# Properties

 Label 2025.4.a.r.1.3 Level $2025$ Weight $4$ Character 2025.1 Self dual yes Analytic conductor $119.479$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$119.478867762$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.7032.1 Defining polynomial: $$x^{3} - x^{2} - 14x + 18$$ x^3 - x^2 - 14*x + 18 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 405) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.52348$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} +O(q^{10})$$ $$q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} -71.3561 q^{11} +51.3935 q^{13} -89.6107 q^{14} -79.8279 q^{16} +33.3191 q^{17} +113.372 q^{19} -251.422 q^{22} -81.9767 q^{23} +181.084 q^{26} -112.281 q^{28} -246.827 q^{29} +222.679 q^{31} -180.215 q^{32} +117.399 q^{34} -22.3910 q^{37} +399.463 q^{38} -434.225 q^{41} +236.850 q^{43} -315.029 q^{44} -288.843 q^{46} +107.984 q^{47} +303.810 q^{49} +226.896 q^{52} +123.961 q^{53} +321.265 q^{56} -869.690 q^{58} +171.091 q^{59} -79.4391 q^{61} +784.604 q^{62} +3.63945 q^{64} +611.506 q^{67} +147.100 q^{68} +511.102 q^{71} +410.012 q^{73} -78.8941 q^{74} +500.524 q^{76} +1814.76 q^{77} -793.597 q^{79} -1529.98 q^{82} -270.081 q^{83} +834.534 q^{86} +901.376 q^{88} +177.400 q^{89} -1307.06 q^{91} -361.918 q^{92} +380.478 q^{94} +881.860 q^{97} +1070.47 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 $$3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 - 58 * q^11 + 47 * q^13 - 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 260 * q^22 - 51 * q^23 - 253 * q^26 - 83 * q^28 - 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 414 * q^37 + 397 * q^38 - 179 * q^41 + 836 * q^43 - 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 1335 * q^52 + 505 * q^53 - 15 * q^56 - 1876 * q^58 - 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 40 * q^67 + 830 * q^68 - 452 * q^71 + 710 * q^73 - 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 - 613 * q^82 + 1734 * q^83 - 460 * q^86 + 768 * q^88 + 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 38 * q^98

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.52348 1.24574 0.622868 0.782327i $$-0.285967\pi$$
0.622868 + 0.782327i $$0.285967\pi$$
$$3$$ 0 0
$$4$$ 4.41489 0.551861
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −25.4325 −1.37322 −0.686612 0.727024i $$-0.740903\pi$$
−0.686612 + 0.727024i $$0.740903\pi$$
$$8$$ −12.6321 −0.558264
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −71.3561 −1.95588 −0.977940 0.208885i $$-0.933017\pi$$
−0.977940 + 0.208885i $$0.933017\pi$$
$$12$$ 0 0
$$13$$ 51.3935 1.09646 0.548231 0.836327i $$-0.315302\pi$$
0.548231 + 0.836327i $$0.315302\pi$$
$$14$$ −89.6107 −1.71068
$$15$$ 0 0
$$16$$ −79.8279 −1.24731
$$17$$ 33.3191 0.475357 0.237678 0.971344i $$-0.423614\pi$$
0.237678 + 0.971344i $$0.423614\pi$$
$$18$$ 0 0
$$19$$ 113.372 1.36891 0.684455 0.729055i $$-0.260040\pi$$
0.684455 + 0.729055i $$0.260040\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −251.422 −2.43651
$$23$$ −81.9767 −0.743188 −0.371594 0.928395i $$-0.621189\pi$$
−0.371594 + 0.928395i $$0.621189\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 181.084 1.36590
$$27$$ 0 0
$$28$$ −112.281 −0.757828
$$29$$ −246.827 −1.58051 −0.790253 0.612781i $$-0.790051\pi$$
−0.790253 + 0.612781i $$0.790051\pi$$
$$30$$ 0 0
$$31$$ 222.679 1.29014 0.645070 0.764124i $$-0.276828\pi$$
0.645070 + 0.764124i $$0.276828\pi$$
$$32$$ −180.215 −0.995557
$$33$$ 0 0
$$34$$ 117.399 0.592169
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −22.3910 −0.0994880 −0.0497440 0.998762i $$-0.515841\pi$$
−0.0497440 + 0.998762i $$0.515841\pi$$
$$38$$ 399.463 1.70530
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −434.225 −1.65401 −0.827007 0.562191i $$-0.809959\pi$$
−0.827007 + 0.562191i $$0.809959\pi$$
$$42$$ 0 0
$$43$$ 236.850 0.839982 0.419991 0.907528i $$-0.362033\pi$$
0.419991 + 0.907528i $$0.362033\pi$$
$$44$$ −315.029 −1.07937
$$45$$ 0 0
$$46$$ −288.843 −0.925817
$$47$$ 107.984 0.335129 0.167564 0.985861i $$-0.446410\pi$$
0.167564 + 0.985861i $$0.446410\pi$$
$$48$$ 0 0
$$49$$ 303.810 0.885744
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 226.896 0.605094
$$53$$ 123.961 0.321272 0.160636 0.987014i $$-0.448646\pi$$
0.160636 + 0.987014i $$0.448646\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 321.265 0.766621
$$57$$ 0 0
$$58$$ −869.690 −1.96889
$$59$$ 171.091 0.377528 0.188764 0.982023i $$-0.439552\pi$$
0.188764 + 0.982023i $$0.439552\pi$$
$$60$$ 0 0
$$61$$ −79.4391 −0.166740 −0.0833700 0.996519i $$-0.526568\pi$$
−0.0833700 + 0.996519i $$0.526568\pi$$
$$62$$ 784.604 1.60717
$$63$$ 0 0
