i	$0.576168\pi$
$618$	0	0
$619$	−30583.2	−1.98585	−0.992927	−	0.118728i	$-0.962119\pi$
$619$	−30583.2	−1.98585	−0.992927	+	0.118728i	$0.962119\pi$
$620$	0	0
$621$	0	0
$622$	−3770.93	−0.243088
$623$	−1218.14	−0.0783365
$624$	0	0
$625$	0	0
$626$	1407.04	0.0898351
$627$	0	0
$628$	−4609.86	−0.292920
$629$	2203.79	0.139699
$630$	0	0
$631$	−16679.8	−1.05232	−0.526159	−	0.850386i	$-0.676368\pi$
$631$	−16679.8	−1.05232	−0.526159	+	0.850386i	$0.676368\pi$
$632$	−8527.29	−0.536705
$633$	0	0
$634$	4959.23	0.310656
$635$	0	0
$636$	0	0
$637$	−7468.75	−0.464557
$638$	−566.293	−0.0351407
$639$	0	0
$640$	0	0
$641$	−4225.31	−0.260358	−0.130179	−	0.991490i	$-0.541555\pi$
$641$	−4225.31	−0.260358	−0.130179	+	0.991490i	$0.541555\pi$
$642$	0	0
$643$	−23703.5	−1.45377	−0.726886	−	0.686758i	$-0.759033\pi$
$643$	−23703.5	−1.45377	−0.726886	+	0.686758i	$0.759033\pi$
$644$	−311.108	−0.0190363
$645$	0	0
$646$	415.234	0.0252898
$647$	−10045.1	−0.610374	−0.305187	−	0.952292i	$-0.598719\pi$
$647$	−10045.1	−0.610374	−0.305187	+	0.952292i	$0.598719\pi$
$648$	0	0
$649$	−5015.23	−0.303336
$650$	0	0
$651$	0	0
$652$	15185.5	0.912130
$653$	−32314.2	−1.93652	−0.968262	−	0.249936i	$-0.919590\pi$
$653$	−32314.2	−1.93652	−0.968262	+	0.249936i	$0.919590\pi$
$654$	0	0
$655$	0	0
$656$	−14060.6	−0.836853
$657$	0	0
$658$	192.309	0.0113936
$659$	12032.4	0.711253	0.355627	−	0.934628i	$-0.384267\pi$
$659$	12032.4	0.711253	0.355627	+	0.934628i	$0.384267\pi$
$660$	0	0
$661$	12864.5	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$661$	12864.5	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$662$	1461.74	0.0858189
$663$	0	0
$664$	579.229	0.0338531
$665$	0	0
$666$	0	0
$667$	2613.00	0.151688
$668$	15811.5	0.915819
$669$	0	0
$670$	0	0
$671$	4854.06	0.279268
$672$	0	0
$673$	−12007.5	−0.687751	−0.343876	−	0.939015i	$-0.611740\pi$
$673$	−12007.5	−0.687751	−0.343876	+	0.939015i	$0.611740\pi$
$674$	8495.16	0.485492
$675$	0	0
$676$	12853.9	0.731332
$677$	−10732.5	−0.609284	−0.304642	−	0.952467i	$-0.598537\pi$
$677$	−10732.5	−0.609284	−0.304642	+	0.952467i	$0.598537\pi$
$678$	0	0
$679$	−796.687	−0.0450280
$680$	0	0
$681$	0	0
$682$	1484.81	0.0833672
$683$	14126.0	0.791383	0.395692	−	0.918383i	$-0.370505\pi$
$683$	14126.0	0.791383	0.395692	+	0.918383i	$0.370505\pi$
$684$	0	0
$685$	0	0
$686$	−561.366	−0.0312435
$687$	0	0
$688$	−8054.59	−0.446335
$689$	4262.42	0.235682
$690$	0	0
$691$	−18467.8	−1.01671	−0.508357	−	0.861147i	$-0.669747\pi$
$691$	−18467.8	−1.01671	−0.508357	+	0.861147i	$0.669747\pi$
$692$	115.954	0.00636981
$693$	0	0
$694$	−9185.17	−0.502398
$695$	0	0
$696$	0	0
$697$	−1680.68	−0.0913345
$698$	−8703.66	−0.471975
$699$	0	0
$700$	0	0
$701$	24876.0	1.34030	0.670152	−	0.742224i	$-0.266229\pi$
