LMFDB - Embedded newform 1875.4.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,4,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1875.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:	$110.628581261$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + \cdots + 42064 $ x^14 - 81x^12 - 7x^11 + 2512x^10 + 517x^9 - 36970x^8 - 12987x^7 + 257291x^6 + 125779x^5 - 718713x^4 - 371750x^3 + 579848x^2 + 394896x + 42064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.73740$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73740 q^{2} +3.00000 q^{3} -4.98145 q^{4} +5.21219 q^{6} +7.66213 q^{7} -22.5539 q^{8} +9.00000 q^{9} +20.4704 q^{11} -14.9443 q^{12} -14.8079 q^{13} +13.3122 q^{14} +0.666425 q^{16} -109.000 q^{17} +15.6366 q^{18} +97.7849 q^{19} +22.9864 q^{21} +35.5652 q^{22} -80.5714 q^{23} -67.6618 q^{24} -25.7272 q^{26} +27.0000 q^{27} -38.1685 q^{28} +59.0555 q^{29} +176.318 q^{31} +181.589 q^{32} +61.4111 q^{33} -189.376 q^{34} -44.8330 q^{36} -230.536 q^{37} +169.891 q^{38} -44.4236 q^{39} +179.464 q^{41} +39.9365 q^{42} -407.459 q^{43} -101.972 q^{44} -139.985 q^{46} +477.166 q^{47} +1.99927 q^{48} -284.292 q^{49} -326.999 q^{51} +73.7647 q^{52} -515.525 q^{53} +46.9097 q^{54} -172.811 q^{56} +293.355 q^{57} +102.603 q^{58} -571.444 q^{59} -52.3159 q^{61} +306.334 q^{62} +68.9591 q^{63} +310.162 q^{64} +106.696 q^{66} -551.814 q^{67} +542.976 q^{68} -241.714 q^{69} +368.658 q^{71} -202.985 q^{72} -406.465 q^{73} -400.533 q^{74} -487.111 q^{76} +156.847 q^{77} -77.1815 q^{78} -654.750 q^{79} +81.0000 q^{81} +311.801 q^{82} +421.856 q^{83} -114.505 q^{84} -707.919 q^{86} +177.166 q^{87} -461.688 q^{88} +425.138 q^{89} -113.460 q^{91} +401.362 q^{92} +528.953 q^{93} +829.027 q^{94} +544.768 q^{96} +590.666 q^{97} -493.928 q^{98} +184.233 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26}+ \cdots - 297 q^{99}+O(q^{100}) $ 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9 - 33 * q^11 + 150 * q^12 - 188 * q^13 - 287 * q^14 + 82 * q^16 - 146 * q^17 - 184 * q^19 - 81 * q^21 - 277 * q^22 - 164 * q^23 + 63 * q^24 + 103 * q^26 + 378 * q^27 + 224 * q^28 + 252 * q^29 - 889 * q^31 - 422 * q^32 - 99 * q^33 - 230 * q^34 + 450 * q^36 - 642 * q^37 - 1833 * q^38 - 564 * q^39 - 164 * q^41 - 861 * q^42 - 696 * q^43 - 6 * q^44 - 419 * q^46 + 92 * q^47 + 246 * q^48 + 81 * q^49 - 438 * q^51 - 1956 * q^52 - 949 * q^53 - 2380 * q^56 - 552 * q^57 - 1041 * q^58 - 81 * q^59 - 496 * q^61 - 1454 * q^62 - 243 * q^63 - 3903 * q^64 - 831 * q^66 - 1926 * q^67 - 685 * q^68 - 492 * q^69 + 2498 * q^71 + 189 * q^72 + 1026 * q^73 - 707 * q^74 - 5704 * q^76 - 5434 * q^77 + 309 * q^78 - 695 * q^79 + 1134 * q^81 + 886 * q^82 - 5315 * q^83 + 672 * q^84 + 3997 * q^86 + 756 * q^87 - 2969 * q^88 - 1424 * q^89 - 1194 * q^91 - 1607 * q^92 - 2667 * q^93 - 629 * q^94 - 1266 * q^96 - 291 * q^97 + 1018 * q^98 - 297 * 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 $

$2$	1.73740	0.614263	0.307131	−	0.951667i	$-0.400631\pi$
$2$	1.73740	0.614263	0.307131	+	0.951667i	$0.400631\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.98145	−0.622681
$5$	0	0
$5$	0	0
$6$	5.21219	0.354645
$7$	7.66213	0.413716	0.206858	−	0.978371i	$-0.433676\pi$
$7$	7.66213	0.413716	0.206858	+	0.978371i	$0.433676\pi$
$8$	−22.5539	−0.996753
$9$	9.00000	0.333333
$10$	0	0
$11$	20.4704	0.561095	0.280548	−	0.959840i	$-0.409484\pi$
$11$	20.4704	0.561095	0.280548	+	0.959840i	$0.409484\pi$
$12$	−14.9443	−0.359505
$13$	−14.8079	−0.315920	−0.157960	−	0.987445i	$-0.550492\pi$
$13$	−14.8079	−0.315920	−0.157960	+	0.987445i	$0.550492\pi$
$14$	13.3122	0.254130
$15$	0	0
$16$	0.666425	0.0104129
$17$	−109.000	−1.55508	−0.777539	−	0.628835i	$-0.783532\pi$
$17$	−109.000	−1.55508	−0.777539	+	0.628835i	$0.783532\pi$
$18$	15.6366	0.204754
$19$	97.7849	1.18071	0.590353	−	0.807145i	$-0.298989\pi$
$19$	97.7849	1.18071	0.590353	+	0.807145i	$0.298989\pi$
$20$	0	0
$21$	22.9864	0.238859
$22$	35.5652	0.344660
$23$	−80.5714	−0.730447	−0.365224	−	0.930920i	$-0.619008\pi$
$23$	−80.5714	−0.730447	−0.365224	+	0.930920i	$0.619008\pi$
$24$	−67.6618	−0.575475
$25$	0	0
$26$	−25.7272	−0.194058
$27$	27.0000	0.192450
$28$	−38.1685	−0.257613
$29$	59.0555	0.378149	0.189075	−	0.981963i	$-0.439451\pi$
$29$	59.0555	0.378149	0.189075	+	0.981963i	$0.439451\pi$
$30$	0	0
$31$	176.318	1.02153	0.510767	−	0.859719i	$-0.329361\pi$
$31$	176.318	1.02153	0.510767	+	0.859719i	$0.329361\pi$
$32$	181.589	1.00315
$33$	61.4111	0.323949
$34$	−189.376	−0.955226
$35$	0	0
$36$	−44.8330	−0.207560
$37$	−230.536	−1.02432	−0.512161	−	0.858889i	$-0.671155\pi$
$37$	−230.536	−1.02432	−0.512161	+	0.858889i	$0.671155\pi$
$38$	169.891	0.725264
$39$	−44.4236	−0.182397
$40$	0	0
$41$	179.464	0.683600	0.341800	−	0.939773i	$-0.388963\pi$
$41$	179.464	0.683600	0.341800	+	0.939773i	$0.388963\pi$
$42$	39.9365	0.146722
$43$	−407.459	−1.44505	−0.722523	−	0.691347i	$-0.757018\pi$
$43$	−407.459	−1.44505	−0.722523	+	0.691347i	$0.757018\pi$
$44$	−101.972	−0.349384
$45$	0	0
$46$	−139.985	−0.448687
$47$	477.166	1.48089	0.740445	−	0.672117i	$-0.234615\pi$
$47$	477.166	1.48089	0.740445	+	0.672117i	$0.234615\pi$
$48$	1.99927	0.00601188
$49$	−284.292	−0.828839
$50$	0	0
$51$	−326.999	−0.897824
$52$	73.7647	0.196718
$53$	−515.525	−1.33609	−0.668046	−	0.744120i	$-0.732869\pi$
$53$	−515.525	−1.33609	−0.668046	+	0.744120i	$0.732869\pi$
$54$	46.9097	0.118215
$55$	0	0
$56$	−172.811	−0.412372
