# Properties

 Label 3332.2.a.g.1.2 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ 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] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.381966 q^{3} -1.38197 q^{5} -2.85410 q^{9} +O(q^{10})$$ $$q-0.381966 q^{3} -1.38197 q^{5} -2.85410 q^{9} +3.23607 q^{13} +0.527864 q^{15} -1.00000 q^{17} +2.47214 q^{19} +5.23607 q^{23} -3.09017 q^{25} +2.23607 q^{27} +2.76393 q^{29} -9.56231 q^{31} +3.70820 q^{37} -1.23607 q^{39} +1.14590 q^{41} +4.85410 q^{43} +3.94427 q^{45} -10.9443 q^{47} +0.381966 q^{51} +12.3262 q^{53} -0.944272 q^{57} -5.23607 q^{59} -12.5623 q^{61} -4.47214 q^{65} -5.09017 q^{67} -2.00000 q^{69} -4.76393 q^{71} +2.85410 q^{73} +1.18034 q^{75} -1.70820 q^{79} +7.70820 q^{81} +1.52786 q^{83} +1.38197 q^{85} -1.05573 q^{87} -13.2361 q^{89} +3.65248 q^{93} -3.41641 q^{95} -3.85410 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 5 q^{5} + q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 5 * q^5 + q^9 $$2 q - 3 q^{3} - 5 q^{5} + q^{9} + 2 q^{13} + 10 q^{15} - 2 q^{17} - 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} + q^{31} - 6 q^{37} + 2 q^{39} + 9 q^{41} + 3 q^{43} - 10 q^{45} - 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} - 6 q^{59} - 5 q^{61} + q^{67} - 4 q^{69} - 14 q^{71} - q^{73} - 20 q^{75} + 10 q^{79} + 2 q^{81} + 12 q^{83} + 5 q^{85} - 20 q^{87} - 22 q^{89} - 24 q^{93} + 20 q^{95} - q^{97}+O(q^{100})$$ 2 * q - 3 * q^3 - 5 * q^5 + q^9 + 2 * q^13 + 10 * q^15 - 2 * q^17 - 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 + q^31 - 6 * q^37 + 2 * q^39 + 9 * q^41 + 3 * q^43 - 10 * q^45 - 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 - 6 * q^59 - 5 * q^61 + q^67 - 4 * q^69 - 14 * q^71 - q^73 - 20 * q^75 + 10 * q^79 + 2 * q^81 + 12 * q^83 + 5 * q^85 - 20 * q^87 - 22 * q^89 - 24 * q^93 + 20 * q^95 - q^97

## 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
$$3$$ −0.381966 −0.220528 −0.110264 0.993902i $$-0.535170\pi$$
−0.110264 + 0.993902i $$0.535170\pi$$
$$4$$ 0 0
$$5$$ −1.38197 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.85410 −0.951367
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 3.23607 0.897524 0.448762 0.893651i $$-0.351865\pi$$
0.448762 + 0.893651i $$0.351865\pi$$
$$14$$ 0 0
$$15$$ 0.527864 0.136294
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 2.47214 0.567147 0.283573 0.958951i $$-0.408480\pi$$
0.283573 + 0.958951i $$0.408480\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.23607 1.09180 0.545898 0.837852i $$-0.316189\pi$$
0.545898 + 0.837852i $$0.316189\pi$$
$$24$$ 0 0
$$25$$ −3.09017 −0.618034
$$26$$ 0 0
$$27$$ 2.23607 0.430331
$$28$$ 0 0
$$29$$ 2.76393 0.513249 0.256625 0.966511i $$-0.417390\pi$$
0.256625 + 0.966511i $$0.417390\pi$$
$$30$$ 0 0
$$31$$ −9.56231 −1.71744 −0.858720 0.512444i $$-0.828740\pi$$
−0.858720 + 0.512444i $$0.828740\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.70820 0.609625 0.304812 0.952412i $$-0.401406\pi$$
0.304812 + 0.952412i $$0.401406\pi$$
$$38$$ 0 0
$$39$$ −1.23607 −0.197929
$$40$$ 0 0
$$41$$ 1.14590 0.178959 0.0894796 0.995989i $$-0.471480\pi$$
0.0894796 + 0.995989i $$0.471480\pi$$
$$42$$ 0 0
$$43$$ 4.85410 0.740244 0.370122 0.928983i $$-0.379316\pi$$
0.370122 + 0.928983i $$0.379316\pi$$
$$44$$ 0 0
$$45$$ 3.94427 0.587977
$$46$$ 0 0
$$47$$ −10.9443 −1.59639 −0.798193 0.602402i $$-0.794211\pi$$
−0.798193 + 0.602402i $$0.794211\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.381966 0.0534859
$$52$$ 0 0
$$53$$ 12.3262 1.69314 0.846569 0.532278i $$-0.178664\pi$$
0.846569 + 0.532278i $$0.178664\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.944272 −0.125072
$$58$$ 0 0
$$59$$ −5.23607 −0.681678 −0.340839 0.940122i $$-0.610711\pi$$
−0.340839 + 0.940122i $$0.610711\pi$$
$$60$$ 0 0
$$61$$ −12.5623 −1.60844 −0.804219 0.594333i $$-0.797416\pi$$