$701$	24876.0	1.34030	0.670152	+	0.742224i	$0.266229\pi$
$702$	0	0
$703$	−33104.8	−1.77606
$704$	3538.91	0.189457
$705$	0	0
$706$	6475.83	0.345214
$707$	−970.752	−0.0516391
$708$	0	0
$709$	21271.1	1.12673	0.563367	−	0.826207i	$-0.309506\pi$
$709$	21271.1	1.12673	0.563367	+	0.826207i	$0.309506\pi$
$710$	0	0
$711$	0	0
$712$	12042.8	0.633881
$713$	−6851.25	−0.359862
$714$	0	0
$715$	0	0
$716$	3648.36	0.190427
$717$	0	0
$718$	2827.09	0.146945
$719$	−19213.9	−0.996603	−0.498302	−	0.867004i	$-0.666043\pi$
$719$	−19213.9	−0.996603	−0.498302	+	0.867004i	$0.666043\pi$
$720$	0	0
$721$	−913.091	−0.0471641
$722$	−1273.79	−0.0656585
$723$	0	0
$724$	−9335.70	−0.479224
$725$	0	0
$726$	0	0
$727$	27454.7	1.40060	0.700302	−	0.713846i	$-0.253049\pi$
$727$	27454.7	1.40060	0.700302	+	0.713846i	$0.253049\pi$
$728$	−277.322	−0.0141184
$729$	0	0
$730$	0	0
$731$	−962.770	−0.0487132
$732$	0	0
$733$	1182.16	0.0595691	0.0297846	−	0.999556i	$-0.490518\pi$
$733$	1182.16	0.0595691	0.0297846	+	0.999556i	$0.490518\pi$
$734$	3057.61	0.153758
$735$	0	0
$736$	4665.58	0.233662
$737$	1463.05	0.0731237
$738$	0	0
$739$	25688.2	1.27869	0.639347	−	0.768918i	$-0.279205\pi$
$739$	25688.2	1.27869	0.639347	+	0.768918i	$0.279205\pi$
$740$	0	0
$741$	0	0
$742$	159.886	0.00791049
$743$	32431.2	1.60132	0.800662	−	0.599116i	$-0.204481\pi$
$743$	32431.2	1.60132	0.800662	+	0.599116i	$0.204481\pi$
$744$	0	0
$745$	0	0
$746$	3056.49	0.150008
$747$	0	0
$748$	508.264	0.0248449
$749$	1822.29	0.0888988
$750$	0	0
$751$	−32093.5	−1.55940	−0.779700	−	0.626154i	$-0.784628\pi$
$751$	−32093.5	−1.55940	−0.779700	+	0.626154i	$0.784628\pi$
$752$	12128.2	0.588127
$753$	0	0
$754$	1125.20	0.0543467
$755$	0	0
$756$	0	0
$757$	1293.36	0.0620977	0.0310488	−	0.999518i	$-0.490115\pi$
$757$	1293.36	0.0620977	0.0310488	+	0.999518i	$0.490115\pi$
$758$	541.323	0.0259390
$759$	0	0
$760$	0	0
$761$	−29013.1	−1.38203	−0.691014	−	0.722841i	$-0.742836\pi$
$761$	−29013.1	−1.38203	−0.691014	+	0.722841i	$0.742836\pi$
$762$	0	0
$763$	1048.74	0.0497599
$764$	−19226.0	−0.910436
$765$	0	0
$766$	−3378.22	−0.159347
$767$	9965.06	0.469123
$768$	0	0
$769$	−31304.1	−1.46795	−0.733976	−	0.679176i	$-0.762337\pi$
$769$	−31304.1	−1.46795	−0.733976	+	0.679176i	$0.762337\pi$
$770$	0	0
$771$	0	0
$772$	9134.55	0.425855
$773$	41913.5	1.95023	0.975113	−	0.221710i	$-0.0711637\pi$
$773$	41913.5	1.95023	0.975113	+	0.221710i	$0.0711637\pi$
$774$	0	0
$775$	0	0
$776$	7876.24	0.364357
$777$	0	0
$778$	4735.89	0.218239
$779$	25246.7	1.16118
$780$	0	0
$781$	−11458.5	−0.524989
$782$	164.287	0.00751263
$783$	0	0
$784$	−17668.4	−0.804867
$785$	0	0
$786$	0	0
$787$	6648.00	0.301113	0.150556	−	0.988601i	$-0.451894\pi$
$787$	6648.00	0.301113	0.150556	+	0.988601i	$0.451894\pi$
