# Properties

 Label 1232.4.a.s.1.3 Level $1232$ Weight $4$ Character 1232.1 Self dual yes Analytic conductor $72.690$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1232.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.6903531271$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.522072.1 Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 5x + 1$$ x^4 - x^3 - 12*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.148103$$ of defining polynomial Character $$\chi$$ $$=$$ 1232.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +O(q^{10})$$ $$q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +11.0000 q^{11} +24.6401 q^{13} -5.11574 q^{15} +17.8800 q^{17} -32.1459 q^{19} -19.4179 q^{21} -14.1248 q^{23} -121.599 q^{25} +128.450 q^{27} -41.5471 q^{29} -175.766 q^{31} -30.5139 q^{33} +12.9093 q^{35} +292.877 q^{37} -68.3513 q^{39} +154.296 q^{41} +277.144 q^{43} -35.6019 q^{45} +52.1450 q^{47} +49.0000 q^{49} -49.5989 q^{51} +82.3907 q^{53} +20.2860 q^{55} +89.1723 q^{57} -712.816 q^{59} -647.078 q^{61} -135.135 q^{63} +45.4408 q^{65} -260.867 q^{67} +39.1821 q^{69} -369.025 q^{71} +1145.77 q^{73} +337.314 q^{75} +77.0000 q^{77} -488.885 q^{79} +164.917 q^{81} -548.982 q^{83} +32.9740 q^{85} +115.251 q^{87} +105.039 q^{89} +172.481 q^{91} +487.572 q^{93} -59.2829 q^{95} -1361.91 q^{97} -212.355 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 $$4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100})$$ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.77399 −0.533854 −0.266927 0.963717i $$-0.586008\pi$$
−0.266927 + 0.963717i $$0.586008\pi$$
$$4$$ 0 0
$$5$$ 1.84418 0.164949 0.0824744 0.996593i $$-0.473718\pi$$
0.0824744 + 0.996593i $$0.473718\pi$$
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ −19.3050 −0.715000
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 24.6401 0.525687 0.262844 0.964838i $$-0.415340\pi$$
0.262844 + 0.964838i $$0.415340\pi$$
$$14$$ 0 0
$$15$$ −5.11574 −0.0880586
$$16$$ 0 0
$$17$$ 17.8800 0.255090 0.127545 0.991833i $$-0.459290\pi$$
0.127545 + 0.991833i $$0.459290\pi$$
$$18$$ 0 0
$$19$$ −32.1459 −0.388146 −0.194073 0.980987i $$-0.562170\pi$$
−0.194073 + 0.980987i $$0.562170\pi$$
$$20$$ 0 0
$$21$$ −19.4179 −0.201778
$$22$$ 0 0
$$23$$ −14.1248 −0.128054 −0.0640268 0.997948i $$-0.520394\pi$$
−0.0640268 + 0.997948i $$0.520394\pi$$
$$24$$ 0 0
$$25$$ −121.599 −0.972792
$$26$$ 0 0
$$27$$ 128.450 0.915560
$$28$$ 0 0
$$29$$ −41.5471 −0.266038 −0.133019 0.991113i $$-0.542467\pi$$
−0.133019 + 0.991113i $$0.542467\pi$$
$$30$$ 0 0
$$31$$ −175.766 −1.01834 −0.509169 0.860667i $$-0.670047\pi$$
−0.509169 + 0.860667i $$0.670047\pi$$
$$32$$ 0 0
$$33$$ −30.5139 −0.160963
$$34$$ 0 0
$$35$$ 12.9093 0.0623448
$$36$$ 0 0
$$37$$ 292.877 1.30131 0.650657 0.759372i $$-0.274494\pi$$
0.650657 + 0.759372i $$0.274494\pi$$
$$38$$ 0 0
$$39$$ −68.3513 −0.280640
$$40$$ 0 0
$$41$$ 154.296 0.587732 0.293866 0.955847i $$-0.405058\pi$$
0.293866 + 0.955847i $$0.405058\pi$$
$$42$$ 0 0
$$43$$ 277.144 0.982887 0.491443 0.870910i $$-0.336470\pi$$
0.491443 + 0.870910i $$0.336470\pi$$
$$44$$ 0 0
$$45$$ −35.6019 −0.117938
$$46$$ 0 0
$$47$$ 52.1450 0.161832 0.0809162 0.996721i $$-0.474215\pi$$
0.0809162 + 0.996721i $$0.474215\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −49.5989 −0.136181
$$52$$ 0 0
$$53$$ 82.3907 0.213533 0.106766 0.994284i $$-0.465950\pi$$
0.106766 + 0.994284i $$0.465950\pi$$
$$54$$ 0 0
$$55$$ 20.2860 0.0497339
$$56$$ 0 0
$$57$$ 89.1723 0.207213
$$58$$ 0 0
$$59$$ −712.816 −1.57289 −0.786447 0.617658i $$-0.788081\pi$$
−0.786447 + 0.617658i $$0.788081\pi$$
$$60$$ 0 0
$$61$$ −647.078 −1.35819 −0.679097 0.734048i $$-0.737629\pi$$
−0.679097 + 0.734048i $$0.737629\pi$$
$$62$$ 0 0
$$63$$ −135.135 −0.270244
$$64$$ 0 0
$$65$$ 45.4408 0.0867114
$$66$$ 0 0
$$67$$ −260.867 −0.475672 −0.237836 0.971305i $$-0.576438\pi$$
−0.237836 + 0.971305i $$0.576438\pi$$
$$68$$ 0 0
$$69$$ 39.1821 0.0683619
$$70$$ 0 0
$$71$$ −369.025 −0.616833 −0.308417 0.951251i $$-0.599799\pi$$
−0.308417 + 0.951251i $$0.599799\pi$$
$$72$$ 0 0
$$73$$ 1145.77 1.83702 0.918509 0.395401i $$-0.129394\pi$$
0.918509 + 0.395401i $$0.129394\pi$$
$$74$$ 0 0