−0.804219 + 0.594333i $$0.797416\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.47214 −0.554700
$$66$$ 0 0
$$67$$ −5.09017 −0.621863 −0.310932 0.950432i $$-0.600641\pi$$
−0.310932 + 0.950432i $$0.600641\pi$$
$$68$$ 0 0
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ −4.76393 −0.565375 −0.282687 0.959212i $$-0.591226\pi$$
−0.282687 + 0.959212i $$0.591226\pi$$
$$72$$ 0 0
$$73$$ 2.85410 0.334047 0.167024 0.985953i $$-0.446584\pi$$
0.167024 + 0.985953i $$0.446584\pi$$
$$74$$ 0 0
$$75$$ 1.18034 0.136294
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.70820 −0.192188 −0.0960940 0.995372i $$-0.530635\pi$$
−0.0960940 + 0.995372i $$0.530635\pi$$
$$80$$ 0 0
$$81$$ 7.70820 0.856467
$$82$$ 0 0
$$83$$ 1.52786 0.167705 0.0838524 0.996478i $$-0.473278\pi$$
0.0838524 + 0.996478i $$0.473278\pi$$
$$84$$ 0 0
$$85$$ 1.38197 0.149895
$$86$$ 0 0
$$87$$ −1.05573 −0.113186
$$88$$ 0 0
$$89$$ −13.2361 −1.40302 −0.701510 0.712659i $$-0.747491\pi$$
−0.701510 + 0.712659i $$0.747491\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.65248 0.378744
$$94$$ 0 0
$$95$$ −3.41641 −0.350516
$$96$$ 0 0
$$97$$ −3.85410 −0.391325 −0.195662 0.980671i $$-0.562686\pi$$
−0.195662 + 0.980671i $$0.562686\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.4721 −1.63904 −0.819519 0.573051i $$-0.805760\pi$$
−0.819519 + 0.573051i $$0.805760\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −10.1803 −0.975100 −0.487550 0.873095i $$-0.662109\pi$$
−0.487550 + 0.873095i $$0.662109\pi$$
$$110$$ 0 0
$$111$$ −1.41641 −0.134439
$$112$$ 0 0
$$113$$ −4.94427 −0.465118 −0.232559 0.972582i $$-0.574710\pi$$
−0.232559 + 0.972582i $$0.574710\pi$$
$$114$$ 0 0
$$115$$ −7.23607 −0.674767
$$116$$ 0 0
$$117$$ −9.23607 −0.853875
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −0.437694 −0.0394655
$$124$$ 0 0
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ −13.3820 −1.18746 −0.593729 0.804665i $$-0.702345\pi$$
−0.593729 + 0.804665i $$0.702345\pi$$
$$128$$ 0 0
$$129$$ −1.85410 −0.163245
$$130$$ 0 0
$$131$$ 2.47214 0.215992 0.107996 0.994151i $$-0.465557\pi$$
0.107996 + 0.994151i $$0.465557\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −3.09017 −0.265959
$$136$$ 0 0
$$137$$ −0.0901699 −0.00770374 −0.00385187 0.999993i $$-0.501226\pi$$
−0.00385187 + 0.999993i $$0.501226\pi$$
$$138$$ 0 0
$$139$$ 11.3820 0.965406 0.482703 0.875784i $$-0.339655\pi$$
0.482703 + 0.875784i $$0.339655\pi$$
$$140$$ 0 0
$$141$$ 4.18034 0.352048
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.81966 −0.317206
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −5.61803 −0.460247 −0.230124 0.973161i $$-0.573913\pi$$
−0.230124 + 0.973161i $$0.573913\pi$$
$$150$$ 0 0
$$151$$ 12.2705 0.998560 0.499280 0.866441i $$-0.333598\pi$$
0.499280 + 0.866441i $$0.333598\pi$$
$$152$$ 0 0
$$153$$ 2.85410 0.230740
$$154$$ 0 0
$$155$$ 13.2148 1.06144
$$156$$ 0 0
$$157$$ −10.2918 −0.821375 −0.410687 0.911776i $$-0.634711\pi$$
−0.410687 + 0.911776i $$0.634711\pi$$
$$158$$ 0 0
$$159$$ −4.70820 −0.373385
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 20.5623 1.59116 0.795580 0.605849i $$-0.207167\pi$$
0.795580 + 0.605849i $$0.207167\pi$$
$$168$$ 0 0
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ −7.05573 −0.539565
$$172$$ 0 0
$$173$$ −6.67376 −0.507397 −0.253698 0.967283i $$-0.581647\pi$$
−0.253698 + 0.967283i $$0.581647\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.00000 0.150329
$$178$$ 0 0
$$179$$ 8.09017 0.604688 0.302344 0.953199i $$-0.402231\pi$$
0.302344 + 0.953199i $$0.402231\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 4.79837 0.354706
$$184$$ 0 0
$$185$$ −5.12461 −0.376769
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.9098 0.789408 0.394704 0.918808i $$-0.370847\pi$$
0.394704 + 0.918808i $$0.370847\pi$$
$$192$$ 0 0
$$193$$ −14.4721 −1.04173 −0.520864 0.853640i $$-0.674390\pi$$
−0.520864 + 0.853640i $$0.674390\pi$$
$$194$$ 0 0