$$64$$ 3.63945 0.00710830
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 611.506 1.11503 0.557517 0.830165i $$-0.311754\pi$$
0.557517 + 0.830165i $$0.311754\pi$$
$$68$$ 147.100 0.262331
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 511.102 0.854318 0.427159 0.904176i $$-0.359514\pi$$
0.427159 + 0.904176i $$0.359514\pi$$
$$72$$ 0 0
$$73$$ 410.012 0.657373 0.328686 0.944439i $$-0.393394\pi$$
0.328686 + 0.944439i $$0.393394\pi$$
$$74$$ −78.8941 −0.123936
$$75$$ 0 0
$$76$$ 500.524 0.755447
$$77$$ 1814.76 2.68586
$$78$$ 0 0
$$79$$ −793.597 −1.13021 −0.565105 0.825019i $$-0.691164\pi$$
−0.565105 + 0.825019i $$0.691164\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1529.98 −2.06047
$$83$$ −270.081 −0.357171 −0.178586 0.983924i $$-0.557152\pi$$
−0.178586 + 0.983924i $$0.557152\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 834.534 1.04640
$$87$$ 0 0
$$88$$ 901.376 1.09190
$$89$$ 177.400 0.211284 0.105642 0.994404i $$-0.466310\pi$$
0.105642 + 0.994404i $$0.466310\pi$$
$$90$$ 0 0
$$91$$ −1307.06 −1.50569
$$92$$ −361.918 −0.410136
$$93$$ 0 0
$$94$$ 380.478 0.417482
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 881.860 0.923086 0.461543 0.887118i $$-0.347296\pi$$
0.461543 + 0.887118i $$0.347296\pi$$
$$98$$ 1070.47 1.10340
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1026.03 1.01083 0.505417 0.862875i $$-0.331339\pi$$
0.505417 + 0.862875i $$0.331339\pi$$
$$102$$ 0 0
$$103$$ 1790.28 1.71263 0.856317 0.516451i $$-0.172747\pi$$
0.856317 + 0.516451i $$0.172747\pi$$
$$104$$ −649.206 −0.612115
$$105$$ 0 0
$$106$$ 436.775 0.400220
$$107$$ 2007.30 1.81358 0.906788 0.421587i $$-0.138527\pi$$
0.906788 + 0.421587i $$0.138527\pi$$
$$108$$ 0 0
$$109$$ −211.654 −0.185989 −0.0929945 0.995667i $$-0.529644\pi$$
−0.0929945 + 0.995667i $$0.529644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2030.22 1.71284
$$113$$ −1.94491 −0.00161913 −0.000809567 1.00000i $$-0.500258\pi$$
−0.000809567 1.00000i $$0.500258\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1089.71 −0.872219
$$117$$ 0 0
$$118$$ 602.835 0.470300
$$119$$ −847.386 −0.652771
$$120$$ 0 0
$$121$$ 3760.70 2.82547
$$122$$ −279.902 −0.207714
$$123$$ 0 0
$$124$$ 983.102 0.711977
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1178.02 0.823089 0.411544 0.911390i $$-0.364989\pi$$
0.411544 + 0.911390i $$0.364989\pi$$
$$128$$ 1454.54 1.00441
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 521.218 0.347626 0.173813 0.984779i $$-0.444391\pi$$
0.173813 + 0.984779i $$0.444391\pi$$
$$132$$ 0 0
$$133$$ −2883.32 −1.87982
$$134$$ 2154.63 1.38904
$$135$$ 0 0
$$136$$ −420.889 −0.265374
$$137$$ 449.532 0.280336 0.140168 0.990128i $$-0.455236\pi$$
0.140168 + 0.990128i $$0.455236\pi$$
$$138$$ 0 0
$$139$$ −2345.45 −1.43121 −0.715605 0.698505i $$-0.753849\pi$$
−0.715605 + 0.698505i $$0.753849\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1800.85 1.06426
$$143$$ −3667.24 −2.14455
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 1444.67 0.818914
$$147$$ 0 0
$$148$$ −98.8537 −0.0549035
$$149$$ −1269.38 −0.697932 −0.348966 0.937135i $$-0.613467\pi$$
−0.348966 + 0.937135i $$0.613467\pi$$
$$150$$ 0 0
$$151$$ 2158.32 1.16319 0.581594 0.813480i $$-0.302429\pi$$
0.581594 + 0.813480i $$0.302429\pi$$
$$152$$ −1432.12 −0.764213
$$153$$ 0 0
$$154$$ 6394.27 3.34588
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2343.31 1.19119 0.595593 0.803286i $$-0.296917\pi$$
0.595593 + 0.803286i $$0.296917\pi$$
$$158$$ −2796.22 −1.40794
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2084.87 1.02056
$$162$$ 0 0
$$163$$ −3399.94 −1.63376 −0.816882 0.576804i $$-0.804300\pi$$
−0.816882 + 0.576804i $$0.804300\pi$$
$$164$$ −1917.06 −0.912786
$$165$$ 0 0
$$166$$ −951.623 −0.444942
$$167$$ 1809.22 0.838332 0.419166 0.907910i $$-0.362323\pi$$
0.419166 + 0.907910i $$0.362323\pi$$
$$168$$ 0 0
$$169$$ 444.292 0.202227
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1045.66 0.463553
$$173$$ 1327.41 0.583358 0.291679 0.956516i $$-0.405786\pi$$
0.291679 + 0.956516i $$0.405786\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5696.21 2.43959
$$177$$ 0 0
$$178$$ 625.063 0.263205
$$179$$ −448.734 −0.187374 −0.0936871 0.995602i $$-0.529865\pi$$
−0.0936871 + 0.995602i $$0.529865\pi$$
$$180$$ 0 0
$$181$$ −3450.55 −1.41700 −0.708502 0.705709i $$-0.750629\pi$$
−0.708502 + 0.705709i $$0.750629\pi$$
$$182$$ −4605.41 −1.87569
$$183$$ 0 0
$$184$$ 1035.54 0.414895
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2377.52 −0.929741
$$188$$ 476.736 0.184944
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1592.09 −0.603141 −0.301570 0.953444i $$-0.597511\pi$$