$$75$$ 337.314 0.519329
$$76$$ 0 0
$$77$$ 77.0000 0.113961
$$78$$ 0 0
$$79$$ −488.885 −0.696251 −0.348125 0.937448i $$-0.613182\pi$$
−0.348125 + 0.937448i $$0.613182\pi$$
$$80$$ 0 0
$$81$$ 164.917 0.226224
$$82$$ 0 0
$$83$$ −548.982 −0.726008 −0.363004 0.931788i $$-0.618249\pi$$
−0.363004 + 0.931788i $$0.618249\pi$$
$$84$$ 0 0
$$85$$ 32.9740 0.0420769
$$86$$ 0 0
$$87$$ 115.251 0.142026
$$88$$ 0 0
$$89$$ 105.039 0.125102 0.0625510 0.998042i $$-0.480076\pi$$
0.0625510 + 0.998042i $$0.480076\pi$$
$$90$$ 0 0
$$91$$ 172.481 0.198691
$$92$$ 0 0
$$93$$ 487.572 0.543644
$$94$$ 0 0
$$95$$ −59.2829 −0.0640242
$$96$$ 0 0
$$97$$ −1361.91 −1.42557 −0.712787 0.701381i $$-0.752567\pi$$
−0.712787 + 0.701381i $$0.752567\pi$$
$$98$$ 0 0
$$99$$ −212.355 −0.215580
$$100$$ 0 0
$$101$$ −1610.32 −1.58647 −0.793234 0.608917i $$-0.791604\pi$$
−0.793234 + 0.608917i $$0.791604\pi$$
$$102$$ 0 0
$$103$$ 123.044 0.117708 0.0588540 0.998267i $$-0.481255\pi$$
0.0588540 + 0.998267i $$0.481255\pi$$
$$104$$ 0 0
$$105$$ −35.8102 −0.0332830
$$106$$ 0 0
$$107$$ 1740.90 1.57289 0.786446 0.617660i $$-0.211919\pi$$
0.786446 + 0.617660i $$0.211919\pi$$
$$108$$ 0 0
$$109$$ 248.938 0.218752 0.109376 0.994000i $$-0.465115\pi$$
0.109376 + 0.994000i $$0.465115\pi$$
$$110$$ 0 0
$$111$$ −812.436 −0.694712
$$112$$ 0 0
$$113$$ −494.465 −0.411641 −0.205820 0.978590i $$-0.565986\pi$$
−0.205820 + 0.978590i $$0.565986\pi$$
$$114$$ 0 0
$$115$$ −26.0488 −0.0211223
$$116$$ 0 0
$$117$$ −475.677 −0.375866
$$118$$ 0 0
$$119$$ 125.160 0.0964151
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −428.016 −0.313763
$$124$$ 0 0
$$125$$ −454.774 −0.325410
$$126$$ 0 0
$$127$$ 979.104 0.684106 0.342053 0.939681i $$-0.388878\pi$$
0.342053 + 0.939681i $$0.388878\pi$$
$$128$$ 0 0
$$129$$ −768.795 −0.524718
$$130$$ 0 0
$$131$$ 1660.85 1.10770 0.553851 0.832616i $$-0.313158\pi$$
0.553851 + 0.832616i $$0.313158\pi$$
$$132$$ 0 0
$$133$$ −225.021 −0.146705
$$134$$ 0 0
$$135$$ 236.884 0.151020
$$136$$ 0 0
$$137$$ −1618.17 −1.00912 −0.504559 0.863377i $$-0.668345\pi$$
−0.504559 + 0.863377i $$0.668345\pi$$
$$138$$ 0 0
$$139$$ −695.736 −0.424544 −0.212272 0.977211i $$-0.568086\pi$$
−0.212272 + 0.977211i $$0.568086\pi$$
$$140$$ 0 0
$$141$$ −144.650 −0.0863950
$$142$$ 0 0
$$143$$ 271.041 0.158501
$$144$$ 0 0
$$145$$ −76.6205 −0.0438826
$$146$$ 0 0
$$147$$ −135.925 −0.0762649
$$148$$ 0 0
$$149$$ −2081.84 −1.14464 −0.572318 0.820032i $$-0.693956\pi$$
−0.572318 + 0.820032i $$0.693956\pi$$
$$150$$ 0 0
$$151$$ −2679.28 −1.44395 −0.721975 0.691919i $$-0.756766\pi$$
−0.721975 + 0.691919i $$0.756766\pi$$
$$152$$ 0 0
$$153$$ −345.173 −0.182390
$$154$$ 0 0
$$155$$ −324.144 −0.167973
$$156$$ 0 0
$$157$$ 2410.32 1.22525 0.612627 0.790372i $$-0.290113\pi$$
0.612627 + 0.790372i $$0.290113\pi$$
$$158$$ 0 0
$$159$$ −228.551 −0.113995
$$160$$ 0 0
$$161$$ −98.8738 −0.0483997
$$162$$ 0 0
$$163$$ −3629.08 −1.74388 −0.871938 0.489616i $$-0.837137\pi$$
−0.871938 + 0.489616i $$0.837137\pi$$
$$164$$ 0 0
$$165$$ −56.2732 −0.0265507
$$166$$ 0 0
$$167$$ 3826.04 1.77286 0.886432 0.462859i $$-0.153176\pi$$
0.886432 + 0.462859i $$0.153176\pi$$
$$168$$ 0 0
$$169$$ −1589.87 −0.723653
$$170$$ 0 0
$$171$$ 620.576 0.277524
$$172$$ 0 0
$$173$$ −4310.67 −1.89442 −0.947209 0.320617i $$-0.896110\pi$$
−0.947209 + 0.320617i $$0.896110\pi$$
$$174$$ 0 0
$$175$$ −851.193 −0.367681
$$176$$ 0 0
$$177$$ 1977.34 0.839696
$$178$$ 0 0
$$179$$ −2491.12 −1.04019 −0.520097 0.854107i $$-0.674104\pi$$
−0.520097 + 0.854107i $$0.674104\pi$$
$$180$$ 0 0
$$181$$ −4315.49 −1.77220 −0.886098 0.463498i $$-0.846594\pi$$
−0.886098 + 0.463498i $$0.846594\pi$$
$$182$$ 0 0
$$183$$ 1794.99 0.725078
$$184$$ 0 0
$$185$$ 540.118 0.214650
$$186$$ 0 0
$$187$$ 196.680 0.0769127
$$188$$ 0 0
$$189$$ 899.147 0.346049
$$190$$ 0 0
$$191$$ −2840.41 −1.07605 −0.538024 0.842930i $$-0.680829\pi$$
−0.538024 + 0.842930i $$0.680829\pi$$
$$192$$ 0 0
$$193$$ −1734.68 −0.646969 −0.323485 0.946233i $$-0.604854\pi$$
−0.323485 + 0.946233i $$0.604854\pi$$
$$194$$ 0 0
$$195$$ −126.052 −0.0462913
$$196$$ 0 0
$$197$$ −3098.42 −1.12057 −0.560287 0.828298i $$-0.689309\pi$$
−0.560287 + 0.828298i $$0.689309\pi$$
$$198$$ 0 0