$57$	293.355	0.681681
$58$	102.603	0.232283
$59$	−571.444	−1.26094	−0.630472	−	0.776212i	$-0.717139\pi$
$59$	−571.444	−1.26094	−0.630472	+	0.776212i	$0.717139\pi$
$60$	0	0
$61$	−52.3159	−0.109809	−0.0549046	−	0.998492i	$-0.517485\pi$
$61$	−52.3159	−0.109809	−0.0549046	+	0.998492i	$0.517485\pi$
$62$	306.334	0.627491
$63$	68.9591	0.137905
$64$	310.162	0.605784
$65$	0	0
$66$	106.696	0.198990
$67$	−551.814	−1.00619	−0.503096	−	0.864231i	$-0.667806\pi$
$67$	−551.814	−1.00619	−0.503096	+	0.864231i	$0.667806\pi$
$68$	542.976	0.968317
$69$	−241.714	−0.421724
$70$	0	0
$71$	368.658	0.616220	0.308110	−	0.951351i	$-0.400304\pi$
$71$	368.658	0.616220	0.308110	+	0.951351i	$0.400304\pi$
$72$	−202.985	−0.332251
$73$	−406.465	−0.651687	−0.325843	−	0.945424i	$-0.605648\pi$
$73$	−406.465	−0.651687	−0.325843	+	0.945424i	$0.605648\pi$
$74$	−400.533	−0.629203
$75$	0	0
$76$	−487.111	−0.735203
$77$	156.847	0.232134
$78$	−77.1815	−0.112040
$79$	−654.750	−0.932470	−0.466235	−	0.884661i	$-0.654390\pi$
$79$	−654.750	−0.932470	−0.466235	+	0.884661i	$0.654390\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	311.801	0.419910
$83$	421.856	0.557887	0.278944	−	0.960307i	$-0.410016\pi$
$83$	421.856	0.557887	0.278944	+	0.960307i	$0.410016\pi$
$84$	−114.505	−0.148733
$85$	0	0
$86$	−707.919	−0.887638
$87$	177.166	0.218325
$88$	−461.688	−0.559273
$89$	425.138	0.506343	0.253171	−	0.967421i	$-0.418526\pi$
$89$	425.138	0.506343	0.253171	+	0.967421i	$0.418526\pi$
$90$	0	0
$91$	−113.460	−0.130701
$92$	401.362	0.454836
$93$	528.953	0.589783
$94$	829.027	0.909655
$95$	0	0
$96$	544.768	0.579168
$97$	590.666	0.618279	0.309139	−	0.951017i	$-0.399959\pi$
$97$	590.666	0.618279	0.309139	+	0.951017i	$0.399959\pi$
$98$	−493.928	−0.509125
$99$	184.233	0.187032
$100$	0	0
$101$	−1533.55	−1.51083	−0.755417	−	0.655244i	$-0.772566\pi$
$101$	−1533.55	−1.51083	−0.755417	+	0.655244i	$0.772566\pi$
$102$	−568.128	−0.551500
$103$	−718.897	−0.687719	−0.343859	−	0.939021i	$-0.611734\pi$
$103$	−718.897	−0.687719	−0.343859	+	0.939021i	$0.611734\pi$
$104$	333.976	0.314895
$105$	0	0
$106$	−895.672	−0.820711
$107$	−944.844	−0.853659	−0.426830	−	0.904332i	$-0.640370\pi$
$107$	−944.844	−0.853659	−0.426830	+	0.904332i	$0.640370\pi$
$108$	−134.499	−0.119835
$109$	1472.33	1.29380	0.646899	−	0.762576i	$-0.276066\pi$
$109$	1472.33	1.29380	0.646899	+	0.762576i	$0.276066\pi$
$110$	0	0
$111$	−691.609	−0.591393
$112$	5.10623	0.00430798
$113$	−820.255	−0.682859	−0.341429	−	0.939907i	$-0.610911\pi$
$113$	−820.255	−0.682859	−0.341429	+	0.939907i	$0.610911\pi$
$114$	509.674	0.418731
$115$	0	0
$116$	−294.182	−0.235466
$117$	−133.271	−0.105307
$118$	−992.826	−0.774551
$119$	−835.170	−0.643360
$120$	0	0
$121$	−911.964	−0.685172
$122$	−90.8935	−0.0674517
$123$	538.393	0.394677
$124$	−878.317	−0.636090
$125$	0	0
$126$	119.809	0.0847101
$127$	1663.86	1.16255	0.581274	−	0.813708i	$-0.302554\pi$
$127$	1663.86	1.16255	0.581274	+	0.813708i	$0.302554\pi$
$128$	−913.841	−0.631038
$129$	−1222.38	−0.834297
$130$	0	0
$131$	−2281.74	−1.52180	−0.760901	−	0.648868i	$-0.775243\pi$
$131$	−2281.74	−1.52180	−0.760901	+	0.648868i	$0.775243\pi$
$132$	−305.916	−0.201717
$133$	749.241	0.488477
$134$	−958.721	−0.618066
$135$	0	0
$136$	2458.37	1.55003
$137$	934.992	0.583078	0.291539	−	0.956559i	$-0.405833\pi$
$137$	934.992	0.583078	0.291539	+	0.956559i	$0.405833\pi$
$138$	−419.954	−0.259049
$139$	−2874.22	−1.75387	−0.876937	−	0.480605i	$-0.840417\pi$
$139$	−2874.22	−1.75387	−0.876937	+	0.480605i	$0.840417\pi$
$140$	0	0
$141$	1431.50	0.854992
$142$	640.505	0.378521
$143$	−303.123	−0.177262
$144$	5.99782	0.00347096
$145$	0	0
$146$	−706.192	−0.400307
$147$	−852.875	−0.478530
$148$	1148.40	0.637826
$149$	2275.13	1.25091	0.625457	−	0.780259i	$-0.284913\pi$
$149$	2275.13	1.25091	0.625457	+	0.780259i	$0.284913\pi$
$150$	0	0
$151$	2396.11	1.29134	0.645670	−	0.763617i	$-0.276578\pi$
$151$	2396.11	1.29134	0.645670	+	0.763617i	$0.276578\pi$
$152$	−2205.44	−1.17687
$153$	−980.997	−0.518359
$154$	272.505	0.142591
$155$	0	0
$156$	221.294	0.113575
$157$	902.424	0.458734	0.229367	−	0.973340i	$-0.426334\pi$
$157$	902.424	0.458734	0.229367	+	0.973340i	$0.426334\pi$
$158$	−1137.56	−0.572782
$159$	−1546.58	−0.771393
$160$	0	0
$161$	−617.348	−0.302198
$162$	140.729	0.0682514
$163$	−4028.26	−1.93569	−0.967846	−	0.251543i	$-0.919062\pi$
$163$	−4028.26	−1.93569	−0.967846	+	0.251543i	$0.919062\pi$
$164$	−893.992	−0.425665
$165$	0	0
$166$	732.931	0.342690
$167$	−982.313	−0.455172	−0.227586	−	0.973758i	$-0.573083\pi$
$167$	−982.313	−0.455172	−0.227586	+	0.973758i	$0.573083\pi$
$168$	−518.433	−0.238083
$169$	−1977.73	−0.900194
$170$	0	0
$171$	880.065	0.393569
$172$	2029.74	0.899802
$173$	1265.10	0.555975	0.277988	−	0.960585i	$-0.410333\pi$
$173$	1265.10	0.555975	0.277988	+	0.960585i	$0.410333\pi$
$174$	307.809	0.134109
$175$	0	0
$176$	13.6420	0.00584262
$177$	−1714.33	−0.728007
$178$	738.634	0.311028
$179$	−1471.61	−0.614488	−0.307244	−	0.951631i	$-0.599407\pi$
$179$	−1471.61	−0.614488	−0.307244	+	0.951631i	$0.599407\pi$
$180$	0	0
$181$	−3742.99	−1.53709	−0.768547	−	0.639794i	$-0.779020\pi$
$181$	−3742.99	−1.53709	−0.768547	+	0.639794i	$0.779020\pi$
$182$	−197.125	−0.0802850
$183$	−156.948	−0.0633984
$184$	1817.20	0.728075
$185$	0	0
$186$	919.001	0.362282
$187$	−2231.26	−0.872547
$188$	−2376.98	−0.922122
$189$	206.877	0.0796197
$190$	0	0
$191$	3953.76	1.49782	0.748911	−	0.662670i	$-0.230577\pi$