−0.301570 + 0.953444i $$0.597511\pi$$
$$192$$ 0 0
$$193$$ −443.299 −0.165334 −0.0826668 0.996577i $$-0.526344\pi$$
−0.0826668 + 0.996577i $$0.526344\pi$$
$$194$$ 3107.21 1.14992
$$195$$ 0 0
$$196$$ 1341.29 0.488807
$$197$$ 208.992 0.0755839 0.0377919 0.999286i $$-0.487968\pi$$
0.0377919 + 0.999286i $$0.487968\pi$$
$$198$$ 0 0
$$199$$ −479.916 −0.170957 −0.0854783 0.996340i $$-0.527242\pi$$
−0.0854783 + 0.996340i $$0.527242\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3615.21 1.25923
$$203$$ 6277.43 2.17039
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6308.00 2.13349
$$207$$ 0 0
$$208$$ −4102.63 −1.36763
$$209$$ −8089.78 −2.67742
$$210$$ 0 0
$$211$$ −2632.14 −0.858786 −0.429393 0.903118i $$-0.641273\pi$$
−0.429393 + 0.903118i $$0.641273\pi$$
$$212$$ 547.275 0.177297
$$213$$ 0 0
$$214$$ 7072.66 2.25924
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −5663.28 −1.77165
$$218$$ −745.759 −0.231693
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1712.38 0.521210
$$222$$ 0 0
$$223$$ −3353.22 −1.00694 −0.503471 0.864012i $$-0.667944\pi$$
−0.503471 + 0.864012i $$0.667944\pi$$
$$224$$ 4583.31 1.36712
$$225$$ 0 0
$$226$$ −6.85286 −0.00201701
$$227$$ −690.764 −0.201972 −0.100986 0.994888i $$-0.532200\pi$$
−0.100986 + 0.994888i $$0.532200\pi$$
$$228$$ 0 0
$$229$$ −668.131 −0.192801 −0.0964003 0.995343i $$-0.530733\pi$$
−0.0964003 + 0.995343i $$0.530733\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3117.94 0.882339
$$233$$ 940.537 0.264449 0.132225 0.991220i $$-0.457788\pi$$
0.132225 + 0.991220i $$0.457788\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 755.347 0.208343
$$237$$ 0 0
$$238$$ −2985.75 −0.813181
$$239$$ 1107.40 0.299715 0.149858 0.988708i $$-0.452118\pi$$
0.149858 + 0.988708i $$0.452118\pi$$
$$240$$ 0 0
$$241$$ 1228.86 0.328456 0.164228 0.986422i $$-0.447487\pi$$
0.164228 + 0.986422i $$0.447487\pi$$
$$242$$ 13250.7 3.51979
$$243$$ 0 0
$$244$$ −350.715 −0.0920172
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5826.58 1.50096
$$248$$ −2812.90 −0.720238
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 738.480 0.185707 0.0928534 0.995680i $$-0.470401\pi$$
0.0928534 + 0.995680i $$0.470401\pi$$
$$252$$ 0 0
$$253$$ 5849.54 1.45359
$$254$$ 4150.72 1.02535
$$255$$ 0 0
$$256$$ 5095.94 1.24412
$$257$$ −6312.57 −1.53217 −0.766084 0.642740i $$-0.777797\pi$$
−0.766084 + 0.642740i $$0.777797\pi$$
$$258$$ 0 0
$$259$$ 569.458 0.136619
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1836.50 0.433051
$$263$$ −5993.62 −1.40526 −0.702628 0.711558i $$-0.747990\pi$$
−0.702628 + 0.711558i $$0.747990\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −10159.3 −2.34176
$$267$$ 0 0
$$268$$ 2699.73 0.615344
$$269$$ −1749.70 −0.396584 −0.198292 0.980143i $$-0.563539\pi$$
−0.198292 + 0.980143i $$0.563539\pi$$
$$270$$ 0 0
$$271$$ −1931.68 −0.432994 −0.216497 0.976283i $$-0.569463\pi$$
−0.216497 + 0.976283i $$0.569463\pi$$
$$272$$ −2659.79 −0.592917
$$273$$ 0 0
$$274$$ 1583.91 0.349225
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3791.55 0.822427 0.411213 0.911539i $$-0.365105\pi$$
0.411213 + 0.911539i $$0.365105\pi$$
$$278$$ −8264.13 −1.78291
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4489.36 0.953071 0.476536 0.879155i $$-0.341892\pi$$
0.476536 + 0.879155i $$0.341892\pi$$
$$282$$ 0 0
$$283$$ −3133.86 −0.658264 −0.329132 0.944284i $$-0.606756\pi$$
−0.329132 + 0.944284i $$0.606756\pi$$
$$284$$ 2256.45 0.471465
$$285$$ 0 0
$$286$$ −12921.4 −2.67154
$$287$$ 11043.4 2.27133
$$288$$ 0 0
$$289$$ −3802.84 −0.774036
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1810.15 0.362778
$$293$$ −2797.50 −0.557787 −0.278894 0.960322i $$-0.589968\pi$$
−0.278894 + 0.960322i $$0.589968\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 282.845 0.0555406
$$297$$ 0 0
$$298$$ −4472.64 −0.869439
$$299$$ −4213.07 −0.814877
$$300$$ 0 0
$$301$$ −6023.67 −1.15348
$$302$$ 7604.77 1.44903
$$303$$ 0 0
$$304$$ −9050.23 −1.70746
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6839.91 1.27158 0.635789 0.771863i $$-0.280675\pi$$
0.635789 + 0.771863i $$0.280675\pi$$
$$308$$ 8011.97 1.48222
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2419.41 0.441132 0.220566 0.975372i $$-0.429210\pi$$
0.220566 + 0.975372i $$0.429210\pi$$
$$312$$ 0 0
$$313$$ −3903.18 −0.704858 −0.352429 0.935838i $$-0.614644\pi$$
−0.352429 + 0.935838i $$0.614644\pi$$
$$314$$ 8256.59 1.48391
$$315$$ 0 0
$$316$$ −3503.64 −0.623718
$$317$$ −9894.73 −1.75313 −0.876567 0.481280i $$-0.840172\pi$$
−0.876567 + 0.481280i $$0.840172\pi$$
$$318$$ 0 0
$$319$$ 17612.6 3.09128
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 7345.99 1.27135