$$199$$ −4497.38 −1.60206 −0.801032 0.598622i $$-0.795715\pi$$
−0.801032 + 0.598622i $$0.795715\pi$$
$$200$$ 0 0
$$201$$ 723.643 0.253940
$$202$$ 0 0
$$203$$ −290.830 −0.100553
$$204$$ 0 0
$$205$$ 284.550 0.0969456
$$206$$ 0 0
$$207$$ 272.680 0.0915582
$$208$$ 0 0
$$209$$ −353.605 −0.117030
$$210$$ 0 0
$$211$$ −1262.32 −0.411857 −0.205929 0.978567i $$-0.566021\pi$$
−0.205929 + 0.978567i $$0.566021\pi$$
$$212$$ 0 0
$$213$$ 1023.67 0.329299
$$214$$ 0 0
$$215$$ 511.105 0.162126
$$216$$ 0 0
$$217$$ −1230.36 −0.384895
$$218$$ 0 0
$$219$$ −3178.35 −0.980700
$$220$$ 0 0
$$221$$ 440.565 0.134098
$$222$$ 0 0
$$223$$ −2931.38 −0.880268 −0.440134 0.897932i $$-0.645069\pi$$
−0.440134 + 0.897932i $$0.645069\pi$$
$$224$$ 0 0
$$225$$ 2347.47 0.695546
$$226$$ 0 0
$$227$$ −4298.37 −1.25680 −0.628399 0.777891i $$-0.716289\pi$$
−0.628399 + 0.777891i $$0.716289\pi$$
$$228$$ 0 0
$$229$$ 698.500 0.201564 0.100782 0.994909i $$-0.467866\pi$$
0.100782 + 0.994909i $$0.467866\pi$$
$$230$$ 0 0
$$231$$ −213.597 −0.0608383
$$232$$ 0 0
$$233$$ 1887.78 0.530783 0.265391 0.964141i $$-0.414499\pi$$
0.265391 + 0.964141i $$0.414499\pi$$
$$234$$ 0 0
$$235$$ 96.1649 0.0266941
$$236$$ 0 0
$$237$$ 1356.16 0.371697
$$238$$ 0 0
$$239$$ 6449.93 1.74565 0.872827 0.488030i $$-0.162284\pi$$
0.872827 + 0.488030i $$0.162284\pi$$
$$240$$ 0 0
$$241$$ 4636.98 1.23939 0.619697 0.784841i $$-0.287256\pi$$
0.619697 + 0.784841i $$0.287256\pi$$
$$242$$ 0 0
$$243$$ −3925.62 −1.03633
$$244$$ 0 0
$$245$$ 90.3650 0.0235641
$$246$$ 0 0
$$247$$ −792.078 −0.204043
$$248$$ 0 0
$$249$$ 1522.87 0.387582
$$250$$ 0 0
$$251$$ −2194.11 −0.551758 −0.275879 0.961192i $$-0.588969\pi$$
−0.275879 + 0.961192i $$0.588969\pi$$
$$252$$ 0 0
$$253$$ −155.373 −0.0386096
$$254$$ 0 0
$$255$$ −91.4695 −0.0224629
$$256$$ 0 0
$$257$$ −2966.97 −0.720133 −0.360067 0.932927i $$-0.617246\pi$$
−0.360067 + 0.932927i $$0.617246\pi$$
$$258$$ 0 0
$$259$$ 2050.14 0.491850
$$260$$ 0 0
$$261$$ 802.066 0.190217
$$262$$ 0 0
$$263$$ −915.810 −0.214720 −0.107360 0.994220i $$-0.534240\pi$$
−0.107360 + 0.994220i $$0.534240\pi$$
$$264$$ 0 0
$$265$$ 151.944 0.0352220
$$266$$ 0 0
$$267$$ −291.376 −0.0667863
$$268$$ 0 0
$$269$$ −164.462 −0.0372768 −0.0186384 0.999826i $$-0.505933\pi$$
−0.0186384 + 0.999826i $$0.505933\pi$$
$$270$$ 0 0
$$271$$ −1502.60 −0.336815 −0.168407 0.985718i $$-0.553862\pi$$
−0.168407 + 0.985718i $$0.553862\pi$$
$$272$$ 0 0
$$273$$ −478.459 −0.106072
$$274$$ 0 0
$$275$$ −1337.59 −0.293308
$$276$$ 0 0
$$277$$ −7500.11 −1.62685 −0.813426 0.581669i $$-0.802400\pi$$
−0.813426 + 0.581669i $$0.802400\pi$$
$$278$$ 0 0
$$279$$ 3393.15 0.728111
$$280$$ 0 0
$$281$$ −2945.12 −0.625235 −0.312618 0.949879i $$-0.601206\pi$$
−0.312618 + 0.949879i $$0.601206\pi$$
$$282$$ 0 0
$$283$$ −5215.29 −1.09547 −0.547733 0.836653i $$-0.684509\pi$$
−0.547733 + 0.836653i $$0.684509\pi$$
$$284$$ 0 0
$$285$$ 164.450 0.0341796
$$286$$ 0 0
$$287$$ 1080.07 0.222142
$$288$$ 0 0
$$289$$ −4593.31 −0.934929
$$290$$ 0 0
$$291$$ 3777.91 0.761049
$$292$$ 0 0
$$293$$ 7407.99 1.47706 0.738531 0.674219i $$-0.235520\pi$$
0.738531 + 0.674219i $$0.235520\pi$$
$$294$$ 0 0
$$295$$ −1314.56 −0.259447
$$296$$ 0 0
$$297$$ 1412.94 0.276052
$$298$$ 0 0
$$299$$ −348.037 −0.0673161
$$300$$ 0 0
$$301$$ 1940.01 0.371496
$$302$$ 0 0
$$303$$ 4467.02 0.846942
$$304$$ 0 0
$$305$$ −1193.33 −0.224033
$$306$$ 0 0
$$307$$ 6850.01 1.27345 0.636727 0.771089i $$-0.280288\pi$$
0.636727 + 0.771089i $$0.280288\pi$$
$$308$$ 0 0
$$309$$ −341.324 −0.0628390
$$310$$ 0 0
$$311$$ 5538.62 1.00986 0.504929 0.863161i $$-0.331519\pi$$
0.504929 + 0.863161i $$0.331519\pi$$
$$312$$ 0 0
$$313$$ 9361.13 1.69049 0.845243 0.534382i $$-0.179456\pi$$
0.845243 + 0.534382i $$0.179456\pi$$
$$314$$ 0 0
$$315$$ −249.214 −0.0445765
$$316$$ 0 0
$$317$$ 219.221 0.0388413 0.0194206 0.999811i $$-0.493818\pi$$
0.0194206 + 0.999811i $$0.493818\pi$$
$$318$$ 0 0
$$319$$ −457.018 −0.0802135
$$320$$ 0 0
$$321$$ −4829.24 −0.839695
$$322$$ 0 0
$$323$$ −574.769 −0.0990123
$$324$$ 0 0
$$325$$ −2996.21 −0.511384
$$326$$ 0 0
$$327$$ −690.551 −0.116781
$$328$$ 0 0
$$329$$ 365.015 0.0611669
$$330$$ 0 0
$$331$$ −3377.63 −0.560880 −0.280440 0.959872i $$-0.590480\pi$$