$788$	10678.6	0.482751
$789$	0	0
$790$	0	0
$791$	−1376.24	−0.0618628
$792$	0	0
$793$	−9644.82	−0.431901
$794$	4660.42	0.208302
$795$	0	0
$796$	6101.88	0.271703
$797$	14798.4	0.657701	0.328850	−	0.944382i	$-0.393339\pi$
$797$	14798.4	0.657701	0.328850	+	0.944382i	$0.393339\pi$
$798$	0	0
$799$	1449.69	0.0641884
$800$	0	0
$801$	0	0
$802$	5591.94	0.246207
$803$	−1761.31	−0.0774038
$804$	0	0
$805$	0	0
$806$	−2950.27	−0.128931
$807$	0	0
$808$	9597.09	0.417852
$809$	−4500.04	−0.195566	−0.0977831	−	0.995208i	$-0.531175\pi$
$809$	−4500.04	−0.195566	−0.0977831	+	0.995208i	$0.531175\pi$
$810$	0	0
$811$	−1909.49	−0.0826773	−0.0413386	−	0.999145i	$-0.513162\pi$
$811$	−1909.49	−0.0826773	−0.0413386	+	0.999145i	$0.513162\pi$
$812$	−602.517	−0.0260396
$813$	0	0
$814$	2838.58	0.122226
$815$	0	0
$816$	0	0
$817$	14462.5	0.619313
$818$	1838.14	0.0785685
$819$	0	0
$820$	0	0
$821$	−10928.6	−0.464569	−0.232285	−	0.972648i	$-0.574620\pi$
$821$	−10928.6	−0.464569	−0.232285	+	0.972648i	$0.574620\pi$
$822$	0	0
$823$	−29278.4	−1.24007	−0.620036	−	0.784573i	$-0.712882\pi$
$823$	−29278.4	−1.24007	−0.620036	+	0.784573i	$0.712882\pi$
$824$	9027.05	0.381641
$825$	0	0
$826$	373.795	0.0157457
$827$	−878.905	−0.0369559	−0.0184779	−	0.999829i	$-0.505882\pi$
$827$	−878.905	−0.0369559	−0.0184779	+	0.999829i	$0.505882\pi$
$828$	0	0
$829$	16130.1	0.675779	0.337890	−	0.941186i	$-0.390287\pi$
$829$	16130.1	0.675779	0.337890	+	0.941186i	$0.390287\pi$
$830$	0	0
$831$	0	0
$832$	−7031.67	−0.293004
$833$	−2111.92	−0.0878436
$834$	0	0
$835$	0	0
$836$	−7635.02	−0.315864
$837$	0	0
$838$	846.838	0.0349088
$839$	39531.8	1.62669	0.813343	−	0.581784i	$-0.197645\pi$
$839$	39531.8	1.62669	0.813343	+	0.581784i	$0.197645\pi$
$840$	0	0
$841$	−19328.5	−0.792507
$842$	−2901.27	−0.118746
$843$	0	0
$844$	6753.77	0.275443
$845$	0	0
$846$	0	0
$847$	−137.079	−0.00556090
$848$	10083.4	0.408331
$849$	0	0
$850$	0	0
$851$	−13097.8	−0.527600
$852$	0	0
$853$	5373.68	0.215699	0.107849	−	0.994167i	$-0.465604\pi$
$853$	5373.68	0.215699	0.107849	+	0.994167i	$0.465604\pi$
$854$	−361.782	−0.0144964
$855$	0	0
$856$	−18015.7	−0.719349
$857$	−45378.4	−1.80875	−0.904374	−	0.426740i	$-0.859662\pi$
$857$	−45378.4	−1.80875	−0.904374	+	0.426740i	$0.859662\pi$
$858$	0	0
$859$	35336.8	1.40358	0.701791	−	0.712383i	$-0.252384\pi$
$859$	35336.8	1.40358	0.701791	+	0.712383i	$0.252384\pi$
$860$	0	0
$861$	0	0
$862$	−1937.09	−0.0765401
$863$	−24595.3	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$863$	−24595.3	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$864$	0	0
$865$	0	0
$866$	4093.05	0.160609
$867$	0	0
$868$	1579.79	0.0617760
$869$	8375.04	0.326932
$870$	0	0
$871$	−2907.02	−0.113089
$872$	−10368.1	−0.402646
$873$	0	0
$874$	−2467.87	−0.0955115