$$323$$ 3777.44 0.650720
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −11979.6 −2.03524
$$327$$ 0 0
$$328$$ 5485.16 0.923377
$$329$$ −2746.29 −0.460207
$$330$$ 0 0
$$331$$ 4163.04 0.691303 0.345651 0.938363i $$-0.387658\pi$$
0.345651 + 0.938363i $$0.387658\pi$$
$$332$$ −1192.38 −0.197109
$$333$$ 0 0
$$334$$ 6374.73 1.04434
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −9704.93 −1.56873 −0.784364 0.620301i $$-0.787011\pi$$
−0.784364 + 0.620301i $$0.787011\pi$$
$$338$$ 1565.45 0.251921
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −15889.5 −2.52336
$$342$$ 0 0
$$343$$ 996.691 0.156899
$$344$$ −2991.90 −0.468931
$$345$$ 0 0
$$346$$ 4677.09 0.726711
$$347$$ 5432.38 0.840420 0.420210 0.907427i $$-0.361956\pi$$
0.420210 + 0.907427i $$0.361956\pi$$
$$348$$ 0 0
$$349$$ 10571.9 1.62150 0.810749 0.585394i $$-0.199060\pi$$
0.810749 + 0.585394i $$0.199060\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 12859.5 1.94719
$$353$$ 9642.98 1.45395 0.726975 0.686664i $$-0.240926\pi$$
0.726975 + 0.686664i $$0.240926\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 783.198 0.116600
$$357$$ 0 0
$$358$$ −1581.10 −0.233419
$$359$$ −3002.18 −0.441362 −0.220681 0.975346i $$-0.570828\pi$$
−0.220681 + 0.975346i $$0.570828\pi$$
$$360$$ 0 0
$$361$$ 5994.17 0.873913
$$362$$ −12157.9 −1.76521
$$363$$ 0 0
$$364$$ −5770.54 −0.830929
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1149.28 0.163465 0.0817326 0.996654i $$-0.473955\pi$$
0.0817326 + 0.996654i $$0.473955\pi$$
$$368$$ 6544.03 0.926986
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3152.64 −0.441178
$$372$$ 0 0
$$373$$ 513.214 0.0712418 0.0356209 0.999365i $$-0.488659\pi$$
0.0356209 + 0.999365i $$0.488659\pi$$
$$374$$ −8377.14 −1.15821
$$375$$ 0 0
$$376$$ −1364.06 −0.187090
$$377$$ −12685.3 −1.73296
$$378$$ 0 0
$$379$$ 12560.0 1.70228 0.851138 0.524942i $$-0.175913\pi$$
0.851138 + 0.524942i $$0.175913\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −5609.70 −0.751355
$$383$$ 273.990 0.0365542 0.0182771 0.999833i $$-0.494182\pi$$
0.0182771 + 0.999833i $$0.494182\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1561.95 −0.205962
$$387$$ 0 0
$$388$$ 3893.31 0.509415
$$389$$ −6846.43 −0.892359 −0.446179 0.894944i $$-0.647216\pi$$
−0.446179 + 0.894944i $$0.647216\pi$$
$$390$$ 0 0
$$391$$ −2731.39 −0.353279
$$392$$ −3837.75 −0.494479
$$393$$ 0 0
$$394$$ 736.377 0.0941577
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 12117.5 1.53189 0.765943 0.642909i $$-0.222273\pi$$
0.765943 + 0.642909i $$0.222273\pi$$
$$398$$ −1690.97 −0.212967
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3016.80 −0.375691 −0.187845 0.982199i $$-0.560150\pi$$
−0.187845 + 0.982199i $$0.560150\pi$$
$$402$$ 0 0
$$403$$ 11444.3 1.41459
$$404$$ 4529.82 0.557839
$$405$$ 0 0
$$406$$ 22118.4 2.70373
$$407$$ 1597.74 0.194587
$$408$$ 0 0
$$409$$ 6535.13 0.790076 0.395038 0.918665i $$-0.370731\pi$$
0.395038 + 0.918665i $$0.370731\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 7903.87 0.945135
$$413$$ −4351.26 −0.518430
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −9261.88 −1.09159
$$417$$ 0 0
$$418$$ −28504.1 −3.33537
$$419$$ 7148.45 0.833471 0.416736 0.909028i $$-0.363174\pi$$
0.416736 + 0.909028i $$0.363174\pi$$
$$420$$ 0 0
$$421$$ 14801.5 1.71350 0.856749 0.515733i $$-0.172480\pi$$
0.856749 + 0.515733i $$0.172480\pi$$
$$422$$ −9274.28 −1.06982
$$423$$ 0 0
$$424$$ −1565.89 −0.179354
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2020.33 0.228971
$$428$$ 8861.98 1.00084
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2284.19 0.255280 0.127640 0.991821i $$-0.459260\pi$$
0.127640 + 0.991821i $$0.459260\pi$$
$$432$$ 0 0
$$433$$ −5529.26 −0.613670 −0.306835 0.951763i $$-0.599270\pi$$
−0.306835 + 0.951763i $$0.599270\pi$$
$$434$$ −19954.4 −2.20701
$$435$$ 0 0
$$436$$ −934.429 −0.102640
$$437$$ −9293.85 −1.01736
$$438$$ 0 0
$$439$$ −11861.6 −1.28958 −0.644788 0.764361i $$-0.723054\pi$$
−0.644788 + 0.764361i $$0.723054\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6033.55 0.649291
$$443$$ 15293.6 1.64023 0.820115 0.572199i $$-0.193910\pi$$
0.820115 + 0.572199i $$0.193910\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −11815.0 −1.25439
$$447$$ 0 0
$$448$$ −92.5602 −0.00976129
$$449$$ 12998.8 1.36626 0.683129 0.730297i $$-0.260619\pi$$
0.683129 + 0.730297i $$0.260619\pi$$
$$450$$ 0 0
$$451$$ 30984.6 3.23506
$$452$$ −8.58657 −0.000893536 0
$$453$$ 0 0
$$454$$ −2433.89 −0.251604
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −14311.9 −1.46495 −0.732477 0.680792i $$-0.761636\pi$$