−0.280440 + 0.959872i $$0.590480\pi$$
$$332$$ 0 0
$$333$$ −5653.98 −0.930439
$$334$$ 0 0
$$335$$ −481.087 −0.0784615
$$336$$ 0 0
$$337$$ 7755.93 1.25369 0.626843 0.779145i $$-0.284347\pi$$
0.626843 + 0.779145i $$0.284347\pi$$
$$338$$ 0 0
$$339$$ 1371.64 0.219756
$$340$$ 0 0
$$341$$ −1933.42 −0.307040
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ 72.2590 0.0112762
$$346$$ 0 0
$$347$$ 831.044 0.128567 0.0642835 0.997932i $$-0.479524\pi$$
0.0642835 + 0.997932i $$0.479524\pi$$
$$348$$ 0 0
$$349$$ 1840.94 0.282359 0.141180 0.989984i $$-0.454911\pi$$
0.141180 + 0.989984i $$0.454911\pi$$
$$350$$ 0 0
$$351$$ 3165.01 0.481298
$$352$$ 0 0
$$353$$ 3409.11 0.514019 0.257009 0.966409i $$-0.417263\pi$$
0.257009 + 0.966409i $$0.417263\pi$$
$$354$$ 0 0
$$355$$ −680.549 −0.101746
$$356$$ 0 0
$$357$$ −347.193 −0.0514716
$$358$$ 0 0
$$359$$ −2199.75 −0.323394 −0.161697 0.986840i $$-0.551697\pi$$
−0.161697 + 0.986840i $$0.551697\pi$$
$$360$$ 0 0
$$361$$ −5825.64 −0.849343
$$362$$ 0 0
$$363$$ −335.653 −0.0485322
$$364$$ 0 0
$$365$$ 2113.01 0.303014
$$366$$ 0 0
$$367$$ 855.008 0.121611 0.0608053 0.998150i $$-0.480633\pi$$
0.0608053 + 0.998150i $$0.480633\pi$$
$$368$$ 0 0
$$369$$ −2978.69 −0.420228
$$370$$ 0 0
$$371$$ 576.735 0.0807078
$$372$$ 0 0
$$373$$ 3193.40 0.443293 0.221646 0.975127i $$-0.428857\pi$$
0.221646 + 0.975127i $$0.428857\pi$$
$$374$$ 0 0
$$375$$ 1261.54 0.173721
$$376$$ 0 0
$$377$$ −1023.72 −0.139853
$$378$$ 0 0
$$379$$ 5614.48 0.760940 0.380470 0.924793i $$-0.375762\pi$$
0.380470 + 0.924793i $$0.375762\pi$$
$$380$$ 0 0
$$381$$ −2716.02 −0.365213
$$382$$ 0 0
$$383$$ 1736.86 0.231721 0.115861 0.993265i $$-0.463037\pi$$
0.115861 + 0.993265i $$0.463037\pi$$
$$384$$ 0 0
$$385$$ 142.002 0.0187977
$$386$$ 0 0
$$387$$ −5350.27 −0.702763
$$388$$ 0 0
$$389$$ 8710.78 1.13536 0.567679 0.823250i $$-0.307841\pi$$
0.567679 + 0.823250i $$0.307841\pi$$
$$390$$ 0 0
$$391$$ −252.552 −0.0326652
$$392$$ 0 0
$$393$$ −4607.17 −0.591352
$$394$$ 0 0
$$395$$ −901.593 −0.114846
$$396$$ 0 0
$$397$$ 11731.6 1.48311 0.741553 0.670894i $$-0.234089\pi$$
0.741553 + 0.670894i $$0.234089\pi$$
$$398$$ 0 0
$$399$$ 624.206 0.0783193
$$400$$ 0 0
$$401$$ −14408.8 −1.79437 −0.897183 0.441659i $$-0.854390\pi$$
−0.897183 + 0.441659i $$0.854390\pi$$
$$402$$ 0 0
$$403$$ −4330.88 −0.535327
$$404$$ 0 0
$$405$$ 304.138 0.0373154
$$406$$ 0 0
$$407$$ 3221.64 0.392361
$$408$$ 0 0
$$409$$ −2155.54 −0.260598 −0.130299 0.991475i $$-0.541594\pi$$
−0.130299 + 0.991475i $$0.541594\pi$$
$$410$$ 0 0
$$411$$ 4488.77 0.538722
$$412$$ 0 0
$$413$$ −4989.71 −0.594498
$$414$$ 0 0
$$415$$ −1012.42 −0.119754
$$416$$ 0 0
$$417$$ 1929.96 0.226644
$$418$$ 0 0
$$419$$ −8443.41 −0.984458 −0.492229 0.870466i $$-0.663818\pi$$
−0.492229 + 0.870466i $$0.663818\pi$$
$$420$$ 0 0
$$421$$ −3070.28 −0.355430 −0.177715 0.984082i $$-0.556871\pi$$
−0.177715 + 0.984082i $$0.556871\pi$$
$$422$$ 0 0
$$423$$ −1006.66 −0.115710
$$424$$ 0 0
$$425$$ −2174.19 −0.248150
$$426$$ 0 0
$$427$$ −4529.55 −0.513349
$$428$$ 0 0
$$429$$ −751.865 −0.0846163
$$430$$ 0 0
$$431$$ 7004.40 0.782808 0.391404 0.920219i $$-0.371990\pi$$
0.391404 + 0.920219i $$0.371990\pi$$
$$432$$ 0 0
$$433$$ 7486.00 0.830841 0.415421 0.909629i $$-0.363634\pi$$
0.415421 + 0.909629i $$0.363634\pi$$
$$434$$ 0 0
$$435$$ 212.544 0.0234269
$$436$$ 0 0
$$437$$ 454.055 0.0497034
$$438$$ 0 0
$$439$$ 13046.2 1.41836 0.709179 0.705029i $$-0.249066\pi$$
0.709179 + 0.705029i $$0.249066\pi$$
$$440$$ 0 0
$$441$$ −945.944 −0.102143
$$442$$ 0 0
$$443$$ 11781.3 1.26354 0.631768 0.775157i $$-0.282329\pi$$
0.631768 + 0.775157i $$0.282329\pi$$
$$444$$ 0 0
$$445$$ 193.711 0.0206354
$$446$$ 0 0
$$447$$ 5774.99 0.611069
$$448$$ 0 0
$$449$$ 7576.58 0.796349 0.398175 0.917310i $$-0.369644\pi$$
0.398175 + 0.917310i $$0.369644\pi$$
$$450$$ 0 0
$$451$$ 1697.26 0.177208
$$452$$ 0 0
$$453$$ 7432.28 0.770859
$$454$$ 0 0
$$455$$ 318.086 0.0327738
$$456$$ 0 0
$$457$$ 11793.0 1.20712 0.603560 0.797318i $$-0.293748\pi$$
0.603560 + 0.797318i $$0.293748\pi$$
$$458$$ 0 0
$$459$$ 2296.68 0.233551
$$460$$ 0 0
$$461$$ 4228.32 0.427185 0.213592 0.976923i $$-0.431484\pi$$
0.213592 + 0.976923i $$0.431484\pi$$
