# Properties

 Label 175.4.a.c.1.2 Level $175$ Weight $4$ Character 175.1 Self dual yes Analytic conductor $10.325$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$10.3253342510$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +O(q^{10})$$ $$q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -18.1005 q^{14} -51.7645 q^{16} +87.2254 q^{17} -44.7696 q^{18} -44.2254 q^{19} -46.5980 q^{21} -98.9188 q^{22} -218.167 q^{23} -160.319 q^{24} +50.0172 q^{26} +64.4802 q^{27} -9.19596 q^{28} -46.9411 q^{29} +194.558 q^{31} -58.8141 q^{32} -254.657 q^{33} -225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -339.362 q^{41} +120.492 q^{42} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +49.0000 q^{49} -580.647 q^{51} +25.4113 q^{52} +209.019 q^{53} -166.732 q^{54} +168.583 q^{56} +294.402 q^{57} +121.380 q^{58} -616.000 q^{59} +320.735 q^{61} -503.087 q^{62} +121.196 q^{63} +566.197 q^{64} +658.488 q^{66} -14.5097 q^{67} -114.589 q^{68} +1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} -824.489 q^{73} +948.603 q^{74} +58.0993 q^{76} +267.784 q^{77} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +61.2162 q^{84} -584.818 q^{86} +312.480 q^{87} +921.301 q^{88} -170.225 q^{89} -135.402 q^{91} +286.607 q^{92} -1295.15 q^{93} +30.1921 q^{94} +391.517 q^{96} -1059.87 q^{97} -126.704 q^{98} +662.333 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 $$2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23} - 496 q^{24} + 216 q^{26} - 86 q^{27} + 140 q^{28} - 26 q^{29} - 120 q^{31} - 672 q^{32} - 498 q^{33} - 24 q^{34} - 136 q^{36} - 564 q^{37} - 320 q^{38} - 14 q^{39} - 328 q^{41} - 56 q^{42} + 260 q^{43} - 1164 q^{44} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 98 q^{49} - 754 q^{51} - 628 q^{52} + 56 q^{53} + 648 q^{54} - 336 q^{56} + 668 q^{57} + 8 q^{58} - 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 84 q^{63} + 2128 q^{64} + 1976 q^{66} + 152 q^{67} - 908 q^{68} + 1332 q^{69} - 1904 q^{71} + 800 q^{72} - 676 q^{73} + 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 756 q^{84} - 768 q^{86} + 410 q^{87} + 4688 q^{88} - 216 q^{89} - 350 q^{91} - 264 q^{92} - 2760 q^{93} - 1928 q^{94} - 2464 q^{96} - 2742 q^{97} - 392 q^{98} + 940 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 56 * q^14 + 168 * q^16 + 50 * q^17 - 16 * q^18 + 36 * q^19 - 14 * q^21 + 184 * q^22 - 244 * q^23 - 496 * q^24 + 216 * q^26 - 86 * q^27 + 140 * q^28 - 26 * q^29 - 120 * q^31 - 672 * q^32 - 498 * q^33 - 24 * q^34 - 136 * q^36 - 564 * q^37 - 320 * q^38 - 14 * q^39 - 328 * q^41 - 56 * q^42 + 260 * q^43 - 1164 * q^44 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 98 * q^49 - 754 * q^51 - 628 * q^52 + 56 * q^53 + 648 * q^54 - 336 * q^56 + 668 * q^57 + 8 * q^58 - 1232 * q^59 + 336 * q^61 + 1200 * q^62 + 84 * q^63 + 2128 * q^64 + 1976 * q^66 + 152 * q^67 - 908 * q^68 + 1332 * q^69 - 1904 * q^71 + 800 * q^72 - 676 * q^73 + 2016 * q^74 + 1768 * q^76 - 98 * q^77 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 756 * q^84 - 768 * q^86 + 410 * q^87 + 4688 * q^88 - 216 * q^89 - 350 * q^91 - 264 * q^92 - 2760 * q^93 - 1928 * q^94 - 2464 * q^96 - 2742 * q^97 - 392 * q^98 + 940 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.58579 −0.914214 −0.457107 0.889412i $$-0.651114\pi$$
−0.457107 + 0.889412i $$0.651114\pi$$
$$3$$ −6.65685 −1.28111 −0.640556 0.767911i $$-0.721296\pi$$
−0.640556 + 0.767911i $$0.721296\pi$$
$$4$$ −1.31371 −0.164214
$$5$$ 0 0
$$6$$ 17.2132 1.17121
$$7$$ 7.00000 0.377964
$$8$$ 24.0833 1.06434
$$9$$ 17.3137 0.641248
$$10$$ 0 0
$$11$$ 38.2548 1.04857 0.524285 0.851543i $$-0.324333\pi$$
0.524285 + 0.851543i $$0.324333\pi$$
$$12$$ 8.74517 0.210376
$$13$$ −19.3431 −0.412679 −0.206339 0.978480i $$-0.566155\pi$$
−0.206339 + 0.978480i $$0.566155\pi$$
$$14$$ −18.1005 −0.345540
$$15$$ 0 0
$$16$$ −51.7645 −0.808820
$$17$$ 87.2254 1.24443 0.622214 0.782847i $$-0.286233\pi$$
0.622214 + 0.782847i $$0.286233\pi$$
$$18$$ −44.7696 −0.586238
$$19$$ −44.2254 −0.534000 −0.267000 0.963697i $$-0.586032\pi$$
−0.267000 + 0.963697i $$0.586032\pi$$
$$20$$ 0 0
$$21$$ −46.5980 −0.484215
$$22$$ −98.9188 −0.958617
$$23$$ −218.167 −1.97786 −0.988932 0.148371i $$-0.952597\pi$$
−0.988932 + 0.148371i $$0.952597\pi$$
$$24$$ −160.319 −1.36354
$$25$$ 0 0
$$26$$ 50.0172 0.377276
$$27$$ 64.4802 0.459601
$$28$$ −9.19596 −0.0620669
$$29$$ −46.9411 −0.300578 −0.150289 0.988642i $$-0.548020\pi$$
−0.150289 + 0.988642i $$0.548020\pi$$
$$30$$ 0 0
$$31$$ 194.558 1.12722 0.563609 0.826042i $$-0.309413\pi$$
0.563609 + 0.826042i $$0.309413\pi$$
$$32$$ −58.8141 −0.324905
$$33$$ −254.657 −1.34334
$$34$$ −225.546 −1.13767
$$35$$ 0 0
$$36$$ −22.7452 −0.105302
$$37$$ −366.853 −1.63001 −0.815003 0.579457i $$-0.803265\pi$$
−0.815003 + 0.579457i $$0.803265\pi$$
$$38$$ 114.357 0.488190
$$39$$ 128.765 0.528688
$$40$$ 0 0
$$41$$ −339.362 −1.29267 −0.646336 0.763053i $$-0.723699\pi$$
−0.646336 + 0.763053i $$0.723699\pi$$
$$42$$ 120.492 0.442676