−0.732477 + 0.680792i $$0.761636\pi$$
$$458$$ −2354.14 −0.240179
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 4290.34 0.433451 0.216725 0.976233i $$-0.430462\pi$$
0.216725 + 0.976233i $$0.430462\pi$$
$$462$$ 0 0
$$463$$ 14510.3 1.45648 0.728240 0.685322i $$-0.240339\pi$$
0.728240 + 0.685322i $$0.240339\pi$$
$$464$$ 19703.7 1.97138
$$465$$ 0 0
$$466$$ 3313.96 0.329434
$$467$$ 7393.05 0.732569 0.366285 0.930503i $$-0.380630\pi$$
0.366285 + 0.930503i $$0.380630\pi$$
$$468$$ 0 0
$$469$$ −15552.1 −1.53119
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −2161.23 −0.210760
$$473$$ −16900.7 −1.64290
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3741.11 −0.360239
$$477$$ 0 0
$$478$$ 3901.91 0.373366
$$479$$ 13509.9 1.28869 0.644344 0.764736i $$-0.277131\pi$$
0.644344 + 0.764736i $$0.277131\pi$$
$$480$$ 0 0
$$481$$ −1150.75 −0.109085
$$482$$ 4329.87 0.409170
$$483$$ 0 0
$$484$$ 16603.1 1.55926
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11140.0 1.03655 0.518277 0.855213i $$-0.326573\pi$$
0.518277 + 0.855213i $$0.326573\pi$$
$$488$$ 1003.48 0.0930849
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2012.74 0.184998 0.0924988 0.995713i $$-0.470515\pi$$
0.0924988 + 0.995713i $$0.470515\pi$$
$$492$$ 0 0
$$493$$ −8224.06 −0.751304
$$494$$ 20529.8 1.86980
$$495$$ 0 0
$$496$$ −17776.0 −1.60920
$$497$$ −12998.6 −1.17317
$$498$$ 0 0
$$499$$ 7352.98 0.659648 0.329824 0.944042i $$-0.393011\pi$$
0.329824 + 0.944042i $$0.393011\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 2602.02 0.231342
$$503$$ −16898.3 −1.49793 −0.748965 0.662609i $$-0.769449\pi$$
−0.748965 + 0.662609i $$0.769449\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 20610.7 1.81079
$$507$$ 0 0
$$508$$ 5200.82 0.454230
$$509$$ −19358.6 −1.68577 −0.842884 0.538095i $$-0.819144\pi$$
−0.842884 + 0.538095i $$0.819144\pi$$
$$510$$ 0 0
$$511$$ −10427.6 −0.902720
$$512$$ 6319.06 0.545440
$$513$$ 0 0
$$514$$ −22242.2 −1.90868
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −7705.30 −0.655471
$$518$$ 2006.47 0.170192
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −146.772 −0.0123420 −0.00617100 0.999981i $$-0.501964\pi$$
−0.00617100 + 0.999981i $$0.501964\pi$$
$$522$$ 0 0
$$523$$ 2872.03 0.240125 0.120062 0.992766i $$-0.461691\pi$$
0.120062 + 0.992766i $$0.461691\pi$$
$$524$$ 2301.12 0.191841
$$525$$ 0 0
$$526$$ −21118.4 −1.75058
$$527$$ 7419.46 0.613277
$$528$$ 0 0
$$529$$ −5446.82 −0.447671
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −12729.5 −1.03740
$$533$$ −22316.4 −1.81356
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −7724.58 −0.622484
$$537$$ 0 0
$$538$$ −6165.03 −0.494039
$$539$$ −21678.7 −1.73241
$$540$$ 0 0
$$541$$ −3810.28 −0.302804 −0.151402 0.988472i $$-0.548379\pi$$
−0.151402 + 0.988472i $$0.548379\pi$$
$$542$$ −6806.24 −0.539397
$$543$$ 0 0
$$544$$ −6004.60 −0.473245
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −5945.59 −0.464744 −0.232372 0.972627i $$-0.574649\pi$$
−0.232372 + 0.972627i $$0.574649\pi$$
$$548$$ 1984.63 0.154707
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −27983.3 −2.16357
$$552$$ 0 0
$$553$$ 20183.1 1.55203
$$554$$ 13359.4 1.02453
$$555$$ 0 0
$$556$$ −10354.9 −0.789828
$$557$$ 10872.2 0.827059 0.413530 0.910491i $$-0.364296\pi$$
0.413530 + 0.910491i $$0.364296\pi$$
$$558$$ 0 0
$$559$$ 12172.5 0.921007
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 15818.2 1.18728
$$563$$ −13575.1 −1.01620 −0.508100 0.861298i $$-0.669652\pi$$
−0.508100 + 0.861298i $$0.669652\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −11042.1 −0.820024
$$567$$ 0 0
$$568$$ −6456.27 −0.476935
$$569$$ 20642.6 1.52088 0.760442 0.649406i $$-0.224982\pi$$
0.760442 + 0.649406i $$0.224982\pi$$
$$570$$ 0 0
$$571$$ −2730.77 −0.200139 −0.100069 0.994980i $$-0.531906\pi$$
−0.100069 + 0.994980i $$0.531906\pi$$
$$572$$ −16190.5 −1.18349
$$573$$ 0 0
$$574$$ 38911.2 2.82948
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −21953.4 −1.58394 −0.791970 0.610560i $$-0.790945\pi$$
−0.791970 + 0.610560i $$0.790945\pi$$
$$578$$ −13399.2 −0.964245
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 6868.82 0.490476
$$582$$ 0 0
$$583$$ −8845.40 −0.628369
$$584$$ −5179.29 −0.366988
$$585$$ 0 0
$$586$$ −9856.92 −0.694856
$$587$$ −11055.3 −0.777348 −0.388674 0.921375i $$-0.627067\pi$$
−0.388674 + 0.921375i $$0.627067\pi$$
$$588$$ 0 0
$$589$$ 25245.5 1.76608
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1787.43 0.124092
$$593$$ −16538.0 −1.14525 −0.572625 0.819817i $$-0.694075\pi$$
−0.572625 + 0.819817i $$0.694075\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5604.18 −0.385161
$$597$$ 0 0