$$195$$ 1.70820 0.122327
$$196$$ 0 0
$$197$$ −2.29180 −0.163284 −0.0816419 0.996662i $$-0.526016\pi$$
−0.0816419 + 0.996662i $$0.526016\pi$$
$$198$$ 0 0
$$199$$ −18.8541 −1.33653 −0.668266 0.743922i $$-0.732963\pi$$
−0.668266 + 0.743922i $$0.732963\pi$$
$$200$$ 0 0
$$201$$ 1.94427 0.137138
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1.58359 −0.110603
$$206$$ 0 0
$$207$$ −14.9443 −1.03870
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −25.5967 −1.76215 −0.881076 0.472974i $$-0.843180\pi$$
−0.881076 + 0.472974i $$0.843180\pi$$
$$212$$ 0 0
$$213$$ 1.81966 0.124681
$$214$$ 0 0
$$215$$ −6.70820 −0.457496
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1.09017 −0.0736669
$$220$$ 0 0
$$221$$ −3.23607 −0.217681
$$222$$ 0 0
$$223$$ 19.8885 1.33184 0.665918 0.746025i $$-0.268040\pi$$
0.665918 + 0.746025i $$0.268040\pi$$
$$224$$ 0 0
$$225$$ 8.81966 0.587977
$$226$$ 0 0
$$227$$ 13.8541 0.919529 0.459765 0.888041i $$-0.347934\pi$$
0.459765 + 0.888041i $$0.347934\pi$$
$$228$$ 0 0
$$229$$ −9.41641 −0.622254 −0.311127 0.950368i $$-0.600706\pi$$
−0.311127 + 0.950368i $$0.600706\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 20.1803 1.32206 0.661029 0.750360i $$-0.270120\pi$$
0.661029 + 0.750360i $$0.270120\pi$$
$$234$$ 0 0
$$235$$ 15.1246 0.986621
$$236$$ 0 0
$$237$$ 0.652476 0.0423829
$$238$$ 0 0
$$239$$ −21.6180 −1.39835 −0.699177 0.714948i $$-0.746450\pi$$
−0.699177 + 0.714948i $$0.746450\pi$$
$$240$$ 0 0
$$241$$ −1.38197 −0.0890203 −0.0445101 0.999009i $$-0.514173\pi$$
−0.0445101 + 0.999009i $$0.514173\pi$$
$$242$$ 0 0
$$243$$ −9.65248 −0.619207
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ −0.583592 −0.0369836
$$250$$ 0 0
$$251$$ −2.18034 −0.137622 −0.0688109 0.997630i $$-0.521921\pi$$
−0.0688109 + 0.997630i $$0.521921\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −0.527864 −0.0330561
$$256$$ 0 0
$$257$$ −10.4721 −0.653234 −0.326617 0.945157i $$-0.605909\pi$$
−0.326617 + 0.945157i $$0.605909\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −7.88854 −0.488289
$$262$$ 0 0
$$263$$ −7.41641 −0.457315 −0.228658 0.973507i $$-0.573434\pi$$
−0.228658 + 0.973507i $$0.573434\pi$$
$$264$$ 0 0
$$265$$ −17.0344 −1.04642
$$266$$ 0 0
$$267$$ 5.05573 0.309406
$$268$$ 0 0
$$269$$ 13.4164 0.818013 0.409006 0.912532i $$-0.365875\pi$$
0.409006 + 0.912532i $$0.365875\pi$$
$$270$$ 0 0
$$271$$ 11.5279 0.700268 0.350134 0.936700i $$-0.386136\pi$$
0.350134 + 0.936700i $$0.386136\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 9.12461 0.548245 0.274122 0.961695i $$-0.411613\pi$$
0.274122 + 0.961695i $$0.411613\pi$$
$$278$$ 0 0
$$279$$ 27.2918 1.63392
$$280$$ 0 0
$$281$$ 14.5066 0.865390 0.432695 0.901540i $$-0.357563\pi$$
0.432695 + 0.901540i $$0.357563\pi$$
$$282$$ 0 0
$$283$$ −25.8541 −1.53687 −0.768433 0.639930i $$-0.778963\pi$$
−0.768433 + 0.639930i $$0.778963\pi$$
$$284$$ 0 0
$$285$$ 1.30495 0.0772987
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 1.47214 0.0862981
$$292$$ 0 0
$$293$$ −21.5967 −1.26170 −0.630848 0.775907i $$-0.717293\pi$$
−0.630848 + 0.775907i $$0.717293\pi$$
$$294$$ 0 0
$$295$$ 7.23607 0.421300
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.9443 0.979913
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.29180 0.361454
$$304$$ 0 0
$$305$$ 17.3607 0.994070
$$306$$ 0 0
$$307$$ −17.4164 −0.994007 −0.497003 0.867749i $$-0.665566\pi$$
−0.497003 + 0.867749i $$0.665566\pi$$
$$308$$ 0 0
$$309$$ 2.29180 0.130376
$$310$$ 0 0
$$311$$ −11.7426 −0.665864 −0.332932 0.942951i $$-0.608038\pi$$
−0.332932 + 0.942951i $$0.608038\pi$$
$$312$$ 0 0
$$313$$ −23.8541 −1.34831 −0.674157 0.738588i $$-0.735493\pi$$
−0.674157 + 0.738588i $$0.735493\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −11.8885 −0.667727 −0.333864 0.942621i $$-0.608352\pi$$
−0.333864 + 0.942621i $$0.608352\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.47214 −0.137553