$$43$$ 226.167 0.802095 0.401047 0.916057i $$-0.368646\pi$$
0.401047 + 0.916057i $$0.368646\pi$$
$$44$$ −50.2557 −0.172189
$$45$$ 0 0
$$46$$ 564.132 1.80819
$$47$$ −11.6762 −0.0362372 −0.0181186 0.999836i $$-0.505768\pi$$
−0.0181186 + 0.999836i $$0.505768\pi$$
$$48$$ 344.589 1.03619
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −580.647 −1.59425
$$52$$ 25.4113 0.0677674
$$53$$ 209.019 0.541717 0.270859 0.962619i $$-0.412692\pi$$
0.270859 + 0.962619i $$0.412692\pi$$
$$54$$ −166.732 −0.420173
$$55$$ 0 0
$$56$$ 168.583 0.402283
$$57$$ 294.402 0.684114
$$58$$ 121.380 0.274792
$$59$$ −616.000 −1.35926 −0.679630 0.733555i $$-0.737860\pi$$
−0.679630 + 0.733555i $$0.737860\pi$$
$$60$$ 0 0
$$61$$ 320.735 0.673212 0.336606 0.941646i $$-0.390721\pi$$
0.336606 + 0.941646i $$0.390721\pi$$
$$62$$ −503.087 −1.03052
$$63$$ 121.196 0.242369
$$64$$ 566.197 1.10585
$$65$$ 0 0
$$66$$ 658.488 1.22810
$$67$$ −14.5097 −0.0264573 −0.0132286 0.999912i $$-0.504211\pi$$
−0.0132286 + 0.999912i $$0.504211\pi$$
$$68$$ −114.589 −0.204352
$$69$$ 1452.30 2.53387
$$70$$ 0 0
$$71$$ −952.000 −1.59129 −0.795645 0.605763i $$-0.792868\pi$$
−0.795645 + 0.605763i $$0.792868\pi$$
$$72$$ 416.971 0.682506
$$73$$ −824.489 −1.32191 −0.660953 0.750427i $$-0.729848\pi$$
−0.660953 + 0.750427i $$0.729848\pi$$
$$74$$ 948.603 1.49017
$$75$$ 0 0
$$76$$ 58.0993 0.0876901
$$77$$ 267.784 0.396322
$$78$$ −332.958 −0.483334
$$79$$ 156.275 0.222561 0.111280 0.993789i $$-0.464505\pi$$
0.111280 + 0.993789i $$0.464505\pi$$
$$80$$ 0 0
$$81$$ −896.706 −1.23005
$$82$$ 877.519 1.18178
$$83$$ 1036.53 1.37077 0.685384 0.728182i $$-0.259634\pi$$
0.685384 + 0.728182i $$0.259634\pi$$
$$84$$ 61.2162 0.0795147
$$85$$ 0 0
$$86$$ −584.818 −0.733286
$$87$$ 312.480 0.385074
$$88$$ 921.301 1.11603
$$89$$ −170.225 −0.202740 −0.101370 0.994849i $$-0.532323\pi$$
−0.101370 + 0.994849i $$0.532323\pi$$
$$90$$ 0 0
$$91$$ −135.402 −0.155978
$$92$$ 286.607 0.324792
$$93$$ −1295.15 −1.44409
$$94$$ 30.1921 0.0331285
$$95$$ 0 0
$$96$$ 391.517 0.416240
$$97$$ −1059.87 −1.10942 −0.554710 0.832044i $$-0.687171\pi$$
−0.554710 + 0.832044i $$0.687171\pi$$
$$98$$ −126.704 −0.130602
$$99$$ 662.333 0.672394
$$100$$ 0 0
$$101$$ −241.833 −0.238251 −0.119125 0.992879i $$-0.538009\pi$$
−0.119125 + 0.992879i $$0.538009\pi$$
$$102$$ 1501.43 1.45749
$$103$$ 1679.58 1.60673 0.803367 0.595484i $$-0.203040\pi$$
0.803367 + 0.595484i $$0.203040\pi$$
$$104$$ −465.846 −0.439230
$$105$$ 0 0
$$106$$ −540.479 −0.495245
$$107$$ −1506.88 −1.36146 −0.680728 0.732537i $$-0.738336\pi$$
−0.680728 + 0.732537i $$0.738336\pi$$
$$108$$ −84.7082 −0.0754727
$$109$$ −1252.41 −1.10054 −0.550271 0.834986i $$-0.685476\pi$$
−0.550271 + 0.834986i $$0.685476\pi$$
$$110$$ 0 0
$$111$$ 2442.09 2.08822
$$112$$ −362.352 −0.305705
$$113$$ −1370.20 −1.14069 −0.570345 0.821405i $$-0.693190\pi$$
−0.570345 + 0.821405i $$0.693190\pi$$
$$114$$ −761.261 −0.625426
$$115$$ 0 0
$$116$$ 61.6670 0.0493589
$$117$$ −334.902 −0.264630
$$118$$ 1592.84 1.24265
$$119$$ 610.578 0.470349
$$120$$ 0 0
$$121$$ 132.432 0.0994984
$$122$$ −829.352 −0.615459
$$123$$ 2259.09 1.65606
$$124$$ −255.593 −0.185104
$$125$$ 0 0
$$126$$ −313.387 −0.221577
$$127$$ −1213.49 −0.847873 −0.423936 0.905692i $$-0.639352\pi$$
−0.423936 + 0.905692i $$0.639352\pi$$
$$128$$ −993.551 −0.686081
$$129$$ −1505.56 −1.02757
$$130$$ 0 0
$$131$$ −1982.42 −1.32217 −0.661087 0.750309i $$-0.729904\pi$$
−0.661087 + 0.750309i $$0.729904\pi$$
$$132$$ 334.545 0.220594
$$133$$ −309.578 −0.201833
$$134$$ 37.5189 0.0241876
$$135$$ 0 0
$$136$$ 2100.67 1.32449
$$137$$ −2210.95 −1.37879 −0.689394 0.724386i $$-0.742123\pi$$
−0.689394 + 0.724386i $$0.742123\pi$$
$$138$$ −3755.34 −2.31649
$$139$$ 528.039 0.322213 0.161107 0.986937i $$-0.448494\pi$$
0.161107 + 0.986937i $$0.448494\pi$$
$$140$$ 0 0
$$141$$ 77.7267 0.0464239
$$142$$ 2461.67 1.45478
$$143$$ −739.969 −0.432722
$$144$$ −896.235 −0.518655
$$145$$ 0 0
$$146$$ 2131.95 1.20851
$$147$$ −326.186 −0.183016
$$148$$ 481.938 0.267669
$$149$$ −328.372 −0.180545 −0.0902727 0.995917i $$-0.528774\pi$$
−0.0902727 + 0.995917i $$0.528774\pi$$
$$150$$ 0 0
$$151$$ 1029.43 0.554793 0.277396 0.960756i $$-0.410528\pi$$
0.277396 + 0.960756i $$0.410528\pi$$
$$152$$ −1065.09 −0.568358
$$153$$ 1510.20 0.797987
$$154$$ −692.432 −0.362323
$$155$$ 0 0
$$156$$ −169.159 −0.0868177
$$157$$ −525.098 −0.266926 −0.133463 0.991054i $$-0.542610\pi$$
−0.133463 + 0.991054i $$0.542610\pi$$
$$158$$ −404.094 −0.203468
$$159$$ −1391.41 −0.694001
$$160$$ 0 0
$$161$$ −1527.17 −0.747562
$$162$$ 2318.69 1.12453
$$163$$ −1002.63 −0.481790 −0.240895 0.970551i $$-0.577441\pi$$
−0.240895 + 0.970551i $$0.577441\pi$$
$$164$$ 445.823 0.212274
$$165$$ 0 0
$$166$$ −2680.24 −1.25317
$$167$$ 359.422 0.166544 0.0832722 0.996527i $$-0.473463\pi$$
0.0832722 + 0.996527i $$0.473463\pi$$
$$168$$ −1122.23 −0.515369
$$169$$ −1822.84 −0.829696
$$170$$ 0 0
$$171$$ −765.706 −0.342427
$$172$$ −297.117 −0.131715
$$173$$ −3293.65 −1.44747 −0.723733 0.690080i $$-0.757575\pi$$