$$598$$ −14844.7 −1.01512
$$599$$ −6403.59 −0.436800 −0.218400 0.975859i $$-0.570084\pi$$
−0.218400 + 0.975859i $$0.570084\pi$$
$$600$$ 0 0
$$601$$ −14210.4 −0.964480 −0.482240 0.876039i $$-0.660177\pi$$
−0.482240 + 0.876039i $$0.660177\pi$$
$$602$$ −21224.3 −1.43694
$$603$$ 0 0
$$604$$ 9528.72 0.641917
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8915.72 0.596174 0.298087 0.954539i $$-0.403651\pi$$
0.298087 + 0.954539i $$0.403651\pi$$
$$608$$ −20431.3 −1.36283
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 5549.66 0.367455
$$612$$ 0 0
$$613$$ 15372.5 1.01287 0.506436 0.862277i $$-0.330963\pi$$
0.506436 + 0.862277i $$0.330963\pi$$
$$614$$ 24100.3 1.58405
$$615$$ 0 0
$$616$$ −22924.2 −1.49942
$$617$$ −17327.5 −1.13060 −0.565298 0.824887i $$-0.691239\pi$$
−0.565298 + 0.824887i $$0.691239\pi$$
$$618$$ 0 0
$$619$$ −28787.3 −1.86924 −0.934621 0.355645i $$-0.884261\pi$$
−0.934621 + 0.355645i $$0.884261\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8524.73 0.549535
$$623$$ −4511.71 −0.290141
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −13752.8 −0.878068
$$627$$ 0 0
$$628$$ 10345.4 0.657369
$$629$$ −746.047 −0.0472923
$$630$$ 0 0
$$631$$ 18091.0 1.14135 0.570675 0.821176i $$-0.306682\pi$$
0.570675 + 0.821176i $$0.306682\pi$$
$$632$$ 10024.8 0.630955
$$633$$ 0 0
$$634$$ −34863.8 −2.18394
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 15613.9 0.971184
$$638$$ 62057.7 3.85092
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 19577.5 1.20634 0.603171 0.797612i $$-0.293904\pi$$
0.603171 + 0.797612i $$0.293904\pi$$
$$642$$ 0 0
$$643$$ 29971.7 1.83821 0.919106 0.394011i $$-0.128913\pi$$
0.919106 + 0.394011i $$0.128913\pi$$
$$644$$ 9204.46 0.563209
$$645$$ 0 0
$$646$$ 13309.7 0.810626
$$647$$ 3635.64 0.220915 0.110457 0.993881i $$-0.464768\pi$$
0.110457 + 0.993881i $$0.464768\pi$$
$$648$$ 0 0
$$649$$ −12208.4 −0.738399
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −15010.3 −0.901610
$$653$$ 19407.3 1.16304 0.581520 0.813532i $$-0.302458\pi$$
0.581520 + 0.813532i $$0.302458\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 34663.3 2.06307
$$657$$ 0 0
$$658$$ −9676.49 −0.573296
$$659$$ −5627.86 −0.332671 −0.166335 0.986069i $$-0.553193\pi$$
−0.166335 + 0.986069i $$0.553193\pi$$
$$660$$ 0 0
$$661$$ −16663.0 −0.980507 −0.490253 0.871580i $$-0.663096\pi$$
−0.490253 + 0.871580i $$0.663096\pi$$
$$662$$ 14668.4 0.861182
$$663$$ 0 0
$$664$$ 3411.68 0.199396
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20234.1 1.17461
$$668$$ 7987.48 0.462642
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5668.47 0.326124
$$672$$ 0 0
$$673$$ 12638.8 0.723906 0.361953 0.932196i $$-0.382110\pi$$
0.361953 + 0.932196i $$0.382110\pi$$
$$674$$ −34195.1 −1.95422
$$675$$ 0 0
$$676$$ 1961.50 0.111601
$$677$$ 25316.0 1.43718 0.718591 0.695433i $$-0.244787\pi$$
0.718591 + 0.695433i $$0.244787\pi$$
$$678$$ 0 0
$$679$$ −22427.9 −1.26760
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −55986.3 −3.14344
$$683$$ −24818.4 −1.39041 −0.695204 0.718812i $$-0.744686\pi$$
−0.695204 + 0.718812i $$0.744686\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 3511.82 0.195455
$$687$$ 0 0
$$688$$ −18907.2 −1.04772
$$689$$ 6370.81 0.352262
$$690$$ 0 0
$$691$$ 19958.0 1.09875 0.549375 0.835576i $$-0.314866\pi$$
0.549375 + 0.835576i $$0.314866\pi$$
$$692$$ 5860.35 0.321932
$$693$$ 0 0
$$694$$ 19140.9 1.04694
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −14468.0 −0.786247
$$698$$ 37250.0 2.01996
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −5624.21 −0.303029 −0.151515 0.988455i $$-0.548415\pi$$
−0.151515 + 0.988455i $$0.548415\pi$$
$$702$$ 0 0
$$703$$ −2538.51 −0.136190
$$704$$ −259.697 −0.0139030
$$705$$ 0 0
$$706$$ 33976.8 1.81124
$$707$$ −26094.6 −1.38810
$$708$$ 0 0
$$709$$ −7715.43 −0.408687 −0.204344 0.978899i $$-0.565506\pi$$
−0.204344 + 0.978899i $$0.565506\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2240.92 −0.117952
$$713$$ −18254.5 −0.958817
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −1981.11 −0.103404
$$717$$ 0 0
$$718$$ −10578.1 −0.549821
$$719$$ 21647.8 1.12285 0.561424 0.827528i $$-0.310254\pi$$
0.561424 + 0.827528i $$0.310254\pi$$
$$720$$ 0 0
$$721$$ −45531.2 −2.35183
$$722$$ 21120.3 1.08867
$$723$$ 0 0
$$724$$ −15233.8 −0.781989
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3071.30 0.156682 0.0783412 0.996927i $$-0.475038\pi$$
0.0783412 + 0.996927i $$0.475038\pi$$
$$728$$ 16510.9 0.840570
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7891.61 0.399291
$$732$$ 0 0
$$733$$ −14652.0 −0.738314 −0.369157 0.929367i $$-0.620354\pi$$
−0.369157 + 0.929367i $$0.620354\pi$$