$$324$$ 0 0
$$325$$ −10.0000 −0.554700
$$326$$ 0 0
$$327$$ 3.88854 0.215037
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.32624 −0.127862 −0.0639308 0.997954i $$-0.520364\pi$$
−0.0639308 + 0.997954i $$0.520364\pi$$
$$332$$ 0 0
$$333$$ −10.5836 −0.579977
$$334$$ 0 0
$$335$$ 7.03444 0.384333
$$336$$ 0 0
$$337$$ 13.4164 0.730838 0.365419 0.930843i $$-0.380926\pi$$
0.365419 + 0.930843i $$0.380926\pi$$
$$338$$ 0 0
$$339$$ 1.88854 0.102572
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2.76393 0.148805
$$346$$ 0 0
$$347$$ −9.88854 −0.530845 −0.265422 0.964132i $$-0.585511\pi$$
−0.265422 + 0.964132i $$0.585511\pi$$
$$348$$ 0 0
$$349$$ 0.472136 0.0252729 0.0126364 0.999920i $$-0.495978\pi$$
0.0126364 + 0.999920i $$0.495978\pi$$
$$350$$ 0 0
$$351$$ 7.23607 0.386233
$$352$$ 0 0
$$353$$ 24.0689 1.28106 0.640529 0.767934i $$-0.278715\pi$$
0.640529 + 0.767934i $$0.278715\pi$$
$$354$$ 0 0
$$355$$ 6.58359 0.349421
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.14590 0.218812 0.109406 0.993997i $$-0.465105\pi$$
0.109406 + 0.993997i $$0.465105\pi$$
$$360$$ 0 0
$$361$$ −12.8885 −0.678344
$$362$$ 0 0
$$363$$ 4.20163 0.220528
$$364$$ 0 0
$$365$$ −3.94427 −0.206453
$$366$$ 0 0
$$367$$ 15.6180 0.815255 0.407627 0.913148i $$-0.366356\pi$$
0.407627 + 0.913148i $$0.366356\pi$$
$$368$$ 0 0
$$369$$ −3.27051 −0.170256
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7.50658 0.388676 0.194338 0.980935i $$-0.437744\pi$$
0.194338 + 0.980935i $$0.437744\pi$$
$$374$$ 0 0
$$375$$ −4.27051 −0.220528
$$376$$ 0 0
$$377$$ 8.94427 0.460653
$$378$$ 0 0
$$379$$ 31.1246 1.59876 0.799382 0.600823i $$-0.205160\pi$$
0.799382 + 0.600823i $$0.205160\pi$$
$$380$$ 0 0
$$381$$ 5.11146 0.261868
$$382$$ 0 0
$$383$$ −25.8885 −1.32284 −0.661421 0.750014i $$-0.730046\pi$$
−0.661421 + 0.750014i $$0.730046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −13.8541 −0.704244
$$388$$ 0 0
$$389$$ −10.5066 −0.532705 −0.266352 0.963876i $$-0.585818\pi$$
−0.266352 + 0.963876i $$0.585818\pi$$
$$390$$ 0 0
$$391$$ −5.23607 −0.264799
$$392$$ 0 0
$$393$$ −0.944272 −0.0476322
$$394$$ 0 0
$$395$$ 2.36068 0.118779
$$396$$ 0 0
$$397$$ 1.67376 0.0840037 0.0420019 0.999118i $$-0.486626\pi$$
0.0420019 + 0.999118i $$0.486626\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 28.9443 1.44541 0.722704 0.691158i $$-0.242899\pi$$
0.722704 + 0.691158i $$0.242899\pi$$
$$402$$ 0 0
$$403$$ −30.9443 −1.54144
$$404$$ 0 0
$$405$$ −10.6525 −0.529326
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 0 0
$$411$$ 0.0344419 0.00169889
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2.11146 −0.103647
$$416$$ 0 0
$$417$$ −4.34752 −0.212899
$$418$$ 0 0
$$419$$ 2.50658 0.122454 0.0612272 0.998124i $$-0.480499\pi$$
0.0612272 + 0.998124i $$0.480499\pi$$
$$420$$ 0 0
$$421$$ −16.8541 −0.821419 −0.410709 0.911766i $$-0.634719\pi$$
−0.410709 + 0.911766i $$0.634719\pi$$
$$422$$ 0 0
$$423$$ 31.2361 1.51875
$$424$$ 0 0
$$425$$ 3.09017 0.149895
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 14.9443 0.719840 0.359920 0.932983i $$-0.382804\pi$$
0.359920 + 0.932983i $$0.382804\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ 1.45898 0.0699528
$$436$$ 0 0
$$437$$ 12.9443 0.619208
$$438$$ 0 0
$$439$$ 22.8541 1.09077 0.545383 0.838187i $$-0.316384\pi$$
0.545383 + 0.838187i $$0.316384\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −24.9443 −1.18514 −0.592569 0.805520i $$-0.701886\pi$$
−0.592569 + 0.805520i $$0.701886\pi$$
$$444$$ 0 0
$$445$$ 18.2918 0.867114
$$446$$ 0 0
$$447$$ 2.14590 0.101497
$$448$$ 0 0
$$449$$ −16.9443 −0.799650 −0.399825 0.916592i $$-0.630929\pi$$
−0.399825 + 0.916592i $$0.630929\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −4.68692 −0.220211
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −21.2705 −0.994992 −0.497496 0.867466i $$-0.665747\pi$$
−0.497496 + 0.867466i $$0.665747\pi$$
$$458$$ 0 0