−0.723733 + 0.690080i $$0.757575\pi$$
$$174$$ −808.007 −0.352039
$$175$$ 0 0
$$176$$ −1980.24 −0.848104
$$177$$ 4100.62 1.74137
$$178$$ 440.167 0.185348
$$179$$ 2978.82 1.24384 0.621921 0.783080i $$-0.286353\pi$$
0.621921 + 0.783080i $$0.286353\pi$$
$$180$$ 0 0
$$181$$ 1462.31 0.600514 0.300257 0.953858i $$-0.402928\pi$$
0.300257 + 0.953858i $$0.402928\pi$$
$$182$$ 350.121 0.142597
$$183$$ −2135.09 −0.862460
$$184$$ −5254.16 −2.10512
$$185$$ 0 0
$$186$$ 3348.97 1.32021
$$187$$ 3336.79 1.30487
$$188$$ 15.3391 0.00595064
$$189$$ 451.362 0.173713
$$190$$ 0 0
$$191$$ −374.923 −0.142034 −0.0710169 0.997475i $$-0.522624\pi$$
−0.0710169 + 0.997475i $$0.522624\pi$$
$$192$$ −3769.09 −1.41672
$$193$$ −733.028 −0.273391 −0.136696 0.990613i $$-0.543648\pi$$
−0.136696 + 0.990613i $$0.543648\pi$$
$$194$$ 2740.61 1.01425
$$195$$ 0 0
$$196$$ −64.3717 −0.0234591
$$197$$ 2093.24 0.757043 0.378521 0.925593i $$-0.376433\pi$$
0.378521 + 0.925593i $$0.376433\pi$$
$$198$$ −1712.65 −0.614711
$$199$$ 2865.04 1.02059 0.510295 0.860000i $$-0.329536\pi$$
0.510295 + 0.860000i $$0.329536\pi$$
$$200$$ 0 0
$$201$$ 96.5887 0.0338948
$$202$$ 625.330 0.217812
$$203$$ −328.588 −0.113608
$$204$$ 762.801 0.261798
$$205$$ 0 0
$$206$$ −4343.03 −1.46890
$$207$$ −3777.27 −1.26830
$$208$$ 1001.29 0.333783
$$209$$ −1691.84 −0.559936
$$210$$ 0 0
$$211$$ 5643.65 1.84135 0.920674 0.390331i $$-0.127640\pi$$
0.920674 + 0.390331i $$0.127640\pi$$
$$212$$ −274.590 −0.0889573
$$213$$ 6337.33 2.03862
$$214$$ 3896.47 1.24466
$$215$$ 0 0
$$216$$ 1552.89 0.489172
$$217$$ 1361.91 0.426048
$$218$$ 3238.46 1.00613
$$219$$ 5488.51 1.69351
$$220$$ 0 0
$$221$$ −1687.21 −0.513549
$$222$$ −6314.71 −1.90908
$$223$$ 6369.16 1.91260 0.956302 0.292381i $$-0.0944477\pi$$
0.956302 + 0.292381i $$0.0944477\pi$$
$$224$$ −411.699 −0.122803
$$225$$ 0 0
$$226$$ 3543.05 1.04283
$$227$$ 1015.67 0.296972 0.148486 0.988914i $$-0.452560\pi$$
0.148486 + 0.988914i $$0.452560\pi$$
$$228$$ −386.758 −0.112341
$$229$$ 4108.35 1.18554 0.592768 0.805373i $$-0.298035\pi$$
0.592768 + 0.805373i $$0.298035\pi$$
$$230$$ 0 0
$$231$$ −1782.60 −0.507733
$$232$$ −1130.50 −0.319917
$$233$$ −608.431 −0.171071 −0.0855357 0.996335i $$-0.527260\pi$$
−0.0855357 + 0.996335i $$0.527260\pi$$
$$234$$ 865.984 0.241928
$$235$$ 0 0
$$236$$ 809.244 0.223209
$$237$$ −1040.30 −0.285126
$$238$$ −1578.82 −0.430000
$$239$$ −5054.44 −1.36797 −0.683985 0.729496i $$-0.739755\pi$$
−0.683985 + 0.729496i $$0.739755\pi$$
$$240$$ 0 0
$$241$$ 4.86782 0.00130109 0.000650547 1.00000i $$-0.499793\pi$$
0.000650547 1.00000i $$0.499793\pi$$
$$242$$ −342.442 −0.0909628
$$243$$ 4228.27 1.11623
$$244$$ −421.352 −0.110551
$$245$$ 0 0
$$246$$ −5841.52 −1.51399
$$247$$ 855.458 0.220370
$$248$$ 4685.60 1.19974
$$249$$ −6900.02 −1.75611
$$250$$ 0 0
$$251$$ −547.921 −0.137787 −0.0688934 0.997624i $$-0.521947\pi$$
−0.0688934 + 0.997624i $$0.521947\pi$$
$$252$$ −159.216 −0.0398003
$$253$$ −8345.92 −2.07393
$$254$$ 3137.83 0.775137
$$255$$ 0 0
$$256$$ −1960.46 −0.478629
$$257$$ 1774.61 0.430729 0.215364 0.976534i $$-0.430906\pi$$
0.215364 + 0.976534i $$0.430906\pi$$
$$258$$ 3893.05 0.939421
$$259$$ −2567.97 −0.616084
$$260$$ 0 0
$$261$$ −812.725 −0.192745
$$262$$ 5126.11 1.20875
$$263$$ 1199.09 0.281138 0.140569 0.990071i $$-0.455107\pi$$
0.140569 + 0.990071i $$0.455107\pi$$
$$264$$ −6132.97 −1.42977
$$265$$ 0 0
$$266$$ 800.502 0.184519
$$267$$ 1133.17 0.259733
$$268$$ 19.0615 0.00434464
$$269$$ 3250.29 0.736706 0.368353 0.929686i $$-0.379922\pi$$
0.368353 + 0.929686i $$0.379922\pi$$
$$270$$ 0 0
$$271$$ −896.143 −0.200874 −0.100437 0.994943i $$-0.532024\pi$$
−0.100437 + 0.994943i $$0.532024\pi$$
$$272$$ −4515.18 −1.00652
$$273$$ 901.352 0.199825
$$274$$ 5717.04 1.26051
$$275$$ 0 0
$$276$$ −1907.90 −0.416095
$$277$$ 386.562 0.0838492 0.0419246 0.999121i $$-0.486651\pi$$
0.0419246 + 0.999121i $$0.486651\pi$$
$$278$$ −1365.40 −0.294572
$$279$$ 3368.53 0.722826
$$280$$ 0 0
$$281$$ −3335.10 −0.708025 −0.354013 0.935241i $$-0.615183\pi$$
−0.354013 + 0.935241i $$0.615183\pi$$
$$282$$ −200.985 −0.0424414
$$283$$ −5412.26 −1.13684 −0.568419 0.822739i $$-0.692445\pi$$
−0.568419 + 0.822739i $$0.692445\pi$$
$$284$$ 1250.65 0.261311
$$285$$ 0 0
$$286$$ 1913.40 0.395601
$$287$$ −2375.54 −0.488584
$$288$$ −1018.29 −0.208345
$$289$$ 2695.27 0.548600
$$290$$ 0 0
$$291$$ 7055.42 1.42129
$$292$$ 1083.14 0.217075
$$293$$ −282.211 −0.0562695 −0.0281347 0.999604i $$-0.508957\pi$$
−0.0281347 + 0.999604i $$0.508957\pi$$
$$294$$ 843.447 0.167316
$$295$$ 0 0
$$296$$ −8835.01 −1.73488
$$297$$ 2466.68 0.481924
$$298$$ 849.099 0.165057
$$299$$ 4220.03 0.816222
$$300$$ 0 0
$$301$$ 1583.17 0.303163
$$302$$ −2661.88 −0.507199
$$303$$ 1609.85 0.305226
$$304$$ 2289.31 0.431910
$$305$$ 0 0
$$306$$ −3905.04 −0.729531
$$307$$ −1919.67 −0.356878 −0.178439 0.983951i $$-0.557105\pi$$
−0.178439 + 0.983951i $$0.557105\pi$$
$$308$$ −351.790 −0.0650815
$$309$$ −11180.7 −2.05841
$$310$$ 0 0
$$311$$ 1213.31 0.221223 0.110612 0.993864i $$-0.464719\pi$$