$$734$$ 4049.45 0.203635
$$735$$ 0 0
$$736$$ 14773.4 0.739886
$$737$$ −43634.7 −2.18087
$$738$$ 0 0
$$739$$ 17610.9 0.876629 0.438314 0.898822i $$-0.355576\pi$$
0.438314 + 0.898822i $$0.355576\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −11108.3 −0.549592
$$743$$ 16788.2 0.828935 0.414468 0.910064i $$-0.363968\pi$$
0.414468 + 0.910064i $$0.363968\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 1808.30 0.0887485
$$747$$ 0 0
$$748$$ −10496.5 −0.513087
$$749$$ −51050.5 −2.49045
$$750$$ 0 0
$$751$$ 23514.2 1.14254 0.571269 0.820763i $$-0.306451\pi$$
0.571269 + 0.820763i $$0.306451\pi$$
$$752$$ −8620.11 −0.418009
$$753$$ 0 0
$$754$$ −44696.4 −2.15882
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −24817.4 −1.19155 −0.595775 0.803151i $$-0.703155\pi$$
−0.595775 + 0.803151i $$0.703155\pi$$
$$758$$ 44254.8 2.12059
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 32517.7 1.54897 0.774485 0.632592i $$-0.218009\pi$$
0.774485 + 0.632592i $$0.218009\pi$$
$$762$$ 0 0
$$763$$ 5382.89 0.255404
$$764$$ −7028.91 −0.332850
$$765$$ 0 0
$$766$$ 965.398 0.0455369
$$767$$ 8792.96 0.413944
$$768$$ 0 0
$$769$$ 18057.8 0.846791 0.423395 0.905945i $$-0.360838\pi$$
0.423395 + 0.905945i $$0.360838\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1957.11 −0.0912411
$$773$$ −38147.2 −1.77498 −0.887491 0.460825i $$-0.847553\pi$$
−0.887491 + 0.460825i $$0.847553\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −11139.7 −0.515326
$$777$$ 0 0
$$778$$ −24123.2 −1.11164
$$779$$ −49228.9 −2.26420
$$780$$ 0 0
$$781$$ −36470.2 −1.67094
$$782$$ −9623.98 −0.440093
$$783$$ 0 0
$$784$$ −24252.5 −1.10480
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4801.76 0.217489 0.108745 0.994070i $$-0.465317\pi$$
0.108745 + 0.994070i $$0.465317\pi$$
$$788$$ 922.673 0.0417118
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 49.4640 0.00222343
$$792$$ 0 0
$$793$$ −4082.66 −0.182824
$$794$$ 42695.6 1.90833
$$795$$ 0 0
$$796$$ −2118.77 −0.0943442
$$797$$ 15471.0 0.687592 0.343796 0.939044i $$-0.388287\pi$$
0.343796 + 0.939044i $$0.388287\pi$$
$$798$$ 0 0
$$799$$ 3597.92 0.159306
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −10629.6 −0.468012
$$803$$ −29256.8 −1.28574
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 40323.6 1.76220
$$807$$ 0 0
$$808$$ −12960.9 −0.564312
$$809$$ −17007.6 −0.739129 −0.369564 0.929205i $$-0.620493\pi$$
−0.369564 + 0.929205i $$0.620493\pi$$
$$810$$ 0 0
$$811$$ −7552.52 −0.327010 −0.163505 0.986543i $$-0.552280\pi$$
−0.163505 + 0.986543i $$0.552280\pi$$
$$812$$ 27714.1 1.19775
$$813$$ 0 0
$$814$$ 5629.58 0.242404
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 26852.1 1.14986
$$818$$ 23026.4 0.984227
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3745.20 0.159206 0.0796031 0.996827i $$-0.474635\pi$$
0.0796031 + 0.996827i $$0.474635\pi$$
$$822$$ 0 0
$$823$$ 30021.0 1.27153 0.635764 0.771884i $$-0.280685\pi$$
0.635764 + 0.771884i $$0.280685\pi$$
$$824$$ −22614.9 −0.956101
$$825$$ 0 0
$$826$$ −15331.6 −0.645828
$$827$$ 7918.68 0.332962 0.166481 0.986045i $$-0.446760\pi$$
0.166481 + 0.986045i $$0.446760\pi$$
$$828$$ 0 0
$$829$$ −6976.17 −0.292271 −0.146135 0.989265i $$-0.546683\pi$$
−0.146135 + 0.989265i $$0.546683\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 187.044 0.00779398
$$833$$ 10122.7 0.421044
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −35715.4 −1.47756
$$837$$ 0 0
$$838$$ 25187.4 1.03829
$$839$$ 8657.91 0.356263 0.178131 0.984007i $$-0.442995\pi$$
0.178131 + 0.984007i $$0.442995\pi$$
$$840$$ 0 0
$$841$$ 36534.7 1.49800
$$842$$ 52152.9 2.13457
$$843$$ 0 0
$$844$$ −11620.6 −0.473930
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −95643.8 −3.88000
$$848$$ −9895.57 −0.400726
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1835.54 0.0739383
$$852$$ 0 0
$$853$$ −18984.8 −0.762047 −0.381024 0.924565i $$-0.624428\pi$$
−0.381024 + 0.924565i $$0.624428\pi$$
$$854$$ 7118.60 0.285238
$$855$$ 0 0
$$856$$ −25356.3 −1.01245
$$857$$ 36434.4 1.45225 0.726124 0.687564i $$-0.241320\pi$$
0.726124 + 0.687564i $$0.241320\pi$$
$$858$$ 0 0
$$859$$ −17485.0 −0.694507 −0.347253 0.937771i $$-0.612886\pi$$
−0.347253 + 0.937771i $$0.612886\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8048.29 0.318011
$$863$$ 10423.6 0.411152 0.205576 0.978641i $$-0.434093\pi$$
0.205576 + 0.978641i $$0.434093\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −19482.2 −0.764472
$$867$$ 0 0
$$868$$ −25002.7 −0.977704
$$869$$ 56628.0 2.21056
$$870$$ 0 0
$$871$$ 31427.4 1.22259
$$872$$ 2673.63 0.103831
$$873$$ 0 0
$$874$$ −32746.7 −1.26736