$$459$$ −2.23607 −0.104371
$$460$$ 0 0
$$461$$ −26.8328 −1.24973 −0.624864 0.780733i $$-0.714846\pi$$
−0.624864 + 0.780733i $$0.714846\pi$$
$$462$$ 0 0
$$463$$ 21.7984 1.01306 0.506528 0.862223i $$-0.330929\pi$$
0.506528 + 0.862223i $$0.330929\pi$$
$$464$$ 0 0
$$465$$ −5.04760 −0.234077
$$466$$ 0 0
$$467$$ 33.2361 1.53798 0.768991 0.639260i $$-0.220759\pi$$
0.768991 + 0.639260i $$0.220759\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 3.93112 0.181136
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −7.63932 −0.350516
$$476$$ 0 0
$$477$$ −35.1803 −1.61080
$$478$$ 0 0
$$479$$ 8.38197 0.382982 0.191491 0.981494i $$-0.438668\pi$$
0.191491 + 0.981494i $$0.438668\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.32624 0.241852
$$486$$ 0 0
$$487$$ 36.9443 1.67410 0.837052 0.547123i $$-0.184277\pi$$
0.837052 + 0.547123i $$0.184277\pi$$
$$488$$ 0 0
$$489$$ 2.29180 0.103639
$$490$$ 0 0
$$491$$ −15.2016 −0.686040 −0.343020 0.939328i $$-0.611450\pi$$
−0.343020 + 0.939328i $$0.611450\pi$$
$$492$$ 0 0
$$493$$ −2.76393 −0.124481
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 27.3050 1.22234 0.611169 0.791500i $$-0.290700\pi$$
0.611169 + 0.791500i $$0.290700\pi$$
$$500$$ 0 0
$$501$$ −7.85410 −0.350895
$$502$$ 0 0
$$503$$ 21.2705 0.948405 0.474203 0.880416i $$-0.342736\pi$$
0.474203 + 0.880416i $$0.342736\pi$$
$$504$$ 0 0
$$505$$ 22.7639 1.01298
$$506$$ 0 0
$$507$$ 0.965558 0.0428819
$$508$$ 0 0
$$509$$ 18.7639 0.831697 0.415848 0.909434i $$-0.363485\pi$$
0.415848 + 0.909434i $$0.363485\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 5.52786 0.244061
$$514$$ 0 0
$$515$$ 8.29180 0.365380
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.54915 0.111895
$$520$$ 0 0
$$521$$ 1.90983 0.0836712 0.0418356 0.999125i $$-0.486679\pi$$
0.0418356 + 0.999125i $$0.486679\pi$$
$$522$$ 0 0
$$523$$ 38.6525 1.69015 0.845077 0.534644i $$-0.179554\pi$$
0.845077 + 0.534644i $$0.179554\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.56231 0.416541
$$528$$ 0 0
$$529$$ 4.41641 0.192018
$$530$$ 0 0
$$531$$ 14.9443 0.648526
$$532$$ 0 0
$$533$$ 3.70820 0.160620
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −3.09017 −0.133351
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 5.34752 0.229484
$$544$$ 0 0
$$545$$ 14.0689 0.602645
$$546$$ 0 0
$$547$$ 27.1246 1.15976 0.579882 0.814700i $$-0.303099\pi$$
0.579882 + 0.814700i $$0.303099\pi$$
$$548$$ 0 0
$$549$$ 35.8541 1.53022
$$550$$ 0 0
$$551$$ 6.83282 0.291088
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1.95743 0.0830882
$$556$$ 0 0
$$557$$ 15.5279 0.657937 0.328968 0.944341i $$-0.393299\pi$$
0.328968 + 0.944341i $$0.393299\pi$$
$$558$$ 0 0
$$559$$ 15.7082 0.664386
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −23.1246 −0.974586 −0.487293 0.873238i $$-0.662016\pi$$
−0.487293 + 0.873238i $$0.662016\pi$$
$$564$$ 0 0
$$565$$ 6.83282 0.287459
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4.09017 −0.171469 −0.0857344 0.996318i $$-0.527324\pi$$
−0.0857344 + 0.996318i $$0.527324\pi$$
$$570$$ 0 0
$$571$$ −2.65248 −0.111003 −0.0555013 0.998459i $$-0.517676\pi$$
−0.0555013 + 0.998459i $$0.517676\pi$$
$$572$$ 0 0
$$573$$ −4.16718 −0.174087
$$574$$ 0 0
$$575$$ −16.1803 −0.674767
$$576$$ 0 0
$$577$$ −18.5836 −0.773645 −0.386823 0.922154i $$-0.626427\pi$$
−0.386823 + 0.922154i $$0.626427\pi$$
$$578$$ 0 0
$$579$$ 5.52786 0.229730
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 12.7639 0.527724
$$586$$ 0 0
$$587$$ 30.0000 1.23823 0.619116 0.785299i $$-0.287491\pi$$
0.619116 + 0.785299i $$0.287491\pi$$
$$588$$ 0 0
$$589$$ −23.6393 −0.974041
$$590$$ 0 0
$$591$$ 0.875388 0.0360087
$$592$$ 0 0
$$593$$ 19.5967 0.804742 0.402371 0.915477i $$-0.368186\pi$$
0.402371 + 0.915477i $$0.368186\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.20163 0.294743
$$598$$ 0 0
$$599$$ 17.0344 0.696008 0.348004 0.937493i $$-0.386859\pi$$