0.110612 + 0.993864i $$0.464719\pi$$
$$312$$ 3101.07 0.562703
$$313$$ 1434.00 0.258960 0.129480 0.991582i $$-0.458669\pi$$
0.129480 + 0.991582i $$0.458669\pi$$
$$314$$ 1357.79 0.244028
$$315$$ 0 0
$$316$$ −205.300 −0.0365475
$$317$$ −6496.95 −1.15112 −0.575560 0.817760i $$-0.695216\pi$$
−0.575560 + 0.817760i $$0.695216\pi$$
$$318$$ 3597.89 0.634465
$$319$$ −1795.72 −0.315176
$$320$$ 0 0
$$321$$ 10031.1 1.74418
$$322$$ 3948.92 0.683432
$$323$$ −3857.58 −0.664524
$$324$$ 1178.01 0.201991
$$325$$ 0 0
$$326$$ 2592.58 0.440459
$$327$$ 8337.11 1.40992
$$328$$ −8172.96 −1.37584
$$329$$ −81.7333 −0.0136964
$$330$$ 0 0
$$331$$ −9683.88 −1.60808 −0.804039 0.594576i $$-0.797320\pi$$
−0.804039 + 0.594576i $$0.797320\pi$$
$$332$$ −1361.70 −0.225099
$$333$$ −6351.58 −1.04524
$$334$$ −929.389 −0.152257
$$335$$ 0 0
$$336$$ 2412.12 0.391643
$$337$$ −29.1319 −0.00470895 −0.00235447 0.999997i $$-0.500749\pi$$
−0.00235447 + 0.999997i $$0.500749\pi$$
$$338$$ 4713.48 0.758520
$$339$$ 9121.24 1.46135
$$340$$ 0 0
$$341$$ 7442.80 1.18197
$$342$$ 1979.95 0.313051
$$343$$ 343.000 0.0539949
$$344$$ 5446.83 0.853701
$$345$$ 0 0
$$346$$ 8516.68 1.32329
$$347$$ 7848.58 1.21422 0.607110 0.794618i $$-0.292329\pi$$
0.607110 + 0.794618i $$0.292329\pi$$
$$348$$ −410.508 −0.0632343
$$349$$ −10269.6 −1.57513 −0.787567 0.616229i $$-0.788659\pi$$
−0.787567 + 0.616229i $$0.788659\pi$$
$$350$$ 0 0
$$351$$ −1247.25 −0.189668
$$352$$ −2249.93 −0.340686
$$353$$ −2799.93 −0.422168 −0.211084 0.977468i $$-0.567699\pi$$
−0.211084 + 0.977468i $$0.567699\pi$$
$$354$$ −10603.3 −1.59198
$$355$$ 0 0
$$356$$ 223.627 0.0332927
$$357$$ −4064.53 −0.602570
$$358$$ −7702.60 −1.13714
$$359$$ −3163.29 −0.465048 −0.232524 0.972591i $$-0.574698\pi$$
−0.232524 + 0.972591i $$0.574698\pi$$
$$360$$ 0 0
$$361$$ −4903.11 −0.714844
$$362$$ −3781.23 −0.548998
$$363$$ −881.583 −0.127469
$$364$$ 177.879 0.0256137
$$365$$ 0 0
$$366$$ 5520.88 0.788472
$$367$$ −3182.85 −0.452706 −0.226353 0.974045i $$-0.572680\pi$$
−0.226353 + 0.974045i $$0.572680\pi$$
$$368$$ 11293.3 1.59974
$$369$$ −5875.62 −0.828923
$$370$$ 0 0
$$371$$ 1463.14 0.204750
$$372$$ 1701.45 0.237139
$$373$$ 2615.14 0.363021 0.181510 0.983389i $$-0.441901\pi$$
0.181510 + 0.983389i $$0.441901\pi$$
$$374$$ −8628.23 −1.19293
$$375$$ 0 0
$$376$$ −281.201 −0.0385687
$$377$$ 907.989 0.124042
$$378$$ −1167.12 −0.158811
$$379$$ −672.434 −0.0911362 −0.0455681 0.998961i $$-0.514510\pi$$
−0.0455681 + 0.998961i $$0.514510\pi$$
$$380$$ 0 0
$$381$$ 8078.03 1.08622
$$382$$ 969.470 0.129849
$$383$$ −1169.86 −0.156075 −0.0780377 0.996950i $$-0.524865\pi$$
−0.0780377 + 0.996950i $$0.524865\pi$$
$$384$$ 6613.92 0.878946
$$385$$ 0 0
$$386$$ 1895.45 0.249938
$$387$$ 3915.78 0.514342
$$388$$ 1392.36 0.182182
$$389$$ −1122.22 −0.146269 −0.0731347 0.997322i $$-0.523300\pi$$
−0.0731347 + 0.997322i $$0.523300\pi$$
$$390$$ 0 0
$$391$$ −19029.7 −2.46131
$$392$$ 1180.08 0.152049
$$393$$ 13196.7 1.69385
$$394$$ −5412.68 −0.692099
$$395$$ 0 0
$$396$$ −870.113 −0.110416
$$397$$ 1985.93 0.251060 0.125530 0.992090i $$-0.459937\pi$$
0.125530 + 0.992090i $$0.459937\pi$$
$$398$$ −7408.38 −0.933037
$$399$$ 2060.81 0.258571
$$400$$ 0 0
$$401$$ −4172.38 −0.519597 −0.259799 0.965663i $$-0.583656\pi$$
−0.259799 + 0.965663i $$0.583656\pi$$
$$402$$ −249.758 −0.0309870
$$403$$ −3763.37 −0.465178
$$404$$ 317.699 0.0391240
$$405$$ 0 0
$$406$$ 849.658 0.103862
$$407$$ −14033.9 −1.70918
$$408$$ −13983.9 −1.69682
$$409$$ −11700.8 −1.41459 −0.707295 0.706919i $$-0.750085\pi$$
−0.707295 + 0.706919i $$0.750085\pi$$
$$410$$ 0 0
$$411$$ 14718.0 1.76638
$$412$$ −2206.47 −0.263848
$$413$$ −4312.00 −0.513752
$$414$$ 9767.22 1.15950
$$415$$ 0 0
$$416$$ 1137.65 0.134082
$$417$$ −3515.08 −0.412791
$$418$$ 4374.72 0.511901
$$419$$ 2733.20 0.318677 0.159339 0.987224i $$-0.449064\pi$$
0.159339 + 0.987224i $$0.449064\pi$$
$$420$$ 0 0
$$421$$ 13549.4 1.56854 0.784272 0.620417i $$-0.213037\pi$$
0.784272 + 0.620417i $$0.213037\pi$$
$$422$$ −14593.3 −1.68339
$$423$$ −202.158 −0.0232370
$$424$$ 5033.87 0.576571
$$425$$ 0 0
$$426$$ −16387.0 −1.86374
$$427$$ 2245.15 0.254450
$$428$$ 1979.60 0.223569
$$429$$ 4925.86 0.554366
$$430$$ 0 0
$$431$$ −6429.25 −0.718530 −0.359265 0.933236i $$-0.616973\pi$$
−0.359265 + 0.933236i $$0.616973\pi$$
$$432$$ −3337.79 −0.371735
$$433$$ −8022.03 −0.890333 −0.445166 0.895448i $$-0.646855\pi$$
−0.445166 + 0.895448i $$0.646855\pi$$
$$434$$ −3521.61 −0.389499
$$435$$ 0 0
$$436$$ 1645.30 0.180724
$$437$$ 9648.50 1.05618
$$438$$ −14192.1 −1.54823
$$439$$ −5569.88 −0.605549 −0.302774 0.953062i $$-0.597913\pi$$
−0.302774 + 0.953062i $$0.597913\pi$$
$$440$$ 0 0
$$441$$ 848.372 0.0916069
$$442$$ 4362.77 0.469493
$$443$$ 5486.21 0.588392 0.294196 0.955745i $$-0.404948\pi$$
0.294196 + 0.955745i $$0.404948\pi$$
$$444$$ −3208.19 −0.342914
$$445$$ 0 0
$$446$$ −16469.3 −1.74853
$$447$$ 2185.92 0.231299
$$448$$ 3963.38 0.417973
$$449$$ −7232.67 −0.760203 −0.380101 0.924945i $$-0.624111\pi$$