$$462$$ 0 0
$$463$$ −14448.8 −1.45031 −0.725154 0.688586i $$-0.758232\pi$$
−0.725154 + 0.688586i $$0.758232\pi$$
$$464$$ 0 0
$$465$$ 899.172 0.0896733
$$466$$ 0 0
$$467$$ −16547.5 −1.63967 −0.819836 0.572599i $$-0.805935\pi$$
−0.819836 + 0.572599i $$0.805935\pi$$
$$468$$ 0 0
$$469$$ −1826.07 −0.179787
$$470$$ 0 0
$$471$$ −6686.21 −0.654107
$$472$$ 0 0
$$473$$ 3048.59 0.296351
$$474$$ 0 0
$$475$$ 3908.91 0.377585
$$476$$ 0 0
$$477$$ −1590.55 −0.152676
$$478$$ 0 0
$$479$$ 2989.34 0.285149 0.142575 0.989784i $$-0.454462\pi$$
0.142575 + 0.989784i $$0.454462\pi$$
$$480$$ 0 0
$$481$$ 7216.51 0.684084
$$482$$ 0 0
$$483$$ 274.275 0.0258384
$$484$$ 0 0
$$485$$ −2511.60 −0.235147
$$486$$ 0 0
$$487$$ 5549.61 0.516379 0.258190 0.966094i $$-0.416874\pi$$
0.258190 + 0.966094i $$0.416874\pi$$
$$488$$ 0 0
$$489$$ 10067.0 0.930976
$$490$$ 0 0
$$491$$ 7751.20 0.712438 0.356219 0.934403i $$-0.384066\pi$$
0.356219 + 0.934403i $$0.384066\pi$$
$$492$$ 0 0
$$493$$ −742.862 −0.0678638
$$494$$ 0 0
$$495$$ −391.621 −0.0355597
$$496$$ 0 0
$$497$$ −2583.17 −0.233141
$$498$$ 0 0
$$499$$ 6841.83 0.613792 0.306896 0.951743i $$-0.400710\pi$$
0.306896 + 0.951743i $$0.400710\pi$$
$$500$$ 0 0
$$501$$ −10613.4 −0.946451
$$502$$ 0 0
$$503$$ 11518.5 1.02105 0.510523 0.859864i $$-0.329452\pi$$
0.510523 + 0.859864i $$0.329452\pi$$
$$504$$ 0 0
$$505$$ −2969.73 −0.261686
$$506$$ 0 0
$$507$$ 4410.27 0.386325
$$508$$ 0 0
$$509$$ 8292.40 0.722110 0.361055 0.932545i $$-0.382417\pi$$
0.361055 + 0.932545i $$0.382417\pi$$
$$510$$ 0 0
$$511$$ 8020.39 0.694327
$$512$$ 0 0
$$513$$ −4129.12 −0.355371
$$514$$ 0 0
$$515$$ 226.917 0.0194158
$$516$$ 0 0
$$517$$ 573.595 0.0487943
$$518$$ 0 0
$$519$$ 11957.8 1.01134
$$520$$ 0 0
$$521$$ 10891.6 0.915868 0.457934 0.888986i $$-0.348590\pi$$
0.457934 + 0.888986i $$0.348590\pi$$
$$522$$ 0 0
$$523$$ −14662.0 −1.22586 −0.612928 0.790139i $$-0.710008\pi$$
−0.612928 + 0.790139i $$0.710008\pi$$
$$524$$ 0 0
$$525$$ 2361.20 0.196288
$$526$$ 0 0
$$527$$ −3142.69 −0.259768
$$528$$ 0 0
$$529$$ −11967.5 −0.983602
$$530$$ 0 0
$$531$$ 13760.9 1.12462
$$532$$ 0 0
$$533$$ 3801.87 0.308963
$$534$$ 0 0
$$535$$ 3210.54 0.259446
$$536$$ 0 0
$$537$$ 6910.33 0.555313
$$538$$ 0 0
$$539$$ 539.000 0.0430730
$$540$$ 0 0
$$541$$ −19825.8 −1.57556 −0.787778 0.615960i $$-0.788768\pi$$
−0.787778 + 0.615960i $$0.788768\pi$$
$$542$$ 0 0
$$543$$ 11971.1 0.946094
$$544$$ 0 0
$$545$$ 459.087 0.0360828
$$546$$ 0 0
$$547$$ −12706.1 −0.993187 −0.496593 0.867983i $$-0.665416\pi$$
−0.496593 + 0.867983i $$0.665416\pi$$
$$548$$ 0 0
$$549$$ 12491.8 0.971109
$$550$$ 0 0
$$551$$ 1335.57 0.103262
$$552$$ 0 0
$$553$$ −3422.19 −0.263158
$$554$$ 0 0
$$555$$ −1498.28 −0.114592
$$556$$ 0 0
$$557$$ −12599.1 −0.958421 −0.479211 0.877700i $$-0.659077\pi$$
−0.479211 + 0.877700i $$0.659077\pi$$
$$558$$ 0 0
$$559$$ 6828.86 0.516691
$$560$$ 0 0
$$561$$ −545.588 −0.0410602
$$562$$ 0 0
$$563$$ −6004.47 −0.449482 −0.224741 0.974419i $$-0.572154\pi$$
−0.224741 + 0.974419i $$0.572154\pi$$
$$564$$ 0 0
$$565$$ −911.884 −0.0678996
$$566$$ 0 0
$$567$$ 1154.42 0.0855046
$$568$$ 0 0
$$569$$ −3145.89 −0.231779 −0.115890 0.993262i $$-0.536972\pi$$
−0.115890 + 0.993262i $$0.536972\pi$$
$$570$$ 0 0
$$571$$ −23549.1 −1.72592 −0.862960 0.505273i $$-0.831392\pi$$
−0.862960 + 0.505273i $$0.831392\pi$$
$$572$$ 0 0
$$573$$ 7879.27 0.574453
$$574$$ 0 0
$$575$$ 1717.57 0.124569
$$576$$ 0 0
$$577$$ 327.335 0.0236172 0.0118086 0.999930i $$-0.496241\pi$$
0.0118086 + 0.999930i $$0.496241\pi$$
$$578$$ 0 0
$$579$$ 4811.98 0.345387
$$580$$ 0 0
$$581$$ −3842.88 −0.274405
$$582$$ 0 0
$$583$$ 906.298 0.0643825
$$584$$ 0 0
$$585$$ −877.235 −0.0619986
$$586$$ 0 0
$$587$$ 13270.8 0.933123 0.466562 0.884489i $$-0.345493\pi$$
0.466562 + 0.884489i $$0.345493\pi$$
$$588$$ 0 0
$$589$$ 5650.14 0.395263
$$590$$ 0 0
$$591$$ 8594.98 0.598224
$$592$$ 0 0
$$593$$ 5098.92 0.353099 0.176549 0.984292i $$-0.443506\pi$$
0.176549 + 0.984292i $$0.443506\pi$$
$$594$$ 0 0
$$595$$ 230.818 0.0159036
$$596$$ 0 0
$$597$$ 12475.7 0.855268
$$598$$ 0 0
$$599$$ −19358.2 −1.32046 −0.660230 0.751064i $$-0.729541\pi$$
−0.660230 + 0.751064i $$0.729541\pi$$
$$600$$ 0 0