0.348004 + 0.937493i $$0.386859\pi$$
$$600$$ 0 0
$$601$$ −32.4721 −1.32457 −0.662283 0.749254i $$-0.730412\pi$$
−0.662283 + 0.749254i $$0.730412\pi$$
$$602$$ 0 0
$$603$$ 14.5279 0.591620
$$604$$ 0 0
$$605$$ 15.2016 0.618034
$$606$$ 0 0
$$607$$ 18.1459 0.736519 0.368260 0.929723i $$-0.379954\pi$$
0.368260 + 0.929723i $$0.379954\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −35.4164 −1.43279
$$612$$ 0 0
$$613$$ 5.85410 0.236445 0.118222 0.992987i $$-0.462280\pi$$
0.118222 + 0.992987i $$0.462280\pi$$
$$614$$ 0 0
$$615$$ 0.604878 0.0243911
$$616$$ 0 0
$$617$$ −3.59675 −0.144800 −0.0723998 0.997376i $$-0.523066\pi$$
−0.0723998 + 0.997376i $$0.523066\pi$$
$$618$$ 0 0
$$619$$ 6.83282 0.274634 0.137317 0.990527i $$-0.456152\pi$$
0.137317 + 0.990527i $$0.456152\pi$$
$$620$$ 0 0
$$621$$ 11.7082 0.469834
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.70820 −0.147856
$$630$$ 0 0
$$631$$ −30.9787 −1.23324 −0.616622 0.787260i $$-0.711499\pi$$
−0.616622 + 0.787260i $$0.711499\pi$$
$$632$$ 0 0
$$633$$ 9.77709 0.388604
$$634$$ 0 0
$$635$$ 18.4934 0.733889
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 13.5967 0.537879
$$640$$ 0 0
$$641$$ −29.7771 −1.17612 −0.588062 0.808816i $$-0.700109\pi$$
−0.588062 + 0.808816i $$0.700109\pi$$
$$642$$ 0 0
$$643$$ −31.6180 −1.24689 −0.623447 0.781866i $$-0.714268\pi$$
−0.623447 + 0.781866i $$0.714268\pi$$
$$644$$ 0 0
$$645$$ 2.56231 0.100891
$$646$$ 0 0
$$647$$ 21.5967 0.849056 0.424528 0.905415i $$-0.360440\pi$$
0.424528 + 0.905415i $$0.360440\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −20.7639 −0.812555 −0.406278 0.913750i $$-0.633173\pi$$
−0.406278 + 0.913750i $$0.633173\pi$$
$$654$$ 0 0
$$655$$ −3.41641 −0.133490
$$656$$ 0 0
$$657$$ −8.14590 −0.317802
$$658$$ 0 0
$$659$$ −21.0902 −0.821556 −0.410778 0.911735i $$-0.634743\pi$$
−0.410778 + 0.911735i $$0.634743\pi$$
$$660$$ 0 0
$$661$$ 31.4164 1.22196 0.610978 0.791647i $$-0.290776\pi$$
0.610978 + 0.791647i $$0.290776\pi$$
$$662$$ 0 0
$$663$$ 1.23607 0.0480049
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 14.4721 0.560363
$$668$$ 0 0
$$669$$ −7.59675 −0.293707
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −16.9443 −0.653154 −0.326577 0.945171i $$-0.605895\pi$$
−0.326577 + 0.945171i $$0.605895\pi$$
$$674$$ 0 0
$$675$$ −6.90983 −0.265959
$$676$$ 0 0
$$677$$ −32.8328 −1.26187 −0.630934 0.775837i $$-0.717328\pi$$
−0.630934 + 0.775837i $$0.717328\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −5.29180 −0.202782
$$682$$ 0 0
$$683$$ −10.1115 −0.386904 −0.193452 0.981110i $$-0.561968\pi$$
−0.193452 + 0.981110i $$0.561968\pi$$
$$684$$ 0 0
$$685$$ 0.124612 0.00476117
$$686$$ 0 0
$$687$$ 3.59675 0.137224
$$688$$ 0 0
$$689$$ 39.8885 1.51963
$$690$$ 0 0
$$691$$ 35.9787 1.36869 0.684347 0.729156i $$-0.260087\pi$$
0.684347 + 0.729156i $$0.260087\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −15.7295 −0.596654
$$696$$ 0 0
$$697$$ −1.14590 −0.0434040
$$698$$ 0 0
$$699$$ −7.70820 −0.291551
$$700$$ 0 0
$$701$$ 39.3050 1.48453 0.742264 0.670108i $$-0.233752\pi$$
0.742264 + 0.670108i $$0.233752\pi$$
$$702$$ 0 0
$$703$$ 9.16718 0.345747
$$704$$ 0 0
$$705$$ −5.77709 −0.217578
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 28.6525 1.07607 0.538033 0.842924i $$-0.319168\pi$$
0.538033 + 0.842924i $$0.319168\pi$$
$$710$$ 0 0
$$711$$ 4.87539 0.182841
$$712$$ 0 0
$$713$$ −50.0689 −1.87509
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.25735 0.308377
$$718$$ 0 0
$$719$$ 22.0902 0.823824 0.411912 0.911224i $$-0.364861\pi$$
0.411912 + 0.911224i $$0.364861\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0.527864 0.0196315
$$724$$ 0 0
$$725$$ −8.54102 −0.317206
$$726$$ 0 0
$$727$$ 0.291796 0.0108221 0.00541106 0.999985i $$-0.498278\pi$$
0.00541106 + 0.999985i $$0.498278\pi$$
$$728$$ 0 0
$$729$$ −19.4377 −0.719915
$$730$$ 0 0
$$731$$ −4.85410 −0.179535