−0.380101 + 0.924945i $$0.624111\pi$$
$$450$$ 0 0
$$451$$ −12982.3 −1.35546
$$452$$ 1800.05 0.187317
$$453$$ −6852.76 −0.710752
$$454$$ −2626.32 −0.271496
$$455$$ 0 0
$$456$$ 7090.16 0.728130
$$457$$ 2900.51 0.296893 0.148446 0.988920i $$-0.452573\pi$$
0.148446 + 0.988920i $$0.452573\pi$$
$$458$$ −10623.3 −1.08383
$$459$$ 5624.31 0.571940
$$460$$ 0 0
$$461$$ 6073.57 0.613611 0.306805 0.951772i $$-0.400740\pi$$
0.306805 + 0.951772i $$0.400740\pi$$
$$462$$ 4609.42 0.464176
$$463$$ 18922.8 1.89939 0.949693 0.313183i $$-0.101395\pi$$
0.949693 + 0.313183i $$0.101395\pi$$
$$464$$ 2429.88 0.243113
$$465$$ 0 0
$$466$$ 1573.27 0.156396
$$467$$ 6776.71 0.671496 0.335748 0.941952i $$-0.391011\pi$$
0.335748 + 0.941952i $$0.391011\pi$$
$$468$$ 439.963 0.0434558
$$469$$ −101.568 −0.00999991
$$470$$ 0 0
$$471$$ 3495.50 0.341962
$$472$$ −14835.3 −1.44672
$$473$$ 8651.96 0.841052
$$474$$ 2689.99 0.260666
$$475$$ 0 0
$$476$$ −802.121 −0.0772377
$$477$$ 3618.90 0.347375
$$478$$ 13069.7 1.25062
$$479$$ 2397.32 0.228677 0.114338 0.993442i $$-0.463525\pi$$
0.114338 + 0.993442i $$0.463525\pi$$
$$480$$ 0 0
$$481$$ 7096.09 0.672669
$$482$$ −12.5871 −0.00118948
$$483$$ 10166.1 0.957711
$$484$$ −173.977 −0.0163390
$$485$$ 0 0
$$486$$ −10933.4 −1.02047
$$487$$ −5586.17 −0.519781 −0.259890 0.965638i $$-0.583686\pi$$
−0.259890 + 0.965638i $$0.583686\pi$$
$$488$$ 7724.35 0.716526
$$489$$ 6674.33 0.617227
$$490$$ 0 0
$$491$$ 537.392 0.0493934 0.0246967 0.999695i $$-0.492138\pi$$
0.0246967 + 0.999695i $$0.492138\pi$$
$$492$$ −2967.78 −0.271947
$$493$$ −4094.46 −0.374047
$$494$$ −2212.03 −0.201466
$$495$$ 0 0
$$496$$ −10071.2 −0.911716
$$497$$ −6664.00 −0.601451
$$498$$ 17842.0 1.60546
$$499$$ 598.965 0.0537342 0.0268671 0.999639i $$-0.491447\pi$$
0.0268671 + 0.999639i $$0.491447\pi$$
$$500$$ 0 0
$$501$$ −2392.62 −0.213362
$$502$$ 1416.81 0.125966
$$503$$ −4426.76 −0.392405 −0.196202 0.980563i $$-0.562861\pi$$
−0.196202 + 0.980563i $$0.562861\pi$$
$$504$$ 2918.79 0.257963
$$505$$ 0 0
$$506$$ 21580.8 1.89601
$$507$$ 12134.4 1.06293
$$508$$ 1594.17 0.139232
$$509$$ 17727.7 1.54374 0.771872 0.635779i $$-0.219321\pi$$
0.771872 + 0.635779i $$0.219321\pi$$
$$510$$ 0 0
$$511$$ −5771.43 −0.499634
$$512$$ 13017.7 1.12365
$$513$$ −2851.66 −0.245427
$$514$$ −4588.77 −0.393778
$$515$$ 0 0
$$516$$ 1977.86 0.168741
$$517$$ −446.671 −0.0379972
$$518$$ 6640.22 0.563233
$$519$$ 21925.4 1.85437
$$520$$ 0 0
$$521$$ 8662.79 0.728453 0.364226 0.931310i $$-0.381333\pi$$
0.364226 + 0.931310i $$0.381333\pi$$
$$522$$ 2101.53 0.176210
$$523$$ 7770.40 0.649667 0.324833 0.945771i $$-0.394692\pi$$
0.324833 + 0.945771i $$0.394692\pi$$
$$524$$ 2604.32 0.217119
$$525$$ 0 0
$$526$$ −3100.60 −0.257020
$$527$$ 16970.4 1.40274
$$528$$ 13182.2 1.08652
$$529$$ 35429.6 2.91194
$$530$$ 0 0
$$531$$ −10665.2 −0.871624
$$532$$ 406.695 0.0331437
$$533$$ 6564.34 0.533458
$$534$$ −2930.12 −0.237451
$$535$$ 0 0
$$536$$ −349.440 −0.0281595
$$537$$ −19829.6 −1.59350
$$538$$ −8404.56 −0.673506
$$539$$ 1874.49 0.149796
$$540$$ 0 0
$$541$$ 21641.0 1.71981 0.859906 0.510453i $$-0.170522\pi$$
0.859906 + 0.510453i $$0.170522\pi$$
$$542$$ 2317.23 0.183642
$$543$$ −9734.41 −0.769325
$$544$$ −5130.09 −0.404321
$$545$$ 0 0
$$546$$ −2330.70 −0.182683
$$547$$ −7489.29 −0.585409 −0.292705 0.956203i $$-0.594555\pi$$
−0.292705 + 0.956203i $$0.594555\pi$$
$$548$$ 2904.54 0.226416
$$549$$ 5553.11 0.431696
$$550$$ 0 0
$$551$$ 2075.99 0.160508
$$552$$ 34976.2 2.69689
$$553$$ 1093.93 0.0841201
$$554$$ −999.566 −0.0766561
$$555$$ 0 0
$$556$$ −693.689 −0.0529118
$$557$$ −25297.9 −1.92443 −0.962214 0.272295i $$-0.912217\pi$$
−0.962214 + 0.272295i $$0.912217\pi$$
$$558$$ −8710.29 −0.660817
$$559$$ −4374.77 −0.331007
$$560$$ 0 0
$$561$$ −22212.5 −1.67168
$$562$$ 8623.85 0.647286
$$563$$ −15661.3 −1.17237 −0.586186 0.810177i $$-0.699371\pi$$
−0.586186 + 0.810177i $$0.699371\pi$$
$$564$$ −102.110 −0.00762343
$$565$$ 0 0
$$566$$ 13994.9 1.03931
$$567$$ −6276.94 −0.464915
$$568$$ −22927.3 −1.69367
$$569$$ −9982.75 −0.735498 −0.367749 0.929925i $$-0.619871\pi$$
−0.367749 + 0.929925i $$0.619871\pi$$
$$570$$ 0 0
$$571$$ −11583.6 −0.848966 −0.424483 0.905436i $$-0.639544\pi$$
−0.424483 + 0.905436i $$0.639544\pi$$
$$572$$ 972.103 0.0710589
$$573$$ 2495.81 0.181961
$$574$$ 6142.63 0.446670
$$575$$ 0 0
$$576$$ 9802.97 0.709127
$$577$$ 595.378 0.0429565 0.0214783 0.999769i $$-0.493163\pi$$
0.0214783 + 0.999769i $$0.493163\pi$$
$$578$$ −6969.39 −0.501537
$$579$$ 4879.66 0.350245
$$580$$ 0 0
$$581$$ 7255.70 0.518102
$$582$$ −18243.8 −1.29936
$$583$$ 7996.00 0.568028
$$584$$ −19856.4 −1.40696
$$585$$ 0 0
$$586$$ 729.738 0.0514423
$$587$$ 15750.3 1.10747 0.553736 0.832693i $$-0.313202\pi$$
0.553736 + 0.832693i $$0.313202\pi$$
$$588$$ 428.513 0.0300537
$$589$$ −8604.42 −0.601934
$$590$$ 0 0
$$591$$ −13934.4 −0.969856
$$592$$ 18990.0 1.31838
$$593$$ 417.878 0.0289379 0.0144690 0.999895i $$-0.495394\pi$$
0.0144690 + 0.999895i $$0.495394\pi$$