$$601$$ 1238.87 0.0840841 0.0420420 0.999116i $$-0.486614\pi$$
0.0420420 + 0.999116i $$0.486614\pi$$
$$602$$ 0 0
$$603$$ 5036.04 0.340105
$$604$$ 0 0
$$605$$ 223.146 0.0149953
$$606$$ 0 0
$$607$$ 14175.6 0.947888 0.473944 0.880555i $$-0.342830\pi$$
0.473944 + 0.880555i $$0.342830\pi$$
$$608$$ 0 0
$$609$$ 806.758 0.0536806
$$610$$ 0 0
$$611$$ 1284.86 0.0850732
$$612$$ 0 0
$$613$$ 6906.67 0.455070 0.227535 0.973770i $$-0.426933\pi$$
0.227535 + 0.973770i $$0.426933\pi$$
$$614$$ 0 0
$$615$$ −789.339 −0.0517549
$$616$$ 0 0
$$617$$ 12104.3 0.789789 0.394894 0.918727i $$-0.370781\pi$$
0.394894 + 0.918727i $$0.370781\pi$$
$$618$$ 0 0
$$619$$ −14945.2 −0.970437 −0.485218 0.874393i $$-0.661260\pi$$
−0.485218 + 0.874393i $$0.661260\pi$$
$$620$$ 0 0
$$621$$ −1814.33 −0.117241
$$622$$ 0 0
$$623$$ 735.271 0.0472841
$$624$$ 0 0
$$625$$ 14361.2 0.919116
$$626$$ 0 0
$$627$$ 980.895 0.0624772
$$628$$ 0 0
$$629$$ 5236.63 0.331953
$$630$$ 0 0
$$631$$ 17711.9 1.11743 0.558717 0.829358i $$-0.311294\pi$$
0.558717 + 0.829358i $$0.311294\pi$$
$$632$$ 0 0
$$633$$ 3501.67 0.219872
$$634$$ 0 0
$$635$$ 1805.65 0.112842
$$636$$ 0 0
$$637$$ 1207.36 0.0750982
$$638$$ 0 0
$$639$$ 7124.02 0.441036
$$640$$ 0 0
$$641$$ −17994.4 −1.10879 −0.554396 0.832253i $$-0.687051\pi$$
−0.554396 + 0.832253i $$0.687051\pi$$
$$642$$ 0 0
$$643$$ −27947.8 −1.71408 −0.857039 0.515251i $$-0.827699\pi$$
−0.857039 + 0.515251i $$0.827699\pi$$
$$644$$ 0 0
$$645$$ −1417.80 −0.0865516
$$646$$ 0 0
$$647$$ 14336.2 0.871122 0.435561 0.900159i $$-0.356550\pi$$
0.435561 + 0.900159i $$0.356550\pi$$
$$648$$ 0 0
$$649$$ −7840.97 −0.474245
$$650$$ 0 0
$$651$$ 3413.00 0.205478
$$652$$ 0 0
$$653$$ 4315.79 0.258637 0.129318 0.991603i $$-0.458721\pi$$
0.129318 + 0.991603i $$0.458721\pi$$
$$654$$ 0 0
$$655$$ 3062.91 0.182714
$$656$$ 0 0
$$657$$ −22119.1 −1.31347
$$658$$ 0 0
$$659$$ −4002.23 −0.236578 −0.118289 0.992979i $$-0.537741\pi$$
−0.118289 + 0.992979i $$0.537741\pi$$
$$660$$ 0 0
$$661$$ −12223.4 −0.719268 −0.359634 0.933094i $$-0.617098\pi$$
−0.359634 + 0.933094i $$0.617098\pi$$
$$662$$ 0 0
$$663$$ −1222.12 −0.0715887
$$664$$ 0 0
$$665$$ −414.980 −0.0241989
$$666$$ 0 0
$$667$$ 586.846 0.0340671
$$668$$ 0 0
$$669$$ 8131.61 0.469935
$$670$$ 0 0
$$671$$ −7117.86 −0.409511
$$672$$ 0 0
$$673$$ −6121.53 −0.350621 −0.175310 0.984513i $$-0.556093\pi$$
−0.175310 + 0.984513i $$0.556093\pi$$
$$674$$ 0 0
$$675$$ −15619.3 −0.890649
$$676$$ 0 0
$$677$$ 9626.46 0.546492 0.273246 0.961944i $$-0.411903\pi$$
0.273246 + 0.961944i $$0.411903\pi$$
$$678$$ 0 0
$$679$$ −9533.34 −0.538816
$$680$$ 0 0
$$681$$ 11923.6 0.670947
$$682$$ 0 0
$$683$$ −11554.4 −0.647315 −0.323658 0.946174i $$-0.604913\pi$$
−0.323658 + 0.946174i $$0.604913\pi$$
$$684$$ 0 0
$$685$$ −2984.19 −0.166453
$$686$$ 0 0
$$687$$ −1937.63 −0.107606
$$688$$ 0 0
$$689$$ 2030.12 0.112251
$$690$$ 0 0
$$691$$ −2991.45 −0.164689 −0.0823444 0.996604i $$-0.526241\pi$$
−0.0823444 + 0.996604i $$0.526241\pi$$
$$692$$ 0 0
$$693$$ −1486.48 −0.0814818
$$694$$ 0 0
$$695$$ −1283.06 −0.0700279
$$696$$ 0 0
$$697$$ 2758.82 0.149925
$$698$$ 0 0
$$699$$ −5236.67 −0.283361
$$700$$ 0 0
$$701$$ 30772.8 1.65802 0.829010 0.559234i $$-0.188905\pi$$
0.829010 + 0.559234i $$0.188905\pi$$
$$702$$ 0 0
$$703$$ −9414.77 −0.505099
$$704$$ 0 0
$$705$$ −266.760 −0.0142507
$$706$$ 0 0
$$707$$ −11272.3 −0.599628
$$708$$ 0 0
$$709$$ −21698.3 −1.14936 −0.574681 0.818377i $$-0.694874\pi$$
−0.574681 + 0.818377i $$0.694874\pi$$
$$710$$ 0 0
$$711$$ 9437.91 0.497819
$$712$$ 0 0
$$713$$ 2482.66 0.130402
$$714$$ 0 0
$$715$$ 499.849 0.0261445
$$716$$ 0 0
$$717$$ −17892.0 −0.931925
$$718$$ 0 0
$$719$$ 12364.2 0.641318 0.320659 0.947195i $$-0.396096\pi$$
0.320659 + 0.947195i $$0.396096\pi$$
$$720$$ 0 0
$$721$$ 861.311 0.0444895
$$722$$ 0 0
$$723$$ −12862.9 −0.661656
$$724$$ 0 0
$$725$$ 5052.09 0.258800
$$726$$ 0 0
$$727$$ 16017.2 0.817119 0.408559 0.912732i $$-0.366031\pi$$
0.408559 + 0.912732i $$0.366031\pi$$
$$728$$ 0 0
$$729$$ 6436.85 0.327026
$$730$$ 0 0
$$731$$ 4955.34 0.250725
$$732$$ 0 0
$$733$$ 10564.4 0.532340 0.266170 0.963926i $$-0.414242\pi$$
0.266170 + 0.963926i $$0.414242\pi$$
$$734$$ 0 0
$$735$$ −250.671 −0.0125798
$$736$$ 0 0