$$732$$ 0 0
$$733$$ 34.2492 1.26502 0.632512 0.774551i $$-0.282024\pi$$
0.632512 + 0.774551i $$0.282024\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 9.61803 0.353805 0.176903 0.984228i $$-0.443392\pi$$
0.176903 + 0.984228i $$0.443392\pi$$
$$740$$ 0 0
$$741$$ −3.05573 −0.112255
$$742$$ 0 0
$$743$$ 35.8885 1.31662 0.658311 0.752746i $$-0.271271\pi$$
0.658311 + 0.752746i $$0.271271\pi$$
$$744$$ 0 0
$$745$$ 7.76393 0.284448
$$746$$ 0 0
$$747$$ −4.36068 −0.159549
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.83282 −0.103371 −0.0516855 0.998663i $$-0.516459\pi$$
−0.0516855 + 0.998663i $$0.516459\pi$$
$$752$$ 0 0
$$753$$ 0.832816 0.0303495
$$754$$ 0 0
$$755$$ −16.9574 −0.617144
$$756$$ 0 0
$$757$$ 14.5623 0.529276 0.264638 0.964348i $$-0.414748\pi$$
0.264638 + 0.964348i $$0.414748\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −53.8885 −1.95346 −0.976729 0.214477i $$-0.931195\pi$$
−0.976729 + 0.214477i $$0.931195\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −3.94427 −0.142605
$$766$$ 0 0
$$767$$ −16.9443 −0.611822
$$768$$ 0 0
$$769$$ 13.5967 0.490311 0.245156 0.969484i $$-0.421161\pi$$
0.245156 + 0.969484i $$0.421161\pi$$
$$770$$ 0 0
$$771$$ 4.00000 0.144056
$$772$$ 0 0
$$773$$ 20.4721 0.736332 0.368166 0.929760i $$-0.379986\pi$$
0.368166 + 0.929760i $$0.379986\pi$$
$$774$$ 0 0
$$775$$ 29.5492 1.06144
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.83282 0.101496
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.18034 0.220867
$$784$$ 0 0
$$785$$ 14.2229 0.507638
$$786$$ 0 0
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ 0 0
$$789$$ 2.83282 0.100851
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −40.6525 −1.44361
$$794$$ 0 0
$$795$$ 6.50658 0.230765
$$796$$ 0 0
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 10.9443 0.387181
$$800$$ 0 0
$$801$$ 37.7771 1.33479
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −5.12461 −0.180395
$$808$$ 0 0
$$809$$ −0.875388 −0.0307770 −0.0153885 0.999882i $$-0.504899\pi$$
−0.0153885 + 0.999882i $$0.504899\pi$$
$$810$$ 0 0
$$811$$ −31.9787 −1.12292 −0.561462 0.827502i $$-0.689761\pi$$
−0.561462 + 0.827502i $$0.689761\pi$$
$$812$$ 0 0
$$813$$ −4.40325 −0.154429
$$814$$ 0 0
$$815$$ 8.29180 0.290449
$$816$$ 0 0
$$817$$ 12.0000 0.419827
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 46.4721 1.62189 0.810944 0.585123i $$-0.198954\pi$$
0.810944 + 0.585123i $$0.198954\pi$$
$$822$$ 0 0
$$823$$ 24.5410 0.855446 0.427723 0.903910i $$-0.359316\pi$$
0.427723 + 0.903910i $$0.359316\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −41.2361 −1.43392 −0.716959 0.697115i $$-0.754467\pi$$
−0.716959 + 0.697115i $$0.754467\pi$$
$$828$$ 0 0
$$829$$ −37.5967 −1.30579 −0.652895 0.757449i $$-0.726446\pi$$
−0.652895 + 0.757449i $$0.726446\pi$$
$$830$$ 0 0
$$831$$ −3.48529 −0.120903
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −28.4164 −0.983390
$$836$$ 0 0
$$837$$ −21.3820 −0.739069
$$838$$ 0 0
$$839$$ 15.0557 0.519781 0.259891 0.965638i $$-0.416313\pi$$
0.259891 + 0.965638i $$0.416313\pi$$
$$840$$ 0 0
$$841$$ −21.3607 −0.736575
$$842$$ 0 0
$$843$$ −5.54102 −0.190843
$$844$$ 0 0
$$845$$ 3.49342 0.120177
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 9.87539 0.338922
$$850$$ 0 0
$$851$$ 19.4164 0.665586
$$852$$ 0 0
$$853$$ −45.7771 −1.56738 −0.783689 0.621154i $$-0.786664\pi$$
−0.783689 + 0.621154i $$0.786664\pi$$
$$854$$ 0 0
$$855$$ 9.75078 0.333470
$$856$$ 0 0
$$857$$ 44.7984 1.53028 0.765142 0.643862i $$-0.222669\pi$$
0.765142 + 0.643862i $$0.222669\pi$$
$$858$$ 0 0
$$859$$ 36.8328 1.25672 0.628360 0.777923i $$-0.283727\pi$$
0.628360 + 0.777923i $$0.283727\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −26.3262 −0.896156 −0.448078 0.893995i $$-0.647891\pi$$
−0.448078 + 0.893995i $$0.647891\pi$$
$$864$$ 0 0
$$865$$ 9.22291 0.313588
$$866$$ 0 0
$$867$$ −0.381966 −0.0129722