$$594$$ −6378.31 −0.440581
$$595$$ 0 0
$$596$$ 431.385 0.0296480
$$597$$ −19072.2 −1.30749
$$598$$ −10912.1 −0.746201
$$599$$ −19997.3 −1.36406 −0.682028 0.731326i $$-0.738902\pi$$
−0.682028 + 0.731326i $$0.738902\pi$$
$$600$$ 0 0
$$601$$ −15992.6 −1.08545 −0.542723 0.839912i $$-0.682607\pi$$
−0.542723 + 0.839912i $$0.682607\pi$$
$$602$$ −4093.73 −0.277156
$$603$$ −251.216 −0.0169657
$$604$$ −1352.37 −0.0911045
$$605$$ 0 0
$$606$$ −4162.73 −0.279042
$$607$$ −14159.2 −0.946793 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$608$$ 2601.08 0.173499
$$609$$ 2187.36 0.145544
$$610$$ 0 0
$$611$$ 225.854 0.0149543
$$612$$ −1983.96 −0.131040
$$613$$ 4629.41 0.305025 0.152512 0.988302i $$-0.451264\pi$$
0.152512 + 0.988302i $$0.451264\pi$$
$$614$$ 4963.87 0.326263
$$615$$ 0 0
$$616$$ 6449.11 0.421821
$$617$$ 23165.3 1.51151 0.755753 0.654857i $$-0.227271\pi$$
0.755753 + 0.654857i $$0.227271\pi$$
$$618$$ 28910.9 1.88182
$$619$$ 12370.6 0.803258 0.401629 0.915803i $$-0.368444\pi$$
0.401629 + 0.915803i $$0.368444\pi$$
$$620$$ 0 0
$$621$$ −14067.4 −0.909028
$$622$$ −3137.36 −0.202245
$$623$$ −1191.58 −0.0766285
$$624$$ −6665.43 −0.427613
$$625$$ 0 0
$$626$$ −3708.02 −0.236745
$$627$$ 11262.3 0.717341
$$628$$ 689.826 0.0438329
$$629$$ −31998.9 −2.02842
$$630$$ 0 0
$$631$$ 13980.2 0.882002 0.441001 0.897507i $$-0.354623\pi$$
0.441001 + 0.897507i $$0.354623\pi$$
$$632$$ 3763.61 0.236880
$$633$$ −37568.9 −2.35897
$$634$$ 16799.7 1.05237
$$635$$ 0 0
$$636$$ 1827.91 0.113964
$$637$$ −947.814 −0.0589541
$$638$$ 4643.36 0.288139
$$639$$ −16482.7 −1.02041
$$640$$ 0 0
$$641$$ −16060.9 −0.989655 −0.494828 0.868991i $$-0.664769\pi$$
−0.494828 + 0.868991i $$0.664769\pi$$
$$642$$ −25938.3 −1.59455
$$643$$ 4502.17 0.276125 0.138063 0.990424i $$-0.455913\pi$$
0.138063 + 0.990424i $$0.455913\pi$$
$$644$$ 2006.25 0.122760
$$645$$ 0 0
$$646$$ 9974.87 0.607517
$$647$$ −29414.8 −1.78735 −0.893675 0.448715i $$-0.851882\pi$$
−0.893675 + 0.448715i $$0.851882\pi$$
$$648$$ −21595.6 −1.30919
$$649$$ −23565.0 −1.42528
$$650$$ 0 0
$$651$$ −9066.03 −0.545815
$$652$$ 1317.16 0.0791164
$$653$$ −13013.6 −0.779882 −0.389941 0.920840i $$-0.627505\pi$$
−0.389941 + 0.920840i $$0.627505\pi$$
$$654$$ −21558.0 −1.28897
$$655$$ 0 0
$$656$$ 17566.9 1.04554
$$657$$ −14275.0 −0.847671
$$658$$ 211.345 0.0125214
$$659$$ 23474.2 1.38759 0.693797 0.720171i $$-0.255937\pi$$
0.693797 + 0.720171i $$0.255937\pi$$
$$660$$ 0 0
$$661$$ −9266.36 −0.545264 −0.272632 0.962118i $$-0.587894\pi$$
−0.272632 + 0.962118i $$0.587894\pi$$
$$662$$ 25040.4 1.47013
$$663$$ 11231.5 0.657914
$$664$$ 24963.0 1.45896
$$665$$ 0 0
$$666$$ 16423.8 0.955572
$$667$$ 10241.0 0.594501
$$668$$ −472.176 −0.0273489
$$669$$ −42398.6 −2.45026
$$670$$ 0 0
$$671$$ 12269.7 0.705909
$$672$$ 2740.62 0.157324
$$673$$ 25067.2 1.43576 0.717882 0.696164i $$-0.245112\pi$$
0.717882 + 0.696164i $$0.245112\pi$$
$$674$$ 75.3288 0.00430498
$$675$$ 0 0
$$676$$ 2394.68 0.136247
$$677$$ 22409.6 1.27219 0.636093 0.771613i $$-0.280550\pi$$
0.636093 + 0.771613i $$0.280550\pi$$
$$678$$ −23585.6 −1.33599
$$679$$ −7419.11 −0.419322
$$680$$ 0 0
$$681$$ −6761.20 −0.380455
$$682$$ −19245.5 −1.08057
$$683$$ 8757.53 0.490626 0.245313 0.969444i $$-0.421109\pi$$
0.245313 + 0.969444i $$0.421109\pi$$
$$684$$ 1005.91 0.0562311
$$685$$ 0 0
$$686$$ −886.925 −0.0493629
$$687$$ −27348.7 −1.51880
$$688$$ −11707.4 −0.648750
$$689$$ −4043.09 −0.223555
$$690$$ 0 0
$$691$$ 8468.42 0.466214 0.233107 0.972451i $$-0.425111\pi$$
0.233107 + 0.972451i $$0.425111\pi$$
$$692$$ 4326.90 0.237694
$$693$$ 4636.33 0.254141
$$694$$ −20294.8 −1.11006
$$695$$ 0 0
$$696$$ 7525.54 0.409849
$$697$$ −29601.0 −1.60864
$$698$$ 26555.1 1.44001
$$699$$ 4050.23 0.219162
$$700$$ 0 0
$$701$$ 15996.9 0.861906 0.430953 0.902374i $$-0.358177\pi$$
0.430953 + 0.902374i $$0.358177\pi$$
$$702$$ 3225.12 0.173397
$$703$$ 16224.2 0.870423
$$704$$ 21659.8 1.15956
$$705$$ 0 0
$$706$$ 7240.03 0.385952
$$707$$ −1692.83 −0.0900503
$$708$$ −5387.02 −0.285956
$$709$$ 19903.0 1.05426 0.527131 0.849784i $$-0.323268\pi$$
0.527131 + 0.849784i $$0.323268\pi$$
$$710$$ 0 0
$$711$$ 2705.70 0.142717
$$712$$ −4099.58 −0.215784
$$713$$ −42446.1 −2.22948
$$714$$ 10510.0 0.550878
$$715$$ 0 0
$$716$$ −3913.30 −0.204256
$$717$$ 33646.7 1.75252
$$718$$ 8179.59 0.425153
$$719$$ −11073.1 −0.574347 −0.287174 0.957879i $$-0.592716\pi$$
−0.287174 + 0.957879i $$0.592716\pi$$
$$720$$ 0 0
$$721$$ 11757.0 0.607288
$$722$$ 12678.4 0.653520
$$723$$ −32.4043 −0.00166685
$$724$$ −1921.06 −0.0986125
$$725$$ 0 0
$$726$$ 2279.58 0.116533
$$727$$ −31652.7 −1.61476 −0.807382 0.590029i $$-0.799116\pi$$
−0.807382 + 0.590029i $$0.799116\pi$$
$$728$$ −3260.92 −0.166013
$$729$$ −3935.94 −0.199967
$$730$$ 0 0
$$731$$ 19727.5 0.998149
$$732$$ 2804.88 0.141628
$$733$$ −16958.3 −0.854528 −0.427264 0.904127i $$-0.640523\pi$$
−0.427264 + 0.904127i $$0.640523\pi$$
$$734$$ 8230.16 0.413870
$$735$$ 0 0
$$736$$ 12831.3 0.642618
$$737$$ −555.065 −0.0277423
$$738$$ 15193.1 0.757813