$191$	3953.76	1.49782	0.748911	+	0.662670i	$0.230577\pi$
$192$	930.485	0.349750
$193$	−4739.26	−1.76756	−0.883780	−	0.467903i	$-0.845010\pi$
$193$	−4739.26	−1.76756	−0.883780	+	0.467903i	$0.845010\pi$
$194$	1026.22	0.379786
$195$	0	0
$196$	1416.19	0.516102
$197$	−1912.59	−0.691708	−0.345854	−	0.938288i	$-0.612411\pi$
$197$	−1912.59	−0.691708	−0.345854	+	0.938288i	$0.612411\pi$
$198$	320.087	0.114887
$199$	−3291.81	−1.17262	−0.586308	−	0.810089i	$-0.699419\pi$
$199$	−3291.81	−1.17262	−0.586308	+	0.810089i	$0.699419\pi$
$200$	0	0
$201$	−1655.44	−0.580925
$202$	−2664.39	−0.928050
$203$	452.491	0.156446
$204$	1628.93	0.559058
$205$	0	0
$206$	−1249.01	−0.422440
$207$	−725.142	−0.243482
$208$	−9.86833	−0.00328964
$209$	2001.69	0.662489
$210$	0	0
$211$	713.739	0.232871	0.116436	−	0.993198i	$-0.462853\pi$
$211$	713.739	0.232871	0.116436	+	0.993198i	$0.462853\pi$
$212$	2568.06	0.831959
$213$	1105.97	0.355775
$214$	−1641.57	−0.524371
$215$	0	0
$216$	−608.956	−0.191825
$217$	1350.97	0.422625
$218$	2558.03	0.794732
$219$	−1219.40	−0.376252
$220$	0	0
$221$	1614.05	0.491281
$222$	−1201.60	−0.363271
$223$	−4536.63	−1.36231	−0.681155	−	0.732140i	$-0.738522\pi$
$223$	−4536.63	−1.36231	−0.681155	+	0.732140i	$0.738522\pi$
$224$	1391.36	0.415019
$225$	0	0
$226$	−1425.11	−0.419455
$227$	49.5501	0.0144879	0.00724396	−	0.999974i	$-0.497694\pi$
$227$	49.5501	0.0144879	0.00724396	+	0.999974i	$0.497694\pi$
$228$	−1461.33	−0.424470
$229$	−2354.73	−0.679498	−0.339749	−	0.940516i	$-0.610342\pi$
$229$	−2354.73	−0.679498	−0.339749	+	0.940516i	$0.610342\pi$
$230$	0	0
$231$	470.540	0.134023
$232$	−1331.93	−0.376921
$233$	−2899.04	−0.815116	−0.407558	−	0.913179i	$-0.633620\pi$
$233$	−2899.04	−0.815116	−0.407558	+	0.913179i	$0.633620\pi$
$234$	−231.545	−0.0646861
$235$	0	0
$236$	2846.62	0.785166
$237$	−1964.25	−0.538362
$238$	−1451.02	−0.395192
$239$	−1054.21	−0.285319	−0.142660	−	0.989772i	$-0.545565\pi$
$239$	−1054.21	−0.285319	−0.142660	+	0.989772i	$0.545565\pi$
$240$	0	0
$241$	6842.40	1.82887	0.914435	−	0.404732i	$-0.132635\pi$
$241$	6842.40	1.82887	0.914435	+	0.404732i	$0.132635\pi$
$242$	−1584.44	−0.420876
$243$	243.000	0.0641500
$244$	260.609	0.0683761
$245$	0	0
$246$	935.403	0.242435
$247$	−1447.99	−0.373009
$248$	−3976.66	−1.01822
$249$	1265.57	0.322096
$250$	0	0
$251$	−6021.53	−1.51424	−0.757122	−	0.653273i	$-0.773395\pi$
$251$	−6021.53	−1.51424	−0.757122	+	0.653273i	$0.773395\pi$
$252$	−343.516	−0.0858710
$253$	−1649.33	−0.409851
$254$	2890.78	0.714110
$255$	0	0
$256$	−4069.00	−0.993408
$257$	−539.470	−0.130939	−0.0654693	−	0.997855i	$-0.520854\pi$
$257$	−539.470	−0.130939	−0.0654693	+	0.997855i	$0.520854\pi$
$258$	−2123.76	−0.512478
$259$	−1766.40	−0.423779
$260$	0	0
$261$	531.499	0.126050
$262$	−3964.28	−0.934787
$263$	−707.750	−0.165938	−0.0829690	−	0.996552i	$-0.526440\pi$
$263$	−707.750	−0.165938	−0.0829690	+	0.996552i	$0.526440\pi$
$264$	−1385.06	−0.322897
$265$	0	0
$266$	1301.73	0.300053
$267$	1275.41	0.292337
$268$	2748.83	0.626536
$269$	−2877.92	−0.652304	−0.326152	−	0.945317i	$-0.605752\pi$
$269$	−2877.92	−0.652304	−0.326152	+	0.945317i	$0.605752\pi$
$270$	0	0
$271$	3569.27	0.800065	0.400033	−	0.916501i	$-0.368999\pi$
$271$	3569.27	0.800065	0.400033	+	0.916501i	$0.368999\pi$
$272$	−72.6401	−0.0161928
$273$	−340.379	−0.0754604
$274$	1624.45	0.358163
$275$	0	0
$276$	1204.09	0.262600
$277$	6956.58	1.50895	0.754477	−	0.656327i	$-0.227891\pi$
$277$	6956.58	1.50895	0.754477	+	0.656327i	$0.227891\pi$
$278$	−4993.67	−1.07734
$279$	1586.86	0.340512
$280$	0	0
$281$	3329.41	0.706819	0.353409	−	0.935469i	$-0.385022\pi$
$281$	3329.41	0.706819	0.353409	+	0.935469i	$0.385022\pi$
$282$	2487.08	0.525190
$283$	−613.980	−0.128966	−0.0644829	−	0.997919i	$-0.520540\pi$
$283$	−613.980	−0.128966	−0.0644829	+	0.997919i	$0.520540\pi$
$284$	−1836.45	−0.383709
$285$	0	0
$286$	−526.645	−0.108885
$287$	1375.08	0.282816
$288$	1634.30	0.334383
$289$	6967.94	1.41827
$290$	0	0
$291$	1772.00	0.356963
$292$	2024.79	0.405793
$293$	−6494.46	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$293$	−6494.46	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$294$	−1481.78	−0.293944
$295$	0	0
$296$	5199.50	1.02100
$297$	552.700	0.107983
$298$	3952.81	0.768390
$299$	1193.09	0.230763
$300$	0	0
$301$	−3122.00	−0.597838
$302$	4162.99	0.793222
$303$	−4600.66	−0.872281
$304$	65.1663	0.0122946
$305$	0	0
$306$	−1704.38	−0.318409
$307$	−967.468	−0.179858	−0.0899289	−	0.995948i	$-0.528664\pi$
$307$	−967.468	−0.179858	−0.0899289	+	0.995948i	$0.528664\pi$
$308$	−781.323	−0.144546
$309$	−2156.69	−0.397055
$310$	0	0
$311$	−1530.33	−0.279026	−0.139513	−	0.990220i	$-0.544554\pi$
$311$	−1530.33	−0.279026	−0.139513	+	0.990220i	$0.544554\pi$
$312$	1001.93	0.181804
$313$	7983.61	1.44173	0.720863	−	0.693078i	$-0.243746\pi$
$313$	7983.61	1.44173	0.720863	+	0.693078i	$0.243746\pi$
$314$	1567.87	0.281783
$315$	0	0
$316$	3261.61	0.580632
$317$	−5281.05	−0.935689	−0.467844	−	0.883811i	$-0.654969\pi$
$317$	−5281.05	−0.935689	−0.467844	+	0.883811i	$0.654969\pi$
$318$	−2687.02	−0.473838
$319$	1208.89	0.212178
$320$	0	0
$321$	−2834.53	−0.492860
$322$	−1072.58	−0.185629
$323$	−10658.5	−1.83609
$324$	−403.497	−0.0691868
$325$	0	0
$326$	−6998.69	−1.18902
$327$	4417.00	0.746975
$328$	−4047.63	−0.681381
$329$	3656.11	0.612667
$330$	0	0
$331$	−2082.60	−0.345830	−0.172915	−	0.984937i	$-0.555319\pi$
$331$	−2082.60	−0.345830	−0.172915	+	0.984937i	$0.555319\pi$