$875$	0	0
$876$	0	0
$877$	8251.06	0.317695	0.158848	−	0.987303i	$-0.449222\pi$
$877$	8251.06	0.317695	0.158848	+	0.987303i	$0.449222\pi$
$878$	4563.52	0.175412
$879$	0	0
$880$	0	0
$881$	3528.36	0.134930	0.0674651	−	0.997722i	$-0.478509\pi$
$881$	3528.36	0.134930	0.0674651	+	0.997722i	$0.478509\pi$
$882$	0	0
$883$	−35459.1	−1.35141	−0.675703	−	0.737174i	$-0.736160\pi$
$883$	−35459.1	−1.35141	−0.675703	+	0.737174i	$0.736160\pi$
$884$	−1009.90	−0.0384238
$885$	0	0
$886$	4900.96	0.185836
$887$	32544.8	1.23196	0.615979	−	0.787763i	$-0.288761\pi$
$887$	32544.8	1.23196	0.615979	+	0.787763i	$0.288761\pi$
$888$	0	0
$889$	1278.98	0.0482514
$890$	0	0
$891$	0	0
$892$	3992.08	0.149848
$893$	−21777.0	−0.816056
$894$	0	0
$895$	0	0
$896$	−1414.93	−0.0527562
$897$	0	0
$898$	−3654.01	−0.135786
$899$	−13268.7	−0.492253
$900$	0	0
$901$	1205.27	0.0445654
$902$	−2164.79	−0.0799108
$903$	0	0
$904$	13605.9	0.500579
$905$	0	0
$906$	0	0
$907$	−20383.4	−0.746219	−0.373110	−	0.927787i	$-0.621709\pi$
$907$	−20383.4	−0.746219	−0.373110	+	0.927787i	$0.621709\pi$
$908$	−5207.46	−0.190326
$909$	0	0
$910$	0	0
$911$	−7783.29	−0.283065	−0.141532	−	0.989934i	$-0.545203\pi$
$911$	−7783.29	−0.283065	−0.141532	+	0.989934i	$0.545203\pi$
$912$	0	0
$913$	−568.887	−0.0206215
$914$	2797.87	0.101253
$915$	0	0
$916$	5957.05	0.214876
$917$	−1769.92	−0.0637382
$918$	0	0
$919$	27294.9	0.979734	0.489867	−	0.871797i	$-0.337045\pi$
$919$	27294.9	0.979734	0.489867	+	0.871797i	$0.337045\pi$
$920$	0	0
$921$	0	0
$922$	−6591.95	−0.235460
$923$	22767.5	0.811920
$924$	0	0
$925$	0	0
$926$	−12607.9	−0.447432
$927$	0	0
$928$	9035.73	0.319625
$929$	46468.7	1.64111	0.820554	−	0.571568i	$-0.193665\pi$
$929$	46468.7	1.64111	0.820554	+	0.571568i	$0.193665\pi$
$930$	0	0
$931$	31724.7	1.11679
$932$	42293.9	1.48646
$933$	0	0
$934$	−1883.29	−0.0659776
$935$	0	0
$936$	0	0
$937$	−6922.45	−0.241352	−0.120676	−	0.992692i	$-0.538506\pi$
$937$	−6922.45	−0.241352	−0.120676	+	0.992692i	$0.538506\pi$
$938$	−109.044	−0.00379575
$939$	0	0
$940$	0	0
$941$	23772.1	0.823536	0.411768	−	0.911289i	$-0.364911\pi$
$941$	23772.1	0.823536	0.411768	+	0.911289i	$0.364911\pi$
$942$	0	0
$943$	9988.80	0.344942
$944$	23573.8	0.812779
$945$	0	0
$946$	−1240.09	−0.0426204
$947$	7177.76	0.246300	0.123150	−	0.992388i	$-0.460700\pi$
$947$	7177.76	0.246300	0.123150	+	0.992388i	$0.460700\pi$
$948$	0	0
$949$	3499.65	0.119709
$950$	0	0
$951$	0	0
$952$	−78.4175	−0.00266967
$953$	−27530.2	−0.935772	−0.467886	−	0.883789i	$-0.654984\pi$
$953$	−27530.2	−0.935772	−0.467886	+	0.883789i	$0.654984\pi$
$954$	0	0
$955$	0	0
$956$	−15124.9	−0.511690
$957$	0	0
$958$	7780.94	0.262412
$959$	−50.9031	−0.00171402
$960$	0	0
$961$	4999.29	0.167812
$962$	−5640.15	−0.189029