$$739$$ −11616.6 −0.578245 −0.289123 0.957292i $$-0.593364\pi$$
−0.289123 + 0.957292i $$0.593364\pi$$
$$740$$ 0 0
$$741$$ −5694.66 −0.282319
$$742$$ −3783.36 −0.187185
$$743$$ −15928.0 −0.786464 −0.393232 0.919439i $$-0.628643\pi$$
−0.393232 + 0.919439i $$0.628643\pi$$
$$744$$ −31191.4 −1.53700
$$745$$ 0 0
$$746$$ −6762.19 −0.331879
$$747$$ 17946.1 0.879003
$$748$$ −4383.57 −0.214277
$$749$$ −10548.2 −0.514582
$$750$$ 0 0
$$751$$ 25571.9 1.24252 0.621260 0.783604i $$-0.286621\pi$$
0.621260 + 0.783604i $$0.286621\pi$$
$$752$$ 604.412 0.0293094
$$753$$ 3647.43 0.176520
$$754$$ −2347.87 −0.113401
$$755$$ 0 0
$$756$$ −592.958 −0.0285260
$$757$$ −6202.41 −0.297794 −0.148897 0.988853i $$-0.547572\pi$$
−0.148897 + 0.988853i $$0.547572\pi$$
$$758$$ 1738.77 0.0833179
$$759$$ 55557.6 2.65693
$$760$$ 0 0
$$761$$ −29199.1 −1.39089 −0.695444 0.718580i $$-0.744792\pi$$
−0.695444 + 0.718580i $$0.744792\pi$$
$$762$$ −20888.1 −0.993037
$$763$$ −8766.87 −0.415966
$$764$$ 492.539 0.0233239
$$765$$ 0 0
$$766$$ 3025.00 0.142686
$$767$$ 11915.4 0.560938
$$768$$ 13050.5 0.613177
$$769$$ 21838.2 1.02407 0.512033 0.858966i $$-0.328892\pi$$
0.512033 + 0.858966i $$0.328892\pi$$
$$770$$ 0 0
$$771$$ −11813.3 −0.551812
$$772$$ 962.985 0.0448945
$$773$$ 25544.8 1.18859 0.594296 0.804246i $$-0.297431\pi$$
0.594296 + 0.804246i $$0.297431\pi$$
$$774$$ −10125.4 −0.470218
$$775$$ 0 0
$$776$$ −25525.2 −1.18080
$$777$$ 17094.6 0.789273
$$778$$ 2901.82 0.133721
$$779$$ 15008.4 0.690286
$$780$$ 0 0
$$781$$ −36418.6 −1.66858
$$782$$ 49206.6 2.25016
$$783$$ −3026.77 −0.138146
$$784$$ −2536.46 −0.115546
$$785$$ 0 0
$$786$$ −34123.8 −1.54854
$$787$$ 37223.2 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$788$$ −2749.91 −0.124317
$$789$$ −7982.19 −0.360169
$$790$$ 0 0
$$791$$ −9591.42 −0.431140
$$792$$ 15951.1 0.715655
$$793$$ −6204.03 −0.277820
$$794$$ −5135.18 −0.229522
$$795$$ 0 0
$$796$$ −3763.83 −0.167595
$$797$$ 40384.6 1.79485 0.897425 0.441168i $$-0.145436\pi$$
0.897425 + 0.441168i $$0.145436\pi$$
$$798$$ −5328.83 −0.236389
$$799$$ −1018.46 −0.0450945
$$800$$ 0 0
$$801$$ −2947.23 −0.130007
$$802$$ 10788.9 0.475023
$$803$$ −31540.7 −1.38611
$$804$$ −126.889 −0.00556598
$$805$$ 0 0
$$806$$ 9731.28 0.425272
$$807$$ −21636.7 −0.943803
$$808$$ −5824.14 −0.253580
$$809$$ −1955.76 −0.0849948 −0.0424974 0.999097i $$-0.513531\pi$$
−0.0424974 + 0.999097i $$0.513531\pi$$
$$810$$ 0 0
$$811$$ −34301.8 −1.48520 −0.742600 0.669735i $$-0.766408\pi$$
−0.742600 + 0.669735i $$0.766408\pi$$
$$812$$ 431.669 0.0186559
$$813$$ 5965.49 0.257342
$$814$$ 36288.7 1.56255
$$815$$ 0 0
$$816$$ 30056.9 1.28946
$$817$$ −10002.3 −0.428319
$$818$$ 30255.8 1.29324
$$819$$ −2344.31 −0.100021
$$820$$ 0 0
$$821$$ −13665.6 −0.580918 −0.290459 0.956887i $$-0.593808\pi$$
−0.290459 + 0.956887i $$0.593808\pi$$
$$822$$ −38057.5 −1.61485
$$823$$ 21519.5 0.911449 0.455724 0.890121i $$-0.349380\pi$$
0.455724 + 0.890121i $$0.349380\pi$$
$$824$$ 40449.7 1.71011
$$825$$ 0 0
$$826$$ 11149.9 0.469679
$$827$$ −35220.6 −1.48094 −0.740471 0.672088i $$-0.765398\pi$$
−0.740471 + 0.672088i $$0.765398\pi$$
$$828$$ 4962.23 0.208272
$$829$$ 31365.5 1.31408 0.657039 0.753857i $$-0.271809\pi$$
0.657039 + 0.753857i $$0.271809\pi$$
$$830$$ 0 0
$$831$$ −2573.28 −0.107420
$$832$$ −10952.0 −0.456362
$$833$$ 4274.04 0.177775
$$834$$ 9089.24 0.377380
$$835$$ 0 0
$$836$$ 2222.58 0.0919491
$$837$$ 12545.2 0.518070
$$838$$ −7067.48 −0.291339
$$839$$ −28287.1 −1.16398 −0.581990 0.813196i $$-0.697726\pi$$
−0.581990 + 0.813196i $$0.697726\pi$$
$$840$$ 0 0
$$841$$ −22185.5 −0.909653
$$842$$ −35035.8 −1.43398
$$843$$ 22201.2 0.907060
$$844$$ −7414.11 −0.302374
$$845$$ 0 0
$$846$$ 522.738 0.0212436
$$847$$ 927.026 0.0376068
$$848$$ −10819.8 −0.438152
$$849$$ 36028.6 1.45642
$$850$$ 0 0
$$851$$ 80035.0 3.22393
$$852$$ −8325.40 −0.334769
$$853$$ −9405.41 −0.377533 −0.188766 0.982022i $$-0.560449\pi$$
−0.188766 + 0.982022i $$0.560449\pi$$
$$854$$ −5805.47 −0.232622
$$855$$ 0 0
$$856$$ −36290.6 −1.44905
$$857$$ 27966.9 1.11474 0.557369 0.830265i $$-0.311811\pi$$
0.557369 + 0.830265i $$0.311811\pi$$
$$858$$ −12737.2 −0.506809
$$859$$ 6281.11 0.249486 0.124743 0.992189i $$-0.460189\pi$$
0.124743 + 0.992189i $$0.460189\pi$$
$$860$$ 0 0
$$861$$ 15813.6 0.625931
$$862$$ 16624.7 0.656890
$$863$$ 4757.13 0.187642 0.0938208 0.995589i $$-0.470092\pi$$
0.0938208 + 0.995589i $$0.470092\pi$$
$$864$$ −3792.35 −0.149327
$$865$$ 0 0
$$866$$ 20743.3 0.813954
$$867$$ −17942.0 −0.702818
$$868$$ −1789.15 −0.0699629
$$869$$ 5978.28 0.233371
$$870$$ 0 0
$$871$$ 280.663 0.0109184
$$872$$ −30162.1 −1.17135
$$873$$ −18350.3 −0.711414
$$874$$ −24949.0 −0.965574
$$875$$ 0 0
$$876$$ −7210.30 −0.278097
$$877$$ 30240.5 1.16437 0.582184 0.813057i $$-0.302198\pi$$
0.582184 + 0.813057i $$0.302198\pi$$
$$878$$ 14402.5 0.553601
$$879$$ 1878.64 0.0720875
$$880$$ 0 0
$$881$$ 44875.5 1.71611 0.858056 0.513556i $$-0.171672\pi$$
0.858056 + 0.513556i $$0.171672\pi$$
$$882$$ −2193.71 −0.0837483