$332$	−2101.45	−0.347386
$333$	−2074.83	−0.341441
$334$	−1706.67	−0.279595
$335$	0	0
$336$	15.3187	0.00248721
$337$	6924.52	1.11930	0.559648	−	0.828730i	$-0.310936\pi$
$337$	6924.52	1.11930	0.559648	+	0.828730i	$0.310936\pi$
$338$	−3436.10	−0.552956
$339$	−2460.76	−0.394249
$340$	0	0
$341$	3609.29	0.573179
$342$	1529.02	0.241755
$343$	−4806.39	−0.756620
$344$	9189.81	1.44035
$345$	0	0
$346$	2197.98	0.341515
$347$	−2159.18	−0.334037	−0.167019	−	0.985954i	$-0.553414\pi$
$347$	−2159.18	−0.334037	−0.167019	+	0.985954i	$0.553414\pi$
$348$	−882.546	−0.135947
$349$	−11809.8	−1.81136	−0.905679	−	0.423963i	$-0.860639\pi$
$349$	−11809.8	−1.81136	−0.905679	+	0.423963i	$0.860639\pi$
$350$	0	0
$351$	−399.813	−0.0607989
$352$	3717.20	0.562862
$353$	8267.86	1.24661	0.623306	−	0.781978i	$-0.285789\pi$
$353$	8267.86	1.24661	0.623306	+	0.781978i	$0.285789\pi$
$354$	−2978.48	−0.447187
$355$	0	0
$356$	−2117.80	−0.315290
$357$	−2505.51	−0.371444
$358$	−2556.77	−0.377457
$359$	4656.29	0.684539	0.342269	−	0.939602i	$-0.388804\pi$
$359$	4656.29	0.684539	0.342269	+	0.939602i	$0.388804\pi$
$360$	0	0
$361$	2702.90	0.394066
$362$	−6503.05	−0.944179
$363$	−2735.89	−0.395584
$364$	565.194	0.0813852
$365$	0	0
$366$	−272.680	−0.0389433
$367$	−519.600	−0.0739044	−0.0369522	−	0.999317i	$-0.511765\pi$
$367$	−519.600	−0.0739044	−0.0369522	+	0.999317i	$0.511765\pi$
$368$	−53.6947	−0.00760606
$369$	1615.18	0.227867
$370$	0	0
$371$	−3950.02	−0.552762
$372$	−2634.95	−0.367247
$373$	−4202.26	−0.583337	−0.291669	−	0.956519i	$-0.594210\pi$
$373$	−4202.26	−0.583337	−0.291669	+	0.956519i	$0.594210\pi$
$374$	−3876.59	−0.535973
$375$	0	0
$376$	−10762.0	−1.47608
$377$	−874.486	−0.119465
$378$	359.428	0.0489074
$379$	2855.16	0.386964	0.193482	−	0.981104i	$-0.438022\pi$
$379$	2855.16	0.386964	0.193482	+	0.981104i	$0.438022\pi$
$380$	0	0
$381$	4991.58	0.671197
$382$	6869.26	0.920057
$383$	−5756.86	−0.768046	−0.384023	−	0.923324i	$-0.625462\pi$
$383$	−5756.86	−0.768046	−0.384023	+	0.923324i	$0.625462\pi$
$384$	−2741.52	−0.364330
$385$	0	0
$386$	−8233.97	−1.08575
$387$	−3667.13	−0.481682
$388$	−2942.37	−0.384990
$389$	10994.1	1.43296	0.716481	−	0.697607i	$-0.245752\pi$
$389$	10994.1	1.43296	0.716481	+	0.697607i	$0.245752\pi$
$390$	0	0
$391$	8782.25	1.13590
$392$	6411.90	0.826148
$393$	−6845.21	−0.878613
$394$	−3322.93	−0.424890
$395$	0	0
$396$	−917.749	−0.116461
$397$	−163.743	−0.0207004	−0.0103502	−	0.999946i	$-0.503295\pi$
$397$	−163.743	−0.0207004	−0.0103502	+	0.999946i	$0.503295\pi$
$398$	−5719.19	−0.720294
$399$	2247.72	0.282022
$400$	0	0
$401$	8096.74	1.00831	0.504154	−	0.863614i	$-0.331804\pi$
$401$	8096.74	1.00831	0.504154	+	0.863614i	$0.331804\pi$
$402$	−2876.16	−0.356841
$403$	−2610.89	−0.322724
$404$	7639.32	0.940768
$405$	0	0
$406$	786.156	0.0960992
$407$	−4719.16	−0.574743
$408$	7375.12	0.894909
$409$	5493.78	0.664181	0.332091	−	0.943247i	$-0.392246\pi$
$409$	5493.78	0.664181	0.332091	+	0.943247i	$0.392246\pi$
$410$	0	0
$411$	2804.98	0.336640
$412$	3581.15	0.428229
$413$	−4378.48	−0.521673
$414$	−1259.86	−0.149562
$415$	0	0
$416$	−2688.95	−0.316915
$417$	−8622.67	−1.01260
$418$	3477.74	0.406942
$419$	11739.7	1.36879	0.684393	−	0.729114i	$-0.260067\pi$
$419$	11739.7	1.36879	0.684393	+	0.729114i	$0.260067\pi$
$420$	0	0
$421$	8506.36	0.984738	0.492369	−	0.870387i	$-0.336131\pi$
$421$	8506.36	0.984738	0.492369	+	0.870387i	$0.336131\pi$
$422$	1240.05	0.143044
$423$	4294.49	0.493630
$424$	11627.1	1.33175
$425$	0	0
$426$	1921.52	0.218539
$427$	−400.851	−0.0454298
$428$	4706.69	0.531557
$429$	−909.368	−0.102342
$430$	0	0
$431$	−15772.5	−1.76272	−0.881362	−	0.472441i	$-0.843373\pi$
$431$	−15772.5	−1.76272	−0.881362	+	0.472441i	$0.843373\pi$
$432$	17.9935	0.00200396
$433$	−5269.35	−0.584824	−0.292412	−	0.956292i	$-0.594458\pi$
$433$	−5269.35	−0.584824	−0.292412	+	0.956292i	$0.594458\pi$
$434$	2347.17	0.259603
$435$	0	0
$436$	−7334.35	−0.805624
$437$	−7878.67	−0.862443
$438$	−2118.58	−0.231117
$439$	6863.61	0.746201	0.373100	−	0.927791i	$-0.378295\pi$
$439$	6863.61	0.746201	0.373100	+	0.927791i	$0.378295\pi$
$440$	0	0
$441$	−2558.63	−0.276280
$442$	2804.25	0.301775
$443$	16108.2	1.72759	0.863795	−	0.503843i	$-0.168081\pi$
$443$	16108.2	1.72759	0.863795	+	0.503843i	$0.168081\pi$
$444$	3445.21	0.368249
$445$	0	0
$446$	−7881.92	−0.836816
$447$	6825.40	0.722215
$448$	2376.50	0.250623
$449$	12908.2	1.35674	0.678370	−	0.734720i	$-0.262687\pi$
$449$	12908.2	1.35674	0.678370	+	0.734720i	$0.262687\pi$
$450$	0	0
$451$	3673.70	0.383565
$452$	4086.06	0.425203
$453$	7188.32	0.745556
$454$	86.0883	0.00889939
$455$	0	0
$456$	−6616.31	−0.679467
$457$	10742.8	1.09962	0.549812	−	0.835288i	$-0.314699\pi$
$457$	10742.8	1.09962	0.549812	+	0.835288i	$0.314699\pi$
$458$	−4091.11	−0.417390
$459$	−2942.99	−0.299275
$460$	0	0
$461$	14421.4	1.45698	0.728492	−	0.685054i	$-0.240222\pi$
$461$	14421.4	1.45698	0.728492	+	0.685054i	$0.240222\pi$
$462$	817.515	0.0823252
$463$	11600.6	1.16442	0.582209	−	0.813039i	$-0.302188\pi$
$463$	11600.6	1.16442	0.582209	+	0.813039i	$0.302188\pi$
$464$	39.3560	0.00393762
$465$	0	0
$466$	−5036.78	−0.500696
$467$	−3093.46	−0.306527	−0.153264	−	0.988185i	$-0.548978\pi$
$467$	−3093.46	−0.306527	−0.153264	+	0.988185i	$0.548978\pi$
$468$	663.882	0.0655726
$469$	−4228.07	−0.416277
$470$	0	0
$471$	2707.27	0.264850
$472$	12888.3	1.25685
$473$	−8340.84	−0.810808
$474$	−3412.69	−0.330696
$475$	0	0