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7873.24 −0.303148 −0.151574 0.988446i $$-0.548434\pi$$
−0.151574 + 0.988446i $$0.548434\pi$$
$$878$$ −41794.1 −1.60647
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −35135.0 −1.34362 −0.671809 0.740725i $$-0.734482\pi$$
−0.671809 + 0.740725i $$0.734482\pi$$
$$882$$ 0 0
$$883$$ −33069.4 −1.26033 −0.630167 0.776460i $$-0.717013\pi$$
−0.630167 + 0.776460i $$0.717013\pi$$
$$884$$ 7559.98 0.287635
$$885$$ 0 0
$$886$$ 53886.7 2.04329
$$887$$ 51479.1 1.94870 0.974351 0.225031i $$-0.0722485\pi$$
0.974351 + 0.225031i $$0.0722485\pi$$
$$888$$ 0 0
$$889$$ −29959.9 −1.13029
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14804.1 −0.555692
$$893$$ 12242.3 0.458761
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −36992.6 −1.37928
$$897$$ 0 0
$$898$$ 45800.9 1.70200
$$899$$ −54963.3 −2.03907
$$900$$ 0 0
$$901$$ 4130.28 0.152719
$$902$$ 109174. 4.03003
$$903$$ 0 0
$$904$$ 24.5683 0.000903904 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 10809.8 0.395737 0.197868 0.980229i $$-0.436598\pi$$
0.197868 + 0.980229i $$0.436598\pi$$
$$908$$ −3049.64 −0.111460
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31539.8 1.14705 0.573523 0.819190i $$-0.305576\pi$$
0.573523 + 0.819190i $$0.305576\pi$$
$$912$$ 0 0
$$913$$ 19271.9 0.698585
$$914$$ −50427.8 −1.82495
$$915$$ 0 0
$$916$$ −2949.72 −0.106399
$$917$$ −13255.9 −0.477368
$$918$$ 0 0
$$919$$ 3592.53 0.128952 0.0644759 0.997919i $$-0.479462\pi$$
0.0644759 + 0.997919i $$0.479462\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 15116.9 0.539966
$$923$$ 26267.3 0.936727
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 51126.7 1.81439
$$927$$ 0 0
$$928$$ 44482.0 1.57348
$$929$$ 31614.7 1.11652 0.558259 0.829667i $$-0.311470\pi$$
0.558259 + 0.829667i $$0.311470\pi$$
$$930$$ 0 0
$$931$$ 34443.5 1.21250
$$932$$ 4152.36 0.145939
$$933$$ 0 0
$$934$$ 26049.2 0.912588
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 20891.5 0.728382 0.364191 0.931324i $$-0.381346\pi$$
0.364191 + 0.931324i $$0.381346\pi$$
$$938$$ −54797.5 −1.90746
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 37769.2 1.30844 0.654219 0.756305i $$-0.272998\pi$$
0.654219 + 0.756305i $$0.272998\pi$$
$$942$$ 0 0
$$943$$ 35596.4 1.22924
$$944$$ −13657.8 −0.470894
$$945$$ 0 0
$$946$$ −59549.1 −2.04663
$$947$$ 10430.8 0.357927 0.178963 0.983856i $$-0.442726\pi$$
0.178963 + 0.983856i $$0.442726\pi$$
$$948$$ 0 0
$$949$$ 21071.9 0.720784
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 10704.2 0.364419
$$953$$ 33865.3 1.15111 0.575553 0.817764i $$-0.304787\pi$$
0.575553 + 0.817764i $$0.304787\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 4889.06 0.165401
$$957$$ 0 0
$$958$$ 47601.7 1.60537
$$959$$ −11432.7 −0.384965
$$960$$ 0 0
$$961$$ 19794.9 0.664461
$$962$$ −4054.65 −0.135891
$$963$$ 0 0
$$964$$ 5425.28 0.181262
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46581.8 1.54909 0.774544 0.632520i $$-0.217979\pi$$
0.774544 + 0.632520i $$0.217979\pi$$
$$968$$ −47505.4 −1.57736
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 13829.5 0.457065 0.228532 0.973536i $$-0.426607\pi$$
0.228532 + 0.973536i $$0.426607\pi$$
$$972$$ 0 0
$$973$$ 59650.5 1.96537
$$974$$ 39251.6 1.29127
$$975$$ 0 0
$$976$$ 6341.46 0.207977
$$977$$ −15943.6 −0.522089 −0.261045 0.965327i $$-0.584067\pi$$
−0.261045 + 0.965327i $$0.584067\pi$$
$$978$$ 0 0
$$979$$ −12658.5 −0.413247
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 7091.85 0.230458
$$983$$ 29111.6 0.944572 0.472286 0.881445i $$-0.343429\pi$$
0.472286 + 0.881445i $$0.343429\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −28977.3 −0.935927
$$987$$ 0 0
$$988$$ 25723.7 0.828318
$$989$$ −19416.1 −0.624265
$$990$$ 0 0
$$991$$ 8745.85 0.280344 0.140172 0.990127i $$-0.455234\pi$$
0.140172 + 0.990127i $$0.455234\pi$$
$$992$$ −40130.1 −1.28441
$$993$$ 0 0
$$994$$ −45800.2 −1.46146
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −45026.9 −1.43031 −0.715153 0.698968i $$-0.753643\pi$$
−0.715153 + 0.698968i $$0.753643\pi$$
$$998$$ 25908.0 0.821748
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2025.4.a.r.1.3 3
3.2 odd 2 2025.4.a.p.1.1 3
5.4 even 2 405.4.a.g.1.1 3
15.14 odd 2 405.4.a.i.1.3 yes 3
45.4 even 6 405.4.e.u.136.3 6
45.14 odd 6 405.4.e.s.136.1 6
45.29 odd 6 405.4.e.s.271.1 6
45.34 even 6 405.4.e.u.271.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.g.1.1 3 5.4 even 2
405.4.a.i.1.3 yes 3 15.14 odd 2
405.4.e.s.136.1 6 45.14 odd 6
405.4.e.s.271.1 6 45.29 odd 6
405.4.e.u.136.3 6 45.4 even 6
405.4.e.u.271.3 6 45.34 even 6
2025.4.a.p.1.1 3 3.2 odd 2
2025.4.a.r.1.3 3 1.1 even 1 trivial