$$883$$ −4892.13 −0.186448 −0.0932238 0.995645i $$-0.529717\pi$$
−0.0932238 + 0.995645i $$0.529717\pi$$
$$884$$ 2216.51 0.0843317
$$885$$ 0 0
$$886$$ −14186.2 −0.537916
$$887$$ −1761.40 −0.0666765 −0.0333382 0.999444i $$-0.510614\pi$$
−0.0333382 + 0.999444i $$0.510614\pi$$
$$888$$ 58813.4 2.22258
$$889$$ −8494.43 −0.320466
$$890$$ 0 0
$$891$$ −34303.3 −1.28979
$$892$$ −8367.22 −0.314075
$$893$$ 516.384 0.0193507
$$894$$ −5652.33 −0.211457
$$895$$ 0 0
$$896$$ −6954.86 −0.259314
$$897$$ −28092.1 −1.04567
$$898$$ 18702.2 0.694988
$$899$$ −9132.79 −0.338816
$$900$$ 0 0
$$901$$ 18231.8 0.674128
$$902$$ 33569.3 1.23918
$$903$$ −10538.9 −0.388386
$$904$$ −32999.0 −1.21408
$$905$$ 0 0
$$906$$ 17719.8 0.649779
$$907$$ −23689.1 −0.867238 −0.433619 0.901096i $$-0.642764\pi$$
−0.433619 + 0.901096i $$0.642764\pi$$
$$908$$ −1334.30 −0.0487669
$$909$$ −4187.03 −0.152778
$$910$$ 0 0
$$911$$ −13877.3 −0.504692 −0.252346 0.967637i $$-0.581202\pi$$
−0.252346 + 0.967637i $$0.581202\pi$$
$$912$$ −15239.6 −0.553325
$$913$$ 39652.2 1.43735
$$914$$ −7500.09 −0.271423
$$915$$ 0 0
$$916$$ −5397.18 −0.194681
$$917$$ −13876.9 −0.499735
$$918$$ −14543.3 −0.522875
$$919$$ 14331.6 0.514426 0.257213 0.966355i $$-0.417196\pi$$
0.257213 + 0.966355i $$0.417196\pi$$
$$920$$ 0 0
$$921$$ 12779.0 0.457201
$$922$$ −15705.0 −0.560971
$$923$$ 18414.7 0.656692
$$924$$ 2341.81 0.0833767
$$925$$ 0 0
$$926$$ −48930.2 −1.73644
$$927$$ 29079.7 1.03032
$$928$$ 2760.80 0.0976592
$$929$$ 16668.4 0.588668 0.294334 0.955703i $$-0.404902\pi$$
0.294334 + 0.955703i $$0.404902\pi$$
$$930$$ 0 0
$$931$$ −2167.04 −0.0762857
$$932$$ 799.300 0.0280922
$$933$$ −8076.82 −0.283412
$$934$$ −17523.1 −0.613891
$$935$$ 0 0
$$936$$ −8065.52 −0.281656
$$937$$ −30384.9 −1.05937 −0.529685 0.848194i $$-0.677690\pi$$
−0.529685 + 0.848194i $$0.677690\pi$$
$$938$$ 262.632 0.00914206
$$939$$ −9545.94 −0.331757
$$940$$ 0 0
$$941$$ −1196.35 −0.0414452 −0.0207226 0.999785i $$-0.506597\pi$$
−0.0207226 + 0.999785i $$0.506597\pi$$
$$942$$ −9038.63 −0.312627
$$943$$ 74037.5 2.55673
$$944$$ 31886.9 1.09940
$$945$$ 0 0
$$946$$ −22372.1 −0.768901
$$947$$ −1788.41 −0.0613681 −0.0306840 0.999529i $$-0.509769\pi$$
−0.0306840 + 0.999529i $$0.509769\pi$$
$$948$$ 1366.65 0.0468215
$$949$$ 15948.2 0.545523
$$950$$ 0 0
$$951$$ 43249.2 1.47471
$$952$$ 14704.7 0.500612
$$953$$ −8578.60 −0.291593 −0.145796 0.989315i $$-0.546574\pi$$
−0.145796 + 0.989315i $$0.546574\pi$$
$$954$$ −9357.70 −0.317575
$$955$$ 0 0
$$956$$ 6640.06 0.224639
$$957$$ 11953.9 0.403776
$$958$$ −6198.95 −0.209059
$$959$$ −15476.6 −0.521133
$$960$$ 0 0
$$961$$ 8061.99 0.270618
$$962$$ −18349.0 −0.614963
$$963$$ −26089.7 −0.873031
$$964$$ −6.39489 −0.000213657 0
$$965$$ 0 0
$$966$$ −26287.4 −0.875552
$$967$$ 55459.3 1.84431 0.922156 0.386818i $$-0.126426\pi$$
0.922156 + 0.386818i $$0.126426\pi$$
$$968$$ 3189.40 0.105900
$$969$$ 25679.3 0.851330
$$970$$ 0 0
$$971$$ 22047.3 0.728662 0.364331 0.931270i $$-0.381298\pi$$
0.364331 + 0.931270i $$0.381298\pi$$
$$972$$ −5554.72 −0.183300
$$973$$ 3696.27 0.121785
$$974$$ 14444.6 0.475191
$$975$$ 0 0
$$976$$ −16602.7 −0.544507
$$977$$ −14402.3 −0.471617 −0.235809 0.971800i $$-0.575774\pi$$
−0.235809 + 0.971800i $$0.575774\pi$$
$$978$$ −17258.4 −0.564277
$$979$$ −6511.94 −0.212587
$$980$$ 0 0
$$981$$ −21683.9 −0.705721
$$982$$ −1389.58 −0.0451561
$$983$$ −7817.11 −0.253639 −0.126819 0.991926i $$-0.540477\pi$$
−0.126819 + 0.991926i $$0.540477\pi$$
$$984$$ 54406.2 1.76261
$$985$$ 0 0
$$986$$ 10587.4 0.341959
$$987$$ 544.087 0.0175466
$$988$$ −1123.82 −0.0361878
$$989$$ −49342.0 −1.58643
$$990$$ 0 0
$$991$$ 24501.6 0.785386 0.392693 0.919670i $$-0.371543\pi$$
0.392693 + 0.919670i $$0.371543\pi$$
$$992$$ −11442.8 −0.366239
$$993$$ 64464.2 2.06013
$$994$$ 17231.7 0.549855
$$995$$ 0 0
$$996$$ 9064.61 0.288377
$$997$$ 50696.0 1.61039 0.805195 0.593010i $$-0.202060\pi$$
0.805195 + 0.593010i $$0.202060\pi$$
$$998$$ −1548.80 −0.0491245
$$999$$ −23654.8 −0.749152
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 175.4.a.c.1.2 2
3.2 odd 2 1575.4.a.z.1.1 2
5.2 odd 4 175.4.b.c.99.2 4
5.3 odd 4 175.4.b.c.99.3 4
5.4 even 2 35.4.a.b.1.1 2
7.6 odd 2 1225.4.a.m.1.2 2
15.14 odd 2 315.4.a.f.1.2 2
20.19 odd 2 560.4.a.r.1.1 2
35.4 even 6 245.4.e.h.226.2 4
35.9 even 6 245.4.e.h.116.2 4
35.19 odd 6 245.4.e.i.116.2 4
35.24 odd 6 245.4.e.i.226.2 4
35.34 odd 2 245.4.a.k.1.1 2
40.19 odd 2 2240.4.a.bo.1.2 2
40.29 even 2 2240.4.a.bn.1.1 2
105.104 even 2 2205.4.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 5.4 even 2
175.4.a.c.1.2 2 1.1 even 1 trivial
175.4.b.c.99.2 4 5.2 odd 4
175.4.b.c.99.3 4 5.3 odd 4
245.4.a.k.1.1 2 35.34 odd 2
245.4.e.h.116.2 4 35.9 even 6
245.4.e.h.226.2 4 35.4 even 6
245.4.e.i.116.2 4 35.19 odd 6
245.4.e.i.226.2 4 35.24 odd 6
315.4.a.f.1.2 2 15.14 odd 2
560.4.a.r.1.1 2 20.19 odd 2
1225.4.a.m.1.2 2 7.6 odd 2
1575.4.a.z.1.1 2 3.2 odd 2
2205.4.a.u.1.2 2 105.104 even 2
2240.4.a.bn.1.1 2 40.29 even 2
2240.4.a.bo.1.2 2 40.19 odd 2