$476$	4160.35	0.400608
$477$	−4639.73	−0.445364
$478$	−1831.59	−0.175261
$479$	−17950.9	−1.71231	−0.856156	−	0.516717i	$-0.827154\pi$
$479$	−17950.9	−1.71231	−0.856156	+	0.516717i	$0.827154\pi$
$480$	0	0
$481$	3413.75	0.323604
$482$	11888.0	1.12341
$483$	−1852.04	−0.174474
$484$	4542.90	0.426644
$485$	0	0
$486$	422.188	0.0394050
$487$	17260.9	1.60609	0.803043	−	0.595921i	$-0.203213\pi$
$487$	17260.9	1.60609	0.803043	+	0.595921i	$0.203213\pi$
$488$	1179.93	0.109453
$489$	−12084.8	−1.11757
$490$	0	0
$491$	8189.74	0.752745	0.376372	−	0.926468i	$-0.377171\pi$
$491$	8189.74	0.752745	0.376372	+	0.926468i	$0.377171\pi$
$492$	−2681.98	−0.245758
$493$	−6437.03	−0.588051
$494$	−2515.73	−0.229126
$495$	0	0
$496$	117.502	0.0106371
$497$	2824.70	0.254940
$498$	2198.79	0.197852
$499$	−10653.8	−0.955767	−0.477884	−	0.878423i	$-0.658596\pi$
$499$	−10653.8	−0.955767	−0.477884	+	0.878423i	$0.658596\pi$
$500$	0	0
$501$	−2946.94	−0.262793
$502$	−10461.8	−0.930144
$503$	6067.02	0.537803	0.268902	−	0.963168i	$-0.413339\pi$
$503$	6067.02	0.537803	0.268902	+	0.963168i	$0.413339\pi$
$504$	−1555.30	−0.137457
$505$	0	0
$506$	−2865.54	−0.251756
$507$	−5933.18	−0.519727
$508$	−8288.43	−0.723897
$509$	−3141.27	−0.273545	−0.136773	−	0.990602i	$-0.543673\pi$
$509$	−3141.27	−0.273545	−0.136773	+	0.990602i	$0.543673\pi$
$510$	0	0
$511$	−3114.39	−0.269613
$512$	241.260	0.0208247
$513$	2640.19	0.227227
$514$	−937.275	−0.0804308
$515$	0	0
$516$	6089.21	0.519501
$517$	9767.76	0.830920
$518$	−3068.94	−0.260311
$519$	3795.30	0.320992
$520$	0	0
$521$	15548.5	1.30747	0.653737	−	0.756722i	$-0.273200\pi$
$521$	15548.5	1.30747	0.653737	+	0.756722i	$0.273200\pi$
$522$	923.426	0.0774277
$523$	19848.9	1.65952	0.829761	−	0.558119i	$-0.188477\pi$
$523$	19848.9	1.65952	0.829761	+	0.558119i	$0.188477\pi$
$524$	11366.3	0.947598
$525$	0	0
$526$	−1229.64	−0.101930
$527$	−19218.6	−1.58857
$528$	40.9259	0.00337324
$529$	−5675.26	−0.466447
$530$	0	0
$531$	−5143.00	−0.420315
$532$	−3732.30	−0.304165
$533$	−2657.49	−0.215963
$534$	2215.90	0.179572
$535$	0	0
$536$	12445.6	1.00292
$537$	−4414.83	−0.354775
$538$	−5000.09	−0.400686
$539$	−5819.56	−0.465058
$540$	0	0
$541$	−13213.7	−1.05010	−0.525048	−	0.851072i	$-0.675953\pi$
$541$	−13213.7	−1.05010	−0.525048	+	0.851072i	$0.675953\pi$
$542$	6201.24	0.491451
$543$	−11229.0	−0.887441
$544$	−19793.2	−1.55997
$545$	0	0
$546$	−591.374	−0.0463525
$547$	−6622.55	−0.517660	−0.258830	−	0.965923i	$-0.583337\pi$
$547$	−6622.55	−0.517660	−0.258830	+	0.965923i	$0.583337\pi$
$548$	−4657.61	−0.363072
$549$	−470.843	−0.0366031
$550$	0	0
$551$	5774.74	0.446483
$552$	5451.61	0.420355
$553$	−5016.78	−0.385778
$554$	12086.3	0.926894
$555$	0	0
$556$	14317.8	1.09210
$557$	−15042.5	−1.14430	−0.572148	−	0.820150i	$-0.693890\pi$
$557$	−15042.5	−1.14430	−0.572148	+	0.820150i	$0.693890\pi$
$558$	2757.00	0.209164
$559$	6033.61	0.456519
$560$	0	0
$561$	−6693.79	−0.503765
$562$	5784.51	0.434173
$563$	−3495.16	−0.261641	−0.130820	−	0.991406i	$-0.541761\pi$
$563$	−3495.16	−0.261641	−0.130820	+	0.991406i	$0.541761\pi$
$564$	−7130.93	−0.532387
$565$	0	0
$566$	−1066.73	−0.0792189
$567$	620.632	0.0459684
$568$	−8314.68	−0.614219
$569$	7467.34	0.550171	0.275085	−	0.961420i	$-0.411294\pi$
$569$	7467.34	0.550171	0.275085	+	0.961420i	$0.411294\pi$
$570$	0	0
$571$	−15504.9	−1.13635	−0.568177	−	0.822906i	$-0.692351\pi$
$571$	−15504.9	−1.13635	−0.568177	+	0.822906i	$0.692351\pi$
$572$	1509.99	0.110377
$573$	11861.3	0.864768
$574$	2389.06	0.173724
$575$	0	0
$576$	2791.45	0.201928
$577$	−25877.8	−1.86708	−0.933542	−	0.358468i	$-0.883299\pi$
$577$	−25877.8	−1.86708	−0.933542	+	0.358468i	$0.883299\pi$
$578$	12106.1	0.871188
$579$	−14217.8	−1.02050
$580$	0	0
$581$	3232.31	0.230807
$582$	3078.66	0.219269
$583$	−10553.0	−0.749675
$584$	9167.39	0.649571
$585$	0	0
$586$	−11283.5	−0.795418
$587$	−498.887	−0.0350788	−0.0175394	−	0.999846i	$-0.505583\pi$
$587$	−498.887	−0.0350788	−0.0175394	+	0.999846i	$0.505583\pi$
$588$	4248.56	0.297972
$589$	17241.2	1.20613
$590$	0	0
$591$	−5737.77	−0.399358
$592$	−153.635	−0.0106662
$593$	16095.6	1.11462	0.557309	−	0.830305i	$-0.311834\pi$
$593$	16095.6	1.11462	0.557309	+	0.830305i	$0.311834\pi$
$594$	960.260	0.0663299
$595$	0	0
$596$	−11333.5	−0.778920
$597$	−9875.44	−0.677010
$598$	2072.87	0.141749
$599$	−998.977	−0.0681421	−0.0340710	−	0.999419i	$-0.510847\pi$
$599$	−998.977	−0.0681421	−0.0340710	+	0.999419i	$0.510847\pi$
$600$	0	0
$601$	−12876.3	−0.873937	−0.436968	−	0.899477i	$-0.643948\pi$
$601$	−12876.3	−0.873937	−0.436968	+	0.899477i	$0.643948\pi$
$602$	−5424.16	−0.367230
$603$	−4966.33	−0.335397
$604$	−11936.1	−0.804093
$605$	0	0
$606$	−7993.18	−0.535810
$607$	−12236.9	−0.818254	−0.409127	−	0.912477i	$-0.634167\pi$
$607$	−12236.9	−0.818254	−0.409127	+	0.912477i	$0.634167\pi$
$608$	17756.7	1.18442
$609$	1357.47	0.0903244
$610$	0	0
$611$	−7065.81	−0.467843
$612$	4886.79	0.322772
$613$	−1839.73	−0.121217	−0.0606085	−	0.998162i	$-0.519304\pi$
$613$	−1839.73	−0.121217	−0.0606085	+	0.998162i	$0.519304\pi$
$614$	−1680.88	−0.110480
$615$	0	0
$616$	−3537.51	−0.231380
$617$	−3461.32	−0.225847	−0.112924	−	0.993604i	$-0.536022\pi$
$617$	−3461.32	−0.225847	−0.112924	+	0.993604i	$0.536022\pi$
$618$	−3747.03	−0.243896
$619$	−4899.75	−0.318155	−0.159077	−	0.987266i	$-0.550852\pi$
$619$	−4899.75	−0.318155	−0.159077	+	0.987266i	$0.550852\pi$
$620$	0	0
$621$	−2175.43	−0.140575
$622$	−2658.80	−0.171396