$963$	0	0
$964$	26270.6	0.877718
$965$	0	0
$966$	0	0
$967$	55407.1	1.84258	0.921288	−	0.388881i	$-0.127138\pi$
$967$	55407.1	1.84258	0.921288	+	0.388881i	$0.127138\pi$
$968$	1355.20	0.0449976
$969$	0	0
$970$	0	0
$971$	55100.7	1.82108	0.910539	−	0.413423i	$-0.135667\pi$
$971$	55100.7	1.82108	0.910539	+	0.413423i	$0.135667\pi$
$972$	0	0
$973$	470.797	0.0155119
$974$	−1530.86	−0.0503615
$975$	0	0
$976$	−22816.3	−0.748290
$977$	44473.9	1.45634	0.728170	−	0.685396i	$-0.240371\pi$
$977$	44473.9	1.45634	0.728170	+	0.685396i	$0.240371\pi$
$978$	0	0
$979$	−11827.8	−0.386127
$980$	0	0
$981$	0	0
$982$	3353.78	0.108985
$983$	−28003.0	−0.908605	−0.454302	−	0.890848i	$-0.650111\pi$
$983$	−28003.0	−0.908605	−0.454302	+	0.890848i	$0.650111\pi$
$984$	0	0
$985$	0	0
$986$	318.170	0.0102765
$987$	0	0
$988$	15170.5	0.488499
$989$	5722.06	0.183974
$990$	0	0
$991$	−40719.5	−1.30524	−0.652622	−	0.757684i	$-0.726331\pi$
$991$	−40719.5	−1.30524	−0.652622	+	0.757684i	$0.726331\pi$
$992$	−23691.6	−0.758274
$993$	0	0
$994$	854.022	0.0272514
$995$	0	0
$996$	0	0
$997$	−18161.1	−0.576898	−0.288449	−	0.957495i	$-0.593140\pi$
$997$	−18161.1	−0.576898	−0.288449	+	0.957495i	$0.593140\pi$
$998$	−1805.91	−0.0572796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.v.1.2		3
3.2	odd	2		825.4.a.q.1.2	yes	3
5.4	even	2		2475.4.a.y.1.2		3
15.2	even	4		825.4.c.n.199.4		6
15.8	even	4		825.4.c.n.199.3		6
15.14	odd	2		825.4.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.2	✓	3	15.14	odd	2
825.4.a.q.1.2	yes	3	3.2	odd	2
825.4.c.n.199.3		6	15.8	even	4
825.4.c.n.199.4		6	15.2	even	4
2475.4.a.v.1.2		3	1.1	even	1	trivial
2475.4.a.y.1.2		3	5.4	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-0.723686 q^{2} -7.47628 q^{4} -1.13288 q^{7} +11.2000 q^{8} +O(q^{10})\)	\(q-0.723686 q^{2} -7.47628 q^{4} -1.13288 q^{7} +11.2000 q^{8} -11.0000 q^{11} +21.8566 q^{13} +0.819851 q^{14} +51.7050 q^{16} +6.18033 q^{17} -92.8393 q^{19} +7.96054 q^{22} -36.7317 q^{23} -15.8173 q^{26} +8.46975 q^{28} -71.1375 q^{29} +186.522 q^{31} -127.018 q^{32} -4.47261 q^{34} +356.581 q^{37} +67.1865 q^{38} -271.940 q^{41} -155.780 q^{43} +82.2391 q^{44} +26.5822 q^{46} +234.566 q^{47} -341.717 q^{49} -163.406 q^{52} +195.018 q^{53} -12.6882 q^{56} +51.4812 q^{58} +455.930 q^{59} -441.278 q^{61} -134.983 q^{62} -321.719 q^{64} -133.005 q^{67} -46.2058 q^{68} +1041.68 q^{71} +160.119 q^{73} -258.053 q^{74} +694.093 q^{76} +12.4617 q^{77} -761.367 q^{79} +196.799 q^{82} +51.7170 q^{83} +112.736 q^{86} -123.200 q^{88} +1075.25 q^{89} -24.7609 q^{91} +274.616 q^{92} -169.752 q^{94} +703.238 q^{97} +247.295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * q^98

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