$$737$$ −2869.54 −0.143420
$$738$$ 0 0
$$739$$ −1760.32 −0.0876242 −0.0438121 0.999040i $$-0.513950\pi$$
−0.0438121 + 0.999040i $$0.513950\pi$$
$$740$$ 0 0
$$741$$ 2197.21 0.108929
$$742$$ 0 0
$$743$$ −11383.4 −0.562067 −0.281034 0.959698i $$-0.590677\pi$$
−0.281034 + 0.959698i $$0.590677\pi$$
$$744$$ 0 0
$$745$$ −3839.29 −0.188806
$$746$$ 0 0
$$747$$ 10598.1 0.519095
$$748$$ 0 0
$$749$$ 12186.3 0.594497
$$750$$ 0 0
$$751$$ 15058.3 0.731670 0.365835 0.930680i $$-0.380783\pi$$
0.365835 + 0.930680i $$0.380783\pi$$
$$752$$ 0 0
$$753$$ 6086.45 0.294558
$$754$$ 0 0
$$755$$ −4941.08 −0.238178
$$756$$ 0 0
$$757$$ 38073.4 1.82801 0.914004 0.405706i $$-0.132974\pi$$
0.914004 + 0.405706i $$0.132974\pi$$
$$758$$ 0 0
$$759$$ 431.003 0.0206119
$$760$$ 0 0
$$761$$ 15232.1 0.725577 0.362788 0.931872i $$-0.381825\pi$$
0.362788 + 0.931872i $$0.381825\pi$$
$$762$$ 0 0
$$763$$ 1742.57 0.0826803
$$764$$ 0 0
$$765$$ −636.563 −0.0300849
$$766$$ 0 0
$$767$$ −17563.8 −0.826850
$$768$$ 0 0
$$769$$ 12013.1 0.563332 0.281666 0.959513i $$-0.409113\pi$$
0.281666 + 0.959513i $$0.409113\pi$$
$$770$$ 0 0
$$771$$ 8230.33 0.384446
$$772$$ 0 0
$$773$$ −14258.5 −0.663443 −0.331721 0.943377i $$-0.607629\pi$$
−0.331721 + 0.943377i $$0.607629\pi$$
$$774$$ 0 0
$$775$$ 21372.9 0.990630
$$776$$ 0 0
$$777$$ −5687.05 −0.262576
$$778$$ 0 0
$$779$$ −4959.99 −0.228126
$$780$$ 0 0
$$781$$ −4059.27 −0.185982
$$782$$ 0 0
$$783$$ −5336.70 −0.243574
$$784$$ 0 0
$$785$$ 4445.08 0.202104
$$786$$ 0 0
$$787$$ 14377.9 0.651227 0.325613 0.945503i $$-0.394429\pi$$
0.325613 + 0.945503i $$0.394429\pi$$
$$788$$ 0 0
$$789$$ 2540.45 0.114629
$$790$$ 0 0
$$791$$ −3461.26 −0.155585
$$792$$ 0 0
$$793$$ −15944.1 −0.713986
$$794$$ 0 0
$$795$$ −421.490 −0.0188034
$$796$$ 0 0
$$797$$ 38515.9 1.71180 0.855899 0.517144i $$-0.173005\pi$$
0.855899 + 0.517144i $$0.173005\pi$$
$$798$$ 0 0
$$799$$ 932.352 0.0412819
$$800$$ 0 0
$$801$$ −2027.77 −0.0894479
$$802$$ 0 0
$$803$$ 12603.5 0.553882
$$804$$ 0 0
$$805$$ −182.341 −0.00798347
$$806$$ 0 0
$$807$$ 456.217 0.0199004
$$808$$ 0 0
$$809$$ 11438.6 0.497106 0.248553 0.968618i $$-0.420045\pi$$
0.248553 + 0.968618i $$0.420045\pi$$
$$810$$ 0 0
$$811$$ −15872.9 −0.687265 −0.343632 0.939104i $$-0.611657\pi$$
−0.343632 + 0.939104i $$0.611657\pi$$
$$812$$ 0 0
$$813$$ 4168.21 0.179810
$$814$$ 0 0
$$815$$ −6692.70 −0.287650
$$816$$ 0 0
$$817$$ −8909.05 −0.381503
$$818$$ 0 0
$$819$$ −3329.74 −0.142064
$$820$$ 0 0
$$821$$ −11988.8 −0.509638 −0.254819 0.966989i $$-0.582016\pi$$
−0.254819 + 0.966989i $$0.582016\pi$$
$$822$$ 0 0
$$823$$ 222.224 0.00941219 0.00470610 0.999989i $$-0.498502\pi$$
0.00470610 + 0.999989i $$0.498502\pi$$
$$824$$ 0 0
$$825$$ 3710.46 0.156584
$$826$$ 0 0
$$827$$ −7524.36 −0.316382 −0.158191 0.987409i $$-0.550566\pi$$
−0.158191 + 0.987409i $$0.550566\pi$$
$$828$$ 0 0
$$829$$ −11807.1 −0.494663 −0.247332 0.968931i $$-0.579554\pi$$
−0.247332 + 0.968931i $$0.579554\pi$$
$$830$$ 0 0
$$831$$ 20805.2 0.868502
$$832$$ 0 0
$$833$$ 876.120 0.0364415
$$834$$ 0 0
$$835$$ 7055.93 0.292432
$$836$$ 0 0
$$837$$ −22577.0 −0.932349
$$838$$ 0 0
$$839$$ 29112.9 1.19796 0.598980 0.800764i $$-0.295573\pi$$
0.598980 + 0.800764i $$0.295573\pi$$
$$840$$ 0 0
$$841$$ −22662.8 −0.929224
$$842$$ 0 0
$$843$$ 8169.73 0.333785
$$844$$ 0 0
$$845$$ −2932.00 −0.119366
$$846$$ 0 0
$$847$$ 847.000 0.0343604
$$848$$ 0 0
$$849$$ 14467.2 0.584819
$$850$$ 0 0
$$851$$ −4136.83 −0.166638
$$852$$ 0 0
$$853$$ −24892.1 −0.999169 −0.499584 0.866265i $$-0.666514\pi$$
−0.499584 + 0.866265i $$0.666514\pi$$
$$854$$ 0 0
$$855$$ 1144.46 0.0457773
$$856$$ 0 0
$$857$$ −27299.2 −1.08812 −0.544062 0.839045i $$-0.683114\pi$$
−0.544062 + 0.839045i $$0.683114\pi$$
$$858$$ 0 0
$$859$$ 6975.13 0.277053 0.138526 0.990359i $$-0.455763\pi$$
0.138526 + 0.990359i $$0.455763\pi$$
$$860$$ 0 0
$$861$$ −2996.11 −0.118591
$$862$$ 0 0
$$863$$ 4481.56 0.176772 0.0883858 0.996086i $$-0.471829\pi$$
0.0883858 + 0.996086i $$0.471829\pi$$
$$864$$ 0 0
$$865$$ −7949.67 −0.312482
$$866$$ 0 0
$$867$$ 12741.8 0.499116
$$868$$ 0 0
$$869$$ −5377.73 −0.209928
$$870$$ 0 0
$$871$$ −6427.80 −0.250055
$$872$$ 0 0
$$873$$ 26291.6 1.01928
$$874$$ 0 0
$$875$$ −3183.42 −0.122993