$623$	3257.46	0.209482
$624$	−29.6050	−0.00189928
$625$	0	0
$626$	13870.7	0.885599
$627$	6005.08	0.382488
$628$	−4495.38	−0.285645
$629$	25128.4	1.59290
$630$	0	0
$631$	−2129.24	−0.134332	−0.0671661	−	0.997742i	$-0.521396\pi$
$631$	−2129.24	−0.134332	−0.0671661	+	0.997742i	$0.521396\pi$
$632$	14767.2	0.929443
$633$	2141.22	0.134448
$634$	−9175.29	−0.574759
$635$	0	0
$636$	7704.19	0.480332
$637$	4209.76	0.261847
$638$	2100.32	0.130333
$639$	3317.92	0.205407
$640$	0	0
$641$	−20797.9	−1.28154	−0.640771	−	0.767732i	$-0.721385\pi$
$641$	−20797.9	−1.28154	−0.640771	+	0.767732i	$0.721385\pi$
$642$	−4924.71	−0.302746
$643$	6395.45	0.392243	0.196121	−	0.980580i	$-0.437165\pi$
$643$	6395.45	0.392243	0.196121	+	0.980580i	$0.437165\pi$
$644$	3075.29	0.188173
$645$	0	0
$646$	−18518.1	−1.12784
$647$	9964.69	0.605491	0.302745	−	0.953071i	$-0.402097\pi$
$647$	9964.69	0.605491	0.302745	+	0.953071i	$0.402097\pi$
$648$	−1826.87	−0.110750
$649$	−11697.7	−0.707510
$650$	0	0
$651$	4052.90	0.244003
$652$	20066.6	1.20532
$653$	29629.0	1.77561	0.887803	−	0.460223i	$-0.152231\pi$
$653$	29629.0	1.77561	0.887803	+	0.460223i	$0.152231\pi$
$654$	7674.09	0.458839
$655$	0	0
$656$	119.599	0.00711825
$657$	−3658.19	−0.217229
$658$	6352.11	0.376339
$659$	18698.2	1.10528	0.552640	−	0.833420i	$-0.313621\pi$
$659$	18698.2	1.10528	0.552640	+	0.833420i	$0.313621\pi$
$660$	0	0
$661$	16576.7	0.975430	0.487715	−	0.873003i	$-0.337831\pi$
$661$	16576.7	0.975430	0.487715	+	0.873003i	$0.337831\pi$
$662$	−3618.30	−0.212431
$663$	4842.16	0.283641
$664$	−9514.50	−0.556076
$665$	0	0
$666$	−3604.80	−0.209734
$667$	−4758.18	−0.276218
$668$	4893.34	0.283427
$669$	−13609.9	−0.786530
$670$	0	0
$671$	−1070.93	−0.0616135
$672$	4174.08	0.239611
$673$	−8889.49	−0.509160	−0.254580	−	0.967052i	$-0.581937\pi$
$673$	−8889.49	−0.509160	−0.254580	+	0.967052i	$0.581937\pi$
$674$	12030.7	0.687542
$675$	0	0
$676$	9851.95	0.560534
$677$	22803.3	1.29454	0.647270	−	0.762261i	$-0.275911\pi$
$677$	22803.3	1.29454	0.647270	+	0.762261i	$0.275911\pi$
$678$	−4275.33	−0.242172
$679$	4525.76	0.255792
$680$	0	0
$681$	148.650	0.00836460
$682$	6270.77	0.352082
$683$	−7599.47	−0.425748	−0.212874	−	0.977080i	$-0.568282\pi$
$683$	−7599.47	−0.425748	−0.212874	+	0.977080i	$0.568282\pi$
$684$	−4384.00	−0.245068
$685$	0	0
$686$	−8350.61	−0.464764
$687$	−7064.20	−0.392308
$688$	−271.541	−0.0150471
$689$	7633.83	0.422099
$690$	0	0
$691$	−12622.0	−0.694883	−0.347441	−	0.937702i	$-0.612949\pi$
$691$	−12622.0	−0.694883	−0.347441	+	0.937702i	$0.612949\pi$
$692$	−6302.03	−0.346195
$693$	1411.62	0.0773780
$694$	−3751.36	−0.205187
$695$	0	0
$696$	−3995.80	−0.217616
$697$	−19561.6	−1.06305
$698$	−20518.3	−1.11265
$699$	−8697.11	−0.470608
$700$	0	0
$701$	−4783.64	−0.257740	−0.128870	−	0.991662i	$-0.541135\pi$
$701$	−4783.64	−0.257740	−0.128870	+	0.991662i	$0.541135\pi$
$702$	−694.634	−0.0373465
$703$	−22543.0	−1.20942
$704$	6349.12	0.339903
$705$	0	0
$706$	14364.6	0.765747
$707$	−11750.3	−0.625056
$708$	8539.86	0.453316
$709$	−33251.4	−1.76133	−0.880664	−	0.473741i	$-0.842903\pi$
$709$	−33251.4	−1.76133	−0.880664	+	0.473741i	$0.842903\pi$
$710$	0	0
$711$	−5892.75	−0.310823
$712$	−9588.53	−0.504699
$713$	−14206.1	−0.746177
$714$	−4353.06	−0.228164
$715$	0	0
$716$	7330.76	0.382630
$717$	−3162.64	−0.164729
$718$	8089.82	0.420487
$719$	12428.5	0.644652	0.322326	−	0.946629i	$-0.395535\pi$
$719$	12428.5	0.644652	0.322326	+	0.946629i	$0.395535\pi$
$720$	0	0
$721$	−5508.28	−0.284520
$722$	4696.01	0.242060
$723$	20527.2	1.05590
$724$	18645.5	0.957119
$725$	0	0
$726$	−4753.33	−0.242993
$727$	8782.11	0.448020	0.224010	−	0.974587i	$-0.428085\pi$
$727$	8782.11	0.448020	0.224010	+	0.974587i	$0.428085\pi$
$728$	2558.97	0.130277
$729$	729.000	0.0370370
$730$	0	0
$731$	44412.9	2.24716
$732$	781.827	0.0394770
$733$	26084.3	1.31438	0.657192	−	0.753723i	$-0.271744\pi$
$733$	26084.3	1.31438	0.657192	+	0.753723i	$0.271744\pi$
$734$	−902.753	−0.0453967
$735$	0	0
$736$	−14630.9	−0.732748
$737$	−11295.8	−0.564569
$738$	2806.21	0.139970
$739$	−23385.1	−1.16405	−0.582027	−	0.813170i	$-0.697740\pi$
$739$	−23385.1	−1.16405	−0.582027	+	0.813170i	$0.697740\pi$
$740$	0	0
$741$	−4343.96	−0.215357
$742$	−6862.75	−0.339541
$743$	26389.6	1.30302	0.651508	−	0.758642i	$-0.274137\pi$
$743$	26389.6	1.30302	0.651508	+	0.758642i	$0.274137\pi$
$744$	−11930.0	−0.587868
$745$	0	0
$746$	−7301.00	−0.358322
$747$	3796.70	0.185962
$748$	11114.9	0.543318
$749$	−7239.52	−0.353172
$750$	0	0
$751$	17725.3	0.861257	0.430629	−	0.902529i	$-0.358292\pi$
$751$	17725.3	0.861257	0.430629	+	0.902529i	$0.358292\pi$
$752$	317.995	0.0154203
$753$	−18064.6	−0.874250
$754$	−1519.33	−0.0733830
$755$	0	0
$756$	−1030.55	−0.0495777
$757$	−4478.94	−0.215046	−0.107523	−	0.994203i	$-0.534292\pi$
$757$	−4478.94	−0.215046	−0.107523	+	0.994203i	$0.534292\pi$
$758$	4960.54	0.237698
$759$	−4947.98	−0.236627
$760$	0	0
$761$	1368.09	0.0651683	0.0325841	−	0.999469i	$-0.489626\pi$
$761$	1368.09	0.0651683	0.0325841	+	0.999469i	$0.489626\pi$
$762$	8672.35	0.412292
$763$	11281.2	0.535265
$764$	−19695.5	−0.932666
$765$	0	0
$766$	−10001.9	−0.471782
$767$	8461.88	0.398358
$768$	−12207.0	−0.573544
$769$	16530.6	0.775176	0.387588	−	0.921833i	$-0.373308\pi$
$769$	16530.6	0.775176	0.387588	+	0.921833i	$0.373308\pi$
$770$	0	0
$771$	−1618.41	−0.0755975
$772$	23608.4	1.10063
$773$	−8002.04	−0.372333	−0.186166	−	0.982518i	$-0.559606\pi$