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −16.4721 −0.558137
$$872$$ 0 0
$$873$$ 11.0000 0.372294
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17.3050 −0.584347 −0.292173 0.956365i $$-0.594378\pi$$
−0.292173 + 0.956365i $$0.594378\pi$$
$$878$$ 0 0
$$879$$ 8.24922 0.278239
$$880$$ 0 0
$$881$$ 21.1591 0.712867 0.356433 0.934321i $$-0.383993\pi$$
0.356433 + 0.934321i $$0.383993\pi$$
$$882$$ 0 0
$$883$$ −14.6869 −0.494254 −0.247127 0.968983i $$-0.579487\pi$$
−0.247127 + 0.968983i $$0.579487\pi$$
$$884$$ 0 0
$$885$$ −2.76393 −0.0929086
$$886$$ 0 0
$$887$$ 11.7426 0.394279 0.197140 0.980375i $$-0.436835\pi$$
0.197140 + 0.980375i $$0.436835\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −27.0557 −0.905385
$$894$$ 0 0
$$895$$ −11.1803 −0.373718
$$896$$ 0 0
$$897$$ −6.47214 −0.216098
$$898$$ 0 0
$$899$$ −26.4296 −0.881475
$$900$$ 0 0
$$901$$ −12.3262 −0.410647
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.3475 0.643133
$$906$$ 0 0
$$907$$ −36.1803 −1.20135 −0.600674 0.799494i $$-0.705101\pi$$
−0.600674 + 0.799494i $$0.705101\pi$$
$$908$$ 0 0
$$909$$ 47.0132 1.55933
$$910$$ 0 0
$$911$$ 50.7214 1.68047 0.840237 0.542220i $$-0.182416\pi$$
0.840237 + 0.542220i $$0.182416\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −6.63119 −0.219220
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −23.6869 −0.781359 −0.390680 0.920527i $$-0.627760\pi$$
−0.390680 + 0.920527i $$0.627760\pi$$
$$920$$ 0 0
$$921$$ 6.65248 0.219207
$$922$$ 0 0
$$923$$ −15.4164 −0.507437
$$924$$ 0 0
$$925$$ −11.4590 −0.376769
$$926$$ 0 0
$$927$$ 17.1246 0.562446
$$928$$ 0 0
$$929$$ 44.7984 1.46979 0.734893 0.678183i $$-0.237232\pi$$
0.734893 + 0.678183i $$0.237232\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 4.48529 0.146842
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −31.4164 −1.02633 −0.513165 0.858290i $$-0.671527\pi$$
−0.513165 + 0.858290i $$0.671527\pi$$
$$938$$ 0 0
$$939$$ 9.11146 0.297341
$$940$$ 0 0
$$941$$ 43.7984 1.42779 0.713893 0.700255i $$-0.246930\pi$$
0.713893 + 0.700255i $$0.246930\pi$$
$$942$$ 0 0
$$943$$ 6.00000 0.195387
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 57.1935 1.85854 0.929269 0.369403i $$-0.120438\pi$$
0.929269 + 0.369403i $$0.120438\pi$$
$$948$$ 0 0
$$949$$ 9.23607 0.299815
$$950$$ 0 0
$$951$$ 4.54102 0.147253
$$952$$ 0 0
$$953$$ −31.0902 −1.00711 −0.503555 0.863963i $$-0.667975\pi$$
−0.503555 + 0.863963i $$0.667975\pi$$
$$954$$ 0 0
$$955$$ −15.0770 −0.487881
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 60.4377 1.94960
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 20.0000 0.643823
$$966$$ 0 0
$$967$$ −36.0902 −1.16058 −0.580291 0.814409i $$-0.697061\pi$$
−0.580291 + 0.814409i $$0.697061\pi$$
$$968$$ 0 0
$$969$$ 0.944272 0.0303344
$$970$$ 0 0
$$971$$ −27.2361 −0.874047 −0.437024 0.899450i $$-0.643967\pi$$
−0.437024 + 0.899450i $$0.643967\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 3.81966 0.122327
$$976$$ 0 0
$$977$$ 19.4508 0.622288 0.311144 0.950363i $$-0.399288\pi$$
0.311144 + 0.950363i $$0.399288\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 29.0557 0.927678
$$982$$ 0 0
$$983$$ 33.2705 1.06116 0.530582 0.847633i $$-0.321973\pi$$
0.530582 + 0.847633i $$0.321973\pi$$
$$984$$ 0 0
$$985$$ 3.16718 0.100915
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 25.4164 0.808195
$$990$$ 0 0
$$991$$ −58.5410 −1.85962 −0.929808 0.368044i $$-0.880028\pi$$
−0.929808 + 0.368044i $$0.880028\pi$$
$$992$$ 0 0
$$993$$ 0.888544 0.0281971
$$994$$ 0 0
$$995$$ 26.0557 0.826022
$$996$$ 0 0
$$997$$ 12.7426 0.403564 0.201782 0.979430i $$-0.435327\pi$$
0.201782 + 0.979430i $$0.435327\pi$$
$$998$$ 0 0
$$999$$ 8.29180 0.262341
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 3332.2.a.g.1.2 2
7.6 odd 2 3332.2.a.o.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.g.1.2 2 1.1 even 1 trivial
3332.2.a.o.1.1 yes 2 7.6 odd 2