$$876$$ 0 0
$$877$$ −7207.96 −0.277532 −0.138766 0.990325i $$-0.544314\pi$$
−0.138766 + 0.990325i $$0.544314\pi$$
$$878$$ 0 0
$$879$$ −20549.7 −0.788536
$$880$$ 0 0
$$881$$ −38413.7 −1.46900 −0.734502 0.678607i $$-0.762584\pi$$
−0.734502 + 0.678607i $$0.762584\pi$$
$$882$$ 0 0
$$883$$ 8705.04 0.331764 0.165882 0.986146i $$-0.446953\pi$$
0.165882 + 0.986146i $$0.446953\pi$$
$$884$$ 0 0
$$885$$ 3646.58 0.138507
$$886$$ 0 0
$$887$$ 44100.0 1.66937 0.834687 0.550725i $$-0.185649\pi$$
0.834687 + 0.550725i $$0.185649\pi$$
$$888$$ 0 0
$$889$$ 6853.73 0.258568
$$890$$ 0 0
$$891$$ 1814.09 0.0682091
$$892$$ 0 0
$$893$$ −1676.25 −0.0628146
$$894$$ 0 0
$$895$$ −4594.08 −0.171579
$$896$$ 0 0
$$897$$ 965.451 0.0359370
$$898$$ 0 0
$$899$$ 7302.55 0.270916
$$900$$ 0 0
$$901$$ 1473.15 0.0544702
$$902$$ 0 0
$$903$$ −5381.57 −0.198325
$$904$$ 0 0
$$905$$ −7958.54 −0.292322
$$906$$ 0 0
$$907$$ −19921.2 −0.729296 −0.364648 0.931145i $$-0.618811\pi$$
−0.364648 + 0.931145i $$0.618811\pi$$
$$908$$ 0 0
$$909$$ 31087.3 1.13432
$$910$$ 0 0
$$911$$ 14446.7 0.525402 0.262701 0.964877i $$-0.415387\pi$$
0.262701 + 0.964877i $$0.415387\pi$$
$$912$$ 0 0
$$913$$ −6038.81 −0.218900
$$914$$ 0 0
$$915$$ 3310.29 0.119601
$$916$$ 0 0
$$917$$ 11625.9 0.418672
$$918$$ 0 0
$$919$$ −44381.1 −1.59303 −0.796516 0.604617i $$-0.793326\pi$$
−0.796516 + 0.604617i $$0.793326\pi$$
$$920$$ 0 0
$$921$$ −19001.8 −0.679839
$$922$$ 0 0
$$923$$ −9092.80 −0.324261
$$924$$ 0 0
$$925$$ −35613.5 −1.26591
$$926$$ 0 0
$$927$$ −2375.37 −0.0841612
$$928$$ 0 0
$$929$$ −3455.30 −0.122029 −0.0610145 0.998137i $$-0.519434\pi$$
−0.0610145 + 0.998137i $$0.519434\pi$$
$$930$$ 0 0
$$931$$ −1575.15 −0.0554494
$$932$$ 0 0
$$933$$ −15364.1 −0.539117
$$934$$ 0 0
$$935$$ 362.714 0.0126866
$$936$$ 0 0
$$937$$ −14962.8 −0.521678 −0.260839 0.965382i $$-0.583999\pi$$
−0.260839 + 0.965382i $$0.583999\pi$$
$$938$$ 0 0
$$939$$ −25967.7 −0.902474
$$940$$ 0 0
$$941$$ −6116.95 −0.211909 −0.105955 0.994371i $$-0.533790\pi$$
−0.105955 + 0.994371i $$0.533790\pi$$
$$942$$ 0 0
$$943$$ −2179.41 −0.0752611
$$944$$ 0 0
$$945$$ 1658.19 0.0570804
$$946$$ 0 0
$$947$$ −53343.7 −1.83045 −0.915225 0.402943i $$-0.867987\pi$$
−0.915225 + 0.402943i $$0.867987\pi$$
$$948$$ 0 0
$$949$$ 28231.9 0.965696
$$950$$ 0 0
$$951$$ −608.117 −0.0207356
$$952$$ 0 0
$$953$$ 1979.62 0.0672887 0.0336443 0.999434i $$-0.489289\pi$$
0.0336443 + 0.999434i $$0.489289\pi$$
$$954$$ 0 0
$$955$$ −5238.24 −0.177493
$$956$$ 0 0
$$957$$ 1267.76 0.0428223
$$958$$ 0 0
$$959$$ −11327.2 −0.381411
$$960$$ 0 0
$$961$$ 1102.58 0.0370104
$$962$$ 0 0
$$963$$ −33608.1 −1.12462
$$964$$ 0 0
$$965$$ −3199.07 −0.106717
$$966$$ 0 0
$$967$$ 38892.0 1.29336 0.646681 0.762761i $$-0.276157\pi$$
0.646681 + 0.762761i $$0.276157\pi$$
$$968$$ 0 0
$$969$$ 1594.40 0.0528582
$$970$$ 0 0
$$971$$ 47826.1 1.58065 0.790325 0.612688i $$-0.209912\pi$$
0.790325 + 0.612688i $$0.209912\pi$$
$$972$$ 0 0
$$973$$ −4870.15 −0.160462
$$974$$ 0 0
$$975$$ 8311.45 0.273005
$$976$$ 0 0
$$977$$ 20840.5 0.682442 0.341221 0.939983i $$-0.389160\pi$$
0.341221 + 0.939983i $$0.389160\pi$$
$$978$$ 0 0
$$979$$ 1155.43 0.0377197
$$980$$ 0 0
$$981$$ −4805.74 −0.156407
$$982$$ 0 0
$$983$$ 54430.5 1.76609 0.883044 0.469290i $$-0.155490\pi$$
0.883044 + 0.469290i $$0.155490\pi$$
$$984$$ 0 0
$$985$$ −5714.05 −0.184837
$$986$$ 0 0
$$987$$ −1012.55 −0.0326542
$$988$$ 0 0
$$989$$ −3914.62 −0.125862
$$990$$ 0 0
$$991$$ 35161.9 1.12710 0.563550 0.826082i $$-0.309435\pi$$
0.563550 + 0.826082i $$0.309435\pi$$
$$992$$ 0 0
$$993$$ 9369.50 0.299428
$$994$$ 0 0
$$995$$ −8293.99 −0.264258
$$996$$ 0 0
$$997$$ 26961.6 0.856451 0.428225 0.903672i $$-0.359139\pi$$
0.428225 + 0.903672i $$0.359139\pi$$
$$998$$ 0 0
$$999$$ 37619.8 1.19143
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 1232.4.a.s.1.3 4
4.3 odd 2 77.4.a.d.1.4 4
12.11 even 2 693.4.a.l.1.1 4
20.19 odd 2 1925.4.a.p.1.1 4
28.27 even 2 539.4.a.g.1.4 4
44.43 even 2 847.4.a.d.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.d.1.4 4 4.3 odd 2
539.4.a.g.1.4 4 28.27 even 2
693.4.a.l.1.1 4 12.11 even 2
847.4.a.d.1.1 4 44.43 even 2
1232.4.a.s.1.3 4 1.1 even 1 trivial
1925.4.a.p.1.1 4 20.19 odd 2