$773$	−8002.04	−0.372333	−0.186166	+	0.982518i	$0.559606\pi$
$774$	−6371.27	−0.295879
$775$	0	0
$776$	−13321.8	−0.616271
$777$	−5299.19	−0.244669
$778$	19101.1	0.880215
$779$	17548.9	0.807131
$780$	0	0
$781$	7546.56	0.345758
$782$	15258.3	0.697742
$783$	1594.50	0.0727749
$784$	−189.459	−0.00863061
$785$	0	0
$786$	−11892.8	−0.539699
$787$	−19432.4	−0.880166	−0.440083	−	0.897957i	$-0.645051\pi$
$787$	−19432.4	−0.880166	−0.440083	+	0.897957i	$0.645051\pi$
$788$	9527.47	0.430713
$789$	−2123.25	−0.0958044
$790$	0	0
$791$	−6284.89	−0.282510
$792$	−4155.19	−0.186424
$793$	774.687	0.0346910
$794$	−284.487	−0.0127155
$795$	0	0
$796$	16398.0	0.730165
$797$	−39528.1	−1.75678	−0.878391	−	0.477942i	$-0.841383\pi$
$797$	−39528.1	−1.75678	−0.878391	+	0.477942i	$0.841383\pi$
$798$	3905.19	0.173236
$799$	−52010.9	−2.30290
$800$	0	0
$801$	3826.24	0.168781
$802$	14067.3	0.619366
$803$	−8320.49	−0.365659
$804$	8246.50	0.361731
$805$	0	0
$806$	−4536.15	−0.198237
$807$	−8633.76	−0.376608
$808$	34587.7	1.50593
$809$	−2308.05	−0.100305	−0.0501526	−	0.998742i	$-0.515971\pi$
$809$	−2308.05	−0.100305	−0.0501526	+	0.998742i	$0.515971\pi$
$810$	0	0
$811$	−4243.29	−0.183726	−0.0918632	−	0.995772i	$-0.529282\pi$
$811$	−4243.29	−0.183726	−0.0918632	+	0.995772i	$0.529282\pi$
$812$	−2254.06	−0.0974162
$813$	10707.8	0.461918
$814$	−8199.07	−0.353043
$815$	0	0
$816$	−217.920	−0.00934894
$817$	−39843.4	−1.70617
$818$	9544.89	0.407982
$819$	−1021.14	−0.0435671
$820$	0	0
$821$	28705.4	1.22025	0.610124	−	0.792306i	$-0.291120\pi$
$821$	28705.4	1.22025	0.610124	+	0.792306i	$0.291120\pi$
$822$	4873.36	0.206786
$823$	28010.8	1.18639	0.593193	−	0.805061i	$-0.297867\pi$
$823$	28010.8	1.18639	0.593193	+	0.805061i	$0.297867\pi$
$824$	16214.0	0.685486
$825$	0	0
$826$	−7607.16	−0.320444
$827$	8868.42	0.372896	0.186448	−	0.982465i	$-0.440302\pi$
$827$	8868.42	0.372896	0.186448	+	0.982465i	$0.440302\pi$
$828$	3612.26	0.151612
$829$	−24216.6	−1.01457	−0.507283	−	0.861779i	$-0.669350\pi$
$829$	−24216.6	−1.01457	−0.507283	+	0.861779i	$0.669350\pi$
$830$	0	0
$831$	20869.7	0.871195
$832$	−4592.83	−0.191380
$833$	30987.7	1.28891
$834$	−14981.0	−0.622003
$835$	0	0
$836$	−9971.34	−0.412519
$837$	4760.58	0.196594
$838$	20396.5	0.840794
$839$	33351.0	1.37235	0.686177	−	0.727435i	$-0.259288\pi$
$839$	33351.0	1.37235	0.686177	+	0.727435i	$0.259288\pi$
$840$	0	0
$841$	−20901.4	−0.857003
$842$	14778.9	0.604888
$843$	9988.24	0.408082
$844$	−3555.45	−0.145004
$845$	0	0
$846$	7461.24	0.303218
$847$	−6987.58	−0.283467
$848$	−343.559	−0.0139126
$849$	−1841.94	−0.0744585
$850$	0	0
$851$	18574.6	0.748214
$852$	−5509.35	−0.221534
$853$	−25395.6	−1.01938	−0.509688	−	0.860359i	$-0.670239\pi$
$853$	−25395.6	−1.01938	−0.509688	+	0.860359i	$0.670239\pi$
$854$	−696.437	−0.0279059
$855$	0	0
$856$	21310.0	0.850887
$857$	28013.4	1.11659	0.558296	−	0.829642i	$-0.311455\pi$
$857$	28013.4	1.11659	0.558296	+	0.829642i	$0.311455\pi$
$858$	−1579.93	−0.0628649
$859$	7094.28	0.281786	0.140893	−	0.990025i	$-0.455003\pi$
$859$	7094.28	0.281786	0.140893	+	0.990025i	$0.455003\pi$
$860$	0	0
$861$	4125.23	0.163284
$862$	−27403.1	−1.08278
$863$	−12168.3	−0.479969	−0.239984	−	0.970777i	$-0.577142\pi$
$863$	−12168.3	−0.479969	−0.239984	+	0.970777i	$0.577142\pi$
$864$	4902.91	0.193056
$865$	0	0
$866$	−9154.96	−0.359236
$867$	20903.8	0.818836
$868$	−6729.78	−0.263161
$869$	−13403.0	−0.523205
$870$	0	0
$871$	8171.20	0.317876
$872$	−33206.9	−1.28960
$873$	5315.99	0.206093
$874$	−13688.4	−0.529767
$875$	0	0
$876$	6074.36	0.234285
$877$	−5576.23	−0.214704	−0.107352	−	0.994221i	$-0.534237\pi$
$877$	−5576.23	−0.214704	−0.107352	+	0.994221i	$0.534237\pi$
$878$	11924.8	0.458364
$879$	−19483.4	−0.747620
$880$	0	0
$881$	44427.2	1.69897	0.849483	−	0.527616i	$-0.176914\pi$
$881$	44427.2	1.69897	0.849483	+	0.527616i	$0.176914\pi$
$882$	−4445.35	−0.169708
$883$	28781.0	1.09689	0.548447	−	0.836186i	$-0.315219\pi$
$883$	28781.0	1.09689	0.548447	+	0.836186i	$0.315219\pi$
$884$	−8040.33	−0.305911
$885$	0	0
$886$	27986.3	1.06119
$887$	13229.3	0.500785	0.250392	−	0.968144i	$-0.419440\pi$
$887$	13229.3	0.500785	0.250392	+	0.968144i	$0.419440\pi$
$888$	15598.5	0.589472
$889$	12748.7	0.480965
$890$	0	0
$891$	1658.10	0.0623439
$892$	22599.0	0.848284
$893$	46659.6	1.74849
$894$	11858.4	0.443630
$895$	0	0
$896$	−7001.96	−0.261071
$897$	3579.27	0.133231
$898$	22426.7	0.833395
$899$	10412.5	0.386293
$900$	0	0
$901$	56192.1	2.07772
$902$	6382.68	0.235610
$903$	−9366.01	−0.345162
$904$	18500.0	0.680642
$905$	0	0
$906$	12489.0	0.457967
$907$	−18849.7	−0.690071	−0.345035	−	0.938590i	$-0.612133\pi$
$907$	−18849.7	−0.690071	−0.345035	+	0.938590i	$0.612133\pi$
$908$	−246.831	−0.00902135
$909$	−13802.0	−0.503611
$910$	0	0
$911$	−20189.5	−0.734258	−0.367129	−	0.930170i	$-0.619659\pi$
$911$	−20189.5	−0.734258	−0.367129	+	0.930170i	$0.619659\pi$
$912$	195.499	0.00709826
$913$	8635.54	0.313028
$914$	18664.6	0.675459
$915$	0	0
$916$	11730.0	0.423111
$917$	−17482.9	−0.629594
$918$	−5113.15	−0.183833
$919$	−9727.34	−0.349157	−0.174579	−	0.984643i	$-0.555856\pi$
$919$	−9727.34	−0.349157	−0.174579	+	0.984643i	$0.555856\pi$
$920$	0	0
$921$	−2902.41	−0.103841
$922$	25055.6	0.894971
$923$	−5459.04	−0.194676
$924$	−2343.97	−0.0834534
$925$	0	0
$926$	20154.9	0.715259
$927$	−6470.07	−0.229240
$928$	10723.8	0.379340
$929$	−39516.5	−1.39558	−0.697790	−	0.716303i	$-0.745833\pi$
$929$	−39516.5	−1.39558	−0.697790	+	0.716303i	$0.745833\pi$
$930$	0	0
$931$	−27799.5	−0.978615
$932$	14441.4	0.507558
$933$	−4591.00	−0.161096
$934$	−5374.57	−0.188288
$935$	0	0
$936$	3005.78	0.104965
$937$	32692.4	1.13982	0.569912	−	0.821706i	$-0.306977\pi$
$937$	32692.4	1.13982	0.569912	+	0.821706i	$0.306977\pi$
$938$	−7345.84	−0.255704
$939$	23950.8	0.832381
$940$	0	0
$941$	−11482.8	−0.397797	−0.198899	−	0.980020i	$-0.563736\pi$
$941$	−11482.8	−0.397797	−0.198899	+	0.980020i	$0.563736\pi$
$942$	4703.61	0.162688
$943$	−14459.7	−0.499334
$944$	−380.825	−0.0131301
$945$	0	0
$946$	−14491.4	−0.498049
$947$	43381.4	1.48860	0.744300	−	0.667845i	$-0.232783\pi$
$947$	43381.4	1.48860	0.744300	+	0.667845i	$0.232783\pi$
$948$	9784.82	0.335228
$949$	6018.89	0.205881
$950$	0	0
$951$	−15843.2	−0.540220
$952$	18836.4	0.641271
$953$	23222.7	0.789358	0.394679	−	0.918819i	$-0.370856\pi$
$953$	23222.7	0.789358	0.394679	+	0.918819i	$0.370856\pi$
$954$	−8061.05	−0.273570
$955$	0	0
$956$	5251.51	0.177663
$957$	3626.66	0.122501
$958$	−31187.8	−1.05181
$959$	7164.03	0.241229
$960$	0	0
$961$	1296.90	0.0435333
$962$	5931.05	0.198778
$963$	−8503.60	−0.284553
$964$	−34085.1	−1.13880
$965$	0	0
$966$	−3217.74	−0.107173
$967$	16334.6	0.543211	0.271605	−	0.962409i	$-0.412445\pi$
$967$	16334.6	0.543211	0.271605	+	0.962409i	$0.412445\pi$
$968$	20568.4	0.682947
$969$	−31975.6	−1.06007
$970$	0	0
$971$	−39066.5	−1.29115	−0.645574	−	0.763698i	$-0.723382\pi$
$971$	−39066.5	−1.29115	−0.645574	+	0.763698i	$0.723382\pi$
$972$	−1210.49	−0.0399450
$973$	−22022.7	−0.725606
$974$	29989.0	0.986559
$975$	0	0
$976$	−34.8646	−0.00114343
$977$	−39942.7	−1.30796	−0.653982	−	0.756510i	$-0.726903\pi$
$977$	−39942.7	−1.30796	−0.653982	+	0.756510i	$0.726903\pi$
$978$	−20996.1	−0.686483
$979$	8702.73	0.284107
$980$	0	0
$981$	13251.0	0.431266
$982$	14228.8	0.462383
$983$	−23347.1	−0.757534	−0.378767	−	0.925492i	$-0.623652\pi$
$983$	−23347.1	−0.757534	−0.378767	+	0.925492i	$0.623652\pi$
$984$	−12142.9	−0.393395
$985$	0	0
$986$	−11183.7	−0.361218
$987$	10968.3	0.353724
$988$	7213.08	0.232266
$989$	32829.5	1.05553
$990$	0	0
$991$	1661.91	0.0532717	0.0266359	−	0.999645i	$-0.491521\pi$
$991$	1661.91	0.0532717	0.0266359	+	0.999645i	$0.491521\pi$
$992$	32017.4	1.02475
$993$	−6247.79	−0.199665
$994$	4907.63	0.156600
$995$	0	0
$996$	−6304.36	−0.200563
$997$	4547.81	0.144464	0.0722320	−	0.997388i	$-0.476988\pi$
$997$	4547.81	0.144464	0.0722320	+	0.997388i	$0.476988\pi$
$998$	−18509.8	−0.587092
$999$	−6224.48	−0.197131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.g.1.10		14
5.4	even	2		1875.4.a.f.1.5		14
25.11	even	5		75.4.g.b.46.5	yes	28
25.16	even	5		75.4.g.b.31.5	✓	28
75.11	odd	10		225.4.h.a.46.3		28
75.41	odd	10		225.4.h.a.181.3		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.5	✓	28	25.16	even	5
75.4.g.b.46.5	yes	28	25.11	even	5
225.4.h.a.46.3		28	75.11	odd	10
225.4.h.a.181.3		28	75.41	odd	10
1875.4.a.f.1.5		14	5.4	even	2
1875.4.a.g.1.10		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73740 q^{2} +3.00000 q^{3} -4.98145 q^{4} +5.21219 q^{6} +7.66213 q^{7} -22.5539 q^{8} +9.00000 q^{9} +20.4704 q^{11} -14.9443 q^{12} -14.8079 q^{13} +13.3122 q^{14} +0.666425 q^{16} -109.000 q^{17} +15.6366 q^{18} +97.7849 q^{19} +22.9864 q^{21} +35.5652 q^{22} -80.5714 q^{23} -67.6618 q^{24} -25.7272 q^{26} +27.0000 q^{27} -38.1685 q^{28} +59.0555 q^{29} +176.318 q^{31} +181.589 q^{32} +61.4111 q^{33} -189.376 q^{34} -44.8330 q^{36} -230.536 q^{37} +169.891 q^{38} -44.4236 q^{39} +179.464 q^{41} +39.9365 q^{42} -407.459 q^{43} -101.972 q^{44} -139.985 q^{46} +477.166 q^{47} +1.99927 q^{48} -284.292 q^{49} -326.999 q^{51} +73.7647 q^{52} -515.525 q^{53} +46.9097 q^{54} -172.811 q^{56} +293.355 q^{57} +102.603 q^{58} -571.444 q^{59} -52.3159 q^{61} +306.334 q^{62} +68.9591 q^{63} +310.162 q^{64} +106.696 q^{66} -551.814 q^{67} +542.976 q^{68} -241.714 q^{69} +368.658 q^{71} -202.985 q^{72} -406.465 q^{73} -400.533 q^{74} -487.111 q^{76} +156.847 q^{77} -77.1815 q^{78} -654.750 q^{79} +81.0000 q^{81} +311.801 q^{82} +421.856 q^{83} -114.505 q^{84} -707.919 q^{86} +177.166 q^{87} -461.688 q^{88} +425.138 q^{89} -113.460 q^{91} +401.362 q^{92} +528.953 q^{93} +829.027 q^{94} +544.768 q^{96} +590.666 q^{97} -493.928 q^{98} +184.233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26}+ \cdots - 297 q^{99}+O(q^{100}) \) 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9 - 33 * q^11 + 150 * q^12 - 188 * q^13 - 287 * q^14 + 82 * q^16 - 146 * q^17 - 184 * q^19 - 81 * q^21 - 277 * q^22 - 164 * q^23 + 63 * q^24 + 103 * q^26 + 378 * q^27 + 224 * q^28 + 252 * q^29 - 889 * q^31 - 422 * q^32 - 99 * q^33 - 230 * q^34 + 450 * q^36 - 642 * q^37 - 1833 * q^38 - 564 * q^39 - 164 * q^41 - 861 * q^42 - 696 * q^43 - 6 * q^44 - 419 * q^46 + 92 * q^47 + 246 * q^48 + 81 * q^49 - 438 * q^51 - 1956 * q^52 - 949 * q^53 - 2380 * q^56 - 552 * q^57 - 1041 * q^58 - 81 * q^59 - 496 * q^61 - 1454 * q^62 - 243 * q^63 - 3903 * q^64 - 831 * q^66 - 1926 * q^67 - 685 * q^68 - 492 * q^69 + 2498 * q^71 + 189 * q^72 + 1026 * q^73 - 707 * q^74 - 5704 * q^76 - 5434 * q^77 + 309 * q^78 - 695 * q^79 + 1134 * q^81 + 886 * q^82 - 5315 * q^83 + 672 * q^84 + 3997 * q^86 + 756 * q^87 - 2969 * q^88 - 1424 * q^89 - 1194 * q^91 - 1607 * q^92 - 2667 * q^93 - 629 * q^94 - 1266 * q^96 - 291 * q^97 + 1018 * q^98 - 297 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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